By

By
VIBRATION ANALYSIS OF A CRACKED SHAFT USING FEM
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology in Mechanical Engineering
By
RAJ KUMAR KISPOTTA
Under the guidance of
Dr. Rabindra Kumar Behera
Department of Mechanical Engineering
National Institute of Technology
Rourkela
2009
VIBRATION ANALYSIS OF A CRACKED SHAFT USING FEM
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology in Mechanical Engineering
By
RAJ KUMAR KISPOTTA
Department of Mechanical Engineering
National Institute of Technology,
Rourkela
Abstract
A strong interest has developed within the past several years in the dynamic behavior of
turbo machinery with cracked shafts. An excellent review of the field of the dynamics of
cracked rotors and of different detection procedures to diagnose fracture damage has been
presented by Wauer . Vibration investigation of
a damaged structure is one approach
for fault diagnosis. Vibration diagnosis, as a non-destructive detection technique,
has
recently become of greater importance.
A crack on a beam element introduces considerable local flexibility due to the strain energy
concentration in the vicinity of the crack tip under load. The vibrational characteristics
of a cracked Timoshenko shaft were investigated by Rajab and Sabeeh . They presented
analytical expressions and derived curves relating the crack depth and location on the shaft
to changes in the first few natural frequencies of the shaft. The element stiffness matrix of
a beam with a crack was derived froman integration of the stress intensity factors and then
a finite element model (FEM) of a cracked beam was established in reference . A similar
approach based on the flexibility matrix developed by Papado-poulos and Dimarogonas is
dealt with by using FEM in the present study for crack detection.
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled “VIBRATION ANALYSIS OF A CRACKED SHAFT
USING FEM” submitted by Sri Raj Kumar Kispotta in partial fulfillment of the requirements for
the award of Bachelor of technology Degree in Mechanical Engineering at the National Institute of
Technology, Rourkela (DeemedUniversity) is an authentic work carried out by him under my
supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any other
University / Institute for the award of any Degree or Diploma.
Date:
Dr. Rabindra Kumar Behera
National Institute of Technology
Rourkela-769008
ACKNOWLEDGEMENT
We deem it a privilege to have been the student of Mechanical Engineering stream in
National Institute of Technology, ROURKELA .
Our heartfelt thanks to Dr.R K Behera , our project guide who helped us to bring out this
project in good manner with his precious suggestion and rich experience.
We take this opportunity to express our sincere thanks to our project guide for co-operation
in accomplishing this project a satisfactory conclusion.
Raj Kumar Kispotta
Roll No. 10403059
Department of Mechanical Engg
National Institute of technology
CONTENTS
SL.NO.
Title
Page.no.
0
Abstract
0
1
Introduction
1
2
Motions Equations of a cracked shaft
3
3
Crack Modeling
8
4
Crack Identification and Estimation method
9
4.1
Forward Problem
9
4.2
Vibration Analysis
12
4.2.1

Effect of the normalized crack location, β
12
4.2.2

Effect of the normalized crack size , α
14
4.3
Inversion Problem
15
5
Experimental Verification
16

5.1
Experimental results
19
6
Discussion and further researches
24
7
Conclusions
27
8
References
28
LIST OF FIGURES
SL NO.
TOPIC
PAGE
1
Model of a rotor element with a transverse edge crack.
4
2
The shaft with a transverse surface crack.
8
3
Simply supported rotor system with a crack in shaft
10
4
Geometry of a cracked section in shaft.
10
5
The first three natural frequency as a function of normalized crack
13
location β for some of the normalized crack size α .
6
The first three natural frequencies as a function of normalized crack
14
size α for same of the normalized crack location.
7
Experimental setup
16
8
Frequency contour plots
23
9
Typical rotor-bearing system considered
24
LIST OF TABLES
SL NO.
1
TOPIC
The experimental and differently elemental results of the first three
PAGE
18
natural frequencies.
