Vibrational characteristics of cracked shafts

Journal of Sound and Vibration (1991) 147(3), 465-473

VIBRATIONAL CHARACTERISTICS OF CRACKED SHAFTS M. D.

RAJAB

AND

A. AL-SABEEH

Department of Mechanical Engineering, Kuwait University, Kuwait (Received 23 September 1987, and accepted in revised form 23 July 1990)

The vibrational characteristics of a cracked Timoshenko shaft are investigated in this paper. 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 are presented. These expressions are obtained by modeling the crack as bending and shear compliances of equivalent incremental strain energy by using the J-integral concept from fracture mechanics. It is shown that knowledge of the changes in the first three natural frequencies relative to the untracked shaft is sufficient to estimate the crack depth and crack location in the shaft. Each additional crack in the shaft requires the knowledge of the changes in two more higher natural frequencies. 1. INTRODUCTION

A method for the detection of cracks in Timoshenko shafts based on measuring changes in the natural frequencies is described in this paper. Cracks may be initiated and subsequently propagated in shafts and 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 [ 11. 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 [ 11. 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 width/depth ratio and for cracks not parallel to the material surface. Analytical expressions for the flexibility effect of a crack were first derived by Irwin [2,3] and later used in structural analysis applications by Liebowitz et al. [4-61. The vibrational characteristics of a rotor containing a crack were studied by Henry and Okah-Avae [7], Mayes and Davies [8], Gasch [9], and Dimarogonas [lo]. Adams et al. [ll] used axial vibration analysis in one-dimensional structures to predict the location and the magnitude of 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 [12] to predict damage in structures. Chondros and Dimarogonas [ 131 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 experimentally from measurements of changes in the natural frequencies for various crack depth values. 465 0022-460X/91/ 120465+09 %03.00/O

@ 1991 Academic Press Limited

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Dimarogonas and Massouros [14] considered the effect of 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. [15] presented a nomogram for identification of a crack on a simple beam. Ju et al. [16] proposed a damage function relating the changes in the natural frequencies of a structure to the crack depth and location. Dimarogonas and Papadopoulos [17,18] studied the dynamic behavior of a rotating shaft with a surface crack. Yuen [ 191, Chen and Chen [20] and Gounaris and Dimarogonas [21] used finite element analysis to study the vibrational characteristics of cracked structures. Papadopoulos and Dimarogonas [22] 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 compliance 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. 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 [23] J(a) d(crack area), where Pi is the load in the same direction as the displacement

Figure 1. The shaft with a transverse

and J(a) is the J-integral

surface crack.

VIBRATIONAL

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SHAFTS

that represents the strain energy density in the presence of the crack. The local flexibility in the presence of a crack is given by

a2

.\(a) d(crack area).

cq =aqaP,

(2)

The J-integral for the loads of Figure 1 is given by J=

(lIE’)(K:b+ KL.),

(3)

where E’ equals E for plane stress and E/( 1 - V) for plane strain; E is the elastic modulus, v is Poisson’s ratio, KIb is the stress intensity factor for the bending load, and Kilo is the stress intensity factor for the shear load. The expressions for the bending and shear stress intensity factors for a strip of unit thickness containing a transverse crack (see Figure 1) are given by [23]

where F, = J(tan p)/p F,, =[1+30-0.65

rO.923 +Oe199( 1 -sin j3)“]/cos p,

((~/~)+0~37(cu/rY)‘+O~28(a/S)~]/~,

cb = 4M/ nR3,

CT,,= V/g-R’

p = ?rcx/2s,

M is the applied bending moment in the y-direction and V is the applied shear force in the z-direction. By using equations (2) and (4), the non-dimensional compliance due to the bending load, ?b, is found to be 0

Eh= ( r2,??d3)Cb = 2048

I

‘“RXco~o~3s~p~O~923+O~~99

costi-l+:

0

1(6) de,

where cb is the bending compliance due to the crack. The parameter p used in equation (6) that is defined following equation (5) can also be expressed in terms of 8 as P=(z-/4){1-(l-a/R)/cos0}. Similarly, the non-dimensional equations (3) and (5) as r, = (T’E’d)c,

= 32

()I[

e*‘0~’ f+

3

+0.28

;

c0S

compliance

e-i+:

[

1

de.

