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APPLICATION NOTE
How to Determine the Modal Parameters of Simple Structures
by Svend Gade, Henrik Herlufsen and Hans Konstantin-Hansen, Brüel & Kjær, Denmark
The modal parameters of simple structures can be easily established by the use of PULSE™, the Multianalyzer System Type 3560. This application note describes how to measure the modal frequencies by
inspection of frequency response functions, how to determine the modal damping with the aid of the
frequency weighting function included in the analyzer, and how to establish the mode shapes by examining
the value of the imaginary part of the frequency response function.
The frequency response function of
a structure can be separated into a
set of individual modes. By using a
PULSE Multi-analyzer System Type
3560, each mode can be identified in
terms of frequency, damping and
mode shape.
Identification of Modes
dB
Frequency
dB
|H|
f 1 f 2 = 205 Hz
|H|
Hz
f3
f 2 = 204.25 Hz
Hz
Mode Shape
Damping
dB
|H| · W(f)
Hz
dB
Imag. H
h(t)
2 nd Bending Mode
8.7 dB
Hz
ms
t = 30.76 ms
z2 =
1
= 0.0254
t · 2pf 2
880212/1e
Introduction
In practice, nearly all vibration problems are related to structural weaknesses, associated with resonance behaviour (that is natural frequencies being excited by operational
forces). It can be shown that the complete dynamic behaviour of a structure (in a given
frequency range) can be viewed as a set of individual modes of vibration, each having
a characteristic natural frequency, damping, and mode shape. By using these so-called
modal parameters to model the structure, problems at specific resonances can be
examined and subsequently solved.
The first stage in modelling the dynamic behaviour of a structure is to determine the
modal parameters as introduced above:
❍
❍
❍
The resonance, or modal, frequency
The damping for the resonance – the modal damping
The mode shape
3560
The modal parameters can be determined from a set of frequency response measurements between a reference point and a number of measurement points. Such a measurement point, as introduced here, is usually called a Degree-of-Freedom (DOF). The
modal frequencies and dampings can be found from all frequency response measurements on the structure (except those for which the excitation or response measurement
is in a nodal position, that is, where the displacement is zero). These two parameters
are therefore called “Global Parameters”. However, to accurately model the associated
mode shape, frequency response measurements must be made over a number of Degrees-of-Freedom, to ensure a sufficiently detailed covering of the structure under test.
In practice, these types of frequency response measurements are made easy by using
a Dual Channel Signal Analyzer such as the standard two-channel configuration of PULSE,
the Multi-analyzer System Type 3560. The excitation force (from either an impact hammer or a vibration exciter provided with a random or pseudorandom noise signal) is
measured by a force transducer, and the resulting signal is supplied to one of the inputs.
If a vibration exciter is used, a generator module should be installed in the analyzer.
The response is measured by an accelerometer, and the resulting signal is supplied to
another input. Consequently, the frequency response represents the structure’s accelerance since the measured quantity is the complex ratio of the acceleration to force,
in the frequency domain. For impact hammer excitation, the accelerometer response
position is fixed and used as the reference position. The hammer is moved around and
used to excite the structure at every DOF corresponding to a DOF in the model. For
vibration exciter excitation, the excitation point is fixed and is used as the reference
position, while the response accelerometer is moved around on the structure. A typical
instrumentation setup is illustrated in Fig. 1.
Fig. 1
An instrumentation
setup, using
broadband pseudorandom force
excitation provided
by a vibration
exciter
#5
#4
Accelerometer
#3
PULSE Multi-analyzer System
3560
#2
#1
4–channel Input
Module
Type 3022
4–channel Input
Module
Type 3022
4–channel Mic.
Module
Type 3028
4–channel Mic.
Module
Type 3028
Input 4
Input 4
Input 4
Input 4
Unit 1
Unit 2
!
!
Active
Sampling Input
Delay Adj.
Active
Input 3
Input 3
!
Active
Input 3
Active
Input 3
!
Active
Active
Active
Active
Sampling Output
Signal Analyzer
Interface Module
Type 7521
Input 2
Input 2
!
Input 2
Input 2
!
Active
Active
Input 1
Input 1
!
Active
Input 1
Active
Input 1
!