2
Crack cases of shaft and identification results.
20
3
The first three natural frequencies under different bearing stiffness
25
1.INRODUCTION
A method for the detection of cracks in Timoshenko shafts based
in the natural frequencies is described in this
on measuring changes
paper . Cracks may be initiated
and
subsequently propagated in shaft sand structures subjected to dynamic loadings. Failure may
result if the history of these cracks is not recorded and precautionary measures are not taken.
At the early stages of crack growth, it is difficult to detect the existence of the crack by
Visual inspection. Other more detailed techniques of non-destructive testing need
to be
used instead. An ultrasonic pulse technique has been used successfully to detect the positions
of cracks in structures and welds
. In some materials, this technique may not work due to
the large attenuation of the signal at all except a particular frequency.
Radiographic techniques have also been used for crack detection in structures . These
Techniques , however , require higher radiation energy input for
Increasing
material
thickness, which increases the cost of operation. In addition, crack detectability is small for
a small crack
Analytical
width/depth
ratio and for cracks not parallel to the material surface.
expressions for the flexibility effect of a crack were first derived by Irwin
and later used in structural analysis applications by Liebowitz .Thevibrational characteristics
of a rotor containing a crack were s tudied by Henry and Okah-Avae , Mayes and Davies,
Gasch , and
Dimarogonas . Adams et al. used axial vibration analysis in one-dimensional
structures to predict the location and the magnitude o f a defect. The effect of the defect was
represented as an elastic stiffness which is determined from a knowledge of the changes in
two natural frequencies of the structure. The method presented would not detect longitudinal
cracks . A finite element analysis was used by Cawley and Adams
to predict damage in
structures. Chondros and Dimarogonas studied the effect of a crack in a welded joint on the
dynamic behavior of beams with various boundary conditions. Crack depth was estimated
from knowledge of changes in the system natural frequencies. A torsional elastic stiffness
was used to model the effect of the crack. The torsional stiffness was
determined experi-
mentally from measurements of changes in the natural frequencies for various crack depth
values. Dimarogonas and Massouros considered the effectof a crack on the torsional
dynamic behavior of a shaft. They used the strain energy release rate to obtain an expression
for
local stiffness
effect due to
a circumferential crack. Anifantis et al.
presented
a nomogram for identification of a c rack on a simple beam . Ju et al. proposed a damage
function
relating the changes in the natural frequencies of a structureto the crack depth and
location. Dimarogonas and
Papadopoulos
studied the dynamic behavior of a rotating
shaft with a surface crack. Yuen Chen and Chen and Gounaris and Dimarogonas used finite
element analysis to study the vibrational characteristics of crackedstructures. Papadopoulos
and Dimarogonas considered the free and forced vibrations of coupled bending and torsional
vibrations of a damaged shaft. The present work is concerned with developing analytical
expressions for the compli-ance effects of a crack in a Timoshenko-type shaft for the purpose
of predicting both the depth and location of the crack on the shaft. A part-through surface
crack is assumed in the analysis, and changes in the shaft natural frequencies are used in the
prediction of crack parameters.
2. MOTIONS EQUATIONS OF A CRACKED SHAFT
The FEM model of a cracked rotor which was derived in the author’s previous work
is used in
present study also. A brief description of the model and equation of motion are given below.
A cracked shaft loaded by three tension forces and three torsion forces is shown in Fig. 1. The
equation of motion of the cracked rotor system is given by
where M, C, G, K0 and Kc are the mass matrix, bearing oil-.lm damping matrix, gyroscopic
matrix, whole stiffness matrix without crack and the whole stiffness decrease caused by crack,
respectively . H is the transformation matrix between rotating co-ordinates Z–z and inertial
co-ordinates y–z, which has
where a is the angle between the two co-ordinates.