(7)

due to the shear load, E,, is obtained from

1.30-O-65(;)

+0.37(;)*+0-28(i)

(8)

The non-dimensional compliances due to bending and shear loads, given by equations (6) and (8), respectively, are plotted in Figure 2 as functions of crack depth.

468

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0.4

Figure

2. The bending

and shear

3. ANALYSIS

OF

AND

A. AL-SABEEH

O-8 0.6 Crack depth, u/R

non-dimension

A SHAFT

compliances

WITH

I.0

12

due to a transverse

A SURFACE

surface

crack.

CRACK

Consider a Timoshenko-type simply supported shaft with a transverse surface crack as shown in Figure 1. The differential equation for the transverse vibration of the shaft is given [ 10,241 (9) where EZ is the bending stiffness, m is the mass per unit length, .Z is the rotary inertia, k is the shape factor for the shaft cross-section, A is the cross-sectional area and G is the shear modulus. For vibration in natural modes, the shaft transverse deflection can be written as ~(x, t) = Y(x) e’“‘.

(10)

Substituting equation (10) into equation (9) and then normalizing results in d4Y/d[4+h;d2Y/d[2-w;Y=0,

(11)

where A;=(L~u~.Z/EZ)(I+E/~G), The shaft deflection

w~=(mo2L4/EZ)(1-.Zu*/kAG),

(=x/L.

is the general solution of equation (11) and is given by

YL(~)=AlrcoshA,~+A2LsinhA,~+AJL~~~A2~+A4~sinA2&, YR(&)= AIR cash A,S+AzR sinh A,&+A3R where YL(e) is the shaft to the right of the crack, A, are parameters to be Similarly, the slope of

cos

A25+A4R sin A&

(12)

deflection to the left of the crack, YR(&) is the shaft deflection A, =JdwG+(Ai/2)*+A$2, A,=~~w~,+(A~/~)~-AZ~/~, and the determined from boundary conditions. the deflection curve is given by

~~(~)=A,~A,sinhA,~+A2,A,coshA,~-A~~AZsinA,~+A4,A2~~~A,~, YR(~)=A~RA~~~~~A,~+A~~A,~~~~A,~-A,R~~S~~A~~+A~R~~COSA~~, where A,=l+A,/A,,

A,= l-A,/A2,

A3= wL/w.

(13)

VIBRATIONAL

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In equations (12) and (13) there are eight boundary conditions to be satisfied. Four of the boundary conditions are associated with the two ends of the shaft where both the deflection and the bending moment vanish. The other four come from continuity of bending moment, continuity of shear force, discontinuity of deflection, and discontinuity of slope across the crack interface between the left and right portions of the shaft. The discontinuity conditions of deflection and slope across the crack are jy

; dx(&I =OR-T,(&),

1

EI d’PL

(&I - VL(Sc) = Yld&)- YdSc),

(14)

where & represents the crack location from the left end of the shaft, and c,~and c,, are the shear and bending compliances, respectively. Using the eight boundary conditions and equations (12) and (13) results in eight homogeneous equations, the solution of which exists if the determinant vanishes. Setting the determinant of the system of equations to zero results in the characteristic equation, which is a function of crack depth, crack location on the shaft, shaft natural frequency, shaft material, shaft length and shaft cross-section geometry. A closed form expression for the characteristic equation is not easily obtained, but a numerical solution was used instead. The final form of the characteristic equation is non-linear. With the characteristic equation in matrix form, a computer program was written that accepts as input the shaft material, shaft cross-section geometry and length, crack depth and crack location on the shaft, and returns the changes in the shaft natural frequencies compared to the virgin shaft system. In the solution process, Brent’s method for the solution of non-linear equations was used [25]. Gaussian quadrature [26] was used for the numerical integration for the bending and shear compliances due to the crack, equations (6) and (8), respectively. The normalized change in the first natural frequency in the presence of a crack is shown in Figure 3 as a function of crack depth for four values of crack locations. In this and the subsequent figures, the change in the natural frequency is defined as the difference between the natural frequency of the untracked shaft and the natural frequency of the shaft with a crack. As expected, the shaft fundamental frequency in the presence of a crack decreases with an increase in crack depth. For a given crack depth, the fundamental frequency decreases as the crack occurs close to the middle of the shaft where, for the simply supported shaft considered, the maximum amplitude of the mode shape occurs.

