TR-87-3069 Rev. 1987
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AFWAL-TR-87-3069
VOLUME I
*|
EXPERIMENTAL MODAL ANALYSIS AND
DYNAMIC COMPONENT SYNTHESIS
VOL I - Summary of Technical Work
Dr. Randall J. Allemang, Dr. David L. Brown
Structural Dynamics Research Laboratory
Department of Mechanical and Industrial Engineering
University of Cincinnati
Cincinnati, Ohio 45221-0072
December 1987
Final Technical Report for Period November 1983 - January 1987
Approved for public release; distribution is unlimited
FLIGHT DYNAMICS LABORATORY
AIR FORCE WRIGHT AERONAUTICAL LABORATORIES
AIR FORCE SYSTEMS COMMAND
WRIGHT-PATITERSON AIR FORCE BASE, OHIO 45433-6553
D,T I
,. ,,
'
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This technical report has been reviewed and is approved for publication.
0T= F. ,ATLRER, Principal Engineer
Strnctural Dynairics Branch
Structures Division
FOR '14E COMMANDER
-•EROME P1FARSON, Chief
Structural Dynamics Branch
Structures Division
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EXPERIMENTAL MODAL ANALYSIS AND DYNAMIC COMPONENT SYNTHESIS
simafrg nf TgIrhbpjjj
Work
12. PERSONAL AUTHORIS)
DR. RANDALL J. ALLEMANG
DR.
FINAL
16. SUPPLEMENTARY NOTATION
DAVID L. BROWN
1l4. DATE OF REPORT (Yr.. Mo.. Day)
13b. TIME COVERED
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FROM NOV 1983ToJAN 198
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15. PAGE COUNT
DECEMBER 1987
138
The computer software contained herein/re theoretical and/or references that in no way
reflect Air Force-owned or developed, eomputer software.
17
COSATI CODES
FIELO
GAOUP
IS. SUBA
T TERMS (Continue on reuerse ifnercesary and identtfy by block number)
SUB. GR.
ul
MODAL TESTING
VIBRATION TESTING
DYNAMICS,
19. ABSTRACT (Continue on re•erse itf neceuary and identify by block number)
This report is one of six reports that represent the final technical report on thework.k
involved with United States Air Force Contract F33615-83-C-3218,9.Experimental Modal
Analysis and Dynamic Component Synthesis.
The reports that are part of the documented
work include the following:,
"AFWAL-TR-87-3069'-*Vol.
I
Summary of Technical Work
Vol. II
Measurement Techniques for Experimental Modal Analysis ,
Vol. III Modal Parameter Estimation I
Vol. IV
System Modeling Techniques
Vol. V
Universal File Formats
Vol. VI
Software User's Guide
For a complete understanding of the research conducted under this contract, all of the
Technical Reports should be referenced.
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EDITION OF I JAN 73 ISOBSOLETE.
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SUMMARY
Volume I introduces the work contained in Volume II through Volume VI of this Technical Report.
This includes a state-of-the-art review in several areas connected with experimental modal analysis
and dynamic component synthesis. It comprehends frequency response measurement techniques,
experimental modal analysis methods, modal parameter estimation, modal modeling, sensitivity
analysis, and component mode synthesis. All discussion and development of this material is
documented using a consistent set of nomenclature. Several new modal parameter estimation
algorithms and a new superelement component dynamic synthesis method were developed as part of
this effort.
With respect to the material contained in Volume II, this report reviews the area of measurement
techniques applicable to experimental modal analysis. Primarily, this is concerned with the accurate
measurement of frequency response functions on linear, time invariant, observable structural systems.
When attempting to experimentally determine the dynamic properties (natural frequency, damping,
and mode shapes) of a structure, one of the most important aspects is to collect and process data that
represent the structure as accurately as possible. These data can then be used as input to a number of
parameter estimation algorithms and could also be used in modal modeling algorithms. Volume I1of
this Technical Report describes in detail the procedures used to collect the data. Many of the
potential errors are discussed as well as techniques to eliminate or reduce the effects of these errors
on the quality of the results. If the steps described in this Technical Report are followed, data can be
collected, as input to modal parameter estimation algorithms, that will yield accurate dynamic
properties of the test structure. With care and attention to theoretical limitations, these dynamic
properties can be used to construct a modal model.
Regarding the material contained in Volume III, this report documents the area of modal parameter
estimation in terms of a review of efforts - over the past twenty-five years - in developing several new
multiple reference methods, and in attempting to provide a common basis and understanding for all
of the modal parameter estimation procedures developed to date. The summary of modal parameter
estimation includes a substantial literature examination and the presentation of earlier methods, such
as the Least Squares Complex Exponential, as special cases of general techniques, such as the
Polyreference Time Domain method. Several new modal parameter estimation methods are
developed and presented using consistent theory and nomenclature. The methods that are described
in this manner include: Polyreference Time Domain, Polyreference Frequency Domain, Multiple
Reference Ibrahim Time Domain, Multiple Reference Orthogonal Polynomial, and Multi MAC.
These techniques, in terms of general characteristics, are also compared to others such as the Least
Squares Complex Exponential, Ibrahim Time Domain, Eigensystem Realization Algorithm, and
Direct Parameter Estimation methods. These methods are all similar in that they involve the
decomposition of impulse response functions (time domain), frequency response functions
(frequency domain), or forced response patterns (spatial domain) into characteristic functions in the
appropriate domain. These characteristic functions are the single degree of freedom information in
the respective domain.
Concerning the material contained in Volume IV, this report lines out the theoretical basis for the
current methods used to predict the system dynamics of a modified structure or of combined
structures based upon a previously determined, modal or impedance, model of the structure(s). The
methods reviewed were: Modal modeling technique, local eigenvalue modification, coupling of
structures using eigenvalue modification, complex mode eigenvalue modification, sensitivity analysis,
impedance modeling technique, building block approach, dynamic stiffness method, and the
frequency response method. The effects of measurement errors, modal parameter estimation error,
and trnicated modes in the application of modal modeling technique are evaluated. Some of the
experimental modal model validation methods are also presented. Several methods to normalize the
measured complex modes were reviewed including both time domain and frequency domain
techniques. A new component mode synthesis method (Superelement Component Dynamic
Synthesis) developed by the University of Dayton Research Institute is presented.
-iii-
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PREFACE
This volume is one of six Technical Reports that represent the final report on the work involved with
United States Air Force Contract F33615-83-C-3218, Experimental Modal Analysis and Dynamic
Component Synthesis. The reports that are part of the documented work include the following:
AFWAL-TR-87-3069
VOLUME
VOLUME
VOLUME
VOLUME
VOLUME
VOLUME
I
II
III
IV
V
VI
Summary of Technical Work
Measurement Techniques for Experimental Modal Analysis
Modal Parameter Estimation
System Modeling Techniques
Universal File Formats
Software User's Guide
For a complete understanding of the research conducted under this contract, all of the Technical
Reports should be referenced.
Accession For
GA&I
NTIS
DTIC TAP
~
U t
"ll
E
/
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ACKNOWLEDGEMENTS
The University of Cincinnati Structural Dynamics Research Laboratory (UC-SDRL) and the
University of Dayton Research Institute (UDRI) would like to acknowledge the numerous people
who have contributed to the work performed under this contract. In particular, we want to
acknowledge the contract monitor, Otto Maurer, for guidance and assistance during the course of the
contract. Also, we want to recognize the technical input of Dr. Havard Vold of SDRC who
participated in an unofficial capacity during the period of the contract. Of the following individuals,
all were members of the UC-SDRL staff during the period of this contract and participated in some
way in the work of the contract. Many of the individuals, such as Jan Leuridan of LMS, continued to
participate after leaving UC-SDRL for positions in their current companies.
Randall Allemang
Mohan Soni
Filip Deblauwe
Greg Hopton
Kenjiro Fukuzono
Mehzad Javidinejad
Max L. Wei
David Brown
Y.G Tsuei
Tony Severyn
Hiroshi Kanda
Lingmi Zhang
Kelley Allen
Alex W. Wang
-VI "
Robert Rost
Jan Leuridan
Stu Shelley
Frans Lembregts
Chih Y. Shih
Vivian Walls
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TABLE OF CONTENTS
Page
Section
. . . . . . . . .
1. OVERVIEW
1.1 Introduction ......................
1.2 Program Objective ....................
1.3 Program Team .....................
1.4 Program Team Experience ..................
1.5 Program Considerations .................
.
.
.
.
. . . . . . .
.......................
.....................
......................
...................
...................
.
2. MEASUREMENT TECHNIQUES - EXPERIMENTAL MODAL
.........................
ANALYSIS ........................
.......................
2.1 Introduction ......................
....................
2.2 Modal Test Objectives ..................
.......................
2.3 Terminology ......................
.......................
2.4 Modal Testing ......................
...................
2.4.1 Test Structure Set-up ...............
....................
2.4.2 Hardware Set-up ..................
...................
2.4.3 Initial Measurements ................
....................
2.4.4 Non-linear Check ..................
......................
2.4.5 Modal Test ....................
...................
2.5 Modal Data Acquisition .................
2.5.1 Digital Signal Processing . . ...................
.................
2.5.2 Transducer Considerations ..............
.................
2.5.3 Error Reduction Methods ..........
....................
2.6 Excitation Techniques .............
..................
2.6.1 Excitation Constraints ...........
....................
2.6.2 Excitation Signals ............
..............
2.7 Frequency Response Function Estimation ......
.......................
2.7.1 Theory ................
..................
2.7.2 Mathematical Models ...........
..................
...........
H
Technique
2.7.2.1 1
..................
2.7.2.2 H 2 Technique ...........
...................
............
Technique
2.7.2.3 H,
...................
............
Technique
H,
2.7.2.4
................
2.7.3 Comparison of H 1 , H 2 , and H, .......
..................
2.8 Multiple Input Considerations .........
................
2.8.1 Optimum Number Of Inputs .........
...................
2.9 Non-linear Considerations ............
2.9.1 Objectives ...........................................
..............
2.9.2 Modal Analysis and Nonlinearities .......
..................
2.9.3 Basic Nonlinear Systems .........
..................
2.9.4 Excitation Techniques ...........
................
2.9.5 Detection of Non-linearities .........
2.10 Summary - Measurement Techniques for Experimental Modal Analysis .....
........................
REFERENCES ..................
................
3. MODAL PARAMETER ESTIMATION ........
.......................
3.1 Introduction .................
.....................
3.2 Historical Overview ..............
.................
3.3 Multiple-Reference Terminology ..........
..................
3.3.1 Mathematical Models ...........
.....................
3.3.2 Sampled Data ..............
....................
3.3.3 Consistent Data .............
......................
3.3.4 Residuals .................
-vii-
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.
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..
..
..
..
..
....
..
..
1
1
1
2
2
2
4
4
4
5
5
6
6
8
8
8
9
9
9
10
13
13
14
19
19
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21
24
26
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27
28
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31
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33
34
35
36
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3.3.5 Global Modal Parameters ...........
3.3.6 Modal Participation Factors .........
3.3.7 Order of the Model ............
3.3.8 Solution Procedure .............
Characteristic Polynomial ............
3.4.1 Differential Theory .............
3.4.2 Difference Theory .............
3.5 Characteristic Space Concepts ..........
3.6 Summary - Modal Parameter Estimation .......
REFERENCES.. . ..........................................
3.4
.................
................
...................
...................
...................
...................
....................
..................
..............
.....................
4. SYSTEM MODELING ...............
........................
4.1 Introduction
......................
4.2 System Modeling ..............
....................
4.3 Boundary Conditions .............
......................
4.4 Modal Modeling ................
...............
4.4.1 Limitations of Modal Modeling .........
............
4.4.2 Validation of Experimental Modal Models ....
..........
4.4.2.1 Frequency Response Function Synthesis ...
..............
4.4.2.2 Modal Assurance Criterion .......
............
4.4.2.3 Detection of Mode Overcomplexity ....
............
4.4.2.4 Mass Additive/Removal Technique ....
..........
4.4.2.5 Improvement of Norms of Modal Vectors ....
.................
4.43 Modal Modeling Summary ..........
.....................
4.5 Sensitivity Analysis ..............
...............
4.5.1 Limitations of Sensitivity Analysis .......
....................
4.6 Impedance Modeling ...............
..............
4.6.1 Limitations of Impedance Modeling ......
.................
4.7 Component Dynamic Synthesis ...........
................
4.7.1 Dynamic Synthesis Methods .........
................
4.7.2 Damping Synthesis Methods ........
............
4.7.3 A Comparison of the Synthesis Methods .....
..................
4.7.4 Superelement Method ...........
...............
4.7.5 Summary - Superelement Method .......
..................
4.8 Summary - System Modeling ..........
REFERENCES ..............................................
..
..
..
..
..
..
..
..
..
..
..
47
47
49
49
50
50
52
54
56
59
64
64
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66
67
68
68
68
69
69
70
70
71
71
72
73
74
74
77
77
78
78
79
80
.................
5. UNIVERSAL FILE STRUCTURE ...........
.......................
5.1 Introduction ................
....................
5.2 Format Development .............
....................
5.3 Universal File Concept ............
....................
5.4 Future Considerations ..............
........................
REFERENCES ..................
84
84
84
85
85
87
.................
6. SOFTWARE DOCUMENTATION ...........
.......................
6.1 Introduction ................
....................
6.2 Software Compatibility ............
..............
6.3 Data Acquisition Hardware Environment ......
...............
6.4 Modal Analysis Hardware Environment .......
..................
6.4.1 Memory Requirements ..........
...................
6.4.2 Disc Requirements .............
...............
6.4.3 Graphics Display Requirements .........
..................
6.4.4 Plotter Requirements ...........
.........
6.5 Modal Analysis Software - Operating System Environment ...
..................
6.5.1 RTE-4-B (Non-session) ..........
...................
6.5.2 RTE-4-B (Session) .............
88
88
88
89
89
90
90
90
90
91
91
91
-viii -
..
..
..
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6.5.3 RTE-6-VM . . .
. .
. .
. .
. .
. .
. .
.
.......................
6.5.4 RTE-A ................
...............
6.5.5 Operating System Requirements ........
6.6 Modal Analysis Software Overview ........
................
...................
6.6.1 Monitor Structure .............
...................
6.6.2 RTE File Structure .............
6.6.2.1 Project Files ............
...................
...................
6.6.2.2 Modal Files .............
................
6.6.2.3 Universal Files .............
....................
..............
6.6.3 Data Acquisition
....................
6.6.4 Graphics Displays ..............
.................
..........
Estimation
Damping
6.7 Frequency
..................
6.7.1 Error and Rank Chart ...........
...............
6.7.2 Measurement Selection Option .........
...................
............
Estimation
6.8 Modal Vector
BIBLIOGRAPHY - Measurement Techniques ........
NOMENCLATURE ................
. .
..
..
..
.......................
91
91
91
92
92
92
93
93
93
93
94
94
95
96
97
98
108
..............
..................
"ix-
. .
...............
BIBLIOGRAPHY - Modal Parameter Estimation ........
BIBLIOGRAPHY - System Modeling ..........
. .
..
116
123
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LIST OF FIGURES
Figure 1. Typical Test Configuration: Shaker ..........
................
Figure 2. Typical Test Configuration: Impact Hammer ......
.............
Figure 3. Multiple Input System Model ............
..................
Figure 4. System Model for H 2 Technique ..........
.................
Figure 5. Evaluation of Linear and Nonlinear Systems ........
Figure 6. Linear Network ................
.............
......................
Figure 7. Frequency Range of Interest ............
..................
.-
o
16
..
17
20
..
25
30
32
46
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LIST OF TABLES
TABLE 1. Transducer Mounting Methods
...............
TABLE 2. Calibration Methods ..............
................
11
....................
TABLE 3. Summary of Excitation Signals ..........
.................
TABLE 4. Comparison of HI,H 2 , and H. .........
.................
TABLE 5. Frequency Response Measurements ........
TABLE 6. Excitation Signal Summary ............
12
18
..
...............
32
..................
TABLE 7. Summary of Modal Parameter Estimation Methods .....
"-xl"
28
34
..........
58
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1. OVERVIEW
1.1 Introduction
Experimental modal analysis is utilized in a variety of applications in the development of aircraft and
aerospace systems. Initially, only a qualitative comparison of the experimentally derived parameters
with the analytically derived parameters was considered sufficient for verification of the analytical
model. Currently, it is desirable to be able to correct, refine, or even define the analytical model
from the experimental data. Experimentally derived structural dynamic models are frequently desired
for the calculation of the vibrational responses or the dynamic stability resulting from known input
forces to the structure or the determination of the input forces or loads, once the operational
displacements, velocities, or accelerations are available. Other applications of experimental modal
analysis involve the prediction of modal parameters of the complete structure when only the modal
parameters of the individual structural components are known from test. This may include structural
and component modifications to obtain desired dynamic properties of the total structure and may
involve active vibration control systems.
Experimentally derived information concerning structural and generalized mass and stiffness for each
degree of freedom is also required for the purpose of verification of the analytical model. These
requirements dictate a compatibility of the experimental modal analysis theory, test configuration,
and modal parameter estimation algorithms with the analytical modeling approach. This concept
necessitates coordination between the test and the analysis. However, the overriding consideration,
in terms of utilizing experimental results in the evaluation of the analytical model is the accuracy of
the experimental modal analysis approach. This requires that the experimental modal analysis
approach be of primary concern and that the analytical modeling approach must conform to the
experimental procedures that result in the best possible accuracy.
This document (Volume I) serves to overview the technical material presented in Volume II through
Volume VI of this Technical Report. Section 2 though Section 6 of this Volume provide an overview
of Volume II through Volume VI, respectively, Complete technical details are provided only in the
individual Volumes.
1.2 Program Objective
The objective of this effort is a refinement of the experimental modal analysis approach with the
particular constraint of applicability to structural modeling approaches including direct dynamic
modeling, model verification, model perturbation, and component synthesis. This refinement of
experimental modal analysis can be based upon experimental parameters with minimized errors and
predictable error bounds. This broad objective has been attempted in relation to four major tasks
and several subtasks as defined originally by the U.S. Air Force.
* Modal Parameter Identification
"*State-of-the-Art Review
"*Experimental Procedure
"*Modal Parameter Estimation
-1-
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a Generalized Parameter Estimation
9 Structural Parameter Estimation
9 Implicit Force Methods
e Experimental-Analytical Coordination
* Dynamic Component Synthesis
"*Modal Synthesis
"*Modal Sensitivity Analysis
"*Structural Modification
"*Modal Truncation
* Software Development
Two specific goals included an updated state-of-the-art bibliography review in each of the technical
areas included in this effort and a development of a consistent nomenclature that would be used to
present and review technical material in all the areas covered by this effort. The results of both of
these goals are summarized at the end of this document.
1.3 Program Team
The University of Cincinnati, in order to perform effectively and respond to all technical
requirements identified under this research effort, decided to approach this effort on a team basis
with another research facility. The program team consists of the University of Cincinnati Structural
Dynamics Research Laboratory (UC-SDRL) and the University of Dayton Research Institute
(UDRI).
1.4 Program Team Experience
The University of Cincinnati Structural Dynamics Research Laboratory (UC-SDRL) has been
involved in numerous investigations involving experimental modal analysis. The previous studies
include involvement as a subcontractor to The Boeing Company on the investigation entitled
Improved Ground Vibration Test Method and involvement as contractor on the investigation entitled
Simultaneous Multiple Random Input Study. The particular interest in almost every study has been
the sensitivity of each portion of the experimental modal analysis approach to refinement and the
evaluation of the sources of error that contribute to invalid estimates of modal parameters.
Particular examples of this approach has led to the development of the modal assurance criterion,
global least squares modal parameter estimation, and the use of multiple inputs in the estimation of
frequency response functions.
1.5 Program Considerations
This research program encompassed investigations of experimental modal analysis and structural
modeling approaches which are optimized in terms of data base organization, error minimization, and
".2-
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accuracy assessment concerned with estimated structural and dynamic parameters. The research
program involved software development as a necessary element of this evaluation. The approaches
that have been investigated will be applicable to full scale aircraft and aerospace structures,
dynamically scaled models, structural sections removed from the total structure with different
boundary conditions, and the modification of the dynamic properties of substructures or components
of the total structure.
The experimental modal analysis approaches investigated will begin to permit the identification ot the
sources of error. Thc aources of error will include all forms of measurement and data processing
error as well as deviations from the theory involved as a basis for the experimental and analytical
methods. One obvious example of this is the identification of nonlinearities which cause significant
inaccuracy due to deviation from the linear models used in almost every experimental or analytical
approach. Particular experimental procedures were reviewed and a new procedure has been
developed to detect the presence of nonlinearities, identify the general characteristic of the
nonlinearities, and to minimize the effects of the nonlinearities in the presence of a linear model.
The amount of time permitted for the experimental test is frequently a limiting factor of the accuracy
of the modal parameters that are estimated with any experimental modal analysis approach. While
this time constraint is often a function of particular experimental test instrumentation and analysis
equipment, methods that allow for shortened or minimal experimental test requirements have been
considered favorably.
Test instrumentation and analysis equipment, while not of primary concern in this study, has been
considered in terms of general criteria that affect the quality of the resulting experimental data. Some
of the criteria that will be considered are as follows: computer word size, analog-to-digital conversion
word size, autoranging transducer amplification, transducer calibration, actual versus effective
dynamic range, system noise, error generation, error reduction, error accumulation, parallel or
multiplexed signal acquisition, input-output flexibility, data processing, analysis and storage
capabilities, software generation, and etc.
The research has been primarily based upon the frequency response fsnction approach to experimental
modal analysis. This approach will include the estimation of frequency response functions from single
or multiple inputs. Other experimental modal analysis approaches that are not based upon frequency
response functions will be investigated, particularly damped complex exponential approaches that can
conveniently be modified to handle impulse response function data.
Applicable software required for evaluation of experimental modal analysis methods, error
determination, accuracy evaluation, and dynamic modeling method will be developed in the
appropriate computer system during the research phases of the proposed work. Software that
contributes to the goals of this research proposal will be made available in the IHP-5451-C Fourier
System, either under Basic Control System (BCS) or Real Time Executive IV (RTE-IV)
environments. Only software that must be available during the actual acquisition of the data will be
d-veloped in the BCS environment. Analysis oriented software will be made available under the
RTE-IV environment compatible with the RTE Modal Program currently in use by UC-SDRL and
by Eglin AFB. While most structural modeling software can also be made available in the RTE-IV
environment, including modal modification and sensitivity software, the structural modeling software
such as component mode synthesis using modal or impedance models may only be operational on
larger computer systems. If this situation occurs, magnetic tape formats for required data bases will
be developed and supplied.
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2. MEASUREMENT TECHNIQUES - EXPERIMENTAL MODAL ANALYSIS
2.1 Introduction
The most fundamental phase of any experimental analysis is to acquire data that are relevant to
defining, understanding, and solving the problem. When attempting to define a structure
dynamically (usually in terms of impedance functions or in terms of natural frequencies, damping
ratios, and modal vectors), this normally involves measuring a force input to the structure and the
system response to that input as either displacement, velocity or acceleration (all of which are related
through differentiation and/or integration).
These data are sometimes observed, measured and analyzed in the time domain using equipment as
simple as a volt meter and an oscilloscope and forcing functions that are well defined such as single
frequency sine waves. The natural frequencies are estimated by observing peaks in the response
amplitude. Damping can be estimated by a log decrement equation and mode vectors are estimated
by measuring the response at various points of interest on the structure. Phase resonance testing, or
forced normal mode testing, used extensively in the aircraft industry, is a refined version of this
approach.
Advances in hardware and software allowed for the computation of the fast Fourier transform (FFT),
the single input, single output frequency response function and the ability to use these measured and
stored frequency response functions as inputs to parameter estimation algorithms which could
"automatically" estimate natural frequency, damping, and mode shapes and even display "animated"
mode shapes on display terminals. The digital computer, mass storage medium, and the FFT allowed
band limited random noise to be used as the forcing function so that the structure could be tested
faster and the data analyzed or re-analyzed at a later time. But these new techniques also caused
many potential errors, particularly signal processing errors.
In recent years, more advances in the speed, size and cost of mini-computers and other test related
S.irdware have made multi-input, multi-output frequency response function testing a desirable testing
technique.
Volume II of this Technical Report is concerned with the measurement techniques that are widely
used in experimental modal analysis. Although some history is presented, a more complete history
can be found by reviewing the literature identified in the Bibliography that is offered as part of this
report. Also, some present research in the areas of frequency response function estimation, multiple
input considerations, and non-linear vibration considerations is cited.
2.2 Modal Test Objectives
The objectives of a modal test are to make measurements that, as accurately as possible represent the
true force input and system response so that accurate frequency response functions are computed.
These frequency response functions are the input to parameter estimation algorithms. If the data
used as input to these algorithms are not accurate, the parameters estimated by the algorithms are
also not accurate.
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2.3 Terminology
Throughout this report, the nomenclature will follow, as close as possible, the nomenclature found at
the end of this report. Any exceptions will be noted at the time they are introduced.
One potential point of confusion is the concept of system degree of freedom versus a measurement
degree of freedom.
A system degree of freedom is the more classical definition of the number of independent
coordinates needed to describe the position of the structure at any time with respect to an absolute
coordinate frame. Therefore, every potential physical point has six (three linear and three rotational)
degrees of freedom. Therefore, the structure has an infinite number of system degrees of freedom.
While the theoretical number of system degrees of freedom is infinite, the number of system degrees
of freedom can be considered to be finite since a limited frequency range will be considered. This
number of system degrees of freedom in the frequency range of interest is referred to in the following
sections as N.
A measurement degree of freedom is a physical measurement location (both in terms of structure
coordinates as well as measurement direction) where data will be collected. Therefore, for a typical
modal test, the number of measurement degrees of freedom will not necessarily be related to the
number of system degrees of freedom. It is apparent that the number of measurement degrees of
freedom must be at least as large as the number of system degrees of freedom. In general, since three
translational motions are measured at every physical measurement location and since these physical
locations are distributed somewhat uniformly over the system being tested, the number of
measurement degrees of freedom will be much larger than the number of system degrees of freedom
expected in the frequency range of interest. This, though, does not guarantee that all modal
information in the frequency range of interest will be found.
