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Harmonic analysis of the vibrations of a cantilevered beam with a closing crack
Pergamon PII: s0045-7949(%)00184-8
Compuars & Swucrures Vol. 61. No. 6. pp. 1057-1074. 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 004%7949196 115.00 + 0.00
HARMONIC ANALYSIS OF THE VIBRATIONS OF A CANTILEVERED BEAM WITH A CLOSING CRACK R. Ruotolo,t C. Surace,? P. Crespu$ and D. Storer$ Departments of jStructural Engineering and SAeronautical and Space Engineering, Politecnico di Torino, Torino, Italy (Received 22 June 1995)
Abstract-In this article, the vibrational response of a cracked cantilevered beam to harmonic forcing is analysed. The study has been performed using a finite element model of the beam, in which a so-called closing crack model, fully open or fully closed, is used to represent the damaged element. Undamaged parts of the beam are modelled by Euler-type finite elements with two nodes and 2 d.f. (transverse displacement and rotation) at each node. Recently the harmonic balance method has been employed by other researchers to solve the resulting non-linear equations of motion. Instead, in this study, the analysis has been extended to employ the first and higher order harmonics of the response to a harmonic forcing in order to characterize the non-linear behaviour of the cracked beam. Correlating the higher order harmonics of the response with the forcing term the so-called higher order frequency response function (FRFs), defined from the Volterra series representation of the dynamics of non-linear systems, can be determined by using the finite element model to simulate the time domain response of the cracked beam. Ultimately the aim will be to employ such a series of FRFs, an estimate of which in practice could be measured in a stepped sine test on the beam to indicate both the location and depth of the crack, thus forming the basis of an experimental structural damage identification procedure. Copyright 0 1996 Elsevier Science Ltd
1. INTRODUCTION Vibration-based inspection of structural behaviour offers an effective tool of non-destructive testing. The analysis of the dynamic response of a structure to excitation forces and the monitoring of alterations which may occur during its lifetime may be employed as a global integrity-assessment technique to detect, for example, play in the joints or the presence of a crack. Indeed it is well known that, in the case of simple structures, crack position and depth can be determined from changes in natural frequencies, modes of vibration or the amplitude of the forced response. In the past many studies have illustrated that a crack in a structure such as a beam may cause the structure to exhibit non-linear behaviour if the crack is open during part of the response and closed in the remaining intervals. This phenomenon has also been detected during experimental testing performed by Gudmundson in which the influence of a transverse closing crack upon the natural frequencies of a cantilevered beam was investigated [ 11. The main result obtained through experimental studies such as these was that the observed decrease in the natural frequencies of the beam due to the presence of the crack is not sufficient to be described by a model of crack which is always open. Therefore it must be concluded that the crack alternately opens and closes, thus giving rise to natural frequencies
falling between those corresponding to the alwaysopen and always-closed (e.g. integral) cases. In fact, if an always-open crack is assumed in the analysis of a beam with a so-called closing crack, i.e. one which both opens and closes during the time interval considered, the reduced decrease in the experimental natural frequencies will lead to an underestimate of the crack depth when determined via a test-model correlation approach. Many researchers have studied the problem of a beam with a closing crack starting from an analytical viewpoint. After demonstrating the bilinear nature of a beam with a closing crack, Zastrau used a finite element method to study the time history and the spectrum of a simply-supported beam with multiple cracks [2]. Qian et al. observed that differences between amplitudes of forced vibrations of cracked and non-cracked beams are reduced when a closing crack model is considered. Furthermore, time series analysis has been applied to identify the modal parameters of cracked beams [3]. Ibrahim et al. applied the bondgraph technique (a numerical tool using lumped parameter elements) to examine the FRFs of a cantilevered beam with a crack [4] and described experimental testing performed to validate their technique [S]. Collins et al. utilized direct numerical integration through the Galerkin method to investigate longitudinal free and forced vibrations of a cracked prismatic bar [6].
1057
1058
R. Ruotolo et al.
Friswell and Penny have simulated the non-linear behaviour of a beam with a closing crack vibrating in its first mode of vibration through a simple 1 d.f. model with bilinear stiffness [7]. The analysis of the FRFs and the response to harmonic loading, obtained by numerical integration, demonstrates the occurrence of harmonics in the response spectra which are integer multiples of the exciting frequency. Shen and Chu simulated, with a bilinear equation of motion for each mode of vibration, the dynamic response of simply-supported beams with a closing crack, analysing the response spectra in order to detect changes which are potentially useful for damage assessment [8]. Later the same authors, in order to reduce the computational effort, used a closed-form solution based on the use of two square-wave functions to model stiffness change for bilinear oscillators, and simulate the cracked beam under low frequency excitation [9]. Krawczuk and Ostachowicz studied the transverse vibrations of a beam with a closing crack simulated through springs with periodically time-variant stiffness [lo]. In a recent paper these authors presented an analysis of the forced vibrations of a cantilevered beam with a closing crack, in which the equations of motion were solved using the harmonic balance technique [ 1I]. The periodically time-variant beam stiffness is represented with a square-wave function with a fundamental frequency equal to the forcing term frequency. This research has shown that when a closing crack is present in a beam, higher harmonic components in the frequency spectrum of the response are generated when excited by a sinusoidal forcing function, indicating that the structure behaves non-linearly. This in itself would permit one to distinguish between the types of crack, i.e. whether closing or always open. However, the aim of the research reported in this article has been to extend the analysis of the harmonic components of the response spectrum using the so-called higher order FRFs which can be defined using the Volterra series for systems in which the non-linearity can be represented by a polynomial-type function [ 121. If measurable in practice, certain features of the higher order FRFs, for example their relatively sensitive dependence on the type and form of non-linearity present, would provide several distinct advantages over other methods for detecting faults in structures, assuming that the faults cause the structure to behave non-linearly, i.e. not as in the case of a beam with a crack which is open throughout the response cycles. Since the FRFs are related to the coefficients of the polynomial function in the governing equations of motion, it should be possible to correlate the amplitude of the peaks in the higher order FRFs with the crack depth and/or position. The intention would be to develop a damage assessment procedure based on a quantitive comparison of the higher order FRFs measured from the structure under study.
As described in this paper, in order to examine the feasibility of such an approach, a series of numerical simulations of a cracked beam has been performed, with “measured” estimates of the FRFs being used in an attempt to distinguish between cracks of varying depths and location along the length of the beam. In order to model accurately the non-linear behaviour of a beam with a closing crack, it is critical to determine the precise moment that the beam changes state, i.e. when the crack opens or closes. In the results presented, it has been assumed that the change between fully opened and fully closed takes place instantaneously giving rise to a bilinear-type stiffness non-linearity (which can be approximated by a high-order polynomial). In the numerical simulation, the change of state is imposed in terms of the beam curvature at the cracked section: the crack is assumed to be open if the curvature is in the positive sense, otherwise it is closed. In this way there is no time-dependent control function for the stiffness of the beam. Using a specific finite element for the section with the crack, it is possible to simulate the free and forced bending vibrations of a cracked cantilevered beam by employing non-linear finite element analysis [ 131. The aim of the paper is to present a harmonic analysis of a cantilevered beam with a closing crack, and to analyse the response of the beam to sinusoidal loads at various forcing frequencies in relation to diverse crack depths and positions. A similar approach is often adopted during experimental “stepped-sine” tests on structures and, as described subsequently, this procedure is particularly appropriate for estimating the higher order FRFs in practice.
2. MATHEMATICAL 2.1. St@ess
MODEL OF THE CRACKED REAM
matrix
The mathematical model used for the Euler beam with a transverse on-edge non-propagating closing crack is based on the finite element model proposed in Refs [3] and [14]. According to the principle of St Venants the stress field is influenced only in the region adjacent to the crack. The element stiffness matrix, with the exception of the terms which represent the cracked element, may be regarded as unchanged under a certain limitation of the element size. The additional stress energy of a crack leads to a flexibility coefficient expressed by a stress intensity factor derived by means of the Castigliano’s theorem in the linear elastic range. The general approach to the problem is that, for the cracked element in the beam, the elements situated on one side can be considered as external forces applied to the cracked element, while the elements on the other side can be regarded as constraints. In this way the flexibility matrix can be
1059
Vibrations of a cantilevered beam calculated easily and therefore, from the conditions of equilibrium, the stiffness matrix of the cracked element can be derived. In this study the element employed is shown in Fig. 1. Neglecting shear action, the strain energy of an element without a crack can be formulated as
and F,(s) = ,/(2/ns)tan(xs/2)[0.923 + 0.199( 1 - sin(rrs/2)“)]/cos(7rs/2), Fir(s) = (3s - 2?)(1.122 - 0.561s
W”’ = AI
+ 0.085~~ + 0.18s’)/J1-s,
’ (M + Pz)’ dz s0
= &I (Ml + P2P/3 + MPP),
(1)
I being the length of the element, and the additional energy due to the crack is
(5)
where s = a/h is the relative depth of the crack, h being thickness of the beam. The term c$” of the flexibility matrix C:) for an element without crack can be written as (0) _
C/k-
a2~o) -=& a~, ap,
i,k=l,2
PI=P,
Pz=M.
(6) u”” = b
‘[(K: s0
+ Kf,)/E’ + (A + v)K&/Ej
da,
(2)
where E’ = E for plane stress, E’ = E/(1 + v) for plane strain, E is the elastic modulus, v is the Poisson ratio and K,, Kn , Ktn are stress intensity factors for opening-type, sliding-type and tearing-type cracks respectively, a is the depth of the crack and 6 the width of the beam. Taking into account only bending:
The term c$’ of the additional flexibility matrix Ci” due to the crack can be formulated: 8W”
&’ = -ap, a~, = CA!? i, k = I,2
p, = p,
pz = M.