Crock depth,

u/f?

Figure 3. The normalized change in the first natural frequency of the shaft.

470

M.

a

D.

RAJAR

AND

AL-SABEEH

I

0.0

0.2

06

04 Crack

Figure

A.

4. The normalized

change

depth,

O-6

IO

I2

a/R

in the second

natural

frequency

of the shaft.

0.15 3" 3 a 0 IO

0.00 0.0

0.2

06

0.4 Crack

Figure

5. The normalized

change

depth,

O-8

IO

12

a/R

in the third natural

frequency

of the shaft.

The normalized changes in the second and the third natural frequencies, respectively, for the shaft with a crack are shown in Figures 4 and 5. The general trend of a decrease in the natural frequency with an increase in crack depth is observed. However, due to the shift of the nodes for the second and third mode shapes, the change in the natural frequency with the crack location is not as monotonic as in the first mode but depends on how close the crack is to the mode shape node. 4. THE CRACK

PREDICTION

TECHNIQUE

In order to predict the presence of one or more cracks in a shaft, it is first necessary to measure an adequate number of shaft natural frequencies and then use the method outlined in this section to predict the crack depth and location on the shaft. The adequate number of natural frequencies to be measured depends on the number of cracks in the shaft. If a programmed maintenance schedule is adopted at the start of service, then only a few cracks would develop between maintenance periods, and hence only the first few system natural frequencies need to be measured. In the method described below, the results of the computer program developed together with the results shown in Figures 3-5 are used.

VIBRATIONAL

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For a shaft with a single crack of an unknown depth and location on the shaft, the first three natural frequencies need to be measured to predict the crack uniquely. Let the changes in the first three natural frequencies from those of the untracked shaft be Am,, Awz, and A+. Then, on Figure 3, a straight line is drawn at the value of the normalized frequency change. The intersection of this line with the various curves gives values of crack depth against crack location on the shaft. Similar results are obtained by using Figures 4 and 5 and corresponding changes in the second and third frequencies, respectively. A computer program was written that gives the values of crack depth us. crack location on the shaft, given the change in a particular natural frequency. The results of this program for normalized changes in the first three natural frequencies of the shaft of 0.12, 0.14 and 0.11 are shown in Figure 6. From Figure 6, it can be concluded that a crack exists at a distance of 30% of the shaft length from the left end with a depth of 0.60. Similar curves are obtained once the changes in shaft frequencies are known.

Second mode

Third mode

Crack position. &

Figure 6. The crack depth and third natural frequencies

us. crack location for normalized of the shaft, respectively.

changes

of 0.12,0.14

and 0.11 in the first, second

For the single crack considered, measurements of the natural frequencies of the first two modes are not generally adequate to identify the crack uniquely, and the third mode helps to confirm the parameters of the crack. Due to the symmetry of the shaft system considered, a symmetric crack is also predicted with the same depth and location from the right end of the shaft. For each additional crack present in the shaft, two more higher natural frequencies need to be measured to identify uniquely all cracks, since each crack introduces two unknowns-crack depth and location. This imposes some practical restrictions on the use of this method to predict more than a few cracks in the shaft.

5. CONCLUSIONS The detection of cracks in shafts by measuring the changes 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

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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 maintenance schedule to record the history of the cracks as they grow, at appropriate time intervals.