The number of measurement degrees of freedom (the number of physical measurement locations
multiplied times the number of transducer orientations at each physical measurement location) is
referred to in the following sections as m. Note that the number of measurement degrees of freedom
can be used to describe input or output characteristics.
2.4 Modal Testing
The basic goal of any modal test is to determine the damped natural frequency, damping, and in most
cases mode shapes, of a test structure. These are known as the modalproperties or dynamic properties
of a system and are unique to the system and the boundary conditions under which it was tested. In
some cases it is also necessary to compute generalized or modal mass and modal stiffness. Therefore,
by measuring these dynamic properties, the system is defined The results from the modal test are
historically used for one of several purposes. Some of these are:
"*Troubleshooting
"*Finite element model verification
"*Finite element model correction
"•Experimental modal modeling
"•Experimental impedance modeling
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* One of the most fundamental aspects of a modal test is tc decide what the purpose of the modal test
is to be before the test. Too often, the purpose of the test is not stated or is stated too broadly so
that the modal test will be compromised from the start.
In all cases, the modal tests start with acquiring data (usually input and output) from the structure.
B&cause of ;he one to one relationship between the time domain and the frequency domain, the data,
which is always measured in the time domain, may be converted to the frequency domain.
In the time domain, free decay data or impulse response functions, h (t), are used in the estimation
the dynamic properties.
In the frequency domain, frequency response functions, H(w), are estimated. The frequency response
function is then the input to a parameter estimation algorithm used to estimate the dynamic
properties.
There are also modal parameter estimation methods that do not require that intermediate functions
be computed; these methods utilize long time records. Due to practical limitations concerning
archival and retrieval of data in this format, these methods are not addressed in this report.
2.4.1 Test Structure Set-up
The first decision that must be made before any data is collected is the test configuration. Since the
modal parameters that are estimated are for the test structure in the configuration in which it is
tested, the test structure should be in a configuration that, as close as possible, represents the desired
data This means that the boundary conditions are an important consideration when setting up the
test. If the structure is in a free-free configuration, then the modal parameters estimated are for the
free-free case. This is especially important when attempting to verify a finite element model. If the
structure is tested in a configuration that is different from the configuration that was modeled, there
is no chance of correlation. Since the modal parameters that are estimated are for that configuration,
a structure may need to be tested more than once to completely define the structure in its various
operating configurations.
Also in this initial phase of the test, the points to be tested are identified, marked, and measured in
physical coordinates. In most cases, the physical points and associated directions where acceleration
(displacement) is to be measured are selected to give physical significance to the animation. But it is
important that any critical points that need to be measured are also identified.
Another factor that may need to be considered at this time is the ability to access the measurement
degrees of freedom that need to be tested. This may require some ingenuity so that the test
configuration is changed as little as possible during the data collection.
2.4.2 Hardware Set-up
In all cases, it is only possible to estimate the dynamic properties of the system. This is directly a
result of only being able to estimate "true" inputs and responses of the system. It is therefore
imperative that the "best" possible data is collected.
For single input, single output frequency response function testing, a force input to the system must
be measured as well as the system response to that input.
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One of the f irst decisions to be made is the frequency range (fj. tof..,) for the test. The frequency
range must, of course, include all important modes that are to be identified. But. becaus of the
constraints of the parameter estimation algorithms, the number of modes (modal density) should be
kept to a minimum. This may mean that more than one test, using different frequency ranges, needs
to be conducted. Many times, the test frequency range cannot be determined until initial
measurements have been made. An important consideration is that, when uŽing modal modeling
techniques, it is important to identify modes that are higher or lower in frequency than the test
frequency. This will yield more accurate modal models.
Next, the type of excitation and the form of the forcing function must be selected. Sometimes, the
structure may determine the type of excitation. Other times, the use of the data may determine the
excitation. If the purpose of the test is troubleshooting, an impact test may be the best form of
excitation. If a modal model is to be built, more precise input must be used.
If an impact test is to be conducted, the size of hammer and hardness of the impact surface muý; he
selected. This will determine the frequency range of the usable frequency response.
If a shaker is to be used to excite the structure, a forcing signal needs to be selected. This could
include sine, pure random, periodic random, or burst random as well as others that may more cosely
match operating conditions. Section 4 of Volume II of this Technical Report presents mans common
excitation signals and their strong and weak points.
The force input location(s) must be selected to excite all the important modes in the frequency range
to be tested. For multi-input testing, there are other constraints that must be satisfied. Section o of
Volume 11 of this technical report has a complete review of these constraints.
In a typical test, load cells are used to measure the force input and accelerometers to measure
acceleration values which can be related to displacement.
The transducers generally have their own power supply and signal conditioning hardware.
Accelerometers need to be selected such that they have sufficient sensitivity but also low mass to
make measurements of acceleration that accurately define the acceleration of the structure at that
point.
These signals, force and acceleration, are then passed through low-pass anti-aliasing filters, analogto-digital converters, and into the analysis computer.
The computer calculates fast Fourier transforms and all necessary auto and cross spectra needed to
compute a single frequency response function. This is normally stored to disc and another output
selected.
In the case of multiple inputs, many of the potential errors arise from the additional hardware needed
to collect the data required to compute frequency response functions. The potential "bookkeeping"
to be certain that the correct auto and cross spectra are being used in the computations can in itself
be bothersome. A two input, 6 response test necessitates the calculation of 8 auto spectra and 13
cross spectra, each with a real and imaginary part, in addition to the 12 frequency response functions
that are estimated in one acquisition session. There are also 2 load cells, 6 accelerometers, 16 cables,
8 transducer power supplies, 2 exciter systems, 2 signal generators, 8 anti- aliasing filters, and 8 ADC
channels that all have the possibility of failure during the test. It is therefore important to have a
technique to check various components used in the test at selected intervals. Most of these errors can
be eliminated by good measurement practice.
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2.4.3 Initial Measure nents
Once the structure is defined and an input point(s) selected, it is necessary to take initial
measurements to be certain that the input point excites the structure reasonably well over the analysis
range and to be certain that all hardware is operating properly.
Usually, the "driving point" is measured first. This is because the form of the driving point
measurement is well known and defined and also because this should be a "clean" measurement. In
a driving point frequency response function, all peaks in the imaginary part should be of the same
sign (positive or negative), each resonance should be followed by an anti-resonance, and all circles in
the Argand plot lie on the same half of the plane. Many potential problems can be averted based on
this one measurement.
Once the driving point measurement is satisfactory, measurements at remote points are made. This
will ensure that the structure is satisfactorily excited at all points for that force level. In a typical test,
the level of excitation is not changed over the duration of the test. In fact, if the structure is highly
non-linear, this would make the analysis overly complicated.
2.4.4 Non-linear Check
Another important step in a successful modal test is to check for linearity. The basic theory of modal
analysis requires a linear structure. Seldom is the structure under test linear over all but a limited
force range. Linearity is easily checked by exciting the structure at various force levels. If a shift in
natural frequency occurs for different force levels, the structure exhibits some form of non-linear
stiffness. If the amplitude of the frequency response function changes, the structure exhibits nonlinear damping. Section 7 of Volume II of this technical report is an extensive review of non-linear
considerations and of non-linear detection methods.
2.4.5 Modal Test
Once the initial set up is complete, the actual testing phase is simply a process of collecting,
processing, and storing the relevant information. This data will then be used in parameter estimation
algorithms and potentially modal modeling algorithms. For an in-depth review of these areas, other
technical reports, found in the preface, should be consulted.
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2.5 Modal Data Acquisition
Acquisition of data that will be used in the formulation of frequency response functions or in a modal
model involves many important technical concerns. One primary concern is the digital signal
processing or the converting of analog signals into a corresponding sequence of digital values that
accurately describe the time varying characteristics of the inputs to and responses from a system.
Once the data is available in digital form, the most common approach is to transform the data from
the time domain to the frequency domain by use of a discrete Fourier transform algorithm. Since
this algorithm involves discrete data over a limited time period, there are large potential problems
with this approach that must be well understood.
2.5.1 Digital Signal Processing
The process of representing an analog signal as a series of digital values is a basic requirement of
modern digital signal processing analyzers. In practice, the goal of the analog to digital conversion
(ADC) process is to obtain the conversion while maintaining sufficient accuracy in terms of
frequency, magnitude, and phase. When dealing strictly with analog devices, this concern was
satisfied by the performance characteristics of each individual analog device. With the advent of
digital signal processing, the performance characteristics of the analog device is only the first criteria
of consideration. The characteristics of the analog to digital conversion now become of prime
importance.
This process of analog to digital conversion involves two separate concepts, each of which are related
to the dynamic performance of a digital signal processing analyzer. Sampling is the part of the process
related to the timing between individual digital pieces of the time history. Quantization is the part of
the process related to describing an analog amplitude as a digital value. Primarily, sampling
considerations alone affect the frequency accuracy while both sampling and quantization
considerations affect magnitude and phase accuracy.
2.5.2 Transducer Considerations
The transducer considerations are often the most overlooked aspect of the experimental modal
analysis process. Considerations involving the actual type and specifications of the transducers,
mounting of the transducers, and calibration of the transducers will often be some of the largest
sources of error.
Transducer specifications are concerned with the magnitude and frequency limitations that
transducer is designed to meet. This involves the measured calibration at the time that
transducer was manufactured, the frequency range over which this calibration is valid, and
magnitude and phase distortion of the transducer, compared to the calibration constant over
range of interest. The specifications of any transducer signal conditioning must be included in
evaluation.
the
the
the
the
this
Transducer mounting involves evaluation of the mounting system to ascertain whether the mounting
system has compromised any of the transducers specifications. This normally involves the possibility
of relative motion between the structure under test and the transducer. Very often, the mounting
systems which are convenient to use and allow ease of alignment with orthogonal reference axes are
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subject to mounting resonance,. which -eSUlt -n substantial relative rnoiotn hertseen the iransdricer
and the structure under test in the frequency range of interest. T'herefore, the rnou1!)!!1g system
which should bie used depends hcavily upon the frcquzencv range of j!chcr:.si ainsýi upon theý test
conditions. Test conditions are factors such as, temperature, roving oi fixeeý triuvduccts, .no surface
h'w in Table 1.
irregularity. A brief review ofrmarty ccmirn on tr'ansducer mounting rrethods
Transducer calibration rejeirs co the actual engineeting) unui per voltO
it(
h
irsuc
n
signal conditioning :;ysttrm. Calibration of tie. compiete rtleasurernent _swin 1em
nccl(ed( lo verify that
the performance of the transducer and signial conditioning system is proper. 0hvin1u,,y, if the
measured calibration (differs widely from the manufacturers specification~s, the us" of "iat pa-ticular
transducer and signal conditioning path should be cliestioned. Also, certain ,1hi1i1;ica!oc-, such as
10>Ihc- Lip of an
impact testing, involve s~ight changes; in the-- transducer sysztrn (suich as addilret
instrumented hammr)uc
that aftfu~t the ausscw;ared t-aiibration ot th.-
'
f-'
ek1rne boh(, !0eiore anrd ah
Ideally, on-sire cuib rauiz shio' I
transdiucer and signal *:;nditioniiig svstemn is; operating as expected. Theal "i
using the same signal processing and data anilysis ,,quipment, ihat vwill be 1, Cci if-,~
'!
that the
pt-rformed
.
'qitor.
cnad :
signal
There are a number of .calibration r.aethods whit h can he used !o calibrat .11
3S, a
'~'naise,
p
condirionineu Some of these miethoCds yiuld a calibration curve. with
urn
M,''ýt uh
function of frequenc%';while otoi inelhods -,inplv esinuwi a calibration
~i,sie
!c 1for field
calibration methods ate reviewvd in labic 2. Nott tlia, some of the merohod
;nclcitrto
nermaneýnt
Vc
are
more
suited
ios.
while
other
methods
calibration
laboratories [4-91.
2.5.3 Eiiror Reduiction Mvethods
There are several factors that contribute to the quality of ;Actuatl mneatured i2e-q'acy tespornse
function estimates. Some of the most comnmon sources of erro; ;4re 'jue to nicasufrement mistakes.
With a proper measurement approach, most of fllin type ot eiror ,uch as oivrloa3diKng the input,
Iivls nec:rbv, etc., can be
extraneous signal pick-up Via gru()llnd loops o! stroog electri! oý ma~grc!,cN
avoided. Violation of test assumptions are ot'ttn the source of anot~her: inaccuracy. an,' can be viewed
as a measurement mistake. For example, freqiuency respor-se u'.d colicrencc itincc'ens have been
defined as parameters of a linear system. Nonlinearities will gencrally shift energy from one
frequency to many new frequencies, in a way which -nay be difficult to recognize. The result will be a
distortion in the estimates of the systemri parameters, which may not be apparent unless the excitation
is changed. One way to reduce the effect of nonlincaiities is,to randomize thesc contributions by
choosing a randomly different input signal for each of the n measurement.. Subsequent averaging will
reduce these contributions in the same manner that random noise is reduced. Another example
involves control of the system input. One of the most obvious requirements is to excite the system
with energy at all frequencies for which measurements are expected. It is important to be sure that
the input signal spectrum does not have "holes" where little energy exisýt. Otherwise, coherence will
be very low, and the variance on the frequency response function wi li be large.
Assuming that the system is linear, the excitation is proper, and obvious measurement mistakes are
avoided, some amount of noise will be present in the measurement process. Noise is a general
designation describing the difference between the true value and the estimated value. A more exact
designation is to view this as the total error comprised of two terms, variance and bias. Each of these
classifications are merely a convenient grouping of many individual errors which causc it specific kind
of inaccuracy in the function estimate. The variance p~ortionl of the errnr essenioally is Gaussian
distributed and can be reduced by any form of synch ro nization in-the meselctor
analysis
process. The bias or distortion portion of the error causes the expected value ot the estimated
function to be different from the true value. Normally, bias errors are remnoved :possible but, if the
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Method
Hand-held
Frequency
range (Hz)
20-1000
Main advantage
"Quick look"
Putty
0-200
Alignment, ease of mounting
Wax
0-2000
Ease of application
Hot glue
0-2000
Magnet
0-2000
Qutck setting time. good
alignment
Quick setup
Adhesive
film
Epoxycement
Stud
mount
0-2000
Quick setup
0-5000
Mounts on irregular surface.
alignment
Accurate alignment if
carefully machined
Approximate freq ranges.
depends on transducer
mass. and contact
conditions
0-10,000
Main disadvantage
Poor measunng quality for long
sample periods
Low-frequency range. creep problem
during measurement
Temperature limitations, frequency
range limited by wax thickness.
alignment
Temperature-sensitive transducers
Requires magnetic matenal,
alignment, bounce a problem during
impact, surface preparation
important
Alignment, flat surface
Long curing time
Difficult setup
TABLE 1. Transducer Mounting Methods
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Remarks
Method
Inversion
-
Constant
Can only be used with
Frequency
response
transducer that has stable
dc output: calibration
against local earth's
gravity
Calibration against reference
transducer
Constant
Calibration against
j•
.
test
Transducer
2
Comparison
method
t
Reference
transducer
Transduce'
Rectprctty
Reciprocity
method
Mass
T-.--
and/or
:
mass-loaded shaker
frequency
Exciter
response
6P
Constant
Drop
Calibration against local
earth and gravity: used
for ac-couplcd
transducers
method
-[
Ratio
method
]
Frequencyresponse
ratio
"
F
F
-
- (Rigid
m
mass)
TABLE 2. Calibration Methods
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Calibration against knovn
a/F for a ngid mass
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form and the source of a specific bias error is known, many techniques may be used to reduce the
magnitude of the specific bias error.
Four different approaches can be used to reduce the error involved in frequency response function
measurements in current fast Fourier transform (FTT) analyzers. The use of averaging can
significantly reduce errors of both variance and bias and is probably the most general technique in the
reduction of errors in frequency response function measurement. Selective excitation is often used to
verify nonlinearities or randomize characteristics. In this way, bias errors due to system sources can
be reduced or controlled. The increase of frequency resolution through the zoom fast Fourier
transform can improve the frequency response function estimate primarily by reduction of the
leakage bias error due to the use of a longer time sample. The zoom fast Fourier transform by itself
is a linear process and does not involve any specific error reduction characteristics compared to a
baseband fast Fourier transform(FFT). Finally, the use of weighting functions(windows) is
widespread and much has been written about their value [1-3,10,1Ill Primarily, weighting functions
compensate for the bias error(leakage) caused by the analysis procedure.
2.6 Excitation Techniques
When exciting a structure to determine its modal properties, it is important to remember that the
form of the excitation will have an effect on the validity of the estimates of the modal properties. If
the frequency response estimates contain errors, then the estimates of the modal properties will also
contain errors. There are many signals that can be used to excite structures for modal testing. Some
have many advantages over others. The accuracy of the estimates of the frequency response functions
and the time to acquire the data are only some of the differences between the signals.
2.6.1 Excitation Constraints
While there is no well developed theory involving the excitation of structures for the purpose of
estimating the frequency response functions, there are a number of constraints that must be
considered in order to yield an estimate of the frequency response function that is unbiased 12-31
The first constraint that is important to the estimation of the frequency response function is
concerned with digital signal processing. Since most modern data acquisition equipment is based
upon digital data acquisition and Discrete Fourier Transforms, unique requirements are placed on
the excitation signal characteristics. This digital approach to processing the input and response
signals, with respect to the frequency domain, assumes that starting at a minimum frequency and
ending at a maximum frequency the analysis is going to proceed only at integer multiples of the
frequency resolution, therefore matching the limits of the Discrete Fourier Transform. Therefore.
this constraint first indicates that any excitation signal should only contain frequency information
between the minimum and maximum frequency. It also implies that, ideally, either the frequency
content should be discrete and located only at integer multiples of the frequency resolution or that
the excitation should be a totally observed transient.
Both of these methods match the Discrete Fourier Transform equally well, but there are advantages
and disadvantages to both. If the data contains information only at multiples of the frequency
resolution, it is impossible to use a zoom Fourier Transform to achieve a smaller frequency resolution
on the same excitation function. If a new excitation function is created that contains information
only at integer multiples in the zoom band, it is possible to zoom. If the data are a transient, the
signal-to-noise ratio may become a problem.
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The second constraint that is important to the estimation of the frequency response function is
concerned with the requirements of the modal parameter estimation algorithms. A fundamental
assumption in modal analysis is that the structure under evaluation is a linear system or at least
behaves linearly for some force level. While this is never absolutely true, parameter estimation
algorithms are written as though this assumption is valid. With the increasing complexity of the
modal parameter estimation algorithms, violation of this characteristic within the frequency response
function data base renders these algorithms impotent. Therefore, the modal parameter estimation
constraint requires that the excitation signal yield the best linear estimate of the frequency response
function even in the presense of small nonlinear characteristics or a significant nonlinear
characteristic being evaluated around an operating point.
An additional restriction that is important when using multiple inputs is the requirement that the
inputs be uncorrelated. This can be achieved by using deterministic signals, such as sinusoids, with
different magnitude, phases, and frequencies for each input during each average involved with the
estimation of the frequency response function. Normally, uncorrelated inputs are achieved by using a
different random excitation signal for each input. Assuming that a significant number of averages is
involved, the use of uncorrelated random signals, is a simple solution to the requirement that the
excitation signals be uncorrelated.
2.6.2 Excitation Signals
Inputs which can be used to excite a system in order to determine frequency response functions
belong to one of two classifications. The first classification is that of a random signal. Signals of this
form can only be defined by their statistical properties over some time period. Any subset of the total
time period is unique and no explicit mathematical relationship can be formulated to describe the
signal. Random signals can be further classified as stationary or non-stationary. Stationary random
signals are a special case where the statistical properties of the random signals do not vary with
respect to translations with time. Finally, stationary random signals can be classified as ergodic or
non-ergodic. A stationary random signal is ergodic when a time average on any particular subset of
the signal is the same for any arbitrary subset of the random signal. All random signals which are
commonly used as input signals fall into the category of ergodic, stationary random signals.
The second classification of inputs which can be used to excite a system in order to determine
frequency response functions is that of a deterministic signal. Signals of this form can be represented
in an explicit mathematical relationship. Deterministic signals are further divided into periodic and
non-periodic classifications. The most common inputs in the periodic deterministic signal designation
are sinusoidal in nature while the most common inputs in the non-periodic deterministic designation
are transient in form.
The choice of input to be used to excite a system in order to determine frequency response functions
depends upon the characteristics of the system, upon the characteristics of the parameter estimation,
and upon the expected utilization of the data. The characterization of the system is primarily
concerned with the linearity of the system. As long as the system is linear, all input forms should give
the same expected value. Naturally, though, all real systems have some degree of nonlinearity.
Deterministic input signals result in frequency response functions that are dependent upon the signal
level and type. A set of frequency response functions for different signal levels can be used to
document the nonlinear characteristics of the system. Random input signals, in the presence of
nonlinearities, result in a frequency response function that represents the best linear representation
of the nonlinear characteristics for a given level of random signal input. For small nonlinearities, use
of a random input will not differ greatly from the use of a deterministic input.
The characterization of the parameter estimation is primarily concerned with the type of
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mathematical model being used to represent the frequency response function. Generally, the model
is a linear summation based upon the modal parameters of the system. Unless the mathematical
representation of all nonlinearities is known, the parameter estimation process cannot properly
weight the frequency response function data to include nonlinear effects. For this reason, random
input signals are prevalently used to obtain the best linear estimate of the frequency response
function when a parameter estimation process using a linear model is to be utilized.
The expected utilization of the data is concerned with the degree of detailed information required by
any post-processing task. For experimental modal analysis, this can range from implicit modal
vectors, needed for trouble-shooting, to explicit modal vectors used in an orthogonality check. As
more detail is required, input signals, both random and deterministic, will need to match the system
characteristics and parameter estimation characteristics more closely. In all possible uses of
frequency response function data, the conflicting requirements of the need for accuracy, equipment
av3ilability, testing time, and testing cost will normally reduce the possible choices of input signal.
With respect to the reduction of the variance and bias errors of the frequency response function,
random or deterministic signals can be utilized most effectively if the signals are periodic with respect
to the sample period or totally observable with respect to the sample period. If either of these criteria
are satisfied, regardless of signal type, the predominant bias error, leakage, will be eliminated. If
these criteria are not satisfied, the leakage error may become significant. In either case, the variance
error will be a function of the signal-to-noise ratio and the amount of averaging.
Many signals are appropriate for use in experimental modal analysis. Some of the most commonly
used signals are described in Volume II of this Technical Report. For those excitation signals that
require the use of a shaker, Figure 1 shows a typical test configuration; Figure 2 shows a typical test
configuration when an impact form of excitation is to be used. The advantages and disadvantages of
each excitation signal are summarized in Table 3.
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FOURIER
ANALYZER
SUGNAL
CHARGE
ANTI.ALIASING
3ENERATOR
AMPUiFIERS
FILTER
DI
\%xa\*TEST SPECIMEN
SHAKER
Figure 1. Typical Test Configuration: Shaker
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FOURIER
ANALYZER
CHARGE
AMPUFIERS
ANTI-ALIASING
FILTERS
ACEEROMETER
TEST SPECIMEN
Figure 2. Typical Test Configuration: Impact Hammer
.17-
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Minimize Leakage
Signal-to-Noise Ratio
Steady
Slate
Sine
No
Very
Type of Excitation
Pure *
Psuedo
Random
Random
Random
I
No
Yes
Yes
Fair
Fair
Fair
RMS-to-Peak Ratio
Test Measurement Time
High
Very
Fair
Good
High
Fast
Sine
Impact
Burst
Sine
Burst
Random
Yes
High
Yes
Low
Yes
High
Yes
Fair
High'
Very
Good
Yes
Fair
Very
Good
Yes
Controlled Frequency Content
Yes
Yes
Fair
Very
Short
Yes
Controlled Amplitude Content
Yes
No
Yes
No
Yes
No
Yes
No
Removes Distortion
Characterize Nonlinearity
No
Yes
Yes
No
No
No
Yes
No
No
Yes
No
No
No
Yes
Yes
No
Long
Fair
Fair
High
Fair
Yes
Yes
Low
Very
Good
No
Requires special hardware
TABLE 3. Summary of Excitation Signals
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2.7 Frequency Response Function Estimation
The theoretical foundation for the estimation of modal parameters has been well documented.
Historically, modal testing was first done using the phase resonance, or forced normal mode testing
method. Using this method, the structure was forced into a normal mode by a number of single
frequency force inputs. The frequency, damping, and modal vector could then be estimated.
With advances in computer technology (both hardware and software), and especially the
development of the Fast Fourier Transform, it became practical to estimate frequency response
functions for random data. The theoretical foundation for the cor utation of frequency response
functions for any number of inputs has be well documented (1-3,22. A single input, single output
frequency response function was estimated for all test points. This greatly reduced test time. But, in
order to insure that no modes had been missed, more than one input location should be used.
Starting in about 1979, the estimation of frequency response functions for multiple inputs has been
investigated [27,.-] The multiple input approach has proven to have advantages over the single
input approach. When large numbers of responses are measured simultaneously, the estimated
frequency response functions are consistent with each other.
2.7.1 Theory
Consider the case of N, inputs and N, outputs measured during a modal test on a dynamic system as
shown in Figure 3. Equation I is the governing equation.
(1)
X(w) = H(w) * F(w)
For simplicity, the w will be dropped from the equations. Since the actual measured values for input
and output may contain noise, the measured values are:
F
v
and
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Therefore, a more general model for the computation of frequency response functions for Nj inputs
and N, outputs could be at response location p:
Xp -
nE H, *
- v.)
(2)
q=1
Where:
F
X
=
=
,YP
= Spectrum of the p-th output, measured
F,
= Spectrum of the q-th input, measured
Hpq
= Frequency response function of output p with respect to input q
i
rip
v- Actual input
1- Actual output
=Spectrum of the noise part of the input
ý-Spectrum of the noise part of the output
=
tVN2
F, +
F2 +
+
FN
H
X,
X2
+h
+
x1
N
+
k2
fix.