Trigger Input
Active
B7/6-'89
K
Brüel & Kjær
B
7/6-'89
Active
B
K
7/6-'89
Brüel & Kjær
7/6-'89
Force Transducer
8200
Ch. 1
7/6-'89
Brüel & Kjær
B
Active
B
K
K
Brüel & Kjær
Active
B
7/6-'89
K
Brüel & Kjær
K
Generator
Ch. 2
Mini-Shaker
4810
Power Amplifier
2706
990052e
For structures defined with a large number of DOFs, the Multi-analyzer System Type
3560 can be equipped with up to eight four-channel modules (without expanding the
physical dimensions of the system) to allow for easier and faster mobility measurements.
Simultaneous measurement of up to 31 response signals and one force input signal can
be performed, thereby greatly reducing the time needed to move the response accelerometer.
2
Simple Structures
Structures which exhibit lightly coupled modes are usually referred to as simple structures. The modes are not closely spaced, and are not heavily damped (see Fig. 2). At
resonance, a simple structure behaves predominantly as a Single-Degree-of-Freedom
(SDOF) system, and the modal parameters can be determined relatively easily with a
suitable configuration of the Brüel & Kjær PULSE Multi-analyzer System Type 3560.
Fig. 2
The frequency
response of simple
structures can be
split up into
individual modes,
each mode behaving
as a single-degreeof-freedom system
H(f)
H(f)
Mode 1
Mode
2
Mode 3
3 dB
ζ1
ζ3
ζ2
f
f1
f2
f3
f
941269e
Determination of the Modal Frequencies
Fig. 3
Magnitude of the
frequency response
function
(Accelerance) for
the test structure
The resonance frequency is the easiest
modal parameter to
determine. A resonance is identified as
a peak in the magnitude of the frequency
response
function,
and the frequency at
which it occurs is
found using the analyzer’s
cursor
as
shown in Fig. 3.
Fig. 4
Zoom
measurement for
achieving higher
frequency
resolution in the
determination of
the resonance
frequency for the
second mode.
Determination of
the damping for
the second mode is
identified by the
half-power points
and is read-out in
the auxiliary cursor
field
The frequency resolution of the analysis determines the accuracy
of
the
frequency
measurement. Greater
accuracy can be attained by reducing the
frequency range of the
baseband
measurement, for resonances
at low frequencies, or
making a zoom measurement around the
frequency of interest,
as shown in Fig. 4.
3
Determination of the Modal Damping
The classical method of determining the damping at a resonance, using a frequency
analyzer, is to identify the half power (–3 dB) points of the magnitude of the frequency
response function (see Fig. 4). For a particular mode, the damping ratio ζr can be found
from the following equation:
∆f
ζ r = ------2f r
where ∆f is the frequency bandwidth between the two half power points and fr is the
resonance frequency. Type 3560 contains a built-in standard cursor reading which calculates the modal damping.
The accuracy of this method is dependent on the frequency resolution used for the
measurement because this determines how accurately the peak magnitude can be
measured. For lightly damped structures, high resolution analysis is required to measure
the peak accurately, consequently, a zoom measurement at each resonance frequency
is normally required to achieve sufficient accuracy. This means that a new measurement
must be made for each resonance.
Fig. 5
The frequency
weighting function
of the PULSE
System allows a
single mode to be
isolated from the
frequency response
function.
4
However, the PULSE
System can be used to
determine the damping ratio by an alternative method which
requires no new measurements. By using a
frequency weighting
function (window) to
isolate a single mode
from the frequency response function (see
Fig. 5), the impulse response function for
that mode alone is
easily produced. The
magnitude (envelope)
of the impulse response function can
then be displayed by
virtue of the Hilbert
transform facility incorporated in the analyzers.
Fig. 6
The impulse
response function
of the isolated
mode can be
represented by its
magnitude and
displayed on a
logarithmic scale to
enable easy
calculation of the
damping
Since a simple structure behaves as a Single-Degree-of-Freedom
system at each resonance, the impulse response function of the
windowed resonance
will show characteristic exponential decay
τ. By displaying the
magnitude on a logarithmic scale the impulse response is
represented
by
a
straight
line
(see
Fig. 6).
The decay rate σr for the isolated mode is related to the time constant τr by:
1
τ r = ----σr
The decay corresponding to time constant τr is given by the factor e–1, or in dB: –20loge
= –8.7 dB.
The damping ratio is related to the decay rate by:
σr
1
ζ r = ---------- = ---------------2πf r
τ r 2πf r
By moving the frequency window through the frequency response function, and looking
at each impulse response function in turn, the modal damping at each resonance can
be determined from a single baseband measurement. Determination of the modal damping is immediately available by use of the main cursor and the reference cursor as
illustrated in Fig. 6.