(7) The term cilrof the total flexibility matrix for the damaged element C, is C,k= cl:) + c$’ .
t+‘(” = b ’ ([(Km + KIP)*+ K&]/E’] da,
(3)
From the equilibrium (f'i
with
Mi
(8)
condition (Fig. 1): M,+I)~=T(P,+,
f's+,
M,+I)‘,
(9)
where K,M = (6M/bh’)&F,(s), K,p = (3Pl/bh2),/&F,
(s),
T=
%IP = (Plbh),/&,(s),
(4)
-1
0
-1
-1
1
0
0
1
,
5+
‘i ui
1a
ui+ ei+l
ct
1 It I
I
/
M
Mi
Y
1 Fig. 1. Schematic diagram of an element.
x
i+l
I.
(10)
1060
R. Ruotolo et al.
Using the principle of the virtual work, the stiffness matrix of the undamaged element can be written: K c = TC””c - ‘F v
r 1261 4P61 K
e = gP
(11)
-12
61
-61
2P
-12
-61
12
-61
61
2P
-61
4P
1
1 (12)
’ 1
while the stiffness matrix of the cracked element may be derived as IG, = TC;‘TT.
(13)
techniques capable of arriving at the steady-state solution of the response whilst avoiding direct integration. Shen and Chu used a bilinear oscillator to simulate each mode of the beam with a closing crack. In Ref. [9] a closed-form solution based on the use of two square-wave functions to model stiffness change was developed. However, this solution is valid only for low frequency excitation up to a ratio of 5-6 between the first natural frequency of the linearized system, wo, and the frequency of the forcing term. Also, Krawczuk and Ostachowicz [ 1l] have used a square-wave functionf(t) to model the beam stiffness, transforming the non-linear equations of motion into a linear periodically time-variant equation: Mii + Di + (K. - AKf(t))u = R.
(17)
2.2. Mass matrix In order to evaluate the dynamic response of the cracked beam when acted upon by an applied force, it is supposed that the crack does not affect the mass matrix M. Therefore, for a single element, the mass matrix can be formulated directly:
r 156
221
-131
-311
54 131 156 -221
- 1321 -3P -221 3 (14) 4P
where m is the mass per unity length of the beam. 2.3. Damping matrix Assuming that the damping matrix D is not affected by the crack, it can be calculated through the inversion of the modeshape matrix relative to the undamaged structure:
In order to apply the harmonic balance method the closing crack function f(t) in Fourier series was expressed as 1 2 2 f(t) = - + - cos(wt) - - cos(3wt) 2 n 3n 2 + j-& cos(5wt) + . . , (18) where w represents the angular frequency of the harmonic input. As shown in Fig. 2, this is the Fourier series approximation to a square-wave function. The solution of eqn (17) was obtained by assuming that 0 R=
D = (dp=)’ d@-‘,
?
cos(ot),
(19)
i.1R0 and that the degrees of freedom of the structure can be expressed as
where
1
d=
9
(16)
in which 6 is the ith modal damping ratio, co, is the ith natural frequency and M, is the ith modal mass relative to the undamaged beam. According to Ref. [ 111, the modal damping ratio ii can be assumed to be approximately equal to 0.01. 3. THE HARMONIC
BALANCE APPROACH
Numerical integration of non-linear equations of motion is usually very computer-time consuming. For this reason some researchers have studied
u = : (ai sin(W) + bi cos(iot)), i=,
(20)
where a, and b, are vectors of the constant variables. This approach seems to be extremely efficient both in terms of computer time and analytically. Furthermore, this approach enables the higher harmonics of the response to be obtained quickly for any excitation frequency. However, a comparison of time history of the steady-state solution of the forced equation of motion obtained through direct numerical integration, as described in Section 5, and the time response obtained from eqn (17) demonstrates that the model described in Ref. [l l] is not valid.
Vibrations of a cantilevered beam
1
0.8
,‘\
I”.
\
1061
:
I
,‘I
‘./’
-. : ‘\
*
’ I
I
;
i
I ; .;.. i j
i I
0.6 ,-
. . ... .
I.
.......
...“‘i.. : i
-5
!....
I
I ..:
‘7
i;
:
f 0.4I-
..
..
:.
: 0.2
,_
al-
I1. .:. i I i I ;f. .
_.
3 _:_
0
0.2
i !i. :I
I
!
./.,;
‘\
i
i
.;
::
ii ,-0.2
,i
.:.
/‘\
/
.:.
,,,,
i
.A
e., i
1:: ;I\.\
‘$ ‘,
.. I :\.
0.4
:‘\/’
0.6
”
0.8
: .,!
1
1
1.2
[=I Fig. 2. The square-wave function and its approximation.
In particular, it is evident that the assumption of a square-wave type stiffness variation (fundamental frequency equal to the forcing frequency) is incorrect. This is especially noticeable when the forcing frequency is equal to 50~. Indeed, in this case thef(t) function imposes that the crack changes its state (open to closed and vice versa) only once per cycle. Instead the Krawczuk and Ostachowicz’s solution of the time response, shown in Fig. 8, is inconsistent with this statement. This is due to the fact that the functionf(t) modelling the stiffness does not consider either the out-of-phase relation between forcing term and structural response, or the fact that the non-linear nature of the system may cause appreci-
A
Compressive
able distortion in the response waveform due to the higher harmonic components which in turn influence the activation of the crack. This point is very important because, when the forcing frequency is close to $0, fo0, &00, etc., significant superharmonic content in the response spectra is manifest. For example, if the forcing frequency is equal to awO, the response contains harmonic components at ioO, w. and so on, while the square wave function developed as in eqn (18) does not contain the harmonic term relative to w. . Instead this aspect has been taken into account by Shen and Chu in Ref. [9], limiting the validity of the closed-form solution to low frequency excitation for which the superharmonic effect can be neglected.
half cycle (K=K “) 4. EQUATION
/ / / /
OF MOTION
When the crack closes and its interfaces are completely in contact with each other, the dynamic response can be determined directly as that of the untracked beam. However, when the crack opens the stiffness matrix of the cracked element should be introduced in replacement at the appropriate rows and columns of the general stiffness matrix. Under the action of the excitation force R, alternate crack opening and closing causes the equations of motion of the cracked beam to be non-linear: Fig. 3. Crack opening-closing process.
Mii+Di+Ku=R,
(21)
1062
R. Ruotolo er al.
where K = K. - 6AK,
(22)
and by denoting the changes in the global stiffness matrix due to the crack: AK=K.-Kd,
where eDis a displacement convergence tolerance, so that eqn (24) is balanced. The second convergence criterion is obtained by applying the out-of-balance load vector:
’ +“‘s = M r+Af#l
+
Dl+Ar,$O
,+A’R
_
+
r+A,F(,-I,,
(27)
(23) and by imposing that
with 6=
1 when the crack is open, when the crack is closed.
i0
Since an exact solution of these equations cannot be determined, a numerical method is adopted in order to simulate the dynamic behaviour of the cracked beam, proceeding step-by-step in time. In such a simulation, to determine the state of the crack, i.e. whether opened or closed, it is sufficient to evaluate the slopes ffi, 8,+, of the response deformation at the “control” nodes i and i + 1 closest to the crack (where i is the node closer to the fixed end of the beam). For a crack on the upper side of the beam, the condition of crack closing is then equivalent to L$+, > 0,. 5. NON-LINEAR FINITE ELEMENT ANALYSIS
According to Bathe [13], it is possible to write an incremental form of the non-linear equations of motion for the cracked beam, which can be solved with an implicit time integration scheme and modified Newton iteration: M’ +Artit
+
,,’
+ A$,‘0
+
‘KAu’”
=
f + AtR
_
t + ArFy’-
I)
(24) where: ‘K is the stiffness matrix at time t; ’ +A’Ris the external load vector at time t + At; ‘+A’F(‘-‘) is the nodal point force vector equivalent to the element stresses at time t + At and at iteration step I - 1; ’ +Arii(‘J,’ +Ar~(‘J are the acceleration and velocity vectors of the nodal points, or the structure at time t + At and iteration step I; A& is the increment of the nodal point displacements of the structure at time t + At from iteration step I - 1 to step I: ! + AI~VI =
I + At&
- 1) +
Au’“.
(25)
To be sure that the accuracy of the solution is satisfied at each iteration, two convergence criteria are used. The first one imposes that the increment Au”’ is negligible with respect to ‘II:
where j/*R11is the amplitude, greater than zero, of the forcing term at a given time 5; CR is a specified tolerance on the force. During the course of this research, the interdependence of these parameters and of the sample time on the number of balancing iterations has emerged. In fact, if an insufficient number of balancing iterations is performed, which happens when the tolerance is not sufficiently low, an incorrect solution for eqn (24) will be obtained. Generally a smaller number of balancing iterations is required if the sample time and/or the tolerance values are reduced. If a general non-linear problem is studied, the main computational expense in the solution of eqn (24) lies in the evaluation of ‘K, its factorization and then the equilibrium iteration [15]. In the analysis of some problems it may be more effective not to calculate a new stiffness matrix at each time-step, but instead to use the original matrix corresponding to the initial time throughout the response analysis by formulating the governing finite element equations at time t + At as M’
+ At@
+
D’
+ Al@
+
O~u’O
=
I +
A'R
_
I +
A,F,, - I)
(29) where OK is the stiffness matrix at the initial time. Applying this assumption to the case under study, a constant effective stiffness matrix is formed: thus only one factorization is needed in the calculation of the dynamic response. In the case studied, the vector ‘+ ArF(‘-I) represents the restoring force of the beam, evaluated at the previous balancing iteration step. Due to the nature of the system it is possible to split this term into two parts:
which is the term corresponding beam, and
to the undamaged
1063
Vibrations of a cantilevered beam which takes into account the bilinear nature of the closing crack. Obviously, if a constant stiffness matrix is used, the number of balancing iterations will increase and thus will be computationally efficient only if the computer time required for the balancing of eqn (29) is less than for eqn (24). A good way to reduce the computational effort in solving eqn (24) is to use mode superposition. This technique involves a coordinate transformation from the n finite element displacements to p modal generalized displacements in which usually p < n. Considering the non-linear characteristics of eqn (24) this change of basis should be performed at each time-step, using the modeshapes corresponding to time t. However, such a procedure would require the solution of the following eigenproblem:
'K9 = WM’@,
(32)
(in which IQ2 is the diagonal matrix of the natural angular frequencies of the system at t) at each time t. Due to the use of eqn (29) in which a constant stiffness matrix is used, the eigenproblem is solved only once at the initiation of the non-linear integration procedure. Assuming ,+A’”
=
O@!+AtX
(33)
where r+Ar~ is the vector of the generalized modal displacements at time t + At, a@ is the modeshapes matrix corresponding to the eigenproblem OK”@ = OfJZM”QI
(34)
and OQ*is the diagonal matrix of the natural angular frequencies of the system, initially
L 1 02 WI
0
. . .