XNj
Figure 3. Multiple Input System Model
If N, - N. = 1, Equation 2 reduces to the classic single input, single output case. With N, not equal 1.
the equation is for the multiple input case.
For the multiple input case, the concept of coherence must be expanded to include ordinary, partial,
and multiple coherence functions [3o,35]. Each of the coherence functions is useful in determining the
validity of the model used to describe the system under test or, as discussed in Section 6 of Volume II
of this Technical Report, to evaluate how well the inputs conform to the theory.
Ordinary coherence is defined as the correlation coefficient describing the possible causal
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rehid
hilj
e
\ t,.,,o ",igiis. Ordinary coherence can be calculated between any two forces
or belý,,_n A lý ib ,, i;i! .miri; c:•.ponse In this calculation, the contribution of ail other signais is
ignored f heiý-tie. tt! niwciretation of the ordinary coherence functions must be made with gieat
care. Uhe use and interprctation of the ordinary coherence function between forces will be discussed
in Sectior tot \Voilme 1Iof this Technical Report. The ordinary coherence function between an
input and an ouirpo is of little use in determining the validity of the model This is because the
output in the- maitipie input case is due to a number of inputs so that the ordinaiy coherence will nor
have the same useful interpretation as in the single input case.
&.='•."•,
Parti.a coherence is defined as the ordinary coherence between any two conditioned signals. The
signals are conditr;nmd by removing, in a systematic manner, the contribution(s) cr other .,igr.,ls.
fhe order of cnditioning has an effect on the degree of correlation. A partial c,,hicence function
can he calculated between conditioned inputs, a conditioned output and a conditiorcd input, oi A;tJi
multiple outrput.., {'.'.en conditioned outputs. Typically, the input and ourn-•ut ar-e w.Onditioned I1v
removing the p,.ten.i l contributions to the output and input from other input(s) Thf-sreremoval of
,- c t:• r ,. ...... ..
,• , ,. frnilulited on a linear least -quares b,_t'
Ih .. ..
,
coherence fuinction tor every input/output combination for all permutations of w.onditioning. The
usefulness of partial coherence with respect to frequency response function c.-.timat~on :s to
determine the degree of correlation between inputs. The use and interpretation of the partial
coherence will be discua-,ed In Section 6.
Multiple coherence Is defined as the correlation coefficient describing the possible ,.:usal relationlhip
between an output aid all known inputs. There will be one multiple coherence luncmion tor eewr
output. Multiple coherence is used similarly to the ordinary coherence in the single input case. The
multiple coherence function should be close to unity throughout the entire frequency, range ol the
estimated freouency response function. A low value of multiple coherence at resonance indi,..ates
possible measurement error, unknown inputs, unmeasured inputs, or signal processing errors such as
leakage. However, a low value of multiple coherence is not expected at an antiresonance since there
should be sufficient signal-to-noise ratio at these frequencies (antiresonance is not a global property
of the system).
2.7.2 Mathemalical Models
l)epcnding oin tile Abere the noise is assumed to enter the measurement proce,,s, there are at least
three diffe!ent mathematical models that can be used to estimate the frequency response fi,,tions.
It is important to remember that the system determines its own frequent:" i24)onse tunction for a
given input/output pair and the boundary conditions for the test. In the limit, if all noise were
removed, any estimation technique must give the same result.
2.7.2.1 H, Technique
Assuming that there are no measurement errors on the input forces, let the measurement errors on
the response signal be represented by {(Y}. The H, least squares technique aims at finding the
solution [H] of Equation 3 that minimizes the Euclidean length of {,i}, the "squared error". This
solution is also called the least squares estimate. Writing Equation 2, using all measured values (the
ha.s been dropped for simplicity) in a form more readily recognized yields -"1:
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[HIN.x
{F}IN, 1 = {X}N0o ,X" {i7}No x1
(3)
The subscripts refer to the size of the matrix. It is well known that the solution HI] can be found as
the solution of the set of "normal equations" formed by post multiplying by {F}HI339].
{ 7 )}{FIH
[H I{F} {F}H ={X} {F}1-
(4)
I I,I I isminimum
Where:
H: complex conjugate transpose (Hermitian)
SII. 112 : Euclidean norm
Equation 4 can be reduced to Equation 5 by assuming that the noise on the outputs are uncorrelated
with the inputs and that with sufficient averages, the normalized noise spectra are close to zero.
[H
.,I, {F,,,.x1{F} 9xN = {X},N. XI{F}xN
xIN.
1
(5)
The elements of the coefficient matrix and right hand matrix in Equation 5 are readily identified,
when expanded, with the auto and cross power spectra of input forces and response signals [3o-35].
When the matrix multiplications of Equation 5 are expanded to form Equations 6 or 7, the form is
more readily recognized as a frequency response function estimation.
[H I[GFF]=[GXF]
(6)
[H ]=[GXF I[GFFI1-
(7)
or
Where:
[H]
=
Frequency response function matrix
[H 11
H
H1 2
H. N
.
21
[H;
.
.
HN.JN
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[GXF ]= Input/output cross spectra matrix
{X}{F}'
li1
=
X2"
complex conjugate
*
[GFF]= Input cross spectra matrix
=
=
{F{F}F
{
l[F
F
...
1 .F
GFFNvI ..
.;
F,]
GFF~v
1
GFFN.Nvj
GFFak= GFFL (Hermitian matrix)
The ordinary coherence function can be formulated in terms of the elements of the matrices defined
previously. The ordinary coherence function between the p-th output and the q-th input can be
computed from Equation 8.
I GXFpq,12
(8)
COHPI = GFFqq GXXp(
Where:
GXXp, = Auto power spectrum of the output
The magnitude of the error vector that corresponds to the least squares solution is a measure of how
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well the response signal is predicted by the input forces. When compared with the magnitude of the
response signal a normalized measure, known as the multiple coherence function, can be defined by
Equation 9-[34'35).
N, N,
MCOHp
E
Hp, GFF, H;
GXXJ,
(9)
-I t=.1
Where:
H,.
H,,
For more
10. Note
functions
equations
= Frequency Response Function for output p and input s
= Frequency Response Function for output p and input t
than 2 inputs, Equation 6 can be expanded, as an example, for six inputs to yield Equation
that Equation 10 has been put in transposed form in which the frequency response
appear as a column instead of a row. Equation 10 is recognized as a set of simultaneous
with ';'requency response functions as the unknowns.
GFF11
GFF12
GFF16.
GFF6 1
HPI .
.
Hp,2
GXFpI
GXFp 2
Hp6
GXFý'
GFFee
(10)
6
Equation 10 could be solved for the frequency response functions by inversion of the [GFFJ matrix
but the computational time and possible dynamic range errors may make the inversion technique
undesirable P14,4o. Computational techniques for solution of the equation are discussed in Volume 11
of this Technical Report.
As before, ordinary coherence functions can be defined between any two forces or any force with the
response giving a total of 21 possible ordinary coherence functions. In a systematic way, 4 partial
coherence functions between forces can also be defined and one multiple coherence function can be
defined by Equation 9. The partial coherence functions are defined and discussed in Section 6 of
Volume II of this Technical Report.
2.7.2.2 H
2
Technique
If all measurement errors are assumed to be confined to the inputs, let the errors associated with the
inputs be represented by {v}. The H 2 least squares technique aims at finding the solution [HI of
Equation 11 that minimizes the length of {(}. The basic model for the H 2 technique is shown in
Figure 4 [41,42]
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[HI•o.N,
(11)
{ {F}N, xr +{v}NlxI} ={X}Nox
F,
t2
F2
V+
F1
F2
H
Figure 4. System Model for H 2 Technique
To find the solution of Equation 11 in a similar manner as in the H, case, postmultiply by {X}li.
[HI{ {F} + {v}
{X}I
(12)
={X} {X}R
If the noise on the input {v} is assumed not correlated with the output and if sufficient averages are
taken so that the noise matrix approachs zero, Equation 12 can be written as:
[H]•No x×N, {F},vN
{X}•to × = {X}
(13)
x {X}•×xNo
The elements of the matrices are now identified as the cross power spectra between inputs and
outputs and the output auto power spectra.
To investigate the potential uses of the H 2 technique for multiple inputs, it will be helpful to expand
the equations for two cases. One is when the number of inputs and the number of outputs are equal
and another when the number of outputs is greater than the number of inputs.
For the case of two inputs and two responses, Nk = Ni = 2, Equation 13 can be solved for the
frequency response functions by inverting the input/output cross spectra matrix at every frequency in
the analysis range and solving the set of simultaneous equations. For the case of two inputs and
three responses, N. - 3 and Ni = 2, Equation 13 suggests that a 3 x 3 matrix must be multiplied by a
2 x 3 matrix. For the equation to be valid, a generalized inverse must be used. Therefore, unique
frequency response functions cannot be estimated from this set of data. Therefore, an added
constraint on the H 2 technique is that the number of outputs must equal the number of inputs [42,43]
For the single input, single output case, this constraint is not a disadvantage. But, for the multiple
input technique, this constraint makes the H 2 technique impractical for many testing situations (for
example a 2 triaxial response test with 2 inputs). Also, the major advantage of the H 2 technique is to
reduce the effects of noise on the input. This can also be accomplished by selective excitation that is
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investigated in Section 4 of Volume II of this Technical Report or by formulating the H, frequency
response estimate which is better suited for multiple inputs. Therefore, the H2 technique was not
heavily investigated for the multiple input case.
2.7.2.3 Hv Technique
Assume now that measurement errors are present on both the input and the response signals,
represented by {v,} for the noise on the input and {r)} for the noise on the output. The H, least
squares technique aims at finding the solution [H] in Equation 2 that minimizes the sum of the
Euclidean lengths of {i,} and {v}, or the "total squared error" [42,44-46]. This solution is referred to as
the Total Least Squared estimate. It is proved in the literature that it can be identified with the
elements of the matrix [GFFX] defined by Equation 14. Again the elements of this matrix are readily
identified with the auto and cross power spectra of input forces and response signals.
[HI { {F}. -{v} } ={x - {,n}
(13)
[GFFX] = [{F} {X}] 0 [{F} {X}]
(14)
[[I
= [GFF]
[GFX
(14)
[GFFX] [[GFXIH GXX(
The matrix [GFFX] is Hermitian; its eigenvalue decomposition is therefore defined by Equation 15.
The Total Least Squared estimate for [H] is then defined by Equation 16.
(15)
[GFFX] = [V] r A j [VIE
Where:
I Aj = diag (A,X,
A2.. ,,)
[VP [VI =I
(.Vlp1
{H}
.
VP+1 i?+1,
=
(16)
-P+1vp+1 +
Notice that the Total Least Squares solution does not exist if Vp,,p
1 l equals 0. This however can only
happen if the submatrix [GFF]of [GFFX] is singular 144): that is, if the input forces are correlated.
Verifying that the input forces are not correlated is therefore sufficient to warrant the existence of
the Total Least Squares solution.
Corresponding to the Total Least Squares estimate, there will be errors on both input forces and
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response signals. The magnitude of the errors on the response signal can be expressed by Equation
17. If this error is substituted into Equation 18, one calculates a measure of how well the response
signal is predicted by the input forces, considering now however also errors on the input forces.
GYM
=
A÷1"+p+1.V+l
p+1
(17)
MCOH = 1-_ 9 G
GXX
(18)
2.7.2.4 H, Technique
In a similar fashion, a "scaled" frequency response function has been proposed by Wicks and Void
[471
Starting with Equation 2 for a single input (the equations can be readily expanded to the multiple
input case):
X - Y= H*(F - v)
(19)
Expanding for the single input case and collecting error terms yields:
Y '7 + (HH*) (uv*) = (HPF
) (HFP -k)
(20)
If the error terms of Equation 22 are equal in magnitude, a least squares minimization can be applied
to Equation 22. To insure that the magnitudes are equal, either the input or the output can be
scaled. Assuming that the input is scaled by S, Equation 21 can be written as:
xk
-,7
= HS(F - v)
If the scaling constant is carried throughout the development, an equation can be written for a
"scaled" frequency response function.
CX•
Hs= -(21)
'" s2
P
) +,(s
2.7.3 Comparison of H,
.-k,)+4s2k"
"k
2SX F
H 2 , and H,
The assumption that measurement errors are confined totally to the input forces or totally to the
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response signals is sometimes unrealistic. But, it is important to understand why H1 ,H2 , and H,
yield different estimates of the same input/output frequency response function for a given system. It
is important also to remember that a linear system has only one theoretical frequency response for
any given input/output pair. Table 4 compares the different assumptions and solution techniques.
TABLE 4. Comparison of H1 .H 2 , and H,
Technique
Solution
Method
H,
H2
LS
LS
TLZ
Assumed location of noise
Force Inputs
Response
no noise
noise
noise
noise
no noise
noise
It is also important to realize that if the noise on the inputs {(7} and the noise on the responses {v}
are eliminated, H, equals H 2 and they are approximately equal to H,. Therefore, it is important to
spend time to acquire data that is noise free and that fits the assumptions of the Discrete Fourier
Transform rather than accept the errors and try to minimize their effect by the solution technique.
From the standpoint of frequency response function estimation, the H1 technique, at resonances,
underestimates the height of the peak amplitude and therefore overestimates the damping. In the
Argand plane, the circles look "flat". The H2 technique, at resonances, overestimates the amplitude
and therefore underestimates damping. The circles look oblong in the H 2 technique. The H.
technique gives, at resonance, an estimate of the frequency response function that is between the H,
and H2 estimates. At antiresonances, the reverse is true, H1 gives the lowest estimate and H 2 gives
the highest estimate with H, in the middle. Away from resonance, all three give the same estimate. It
is important to remember that in all three cases, the value computed is only an estimate of the
theoretical frequency response function. If other measurement errors or violation of system
assumptions are present, all three estimators will give erroneous results. It is therefore important to
spend as much time as possible to reduce known errors before data acquisition begins.
2.8 Multiple Input Considerations
From the theory of multiple input frequency response function estimation, the equations for the
computation of the fre uency response functions all require that the input cross spectra matrix
[GFF] not be singular [T-"2' -1. Unfortunately, there are a number of situations where the input
cross spectra matrix [GFFI may be singular at specific frequencies or frequency intervals. When this
happens, the equations for the frequency response functions cannot be used to solve for unique
frequency response functions at those frequencies or in those frequency intervals even though the
equations are still valid.
One potential reason for the input cross spectra matrix [GFF] to be singular is when one or more of
the input force auto power spectrum is zero at some frequency or some frequency interval. If an input
has a zero in the auto power spectrum, the associated cross spectrums calculated with that force will
also have zeros at the same frequency or frequency interval. The primary reason for this to occur
would be because of an impedance mismatch between the exciter system and the system under test.
Unfortunately, this situation occurs at system poles that have a low value of damping where a good
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estimate of the frequency response is desired. Therefore, it is imperative to check the input cross
spectra matrix for zeros. For the two input case where the determinant is calculated, a good check is
to be sure that the determinant does not have zeros in it.
Another way that the input cross spectra matrix may be singular is if two or more of the inputs are
totally coherent at some frequency or over some frequency interval. A good method to check for
coherent forces is to compute the ordinary and conditioned partial coherence functions among the
inputs [t53o]. A technique is also presented in Section 6.2.2 of Volume II of this Technical Report
that computes the principal auto power spectra of the input forces t481. This technique uses an
eigenvalue decompnsition to determine the dimensionality of the input cross spectra matrix [GFFI. If
two of the inputs are fully coherent, then there are no unique frequency response functions
associated with those inputs at those frequencies even though the Equation 7 is still valid. This is
because the frequency response is now estimated using a singular matrix that will yield infinite
solutions that are combinations of each other. Although the signals used as inputs to the exciter
system are uncorrelated random signals, the response of the structure at resonance, combined with
the inability to completely isolate the exciter systems from this response will result in the ordinary or
conditioned partial coherence functions to have values other than zero, particularly, at the system
poles. As long as the coherence functions are not unity at any frequency, the equations will give a
correct estimate of the frequency response function. It is therefore necessary to have a method to
evaluate the inputs to assure that there are neither holes in the auto power spectrum nor perfectly
coherent inputs.
2.8.1 Optimum Number Of Inputs
When considering the estimation of frequency response functions in the presence of multiple inputs,
more time must be spent to determine the number of inputs, the input directions, and the input
locations.
An advantage of the multiple input technology is that, for most structures, all important modes can
be excited in one test cycle. For example, in a typical test of an aircraft structure, if existing single
input technology is used, at least two complete tests must be conducted in order to get sufficient
energy into both the vertical and lateral fuselage modes. If two symmetric, correlated inputs with zero
or 180 degree phase difference are used, even though the number of degrees of freedom that the
parameter estimation algorithm must deal with is reduced, at least two complete tests must also be
conducted to define all the modes of the structure. With uncorrelated random multiple inputs, since
there is no constraint on the input directions, one input could be vertical and the other horizontal. In
this way, both the vertical and lateral modes will be excited in the same test cycle. By exciting at
symmetric locations, the frequency response estimates can be added or subtracted to enhance in
phase and out of phase modes. Since the original frequency response estimates are not destroyed,
effectively, four pieces of useful information have been estimated for the structure under test in one
test cycle.
But, as the number of inputs is increased, so too is the potential for problems with the excitation
forces. One such problem is that, due to the structural response, the inputs may be correlated by one
or more exciters driving the other exciters. This happens most often if the exciters are placed at
locations that have a high amplitude of motion particularly at resonance. Also, depending on the size
of the structure, there is a diminishing return on more inputs. The advantage of two inputs to one
input has been apparent in almost every test case. For more than two inputs, particularly on smaller
structures, the added inputs mean that more averages must be taken to compute "clean" frequency
response functions. In practice, fighter aircraft have been tested with as many as six inputs with no
adverse effects. For automobiles, three inputs appears to be a practical limit.
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2.9 Non-linear Considerations
The vibration of structures is a very natural phenomenon and although much work has been
dedicated to the analysis and understanding of it, there exist an infinite number of vibrational
problemb which cannot be predicted. Because of this, experimental testing is needed to describe the
vibrational characteristics of many structures. In this research, a linear system will be defined as a
system in which the responses are linearly proportional to the input forces. A nonlinear system will
be defined as having responses that are not linearly proportional to the input forces. In the analysis
of linear systems, the responses can be predicted and an explicit mathematical model can be
generated to represent the physical characteristics of the system. In the analysis of nonlinear systems,
the responses can not adequately predict the measured dynamic characteristics.
It is accepted that most real structures exhibit nonlinear characteristics. Practical experience suggest
that in many cases, this nonlinear term is negligible and a linear system can be assumed. However, as
structures become more complicated and more accurate results are required, the nonlinear
component is no longer negligible.
In modal analysis, frequency response functions are calculated based on a linear model of the
structure. Thus, it is important to first accurately determine the contribution of the nonlinear
components to the system. As described in Figure 5, the contribution of nonlinear components varies
from system to system.
Linear System
Application of
Nonlinear Models
Negligible
Errors
'Weak'
Nonlnearities
'Strong'
Nonlinearities
Application of
Linear Models
Application of Perturbed
Linear Modal
A
B
C
Figure 5. Evaluation of Linear and Nonlinear Systems
At position "A", there exists a totally linear system having no contamination of nonlinear
components. Between positions "A" and "B" the system is composed of linear and nonlinear terms;
however, the nonlinear component in this case is small enough to be negligible. Between positions
"B" and "C", the nonlinear term can no longer be negligible - the system should not be assumed to
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be linear. Finally, past position "C" the system is considered to highly nonlinear [49]
2.9.1 Objectives
One of the primary objectives of this research is to investigate previous studies dealing with
nonlinearities as related to the modal analysis field. A review of current studies which deal with
nonlinearities was made in terms of a literature search. Because of the difficulty involved in creating
a physical structure with a known nonlinearity, most of the past research has only dealt with basic
mathematical nonlinear structures having only one or two degrees of freedom.
Because of the increasing utilization of the multiple input estimation technique, it is the ultimate goal
of this research to investigate a method for detecting nonlinearities in a system when using the
multiple input estimation technique combined with random excitation signals. What is eventually
desired is a detection method which could be programmed into the modal acquisition software and
accessed to evaluate the degree of nonlinearity within a test structure.
2.9.2 Modal Analysis and Nonlinearities
In the field of experimental modal analysis there are three basic assumptions that are made about a
structure. First, the structure is assumed to be time invariant. This means that the modal parameters
that are to be determined will be constants of the structure. In general, a structure will have
components whose mass, stiffness, or damping depend on factors which are not measured or included
in the model. For example, in some structures, the components could be temperature dependent. In
this case, temperature could be described as a time varying parameter; therefore, each of the
temperature dependent components would be considered to be time varying. Thus, for a time
varying structure, the same measurements made at different times would be inconsistent.
The second basic assumption is that the structure is observable. By observable, it is meant that the
input/output measurements that are made contain enough information to generate an adequate
behavioral model of the structure. For example, if a structure has several degrees of freedom of
motion that are not measured, then the structure is considered to be not observable. Such a case
would be that of a partially filled tank of liquid when sloshing of the fluid occurs
Finally, the third basic assumption is that the structure is either linear or can be approximated as
linear over a certain frequency range. This essentially means that the response of the structure due to
the simultaneous application of two or more excitation forces is a linear combination of the responses
from each of the input forces acting separately. This relationship is shown in Figure 6. If a particular
input signal, a (t), causes an output signal, A (t), and a second input signal, b (t), causes a different
output signal, B (t); then, if both input signals, a (t) and b (t), are applied to a linear system, the
output signal will be the summation of the individual signals, A (t)+B (t) [5I]
2.9.3 Basic Nonlinear Systems
* A linear system with several degrees of freedom can be modeled completely by a frequency response
function which can be defined as the Fourier transform of the output signal divided by the Fourier
transform of the input signal. In mechanical systems, the input signal is a type of force while the
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a(t)
> Network
b(t)
Network
a(t) + b(t)
-
;A(t)
.B(t)
Network --------
A(t) + B(t)
Figure 6. Linear Network
output signal is a quantity such as displacement, velocity, or acceleration. The frequency response
function of these different output quantities are referred to as receptance, mobility, or compliance
respectively, see Table 5.
TABLE 5. Frequency Response Measurements
Acceleration
Force
Receptance
Force
Acce
Effective Mass
Acceleration
Mobility
Velocity
Force
Force
Voci
Impedance
Velocity
Displacement
Force
Force
Dslce
Displacement
Dynamic Compliance
Dynamic Stiffness
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As kaJ pry u,>IV stated, a linear structure can be characterized by its frequency response function
and, ;i, long as the structure does not physically change, this function will remain constant. This is not
the c;i.se tor a nonlinear structure; the system is no longer characterized by a single response function.
In this c,.-c, the structure is very dependent to the time varying variables in the inputs of the system.
Altha1.1n,, there exist very linear systems, most real structures have nonlinear components. Practical
eXpCr.nLe Ihows that the degree of nonlinearity of a structure varies according to the characteristics
of the oystemn. ihat is, welded structures will usually exhibit a linear response; where a riveted or spot
welded sti ucwure exhibits a very nonlinear response
f-owevci. thi-s inear characteristic property is not found in many systems. In a simple coil spring, a
nnlinear behavior will occur when the spring is overly compressed or extended. In either case, the
e!asti, 'ýprm1 will exhibit a nonlinear cha.,•,aeristic such that the spring force incrca.ses more rapidly
than th•e spiing deformation; this is referred to as a hardening spring. On the ether hand, certain
systems -uch as the -imple pendulum exhibit a softening characteristic.
Nonlinear behavior in structures can be related to such characteristics as backlash, nonlinear stiffness,
nonlinear damping, nonlinear material properties, or friction. These nonlinearities can be classified as
"limited" or "nonlimited" nonlinearities. In the "limited" case, the nonlinearity is limited within a
partic•ar force level range. Nonlinearities due to backlash could be classified as a "limited"
norlinci•,i. "Nonlirnited" nonlinearities refer to those nonlinearities which are independent of force
level. N,;nlirear damping is an example of a "nonlimited" nonlinearity [54].
2.9.4 Excitation Techniques
When a structure is to be tested to determine the modal parameters, one of the most important
consideiations is the excitation method to be used. For a linear structure, the frequency response
function is independent of the amplitude and type of the excitation signal. This is not the case for a
iionlincai structuie: the selection is crucial since the method of excitation can either minimize or
enhanice the nonlinear behavior of the structure. The different excitation signals can be divided into
two classifications; the determinististic excitation signals and the random excitation signals.
The deterministic signal is one which can be described by an explicit mathematical relationship. These
signals are then divided into two other classifications, periodic or non-periodic. A signal is periodic if
it repeats itself at equal time intervals. Frequency response functions that result from deterministic
signals are dependent upon the signal level and type. Therefore, these signals are very useful in
detecting nonlinearities in structures. Table 6 gives a summary of the different excitation signals and
their classification.
The use of a sine wave, which is a deterministic signal, to excite a structure is very common [55). The
main advantage of a swept sine test is that the input force can be precisely controlled. It is this
characteristic that makes this method particularly useful when trying to identify nonlinear systems. If
a particular system is nonlinear, by varying the input force levels, one can compare several frequency
response functions and identify inconsistencies. A major disadvantage with this method is that it
gives a very poor linear approximation of a nonlinear system. This causes a serious problem if the
data is to be used to estimate modal parameters. Therefore, this method is adversely affected by
nonlinearities.