F is the excitation matrix including the unbalance forces and gravity forces. The stiffness
matrix of cracked
where
element [K]crack can be written as
The .exibility matrix of cracked element can be expressed as
where [c](0) is the .exibility matrix of element without crack, [c](1) is the additional .exibility
matrix due to crack, which has
The expressions of cij can be found in Ref. [15].
The switch function of cracks used here is same as that in Ref. [15]:
where me is the mass unbalance.
For the state of a crack propagating during the startup process , the stiffness of cracked shaft
varies as a function of time and propagation rate. Because crack propagating is a speci.c complex
process, following two assumptions are adopted in present study:
(1) The crack tip growth occurs at the beginning of a cyclic.
(2) The crack tip line after growth parallels to the initial crack tip line.
The dynamic response can be obtained using numerical integration method for non-linear
system— Newmark method [22]. The computational algorithm is as follows:
According to the implicit integration formula of the Newmark integral method , we have
(1) Set i =0
(2) Calculate the residual force
(3) Calculate the effective stiffness matrix
.
(4) Solve the following equation:
(5) Revise displacement, velocity, acceleration and i:
and then reduplicate the above process until
3. CRACK MODELING
The presence of a crack in a structure tends to modify the dynamic characteristics of
the structure, such as the natural frequencies and mode shapes. This fact can be used
inversely to predict the crack parameters from measurements of the changes in the natural
frequencies and mode shapes of the structure once a functional relationship between the
crack parameters and the changes in the structure dynamic characteristics has
been
determined . The crack parameters of interest are the crack depth and crack location in
the structure. To this end, an analytical solution for the functional relationship between
the system natural frequencies and the crack parameters is attempted for a Timoshenkotype shaft with a transverse part-through surface crack .
For a shaft with a transverse surface crack and loaded with bending moment and shear
force as shown in Figure 1, the displacement in the i direction is given by
where Pi is the load in the same direction as the displacement and J(a) is the J-integral
Figure 2 . The shaft with a transverse surface crack.
3. CRACK IDENITIFICATION AND ESTIMATION METHOD
3.1. Forward problem
Friswell and Penny compared the different approaches to crack modeling, and demonstrate that
for structural health monitoring using low-frequency vibration, simple models based on beam
elements are able to model the effects of an open crack. Because the natural frequencies can be
easily and cheaply acquired in practice and the linear rotational spring model can effectively
describe open crack, the present work is based on the open transverse crack in shafts and using
the first three natural frequencies to identify crack location and size.
Fig. 3 shows a simply supported rotor system with a crack in shaft (suppose the crack occurred
on segment L2). A transverse crack of depth δ is considered on a shaft of diameter d1 (the
corresponding radius is r1) as shown in Fig. 4. The shaft has local flexibility due to the crack, in
many directions, depending on the direction of the applied forces. Here, only bending
deformation will be considered. Axial forces which give coupling with transverse motions of the
cracked shaft will not be considered. Therefore, the shaft is bent by a pure bending moment and
the additional angular deflection of the shaft end relative to the other will be computed. The local
stiffness kt due to the crack is [7]
…………………..(34)
where E is the Young's modulus, μ is the Poisson's ratio, α=δ/2r1 denotes normalized crack size,
,
and the function F(η/H) can be given by
the experimental formula [37].
F(η/H)=1.122-1.40(η/H)+7.33(η/H)2-13.08(η/H)3+14.0(η/H)4................................ (35)
Eq. (34) is a function of normalized crack size α only and can be computed by numerical
integration. β=e/L2 denotes normalized crack location.
Fig. 3. Simply supported rotor system with a crack in shaft.
Fig. 4. Geometry of a cracked section in shaft.
The physical model and rotational spring model with stiffness kt , are shown in Fig. 5.
The
corresponding
DOF
of
the
left
neighboring
element
around
crack
are
and the corresponding DOF of the right neighboring element are
wright={wj+1θj+1 }T. If we change the two entries of
row and column entries of the global stiffness matrix
to wleft={ θjwj}T, the corresponding
and mss matrix
should be exchanged
their locations each other. The DOF of wleft and wright can be assembled to { θjwj+1θj+1,…}T.