0
. 0
. . .:
w2
1.’
. 0;; 0.
0
op =
. (j 0
2
In this study it has been supposed that at time t = 0 the crack is closed; thus it can be assumed that OK= K.. The substitution of eqn (33) into eqn (29) gives
- (/I - aoaT dK”@)‘+A’x”- ‘1, (35)
O@TOKO@
=
OQ2
OcBTDo@ = A,
and I is the identity matrix. Assuming OK= K., the solution of eqn (35) is performed in the modal space spanned by the eigenvectors of the untracked beam. It should be noted that when 6 = 0, i.e. the crack is closed, eqn (35) are uncoupled; thus the solution is found with few balancing iterations, without loss in accuracy. When 6 = 1 (crack open), eqn (35) are coupled; thus the solution of the p equations must be performed simultaneously. In this situation a loss of accuracy can result if the tolerances tn and ERare not sufficiently low. Therefore a greater number of iterations would be required, although this number is usually less than 10 for fR = IO-’ and tD = IO--‘. Figure 4 shows the flow-chart of the MATLAB”. program which implements the non-linear procedure described above.
6. HIGHER ORDER FREQUENCY RESPONSE FUNCTIONS
Recently, investigations into the behaviour of structures with polynomial-type non-linearities have used the concepts of higher order frequency response functions (FRFs) defined from the Volterra series [12]. These can be used to represent the linear and non-linear characteristics of the system providing the system can be considered to exhibit certain properties. It has been demonstrated that higher order FRFs, if measurable, would be a valuable tool when investigating the effect of non-linearities in a structure, such as computing a mathematical model to be used for prediction purposes or as means of detecting the location of sources of non-linearity in a structure. The Volterra series provides the basis for a rigorous definition of higher order FRFs, and is an established methodology for the analysis of nonlinear ditferential equations [ 161. The Volterra series extends the familiar concept of the convolution integrals for linear systems to a series of multi-dimensional convolution integrals necessary instead for polynomial-type non-linearities. For a general polynomial or in situations where the system is to be identified and is not known in advance, it is usual to consider a Volterra series with an infinite number of terms to represent a polynomial of any order. For any linear system, with input r(t) and output u(t), the convolution integral is written
where it is assumed that u(t) = ‘QiTMo@= I,
h(t)
r(t - T) . dz.
(36)
1064
R. Ruotolo et al.
Correspondingly, for a non-linear the Volterra series is written
Volterra system 1=x=x=x=0
( s=o
3L u(t) =
r(t - 71) . dr,
hl(r~) s --01 3c1 cc +
h2(5,,72) SI-co
r(t
-
-
71)
. dz, . dt2
7*)
m
m
a
+
h3(7,,72,73) -cc sss
. r(t
+
. r(t
--3o
-
-m
TV)
m
r(t
-
7,)
---L
r(t
-
dz, . dr2. dz3 +
TV).
.
. co hn(T,,...,T,) s--z
s-cc
.;fir(t-i&d7
I,...,
dr..
Fig. 4. Flow-chart of the program which implements the non-linear finite element procedure.
(37)
In the analysis of linear systems, the relationship between the impulse response function, h(r), and the FRF H(jc~) is well known. An equivalent relationship can be defined for the corresponding Volterra kernels, h(r,, . . . , T,), in the Volterra series by using multi-dimensional Fourier transforms (MFTs). The first term in the series has exactly the same form as the convolution integral, and the first order FRF is realized with the one-dimensional Fourier transform (FT):
h, (7,)
d7,.
e-@‘171
Higher order FRFs can be defined simply by extending this approach using MFTs:
H,(iw,,...,jw,)=
e-~rw,r,+ ..+~nrnl. dr, . . . d7,.
The multi-dimensional higher order FRFs defined in this way provide a very general representation of this class of polynomial non-linear systems and can be used to explain how systems excited by, for example,
(38)
Rosin o
/
/
t ~ 4
/ I
,
I
I
I K-1
I
I
I
I
I
I
l
l
l
l
l
l
l
l
/ _p_
Y
L=700
(39)
mm
Fig. 5. The finite element mesh for the beam analysed.
4
I
si
I
‘b(
1065
Vibrations of a cantilevered beam
a broadband random signal tend to distribute energy between frequencies in a way that reflects the type of non-linearity present in the system. However, visualization of these general FRFs are difficult for third order and above, and measurement of these FRFs is difficult for random input. On the other hand the simplifications afforded by considering a simple single harmonic input are significant. Only the leading diagonals of the higher order FRFs need to be considered where a, = a2 =. . = (I&, and since these functions are one-dimensional in frequency they can be visualized easily. This development leads to a simple result for systems in this class excited by a single harmonic r(t) = R . dw’. A subset of single-dimensional FRFs, which nevertheless characterize the non-linear behaviour of the system, may be defined using only the one-dimensional FTs of the input r(t) and the output u(t) to formulate the following:
&(jw) =#,
H2( jo,jo)=
H.(jo,
2A::$),
. . , jw) = ‘i:!).
(40)
The term U( jw) is the fundamental output term, and U( j2w), U( j3w), . . . , U( jnw) are the higher harmonic terms in the spectrum of the output, which can be written as u(t)=
H,(jw).Re““‘+
H~(jw,jo).R’e+
+...+H,(jw,...,jw).R”e”“‘+... In practice, it is difficult to measure higher order FRFs of a system directly [12]. Several techniques have been developed for measuring higher order transfer functions (TFs) of a system which can be related to the ideal higher order FRFs [17]. The most fundamental technique uses a single sinewave input. This straightforward approach for measuring the TFs can be implemented in a practical testing procedure and the relationship between TF and FRF can be explained and interpreted. Table 1. Case studied vs crack size and position Case number
Size (mm)
Position (mm)
1 2 3
4 8 8
150 150 250
Considering the output time signals in terms of the displacement u(t) response to a sinusoidal forcing r(t), the tirst and higher order TFs determined using the single, dimensional FT of the time signals can be written as
TF2(
jm)
=
2 UWw) R(jw)*
TF,(
ja)
=
4. WW R(jw))
TF.( jo) =
’
’
2” - ’ U( jnw)
R(jw)”
.
(41)
The term U(iw) is the fundamental output term at the input frequency o, and U(j2w), U(j3w), . . . , U(jno) are-the higher harmonic terms in the spectrum of the output. Each of the terms in the spectra are complex quantities, and the TFs convey both gain and phase information regarding the transfer of energy between frequencies. A stepped-sine test is a convenient way to measure these TFs both in simulations and in practical testing. Comparison of eqns (40) and (41) reveals the close relationship between the Volterra series defined FRF diagonals, which are unique for the system, and the higher order TFs from a stepped-sine test. However, an important difference does arise in the fact that the TFs are determined by inputting a sinusoid to the system r(t) = R cos(wt) = R/2(P’ + e-h’) rather than an ideal harmonic r(t) = R fl’. The two harmonic terms present in the sinewave can interact in a non-linear system and give rise to “degenerative” effects of higher order terms which influence the measurement of lower order TFs. In fact these effects are thought to cause the classical distortion phenomenon observed on transfer functions measured during stepped-sine tests on non-linear structures [ 171. 7. RESULTS OF NUMERICAL SIMULATIONS
In order to investigate the potential of using harmonic distortion and higher order TFs as a means of assessing damage in structures, a simulation of a cantilevered steel beam with length of 0.7 m, and cross-section of 20 x 20 mm’ was performed (Fig. 5). For the beam material, a Young’s modulus of 2.06 x 10” N m-’ and a density of 7.85 kg m-3 was assumed, and the response at the free-end of the beam subjected to a transverse, harmonic force &, sin(wt) was calculated. The higher order TFs were determined by using the non-linear model described in a previous section to simulate the time domain response of the cracked beam. The frequency of the harmonic exciting force
R. Ruotolo et al.
1066
o was varied in the range from 0.2 to 1.4 w0 (where w. is the first natural frequency of the linearized system) in order to simulate a stepped-sine test. When determining by simulation (or measurement) the higher order TFs using eqn (41), it is necessary to first attain the steady-state response; due to the light (a) x 1 o-3
1
damping present in a structure like a steel beam, the transient has a long duration which is very time consuming to simulate. Each integration for a given value of the forcing term takes about 1 h on a Pentium 90 computer employing a routine written in MATLAB” to implement the non-linear finite element procedure previously described. Case
I
I
n.1
I
,
I
I
I
.._
_; ..;_._.._.~._..._..._.._.:
(b) 1.5
1
Case
. 1 o-3 I
I
.,.....,......
.,..
Y
.-.-
+___
I
..
n.2
I
i i
1.18
1.16
I
I
..I..........
.-
0.5
I I
I
I 1.14
1.12
1 .l
1.08
1.06
I
I
I
I
-1’
.‘.
““”
:
I’-.-
:
;.....
i i z E
a
-..-..
___
.._ - _-...
-. .-.-
~~
I$
-0.5
:.
-1
-1.= 1 7 88
IF A...
I
0.9
I
0.92
I
I
0.94
0.96
I
0.98
:
I
1
lsecl
Fig. 6. Time histories of the free-end response obtained with the non-linear finite element procedure. (Continued opposite.)