Another deterministic excitation signal is an impact signal. The impact testing technique is very useful
for trouble-shooting and preliminary modal surveys. However, this technique should not be utilized
with nonlinear structures because of the difficulties in controlling the impact force and insufficient
energy to properly excite the structure [561
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TABLE 6. Excitation Signal Summary
Deterministic Excitation
Random Excitation
Slow Sinusoidal Sweep
Fast Sinusoidal Sweep
Periodic Chirp
Impulse (Impact)
Step Relaxation
True (Pure) Random
Pseudo Random
Periodic Random
Burst Random (Random Transient)
The random signal is one which can only be described by its statistical properties over a given time
period; no explicit mathematical relationship exists. For structures with small nonlinearities, the
frequency response functions from a random signal will not differ greatly from that of a deterministic
signal. However, as the nonlinearity in the structure increases, the random excitation gives a better
linear approximation of the system, since the nonlinearities tend to be averaged out. The random
excitation signal has been increasingly useful since it enables the structure to be investigated over a
wide frequency range, unlike the sine wave [9]
2.9.5 Detection of Non-linearities
It is apparent that in physical mechanical systems, there exists a linear and nonlinear response due to
some force input. In many cases, this nonlinear behavior can be neglected. However, in other cases,
the nonlinear response cannot be ignored. It is and has been essential to perform some type of
linearity check in order to make this evaluation. As the use of dynamic models based on experimental
data becomes more extensive, the detection and eventually the characterization of nonlinearities
becomes even more important.
The work reviewed in Volume TI of this Technical Report provides several nonlinearity detection
methods which are currently being implemented. Each of these different techniques has certain
limitations which effect the accuracy of the detection method. As an alternative detection method,
this work researched the possibility of detecting nonlinearities by utilizing higher order terms in
conjunction with the multiple input/output estimation theory. By using higher order terms of a
measured input force, the nonlinear behavior of a system can be detected. This alternative technique
was researched as a fast and valid method to give an indication of the linearity of a system when
implementing a random type of excitation signal.
This research demonstrated that for a theoretical single degree-of-freedom system, as the amount of
nonlinearity increases, the nonlinear detection functions become more predominant. This preliminary
study indicates that this nonlinear detection technique is sensitive to different types of nonlinearities
if the correct number of higher order nonlinear terms are utilized. Further research is needed to
determine the actual number of higher order terms necessary to accurately model a particular
nonlinear system. To further validate this detection technique, the frequency response functions
which are estimated utilizing the multiple input theory should be compared to the frequency response
functions estimated by a single input/output algorithm. Although initial investigations demonstrated
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that the utilized forces were uncorrelated in the frequency domain, additional research is needed to
study the relationship between these forces. As in most cases, this detection method is only valid for
the measured response and input force points; it is still necessary to check several critical points of
the structure for linearity and then assume the entire system behaves accordingly. The higher order
detection technique does however eliminate having to perform a linearity check at different force
levels for comparison purposes. The technique will give an indication of the amount of nonlinearity
present in the structure at a particular force level. Further research is needed to evaluate an
acceptable level at which linearity within the system can be assumed.
2.10 Summary - Measurement Techniques for Experimental Modal Analysis
In the material contained in Volume II of this Technical Report, the area of measurement techniques
applicable to experimental modal analysis is discussed in some detail. This review is primarily
concerned with the accurate measurement of frequency response functions on linear, time invariant.
observable structural systems. Much effort was spent on the understanding of the many diffcrent
algorithms used to estimate frequency response functions. It is most important to understand that if
the noise in the measurement problem is reduced to zero (variance and bias errors) all of the
frequency response functions reduce to the same form. When attempting to experimentally
determine the dynamic properties (natural frequency, damping, and mode shapes) of a structure, one
of the most important aspects is to collect and process data that represent the structure as accurately
as possible. These data can then be used as input to an number of parameter estimation algorithms
and could also be used in modal modeling algorithms. Volume II of this Technical Report describes
in detail the procedure used to collect these data. Many of the potential errors are discussed as well
as techniques to eliminate or reduce the effects of these errors on the quality of the results. If the
procedures described in this Technical Report are followed, data can be collected, as input to modal
parameter estimation algorithms, that will yield accurate dynamic properties of the test structure.
With care and attention to theoretical limitations, these dynamic properties can be used to construct
a modal model.
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3. MODAL PARAMETER ESTIMATION
3.1 Introduction
Modal parameter estimation is the determination of frequency, damping, and modal coefficients from
the measured data which may be in: (1) relatively raw form in terms of force and response data in the
time or frequency domain, or (2) in a processed form such as frequency response or impulse response
functions. Most modal parameter estimation is based upon the measured data being the frequency
response function; or the equivalent impulse response function, typically found by inverse Fourier
transforming the frequency response function. Regardless of the form of the measured data, the
modal parameter estimation techniques have traditionally been divided into two categories: (1)
single degree-of-freedom (SDOF) approximations, and (2) multiple degree-of-freedom (MDOF)
approximations. Since the single degree-of-freedom equations are simply special cases of the multiple
degree-of-freedom equations, all theoretical discussion is made only in terms of the multiple degreeof-freedom case.
The current effort in the modal parameter estimation area is concerned with a unified theory that
explains any previously conceived modal parameter estimation method as a subset of a general
theory. This unified theory concept would eliminate the confusing nomenclature that currently exists
and simplify the understanding of the strengths and weaknesses of each method. The modal
parameter estimation methods that have been developed over the past several years involve multiple
measurement, multiple reference concepts that can be viewed as an interaction between the temporal
domains (time, frequency, etc.) and the spatial domains (physical coordinates, modal coordinates,
etc.) in order to achieve the "best" estimate of the modal parameters. Volume III of this Technical
Report presents this background and provides a complete development, using a consistent set of
nomenclature, of most multiple reference modal parameter estimation algorithms in use at the
present time.
3.2 Historical Overview
While engineers have tried to estimate the vibration characteristics of structures since the turn of the
century, the actual history of experimental modal parameter estimation is normally linked to the work
by Kennedy and Pancu[1 ] in 1947. Until this time, the instrumentation that had been available was
not sufficiently refined to allow for detailed study of experimental modes of vibration. As the
instrumentation and analysis equipment has improved over the last forty years, major improvements
in modal parameter estimation techniques have followed. Specifically, the development of accurate
force and response transducers, the development of test equipment based upon digital computers and
the development of the fast Fourier transform (FFT) have been the key advances that have initiated
bursts of development in the area of modal parameter estimation.
During this time period, efforts in modal parameter estimation have involved two concepts. The first
concept involved techniques oriented toward the forced normal mode approach to modal parameter
estimation. This approach to the estimation of modal parameters involves exciting the system into a
single mode of vibration by using a specific sinusoidal forcing vector. Since the success of this
method is determined by the evaluation of the phase characteristics with respect to the characteristics
occurring at resonance, this approach can be broadly classified as the phase resonance method. In
order to refine the phase resonance method, particularly the force appropriation aspect of the
method, efforts began to estimate the modal information on a mode by mode basis, using measured
impedance, or frequency response functions. Much of the early work on this concept centered on
using the phase information as a means of identifying the effects of separate modes of vibration in the
measurement. For this reason, this concept has become known as the phase separation method.
Most methods that are in use today can be classified as phase separation methods since no effort is
made to excite only one mode of vibration at a time.
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With respect to the phase resonance methods, the basic theory was first documented by Lewis and
Wrisley[2] in 1950. This theory was refined and presented in a more complete manner by F. de
Veubeke[ 3] in 1956. Significant advances in the approach to force appropriation were documented by
Trail-Nash[ 4 1 and Asher[ 5] in 1958. Significant improvements and refinements in the phase
resonance methods have taken place in the last thirty years, particularly in the automation of the
force appropriation and the use of digital computers, but the basic theoretical concept has not
changed since 1950.
With respect to the phase separation methods, much effort has occurred over the last forty years and
continues to the present day. Kennedy and Pancu[1] in 1947 documented that the presence of two
modes of vibration could be detected by observing the rate of change of the phase in the area of
resonance. Since this method was developed based upon a plot of the real part of the impedance
function versus the imaginary part of the impedance function, this method is referred to as a circle-fit
method based upon these characteristics in the Argand plane. Broadbent[ 6] applied this concept to
flight flutter data in 1958. Sin-e the data acquisition process was largely analog until 1970, most of
the work until that time was oriented towards trying to fit a single degree of freedom model to
portions of the analog data. The significant contributors during this time period began with Stahle [7]
in 1958, and continued with Bishop and Gladwell [8] and PenGcred and Bishop [9-1] in 1963, and
Mahalingham [12] in 1967. Once data began to be collected and stored in a digital fashion, the phase
separation methods migrated to multiple degrees of freedom approaches. The initial work involving
multiple degree of freedom models was documented by Klosterman [13] in 1971 Richardson and
Potter [141 in 1974 and Van Loon [15] in 1974. While the work during this period evolved the basic
polynomial and partial fraction models that are the basis of modem experimental modal parameter
estimation methods, the algorithms were basically unstable, iterative approaches to the solution for
the unknown parameters. Also, these methods used only one measurement at a time in the
estimation of the modal parameters. In 1978, Brown [16] documented work on the Least Squares
Complex Exponential method that was a two stage approach to the estimation of modal parameters
using all of the available data. In the first stage, the frequency and damping values are estimated; in
the second stage, the modal coefficients are estimated. Ibrahim (17], also in 1977, documented the
initial version of the Ibrahim Time Domain Method, which formulated the solution for the modal
parameters into an eigenvalue-eigenvector solution approach. These last approaches represent
conceptual approaches that have been extended today into similar methods involving multiple
references. The significant advances in the multiple reference, or polyreference, methods used and
being developed at the present time were first documented by Void [181 in 1982 with the
Polyreference Time Domain method. Since that time, several other polyreference methods have
been developed. Detailed documentation of the multiple reference methods is contained in later
sections.
In summary, over the last forty years, many experimental modal parameter estimation methods have
been developed that can be classified as either phase resonance or phase separation methods. Often,
it seems that these methods are very different and unique. In reality, the methods all are derived
from the same equation and are concerned with the decomposition of a composite function into its
constituent parts. This decomposition may occur in the time domain in terms of damped complex
exponentials, in the frequency domain in terms of single degree-of-freedom functions, or in the
modal domain in terms of modal vectors. This decomposition may occur during the test, as in the
phase resonance methods, or occur during analysis, as in the phase separation methods. The various
modal parameter estimation methods are enumerated in the following list:
"*Forced Normal Mode Method
"*Quadrature Amplitude [7,8,111
* Kennedy-Pancu Circle Fit
[2-5,19,20]
[1,13,21-23]
"*Single Degree-of-Freedom Polynomial
[14,21,22,24,25]
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"*Nonlinear Frequency Domain
[13,14,21,221
* Complex Exponential [26,27]
"*Least-Squares Complex Exponential (LSCE)
[16]
"*Ibrahim Time Domain (ITD) [17,28-32]
"*Eigensystem Realization Algorithm [33-35]
"*Orthogonal Polynomial
[36,37]
"*Global Orthogonal Polynomial [381
"*Polyreference Time Domain [18,39,40]
"*Polyreference Frequency Domain [40- ]
"*Direct Parameter Identification: Time Domain [40]
"*Autoregressive Moving Average (ARMA) [44-48]
"•Direct Parameter Identification: Frequency Domain
[40,49]
3.3 Multiple-Reference Terminology
3.3.1 Mathematical Models
The most general model that can be used is one in which the elements of the mass, damping, and
stiffness matrices are estimated, based upon measured forces and responses. Thus, the model that is
used is based upon a matrix differential equation transformed into the domain of interest.
Time domain:
[MI {ji(t)} + [C] {i(t)} + [K] {x(t)} = {F(t)}
(24)
Frequency domain:
-'?[M] {X(w)} +jw[C] {X(w)} + [K] {X(w)} = {F(w)}
(25)
Laplace domain:
s 2 [M! {X(s)} + s[C] {X(s)} + [K] {X(s)} = {F(s)}
(26)
If Eq. (24), (25), or (26) is used as the model for parameter estimation, the elements of the unknown
matrices must first be estimated from the known force and response data measured in the time,
frequency or Laplace domain. Once the matrices have been estimated, the modal parameters can be
found by the solution of the classic eigenvalue-eigenvector problem [ ,42,49]. Due to truncation of the
data in terms of frequency content, limited numbers of degrees-of-freedom, and measurement errors,
the matrices found by Eq. (24), (25), or (26) are, in general, not directly comparable to matrices
determined from a finite element approach. Instead, the matrices that are estimated simply yield
valid input-output relationships and valid modal parameters. This is because there is an infinite
number of sets of mass, damping, and stiffness matrices that yield the same modal parameters over a
reduced frequency range limited to the dynamic range of the measurements. For this reason, Eqs.
(24), (25), and (26) are often pre-multipled by the inverse of the mass matrix so that the elements of
the two matrices, the dynamic damping matrix [D] and dynamic stiffness matrix [EJ are estimated:
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Time domain:
[I] {x(t)} + [D] {x(t)} + [E] {x(t)} ={F'(t)}
(27)
Frequency domain:
-- ,2[I] {X(w)}
+jw[D] {X(w)} + [E] {X(w)} ={F'(w)}
(28)
Laplace domain:
s2[1] {X(s)} + s[D] {X(s)} + [E] {X(s)} = {F (s)}
(29)
Existing modal parameter estimation methods used in commercial modal analysis systems most often
employ a model based upon measured impulse response (time domain) or frequency response
(frequency domain) functions. While the exact model used as the basis for modal parameter
estimation varies, almost all models used in conjunction with frequency response function data can be
described by a general model in the time domain, frequency domain, or Laplace domain. The general
model in the time domain is a damped complex exponential model (often the impulse response
function) while the general model in the frequency domain is the frequency response function. The
general model in the Laplace domain is the transfer function. For general viscous damping, the
mathematical models for each domain for a multiple degree degree-of-freedom mechanical system
can be stated as:
Time Domain:
h
= NAp. e 't +A,
-,t)
e
t
(30)
M.I
FrequencyDomain:
N A_.._ + A W
H,(w) = E j-(31)
1w-A,
'=.1 1wp-A,
Laplace Domain:
Hp.(s) =
NA
A
s-, + s(32)
SA
ML~ s-A,
where:
s
=
s
=
ar
W
p
q
r
N
AM,
=
=
=
=
=
=
=
AW =
AW,
=
Q,
=
Laplace variable
a +jw
angular damping variable (rad/sec)
angular frequency variable (rad/sec)
measured degree-of-freedom (response)
measured degree-of-freedom (input)
modal vector number
number of modal frequencies
residue
Q44vrq.
,OVLV
complex modal scaling coefficient for mode r
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L
A,
A,
=
modal coefficient for measured
degree-of-freedom p and mode r
=
modal participation factor for reference
degree-of-freedom q and mode r
system pole
a, +j4
=
=
The models described in Eqs. (30) through (32) have many other equivalent forms based upon
expansion of the terms under the summation. Also, the models take on slightly different forms under
assumptions concerning specific physical damping mechanisms (hysteretic, etc.) [ 3,I4,4] Other
forms of these models are also used where certain assumptions or mathematical relationships are
utilized. For example, an equivalent model can be found when the common denominator of Eq. (31)
is formed yielding a polynomial numerator and polynomial denominator of maximum order "2N"
.13,14,22]
The denominator polynomial is then a function of the system poles. Often, an assumption is
made concerning the modal vectors being normal (real) rather than complex. This reduces the
number of unknowns that must be estimated from "2N" to "N".
3.3.2 Sampled Data
The mathematical models described in the previous section are all developed based upon the concept
that the temporal variable (time or frequency) is continuous. In reality the temporal variable must be
thought of as sampled in each domain. This restriction requires special consideration when applying
the models developed in Eqs. (24) through (32). Differential equations must now be thought of as
finite difference equations; continuous integral transforms are replaced by discrete transforms such
as the Fast Fourier Transform (FFT) and the Z Transform. The concepts affecting the numerical
processing of sampled data with respect to the continuous models represented in Eqs. (24) through
(32) are exactly the same as the concepts that are the basis of the area of digital signal analysis with
respect to the measurement of the data. The limitation of the frequency information creates special
processing problems that are related to Shannon's Sampling Theorem; the limitations of the dynamic
range of the measured data and of the computer precision yield special numerical problems with
respect to the solution algorithm.
In general, the numerical considerations often determine which mathematical model will be most
effective in the estimation of modal parameters. Time domain models tend to provide the best
results when a large frequency range or large numbers of modes exist in the data. Frequency domain
models tend to provide the best results when the frequency range of interest is limited and when the
number of modes is small. While these are general considerations, the actual numerical
implementation determines the ability of the algorithm to estimate modal parameters accurately and
efficiently.
3.3.3 Consistent Data
Modal parameter estimation methods all assume that the system that is being investigated is linear
and time invariant. While this is often nearly true, these assumptions are never exactly true.
Consistent data refers to the situation where the data is acquired so as to best satisfy these two
assumptions. Problems associated with nonlinearity can be minimized by maintaining a prescribed
force level and/or using excitation methods that give the best linear approximation to the nonlinear
characteristic (random excitation). Problems associated with the time invariance constraint can be
minimized by acquiring all of the data simultaneously using multiple excitations [5o-54]. This reduces
mass loading and boundary condition variations that can be caused by moving a transducer around
the structure or by changing the location of the excitation.
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3.3.4 Residuals
With respect to spatial geometry, continuous systems have an infinite number of degrees-of-freedom
but, in general, only a finite number of modes can be used to describe the dynamic behavior of a
system. The theoretical number of degrees-of-freedom can be reduced by using a finite frequency
range (f.,fb). Therefore, for example, the frequency response function can be broken up into three
partial sums, each covering the modal contribution corresponding to modes located in the frequency
ranges (O,f.), (fafb), and (fb,oo) as shown in Figure 7.
I0I.
ios
Frequency Range
of Intrest
0~
I
'
101'
1Ot
0.0
1000.0
2000.0
FREQTMNCY, HZ
Figure 7. Frequency Range of Interest
In the frequency range of interest, the modal parameters can be estimated to be consistent with Eq.
(33). In the lower and higher frequency ranges, residual terms can be included to handle modes in
these ranges. In this case, the general frequency response function model can be stated:
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N
H()
A
A;,
Rip,+
()+
+RJ.,
(33)
where:
RIP, (w)
Rrp
=
=
residual effect of lower frequency modes
residual effect of higher frequency modes (constant with W)
In many cases the lower residual is called the inertia restraint, or residual inertia, and the upper
residual is called the residualflexibility [13]. In this common formulation of residuals, both terms are
real-valued quantities. The lower residual is a term reflecting the inertia or mass of the lower modes
and is an inverse function of the frequency squared. The upper residual is a term reflecting the
flexibility of the upper modes and is constant with frequency. Therefore, the form of the residual is
based upon a physical concept of how the combined system poles below and above the frequency
range of interest affect the data in the range of interest. As the system poles below and above the
range of interest are located in the proximity of the boundaries of the frequency range of interest,
these effects are not the simple real-valued quantities noted in Eq. (33). In these cases, residual or
computational modes may be included in the model to partially account for these effects. When this
is done, the modal parameters that are associated with these computational poles have no physical
significance because the poles are not structural modes of the system, but may be required in order to
compensate for strong dynamic influences from outside the frequency range of interest. Using the
same argument, the lower and upper residuals can take on any mathematical form that is convenient
as long as the lack of physical significance is understood. Power functions of frequency (zero, first,
and second order) are commonly used within such a limitation. In general, the use of residuals is
confined to frequency response function models. This is primarily due to the difficulty of formulating
a reasonable mathematical model and solution procedure in the time domain for the general case
that includes residuals.
3.3.5 Global Modal Parameters
Theoretically, modal parameters are considered to be unique based upon the assumption that the
system is linear and time invariant. Therefore, the modal frequencies can be determined from any
measurement and the modal vectors can be determined from any reference condition. If multiple
measurements or reference conditions are utilized, the possibility of several, slightly different, answers
for each modal parameter exists. The concept of global modal parameters, as it applies to modal
parameter estimation, means that there is only one answer for each modal parameter and that the
modal parameter estimation solution procedure enforces this constraint. Every frequency response
or impulse response function measurement theoretically contains the information that is represented
by the characteristic equation (modal frequencies and damping). If individual measurements are
treated in the solution procedure independent of one another, there is no guarantee that a single set
of modal frequencies and damping are generated. In a like manner, if more than one reference is
measured in the data set, redundant estimates of the modal vectors can be estimated unless the
solution procedure utilizes all references in the estimation process simultaneously. Most modal
parameter estimation algorithms estimate the modal frequencies and damping in a global sense but
few estimate the modal vectors in a global sense.
3.3.6 Modal Participation Factors
A modal participationfactor is a complex-valued scale factor that is the ratio of the modal coefficient
at one reference degree-of-freedom to the modal coefficient at another reference degree-of-freedom.
A more general view of the modal participation factor is that it represents the relationship between
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the residue and the cigenvector coefficient as in the following equations:
=- Qr010q9,
A,
S=
AW,
(34)
Q",Ob
(35)
OWLqr
(36)
=
where:
p
q
measured degree-of-freedom (response)
measured degree-of-freedom (reference)
r
A_
-
modal vector number
residue
-
complex modal scaling coefficient for mode r
=
Q,
ipr,
modal coefficient for measured
degree-of-freedom p and mode r
modal coefficient for reference
degree-of-freedom q and mode r
L,
modal participation factor for reference
=
degree-of-freedom q and mode r
From a mathematical standpoint, the modal participation matrix is equal to the left eigenvectors of
the transfer matrix [H] as shown in Eq. (37):
[H]= [11 fA IL[L](37)
where:
[H]
[
=
I] =
[A]
=
transfer function matrix
complex modal vector matrix
diagonal matrix with poles
Note that for Eq. (35) the modal participation factor represents the product of a modal scaling
coefficient and another term from the right eigenvector for reference degree-of-freedom q. This will
always be true for reciprocal systems since the left and right eigenvectors for a given mode are equal.
For non-reciprocal systems, the modal participation factor is the appropriate term from the left
eigenvector. Note also that the modal participation factor, since it is related to the eigenvector, has
no absolute value but is relative to the magnitudes of the other elements in the eigenvector.
Modal participation factors reflect the interaction of the spatial domain with the temporal domain
(time or frequency). Modal participation factors can be computed any time that multiple reference
data are measured and such factors are used in multiple reference modal parameter estimation
algorithms. Modal participation factors relate how well each modal vector is excited from each of the
reference locations. This information is often used in a weighted least squares error solution
procedure to estimate the modal vectors in the presence of multiple references. Theoretically, these
modal participation factors should be in proportion to the modal coefficients of the reference degrees
of freedom for each modal vector. Modal participation factors in a solution procedure enforce the
constraint concerned with Maxwell's reciprocity between the reference degrees of freedom. Most
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multiple reference, modal parameter estimation methods estimate modal participation factors as part
of the first stage estimation of global modal frequencies and damping.
3.3.7 Order of the Model
The order of the model equals the number of unknowns that must be estimated in the model. In the
modal parameter estimation case, this refers to the frequency, damping, and complex modal
coefficient for each mode of vibration at every measurement degree-of-freedom plus any residual
terms that must be estimated. Therefore, the order of the model is directly dependent on the number
of modal frequencies, "N", that are to be estimated. For example, for a system with "N" modes of
vibration, assuming that no residuals were required, "4N" unknowns must be estimated. For cases
involving measured data, the order of the model is extremely important. Estimates of modal
parameters are affected by the order of the model. A problem arises from the inability to be certain
that the correct order of the model has been chosen during the initial estimation phase. If the
number of modes of vibration is more or less than "N", modes of vibration will be found that do not
exist physically or modes of vibration will be missed that actually do occur. In addition, the values of
frequency, damping, and complex modal coefficient for the actual modes of vibration will be affected.
The number of modes of vibration is normally chosen between one and an upper limit, dependent on
the memory limitations of the computational hardware. The true number of system poles is a
function of the frequency range of the measurements used to estimate the modal parameters. By
observing the number of peaks in the frequency response function, the minimum number of system
poles can be estimated. This estimate is normally low, based upon poles occurring at nearly the same
frequency (pseudo-repeated roots), limits on dynamic range, and poorly excited modes. For these
reasons, the estimate of the correct order of the model is often in error. When the order of the model
is other than optimum, the estimate of the modal parameters will be in error.
Many of the parameter estimation techniques that are used assume that only one mode exists in a
limited range of interest and all of the other modes appear as residual terms. For this case Eq. (33)
can be rewritten as:
Hp,
p
)w-.-A,
M(+
w-
(38)
+RFr,
3.3.8 Solution Procedure
Equations (30) through (32) are nonlinear in terms of the unknown modal parameters. This can be
noted from the unknowns in the numerator and denominator of Eq. (31) and the unknowns as the
argument of the transcendental functions of Eq. (30). The nonlinear aspect of the model must be
treated in one of two ways: (1) by the use of an iterative solution procedure to solve the nonlinear
estimation problem, allowing all modal parameters to vary according to a constraint relationship until
an error criterion reaches an acceptably low value, or (2) by separating the nonlinear estimation
problem into two linear estimation problems. For the case of structural dynamics, the common
technique is to estimate "2N" frequencies and damping values in a first stage and then to estimate the
"4N" modal coefficients plus any residuals in a second stage.
In the iterative technique in the solution of the nonlinear estimation approach, a set of starting values
must be chosen to initiate the sequence. The number and value of these starting values affect the final
result. Poor initial estimates can lead to problems of convergence, as a result of which, close operator
supervision usually is required for a successful use of this technique.
An alternative method is to reformulate the nonlinear problem into a number of linear stages so that
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each stage is stable. The actual data that are used in the estimate of the modal parameters also affect
the results. Based on the choice of the order of the model, "N", there are "4N" modal parameters to
be estimated. If residuals, in one form or another, are also included, the number of modal
parameters to be estimated will be slightly higher. The common method of solving for these
unknown modal parameters is to find an equation involving known information for every unknown to
be found. In this case, the measured frequency response function or impulse response function
provides the known information and Eq. (30) or Eq. (31) can be repeated for different frequencies or
time values in order to obtain a sufficient number of equations. These equations, for the linear case,
can then be solved simultaneously for the unknown modal parameters. As an illustration of this
relationship, consider a common modal parameter situation in which the number of modes in the
frequency range of interest is between I and 30. Assuming the highest ordered model means that
slightly more than 120 modal parameters must be estimated. From a single frequency response
measurement, 1024 known values of the function will be available (512 complex values at successive
values of frequency). Many more equations, based on the known values of frequency response, can
be formed than are needed to find the unknown modal parameters. An obvious solution is to choose
enough equations to solve for the modal parameters. The problem arises in determining what part of
the known information is to be involved in the solution. As different portions of the known data
(data near a resonance compared to an anti-resonance, for example) are used in the solution, the
estimates of modal parameters vary. As the quality of the data becomes marginal, this variance can
be quite large. When the modal parameters that are estimated appear to be non-physical, this is
often the reason.
