频响函数用于转子振动信号诊断

A frequency response function-based structuraldamage identification methodUsik Lee *,Jinho ShinDepartment of Mechanical Engineering,Inha University,253Yonghyun-Dong,Nam-Ku,Incheon 402-751,South KoreaReceived 9March 2001;accepted 9October 2001AbstractThis paper introduces an frequency response function (FRF)-based structural damage identification method (SDIM)for beam structures.The damages within a beam structure are characterized by introducing a damage distribution function.It is shown that damages may induce the coupling between vibration modes.The effects of the damage-induced coupling of vibration modes and the higher vibration modes omitted in the analysis on the accuracy of the predicted vibration characteristics of damaged beams are numerically investigated.In the present SDIM,two feasible strategies are introduced to setup a well-posed damage identification problem.The first strategy is to obtain as many equations as possible from measured FRFs by varying excitation frequency as well as response measurement point.The second strategy is to reduce the domain of problem,which can be realized by the use of reduced-domain method in-troduced in this study.The feasibility of the present SDIM is verified through some numerically simulated damage identification tests.Ó2002Elsevier Science Ltd.All rights reserved.Keywords:Structural damage;Damage identification;Beams;Frequency response function;Damage-induced modal coupling;Reduced-domain method1.IntroductionExistence of structural damages within a structure leads to the changes in dynamic characteristics of the structure such as the vibration responses,natural fre-quencies,mode shapes,and the modal dampings.Therefore,the changes in dynamic characteristics of a structure can be used in turn to detect,locate and quantify the structural damages generated within the structure.In the literature,there have been appeared a variety of structural damage identification methods (SDIM),and the extensive reviews on the subject can be found in Refs.[1–3].The finite element model (FEM)update techniques have been proposed in the literature [4–9].As a draw-back of FEM-update techniques,the requirement of reducing FEM degrees of freedom or extending the measured modal parameters may result in the loss of physical interpretability and the errors due to the stiff-ness diffusion that smears the damage-induced localized changes in stiffness matrix into the entire stiffness matrix.Thus,various experimental-data-based SDIM have been proposed in the literature as the alternatives to the FEM-update techniques.The experimental-data-based SDIM depends on the type of data used to detect,locate,and/or quantify structural damages.They include the changes in modal data [10–18],the strain energy [19,20],the transfer function parameters [21],the flexibility matrix [22,23],the residual forces [24,25],the wave characteristics [26],the mechanical impedances [27,28],and the frequency response functions (FRFs)[29–31].Most of existing modal-data-based SDIM have been derived from FEM model-based eigenvalue problems.As discussed by Banks et al.[32],the modal-data-based SDIM have some shortcomings.First,themodal*Corresponding author.Tel.:+82-32-860-7318;fax:+82-32-866-1434.E-mail address:ulee@inha.ac.kr (U.Lee).0045-7949/02/$-see front matter Ó2002Elsevier Science Ltd.All rights reserved.PII:S 0045-7949(01)00170-5data can be contaminated by measurement errors as well as modal extraction errors because they are indirectly measured test data.Second,the completeness of modal data cannot be met in most practical cases because they often require a large number of sensors.On the other hand,using measured FRFs may have certain advan-tages over using modal data.First,the FRFs are less contaminated because they are directly measured from structures.Second,the FRFs can provide much more information on damage in a desired frequency range than modal data are extracted from a very limited number of FRF data around resonance[30].Thus,the use of FRFs seems to be very promising for structural damage identification.How to minimize the experimental measurement errors,structure model errors,and the damage identifi-cation analysis errors has been an important issue in most structural damage identification researches.To develop or to choose a proper reliable SDIM,one needs to well understand the degree of damage effects on the dynamics of a structure as well as the aforementioned errors.Some researchers[13,16,32–37]have investigated the damage-induced changes in natural frequencies, mode shapes,and curvature