1067
Vibrations of a cantilevered beam Case
-1
I 0.92
I 0.9
n.3
I 0.96
I 0.94
b
I
I 0.98
1
I
!
I
I
1.02
1 set 1 Fig. 6-Continued. Performing a simulation of a stepped-sine test enables both the so called response envelope function (REF) and the higher order TFs of the system to be determined: (42) and the ith order TF as TF,(jo)
= “‘;;($).
(43)
Table 1 shows the three cases which have been analysed. In Fig. 6, the time histories of the response at the free-end corresponding to the cases 1, 2 and 3 are shown. In this example, a forcing frequency equal to foe is assumed in order to demonstrate the non-linear effect due to the closing crack. In the diagrams the continuous line represents the free-end response while the dotted one corresponds to the opening and closing process of the crack; from these data it is clear that the state of the crack is a function of the response of the beam as opposed to the excitation. In case 1 the non-linearity of the beam is not particularly evident; thus the linear part of the response remains dominant. In case 2 the non-linear behaviour is so distinct that the second order becomes very well defined. Furthermore, in this latter case the crack changes state twice per cycle. Figures 7 and 8 illustrate comparisons between the REFs and the time histories obtained with the
non-linear finite element procedure described in this paper and that obtained with the Krawczuk and Ostachowicz’s model. Figure 7 shows the point-to-point REFs for cases 2 and 3 at the free-end of the beam; the continuous line represents the results obtained with the model described herein while the dotted one represents the results obtained by Krawczuk and Ostachowicz. In both cases it is possible to see some discrepances between the results. Figure 8 illustrates the time histories for one period of the exciting harmonic force determined with the same two models. In case 2 the two time histories are similar, while in case 3 the results obtained by Krawczuk and Ostachowicz are incorrect due to the relatively high non-linearity of the beam. Furthermore, in case 2 it is evident that Krawczuk and Ostachowicz’s results also represent a correct double opening-closing process of the crack in one period of the load which is not considered by the square-wave function f(t). In general, the higher order TFs give an accurate and extremely sensitive indicator of non-linear behaviour of a system. Thus the evolution of TFs for varying the crack size of location may be very useful for structural damage assessment. Figure 9 shows the first to the fourth direct higher order TFs of the beam at the free-end; the continuous line refers to case 1, while the dotted to case 2. As would be expected, the first-order TF does not change with the increase in the depth of the crack, while the second and the fourth exhibit clear dependence on the size of the crack. Indeed the second harmonic of
1068
R. Ruotolo et W
10”: . .
.I:.
!...
................
al.
Case n.2 .“’ ‘:’I ._ .._ ,,...
1
.
I
..~
.
_
...
.
........... .:
....
............
.:
I...‘..’
.:
f ...
......
.
.....
_
1.
1o-5
5
. . ..i.....-
.
.......i......
.
I
I
1
I
,
I
I
10
15
20
25
30
35
40
45
[Hz1
(b)
Case n.3
.........
t :‘:::.:::::::..:..
::.:.: ,::::::.::I:.::::.:,
1
.....
...............
10+t
5
.....
....
.
I
I
1
10
15
20
:: :::.:.,:::
: ::: .:::
.... ............
........
.......................
I
$1
.... .......
f
I
I
30
35
40
I 45
Fig. 7. Comparison between non-linear finite element (-) and Krawczuk and Ostachowicz’s response envelope functions (-.-)---cases 2 and 3. the response
increases
by a factor
of approximately
five between cases 1 and 2, and the fourth harmonic increases by a factor of three. Figure 10 shows the first to the fourth direct higher
at the free-end. Due to the size of the crack, the extent of the non-linearity is greater in case 3 than in case 1, as demonstrated by comparing Fig. 9 with Fig. 10. order TFs of the beam measured
1069
Vibrations of a cantilevered beam
(4 1.5
Case n.2
x 1om3
I
1
-1.5
o;9
I
8
I
I
I
I
I
I
I
0.91
0.92
0.93
0.94
0.95
0.96
[=I
(b) x 1 o-3 ‘rI
Case n.3
I
I
I
I
I
0.94
0.95
0.96
0.97
[=I Fig. 8. Comparison between non-linear finite element (-) and Krawczuk and Ostachowicz’s results (- . -)--cases 2 and 3.
1070
R. Ruotolo et al.
(a) 1 st Order TF
I
I
I
I
2-
I
1 .5E E lu l-
0 .5-
O5
I
I
I
I
I
I
I
10
15
20
25
30
35
40
45
I Hz I
(b) 1.2
x 1o-5
2nd Higher Order TF
I
r
I
I
I
I
L
.
‘i’ i! t’ I!
‘r’ ‘. ii ‘: I ;: , ii.!.
.
_..~ ..,
,
i. I I
J -5
10
15
20
Fig. 9. Higher order TFs for cases 1 (-)
30
35
and 2 (- . -). (Conrimed
40 opposite.)
45
Vibrations of a cantilevered beam
1071
(c) 1.6
x 10.’
I
I
3rd Higher Order TF I
I
,
I
I
I
.................
..
.................
:.
Ii
:.
I’‘; ’
)
.........
.
.... .
..... _
.........
.._
I
. . ... . . . . . . . . . .
I
,
.............
.
.........
.
......
.
I I .:
........
._
I I
i
:
i!
I
il...
.I ,.....
.,..
.“’ t.
j ,,..
.;.lp+fi
_)
.........
. ......
....
-
:,:!
I I
!I’.
,
..I...
\ “’ ,I!’ : ~~.~ \.I*
! .,h. ii, __!I. ~,_~:;~~,~~~~~~,.~
. . .. . ..
‘\
:
I’<.
. . .. ..
1
. . . . ..
i,
:
15
1
.;.I
30
20
35
40
45
[El
(4 4th Higher Order TF
o x 1o-B
I
“I
i 6-
i...
.i
~
1
. . . ..
i 7-
j...
.._...
.__:
.._.
II i
E 5
6-
ii..
5-
ii ii ii.’
4-..
.,,....
..,.
: .;
j ...,,
ii : .;.i... ;
..,.i
iI 3__
.,,.)..,
I
,...,........;....
I
Ii ‘,.;I ii : ‘. , _ .;,.I, !;. .I..
2-
0 5
. . . . .. . . ....I. ‘-.-.-.
10
: 15
._,
._,_’ 20
30 $1
Fig. 9-Continued.
35
40
45
1072
et al.
R. Ruotolo (4 x 1o-3 2.51
I
I
I
1 st Order TF I
I
1
..,
I
,.
.:.
..:..
,.
0
5
L
I
I
I
10
15
20
25
30
35
40
I
I
[Hz1
lb) 1.2
x 1o-5
2nd Higher Order TF
I
I
I
,
I
i
, t
I :...
0.8
0,4-.
...*
I
t
Erp_.
.,
........
. .. . . . . . . . .
.;.
.I
.
... ..... f.
0.2_ ..........
Fig. 10. Higher
order
TFs for cases 2 (-)
and 3 (-
-). (Continued opposite.)
45
Vibrations of a cantilevered beam
1073
(c) 3rd Higher Order TF
x 10”
I
1.6r
I
I
I
,
.:
.,...
.
.,
‘:““‘.
“’
I
4:
[Hz1
(d) 4th Higher Order TF
x 1o’8 91
I
I
I
I
I
I
1
.
I
8--.
. . ...
7-..
..
..I
.
...)
6-
.
5--.
.
:.
B 5
.:
.._
._
..I
4-,-
\
!-_L_; !;
3-2-
.._......,,.....:.
.,....,;
‘,
;
,!.I... j ,. \
’
.I
0’
5
..,
;
_
A;
i
‘. : FL._.
10
%.;,,i\,
15
20
30 $1
Fig.
i
IO-Continued.
;
e:
I
-._.
35
:
.-
40
45
1074
R. Ruotolo et al. 8. CONCLUSIONS
AND FURTHER WORK
This paper presents an analysis of bending harmonic forced vibrations of a cantilevered beam with a transverse one-edge non-propagating closing crack. To simulate the harmonic forced vibrations of the beam, a non-linear finite element procedure has been developed. The comparison of the numerical results with that obtained by Krawczuk and Ostachowicz in a recent paper has highlighted some differences. These are due to some inaccuracies of the Krawczuk and Ostachowicz’s model and in this article, a correction for the model is indicated. The use of higher order FRFs has been suggested in order to study the non-linearity due to the closing crack. The analysis of such functions shows that they are highly dependent upon the size and the position of the crack, and thus can be used as damage indicators. Furthermore, correlating the shape and the value of these functions to the crack size and position, it may be possible to form the basis for an experimental structural damage-identification procedure. The numerical procedure described in this article is similar to the experimental approach in which a stepped-sine test is performed on a cracked beam. To validate the use of higher order FRFs as a damage-identification procedure, an experimental program has already been initiated in which a set of cracked beams with one-edge closing crack are being tested. The conclusion of these tests will be the subject of future articles into this topic. Acknowledgements-The authors would like to thank their research supervisors past and present. in particular Prof. Albert0 Carpinteri, Prof. Giuseppe Surace of the Polite&co di Torino, Italy and Prof. Geof Tomlinson of the University of Manchester, England. The research was financed by Consiglio Nazionale delle Ricerche (CNR) funds.
REFERENCES I.
P. Gudmundson, The dynamic behaviour of slender structures with cross-sectional cracks. J. Mech. Phys. Solids 31, 329-345 (1983).
2. B. Zastrau, Vibrations of cracked structures. Arch. Mech. 37(6), 731-743 (1985). G. L. Qian, S. N. Gu and J. S. Jiang, The dynamic
behaviour and crack detection of a beam with a crack. J. Sound Vibr. 138(2). . ,. 233-243 (1990). 4. A. Ibrahim, F. Ismail and H. RI Martin, Modeling of
dynamics of continuous beam including nonlinear fatigue crack. fnr. J. expl Modal Anal. 2(2), 7682 (1987).