To solve this problem, all or a large portion of the data can be used if a pseudo-inverse type of
solution procedure is used. One procedure that is used is to formulate the problem so as to minimize
the squared error between the data and the estimated model. This least-squares error method to the
solution is the most commonly used technique in the area of modal parameter estimation. If there
are many more known pieces of information than unknowns that must be estimated, many more
equations can be formed than are needed to solve for the unknowns. The least-squares error method
to the solution allows for all of these redundant data to be used to estimate the modal parameters in
a computationally efficient manner. The least-squares error method usually can be derived directly
from the linear equations using a normal equations approach. In general, this procedure does not
significantly increase the memory or computational requirements of the computational hardware.
Any solution procedure that can be used is only estimating a "best" solution based upon the choice of
the model, the order of the model, and the known, measured data used in the model.
3.4 Characteristic Polynomial
The impetus of this section is to show that for discrete data, a difference characteristic equation can
be formulated in order to solve for the poles of the system. Further, it will be shown that the
difference equation can be formulated directly from the impulse response function data. By solving
for the polynomial coefficients and the roots of the polynomial equation, the modal parameters,
frequency and damping, are determined. The characteristic polynomial will be formulated for the
continuous case, as a differential function, and then extended to the discrete case, as a difference
function.
3.4.1 Differential Theory
The homogeneous differential equation for a single degree of freedom system is:
m i(t) +c .i(t) + k x (t) = 0
(39)
In order to solve the differential equation assume a solution of the of the form x (t) = X c' ', where X
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is a scalar value. Substituting the appropriate derivatives of the assumed solution into Eq.(39):
(m s2 + c s + k ) X e
=0
(40)
Thus, the differential equation is transformed into an algebraic polynomial equation, called the
characteristic equation.
m s 2 +c s + k = 0
(41)
The complex valued roots of the characteristic equation will yield the characteristic solutions, A, and
A 2 . The real part is the damping and the imaginary part is the eigenfrequency, or damped natural
frequency. Thus, the solution to the governing differential equation is:
2
X(t) = E X, ex
(42)
r--.
The scalar magnitudes, X, and X 2 , are determined from the system initial conditions. Note that any
exponential function will satisfy the differential equation. One such function of particular interest, is
the impulse response function.
2
h(t)-=
A, e"'
(43)
r=1
A system with N degrees-of-freedom can be described by a set of N, coupled, second order
differential equations. The characteristic equation for this system is represented by the following
polynomial:
'I + a 2M-2 S2N-2
a2N s•
2N+ a2m1 s
•••÷
+ a, s + a0 = 0
(44)
Solving this polynomial equation will yield 2N complex valued roots, or, characteristic solutions A,.
Then the solutions to the differential equations will be complex exponentials of the form:
x•j(t) = 2N, X, e•'
(45)
r=1
where:
"*p = response location degree-of-freedom
"*q = reference location degree-of-freedom
Thus, impulse response functions,
2N
h,(t) = , A,
e"rt
(46)
will also satisfy the differential equations. Consider a few impulse response functions for various
reference and response points.
hil(t) = EAII, e*"t
(47)
r=1
2N
h12(0)=
rA
12,
e 't
(48)
r-i
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h13(1) =
2N
E'A13, ex''
(49)
r-=1
2N
(50)
hQ(t) = F, A,,, eA't
r=:1
The common characteristic in each of the above equations is that every impulse response function is a
linear superposition of identical damped complex exponentials, eArt for r = 1 -. 2N. That is, the roots
of the characteristic polynomial are common to all reference and response locations. Thus, the
characteristic solutions are global system parameters, since they are independent of reference or,
response location. The important result is that since each eArt is a characteristic solution to the
homogeneous linear differential equation,
+a 2 m. 2 D'2N-2(AP, eAr )
+
e
+art)•.1
a2ND2N(AP,e-%rt)
+...
x, +ao =0
+aD(A ert
where,
D"[f (t)] = dm[f (t)] (differential operator)
dtf
that a linear superposition of characteristic solutions will also be a solution. That is, hM(t) will also
satisfy the differential equation. Actually, a set of N second order linear differential equations must
be satisfied, but, a differential equation of order 2N can be found that will have the same roots as the
set of N second order equations.
Substituting a few impulse response functions for various reference and response points, a number of
differential equations are obtained.
a2NDa(hii(t) +a2. 1 DO2N'h110(t)) +aN-2 D
a22.1 DWN h 12 (t)
+
a~v- D 2" 1
a2N D2hNs(t)) + a2N-1 D2'N
(h 1 2(tW)
+
'2N-hi(t)2
+. ..
a ZV-2 D2M2(h 1 2 (0))
h1 3 (t)0 + a 2•.• 2 D
+
+...
'2-2(hi(t)
(t)) + a2 x-. D2N-1 hf(t)) +aa
aD' D2N hMt
2 •. 2 D2N-2 h(t))
+a, D hii(t)) +ao = 0
a, D h 12 (t) +.ao= 0
+a 1
D hh13 (t))+ ao
=
O
+... + a1 D( hl,(t)) + ao = 0
Note that the coefficients a0 to a 2 do not vary with reference or response location and thus, can be
estimated from a combination of various number of reference and response points.
3.4.2 Difference Theory
From an experimental standpoint, the data are sampled, which means instead of continuous
knowledge of the system, the values obtained are for distinct discrete temporal points. The impulse
response functions are generally obtained by inverse Fourier transforming the frequency response
functions. Thus, from the discrete impulse response functions the pole information, frequency and
damping, is determined. The model for the discrete impulse response function is:
2N
h,,(t,)= EA,
r--•
2M
(51)
e""tk = EApz,,.
r--1
where,
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"•tý
=kAt
"•A t is the sample interval
"*k = I -. blocksize
It should be noted, for discrete data, that the sample interval, At, limits the frequency for which valid
information can be determined, whereas, in the analysis of continuous data, there are no frequency
constraints. In other words, theoretically, characteristics can be determined to infinite frequency for
continuous functions, but, the process of digitally sampling continuous data causes a maximum
frequency for which characteristics can be determined. The frequencies above this maximum will
alias back into the sampled bandwidth, and thus bias the results. For this reason, low-pass filters are
used to exclude information above the maximum frequency.
Recall the characteristic equation for an N degree-of-freedom system:
a 0 s 0 + a I 1s + a 2 s 2 +... + a2N
ss'
2-i
+ aN gasy
(52)
=o
will have 2N characteristic solutions, A, for r = 1 -- 2N. The characteristic polynomial is not unique
in that, many polynomials can be constructed that will yield the same characteristic solutions, even
though the coefficients will be different. For this reason, another polynomial can be formulated that
will have characteristic solutions that are related to the characteristic solutions of Eq.(52). The
polynomial has the form:
a' zo+a, z' +a; z 2 +... +a' -1z 2N' +a* z2
--
(53)
.
The relationship between z and s is z = e'61. Analogous to Eq.(52), there are also 2N characteristic
solutions of Eq.(53), z, for r = 1 -- 2N. The roots of the two equations are related by z, = e'At
where, z, are precisely the values of z for which the characteristic equation, Eq.(53), is zero. Note that
z, is simply the sampled form of the continuous exponential solution in the differential case. Thus, by
knowing the system characteristics, z2, the desired parameters, A,, can be determined. If the
coefficients are known, Eq.(53) could be solved, but, from an experimental aspect, both the
coefficients and the system characteristics are unknown. Thus, in order to determine the system
characteristics, the a' coefficients must be determined first. This is accomplished by substituting a
characteristic solution of the system, z,, into Eq.(53).
•=0(4
mz"2N-1
•A" 2N~~
a0 AP, z° +a; Amr zi +a; AP, z2
(54)
APa II+ a* A,,q,4=O0
aoAN.j Ar 4
Substituting z, = e)'•,A into the above equation,
aoAm'(eA'A)° +a Ap. (e'%6t)' +aý Ap.(eA'A% )2+...+aa
A"A('(e-rA£')2-=O
(55)
or,
a' AM"e +a1 Am, e %at + aý AP, e,.2, t +...+a' AM" eAr•NAt-
.
(56)
The important result is that since each ek' At is a characteristic solution to the homogeneous linear
difference equation, that a linear superposition of characteristic solutions will also be a solution of
Eq.(56), which means that, in general, Eq.(51) can be substituted into Eq.(56). Once again, a set of N
second order linear difference equations must be satisfied, but, a difference equation of order 2N can
be found that will have the same roots as the set of N second order equations.
Consider a number of equations for various reference and response locations:
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ao h ,(to) + a, hQ(tI) + a' h11 (t 2 ) +...
+
ao h 12 (10 ) + a h 12(t) +a7 h1 2 (t 2 )
+ a'
0 h1
...
+ a I h1 3 (t 1 ) + a h1 3 (t 2 ) +.
+
a
a
ao hN(to) + a, hp(tl) + aý h,(t2 ) +... + a'.-
h 1 I (t 2,- 1 ) + a2 h 1I(t2N) = 0
(57)
-, h 12 0(t
2 NV.) +a• h 1 2(t2) = 0
(58)
0
(59)
hls(t2Nl) +
+(tO)
a' h13 (t•)
hN(t,-.) + ad2 hp,(t2N)
=
=
0
(60)
Note that the coefficients ao to ad2 do not vary with reference or response location and thus, can be
estimated from a combination of various number of reference and response points. Once the a'
coefficients are estimated from a set of equations similar to the ones above, the poles, z,, and hence
A,, can be estimated from the 2N solutions of the characteristic equation,
a' z° +a' zi +aýz 2 +... + a' .1 z 2K•' +a'u. z2 = 0
(61)
where:
s z, = e
In summary, a series of 2N linear difference equations with constant coefficients are formed from the
sampled impulse response function data in order to solve for the common constant coefficients.
These coefficients are then used in the characteristic equation to solve for the system characteristics,
z,, which contain the desired parameters, A,.
Note that the characteristic polynomial for the continuous, or discrete case, is of order 2N, that is,
twice the number of modes. This results in a time domain differential, or difference equation of
order 2N. For this reason, from a numerical analysis concept, for large numbers of modes, N, or
large differences in modal frequency (A, compared to AN), time domain methods are numerically
better conditioned.
3.5 Characteristic Space Concepts
A new way of conceptualizing the area of parameter identification was developed during the course
of the work under this contract. One of the objectives of the contract was to summarize existing
modal parameter estimation methods and develop new ones. In the process of performing this task,
it became obvious that most of the current algorithms could be described conceptually in terms of a
three-dimensional complex space of the system's characteristics. Modal parameter estimation is the
process of deconvolving measurements defined by this space into the system's characteristics.
The frequency and/or unit impulse response function matrix which describes a system, can be
expressed in terms of the convolution of three fundamental characteristic functions; two complex
spatial, and one complex temporal. The spatial characteristics are a function of geometry and the
temporal corresponds to either time or frequency. Mathematically the frequency response matrix and
the impulse response matrix can be expressed as follows:
IH(wk)] =F I rAkJ IL]
[h(th)]=
I[eAtk] [L]
where:
"*[H (wk)]No
"*[h (t k)]N
xN, =
frequency response matrix (element HM(wk))
xN, = unit impulse response matrix (element h,(t h))
-54-
(62)
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I'k IN. 2N = modal vector matrix (function of spatial variable p, element 0,,)
IIL ]2 xN, = modal participation factor (function of spatial variable q, element L.)
1
[*k]rNli2 = diagonal matrix of characteristic
roots (element j
*[
e"t2Jzv
W
=
rt =
1,,
-,
= diagonal matrix of characteristic roots (element eArth)
frequency temporal variable (k = 1 - blocksize/2, may be unequally spaced)
time temporal variable (tk
=k
A t)
* p = response degree-of-freedom spatial variable
* q = reference degree-of-freedom spatial variable
* r = temporal degree-of-freedom variable
"*N
"*N,
"*N,
=
number of modes (system poles, indexed by r)
=
number of responses (indexed byp)
=
number of references (indexed by q)
The frequency response function matrix consists of a three-dimensional complex space, which for a
real system is a continuous function of the three characteristic variables (p,q,w). However, in terms
of measurements the functions consist of sampled data where, p,q and WAkare sampled characteristic
variables. In other words, the frequency response function is measured at discrete input, or reference
points (q), output response points (p), and discrete frequency (W/k), or time points (t k).
A summary of the characteristic vectors are:
"*The response characteristic functions consist of a set of vectors which are proportional to the
eigenvectors of the system. The eigenvectors are indexed by r and the elements of the vectors are
indexed byp.
"*The reference characteristic functions consist of a set of vectors which are proportional to the
modal participation factors, which are in turn proportional to the system eigenvectors at the
reference degrees-of-freedom. The modal participation vectors are indexed by r and the elements
of the vectors are indexed by q.
"•The temporal characteristic functions consist of vectors which are equivalent to sampled single
degree-of-freedom frequency response functions, or unit impulse response functions. The index
on the vector is r and the index on the sampled element of each vector is Wk, or tk.
The variable r is the index on the characteristic. For a given r there is a discrete characteristic space.
The summation, or superposition with respect to r defines; the measured, or sampled frequency
response, or impulse response matrix, or, in other words, the three-dimensional complex space.
This concept is difficult to visualize, since the matrix is represented by three-dimensional complex
characteristic space. The easiest method is to describe the variation along lines parallel to axes of the
space. Lines parallel to the temporal axis correspond to individual frequency response functions, or
unit impulse response functions. These frequency response functions consist of a summation of the
temporal characteristics, weighted by the two spatial characteristics, which define the other two axis
of the characteristic space.
Lines parallel to the response axis correspond to forced modes of vibration. These forced modes
consist of a summation of the system eigenvectors weighted by the input characteristic and the
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temporal characteristic.
Likewise, lines parallel to the input, or reference axis consist of a summation of the system
eigenvectors weighted by the response characteristic and the temporal characteristic. The variation
along these lines are referred to in the literature as the m )dal participation factors.
Modal parameter estimation is the process of deconvolving this sampled space into the basic
characteristic functions which describe the space. In practice, there are many more measured, or
sampled points in the space than there are elements in the three characteristic vectors, therefore, the
parameter estimation process is over determined. As a result, one of the important steps in the
process has been the reduction of the data to match the number of unknowns in the parameter
identification process. This data reduction has historically been done by using a pseudo inverse, or a
principal component method, with least squares being the most common pseudo inverse method.
The early single degree-of-freedom (SDOF) and multiple degree-of-freedom (MDOF) modal
parameter estimation methods used subsets of the sampled data and extracted one of the
characteristic functions at a time, normally the temporal characteristic. For example, the very early
methods like the complex exponential were used to fit individual frequency response, or unit impulse
response functions for the temporal characteristics (eigenvalue) and the residues (convolution of the
response and input characteristics). For these cases, each frequency response measurement gave a
different estimation of the system eigenvalues, or temporal characteristics. Since the measurements
were taken one function at a time some of this variation was due to inconsistencies in the data base
and the rest of the variation due to noise and distortion errors.
Later methods started to use either, least square, or principal component methods to condense the
data over a number of sampled frequency response functions, into small subsets parallel to the
temporal axis (for example the Least Square Complex Exponential and/or the Polyreference Time
Domain methods). These methods then give global estimates of the eigenvalues, or temporal
characteristic functions. The Least Squares Complex Exponential parameter estimation algorithm
reduced the information to a single function parallel to the temporal axis and as a result, only
estimated the temporal characteristic in a global sense. The Polyreference Time Domain algorithm
estimates several functions parallel to the temporal axis at the input, or reference points. As a result
this method also gives global estimates of the input characteristic functions, or modal participation
factors.
The more recent methods use larger subsets of the sampled data and utilize simultaneous data from
all three axis resulting in global estimates of all three characteristics. In order to use these global
methods, it is important that a consistent data base be measured.
3.6 Summary - Modal Parameter Estimation
One of the conclusions reached in a previous Air Force Contract (F33615-77-C3059) was that the
area of modal parameter estimation will, in the future, advance rapidly due to technology transfer
from other fields involved in parameter estimation. This certainly has occurred as indicated by the
drastic increase in the number of parameter estimation algorithms which have been described in the
literature in the last five years. This effort has been international in scope, with many of the newer
techniques being variations of each other. Volume III of this Technical Report reviews multiple
reference modal parameter estimation in detail.
These methods range from single reference single degree-of-freedom (SDOF) methods to
sophisticated multi-reference, multi-response, multiple degree-of-freedom (MDOF) methods. The
algorithm of choice depends upon a number of conditions:
-56-
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"*Modal Application
* Trouble Shooting--For many of the problem solving, or trouble shooting applications, the
simplier SDOF, or single reference MDOF methods are used, since simple test procedures
and a quick look are desirable.
• Model Verification--There has been increased emphasis on finite element verification. These
applications require a higher level test and parameter identification procedures.
. Model Generation, or Correction--There is also increased emphasis on; the generation of
modal models based upon experimental data, and/or the correction of existing models. These
applications require the highest level of test and parameter identification procedures.
"*Equipment Considerations
The parameter identification methods reviewed in this report depend heavily upon the testing
methods (single input, or multiple input) and testing equipment. These new algorithms place a
severe requirement upon the testing methods to obtain consistent data bases, particularly for the
more advanced multi-input multi-output methods.
"*Wideband vs Narrowband
Wideband verses narrowband refers to the frequency bandwidth of the frequency response
measurements. In general, for very broad frequency range measurements, time domain
algorithms work well, while frequency domain algorithms seem to perform well for the narrow, or
zoom bands. Recently, there has been increased emphasis in sine testing. Sine testing, not in the
classical sense, but in terms of multi-input multi-output test and parameter estimation methods.
This emphasis will provide the impetus to refine the frequency domain algorithms to efficiently
use the increased spatial information that multi-input multi- output sine testing yields.
"*Modal Density
The choice of the parameter estimation method depends heavily upon the modal density. For
cases with low modal density, single input SDOF or MDOF methods work well. For the high
modal density cases the multi-input methods, especially ones which use spatial information, are
the methods of choice. It should be noted that the advanced methods require consistent data and
place additional constraints on the testing methods.
A summary of the characteristics of the modal parameter identification methods is shown in Table 7.
All of the methods which were discussed in detail in Volume II of this Technical Report are briefly
summarized in this Table.
It should be again noted that all of the methods covered in this report can be described in terms of a
characteristic space, where a particular parameter identification algorithm uses as input, measured
values in this characteristic space, to deconvolve the systems characteristics. The more advanced
methods use information from all three axes of the characteristic space simultaneously. From the
measurement standpoint, it is increasingly more important that the measured data be consistent.
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TABLE 7. Summary of Modal Parameter Estimation Methods
Modal Parameter Estimation Characteristics
Quadrature Amplitude
Kennedy-Pancu Circle Fit
SUOF Polynomial
Non-Linear Frequency Domain
Complex Exponential
Least Squares Complex Exponential (LSCE)
Ibrahim Time Domain (lID)
Multi-relerence Ibrahim Tune Domain (MITID)
Eigensystem Realization Algorithm (ERA)
Orthogonal Polynomial
Mulll-rf•rience O)rthogonal Polynomial
I'olyreteFence lIime Domain
Polyreference Frequency Domain
Time Domain Direct Parameter Identification
Frequency Domain Direct Parameter Identification
Multi-MAC
Multi-MAC / CMIF I Enhanced FRF
Time.
Frequency.
or Spatial
Domain
Frequency
Frequency
Frequency
Frequency
Time
Time
Time
Time
Time
Frequency
Frequency
Time
Frequency
Time
Frequency
Spatial
Spatial
Single or
Multiple
Degrees-ofFreedom
SDOF
SDOF
SDOF
MDOF
MDOF
MDOF
MDOF
MDOF
MDOF
MDOF
MI)OF
MDOF
MDOF
MDOF
MDOF
SDOF
MDOF
-58-
Global Modal
Frequencies
and Damping
Factors
No
No
Yes/No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
Repeated Modal
Frequencies
and Damping
Factors
No
No
No
No
No
No
No
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Global
Modal
Vectors
No
No
No
No
No
No
Yes
Yes/No
Yes
No
Yes
No
Yes
Yes
Yes
Yes
Yes
Global
Modal
Participation
Factors
No
No
No
No
No
No
No
No/Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
No
Yes
Residuals
No
Yes
No
Yes
No
No
No
No
No
Yes
Yes
No
Yes
No
Yea
No
No
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Allemang, RJ., Brown, D.L., "A Correlation Coefficient for Modal Vector Analysis,"
Proceedings, International Modal Analysis Conference, pp.110-116, 1982.
[781 Allemang, RJ., Brown, D.L., Zimmerman, R.D., "Determining Structural Characteristics from
Response Measurements," University of Cincinnati, College of Engineering, Research Annals,
Volume 82, Number MIE-110, 39 pp., 1982.
[791 Pappa, R.S., et. al., "Modal Identification Using the ERA and Polyreference Techniques,"
University of Cincinnati, Project Report for Mechanical Vibrations and Fourier Transform
Techniques, Spring Quarter, 1985.
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4. SYSTEM MODELING
4.1 Introduction
The goal of Volume IV of this Technical report is to document the review of the current methods
Used to predict the system dynamics of an altered structure or of combined structures based upon a
previously defined, modal or impedance, model of the structure(s). Of particular interest is the
performance of such modeling methods with respect to experimentally based models.
Volume IV investigates several system modeling techniques to determine their capabilities and
limitations from a theoretical and practical viewpoint. Several experimental techniques, practical
aspects of the analytical and approximate techniques, test results, modeling results, and analysis of
the results are presented to compare and evaluate the various modeling methods. This study
presents all of the techniques in a consistent manner from the same origin, using consistent
nomenclature, to clearly highlight the similarities and differences inherent in their development
which form the basis of the strengths and weaknesses of each technique. To gain practical
insights, all of the techniques presented in Sections 2 through 4 of Volume IV of this Technical
Report are compared with experimental results. Section 5 of Volume IV presents the new
superelement method of dynamic component synthesis as developed by the University of Dayton
Research Institute (UDRI).
4.2 System Modeling
System modeling is a computer based technique that is used to represent the dynamic
characteristics of a structure. This representation takes the form of either experimental data, modal
data, or analytical data Once the dynamic characteristics of a structure are used to form a model or
system model, several uses of the model are possible. First, the effects of design changes or
hardware changes to the original structure could be studied. Second, the structure could be coupled
with another structure to determine the overall resultant dynamic behavior. Finally, the model can
be used analytically to apply forces and determine the forced response characteristics ot the
structure.
The main objective of system modeling is to use a mathematical representation of the
dynamic characteristics of a structure in a computer environment to effectively develop a design
or trouble shoot a particular problem of a design. Several techniques of various origin have
evolved with the advancement of computer technology. Depending on the situation, each is very
effective if properly utilized. Design development is generally considered an extensive long range
process that results in an optimally designed structure given the constraints of the project.
Trouble shooting involves the evaluation of failures or design flaws which must be corrected quickly.
The most obvious way to classify system models is into analytical and experimental methods.
The primary approach to analytical modeling is commonly known as finite element Analysis. Finite
element analysis is analytical in nature because only knowledge of the physical properties of the
structure is used to build a dynamic representation. This is done by subdividing the structure into
discrete elements and assembling the linear second order differential equations by estimating the
mass, stiffness, and damping matrices from the physical coordinates, material properties, and
geometric properties.
Finite element analysis is extremely useful because no physical test object is necessary to compute
resonant frequencies and mode shapes, forced response, or hardware modifications. Therefore,
this method comes in very handy in the development cycle, where it can be used to correct major
flaws in the dynamic characteristics of a structure before a prototype is built. Since finite
element analysis is an approximate analytical technique, experimental modal analysis data is
obtained as soon as possible so the analytical model can be validated. Detailed finite element
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analysis method is not covered in this report.
Experimental modeling techniques are further subdivided into two groups. They are modal models
and impedance models. Modal modeling is an experimentally based technique that uses the
results of an experimental modal analysis to create a dynamic model based on the estimated poles
of the system. The result is a computationally efficient model that is uncoupled due to
transformation of the physical coordinates into the modal coordinates which renders the
system into a number of lumped mass, single degree of freedom components. This concept is
fundamental to modal analysis. This model is then used to investigate hardware changes, couple
structures, or compute the forced response. This method is developed fully in Section 2 of Volume
IV of this Technical Report.
Impedance modeling uses measured impedance functions or frequency response functions to
represent the dynamic characteristics of a structure in the frequency domain. This method uses
the experimentally measured functions to compute the effects of hardware changes or to couple
together several components. To use this modeling technique, measurements at the constraint or
connection points, driving point, and cross measurement- oetween the two are needed to
compute a modified frequency response function. Impedance modeling is fully developed for the
compliance method and stiffness method in Section 4 of Volume IV of this Technical Report.
Both of the experimental modeling methods are quick and easy to use in their basic
implementations. Therefore, they are extremely useful in trouble shooting situations but have
limited application in the design cycle.
The final classification of system models is experimental/analytical models or mixture methods. Two
techniques are considered mixture methods. The first is sensitivity analysis. Sensitivity analysis is
an approximate technique that uses the first term or first two terms of a power series expansion to
determine the rate of change of eigenvalues or eigenvectors with respect to physical changes (mass,
stiffness, and damping). Therefore, this method is used for trend analysis, selection of hardware
modification location, and design optimization. This technique is considered a mixture method
because it computes sensitivity values for modal parameters which result from either
experimental modal analysis or finite element analysis.
The other experimental/analytical method is component mode synthesis. Component mode
synthesis and the building block approach are techniques that use experimental or analytical modal
representations of the components of a large or complex structure to predict the resultant dynamic
characteristics of the entire structure. Furthermore, this technique has evolved to the point where
components are combined in either physical or modal coordinates. Therefore, this technique is
truly a mixture method where experimental and analytical data are used to optimize a design. This
method is very useful in the design cycle of industries that produce large structures, such as the
automotive and aerospace industries. A new component mode synthesis method (not a classical
mode synthesis method such as SYSTAN) is discussed in detail in Section 5 of Volume IV and the
building block approach is discussed in Section 4 of Volume IV along with the development of the
impedance modeling technique.