Hence, we can assemble cracked stiffness submatrix KS into the global stiffness matrix easily.
The global mass matrix of cracked rotor system is equal to the uncracked one. From now on, the
cracked rotor system finite element model is constructed by using BSWI beam element. The
solution of the eigenvalue problem can then proceed as usual.
For the determination of the natural frequencies ω for a given crack location (determine the
location of cracked stiffness submatrix in global stiffness) and size (determine Kt), the
normalized crack location β and size α are given as input. The relationship between the natural
frequencies and the crack parameters is
…………………………(36)Because the functions Fj(j=1,2,3,…) are
unknown and the discrete values can be obtained by solving WFEM model, the surface-fitting
techniques are needed for the 3D plots of Eq. (36). Then the crack identification database of forward
problem for cracked rotor system has been built up.
3.2. Vibration analysis
In the simulation, the rotor geometries and the material properties are as follows:
,
,
,
,
,
,
,
, μ=0.3.
13 BSWI43 Rayleigh–Euler beam and 1 BSWI43 Rayleigh–Timoshenko beam elements are
employed to the vibration analysis of rotor system. Fig. 6 shows the relationship between
ωi(i=1,2,3) and all possible normalized crack location β=e/L2 and size α=δ/d1 using surfacefitting techniques (here, α,β [0.1,0.9]).
3.2.1. Effects of the normalized crack location β
Fig. 7 shows the first three natural frequencies as a function of normalized crack location β for
some of the normalized crack size α. It can be seen from Fig. 7(a)–(c) that the change in the first
three natural frequencies were affected when the crack was located at every normalized location
β. For a certain normalized crack size α, Fig. 7(a) show the fundamental frequency would be
decreased significantly with respect to the larger normalized crack location β. While a correlation
between normalized crack location and size was given, as shown in Fig. 7(b) and (c). Taking
α=0.8, for example, the second natural frequency was mostly affected when the crack was
located at the center of the shaft segment L2, and the third natural frequency was mostly affected
when the crack was located at β=0.25. Moreover, Fig. 7(c) shows that the third natural frequency
was almost unaffected for a crack located at β=0.75; the reason for this zero influence was that
the nodal point for the third mode was located at that position.
Fig. 5. The first three natural frequencies as a function of normalized crack location β for some
of the normalized crack size α.
Unlike to the effects of the normalized crack location of simply supported shaft without mass
disc, as mentioned by Dong, the changes in the first three natural frequencies of a cracked rotor
system with a mass disc, would not monotonically decrease with the increment of the normalized
crack location. The reason is that only the crack of shaft segment L2 is considered in the present
study, and this is not affecting the crack identification results.
3.2.2. Effects of the normalized crack size α
The first three natural frequencies as a function of normalized crack size α for some of the
normalized crack location β are shown in Fig. 8. It is noticed that the change in the first three
natural frequencies were affected when the crack was occurred at every normalized size α. For a
certain normalized crack location β, Fig. 8(a)–(c) show the first three frequencies would
monotonically decrease if the larger normalized crack size α was given.
Fig. 6. The first three natural frequencies as a function of normalized crack size α for some of the
normalized crack location
From the above observations, it could be stated that the change in frequencies is not only a function of the
normalized crack location β but also the normalized crack size α. In the procedures of constructing crack
identification database of forward problem for cracked rotor system, the influence of the normalized crack
size α and location β was considered through the computation of rotational spring stiffness kt and the
location of inserting cracked stiffness submatrix into global stiffness, respectively.
3.3. Inverse problem
The crack identification inverse problem can be described by
………………(37)
The measurements of any two natural frequencies enable us to define the normalized location and size of
a crack if Eq. (36) has been determined, i.e. the crack identification database of forward problem for
cracked rotor system has been constructed. However, when we use the method of frequencies contour
plots for crack identification of a rotor system, two natural frequencies contour plots may intersect at
more than one point. Therefore a minimum of three frequencies is required to identify the two unknown
parameters, i.e. the normalized crack location β and size α. Because the first three frequencies can be
measured easily and accurately; they are usually served as inputs to solve the inverse problem in
structural damage identification.