F. Ismail, A. Ibrahim and H. R. Martin, Identification of fatigue cracks from vibration testing. J. Sound Vibr. 140(2), 305-317 (1990). K. R. Collins, P. H. Plaut and P. H. Wauer, Free and
forced longitudinal vibrations of a cantilevered bar with a crack. J. Vib. Acoust. 114, 171-177 (1992). M. I. Friswell and J. E. T. Penny, A simple nonlinear modal of a cracked beam. In: 10th Int. Modal Analysis Conf., San Diego, Calif., pp. 516-521 (1992). M.-H. H. Shen and Y. C. Chu, Vibrations of beams with a fatigue crack. Comput. Struct. 45(l), 79-93 (1992). Y. C. Chu and M.-H. H. Shen, Analysis of forced bilinear oscillators and the application of cracked beam dynamics. AIAA J. 30(10), 2512-2519 (1992). 10. W. Ostachowicz and M. Krawczuk, Vibration analysis of a cracked beam. Comput. Struct. 36,245-250 (1990). 11. M. Krawczuk and W. Ostachowicz, Forced vibrations of a cantilever Timoshenko beam with a closing crack. ZSMA 19. 1067-1078 (1994).
12. S. J. Gifford and G. R: Tomlinson, Recent advances in the application of functional series to non-linear structures, J. Sound Vibr. 135. 289-317 (1989). 13. K. J. Bathe, Finite Element Procedure in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ (1982). 14. G. Gounaris and A. D. Dimarogonas, A finite element of a cracked prismatic beam for a structural analysis. Comput. Struct. 28(3), 3099313 (1988).
15. K. J. Bathe and S. Gracewski, On nonlinear dynamic analysis using substructuring and mode superposition. Conrput. Struct. 13, 699-707 (1981). 16. M. Schetzen, The Volterra and Wiener Theories of Non-Linear Svstems. Wiley. New York (1980). 17. D. M. Storer,‘Dynamic analysis of non-linear structures using higher order frequency response functions. Ph.D. thesis, University of Manchester (1991). 18. M. Krawczuk, Coupled longitudinal and bending forced vibration of Timoshenko cantilever beam with a closing crack. J. theoret. appl. Mech. 2(32), 463482 (1994). 19. D. M. Storer and G. R. Tomlinson,
Recent developments in the measurement and interpretation of higher order transfer functions from non-linear structures. Mech. Systems Signal Process. 7(2), 173-189 (1993).
Compuars & Swucrures Vol. 61. No. 6. pp. 1057-1074. 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 004%7949196 115.00 + 0.00
HARMONIC ANALYSIS OF THE VIBRATIONS OF A CANTILEVERED BEAM WITH A CLOSING CRACK R. Ruotolo,t C. Surace,? P. Crespu$ and D. Storer$ Departments of jStructural Engineering and SAeronautical and Space Engineering, Politecnico di Torino, Torino, Italy (Received 22 June 1995)
Abstract-In this article, the vibrational response of a cracked cantilevered beam to harmonic forcing is analysed. The study has been performed using a finite element model of the beam, in which a so-called closing crack model, fully open or fully closed, is used to represent the damaged element. Undamaged parts of the beam are modelled by Euler-type finite elements with two nodes and 2 d.f. (transverse displacement and rotation) at each node. Recently the harmonic balance method has been employed by other researchers to solve the resulting non-linear equations of motion. Instead, in this study, the analysis has been extended to employ the first and higher order harmonics of the response to a harmonic forcing in order to characterize the non-linear behaviour of the cracked beam. Correlating the higher order harmonics of the response with the forcing term the so-called higher order frequency response function (FRFs), defined from the Volterra series representation of the dynamics of non-linear systems, can be determined by using the finite element model to simulate the time domain response of the cracked beam. Ultimately the aim will be to employ such a series of FRFs, an estimate of which in practice could be measured in a stepped sine test on the beam to indicate both the location and depth of the crack, thus forming the basis of an experimental structural damage identification procedure. Copyright 0 1996 Elsevier Science Ltd
1. INTRODUCTION Vibration-based inspection of structural behaviour offers an effective tool of non-destructive testing. The analysis of the dynamic response of a structure to excitation forces and the monitoring of alterations which may occur during its lifetime may be employed as a global integrity-assessment technique to detect, for example, play in the joints or the presence of a crack. Indeed it is well known that, in the case of simple structures, crack position and depth can be determined from changes in natural frequencies, modes of vibration or the amplitude of the forced response. In the past many studies have illustrated that a crack in a structure such as a beam may cause the structure to exhibit non-linear behaviour if the crack is open during part of the response and closed in the remaining intervals. This phenomenon has also been detected during experimental testing performed by Gudmundson in which the influence of a transverse closing crack upon the natural frequencies of a cantilevered beam was investigated [ 11. The main result obtained through experimental studies such as these was that the observed decrease in the natural frequencies of the beam due to the presence of the crack is not sufficient to be described by a model of crack which is always open. Therefore it must be concluded that the crack alternately opens and closes, thus giving rise to natural frequencies
falling between those corresponding to the alwaysopen and always-closed (e.g. integral) cases. In fact, if an always-open crack is assumed in the analysis of a beam with a so-called closing crack, i.e. one which both opens and closes during the time interval considered, the reduced decrease in the experimental natural frequencies will lead to an underestimate of the crack depth when determined via a test-model correlation approach. Many researchers have studied the problem of a beam with a closing crack starting from an analytical viewpoint. After demonstrating the bilinear nature of a beam with a closing crack, Zastrau used a finite element method to study the time history and the spectrum of a simply-supported beam with multiple cracks [2]. Qian et al. observed that differences between amplitudes of forced vibrations of cracked and non-cracked beams are reduced when a closing crack model is considered. Furthermore, time series analysis has been applied to identify the modal parameters of cracked beams [3]. Ibrahim et al. applied the bondgraph technique (a numerical tool using lumped parameter elements) to examine the FRFs of a cantilevered beam with a crack [4] and described experimental testing performed to validate their technique [S]. Collins et al. utilized direct numerical integration through the Galerkin method to investigate longitudinal free and forced vibrations of a cracked prismatic bar [6].
1057
1058
R. Ruotolo et al.
Friswell and Penny have simulated the non-linear behaviour of a beam with a closing crack vibrating in its first mode of vibration through a simple 1 d.f. model with bilinear stiffness [7]. The analysis of the FRFs and the response to harmonic loading, obtained by numerical integration, demonstrates the occurrence of harmonics in the response spectra which are integer multiples of the exciting frequency. Shen and Chu simulated, with a bilinear equation of motion for each mode of vibration, the dynamic response of simply-supported beams with a closing crack, analysing the response spectra in order to detect changes which are potentially useful for damage assessment [8]. Later the same authors, in order to reduce the computational effort, used a closed-form solution based on the use of two square-wave functions to model stiffness change for bilinear oscillators, and simulate the cracked beam under low frequency excitation [9]. Krawczuk and Ostachowicz studied the transverse vibrations of a beam with a closing crack simulated through springs with periodically time-variant stiffness [lo]. In a recent paper these authors presented an analysis of the forced vibrations of a cantilevered beam with a closing crack, in which the equations of motion were solved using the harmonic balance technique [ 1I]. The periodically time-variant beam stiffness is represented with a square-wave function with a fundamental frequency equal to the forcing term frequency. This research has shown that when a closing crack is present in a beam, higher harmonic components in the frequency spectrum of the response are generated when excited by a sinusoidal forcing function, indicating that the structure behaves non-linearly. This in itself would permit one to distinguish between the types of crack, i.e. whether closing or always open. However, the aim of the research reported in this article has been to extend the analysis of the harmonic components of the response spectrum using the so-called higher order FRFs which can be defined using the Volterra series for systems in which the non-linearity can be represented by a polynomial-type function [ 121. If measurable in practice, certain features of the higher order FRFs, for example their relatively sensitive dependence on the type and form of non-linearity present, would provide several distinct advantages over other methods for detecting faults in structures, assuming that the faults cause the structure to behave non-linearly, i.e. not as in the case of a beam with a crack which is open throughout the response cycles. Since the FRFs are related to the coefficients of the polynomial function in the governing equations of motion, it should be possible to correlate the amplitude of the peaks in the higher order FRFs with the crack depth and/or position. The intention would be to develop a damage assessment procedure based on a quantitive comparison of the higher order FRFs measured from the structure under study.
As described in this paper, in order to examine the feasibility of such an approach, a series of numerical simulations of a cracked beam has been performed, with “measured” estimates of the FRFs being used in an attempt to distinguish between cracks of varying depths and location along the length of the beam. In order to model accurately the non-linear behaviour of a beam with a closing crack, it is critical to determine the precise moment that the beam changes state, i.e. when the crack opens or closes. In the results presented, it has been assumed that the change between fully opened and fully closed takes place instantaneously giving rise to a bilinear-type stiffness non-linearity (which can be approximated by a high-order polynomial). In the numerical simulation, the change of state is imposed in terms of the beam curvature at the cracked section: the crack is assumed to be open if the curvature is in the positive sense, otherwise it is closed. In this way there is no time-dependent control function for the stiffness of the beam. Using a specific finite element for the section with the crack, it is possible to simulate the free and forced bending vibrations of a cracked cantilevered beam by employing non-linear finite element analysis [ 131. The aim of the paper is to present a harmonic analysis of a cantilevered beam with a closing crack, and to analyse the response of the beam to sinusoidal loads at various forcing frequencies in relation to diverse crack depths and positions. A similar approach is often adopted during experimental “stepped-sine” tests on structures and, as described subsequently, this procedure is particularly appropriate for estimating the higher order FRFs in practice.