In summary, system modeling is a diverse field that involves many aspects of structural dynamics.
One of the primary goals of Volume IV of this Technical Report is to present this material in a
concise and consistent manner to reduce unnecessary confusion and better relate the various
factions involved. Furthermore, each technique is applied to a structure to gain further insight into
the practical aspects of system modeling.
4.3 Boundary Conditions
In the application of the system modeling techniques, there are three test configurations used to
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obtain the frequency response functions or the derived modal data base. These three test
configurations involve the boundary condition and can be summarized as free-free, constrained, and
actual operating boundary conditions. In terms of analytical modeling technique, such as finite
element analysis, various boundary conditions can be easily simulated in the mathematical model to
predict the dynamic characteristics of a system. Therefore, this gives the analysts the luxury to
evaluate the model using desired boundary condition. In terms of experimental modeling technique,
in a laboratory environment, usually it is very difficult and too costly to implement the test fixture to
simulate the actual operating condition of a complete system, or the constrained boundary condition
of a component. Therefore, most of the modal tests are performed under an environment simulating
free-free boundary conditions.
Besides the boundary conditions mentioned above, modal tests can be performed on the mass-loaded
structures to predict the shifted dynamic characteristics of the original structure. The advantages of
adding lumped masses at the connecting or attachment points of a component under testing are: (1)
modal coefficients associated with those connecting degrees of freedom can be more accurately
excited and described under the mass loading effect, (2) rotational degrees of freedoms at the
connecting degrees of freedom can be computed using rigid body computer programs if sufficient
number of accelerometers are mounted on the additive masses, (3) analytically, added masses can be
removed from the mass-loaded testing configuration, and the enhanced modal parameters of the
original structure can be obtained, (4) if the dynamic characteristics of the original structures are
available through analytical or experimental method, then more accurate generalized masses can be
obtained through these two sets of data. Examples of applying the mass additive technique can be
found in References [11 and [2].
4.4 Modal Modeling
This section reviews the system modeling technique known as modal modeling. Modal modeling is
also known as the Snyder Technique, Local Eigenvalue Modification, Structural Modifications,
Dual Modal Space Structural Modification Method, and Structural Dynamic Modification [316].
The common thread of the research mentioned is that all utilize a model in generalized or modal
coordinates from experimental data upon which to investigate structural changes. Structural
changes are transformed into modal coordinates and added to the structure and the result is resolved
to yield the modified modal parameters.
The modal modeling technique was initially published by Kron [3] in 1962 and extended by
Weissenberger [4], Simpson and Taborrok [5] , and Hallquist, Pomazal and Snyder [6]. Early
researchers in this area restricted themselves to a local modification eigensolution technique.
Several software packages have been developed employing this technique since the implementation
and widespread use of digital Fourier analysis. Notably, Structural Measurement Systems, Inc. first
released Structural Dynamics Modification [7] using the local modification procedure. The local
modification procedure allows only simple mass, stiffness and damping changes between two
general points.
Recent research has progressed in several areas. First, Hallquist and Snyder [8], Luk and Mitchell [9]
and SMS [71 have used the local modification technique for coupling two or more structures
together. Hallquist [101, O'Callahan [11] and De Landseer [12] have expanded
the method to include
complex modes. O'Callahan [13] and SMS [14] have found methods to approximate more realistic
modifications using local modification for trusses and beams. Mitchell and Elliot [15] and
O'Callahan and Chou [161 have developed different methods that use full six degree of freedom
reirc•scntaIintin thitt depend on the i,,iproxlmntion or mcniaurement of tle rolintional (legicel. of
freedom to make beam or plate modifications. Complete technical details of modal modeling
methods can be found in Section 2 of Volume IV of this Technical Report.
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4.4.1 Limitations of Modal Modeling
Since the structural dynamic modification method uses the modal parameters of the unmodified
structure to predict the effects of the modification, it is apparent that the accuracy of the results is
dependent on the validity of the data base supplied. For this reason, it is important that all errors
associated with the data acquisition and processing of the unmodified structure are minimized.
Among the sources of error that must be addressed are nonlinearity, standard FFT errors (aliasing
and leakage), scaling errors, as well as truncation errors of the modal model itself. Only the most
serious of these errors will be addressed at this time.
One of the assumptions in the experimental modal analysis is linearity of the system. Therefore, all
methods discussed here are based on this assumption. In reality, inaccurate results will arise when
linear system coupling algorithm is used to predict the dynamic characteristics of structure(s) coupled
together by nonlinear joints.
Leakage is a measurement error that arises from the processing of signals that are not periodic in the
time window of the signal analyzer. Because of this truncation in the time domain, the Fourier
coefficients of the sampled signal do not lie on the Af of the analyzer. This causes energy at a specific
frequency to spread out into adjacent frequency bands, and results in an amplitude distortion at the
actual frequency. It is this amplitude error that causes scaling errors in the modal mass and stiffness
estimates associated with each mode. This in turn affects the modification process by scaling the
predicted modes of the modified structure by an amount that is proportional to the amplitude error
of the original data.
This leakage problem can be reduced by: (1) using periodic excitation, or transient excitation signals
such as burst random, or, (2) using the H, frequency response function procedure, or, (3) using cyclic
averaging technique in the measurement stage, or, (4) taking data with smaller Af, or, (5) using
windows on the time domain measurement, such as, impact with exponential window applied to the
response signals. If exponential window is used, it must be accounted for in the parameter estimati on
because it adds artificial damping to the structure. Care must be taken not to overcompensate for
this damping allowing the poles of the system to become negatively damped.
Modification errors often arise from using a modal data base that is not properly scaled relative to
the system of units used by the modeling software. This type of error will occur if the modal
parameters are estimated using frequency response functions which were measured using improper
transducer calibrations. This improper scaling once again results in improper estimates of modal
mass and stiffness used by the modeling software.
Up to this point, it has been shown that the accuracy of the results is dependent on the amount of
error in the data base. Another concern that needs to be addressed is the validity of the modal
model. Because the effects of a structural modification are calculated in modal space, if an
insufficient number of modes are included in the original data base, there will be a limit to the
number of modal vectors that can be predicted. This phenomena is known as modal truncation and
should be considered in choosing a frequency range for the analysis. It may be desirable in some
cases to extend the frequency range of data acquisition above the actual frequency range of interest
for the data base to include a few extra modes. This extended frequency range will improve the
calculation of out of band residuals, and may help for the case where these out of band modes are
shifted into the frequency range of interest by the modification. Care must be taken in extending the
frequency range, to prevent an excessive loss of frequency resolution.
Another concern in the development of the modal model is the number of degrees of freedom to be
included in the analysis. By definition, the number of degrees of freedom must be equal to or greater
than the number of modes in the frequency range of interest. Realistically, the number of DOF
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should be much larger than the number of modes of interest to accurately define the individual
modal vectors. Since modal coefficients exist only at points where data has been taken, it is possible
to miss nodal lines of the structure if too few points are included in the analysis. This generally
becomes a more significant problem for higher order mode shapes.
Once the original data base has been established, the modal parameters can be estimated using any of
several existing parameter estimation algorithms. All of these algorithms attempt to yield a best
estimate of the actual parameters. Because there will always be some degree of experimental error in
the data, the resulting estimates of modal parameters will be subject to error. In order to minimize
this error, it is advantageous to use some sort of least squares implementation to yield a best estimate
of modal parameters.
The use of SDOF versus MDOF parameter estimation algorithms is determined by the modal density
of the structure being analyzed. If a SDOF method is used for a structure with closely coupled
modes, poor estimates of modal mass and stiffness are obtained, and the modification routine will
yield poor results.
The various errors mentioned in the previous paragraphs are commonly committed, and easily
overlooked when performing a modal test. This is not intended to be an exhaustive list of errors
affecting modal modeling, but an indication of the types of things that must be kept in mind when
establishing a valid modal data base. Without good estimates of the original structures modal
parameters, there can be no serious attempt at accurately predicting the characteristics of the
modified structure.
4.4.2 Validation of Experimental Modal Models
As mentioned in the previous section, the accuracy of the modal modeling or structural modification
results is dependent on the validity of the modal model supplied. There will always be some degree of
experimental and modal parameter estimation errors in the data base. Therefore, it is important for
the users to qualitatively, and if possible, quantitatively, examine the validity and errors of the modal
model before it is used to predict the system modeling or modification results. Although perfect
results should not be desired in the application of modal modeling technique, it is important to
realize that any modal model obtained experimentally is far from being perfect. Therefore, it is
suggested that the experimental modal model be validated, or optimized , before the model is input
to any modal modeling algorithm. Some of the validation methods are briefly summarized in the
following sections.
4.4.2.1 Frequency Response Function Synthesis
Synthesizing a frequency response function (not used in the estimation of modal parameters) using
extracted modal parameters and compared with a measured frequency response function at the
synthesized measurement degree of freedom is, in general, a common practice during the modal
parameter estimation process. If a good match exists between these two sets of frequency response
functions, then it is a good indication that the extracted modal parameters are agreeable with the
measurement data. But this doe s not guarantee that there is no error exist in the modal data base.
4.4.2.2 Modal Assurance Criterion
• Modal Assurance Criteria (MAC) 1291 is commonly used to check the consistency of the extracted
modal parameters, when more than one estimate of each mode is available.
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4.4.2.3 Detection of Mode Overcomplexity
This method qualifies each mode by a number called the Mode Overcomplexity Value (M.O.V.) and
the global Modal Model by the Mode Overcomplexity Ratio (M.O.R.) [30]. The basic idea of the
Mode Overcomplexity test is that, for good modal models with complex modes, the frequency
sensitivity for an added mass change should be negative. If it happens that the sensitivity is positive,
it is caused by either an incorrect scale factor (modal mass) or by the fact that the phase angle of the
complex modes compared to the normal mode phase angle exceeds a certain limit; in other words, it
is due to an overcomplexity of the mode shape.
The MOV is defined as the ratio of the number of positive frequency sensitivities over the number of
all the frequency sensitivities for a particular mode. To give more weight to points with a high modal
displacement compared to points with a small modal displacement, a weighted sum is introduced to
give a more general evaluation of the modal model. The value of MOV is between I and 0, the
bigger the value, the modal model is more overcomplex.
The MOR is defined as the ratio of ,7,MOVi over ( 1 - ,E4OV') which gives a one figure assessment
of the modal model with respect to its overcomplexity. The MOR ranges from zero to infinity. A
low MOR value indicates good modal data, while a large MOR indicates a scale factor problem or a
overcomplexity problem.
4.4.2.4 Mass Additive/Removal Technique
This technique employs a mass additive or removal procedure to verify or validate experimental
modal model in the application of modal modeling technique. Modal model can be obtained from
either the original structure, or, mass-loaded structure [2]. If a modal model is obtained from the
original structural configuration, then, comparisons can be made between the analytically predicted
dynamic characteristics of the mass-loaded structure - from the modal modeling algorithm - and the
test results (such as modified resonant frequencies) obtained from the physically modified massloaded structure. If there is no good agreement between these two sets of results for the mass-loaded
structure, then this is an indication that global or local scaling errors, or overcomplexity of some
measured complex modes, exist in the experimental modal model. If high quality data are desired in
predicting the system dynamics of the altered structure or combined structure(s), then the previously
determined experimental modal model needs to be validated, if possible, or, a new set of data needs
to be recollected before any modeling application is attempted.
For the second case, i.e., if a modal model is obtained from the physically mass-loaded structure, then
comparisons can be made between the analytically predicted and experimentally measured dynamic
characteristics of the mass-removed structure. Similarly, if there is no good agreement between these
two sets of data, this indicates some errors exist in the original modal model. In Reference [2], using
approximated real modes from the measured complex modes, a modal scaling procedure can be used
to correct the global scaling errors in the experimental modal model.
The number of masses and the size/weight of each additive mass that can be added to the structure is
dependent on the total mass and size of the structure(s). In general, the following rules can be used
as guidelines in considering the number and size(s) of the additive mass(es):
* The added mass(es) can be considered rigid in the frequency range of interest.
. With small amount of mass(es) added or removed to or from the structure, the mode shapes can
be considered unchanged before and after the modification.
* Sensitivity of the change of system dynamics is dependent on the location(s) of the added
mass(es). In other words, if there is only one mass added to a large structure, then some of the
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modes may not be sensitive enough to alter their frequencies due to the fact that the added single
mass is near the nodal points of such modes.
* Rotational degrees freedom, if permitted, can be extracted from the rigid body motion of the
lumped mass(es). This information is very useful if the mass mounting point(s) is(are) the
connection or coupling point(s) of the structure(s).
4.4.2.5 Improvement of Norms of Modal Vectors
Zhang and Lallement [31] proposed a method to improve the norms of the measured n'odal vectors
and then calculate the generalized modal masses of the original structure. This method will correct
inodal scaling errors in the modal data base. This method requires a set of modal data from the initial
structure and a set of data from the mass loaded (perturbated) structure.
4.4.3 Modal Modeling Summary
In summary, modal modeling has been discussed from its inception by Kron through present day
research inolving beam modifications in the modal domain. Modal modeling is a technique that is
very quick, because the generalized coordinates have a reduced number of degrees of freedom.
Therefore, many modifications can be investigated in a short time. Earlier, this technique was
prese.;ted mainly as a trouble shooting technique. In fact, researchers (16] have found this method
to be three to six times faster than analytical approaches. This ratio increases with the size of the
problem. The speed of this technique and its interactive implementation make it well-suited for onsite problem solving and initial design cycle work.
Many limitations are apparent in the development. First, if experimental data are used, the
frequency response functions must be carefully calibrated. This technique is extremely sensitive to
experimental errors in general. Data must be carefully acquired to avoid bias errors such as leakage
and aliasing. Errors made in the estimation of the frequency response functions translate into errors
in the modal model and modal matrix [*]. Modal parameter estimation is extremely critical in
modal modeling. Parameter estimation is a two-stage process that estimates eigenvalues which are
used to compute the modal model and the modal vectors which make up the transformation
matrix.
Recall that a convenient form of the model is for unity modal mass or unity scaling. Examination
of the modal stifflness and damping matrix reveals that the estimate of the damping ratio " is
involved in both matrices. Unfortunately. damping is a difficult parameter to estimate. This is one
of the major limitations of the accuracy of an experimental modal model. Fortunately, if great care is
taken in the measurements, the magnitudes of this error are not great enough to cause more
variation than found in normal experimental error.
Another source of error is truncation. Errors occur in two forms: geometry and modal truncation.
Geometry truncation is a problem that occurs when not enough physical coordinates are defined
to adequately describe the dynamics of the structure. Higher order mode shape patterns are not
properly defined unless enough points are defined along the shape to describe it. A good rule of
thumb is to apply Shannon's sampling theorem to the highest order mode expected. Geometry
truncation also occurs when all pertinent translational and rotational degrees of freedom are not
measured. If a structure exhibits motion in all translational degrees of freedom and only one is
measured, the associated error is defined as geometry truncation. In general, the number of data
points should be much greater than the number of modes of interest to avoid geometry truncation.
Modal truncation refers to the number of modes included in the data set. Since the modified mode
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shapes are a linear combination of the original mode shapes, the rank of the original modal
matrix limits the possible dynamic changes that can be calculated. The lower limit of the number of
modes required for even simple structures is six [32], to have sufficient rank to accurately predict the
results for the first few modes. A good rule of thumb is to include several modes beyond the
frequency range of interest to insure the validity of the results within the frequency range of interest.
Another serious modal truncation error occurs when the rigid body modes of a free-free structure
are not included when that structure is tied to ground.
Based on the preceding discussion it is apparent that the use of modal modeling programs with
experimental data requires carefully acquired data and good parameter estimation results. These
problems can be overcome by carefully designing the modal test and using the proper parameter
estimation algorithms for the given data [33). This technique works equally well with analytical data
and has been implemented in this manner by Structural Measurement Systems [34]
The issue of complex versus real modes has been debated greatly in recent years. To be
completely accurate the complex form of the modal modeling technique should be used when
nonproportional or heavy damping exists in a structure. Using a real normal mode in this case, will
cause erroneous results 1]. One compromise is to use a normalized set of real modes derived from
the measured complex modes described in Section 2.7 of Volume IV of this Technical Report.
The use of beam modifications greatly increases the capability of modal modeling. Simple
scalar modifications and lumped masses are limiting and unrealistic. Beam modifications require
rotational information at the modification points. This information is not readily available but can
be obtained with some effort experimentally or analytically. Once rotational information is
readily available from experimental sources, modal modification will become a more powerful trouble
shooting tool.
4.5 Sensitivity Analysis
Sensitivity analysis is an approximate
technique
that determines
the rate of change of
eigenvalues and eigenvectors using a Taylor expansion of the derivatives. This technique was
developed by Fox and Kappor [35 and Garg [-6] initially in the late Sixties and early Seventies.
Van Belle and VanHonacker [20,371 further developed its use with mechanical structures and
implemented it for use directly on modal parameters. This technique is approximate because only
one term (differential) or two terms (difference) of the series expansion are used to approximate
the derivative.
Sensitivity analysis is useful in two ways. First, if a certain type of modification of a structure is
required, Sensitivity analysis determines the best location to make effective structural changes.
Sensitivity analysis also is used to predict the amount of change by linearly interpolating the amount
of change from the sensitivity to achieve the desired dynamic behavior. This last method is very
time consuming, especially when using difference sensitivities to maintain accuracy.
4.5.1 Limitations of Sensitivity Analysis
The detailed development of the sensitivity analysis equations is provided in Section 3 of Volume IV
of this Technical Report. Different expressions are developed for the sensitivity of modal parameters
to changes in mass, stiffness and damping. For the calculation of the sensitivity of an eigenvalue, >*,
only the corresponding eigenvector is necessary. Calculation of eigenvalue derivations do not
require complete information on the dynamics of the structure [38]
Finite difference or difference sensitivities use a second term in the approximation to account for the
amount of the physical parameter change. Nevertheless, an improved estimation is obtained only
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when the change in the physical parameter is small, If the magnitude of change is increased beyond
a certain value, the result will be even worse. The difference sensitivity equations involve a term,
1
(A-) that is involved with the second order derivative of an eigenvalue. If there are two close
modes, this term will become large so that the second order derivative contribution dominates the
approximation. Further more, if two adjacent modes are very close to each other, the term will
diverge so the result will often be unacceptable. Hence, care must be taken when a set of modal
data shows repeated eigenvalues. The current sensitivity analysis method does not attempt to
account for repeated eigenvalues. Since, for repeated eigenvalues no unique definition of the modal
vectors associated with the repeated roots exists, sensitivity analysis cannot be used for modal data
containing repeated eigenvalues unless a normalization for the repeated eigenvalue case can be
defined.
From this discussion, it is seen that the expressions used to compute differential or difference
sensitivity from modal parameters are in the form of the transfer function matrix. Because only
one or two terms are used from the Taylor expansion, the technique is an approximate one. Sincc
only modal parameters are necessary, this technique is equally 3a plicable to experimental or
analytical data. Currently it is implemented with experimental data [ .
VanHonacker [381 has shown this method to be accurate for only small incremental changes. The
differential method is far less accurate than the difference method. For small changes of mass,
stiffness, or damping the differential technique will accurately predict the eigenvalue shift. For
more significant parameter changes, the difference technique is recommended. Therefore,
sensitivity analysis has only limited application in the prediction of the effects of structural
modifications.
Sensitivity is extremely useful as a preprocessor to Modal Modeling or Finite Element Analysis
techniques.
The sensitivities of a structure can be computed rapidly from the modal parameters
to determine the optimal location at which to investigate a modification. Furthermore, the
sensitivity value is useful in determining how much of a modification is required. Therefore,
Sensitivity Analysis is a valuable tool in the optimization of a design.
This technique has several limitations. First, the results are only as good as the modal parameters
used in the calculations. Therefore, all of the experimental errors and parameter estimation
limitations which hinder other modeling methods apply to sensitivity techniques as well. Notably, a
limited number of modes are available from zero to the maximum frequency measured. Although
not currently implemented with analytical data, any inaccuracies in an analytical model would
similarly deteriorate the calculations when used with modal data. In the experimental case, geometry
truncation errors are significant due to the exclusion of rotational degrees of freedom.
As a preprocessor to other modeling techniques, sensitivity has advantages. The computations are
fast and stable, especially when compared with a complete eigensolution. It is intuitive in nature
because it provides rates of change that allow the selection of the best type and location of
modification as well as a comparison of different modifications. This provides a large amount
of information that offers much insight into the dynamic behavior of a structure.
4.6 Impedance Modeling
The general impedance method was first introduced by Klosterman and Lemon [39] in 1969.
Due to the state of measurement equipment at that time the method was not pursued further.
As the ability of Fourier analyzers to accurately measure frequency response functions improved
in the late Seventies, the interest in General Impedance Techniques was renewed. Two
techniques are developed in this chapter using experimental data. Both methods use measured
frequency response functions or synthesized frequency response functions.
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The general impedance technique is formulated in two ways: frequency response and dynamic
stiffness. The dynamic stiffness approach was initially developed by Klosterman [4] and
implemented by Structural Dynamics Research Corporation as SABBA (Structural Analysis using
the Building Block Approach) [411. This technique is primarily used to couple together
structures to predict the total dynamic characteristics using the concept of superposition. Thus, the
phrase building block approach was applied to this technique.
The frequency response method was published and implemented by Ciowley and Klosterman [421 at
the Structural Dynamics Research Corporation in 1984 and referred to as SMURF (Structural
Modifications Using Response Functions). It is primarily a trouble shooting technique. It
operates on frequency response functions; therefore, no modal model is necessary to investigate
structural modifications. Problems may be solved by acquiring several frequency response
functions and investigating the effects of modifications. The modified frequency response
function is computed as a function of frequency by simple block operation using the original
frequency response function and a frequency representation of the structural modification.
4.6.1 Limitations of Impedance Modeling
The general impedance technique is a method that employs the use of frequency response data to
investigate coupling of structures and structural modifications. The dynamic stiffness approach is
the more powerful of the two. The advantage of this technique is that a large array of
investigations may be conducted. Also, the method is not as cumbersome as the component
modal synthesis technique because it has a reduced number of degrees of freedom.
Due to the dynamic stiffness formulation, modal models of experimental or analytical origin may
be combined or modified using an array of mass, stiffness, damping, beam, or matrix elements
represented in impedance form. This brings more analyical capability directly to an engineer in
a mini-computer environment as implemented currently [41] when compared with modal modeling
techniques.
Due to numerical problems, Klosterman recommends use of synthesized frequency response
functions to build the dynamic stiffness matrix. This introduces errors made in modal parameter
estimation, but reduces numerical problems associated with noisy frequency response functions
measurements because the parameter estimation process fits a smooth curve through the measured
data Modeling of this type requires carefully acquired, and properly calibrated, data to obtain
the best modal model possible.
One of the major problems with the dynamic stiffness approach is the determination of the
[H 1-1 matrices. The inverse of the matrix must exist. This problem forces the use of modal data
because the inversion process is numerically unstable [43]. The number of modes must be much
greater than the number of constraint points to insure the existence of the inverse.
The most serious limitation of the dynamic stiffness approach is the computational speed and
stability. When implemented initially, only individual frequency response functions were computed
for the resultant structure. This implementation is efficient, but the computations are somewhat
unstable unless synthesized frequency response functions are used. This led to the application of
a determinant search algorithm to compute the resultant eigenvalues and eigenvectors. This
algorithm is not computationally efficient because the equations are solved frequency by
frequency. This fact has led to more widespread use of component modal synthesis techniques that
are described in Section 5 of Volume IV of this Technical Report.
The frequency response method is implemented for single constraint situations to avoid the
matrix inversion problem. This makes it useful for trouble shooting situations. Since measured
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frequency response functions are used in the calculations, no modal model is necessary. Therefore,
if the necessary frequency response functions exist, modifications may be made directly to obtain the
modified frequency response. This makes the frequency response technique the fastest trouble
shooting technique, but only modified frequency response functions are computed not modal data.
This technique is slower than modal modeling when modal data is desired.
Data is required for driving points and cross measurements at the constraints, response point, and
driving point to compute the modified frequency response. Therefore, the impact testing or
multiple reference techniques are most convenient for the frequency response method.
Synthesized frequency response functions can be used in this technique when the desired frequency
responses are not available, but the advantage of avoiding modal parameter estimation is lost.
Modal modeling is a better alternative at this point because the entire set of modified modal
parameters is computed. Only individual modified frequency response functions result from the
frequency response method.
The frequency response technique is sensitive to measurement errors such as leakage, aliasing, and
random noise. Wang t441 has found the errors largest at anti-resonance. This is due to the fact that
the signal to noise ratio is poor at anti-resonance. Therefore, as the magnitude of modification or
constraint increases, the more the accuracy of the calculations will deteriorate because frequencies
will shift closer to or past anti-resonances.
4.7 Component Dynamic Synthesis
Component dynamic synthesis is an analytical procedure for modeling dynamical behavior of complex
structures in terms of the properties of its components or substructure. The procedure involves
explicitly every component in the structure the advantages of which are many-fold: analysis and
design of different components of the structure can proceed independently, component properties
can be obtained from tests and/or analysis, and the size of the built-up structure analysis problem can
be reduced to manageable proportions.
Component synthesis with static condensation [45,461 has long been used for improving efficiency of
static analysis. In this method, known as substructuring technique, unique or functionally distinct
components of a structural system are analyzed separately, condensed, and then combined to form a
reduced model. This reduced model, having fewer degrees of freedom, is generally more economical
to analyze than the original structural model. The static condensation is an exact reduction
procedure.
In dynamic analysis, exact reduction of an indMdual component is dependent upon the natural
frequencies of the total structural system which are yet unknown at the component level. Frequency
independent or iterative reduction methods must therefore be used, which introduce approximations.
The various reduction methods are collectively known as component dynamic synthesis or modal
synthesis (CMS).