Applying pseudo-random excitation via a vibration exciter and using the above mentioned method, the damping value will be correct even though the resolution is low
compared to the width of the resonance peak. See Ref. [1] for more details on this
subject.
The decay rate, calculated from a frequency response function found by using hammer
excitation, will be modified by the effective damping of the exponential weighting function applied to the response channel. This weighting function was added to curtail the
structural ringing excited by the hammer impact. However, this error can easily be
compensated for by using the following correction:
1 1
σ r = ---- – ----τr τu
where τu is the time constant of the exponential weighting function.
5
Determination of the Mode Shape
Fig. 7
The first three
modes of vibration
for the test
structure. The
modal
displacements are
found from the
imaginary part of
the frequency
response function
#5
#4
#3
Im[Hj]
#2
e
od
M
cy
n
e
u
q
re
F
#3
39
0
M
H
z
od
e
#2
20
4
M
H
z
od
e
#1
73
H
z
DOF
#1
941270e
The simplest way of determining the mode shape for a structure is to use a technique
called Quadrature Picking. Quadrature Picking is based on the assumption that the
coupling between the modes is light. In practice, mechanical structures are often very
lightly damped (<1%). This implies that the modes are lightly coupled. At any frequency,
the magnitude of the frequency response function is the sum of the contributions (at
the particular frequency) from all modes. When there is little modal coupling between
the modes, the structural response at a modal frequency is completely controlled by
that mode, and so Quadrature Picking can be used to unravel the mode shapes.
For Single-Degree-of-Freedom systems, the frequency response function (accelerance)
at resonances is purely imaginary. As a result, the value of the imaginary part of the
frequency response function at resonance, for structures with lightly coupled modes,
is proportional to the modal displacement. Consequently, by examining the magnitude
of the imaginary part of the frequency response function at a number of points on the
structure, the relative modal displacement at each point can be found. From these
displacements, the mode shapes can be established. The procedure can then be repeated to determine all the required mode shapes. By making an excitation and response
measurement at the same point and in the same direction, the mode shape can be
scaled in absolute units.
6
Fig. 8
The 3-D map of the
imaginary part of
the frequency
response functions.
The second mode
of vibration is
extracted by using
the cursor
The mode shapes can
be drawn by hand, or
even simpler, the frequency response functions can be stored in
a multi-buffer PULSE
System and the mode
shapes subsequently
displayed by making
slices in the multibuffer as shown in
Fig. 8. This technique
is only meaningful
when the structure
under test can be represented by DOFs on
a straight line (i.e., a
one
dimensional
structure).
Conclusions
PULSE, the Multi-analyzer System Type 3560, in its different configurations, is an ideal
instrument for enabling the engineer to determine the modal parameters of simple
structures. The modal frequencies are determined from the frequency response function.
The modal dampings are found from the magnitude of the impulse response function,
which is produced by isolating a single mode from the frequency response function,
using a frequency weighting function. To obtain the mode shape, a technique called
Quadrature Picking is used to evaluate the modal displacements at each point of interest.
The modal displacements are found from the imaginary part of the frequency response
function. The frequency response functions can be stored in a multi-buffer of the PULSE
System and the mode shapes can subsequently be displayed by making slices in the
buffer.
By adding the necessary software* to the system, the modal parameters can be extracted using curve fitting techniques. Geometry models can be developed and mode shapes
can be shown animated on the PC screen. The dynamic response due to excitation
forces can be simulated and, furthermore, if the vibration characteristics of prototypes
tested, using the PC based modal system, are unsatisfactory, then the influences of
actual mass, stiffness and damping modifications can be simulated. In this way, a PULSE
Multi-analyzer System Type 3560 can be expanded into a fully documented modal
analysis system.
Data export is possible in a number of formats, such as Universal File ASCII, Universal
File Binary, Standard Data Format and STAR binary. PULSE Bridge to ME’scope software
Type 7755 A is available, which facilitates export from PULSE to ME’scope modal software (Type 7754), although the greatest testing and export flexibility is provided by
Modal Test Consultant Type 7753.
* Brüel & Kjær offers a complete palette of advanced Modal Analysis software packages to run on a PC. This
palette covers software modules to be used for simple two channel modal analysis as well as software
modules to be used for multichannel modal analysis. Add-on modules for prediction of dynamic response
due to excitation forces and prediction of structural modifications are also available.
7
References
BO 0428 – 12
99/05
[1] S. Gade and H. Herlufsen, “Digital Filter Technique versus FFT Technique for damping
measurements”, Brüel & Kjær Technical Review No.1, 1994
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