Supposing the first three frequencies are known, the frequency contour plots of the crack identification
database of forward problem for cracked rotor system, can be acquired and plotted on the same axes. The
common intersection of all the three contour lines indicates the normalized crack location and size. This
intersection becomes unique due to the fact that any cracked structural natural frequency can be
represented by a frequency equation ( Eq. (36)) that is dependent on normalized crack parameters [20].
4. EXPERIMENTAL VARIFICATION
4.1. Experimental setup
Fig. 9 shows the experimental setup used for measuring the first three frequencies of the cracked
rotor system with a single mass disc using the Doppler signal laser vibrometer. A Polytec
Doppler laser vibrometer OFV-505/5000 was used to measure the velocities of one point in the
shaft. We point out here that to measure the first three frequencies only requires one
measurement point. The reason is that for the simple structure, single-input and single-output
(SISO) modal analysis by using a hammer as excitation, which is a usually used method. To
reduce the reflection of the laser beam and spectral noise, retro-reflective tapes were put on the
measurement point in the shaft.
Fig. 7. The experimental setup
The laser vibrometer OFV-5000 uses the principle of the heterodyne interferometer to acquire
the characteristics of mechanical vibrations. For each cracked shaft, the high-metrical
frequencies can be obtained by using the standard FFT program of the software Matlab.
Taking the uncracked rotor system with the same geometries and the material properties, as
given by Section 3.2, and the corresponding impulse response signal and power spectrum are
shown in Fig. 10. In the experimental study, the sampling frequency fs is 5000.
Different elements are employed to estimate the first three frequencies, respectively, i.e.:
1. 5 BSWI43 beam elements (not consider rotatory inertia effect);
2. 200 traditional beam elements (not consider rotatory inertia effect);
3. 5 BSWI43 Rayleigh beam elements (consider rotatory inertia effect);
4. 200 BEAM3 (Software Ansys) elements (consider rotatory inertia effect).
The results are shown in Table 1. Compared with the experimental frequencies, both 5 BSWI43
beam element (not consider rotatory inertia effect) and 200 traditional beam elements (not
consider rotatory inertia effect) have the same computational precision. The relative errors of the
second frequencies arrive at 61.5%, which can not be used to the model-based crack
identification method. However, if we consider the rotatory inertia effect, both 5 BSWI43
Rayleigh beam elements (consider rotatory inertia effect) and 200 BEAM3 (Software Ansys)
elements (consider rotatory inertia effect) can achieve high-computational precision. And the
relative errors of the first three frequencies are less than 0.3%. Moreover, the elemental DOF of
5 BSWI43 Rayleigh beam elements (consider rotatory inertia effect) are 58, which are smaller
than 402 DOF of BEAM3 elements. The results of this example indicate the accuracy and
efficiency of the BSWI Rayleigh beam element, which has been constructed herein. Meanwhile,
the BSWI Rayleigh beam element can be employed to build up the crack identification database
for cracked rotor system.
Table 1.
The experimental and differently elemental results of the first three natural frequencies
Computational
Method frequencies/Hz
f1
f2
f3
1
86.33
900.95
2
86.34
901.00
Metrical
Computational
frequencies/Hz
errors/%
f1
f2
f3
f1
f2
f3
1732.07
0.26
61.51
0.26
1732.25
0.27
61.52
0.27
86.11 557.82 1727.54
3
86.33
558.17
1728.25
0.26
0.06
0.04
4
86.33
558.17
1728.20
0.26
0.06
0.04
4.2. Experimental results
We tested six cracked shafts each having an open crack at shaft with four crack cases as shown
in Table 2. The material of workpiece for experiment was 40Cr steel, and the rotor geometries
and the material properties are as follows:
,
density
,
,
,
,
, Young's modulus E=2.06×1011N/m2, material
, Poisson's ratio μ=0.3. The cracks actually are slots cut by a wire
electrical discharge machine, the crack width is 0.2 mm, and the crack depths δ are 3.8 and
7.2 mm. In other words, the normalized crack sizes α=δ/d1 are 0.2 and 0.4. The crack locations e
are 75.2, 112.8 and 150.4 mm, respectively, and the corresponding normalized crack locations
β=e/L2 are 0.4, 0.6 and 0.8, respectively.