2. MATHEMATICAL 2.1. St@ess
MODEL OF THE CRACKED REAM
matrix
The mathematical model used for the Euler beam with a transverse on-edge non-propagating closing crack is based on the finite element model proposed in Refs [3] and [14]. According to the principle of St Venants the stress field is influenced only in the region adjacent to the crack. The element stiffness matrix, with the exception of the terms which represent the cracked element, may be regarded as unchanged under a certain limitation of the element size. The additional stress energy of a crack leads to a flexibility coefficient expressed by a stress intensity factor derived by means of the Castigliano’s theorem in the linear elastic range. The general approach to the problem is that, for the cracked element in the beam, the elements situated on one side can be considered as external forces applied to the cracked element, while the elements on the other side can be regarded as constraints. In this way the flexibility matrix can be
1059
Vibrations of a cantilevered beam calculated easily and therefore, from the conditions of equilibrium, the stiffness matrix of the cracked element can be derived. In this study the element employed is shown in Fig. 1. Neglecting shear action, the strain energy of an element without a crack can be formulated as
and F,(s) = ,/(2/ns)tan(xs/2)[0.923 + 0.199( 1 - sin(rrs/2)“)]/cos(7rs/2), Fir(s) = (3s - 2?)(1.122 - 0.561s
W”’ = AI
+ 0.085~~ + 0.18s’)/J1-s,
’ (M + Pz)’ dz s0
= &I (Ml + P2P/3 + MPP),
(1)
I being the length of the element, and the additional energy due to the crack is
(5)
where s = a/h is the relative depth of the crack, h being thickness of the beam. The term c$” of the flexibility matrix C:) for an element without crack can be written as (0) _
C/k-
a2~o) -=& a~, ap,
i,k=l,2
PI=P,
Pz=M.
(6) u”” = b
‘[(K: s0
+ Kf,)/E’ + (A + v)K&/Ej
da,
(2)
where E’ = E for plane stress, E’ = E/(1 + v) for plane strain, E is the elastic modulus, v is the Poisson ratio and K,, Kn , Ktn are stress intensity factors for opening-type, sliding-type and tearing-type cracks respectively, a is the depth of the crack and 6 the width of the beam. Taking into account only bending:
The term c$’ of the additional flexibility matrix Ci” due to the crack can be formulated: 8W”
&’ = -ap, a~, = CA!? i, k = I,2
p, = p,
pz = M.
(7) The term cilrof the total flexibility matrix for the damaged element C, is C,k= cl:) + c$’ .
t+‘(” = b ’ ([(Km + KIP)*+ K&]/E’] da,
(3)
From the equilibrium (f'i
with
Mi
(8)
condition (Fig. 1): M,+I)~=T(P,+,
f's+,
M,+I)‘,
(9)
where K,M = (6M/bh’)&F,(s), K,p = (3Pl/bh2),/&F,
(s),
T=
%IP = (Plbh),/&,(s),
(4)
-1
0
-1
-1
1
0
0
1
,
5+
‘i ui
1a
ui+ ei+l
ct
1 It I
I
/
M
Mi
Y
1 Fig. 1. Schematic diagram of an element.
x
i+l
I.
(10)
1060
R. Ruotolo et al.
Using the principle of the virtual work, the stiffness matrix of the undamaged element can be written: K c = TC””c - ‘F v
r 1261 4P61 K
e = gP
(11)
-12
61
-61
2P
-12
-61
12
-61
61
2P
-61
4P
1
1 (12)
’ 1
while the stiffness matrix of the cracked element may be derived as IG, = TC;‘TT.
(13)
techniques capable of arriving at the steady-state solution of the response whilst avoiding direct integration. Shen and Chu used a bilinear oscillator to simulate each mode of the beam with a closing crack. In Ref. [9] a closed-form solution based on the use of two square-wave functions to model stiffness change was developed. However, this solution is valid only for low frequency excitation up to a ratio of 5-6 between the first natural frequency of the linearized system, wo, and the frequency of the forcing term. Also, Krawczuk and Ostachowicz [ 1l] have used a square-wave functionf(t) to model the beam stiffness, transforming the non-linear equations of motion into a linear periodically time-variant equation: Mii + Di + (K. - AKf(t))u = R.
(17)
2.2. Mass matrix In order to evaluate the dynamic response of the cracked beam when acted upon by an applied force, it is supposed that the crack does not affect the mass matrix M. Therefore, for a single element, the mass matrix can be formulated directly:
r 156
221
-131
-311
54 131 156 -221
- 1321 -3P -221 3 (14) 4P
where m is the mass per unity length of the beam. 2.3. Damping matrix Assuming that the damping matrix D is not affected by the crack, it can be calculated through the inversion of the modeshape matrix relative to the undamaged structure:
In order to apply the harmonic balance method the closing crack function f(t) in Fourier series was expressed as 1 2 2 f(t) = - + - cos(wt) - - cos(3wt) 2 n 3n 2 + j-& cos(5wt) + . . , (18) where w represents the angular frequency of the harmonic input. As shown in Fig. 2, this is the Fourier series approximation to a square-wave function. The solution of eqn (17) was obtained by assuming that 0 R=
D = (dp=)’ d@-‘,
?
cos(ot),
(19)
i.1R0 and that the degrees of freedom of the structure can be expressed as
where
1
d=
9
(16)
in which 6 is the ith modal damping ratio, co, is the ith natural frequency and M, is the ith modal mass relative to the undamaged beam. According to Ref. [ 111, the modal damping ratio ii can be assumed to be approximately equal to 0.01. 3. THE HARMONIC
BALANCE APPROACH
Numerical integration of non-linear equations of motion is usually very computer-time consuming. For this reason some researchers have studied
u = : (ai sin(W) + bi cos(iot)), i=,
(20)
where a, and b, are vectors of the constant variables. This approach seems to be extremely efficient both in terms of computer time and analytically. Furthermore, this approach enables the higher harmonics of the response to be obtained quickly for any excitation frequency. However, a comparison of time history of the steady-state solution of the forced equation of motion obtained through direct numerical integration, as described in Section 5, and the time response obtained from eqn (17) demonstrates that the model described in Ref. [l l] is not valid.
Vibrations of a cantilevered beam
1
0.8
,‘\
I”.
\
1061
:
I
,‘I
‘./’
-. : ‘\
*
’ I
I
;
i
I ; .;.. i j
i I
0.6 ,-
. . ... .
I.
.......
...“‘i.. : i
-5
!....
I
I ..:
‘7
i;
:
f 0.4I-
..
..
:.
: 0.2
,_
al-
I1. .:. i I i I ;f. .
_.
3 _:_
0
0.2
i !i. :I
I
!
./.,;
‘\
i
i
.;
::
ii ,-0.2
,i
.:.
/‘\
/
.:.
,,,,
i
.A
e., i
1:: ;I\.\
‘$ ‘,
.. I :\.
0.4
:‘\/’
0.6
”
0.8
: .,!
1
1
1.2
[=I Fig. 2. The square-wave function and its approximation.
In particular, it is evident that the assumption of a square-wave type stiffness variation (fundamental frequency equal to the forcing frequency) is incorrect. This is especially noticeable when the forcing frequency is equal to 50~. Indeed, in this case thef(t) function imposes that the crack changes its state (open to closed and vice versa) only once per cycle. Instead the Krawczuk and Ostachowicz’s solution of the time response, shown in Fig. 8, is inconsistent with this statement. This is due to the fact that the functionf(t) modelling the stiffness does not consider either the out-of-phase relation between forcing term and structural response, or the fact that the non-linear nature of the system may cause appreci-
A
Compressive
able distortion in the response waveform due to the higher harmonic components which in turn influence the activation of the crack. This point is very important because, when the forcing frequency is close to $0, fo0, &00, etc., significant superharmonic content in the response spectra is manifest. For example, if the forcing frequency is equal to awO, the response contains harmonic components at ioO, w. and so on, while the square wave function developed as in eqn (18) does not contain the harmonic term relative to w. . Instead this aspect has been taken into account by Shen and Chu in Ref. [9], limiting the validity of the closed-form solution to low frequency excitation for which the superharmonic effect can be neglected.
half cycle (K=K “) 4. EQUATION
/ / / /
OF MOTION
When the crack closes and its interfaces are completely in contact with each other, the dynamic response can be determined directly as that of the untracked beam. However, when the crack opens the stiffness matrix of the cracked element should be introduced in replacement at the appropriate rows and columns of the general stiffness matrix. Under the action of the excitation force R, alternate crack opening and closing causes the equations of motion of the cracked beam to be non-linear: Fig. 3. Crack opening-closing process.
Mii+Di+Ku=R,
(21)
1062
R. Ruotolo er al.
where K = K. - 6AK,
(22)
and by denoting the changes in the global stiffness matrix due to the crack: AK=K.-Kd,
where eDis a displacement convergence tolerance, so that eqn (24) is balanced. The second convergence criterion is obtained by applying the out-of-balance load vector:
’ +“‘s = M r+Af#l
+
Dl+Ar,$O
,+A’R
_
+
r+A,F(,-I,,
(27)
(23) and by imposing that
with 6=
1 when the crack is open, when the crack is closed.
i0
Since an exact solution of these equations cannot be determined, a numerical method is adopted in order to simulate the dynamic behaviour of the cracked beam, proceeding step-by-step in time. In such a simulation, to determine the state of the crack, i.e. whether opened or closed, it is sufficient to evaluate the slopes ffi, 8,+, of the response deformation at the “control” nodes i and i + 1 closest to the crack (where i is the node closer to the fixed end of the beam). For a crack on the upper side of the beam, the condition of crack closing is then equivalent to L$+, > 0,. 5. NON-LINEAR FINITE ELEMENT ANALYSIS
According to Bathe [13], it is possible to write an incremental form of the non-linear equations of motion for the cracked beam, which can be solved with an implicit time integration scheme and modified Newton iteration: M’ +Artit
+
,,’
+ A$,‘0
+
‘KAu’”
=
f + AtR
_
t + ArFy’-
I)
(24) where: ‘K is the stiffness matrix at time t; ’ +A’Ris the external load vector at time t + At; ‘+A’F(‘-‘) is the nodal point force vector equivalent to the element stresses at time t + At and at iteration step I - 1; ’ +Arii(‘J,’ +Ar~(‘J are the acceleration and velocity vectors of the nodal points, or the structure at time t + At and iteration step I; A& is the increment of the nodal point displacements of the structure at time t + At from iteration step I - 1 to step I: ! + AI~VI =
I + At&
- 1) +
Au’“.
