Crack Detection in a Cantilever Beam using Harmonic Probing and Free Vibration Decay

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IINNDDIIAA--444400001111 EEmmaaiill:: aacchhaatttteerrjjeeee@@mmeecc..vvnniitt..aacc..iinn

Nomenclature

)(tx System response )(tf Excitation force )(txn nth order response component

),...,( 1 nnh nth order Volterra kernels )(, qpnH Higher order FRFs

2k Square nonlinear parameter

3k Cubic nonlinear parameter )( nX nth harmonic response amplitude

Bilinear parameter ABSTRACT Vibration testing forms one of the most effective and recent technique of non-destructive testing of structures and various engineering components. There are mainly two approaches to crack detection through vibration testing; open crack model and breathing crack model. The present study is based on breathing crack model, in which the bilinear stiffness characteristic of a cracked cantilever beam is approximated by a polynomial series and a nonlinear dynamic model of the cracked structure is developed using higher order Frequency Response Functions. Effect of crack severity is investigated through measurement of response harmonic amplitudes and a procedure is suggested to estimate the crack severity through nonlinear response analysis. 1. Introduction Failures in structures and machine elements can be prevented through early detection of fatigue cracks using various non-destructive testing methods. Traditional non-destructive testing methods such as dye-penetrant test, magnetic particle inspection, ultrasonic test etc have their own limitations and are often very expensive and inconclusive. Vibration testing forms one of the most effective and recent technique of non-destructive testing of structures and various engineering components. There are mainly two approaches to crack detection through vibration testing; open crack model and breathing crack model. In the first approach, crack is considered to be always open and the focus is on the changes in natural frequency and other modal parameters.[1-3]. However, the changes in natural frequency have been found to be significant only for a large crack size [4] Recently, researchers have shown that a structure with a breathing crack behaves more like a nonlinear system, similar to that of a bilinear oscillator and the nonlinear response characteristics can very well be investigated to identify the presence of the crack (Shen and Chu [5], Rivola and White [6], Chondros and

Proceedings of the IMAC-XXVIIFebruary 9-12, 2009 Orlando, Florida USA

2009 Society for Experimental Mechanics Inc.

Dimarogonas [7]). The effect of crack on a continuous structure, have been modeled as a local change in the stiffness matrix by Kisa and Brandon [8], Saavedra and Cuitino [9]. These research works mainly provide qualitative understanding of the nonlinear response characteristics of a cracked beam and very few have attempted quantitative assessment of the crack. The present work investigates both forced response and free vibration decay of the nonlinear model of a cracked beam. In the forced response analysis, harmonic excitation is used and Volterra series response representation is employed for developing a quantitative damage assessment technique. Volterra series [10] represents a nonlinear system through a set of first and higher order frequency response functions (FRFs). The bilinear restoring force function of a cracked cantilever beam is approximated by a finite term polynomial series and the response amplitudes under harmonic excitation are obtained using Volterra series response representation. A procedure is then suggested for estimating the structural damage through measurement of the first and second harmonic amplitudes. In free vibration analysis, the response decay of the bilinear stiffness model is investigated in each half cycle and the stiffness degradation due to a crack is estimated through measurement of the asymmetric amplitude decay parameters. 2 Volterra Series Response Representation Volterra series represents the input-output mapping of a physical system, with )(tf as input excitation and )(tx as output response, in a form of functional series given by

++

+=

....)()()(),,(

)()(),()()()(

3213213213

21212121111

dddtftftfh

ddtftfhdtfhtx

(1)

...)(...)()( 21 ++++= txtxtx n )( 11 h , is the familiar impulse response function of a linear system and ),...,( 1 nnh are the nth order Volterra

kernels. Higher Order Frequency Response Functions or Volterra kernel transforms can be defined as the multi-dimensional Fourier transforms of the higher order Volterra kernels as

== n

n

i

jnnnn dddehH ii ...),...,,(...),...,,( 21

12121 (2)

For a system with polynomial form of stiffness nonlinearity, given by the equation of motion )(...)()()()()( 33

221 tftxktxktxktxctxm =+++++ &&& (3)

with harmonic excitation tjtj eAeAtAtf +==

22 cos )( , (4)

the response, using Volterra series (1), is obtained as

= =+

=1

, , )( 2

)(n nqp

tjqpnq

nn

qpeHCAtx (5)

Where )(, qpnH are the higher order FRFs given by ),...,,,...,()(

times times

,43421321

qpn

qpn HH = )(, qpqp = (6)