Table 2.
Crack cases of shaft and identification results
Metrical frequencies/Hz
Case β
α
Predicted β* (Error/%) Predicted α* (Error%)
f1/Hz
f2/Hz
f3/Hz
1
0.4 0.2 98.2
582.04 1049.88 0.415 (3.75)
0.196 (2)
2
0.4 0.4 96.62 568.43 1016.25 0.41 (2.5)
0.42 (5)
3
0.6 0.2 98.12 583.13 1055.80 0.585 (2.5)
0.17 (14)
4
0.6 0.4 95.76 572.14 1053.12 0.605 (0.83)
0.385 (3.75)
5
0.8 0.2 97.34 583.84 1051.64 0.785 (1.88)
0.225 (12.5)
6
0.8 0.4 93.04 580.87 1032.62 0.794 (0.75)
0.425 (6.25)
In most cases, however, the three lines do not accurately intersect at one point because of
inaccuracies in the modeling as compared to measured results. For this purpose, the ‘zerosetting’ procedure that described by Adams [38] is used. In this procedure, Young's modulus of
the structure is changed by using the undamaged natural frequencies of the structure to determine
an effective value, and given by the following iterative approach:
…………………………………(38)
where Em is the corrected value of Young's modulus E, which can be acquired through solving
Eq. (38) for each frequency. It should be noted that the physical signification for the correction
of Young's modulus is not to change the value of E but to make the metrically uncracked natural
frequencies match the computational ones. This procedure can greatly reduce the error between
theoretical analysis and the experimental studies, which are caused by boundary conditions and
material parameters.
In this section, the first three metrical frequencies are employed as the inputs of inverse problem
for crack quantitative identification. Fig. 11 shows the crack identification results in a shaft by
using the frequency contour plots. The intersection point A of three lines indicates the
normalized crack location β and size α. In the experimental studies, when the three lines do not
meet exactly, the centroid of the three pairs of intersections is taken as the normalized crack
location and size [13]. Table 2 shows the comparison of actual normalized crack parameters β
and α and the predicted crack parameters β* and α*. For the given cases, the relative errors of β*
are not more than 4% while the relative errors of α* arrive at 15%. Hence, the proposed modelbased crack identification method by using BSWI Rayleigh beam element is considered to be
valid for actual application to detect open cracks in rotor systems. We point out here that the
identification precisions of the crack sizes are much larger than those of the crack location. The
probable reason is that the effects of the normalized crack size α are much complicated than
those of the normalized crack location β. Therefore, for the identification of α, the tiny metrical
errors would be significantly enlarged.
Fig. 8. Frequency contour plots (- - -: 1st frequency; -.-.-: 2nd frequency; - - -: 3rd frequency): (a)
case 1; (b) case 2; (c) case 3; (d) case 4; (e) case 5 and (f) case 6.
5. DISCUSSION AND FURTHER RESEARCHES
Except the open crack, there are many other types of open and closed cracks and defects in rotor
systems, and different types of defects have different controller equations and because damage
estimation is essentially an inverse problem, it is difficult to obtain a unique or accurate
estimation of cracks, especially for on-line crack identification of rotor systems. On-line
structural crack detection is challenging because actual rotor system dynamic responses are
always contaminated by unknown external loads as well as other harmless conditions, such as
material composition uncertainly, geometric variation, sensor noise, and so on. Therefore, after
the possible existence of damage is detected by an on-line damage detection method as
mentioned by Green, a reliable inspection of the stopped system to find crack location and size is
necessary in order to assure the existence of damage, and the model-based crack identification
method by using BSWI Rayleigh beam element is intended for such use.