(25)
To be sure that the accuracy of the solution is satisfied at each iteration, two convergence criteria are used. The first one imposes that the increment Au”’ is negligible with respect to ‘II:
where j/*R11is the amplitude, greater than zero, of the forcing term at a given time 5; CR is a specified tolerance on the force. During the course of this research, the interdependence of these parameters and of the sample time on the number of balancing iterations has emerged. In fact, if an insufficient number of balancing iterations is performed, which happens when the tolerance is not sufficiently low, an incorrect solution for eqn (24) will be obtained. Generally a smaller number of balancing iterations is required if the sample time and/or the tolerance values are reduced. If a general non-linear problem is studied, the main computational expense in the solution of eqn (24) lies in the evaluation of ‘K, its factorization and then the equilibrium iteration [15]. In the analysis of some problems it may be more effective not to calculate a new stiffness matrix at each time-step, but instead to use the original matrix corresponding to the initial time throughout the response analysis by formulating the governing finite element equations at time t + At as M’
+ At@
+
D’
+ Al@
+
O~u’O
=
I +
A'R
_
I +
A,F,, - I)
(29) where OK is the stiffness matrix at the initial time. Applying this assumption to the case under study, a constant effective stiffness matrix is formed: thus only one factorization is needed in the calculation of the dynamic response. In the case studied, the vector ‘+ ArF(‘-I) represents the restoring force of the beam, evaluated at the previous balancing iteration step. Due to the nature of the system it is possible to split this term into two parts:
which is the term corresponding beam, and
to the undamaged
1063
Vibrations of a cantilevered beam which takes into account the bilinear nature of the closing crack. Obviously, if a constant stiffness matrix is used, the number of balancing iterations will increase and thus will be computationally efficient only if the computer time required for the balancing of eqn (29) is less than for eqn (24). A good way to reduce the computational effort in solving eqn (24) is to use mode superposition. This technique involves a coordinate transformation from the n finite element displacements to p modal generalized displacements in which usually p < n. Considering the non-linear characteristics of eqn (24) this change of basis should be performed at each time-step, using the modeshapes corresponding to time t. However, such a procedure would require the solution of the following eigenproblem:
'K9 = WM’@,
(32)
(in which IQ2 is the diagonal matrix of the natural angular frequencies of the system at t) at each time t. Due to the use of eqn (29) in which a constant stiffness matrix is used, the eigenproblem is solved only once at the initiation of the non-linear integration procedure. Assuming ,+A’”
=
O@!+AtX
(33)
where r+Ar~ is the vector of the generalized modal displacements at time t + At, a@ is the modeshapes matrix corresponding to the eigenproblem OK”@ = OfJZM”QI
(34)
and OQ*is the diagonal matrix of the natural angular frequencies of the system, initially
L 1 02 WI
0
. . .
0
. 0
. . .:
w2
1.’
. 0;; 0.
0
op =
. (j 0
2
In this study it has been supposed that at time t = 0 the crack is closed; thus it can be assumed that OK= K.. The substitution of eqn (33) into eqn (29) gives
- (/I - aoaT dK”@)‘+A’x”- ‘1, (35)
O@TOKO@
=
OQ2
OcBTDo@ = A,
and I is the identity matrix. Assuming OK= K., the solution of eqn (35) is performed in the modal space spanned by the eigenvectors of the untracked beam. It should be noted that when 6 = 0, i.e. the crack is closed, eqn (35) are uncoupled; thus the solution is found with few balancing iterations, without loss in accuracy. When 6 = 1 (crack open), eqn (35) are coupled; thus the solution of the p equations must be performed simultaneously. In this situation a loss of accuracy can result if the tolerances tn and ERare not sufficiently low. Therefore a greater number of iterations would be required, although this number is usually less than 10 for fR = IO-’ and tD = IO--‘. Figure 4 shows the flow-chart of the MATLAB”. program which implements the non-linear procedure described above.
6. HIGHER ORDER FREQUENCY RESPONSE FUNCTIONS
Recently, investigations into the behaviour of structures with polynomial-type non-linearities have used the concepts of higher order frequency response functions (FRFs) defined from the Volterra series [12]. These can be used to represent the linear and non-linear characteristics of the system providing the system can be considered to exhibit certain properties. It has been demonstrated that higher order FRFs, if measurable, would be a valuable tool when investigating the effect of non-linearities in a structure, such as computing a mathematical model to be used for prediction purposes or as means of detecting the location of sources of non-linearity in a structure. The Volterra series provides the basis for a rigorous definition of higher order FRFs, and is an established methodology for the analysis of nonlinear ditferential equations [ 161. The Volterra series extends the familiar concept of the convolution integrals for linear systems to a series of multi-dimensional convolution integrals necessary instead for polynomial-type non-linearities. For a general polynomial or in situations where the system is to be identified and is not known in advance, it is usual to consider a Volterra series with an infinite number of terms to represent a polynomial of any order. For any linear system, with input r(t) and output u(t), the convolution integral is written
where it is assumed that u(t) = ‘QiTMo@= I,
h(t)
r(t - T) . dz.
(36)
1064
R. Ruotolo et al.
Correspondingly, for a non-linear the Volterra series is written
Volterra system 1=x=x=x=0
( s=o
3L u(t) =
r(t - 71) . dr,
hl(r~) s --01 3c1 cc +
h2(5,,72) SI-co
r(t
-
-
71)
. dz, . dt2
7*)
m
m
a
+
h3(7,,72,73) -cc sss
. r(t
+
. r(t
--3o
-
-m
TV)
m
r(t
-
7,)
---L
r(t
-
dz, . dr2. dz3 +
TV).
.
. co hn(T,,...,T,) s--z
s-cc
.;fir(t-i&d7
I,...,
dr..
Fig. 4. Flow-chart of the program which implements the non-linear finite element procedure.
(37)
In the analysis of linear systems, the relationship between the impulse response function, h(r), and the FRF H(jc~) is well known. An equivalent relationship can be defined for the corresponding Volterra kernels, h(r,, . . . , T,), in the Volterra series by using multi-dimensional Fourier transforms (MFTs). The first term in the series has exactly the same form as the convolution integral, and the first order FRF is realized with the one-dimensional Fourier transform (FT):
h, (7,)
d7,.
e-@‘171
Higher order FRFs can be defined simply by extending this approach using MFTs:
H,(iw,,...,jw,)=
e-~rw,r,+ ..+~nrnl. dr, . . . d7,.
The multi-dimensional higher order FRFs defined in this way provide a very general representation of this class of polynomial non-linear systems and can be used to explain how systems excited by, for example,
(38)
Rosin o
/
/
t ~ 4
/ I
,
I
I
I K-1
I
I
I
I
I
I
l
l
l
l
l
l
l
l
/ _p_
Y
L=700
(39)
mm
Fig. 5. The finite element mesh for the beam analysed.
4
I
si
I
‘b(
1065
Vibrations of a cantilevered beam
a broadband random signal tend to distribute energy between frequencies in a way that reflects the type of non-linearity present in the system. However, visualization of these general FRFs are difficult for third order and above, and measurement of these FRFs is difficult for random input. On the other hand the simplifications afforded by considering a simple single harmonic input are significant. Only the leading diagonals of the higher order FRFs need to be considered where a, = a2 =. . = (I&, and since these functions are one-dimensional in frequency they can be visualized easily. This development leads to a simple result for systems in this class excited by a single harmonic r(t) = R . dw’. A subset of single-dimensional FRFs, which nevertheless characterize the non-linear behaviour of the system, may be defined using only the one-dimensional FTs of the input r(t) and the output u(t) to formulate the following:
&(jw) =#,
H2( jo,jo)=
H.(jo,
2A::$),
. . , jw) = ‘i:!).
(40)
The term U( jw) is the fundamental output term, and U( j2w), U( j3w), . . . , U( jnw) are the higher harmonic terms in the spectrum of the output, which can be written as u(t)=
H,(jw).Re““‘+
H~(jw,jo).R’e+
+...+H,(jw,...,jw).R”e”“‘+... In practice, it is difficult to measure higher order FRFs of a system directly [12]. Several techniques have been developed for measuring higher order transfer functions (TFs) of a system which can be related to the ideal higher order FRFs [17]. The most fundamental technique uses a single sinewave input. This straightforward approach for measuring the TFs can be implemented in a practical testing procedure and the relationship between TF and FRF can be explained and interpreted. Table 1. Case studied vs crack size and position Case number
Size (mm)
Position (mm)
1 2 3
4 8 8
150 150 250
Considering the output time signals in terms of the displacement u(t) response to a sinusoidal forcing r(t), the tirst and higher order TFs determined using the single, dimensional FT of the time signals can be written as
TF2(
jm)
=
2 UWw) R(jw)*
TF,(
ja)
=
4. WW R(jw))
TF.( jo) =
’
’
2” - ’ U( jnw)
R(jw)”
.