The objectives of this section are to review the state of the art in component dynamic synthesis and to
develop and implement an improved dynamics synthesis procedure.
4.7.1 Dynamic Synthesis Methods
In order to review the existing dynamic synthesis procedures, it is necessary to define certain
frequently-used terms. A component or substructure is one which is connected to one or more
adjacent components by redundant interfaces. Discrete points on the connection interface are called
boundary points and the remainder are called interior points. The following classes of modes are
commonly used as basis components in component dynamic model definition. Details of these mode
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sets is given later in this report.
1. Normal Modes: These are free vibration eigenmodes of an elastic structure that result in a
diagonal generalized mass and stiffness matrix. The normal modes are qualified as free or fixed
interface modes, depending on whether the connection interfaces are held free or fixed. Loaded
interface normal modes simulate intermediate fixity of the interfaces.
2. Constraint Modes: These are static deflection shapes resulting from unit displacements imposed
on one connection degree of freedom and zero displacements on all the remaining degrees of
freedom.
3. Attachment Modes: The attachment modes are static deflection shapes defined by imposing a unit
force on one connection degree of freedom while the remaining connection degrees-of-freedom
(DOFs) are force free. If the structure is unrestrained, this mode set will consist of inertia relief
modes. Attachment modes are also static modes.
4. Rigid-body Modes: These are displacement shapes corresponding to rigid body degrees of
freedom. They may be considered a subset of normal modes corresponding to null eigenvalues or
else defined directly by geometrical consideration.
5. Admissible Shape Functions: These are any general distributed coordinates or space functions,
linear combinations of which simultaneously approximate the displacement of all points of an elastic
structure. The only requirements are that the admissible functions satisfy geometry boundary
conditions of the component over which they are defined, and satisfy certain differentiability
conditions. These are analogous to finite element shape functions.
Static condensation or Guyan Reduction [46] is the simplest of all component dynamic synthesis
techniques. The approach is a direct extension of static condensation. The transformation matrix of
static constraint modes which is used to reduce the order of the stiffness matrix in static analysis is
also used to reduce the order of the structure mass matrix. The kinetic energy of the interior nodes is
represented by only static mode shapes. Drawbacks of this approach are obvious. The static modes
are not the best Ritz modes for component dynamic representation.
The concept of Component Modal Synthesis (CMS) was first proposed by Hurty [47] Component
members were represented by admissible functions (low- order polynomials) to develop a reduced
order model. This procedure is essentially the application of the Rayleigh-Ritz procedure at the
component level. Hurty extended the method to include discrete finite element models[481 . This
method proposed that the connect DOF of a component were fixed or had a zero displacement.
Hurty then partitioned the modes of the structure into rigid body modes, constraint modes, and
normal modes. The constraint and rigid body modes were found by applying unit static load to each
of the connection points individually to obtain static deformation shapes of the structure. These
modes were added to the constrained normal modes to form a truncated mode set used in the
synthesis of the entire structure. A simplification of Hurty's fixed interface method was presented by
Craig and Bampton [49]. Substructure component modes were divided into only two groups:
constraint modes and normal modes. This resulted in a procedure which is conceptually simpler,
easier to implement in analysis software. Bamford [501 further increased the accuracy of the method
by adding attachment modes which improve the convergence of the method. The attachment modes
are the displaced configurations of a component when a unit force is applied to one boundary degree
of freedom while all other boundary DOF remain free of loads.
Goldman [511 introduced the free interface method, employing only rigid body m•des and free-free
normal modes in substructure dynamic representation. This technique eliminates the computation of
static constraint modes, but their advantage is negated by the poor accuracy of the method. Hou [52]
presented a variation of Goldman's free-interface method in which no distinction is made between
rigid body modes and free-free normal modes. Hou's approach also includes an error analysis
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procedure to evaluate convergence.
Gladwell [53] developed "branch mode analysis" by combining free interface and fixed interface
analysis to reduce the order of the stiffness and mass matrices for individual substructures. The
reduction procedure depends upon the topological arrangement of the substructures in the model.
Thus, reduction of any one substructure requires knowledge of the arrangement of all substructures
in the model.
Bajan, et al. [54] developed an iterative form of the fixed interface method. He showed that
significant improvements in synthesis accuracy can be achieved by repeating the reduction, based on
updated estimates of system frequencies and mode shapes.
Benfield and Hruda [55] introduced inertia and stiffness loading of component interfaces to account
for adjacent substructures. The use of loaded interface modes is shown to have superior convergence
characteristics.
Motivated by the need to use experimental test data, MacNeal (56] introduced the use of hybrid
modes and inertia relief modes for component mode synthesis. Hybrid modes are substructure
normal modes computed with a combination of fixed and free boundary conditions. Inertia relief
attachment modes are attachment modes for components with rigid body freedoms. MacNeal also
included residual inertia and flexibility to approximate the static contribution of the truncated higher
order modes of a component. Rubin [57] extended the residual flexibility approach for free interface
method by introducing higher order corrections to account for the truncated modes. Klosterman [581
more fully developed the combined experimental and analytical method introduced by MacNeal.
Hintz discussed the implications of truncating various mode sets and developed guidelines for
retaining accuracy with a reduced size model 59 ].
Many authors have compared the techniques discussed. No method clearly appears to be superior to
the other. The constrained interface method of Craig and Bampton and Hurty is expected to be the
most accurate when the connect degrees of freedom have little motion. The free interface method
with the use of residuals as proposed by Rubin appears to be more accurate than the constrained
approach.
Recent research has centered on comparisons of the various methods. Baker [60, for example,
compares the constrained and free-free approach using experimental techniques and also investigates
using mass additive techniques and measured rotational DOF
.45].This investigation was motivated
by a need to find the best method for rigidly connected flexible structures. In this connection, the
rconstrained method produced the best result. Klosterman [58] has shown the free-free method to be
accurate for relatively stiff structures connected with flexible elements. This supports Rubin's
conclusion [57] that the free-free method is at least as accurate when residual effects are accounted
for. These conclusions are intuitive because the type of boundary condition imposed in the analysis
that best represents the boundary of the assembled structure provides the best accuracy in the modal
synthesis.
Meirovitch and Hale 161] have developed a generalized synthesis procedure by broadening the
definition of the admissible functions proposed by Hurty [47]. This technique is applicable to both
continuous and discrete structural models. The geometric compatibility conditions at connection
interfaces are approximately enforced by the method of weighted residuals.
The method due to Klosterman 158] has been implemented in an interactive computer code SYSTAN
1621 and that due to Herting [63] is available in NASTRAN. The latter is the moot general of the
modal synthesis techniques. It allows retention of an arbitrary set of component normal modes,
inertia relief modes, and all geometric coordinates at connection boundaries. Both the fixedinterface method of Craig and Bampton, and the MacNeal's residual flexibility method, are special
cases of the general method. Other analyses presented in the literature based on modal synthesis
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techniques are not incorporated into general structural analysis codes. In general, there is a lack of
sophistication in available software.
4.7.2 Damping Synthesis Methods
The methods of dynamic synthesis are particularly useful and sometimes the only alternative available
in damping prediction for built-up structures. Most frequently, damping has been synthesized in the
manner analogous to stiffness and mass synthesis with the assumption of proportional damping.
Hasselman and Kaplan [64] used complex modes of components with nonproportional damping.
Obtaining damping matrices in the presence of general energy dissipating mechanisms in a complex
structure is one of the complicating factors, however. In such cases an average damping behavior can
be obtained from tests in the form of cyclic energy dissipated versus peak stored energy correlation or
damping law. Kana et al. [65] synthesized system damping based on substructure stored energy at the
system modal frequency. Soni 1661 developed a substructure damping synthesis method applicable to
cases where substructure damping varies greatly and irregularly from mode to mode. The procedure
has been validated in experimental studies [6. Jezequel 168 employed fixed interface component
modes together with mass loaded interface modes replacing the static constraint mode in his damping
synthesis method. Mass loading results in an improved representation of interface flexibility and
dissipation; however, the use of constrained interface modes make it difficult to implement it in
experimental testing.
The subject of component dynamic synthesis has received increasing attention in recent years.
Reference (69] presents several detailed reviews, applications, and case histories, with particular
emphasis on experimental characterization of component dynamics.
4.7.3
A Comparison of the Synthesis Methods
While differing in their detailed treatment, all the synthesis methods have the following objectives:
(1) to efficiently predict the dynamics of a structure within required accuracy for a minimum number
of DOFs; (2) to analyze the components totally independent of other components so the design
process is uncoupled, and (3) to use component properties derived from tests and/or analyses. The
various methods discussed in the preceding paragraphs only partially satisfy the three basic
requirements. Common to all modal synthesis methods discussed in the preceding is the complexity
of the matrix manipulations involved in setting up the coupling and assembly procedure to obtain the
final reduced equation system.
The major limitation on the use of the existing modal synthesis methods is their lack of compatibility
with practical experimental procedures. Although various types of component dynamic
representations have been proposed, only those requiring normal modal properties are practical.
Test derived modal representation is, in general, incomplete; the component normal modes obtained
assuming any type of support conditions at the interfaces are, in general, different from those
occurring when the components are acting within the compound of the total system. Since only a
limited set of modal data is obtainable, the interface flexibility is not adequately modeled. Depending
upon the synthesis method used, additional information is therefore usually required to approximate
the effect of interface condition or modal truncation.
Fixed interface mode synthesis methods employ static deflection shapes. In an experimental setup,
constraining interface degrees of freedom proves impractical, particularly when large dimensions or a
large number of connection points are involved. Also, damping data associated with static modes is
unavailable. For these reasons the free interface based modal synthesis methods are best suited for
achieving test compatibility. These methods also lend themselves to accuracy improvements via the
artifice of interface loading or by augmenting the normal modal data with residual flexibility and
inertia effects of truncated modes.
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4.7.4 Superelement Method
The objective of this work is to investigate and develop component dynamic synthesis procedures and
associated computer software which (a) combines component dynamic characteristics obtained from
modal tests or analyses or both; (b) accounts for the effects of differences in interface boundary
constraints of the component structures in the modal test and in the comparison of total structure;
and c) reduces inaccuracies due to modai truntation.
In view of the above objectives, and the assessment of existing synthesis methods, reviewed in the
preceding section, the free interface modal synthesis methods are studied further in this work. For
completeness, the fLxed interface and the discrete element representations are also considered. For
certain components the use of constrained interface conditions may be unavoidable. Structural
components such as panels. stringers, simple masses, vibration control devices, etc. are conveniently
input to the synthesis procedure via discrete elements.
A principal feature of the work developed here is the component dynamic model reduction
procedure that leads to an exact and numerically stable synthesis. In order to affect component
coupling, neither the specification of external coupling springs nor an user-specified selection of
independent coordinate is required. Existing synthesis procedures suffer from these drawbacks.
Component dynamic models considered include free-free normal modes with or without interface
loading, up to second order stiffness and inertia connections accounting for the effect of modal
truncation, fixed interface modes, and also the physical coordinate components. The modal
reduction procedure involves interior boundary coordinate transformations which explicitly retain
connection interface displacement coordinates in the reduced component dynamic representations.
Interior coordinates may include physical. modal, or any admissible coordinates. Components in this
reduced form are termed "superelement" because they are a generalization of the conventional finite
elements of structural mechanics. The problem of component dynamic synthesis is then reduced to
the assembly of the superelements. The direct stiffness approach and all subsequent processing
operations of the finite element method are then applicable.
In order to develop an improved component dynamics synthesis procedure, there are two key issues
to focus on: the modeling of component dynamics and the coupling of component coordinates. As
seen in the review, no one method of component dynamic modeling is shown to be superior to any
other. The methods of synthesis developed in the literature use one or other type of component
representation. Aerospace structures involve a wide variety of components and any single component
dynamic modeling method may not be uniformly suitable to all the components. With this in mind, a
generalized synthesis method was developed which permits different types of component models and
an associated coupling procedure. This material is reviewed in detail in Section 5.2 of Volume IV of
this technical Report.
4.7.5 Summary - Superelement Method
A set of consistent Ritz transformations was derived that lead to an exact, efficient and unified
procedure for coupling component dynamic models. A broad class of test and/or analysis derived
component dynamic models were considered in this work. These dynamic models are compatible
with the state of the art experimental modal testing as well as analytical procedures and permit
improvement of synthesis accuracy through the inclusion of flexibility, inertia and damping
corrections of truncated high frequency modes in the component dynamic models. The synthesis
procedure developed in this work may be considered as a generalization of the Craig-Bampton
method to include free interface normal modes, residual flexibility attachment modes, loaded
interface normal modes, and any general type of component modes or admissible shapes that
adequately represent the dynamics of a component. Several existing methods such as MacNeal's
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method, and Rubin's method are shown to be the particular cases of the generalization presented in
this work. As a result of this generalization, the different components of a built-up system may be
characterized using any type of dynamic model which is most convenient regardless of the manner in
which the other components are characterized. Furthermore the component coupling is exact and
computationally efficient; no artificial coupling element or the user specification of independent
coordinates is required. The developed procedure is implemented in a stand alone computer
program COMSYN which is documented in Appendix C of Volume IV of this Technical Report.
The component dynamics reduction method developed in this work transforms the component
dynamic coordinates to the superelement coordinates containing the physical coordinates of the
connection interfaces as well as any desired noninterface points. As a result of this reduction
procedure the component subsystems take the form of a finite element. It is therefore possible to
obtain system synthesis even with nonlinear components. The addition of the necessary data
handling and solution algorithms to treat nonlinear components will greatly enhance the capabilities
of COMSYN. It is recommended as a further work.
4.8 Summary - System Modeling
Experimental modal analysis developed in the past decade can provide a valid data base used in the
application of system modeling techniques. The success of applying system modeling techniques in
improving the engineering quality of the industrial products through a design cycle, is dependent on
the quality of the experimental data, and the accuracy of the system modeling algorithm used to
predict the altered system dynamics of a structure or combined structure(s).
Generally speaking, all system modeling techniques, which include modal or impedance modeling
method, sensitivity analysis, and the component mode synthesis method, can predict satisfactory
results if a complete and perfect experimental model can be obtained from testing and used as data
base for system modeling predictions. In reality, there exist many uncertainties and difficulties in
obtaining a complete modal or impedance model representing a physical structure. Difticulties in
simulating actual boundary conditions in the testing laboratory, lack of rotational degrees of freedom
measurement, incomplete modal model due to limited testing frequency range, nonlinearities existing
in the structure under test, scaling errors, and mode overcomplexity, could seriously affect the quality
and completeness of the experimentally-derived modal or impedance model. These deficiencies in
obtaining a reliable and complete experimental model make the system modeling technique a much
less powerful tool in the application of engineering design. In other words, currently, the weakness of
applying system modeling techniques comes from those limitations and uncertainties to obtain a
desired modal or impedance model of physical structure(s).
At the present time, many efforts have been dedicated by researchers to overcome those deficiencies
in obtaining a desired experimental modal or impedance model, such as the development of
rotational transducers. Further research and practices are still needed to develop a well-defined
engineering procedure and criterion to make the use of the system modeling techniques a more
powerful and reliable tool in engineering practices.
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Method, AIAA/ASMEIAHCE/AHS 20th Structural Dynamics and Materials Conference, (St.
Louis, MO: 1979)
[64]
Hasselman, T. K. and A. Kaplan, Dynamic Analysis of Large Systems by Complex Mode
Synthesis, Journal of Dynamical Systems, Measurement and Control (Sept. 1974), pp. 327-333.
1651
Kana, D. D. and J. F. Unruh, SubstructureEnergy Methods for Prediction of Space Shuttle Modal
Damping Journal of Spacecraft, Vol. 12 (1975), pp. 294-301
[66]
Soni, M. L., Prediction of Dampingfor Flexible Spacecraft Appendages, Proceedings of the 2nd
International Modal Analysis Conference, 1984
[67]
Soni, M. L., M. Kluesener and M. L. Drake, Damping Synthesis and Damped Design for Flexible
Spacecraft Structures, Computers and Structures, Vol. 20, No. 1, 1985, pp. 53-574
[681
Jezequel, L., A Method of Damping Synthesis from Substructure Tests, Journal of Mechanical
Design Trans. ASME, Vol. 102, April 1980, pp. 286-294
[69]
Combined Experimental/Analytical Modeling of Dynamic Structural Systems, Papers presented
at ASCE/ASME Mechanics Conference, Albuquerque, NM, June 24-26, 1985, ASME
publication AMD VOL 67, edited by D. R. Martinez and A. K. Miller.
(701
Walton, W. C. Jr. and E. C. Steeves, A New Matrix Theorem and Its Application for Establishing
Independent Coordinatesfor Complex Dynamic Systems with Constraints, NASA TR R-326,
1969.
[71]
Craig, R. Jr., StructuralDynamics: An Introduction to Computer Methods, John Wiley & Sons,
1981
[72]
Lamontia, M. A., On the Determination of Residual Flexibilities, Inertia Restraints, and Rgid
Body Modes, Proceedings of International Modal Analysis Conference, pp. 153-159
[731
Kramer, D. C. and M. Baker, A Comparison for the Craig-Bampton and Residual Flexibility
Methods for Component SubstructureRepresentation, AIAA Paper 85-0817
[741
O'Callahan, J. C. and C. M. Chou, Study of a Structural Modification Procedure with Three
DimensionalBeam Elements Using a Local Eigenvalue Modification Procedure, Proceedings of
the 2nd International Modal Analysis Conference, pp. 945-952
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5. UNIVERSAL FILE STRUCTURE
5.1 Introduction
One of the significant
problems
experimental
and analytical structural analysis involves combining,
comparing,
and correlating
data of
that
exists in different
formats, in different software and in different
hardware. This problem is not a technological problem so much as it is a logistical problem. In order
to address this problem, a standardized data base structure needs to be identified and supported by
all of the organizations operating in the structural dynamics area. While this goal cannot be
accomplished, ultimately, until an official standard exists, it is possible to alleviate the problem by
identifying the basis for a data base structure and providing this information to the organizations that
would eventually be involved in the development of an official standard. The objective of Volume V
of this Technical Report is to document a data base standard that can provide a means for data
exchange.
R
týN
The requirements for a data base standard that can be applicable to different software and hardware
environments must be very general so that any level of user can support the data base standard. For
this reason, an eighty character per record, ASCII format is the only basis for the data base structure
that can be supported in the required environments. It is important to note that this data base
format is not intended to be used as an internal format within software or as the basis of a hardware
format. This sort of format is only useful as a mechanism for input and output to media that are
compatible with the different environments that may need to be utilized.
5.2 Format Development
In order to develop the data base structure, the types of formats or capabilities that were needed
were first identified. The basic requirements included a file structure that could define the geometry
of the nodal degrees of freedom, measurement data at the nodal degrees of freedom, and modal
parameters associated with the nodal degrees of freedom. In addition to these basic requirements,
information concerned with the source of the file information and the units of the data is needed to
qualify the information in the files that belong to a specific data base.
Once the basic requirements were identified, existing data base structures were evaluated to
determine whether a current format would be sufficient or could be modified to meet the basic
requirements. In this regard, consideration was given to the basic requirement that the format be
ASCII, to whether the data base already included the required formats, to whether the data base is
being utilized at the present time, etc. Several possibilities existed with respect to an internal data
base developed at the University, to data bases utilized by finite element programs, and to data bases
utilized by experimentally based programs. For example, the University of Cincinnati Structural
Dynamics Laboratory (UC-SDRL) had developed an ASCII format data base in order to compare
finite element and experimental test data. This format was limited to nodal geometry and modal
parameters and would have to be expanded in order to service all of the needs that exist in the
analytical and experimental structural dynamics area.
As a result of this review and deliberation, the Universal File 1 '"2' concept utilized by Structural
Dynamics Research Corporation (SDRC) was adopted as the basis for the data base structure. In
general, this Universal File concept addressed the needs of both the analytical and experimental
aspects of the structural dynamics area. Also, there is considerable experience and history of the use
of this Universal File structure in both the analytical and experimental programs that SDRC has
developed. The structure of the Universal File is documented very well and has already been
adopted by other organizations as the basis for internal data base structures. Additionally, SDRC
supported the concept of a wider application of the Universal File concept and has added Universal
Ai
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File structures to address potential needs that previously had not been identified. For example, the
Units File (File Type 156) has been added to facilitate the different units that occur when data
originates from different hardware and software vendors.
5.3 Universal File Concept
A Universal File is a physical file, card deck, magnetic tape, paper tape, etc. containing symbolic data
in physical records with a maximum record length of 80 characters.
On the physical file, data is contained in logical data sets with the following characteristics:
a.
The first record of the data set contains "-l" right justified in columns 1 through 6.
Columns 7 through 80 of the physical record are blanks.
b.
The second record of the data set contains the data type number, numeric range 1 through
32767, right justified in columns 1 through 6. Columns 7 through 80 of this physical record
are blanks.
c.
The last record of the data set contains "-1" right justified in columns 1 through 6.
Columns 7 through 80 of the physical record are blanks.
d.
The specification of data on the remaining records of the data set are totally dependent on
the data set type.
For example:
-1
xxx
(data pertaining to the data set type)
-1
Although the data organization is built ar•und 80 character records, the capacity for data record
blocking has been provided. Its principle use would be for magnetic tapes where the overhead
associated with 80 character records is excessive. As such, a preferred physical/logical record blocking
of 80 logical records per physical record is recommcnded. This improves system capacity and
response dramatically.
5.4 Future Considerations
If further data base structures become necessary, several options can be pursued. First of all, the
Universal Files documented in later sections of this report are a subset of the Universal Files
supported bySWRC. Other Universal File formats may already exist which satisfy future
requirements.[ I' If another Universal File format does not already exist to service the intended
needs, a new format can be developed as long as the Universal File format number is unique.
Another future consideration is the development of other similar formats. A spikir concept to
Universal Files is being developed in Europe, called Neutral Files and Meta Files,t-' to serve the
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same purpose. If future standards are developed and adopted, conversion programs to convert from
the Universal File format to the new formats should be straight forward.
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REFERENCES
[I] SDRC J-deas Level 3 User's Guide, "Section VI, Universal File Datasets", 1986, pp. 306-470.
[21 Reference Manual for Modal-Plus 9.0, "Appendix A, SDRC Universal File Formats", SDRC
GE-CAE International, 1985, 26 pp.
[31 Ghijs, C., Helpenstein, H., Splid, A., Maanen, J.; Design of Neutral File I to 8, Rutherford
Appleton Laboratory, CAD*I Paper RAL-012-85, 1985, 10 pp.
[4] Leuridan, J.. Contents of the Common Database for Experimental Modal Analysis, Leuven
Measurement and Systems, CAD*I Paper LMS-007-85, 1985.
[51 Proposalfor ESPRIT CAD Interfaces, ESPRIT Project Reference Number 5.2.1, Technical
Annex, 1984.
[6]
Heylen, W., PreliminaryList of Keywords for Neutral Files 7 and 8, CAD*I Paper KUL-017-85.
1985.
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6. SOFIWARE DOCUMENTATION
6.1 Introduction
Volume VI of this Technical Report describes briefly the history and current state of development of
the Real Time Executive (RTE) Modal Program at the University of Cincinnati Structural Dynamics
Research Laboratory (UC-SDRL). The RTE Modal Program serves as the kernal for much of the
software development under this research effort. The purpose of Volume VI is to provide a
reference for the operation of the RTE Modal Program and to provide a reference for future
program development.
The RTE Modal Program has been developed as a replacement of an earlier program (User Program
9) that was written for the HP-5451-B Fourier system. The original concept of an RTE based
program began in 1978 but was not realized in a working form until early in 1981. Based on the
operating system of the HP-5451-B, Basic Control System (BCS), continued expansion of that
software is prohibitive due to the inflexible programming environment and the memory limitations.
To address these problems, the RTE Modal Program utilizes the overhead functions of the File
Management Program under (RTE), an operating system available on Hewlett Packard computers,
to provide flexibility that does not have to be built into the modal software The emphasis of the
modal software development in the RTE environment is toward supportability rather than efficiency.
For future development reasons and based upon the research nature of the Structural Dynamics
Research Laboratory, the ability of graduate students to extend and enhance the current software is
always the primary consideration. In this way, the modal software can eventually support any type of
data acquisition system as well as interface through file structures to related software such as finite
element analysis packages.
Future development of this software will be based upon a graphics workstation concept, utilizing a
Unix operating system. This project has already been initiated in order to allow the developments in
modal analysis software to be more readily available in several hardware configurations.
6.2 Software Compatibility
The UC-SDRL and the UDRI believe that the success of this effort in providing an efficient and
user-oriented analytical tool is highly dependent upon the program development philosophy which
was adopted during this effort. The more important guidelines that were followed during this effort
will include the following:
" Programming Language
All program development is compatible with the current version of ANSI standard FORTRAN
(1977), when possible. Exceptions to this would be assembly level software required by particular
hardware or software operating systems.
" Structured Programming
All software developed during the research effort has been written in "structured" FORTRAN
and therefore will be arranged in short modular subroutines for faster compilation, less memory
requirements, and easier modification.
"*Internal Documentation
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The programs are internally documented containing a "block" of comments at the top of each
module and "step comments" within each module. The block comments provide a concise
statement of the function of the subroutine, algorithm used, input and output arguments, names
and meaning of the important variables, subprograms called, and any peculiar features of the
subroutines. Included is an identification of system dependent code with an explanation of the
purpose of the system dependent operation.
" External Documentation
External documentation in the form of a User's Manual is provided. This consists of an appendix
in Volume IV and all of Volume VI of this Technical Report. All units, modules, programs,
systems and interactions between them will be complete. Information sufficient for a user of the
program to prepare data, run the program, and assess the results are included in the manual.
Also included will be statement of the program function, names and functions of the principal
modules, call sequence of modules, list of modules called by each module and name and purpose
of major variables.
" Compatibility with Modal Analyzers (Data Acquisition)
All programs are compatible with the data base generated by HP-5451-B and HP-5451-C Fourier
Systems. Through the use of Universal Files, almost any Fourier analyzer can be made to be
compatible. The component synthesis program, in particular, is also compatible with the format
generated by the NASTRAN finite element program. The format for specification of component
data to the synthesis program is described in an appendix to Volume IV of this Technical Report.
for ease of developing interface with any other modal analysis software. All magnetic tape
formats that are developed will be based upon an 80 ASCII character record (card image) format.