REFERENCES 1. UNITED KINGDOM ATOMIC ENERGY AUTHORITY RESEARCH GROUP 1969 Non-Destructive Testing: Views, Reviews, Previews. Harwell Post-Graduate Series. Oxford University Press. 2. G. R. IRWIN 1957 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 24, 361-364. Analysis of stresses and strains near the end of a crack transversing a plate. 3. G. R. IRWIN 1957 IXth International Congress of Applied Mechanics, University of Brussels VIII (Paper No. 101 (II)), 245-251. Relation of stresses near a crack to the crack extension force. 4. 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. 5. H. OKAMURA, H. W. LIU, C. CHU and H. LIEBOWITZ 1969 Enginering Fracture Mechanics 1, 547-564. A cracked column under compression. 6. H. LIEBOWITZ 1969 in Fracture, Volume IV (H. Liebowitz, editor), 113-171. New York: Academic Press. Chapter 4: Fracture and carrying capacity of notched column. 7. M. A. HENRY and B. E. OKAH-AVAE 1976 Publication: Vibrations in Rotating Machinery, Institution of Mechanical Engineers Conference, Paper No. C162/76. Vibrations in cracked shafts. 8. I. W. MAYES and W. G. DAVIES 1976 Institution of Mechanical Engineers Conference Publication, Vibrations in Rotating Machinery, Paper No. C168/76. The vibrational behavior of a rotating shaft system containing a transverse crack. 9. R. GASCH 1976 Institution of Mechanical Engineers Conference Publication, Vibrations in Rotating Machinery, Paper No C178/76. Dynamic behavior of a simple rotor with a cross-sectional crack. 10. A. D. DIMAROGONAS 1976 Vibration Engineering, St. Paul, Minnesota: West Publishers. 11. R. ADAMS, P. CAWLEY, C. PYE and B. STONE 1978 Journal of Mechanical Engineering Science 20 (2), 93-100. A vibration technique for non-destructively assessing the integrity of structures. 12. P. CAWLEY and R. D. ADAMS 1979 Journal of Strain Analysis 14 (2), 49-57. The location of defects in structures from measurements of natural frequencies. 13. T. G. CHONDROS and A. D. DIMAROGONAS 1980 Journal of Sound and Vibration 69,531-538. Identification of cracks in welded joints of complex structures. 14. A. DIMAROGONAS and G. MASSOUROS 1980 Engineering Fracture Mechanics 15 (3-4), 439-444. Torsional vibration of a shaft with a circumferential crack. 15. A. ANIFANTIS, P. RIZOS and A. DIMAROGONAS 1987 American Society of Mechanical Engineers, Design Technology Conference, Boston, Massachusetts, U.S.A. Identification of cracks on beams by vibration analysis. 16. F. Ju, M. AKGAN, E. WOND and T. LOPEZ 1982 American Society of Mechanical Engineers, Meeting, Phoenix, Arizona, U.S.A., Nouember 1982. Modal method in diagnosis of fracture damage in simple structures. 17. A. D. DIMAROGONAS and C. A. PAPADOPOULOS 1983 Journal of Sound and Vibration 91 (4), 583-593. Vibration of cracked shafts in bending. 18. C. A. PAPADOPOULOS and A. D. DIMAROGONAS 1987 Journal of Sound and Vibration 117 (l), 81-93. Coupled longitudinal and bending vibrations of a rotating shaft with an open crack. 19. M. M. F. YUEN 1985 Journal of Sound and Vibration 103 (3), 301-310. A numerical study of the eigenparameters of a damaged cantilever. 20. L. CHEN and C. CHEN 1988 Computers and Structures, 28 (l), 67-74. Vibration and stability of cracked thick rotating blades. 21. G. GOUNARIS and A. DIMAROGONAS 1988 Computers and Structures 28(3), 309-313. A finite element of a cracked prismatic beam for structural analysis. 22. C. A. PAPADOPOULOS and A. D. DIMAROGONAS 1987 ZngenieurArchiu 57,257-266. Coupling of bending and torsional vibration of a cracked Timoshenko shaft.

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23. H. TADA 1973 TheStress AnalysisofCrack.sHandbook Hellertown, Pennsylvania: Del Research Corporation. 24. W. THOMSON 1988 Theoryof Vibrations with Applications (third edition). Englewood Cliffs, New Jersey: Prentice-Hall. 25. R. P. BRENT 1973 Algorithms for Minimization without Derivatives. Englewood Cliffs, New Jersey: Prentice-Hall. 26. S. D. CONTE and C. DEBOOR 1980 Elementary Numerical Analysis: An Algorithmic Approach (third edition). New York: McGraw-Hill.