(41)
The term U(iw) is the fundamental output term at the input frequency o, and U(j2w), U(j3w), . . . , U(jno) are-the higher harmonic terms in the spectrum of the output. Each of the terms in the spectra are complex quantities, and the TFs convey both gain and phase information regarding the transfer of energy between frequencies. A stepped-sine test is a convenient way to measure these TFs both in simulations and in practical testing. Comparison of eqns (40) and (41) reveals the close relationship between the Volterra series defined FRF diagonals, which are unique for the system, and the higher order TFs from a stepped-sine test. However, an important difference does arise in the fact that the TFs are determined by inputting a sinusoid to the system r(t) = R cos(wt) = R/2(P’ + e-h’) rather than an ideal harmonic r(t) = R fl’. The two harmonic terms present in the sinewave can interact in a non-linear system and give rise to “degenerative” effects of higher order terms which influence the measurement of lower order TFs. In fact these effects are thought to cause the classical distortion phenomenon observed on transfer functions measured during stepped-sine tests on non-linear structures [ 171. 7. RESULTS OF NUMERICAL SIMULATIONS
In order to investigate the potential of using harmonic distortion and higher order TFs as a means of assessing damage in structures, a simulation of a cantilevered steel beam with length of 0.7 m, and cross-section of 20 x 20 mm’ was performed (Fig. 5). For the beam material, a Young’s modulus of 2.06 x 10” N m-’ and a density of 7.85 kg m-3 was assumed, and the response at the free-end of the beam subjected to a transverse, harmonic force &, sin(wt) was calculated. The higher order TFs were determined by using the non-linear model described in a previous section to simulate the time domain response of the cracked beam. The frequency of the harmonic exciting force
R. Ruotolo et al.
1066
o was varied in the range from 0.2 to 1.4 w0 (where w. is the first natural frequency of the linearized system) in order to simulate a stepped-sine test. When determining by simulation (or measurement) the higher order TFs using eqn (41), it is necessary to first attain the steady-state response; due to the light (a) x 1 o-3
1
damping present in a structure like a steel beam, the transient has a long duration which is very time consuming to simulate. Each integration for a given value of the forcing term takes about 1 h on a Pentium 90 computer employing a routine written in MATLAB” to implement the non-linear finite element procedure previously described. Case
I
I
n.1
I
,
I
I
I
.._
_; ..;_._.._.~._..._..._.._.:
(b) 1.5
1
Case
. 1 o-3 I
I
.,.....,......
.,..
Y
.-.-
+___
I
..
n.2
I
i i
1.18
1.16
I
I
..I..........
.-
0.5
I I
I
I 1.14
1.12
1 .l
1.08
1.06
I
I
I
I
-1’
.‘.
““”
:
I’-.-
:
;.....
i i z E
a
-..-..
___
.._ - _-...
-. .-.-
~~
I$
-0.5
:.
-1
-1.= 1 7 88
IF A...
I
0.9
I
0.92
I
I
0.94
0.96
I
0.98
:
I
1
lsecl
Fig. 6. Time histories of the free-end response obtained with the non-linear finite element procedure. (Continued opposite.)
1067
Vibrations of a cantilevered beam Case
-1
I 0.92
I 0.9
n.3
I 0.96
I 0.94
b
I
I 0.98
1
I
!
I
I
1.02
1 set 1 Fig. 6-Continued. Performing a simulation of a stepped-sine test enables both the so called response envelope function (REF) and the higher order TFs of the system to be determined: (42) and the ith order TF as TF,(jo)
= “‘;;($).
(43)
Table 1 shows the three cases which have been analysed. In Fig. 6, the time histories of the response at the free-end corresponding to the cases 1, 2 and 3 are shown. In this example, a forcing frequency equal to foe is assumed in order to demonstrate the non-linear effect due to the closing crack. In the diagrams the continuous line represents the free-end response while the dotted one corresponds to the opening and closing process of the crack; from these data it is clear that the state of the crack is a function of the response of the beam as opposed to the excitation. In case 1 the non-linearity of the beam is not particularly evident; thus the linear part of the response remains dominant. In case 2 the non-linear behaviour is so distinct that the second order becomes very well defined. Furthermore, in this latter case the crack changes state twice per cycle. Figures 7 and 8 illustrate comparisons between the REFs and the time histories obtained with the
non-linear finite element procedure described in this paper and that obtained with the Krawczuk and Ostachowicz’s model. Figure 7 shows the point-to-point REFs for cases 2 and 3 at the free-end of the beam; the continuous line represents the results obtained with the model described herein while the dotted one represents the results obtained by Krawczuk and Ostachowicz. In both cases it is possible to see some discrepances between the results. Figure 8 illustrates the time histories for one period of the exciting harmonic force determined with the same two models. In case 2 the two time histories are similar, while in case 3 the results obtained by Krawczuk and Ostachowicz are incorrect due to the relatively high non-linearity of the beam. Furthermore, in case 2 it is evident that Krawczuk and Ostachowicz’s results also represent a correct double opening-closing process of the crack in one period of the load which is not considered by the square-wave function f(t). In general, the higher order TFs give an accurate and extremely sensitive indicator of non-linear behaviour of a system. Thus the evolution of TFs for varying the crack size of location may be very useful for structural damage assessment. Figure 9 shows the first to the fourth direct higher order TFs of the beam at the free-end; the continuous line refers to case 1, while the dotted to case 2. As would be expected, the first-order TF does not change with the increase in the depth of the crack, while the second and the fourth exhibit clear dependence on the size of the crack. Indeed the second harmonic of
1068
R. Ruotolo et W
10”: . .
.I:.
!...
................
al.
Case n.2 .“’ ‘:’I ._ .._ ,,...
1
.
I
..~
.
_
...
.
........... .:
....
............
.:
I...‘..’
.:
f ...
......
.
.....
_
1.
1o-5
5
. . ..i.....-
.
.......i......
.
I
I
1
I
,
I
I
10
15
20
25
30
35
40
45
[Hz1
(b)
Case n.3
.........
t :‘:::.:::::::..:..
::.:.: ,::::::.::I:.::::.:,
1
.....
...............
10+t
5
.....
....
.
I
I
1
10
15
20
:: :::.:.,:::
: ::: .:::
.... ............
........
.......................
I
$1
.... .......
f
I
I
30
35
40
I 45
Fig. 7. Comparison between non-linear finite element (-) and Krawczuk and Ostachowicz’s response envelope functions (-.-)---cases 2 and 3. the response
increases
by a factor
of approximately
five between cases 1 and 2, and the fourth harmonic increases by a factor of three. Figure 10 shows the first to the fourth direct higher
at the free-end. Due to the size of the crack, the extent of the non-linearity is greater in case 3 than in case 1, as demonstrated by comparing Fig. 9 with Fig. 10. order TFs of the beam measured
1069
Vibrations of a cantilevered beam
(4 1.5
Case n.2
x 1om3
I
1
-1.5
o;9
I
8
I
I
I
I
I
I
I
0.91
0.92
0.93
0.94
0.95
0.96
[=I
(b) x 1 o-3 ‘rI
Case n.3
I
I
I
I
I
0.94
0.95
0.96
0.97
[=I Fig. 8. Comparison between non-linear finite element (-) and Krawczuk and Ostachowicz’s results (- . -)--cases 2 and 3.
1070
R. Ruotolo et al.
(a) 1 st Order TF
I
I
I
I
2-
I
1 .5E E lu l-
0 .5-
O5
I
I
I
I
I
I
I
10
15
20
25
30
35
40
45
I Hz I
(b) 1.2
x 1o-5
2nd Higher Order TF
I
r
I
I
I
I
L
.
‘i’ i! t’ I!
‘r’ ‘. ii ‘: I ;: , ii.!.
.
_..~ ..,
,
i. I I
J -5
10
15
20
Fig. 9. Higher order TFs for cases 1 (-)
30
35
and 2 (- . -). (Conrimed
40 opposite.)
45
Vibrations of a cantilevered beam
1071
(c) 1.6
x 10.’
I
I
3rd Higher Order TF I
I
,
I
I
I
.................
..
.................
:.
Ii
:.
I’‘; ’
)
.........
.
.... .
..... _
.........
.._
I
. . ... . . . . . . . . . .
I
,
.............
.
.........
.
......
.
I I .:
........
._
I I
i
:
i!
I
il...
.I ,.....
.,..
.“’ t.
j ,,..
.;.lp+fi
_)
.........
. ......
....
-
:,:!
I I
!I’.
,
..I...
\ “’ ,I!’ : ~~.~ \.I*
! .,h. ii, __!I. ~,_~:;~~,~~~~~~,.~
. . .. . ..
‘\
:
I’<.
. . .. ..
1
. . . . ..
i,
:
15
1
.;.I
30
20
35
40
45
[El
(4 4th Higher Order TF
o x 1o-B
I
“I
i 6-
i...
.i
~
1
. . . ..
i 7-
j...
.._...
.__:
.._.
II i
E 5
6-
ii..
5-
ii ii ii.’
4-..
.,,....
..,.
: .;
j ...,,
ii : .;.i... ;
..,.i
iI 3__
.,,.)..,
I
,...,........;....
I
Ii ‘,.;I ii : ‘. , _ .;,.I, !;. .I..
2-
0 5
. . . . .. . . ....I. ‘-.-.-.
10
: 15
._,
._,_’ 20
30 $1
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R. Ruotolo et al. 8. CONCLUSIONS
AND FURTHER WORK
This paper presents an analysis of bending harmonic forced vibrations of a cantilevered beam with a transverse one-edge non-propagating closing crack. To simulate the harmonic forced vibrations of the beam, a non-linear finite element procedure has been developed. The comparison of the numerical results with that obtained by Krawczuk and Ostachowicz in a recent paper has highlighted some differences. These are due to some inaccuracies of the Krawczuk and Ostachowicz’s model and in this article, a correction for the model is indicated. The use of higher order FRFs has been suggested in order to study the non-linearity due to the closing crack. The analysis of such functions shows that they are highly dependent upon the size and the position of the crack, and thus can be used as damage indicators. Furthermore, correlating the shape and the value of these functions to the crack size and position, it may be possible to form the basis for an experimental structural damage-identification procedure. The numerical procedure described in this article is similar to the experimental approach in which a stepped-sine test is performed on a cracked beam. To validate the use of higher order FRFs as a damage-identification procedure, an experimental program has already been initiated in which a set of cracked beams with one-edge closing crack are being tested. The conclusion of these tests will be the subject of future articles into this topic. Acknowledgements-The authors would like to thank their research supervisors past and present. in particular Prof. Albert0 Carpinteri, Prof. Giuseppe Surace of the Polite&co di Torino, Italy and Prof. Geof Tomlinson of the University of Manchester, England. The research was financed by Consiglio Nazionale delle Ricerche (CNR) funds.
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