While this format produces a somewhat longer data tape, the ability of most computer systems to
read such a format with standard 1/0 subroutines is a stronger consideration.
6.3 Data Acquisition Hardware Environment
The IP-5451-C Fourier System was originally the primary target for the initial version of the RTE
Modal Program. This system provides a BCS programming environment for the estimation of
frequency response functions and the storage of the frequency response functions to disc media
compatible with the RTE environment. Current software is compatible with HP-1000 systems with
either 21-MX-E or 21-MX-F processors or HP A Series computers such as the A-700 or A-900. In
this mode of operation, data acquisition will be provided by a HP-5451-B/C, a HP-5420-A, a HP5423-A an S/K-LMS FMON, or a Genrad 2515 Fourier System. Data will be available on disc media
via the FMTXX structure defined by the HP-5451 Fourier Systems. Compatibility of data from these
as well as other Fourier systems is always available through the Universal File Structure supported by
SDRC and UC-SDRL Documentation on this file structure may be found in Volume VI of this
Technical Report.
6.4 Modal Analysis Hardware Environment
The RTE Modal Program is designed to be executed on an HP-5451-C Fourier System with multiple
HP-7900 Discs, an HP-7906 Disc or an HP-7925 Disc. The minimum memory configuration is 128K
words but portions of the RTE Modal Program will run more efficiently if more memory is available
(256K words or larger). At the present time, the Extended Memory Area (EMA) and the Vector
Instruction Set (VIS) are not utilized in any of the primary programs. These capabilities are utilized
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in some of the advanced parameter estimation and modal animation programs. Due to the increasing
memory requirements and computational load of many of the parameter estimation algorithms
currently under evaluation, these options will be utilized even more in the future.
6.4.1 Memory Requirements
The RTE Modal Program involves the operation of multiple programs through a series of monitors.
Programs may be suspended as other programs are executed or multiple programs may be executed
simultaneously. For this reason, the optimum memory size currently would require five partitions of
28K words available to the RTE Modal Program at one time. This allows all dormant, suspended
programs as well as active programs to be memory resident and reduces the program swapping time.
If this much memory is not available, dormant programs will be swapped to disc to allow active
programs to be executed. Therefore, in this situation, more work track area will be required on the
system discs to swap dormant programs.
6.4.2 Disc Requirements
The RTE Modal Program is designed to run most efficiently on a multiple HP-7900 Disc system, a
HP-7906 Disc, or a HP-7925 Disc, all of which are supported as BCS environment options on the
HP-5451-C Fourier System. The RTE Modal Program will run on a HP-5451-C Fourier System with
only one HP-7900 Disc but file storage is minimal.
6.4.3 Graphics Display Requirements
Originally, the HP-5460-A Display Unit was the primary graphics vector display that was supported as
part of the RTE Modal Program for data evaluation and modal vector animation. Additionally,
several other graphics vector display devices are currently supported. The IP-1351 Vector Graphics
Generator is supported as an optional display for the HP-1000 systems that do not normally include a
high speed vector display. Both the HP-5460 and the HP-1351 displays are controlled from RTE
using the Universal Interface Driver (DVM72) supported by Hewlett-Packard as part of the RTE
operating system. Both displays are interfaced via the Data Control Interface Card (HP-0546060025). The HP-1351 Vector Graphics Generator requires the 16 Bit Parallel Interface (Option 002)
to operate in this format. Operation of the HP-1351 Graphics Vector Generator also requires the
maximum amount of memory available for the unit.
In addition to these two displays, support of the HP-134x displays has recently been added. Support
for the HP-1345 involves a 16 bit parallel interface with the use of the Universal Interface Driver and
support for the HP-1347 involves an IEEE-488 interface with the use of the appropriate HP-IB
driver.
6.4.4 Plotter Requirements
All HP plotters interfaced via the HP-IB, the HP-7210 Digital Plotter, and all Tektronix 40xx
Terminals will operate with the current software. Logical units have been defined within the RTE
Modal Program to include up to five plotter logical units to allow for future plot flexibility. The
tentative plan is to eventually include the HP-264X Graphics Terminal.
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6.5 Modal Analysis Software - Operating System Environment
The RTE Modal Program currently runs in any revision of RTE later than Revision 2140 of RTE-4B. RTE software is not part of the standard HP-5451-C Fourier System. Therefore, any group or
facility that would wish to run the RTE Modal Program in this environment must purchase this
software from Hewlett-Packard. This software can be generated on either a session or non-session
basis. The non-session structure is for a limited number of users with no accounting feature. The
session structure is for multiple users and uses an account structure to restrict access to portions of
the system. In the session type of environment, the RTE Modal Program runs in a multi-user
situation, allowing multiple copies of a program to run and managing resources such as modal
animation devices and data logical units based upon the workstation that is in use.
6.5.1 RTE-4-B (Non-session)
RTE-4-B (Non-Session) is an RTE environment that is currently supported by Hewlett-Packard.
This is compatible with the FSDS systems that are supported with the HP-5451-C systems but
includes a newer revision operating system and the loader program.
6.5.2 RTE-4-B (Session)
RTE-4-B (Session) is an RTE environment for multiple users that is currently supported by
Hewlett-Packard. While this operating system is not the same as RTE-4-B (Non-Session), the RTE
Modal Program will currently run in this environment.
6.5.3 RTE-6-VM
RTE-6-VM is the virtual memory RTE environment which is available as of Revision 2201. While
this is not a true virtual memory operating environment, this system is expected to reduce the
overhead of working with large arrays. It is expected that conversion to the RTE-6-VM will require
changes that will not be downward compatible but, due to the attractive characteristics of the
operating system, the eventual target environment will most likely be RTE-6-VM.
6.5.4 RTE-A
RTE-A is the virtual memory RTE environment available for the A Series Hewlett Packard
computers. This operating system is very similar to the RTE-6-VM operating system.
6.5.5 Operating System Requirements
Within the structure of the RTE Operating System, certain system capabilities must be available.
First of all, the RTE Modal Program makes use of a minimum of 432 blocks of 128 words as a
temporary area for the storage of arrays during program execution. This working space is located on
disc and serves as the database for the RTE Modal Program. Therefore, if sufficient disc space is not
available, the program will terminate execution at the initialization stage. Additionally, if memory is
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at a minimum, more disc space will be required by the RTE Operating System to swap dormant
programs to the disc in order to run active programs. If sufficient disc space is not available, a
currently active program will not be able to schedule a son program without suspending the RTE
Modal Program while waiting for disc space to become available. Unfortunately, it is unlikely that
any activity, except for the removal of a dormant program from the program stack with the
'OF,NAMR, I', will ever release disc space so that the suspended program can continue. Therefore,
in minimum memory configurations, more disc space must be made available so the RTE Modal
Program cannot be suspended. The current version of the software requires a minimum of 25 work
tracks for operation in a 96K word RTE Operating System.
The only other system capability that is used by the RTE Modal Program is the System Available
Memory (SAM). This buffer in the system must be at least 3000 words in length for class I/O data
transfers used by the RTE Moda! Program.
6.6 Modal Analysis Software Overview
The RTE Modal Program development is structured to emphasize simplicity rather than efficiency.
For this reason, approximately 90% of the software code is in Fortran, ANSI 1966 or ANSI 1977.
Many operations could proceed faster or more efficiently if written in Assembly language but, as the
software and hardware changes in the future, the overhead required to recode these operations is not
efficient in the long term sense and would not be efficient with regards to the long term goals of the
research program at the University of Cincinnati.
Much of the function of the RTE Modal Program is designed to facilitate access to other related
programs and their data sets as well as to provide other programs access to the data sets created from
the RTE Modal Program. In this way, the RTE Modal Program can use or provide information
from/to a finite element program or alternate experimental data analysis techniques.
The structure of the monitor and commands within the RTE Modal Program is intended to facilitate
a tutorial approach to the use of the program. Each monitor has a help feature where the available
commands can be determined as well as a short description covering the use of each command. The
individual commands often involve multiple optional parameters which provide the experienced user
with the ability to streamline the use of the command and answer a minimum number of questions.
6.6.1 Monitor Structure
The RTE Modal Program is structured as a nested set of monitors where each monitor exits to the
next higher monitor until the File Manager (FMGR) monitor is reached. At the current time, no
capability of sequencing commands either within or among the monitors in an automatic way is
provided. In the future, this type of programming is an obvious extension to the current capability.
Each monitor contains a user help feature that gives the user access to an on-line user manual. This
help feature can be accessed in each monitor to determine what commands are available and
specifically how to exercise the command.
6.6.2 RTE File Structure
The RIE Modal Program generates and uses two types of FMGR files in order to facilitate data
storage and retrieval as well as to provide data sets to other programs. The two file types are
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designated as Project Files and Modal Files. The use of Project Files is intended to provide data
storage and retrieval for the RTE Modal Program while the use of Modal Files is to create a file
format that is documented (Appendix D of Volume VI of this Technical Report) to be used to
transfer modal data files between the RTE Modal Program and other programs. Modal Files are
also convenient for storing only a small portion of the total modal data set. Component definition
information, coordinates, display sequence, frequency and damping information or a subset of the
modal vectors may individually stored in a modal file. Refer to the File Store Command for details.
6.6.2.1 Project Files
Project Files are binary files consisting of 128 word records. Within the FMGR concept, this i,ý
designated as a Type 1 File. The Project File is a block image of the data storage area managed by the
RTE Modal Program. Note that a block is defined as 128 words of storage either in memory or on
disc. Effectively, this data area contains the current state of all important variables and data arrays so
that the operation of the program can be restarted in a given state very easily.
6.6.2.2 Modal Files
Modal Files are binary files consisting of 16 word records. Within the FMGR concept, this is
designated as a Type 2 File. The Modal File is a structured copy of a specific part of the modal data
set that exists at the time the file is created. Within the RTE Modal Program, five Modal Files have
been defined currently which can be stored in this manner.
6,6.2.3 Universal Files
Data can be written to or read from other system types and other programs by means of universal
files. Universal files are ASCII files with defined formats for storing data, including modal
parameters, structure geometry, display sequences, frequency response functions and general
measurements. For a complete description of available universal file formats see Appendix I of
Volume VI of this Technical Report.
This concept thus allows communication between any programs supporting universal files such as
data acquisition, parameter estimation, modal modification and finite element programs.
These universal file formats were originally developed at Structural Dynamics Research Corporation.
6.6.3 Data Acquisition
Data acquisition was originally expected to take place on a HP-5451-B/C Fourier System. The
resulting frequency response function data is placed on a data disc according to a table contained
within the subroutine FMTXX. This table, DIFS, is used by the BCS operating environment to
determine where any record of any of nine file types is located on the data disc. This same
subroutine, FMTXX, is loaded with the RTE Modal Program so that the same DIFS table is
available to the RTE Modal Program. This table can be altered at any time thru use of the
Measurement Format Command to accomodate users with multiple FMTXX structures.
Data acquisition is also now supported on several other devices. First of all, any device that supports
the Universal File structure can serve as a source of modal data using File Type 58. This Universal
File Structure is documented in Appendix I. In addition to this possible form of support, data
acquired from the HP-5423-A, data acquired and coded from SMS modal software, and data
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acquired from the S/K-LMS Fourier System (FMON) is supported by way of the ,
surement
Format Command and the Measurement. Header Command. Data acquisition can take place on a
HP-5420-A or a HP-5423-A if the data can be moved to the data disc in a format compatible with the
HP-5451 Fourier System. User programs exist for the HP-5451-C Fourier system to do this in a BCS
operating environment. The programs for the HP-5423-A are User Program 80 and 81 while the
programs for the HP-5420-A are User Programs 82 and 83. The standard versions of these programs
do not provide any modal information in the header of the resulting HP-5451-C Fourier System data
record. This information must be added using the Data Setup Command. The versions of the User
Programs 80 and 81 in use at the University of Cincinnati for the HP-5423-A automatically insert the
63 header words from the HP-5423-A in words 14 through 76, inclusive, of the 128 word header of
the HP-5451-C Fourier System data record. In this way, modal data taken on a HP-5423-A can
immediately be processed by choosing the proper format using the Measurement Source Command.
6.6.4 Graphics Displays
Within the RTE Modal Program, all data and display animations occur on one of several graphics
vector displays. Graphics vector displays are used due to the higher quality of the vector displays
compared to raster scan displays. Currently, several graphics displays (HP-5460, HP-1345, HP-1347,
HP-1351) are supported. Any number of graphics vector displays in any combination may be present
in the system at any time in order to support multiple display requirements as well as multiple users.
The user is often required to interact with the RTE Modal Program by providing information based
upon the data currently displayed on the graphics vector display unit. This interaction normally
occurs via control of the cursor, mode, and scaling functions of the graphics vector display unit.
6.7 Frequency - Damping Estimation
The task of determining damped natural frequencies can be performed using one of the following
methods:
"*Manual (spectral line)
"*Cursor (spectral line)
"*Least Squares Complex Exponential (frequency and damping)
"*Polyreference Time Domain (frequency and damping)
"*Polyreference Frequency domain (frequencydamping and modal vectors)
"*Orthogonal Polynomial (frequency and damping)
"*Multi-Mac (frequency and modal vectors)
"*Modified Ibrahim Time Domain (frequency and damping)
The first two methods, manual and cursor, are single degree-of-freedom (SDOF) approximation
methods. With these methods, only one frequency response function can be used at a time.
Therefore, it is wise to scan at least one frequency response from all major structure components so
that no important modes are inadvertently missed. Operation of the cursor automatically stores the
spectral line and frequency with the designated mode.
The remaining methods; Least Squares Complex Exponential (LSCE), Polyreference Time Domain
(PTD), Polyreference Frequency Domain (PFD), Orthogonal Polynomial (OP), Multi-Mac (MM),
and Modified Ibrahim Time Domain (MITD), are all multiple degree-of-freedom methods. In
addition, the last five methods are multi-reference methods. However, they can also be used on single
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reference data.
The Least Squares Complex Exponential and the Polyreference Time Domain algorithm are basically
the same methods. The last one is an extension of the first one to multiple references. They are both
linear least squares time domain methods based upon complex exponentials. In the process of
determining the frequency and damping, any and/or all of the measurements can be involved. An
additional feature of the Polyreference Time Domain, as compared with the Least Squares Complex
Exponential, is that the poles in the frequency range of interest can be determined based on different
numbers of degrees-of-freedom (DOF), which can be sometimes advantageous.
The Polyreference Frequency Domain, Orthogonal Polynormial, and Multi-Mac methods are
frequency domain methods. They have the advantage that any arbitrary frequency window can be
selected out of the measured frequency range. They can also handle frequency response function
data with variable frequency spacing. The disadvantage of these methods is that they become
numerically unstable for wide frequency ranges and for high numbers of modes. The Polyreference
Frequency Domain algorithm estimates the damping and damped natural frequency as well as the
associated modal vectors in a single process. So this technique is a one-stage technique, while for all
other methods, with the exception of Multi-Mac, the modal vectors are obtained in a second stage.
Multi-Mac is the only method of these three methods that does not calculate the damping. Similar to
the Least Squares Complex Exponential and Polyreference Time Domain, in the Polyreference
Frequency Domain all measurements, or a subset of the measurements, can be included in the
estimation of frequency and damping.
The Modified Ibrahim Time Domain algorithm is similar to the Polyreference Time Domain
technique. Specifically, both are time domain techniques based upon complex exponentials, but the
Modified Ibrahim Time Domain has the advantage of computing fewer computational poles.
However, due to the fact that more memory is needed to calculate the frequency and damping values,
the algorithm may not be able to simultaneously process all measurements. Therefore, data sets
containing many measurements may have to be reduced to a subset, in order to use this method.
For all of the algorithms, the location of the poles in the frequency range of interest is very important.
In general, poor damping values are estimated for poles too close to the edges of the frequency range.
An exception to the previous constraint is the Orthogonal Polynomial algorithm.
A difficult task in modal parameter estimation is the determination of the order of the model, or the
number of degrees of freedom of the system, such that, the estimating algorithm will find all
structural poles. Three features are implemented to help in the process of deciding this value; an
error chart, a stabilization diagram, and a rank estimate chart. These features will provide
approximate values for the order, or degree of freedom of the system, but, in general, some
judgement is still necessary to determine the "best" number for acceptable frequency/damping
estimates.
The time domain algorithms tend to produce more computational poles than the frequency domain
algorithms. On the other hand, frequency domain methods like Multi-Mac and Polyreference
Frequency Domain, which force the modal vectors to be orthogonal, tend to have difficulties
estimating the correct pole values for closely coupled poles, or for very local modes.
6.7.1 Error and Rank Chart
Most of the advanced algorithms use an error chart and/or a rank estimate chart, to aid the user
when a decision has to be made about the order of the model. An error chart basically explains what
the error will be in predicting the next point in an impulse response function, based on the
information of the previous points. The number of previous points used is, in this case, related to (2
or 4 times) the estimated order, or degree-of-freedom of the model. The error chart may be
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interpreted in the following way. in general, the error chart will have an area where the error rolls off
drastically with increasing degro-e-of-freedom. This area can be approximated by a straight line with a
slope equal to the roll off. In addition, there will be a second part in the error chart where the error
will stabilize. This range can be approximated by another straight line. The two lines will intersect
each other i the approximate order of the model. For the frequency domain methods this is
approximately the number of degrees-of-freedom that has to be entered in order to get a good
estimate of the poles in the frequency range of interest. For the time domain methods, this value will
generate, in geneiral, a reasonabhe !stimate for the frequency values of the poles in the frequency
range of interest. P .. cver, quite*often a poor estimate of the damping value of the poles will be
obtained for thiv f4 gree-of-freedom. But, by entering this number of degree-of-freedom an idea is
obtained abc',,, the number of effective poles in the frequency range of interest. This can be helpful
later on, to distinguish the real poles from the computational poles when a higher degree-of-freedom
is entcred in the algorithm. For the time domain methods, the best pole estimates are obtained when
,tie number of degrees-of-freedom chosen is equal to 1.5 to 2 times the estimated order of the model.
Some algorithms provide a rank estimate chart. This chart comes from a singular-value
decomposition of a matrix, which is related, or equivalent, to the system matrix. The rank of this
matrix is once again equal to the order of the model. The rank estimate chart is interpreted in much
the same way as the error chart (see previous paragraph).
6.7.2 Measurement Selection Option
A subset of the data set can be selected in the frequency/damping estimation phase. At times it may
be desirable to exclude some measurements from the data set in the frequency/damping estimation
process. For example, the estimation of a mode local to a specific direction, component, or set of
points would be enhanced if only the direction, component, or points active in that mode are
included in the estimation process. If all measurements are included, the local mode may be
dominated by another structural mode and the algorithm might be unable to detect the local mode,
or estimate it accurately. In the case of multiple references, a single reference may be excluded from
the estimation process and instead used to synthesize frequency response functions in order to verify
the modal model. For these and many other reasons, the measurement selection option is
implemented. The measurement selection consists of the following options:
"*Measuremeni Direction
"*Components
"*Point Numbers
"*References
If a subset of the measurements is desired, one of the four options can be invoked. With the first
three options, parameter" (:an be input individually (NI), or sequentially (NI,N2) for all
frequency/damping methods. The selection of references to be used is somewhat different for the
multiple reference algorithms, but similar to the first three options for single degree-of-freedom and
the Least-Squares Time Domain methods. In all cases, only the parameters entered for the option
chosen are used to form the subset and the other options remain unchanged, unless they too are
invoked. In other words, if the point number option is selected, only the point numbers entered
would be used to form the subset (all other point numbers are excluded), but all directions,
components and references remain active. To exit an option, zero is entered. "Continue" is selected
after selecting the desired subset.
By using the measurement selection option, a subset of the measurements defined in the
measurement directory can be selected for the estimation of frequency and damping values. This
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subset remains active only for the Frequency/Damping Estimation Monitor and all measurements in
the measurement directory remain active for the estimation of modal coefficients, except for the
Polyreference Frequency Domain method. For this method, the modal vectors will be determined
ONLY for the same subset, since all modal parameters are determined in a single solution process.
6.8 Modal Vector Estimation
The task of estimating modal coefficients can be performed by one of the following methods:
"*Complex Magnitude
"*Real Part of Frequency Response Function
"•Imaginary Part of Frequency Response Function
"*Real Circle Fit
"*Complex Circle Fit
"*Least-Squares Frequency Domain
"*Polyreference Time Domain
"*Polyreference Frequency Domain
The first five methods, complex magnitude, real part, imaginary part, real circle fit and complex circle
fit, are single degree-of-freedom methods. The Least-Squares frequency domain method is a
multiple degree-of-freedom method, but similar to the first five methods, does not estimate global
modal vectors. The two polyreference methods are multiple degree-of-freedom, multiple reference
methods and estimate global modal vectors.
At the present time, the RTE Modal Program is capable of estimating complex modal coefficients
using a floating point word for the real part and a floating point word for the imaginary part. The
modal vectors are actually stored, regardless of the method used to estimate the modal coefficients,
as the diameter of the complex circle that can be used to describe the single degree of freedom and
with the units of the data from which the modal vectors were estimated. Within the RTE Modal
Program, if the modal vectors are rescaled, the actual values of the modal vectors are never altered;
a complex scale factor is altered from unity to account for any scaling required. All values that are
output from the RTE Modal Program include this complex scale factor in a transparent manner.
The ability to animate the modal vectors is possible in any of four formats. The possibilities allow the
user to view the modal vectors in complex or one of three real formats. Options are available in the
real formats to view the complex magnitude, real component, or imaginary component so that all
data types (D/F,V/F,AIF,D/D,V/V,A/A) can be used to determine modal vectors. This also gives the
user the possibility to view the out-of-phase portion of the modal vector to determine whether a
complex modal vector is a function of reasonable structure characteristics or a function of poor
excitation energy distribution.
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NOMENCLATURE
Matrix Notation
{..}
t.. IT
[.1
[..]H
[..]T"
[..1'[..]+
[.]9 XP
r.i
braces enclose column vector expressions
row vector expressions
brackets enclose matrix expressions
complex conjugate transpose, or Hermitian transpose, of a matrix
transpose of a matrix
inverse of a matrix
generalized inverse (pseudoinverse)
size of a matrix: q rows, p columns
diagonal matrix
Operator Notation
A"
F
F"•
H
H-1
In
L
V"1
Re +jim
x
iisecond
complex conjugate
Fourier transform
inverse Fourier transform
Hilbert transform
inverse Hilbert transform
natural logrithm
Laplace transform
inverse Laplace transform
complex number: real part "Re", imaginary part "Im"
first derivative with respect to time of dependent variable x
derivative with respect to time of dependent variable x
mean value of y
estimated value of y
summation of A, Bi from i = I to n
det[..]
1
.-1 12
partial derivative with respect to independent variable "t"
determinant of a matrix
Euclidian norm
Roman Alphabet
Apr
C
COH
COHNA
COWH
e
F
residue for response location p, reference location q, of mode r
damping
ordinary coherence functiont
ordinary coherence function between any signal i and any signal kt
conditioned partial coherencet
base e (2.71828...)
input force
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GXF
GXX
GXX"W
spectrum of q* referencet
auto power spectrum of referencet
auto power spectrum of reference qt
cross power spectrum of reference i and reference kt
reference power spectrum matrix augmented with the response/reference cross
power spectrum vector for use in Gauss elimination
cross power spectrum of response/referencet
auto power spectrum of responset
auto power spectrum of response pt
h (t)
impulse response functiont
h,3 (1)
H (s)
H (w)
HM(w)
impulse response function for response location p, reference location q t
transfer functiont
frequency response function, when no ambiguity exist, H is used instead of H (w)t
frequency response function for response location p, reference location q, when no
ambiguity exist, Hp, is used instead of Hp,(w)t
frequency response function estimate with noise assumed on the response, when no
ambiguity exist, H, is used instead of HI (w)t
frequency response function estimate with noise assumed on the reference, when no
ambiguity exist, H2 is used instead of H2(w)t
scaled frequency response function estimate, when no ambiguity exist, Ms is used
instead of Hs(w)t
frequency response function estimate with noise assumed on both reference and
response, when no ambiguity exist, H,, is used instead of H,,(w)t
identity matrix
F,
GFF
GFFgq
GFFa
[GFFX]
H, (w)
H2(wo)
Hs(w)
H,,(w)
[I]
i
V/7-
K
L
stiffness
modal participation factor
M
mass
MI,
MCOH
N
N,
NI
P
modal mass for mode r
multiple coherence functiont
number of modes
number of references (inputs)
number of responses (outputs)
output, or response point (subscript)
input, or reference point (subscript)
mode number (subscript)
residual inertia
residual flexibility
Laplace domain variable
independent variable of time (sec)
discrete value of time (sec)
9
R,
Rr
s
t
tA;
t4
T
x
X
XP
z
=kAt
sample period
displacement in physical coordinates
response
spectrum ofp* responset
Z domain variable
Greek Alphabet
6(t)
Af
Dirac impulse function
discrete interval of frequency (Hertz or cycles/sec)
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At
e
A,
discrete interval of sample time (sec)
small number
noise on the output
rA complex eigenvalue, or system pole
[ Aj
diagonal matrix of poles in Laplace domain
A,=a,+
j4,
V
noise on the input
w
w
variable of frequency (rad/sec)
imaginary part of the system pole, or damped natural frequency, for mode r
(rad/sec)
, "n, VT7 C
C),.
undamped natural frequency (rad/sec)
Vf
00,
{0},
[PC
{ 0}
or,
{ 1},
[qf]
a
a,
t
+42~
-,
scaled p6 response of normal modal vector for mode r
scaled normal modal vector for mode r
scaled normal modal vector matrix
scaled eigenvector
scaled p* response of a complex modal vector for mode r
scaled complex modal vector for mode r
scaled complex modal vector matrix
variable of damping (rad/sec)
real part of the system pole, or damping factor, for mode r
damping ratio
damping ratio for mode r
vector implied by definition of function
1N
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