Response amplitude for the first three harmonics, 3 and 2, , from Eqn. (5), can be expressed as ),,(

43)()( 3

3

1 += HAAHX + higher order terms (7a)

),,,(2

),(2

)2( 44

2

2 += HAHAX + higher order terms (7b)

),,(4

)3( 33

HAX = + higher order terms (7c)

f(t)

m

(1-) k

k

c Y u Crack

The higher order FRFs can be synthesized [11] from the first order FRFs as )2(*)(),( 1

2122 HHkH = (8a) [ ]31221313 )2(2 *)3(*)(),,( kHkHHH = (8b)

{ }

+= 311221313 )0(2)2(32 )(-)(),,( kHHkHHH (8c)

{ }

++

++

=),()()(6),()(6

),(),(4),,()(6),,()(2

4)2(

),,,(

2112213

22

31312

14

HHHHHk

HHHHHH

k

HH (8d)

3. Modeling of a cracked cantilever beam as bilinear oscillator

A cantilever beam with a breathing edge crack (Fig. 1) exhibits bilinear stiffness characteristics depending on whether the crack is open or closed. Using Finite Element method, the equation of motion can be expressed in terms of the mass, stiffness and damping matrices as [ ]{ } [ ]{ } [ ]{ } { })()()()( tFtuKtuCtuM =++ &&& when the crack is closed (9) and [ ]{ } [ ]{ } [ ]{ } { })()()()( tFtuKKtuCtuM =++ &&& when the crack is open (10) Using the coordinate transformation { } [ ]{ })()( tqtu = and modal analysis, one can obtain the single-degree-of-freedom equation for the fundamental mode of vibration as

)()()(2)( 11211111 tftqtqtq =++ &&& (11)

where 1 and 1 are respectively modal damping and natural frequency of the un-cracked beam in the first mode and

[ ]21

11

KK

T = when the crack is open and 1= , when the crack is closed. The term K in above equation represents the reduction in the stiffness matrix due to the crack and is a function of the crack size as well as of crack location Eq. (11) for the cracked beam is similar to that of a bilinear oscillator (Fig. 2) given by

)()]([)()( tftxgtxctxm =++ &&& (12) where, )( )]([ tkxtxg = with 0)(for 1 ptx= and 0)(for 1 txp Thus the nonlinear vibration response of a beam with breathing crack be very well obtained through analyzing the response of an equivalent bilinear oscillator.

Figure 1: A cantilever beam with an edge Figure 2: Bilinear oscillator model of a crack spring mass damper system

L

4. Crack severity assessment through forced vibration analysis To obtain the response harmonic amplitudes through Volterra series, restoring force )]([ txg due to the bilinear stiffness of the cracked beam is approximated with a polynomial form )]([ txg given as

...)()()()]([ 332

210 ++++= txktxktxkgtxg (13) For the sake of simplicity here, the polynomial is considered up to the cubic power term and the coefficients,

, , 321 kkk are computed by minimizing the error function E , given as

{ }

=X

XdxtxgtxgkkkE 2321 )]([)]([),,( . (14)

Applying 3,2,1for ,0 == ikE

i the coefficients are obtained as

kk2

)1(1

+= , kX

k32

)1(152

= , 03 =k (16) The first and second harmonic response amplitude, following Eqs. (8a-d), then can be expressed as

{ } )(-)()0(2)2(2

)()( 13111

22

3

1 HHHHkAAHX ++= + higher order terms (17a)

)2()(2

)2( 1212

2 HHkAX = + higher order terms (17b)

As decreases for a growing crack, the nonlinear parameter 2k increases (Eq. 16) and thus the harmonic amplitude )2( X which is directly related to 2k (Eq. 17b ) also increases. Thus 1- can be used as an indicator of the crack severity and is termed as Crack Severity Index (C.S.I). Fig. 3 shows the variation in the first and second harmonic amplitudes for different crack depth depicted by crack severity index varying between 0.01 to 0.10 for typical excitation frequencies, 0.4 and .3 0 // =mk . It can be seen that the second harmonic amplitude )2( X is much smaller than the first harmonic amplitude )(X when the index 1- is small. In such cases an excitation frequency close to half the bilinear natural frequency, such as 0.4 // =mk will give higher signal strength )2( X than an excitation frequency, 0.3 // =mk . The bilinear frequency is defined as

010

10

122

+=+=b (18)

where 0 and 1 are respectively the natural frequency of the beam in un-cracked and cracked conditions.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

10-3

10-2

10-1

100

101

Crack Severity Index

Res

pons

e A

mpl

itude

Freq. = 0.4Freq. = 0.3

First harmonic

Second harmonic

Figure 3: Harmonic amplitudes at different damage levels.