The effect of the bearing on the fault diagnosis of the rotor system is very obvious for the
rotating shafts. In this section, the bearing stiffness is considered, as shown in Fig. 12. The
stiffness matrix considers the stiffness of the shaft elements including cracked shaft element and
the bearing stiffness kb.
Fig. 9. Typical rotor-bearing system considered.
The vibration analysis has been carried out using BSWI Rayleigh beam elements for the rotor
system with the same geometrical and physical parameters as shown in Section 4.2. The bearing
stiffness kb is varying from 1×105 to
, supposing the normalized crack location
β=0.4 and crack depth α=0.4. The first three natural frequencies under different bearing stiffness
are shown in Table 3. The first three natural frequencies influenced by bearing stiffness are
shown in Fig. 13. To describe clearly, we adopt the logarithm coordinates. From Table 3 and Fig.
13, the influence of bearing stiffness can be neglected when kb>1×108. Both simply supported
rotor system as shown in Fig. 3 and the bearing stiffness is considered as shown in Fig. 12 have
the same computational results.
Table 3.
The first three natural frequencies under different bearing stiffness
Simply
Bearing stiffness kb (N/m)
Frequencies
supported
1×105
5×105
1×106
1×107
1×108
1×109
5×109
84.77
90.31
96.25
96.90
96.96
96.97
567.83
569.002 569.11
f1/Hz
96.97
60.25
f2/Hz
569.13
225.03 398.18 464.15 556.38
f3/Hz
1019.25
452.90 693.05 827.37 1001.13 1017.48 1019.08 1019.22
In the present experimental studies of this paper, only static (non-rotating) rotor system is
studied, the bearing stiffness nearly to
. Hence the simply supported model of
rotor system can give reasonable results. The bearing damping and other factors are not
considered. It obviously introduces tiny errors of the wavelet-based finite element model. So the
‘zero-setting’ procedure is introduced to eliminate those influences. However, it is worth to point
out here that the present method is only a reliable inspection of the stopped system to find crack
location and size for the sake that the bearing effect is not considered. The further research is that
a more complicated rotating rotor-bearing model which includes bearing stiffness, bearing
damping, gyroscopic effect, etc., will be studied for on-line crack identification use.
6. CONCLUSIONS
The detection of cracks in shafts by measuring the change s in an adequate number of
The natural frequencies has been considered in this paper. A crack is known to introduce
local flexibility in the shaft. The local flexibility due to a crack in the presence of bending
moment and shear loads is modeled by using fracture mechanics concepts . The natural
frequencies of the cracked shaft are determined numerically by solving the characteristic
equation of the shaft . Cracks are then predicted by measuring an adequate number of
shaft natural frequencies and constructing curves similar to Figure 6. The adequate number
of natural frequencies that needs to be measured depends on the number of cracks present.
The first three natural frequencies need to be measured for a single crack, and each
additional crack requires the measurement of two more higher natural frequencies. This
may limit the application of this method to shafts with not more than three or four cracks
and with no closely coupled modes. The technique presented is most suited as a preventive
maintenance tool from the initial installation of the shaft
through a programmed mainten-
ance schedule to record the history of the cracks as they grow, at appropriate time intervals.
7. REFERENCES
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(2).G. R. IRWIN 1957 Transactions of the American Society of Mechanical Engineers,
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crack
(3)H. LIEBOWITZ, H. VANDERVELT, and D. W. HARRIS 1967 International of Journal
of Solids and Structures 3, 489-500. Carrying capacity of a notched column.
(4) I. W. MAYES and W. G. DAVIES 1976 Institution of Mechanical Engineers
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