Fig. 4 shows the variation in spectral amplitude ratio )(/)2( XX for different crack depth for a typical excitation frequency, 0.3 // =mk . It is significant to note that the ratio is almost proportional to the severity index and thus can be potentially used as a measure of the damage. Neglecting the higher order terms in the Eqs. (17a,b) an expression for the crack severity index can be obtained as

)2()()2()0(

15641

*

XXXX= (19)

where )0(X is the static deflection which can be experimentally obtained with a static load. The value of the Crack Severity Index (CSI), i.e., 1 , can be easily estimated from experimental measurements using above Eq. (19). Figure 5 shows the accuracy of estimation of the crack severity index by the proposed technique, through measurement of first and second harmonic amplitudes.

0 0.02 0.04 0.06 0.08 0.1

0

0.01

0.02

0.03

0.04

0.05

Crack Severity Index

Res

pons

e am

plitu

de ra

tio

Figure 4: Amplitude ratio )(/)2( XX for different damage levels with 3.0// =mk

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Crack Severity Index

Est

imat

ed v

alue

Exact valueEstimated value

Figure 5: Estimation of crack severity with 3.0// =mk

5. Crack severity assessment through free vibration analysis Free vibration response of a general harmonic oscillator is given by

+= txtxetx dd

dtn

sin)0(cos)0()( & (20)

Starting with 0)0( =x , the response will be given by

= txetx d

d

tn sin)0()( & (21)

First cycle of decay can be considered in two halves; positive half and the negative half. Natural frequency will be different in positive half and negative half.

mk

n =1 when 0fx and m

kn =2 , when 0x (22)

If 321 and , XXX respectively represent the free vibration peak amplitudes in first positive half, first negative half and the second positive half, then one can obtain

21/

1

2

2

1 = e

XX

n

n and 21/

2

1

3

2 = e

XX

n

n (23)

Which gives

==

22

21

31

22

n

nXX

X (24)

The crack severity index thus can be obtained as

31

2211XX

X= (25)

6 Conclusion The nonlinear response of a cracked beam under both forced and free vibration is investigated for assessment of crack severity. The bilinear restoring force due to crack open and crack closed modes is approximated by a polynomial series and the first and second order Frequency Response Functions are developed in terms of the bilinear parameter. It is shown that the crack severity can be estimated through measurement of the first and second harmonic amplitudes. The stiffness degradation induced by a crack can also be estimated through measurement of asymmetric response amplitude decay in a free vibration test.

Reference

[1] Chondros T G, Dimarogonas A D, Vibration of a cracked cantilever beam, J Vibration Acoustics, 120: 742- 746, 1998. [2] Chinchalkar S, Determination of crack location in beams using natural frequencies, J.Sound Vib., 247(3), 417-429, 1998. [3] Panteliou S D, Chandros T G, Argyrakis V C, Dimaragonas A D, Damping factor as an indicator of crack severity, J. Sound Vib., 241, 235-245, 2001. [4] Cheng S M, Wu X J, Wallace W, Vibrational response of a beam with a breathing crack, J. Sound Vib.,

225(1), 201-208, 1996. [5] Shen M.-H.H, Chu Y C, Vibrations of beam with a fatigue crack, Compt. Struct., 45(1), 79-93, 1992. [6] Rivola A, White P R, Bispectral analysis of the bilinear oscillator with application to the detection of fatigue cracks, J. Sound Vib., 216(5): 889-910, 1998.

[7] Chondros T G, Dimarogonas A D, Vibration of a beam with a breathing crack, J. Sound Vib., 239 (1): 57-67, 2001 [8] Kisa M, Brandon J A, The effects of closure of cracks on the dynamics of a cracked on the dynamics of a cracked cantilever beam, J. Sound Vib. ; 238 (1): 1-18, 2000. [9] Saavedra P N, Cuitino L A, Crack detection and vibration behavior of cracked beams, Comp. Struct.; 79, 1451-1459, 2001. [10] Schetzen M, The Volterra and Wiener Theories of Nonlinear Systems, John Wiley and Sons, New York,1980. [11] Chatterjee, A. and Vyas, N.S., Nonlinear Identification through Structured Response Component Analysis using Volterra Series, Mechanical Systems and Signal Processing, 15(2), pp. 323-336, 1999.

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