mode shapes with varying the location and severity of a damage.However,very few attentions have been given to the effects of the damage-induced coupling of vibration modes(simply, damage-induced modal coupling)and the higher vibra-tion modes omitted in the analysis on the accuracy of predicted vibration characteristics of the damaged beam, from a damage identification viewpoint.The purposes of the present paper are:to develop an FRF-based SDIM,in which an efficient reduced-domain method of damage identification can be used,to inves-tigate the effects of the damage-induced modal coupling and the omitted higher vibration modes on the accuracy of predicted vibration characteristics of the damaged beam,andfinally to verify the feasibility of the present SDIM through some numerically simulated damage identification tests.2.Dynamics of damaged beams2.1.Dynamic equation of motion for damaged beamsThough the FRF-based SDIM developed in this paper can be readily extended to the higher order structures including Timoshenko beams and plate structures,the Bernoulli–Euler beam is considered in this paper as an example structure,for simplicity.The beam has the length L,the mass density per length q A, and the intact Young’s modulus E.For small amplitude vibrations,the dynamic equation of motion for the beams in an intact state is given by[38]o2o x2EIo2wo x2þq A€w¼fðx;tÞð1Þwhere wðx;tÞis theflexural deflection,fðx;tÞthe external force,and EI is the bending stiffness for the intact beam. In Eq.(1),dot(Á)indicates the partial derivative with respect to time t.For most practical vibration monitoring problems,it might be difficult to assign a definitive representation for the stiffness of damaged area because the location,sizes, and geometry of the damage are not known in prior. Thus,one of the simplest approaches is to represent the damage-induced change in stiffness at damage location by the degradation of elastic modulus as follows[16, 32,35]:E dðxÞ¼E1½ÀdðxÞ ð2Þwhere E d is the effective Young’s modulus in the dam-aged state,and dðxÞis the damage distribution function which may characterize the state of damage.The case dðxÞ¼0indicates the intact state,while dðxÞ¼1indi-cates the complete rupture of material due to damage.It seems to be reasonable to assume that the damage-in-duced changes in mass distribution are negligible be-cause the damage does not result in complete breakage with a loss of mass[13,17,18,35].Assume that the damages in a beam are uniform through the thickness of beam(i.e.,thickness-through damages).Then,the intact Young’s modulus E in Eq.(1)can be replaced with the effective Young’s modulusE d to derive the dynamic equation of motion for the beams in the damaged state as follows:o2o x2EIo2wo x2Ào2o x2EI Do2wo x2þq A€w¼fðx;tÞð3Þwhere EI D is the effective reduction of bending stiffness due to the presence of damages:EI DðxÞ¼ZAEdðxÞy2d Að4ÞThe second term in the left side of Eq.(3)should vanish for the intact state.In this study,it is assumed that there are no damages on the boundaries of beam. Thus,the boundary conditions applied to a beam in the intact state can be equally applied to the beam in the damaged state.2.2.Dynamic response of the intact beamThe dynamic equation of motion for uniform intact beams(i.e.,EI¼constant)is reduced from Eq.(1)as118U.Lee,J.Shin/Computers and Structures80(2002)117–132EI o4wo x4þq A€w¼fðx;tÞð5ÞForced vibration response can be obtained by su-perposing M normal modes aswðx;tÞ¼X MmW mðxÞq mðtÞð6Þwhere q mðtÞare the modal(or generalized)coordinates and W mðxÞare the normal modes satisfying the eigen-value problemEIW0000m Àq A X2mW m¼0ðm¼1;2;...;MÞð7Þand the orthogonality propertyZ Lq AW m W n d x¼d mnð8ÞZ L 0EIW00mW00nd x¼X2md mnð9Þwhere X m are the natural frequencies for the intact beam and d mn is the Kronecker symbol.Substituting Eq.(6)into Eq.(5)and then applying Eqs.(8)and(9)yields the modal equations as€q mþX2mq m¼f mðtÞðm¼1;2;...;MÞð10Þwhere f mðtÞare the modal(or generalized)forces defined byf mðtÞ¼Z Lfðx;tÞW m d xð11ÞAssume that a harmonic point force is applied at x¼x F asfðx;tÞ¼F0d xðÀx FÞe i x tð12Þwhere F0is the amplitude of the harmonic point force and x is the excitation(circular)frequency.Substituting Eq.(12)into Eq.(11)givesf mðtÞ¼W mðx FÞF0e i x tð13ÞSolving Eq.(10)for q m yieldsq mðtÞ¼W mðx FÞX2mÀx2F0e i x t Q m e i x tð14ÞThe vibration response of the intact beam can be readily obtained by substituting Eq.(14)into Eq.(6).2.3.Dynamic response of the damaged beamThe dynamic equation of motion for damaged uni-form beams can be reduced from Eq.(3)as EIo4wo x4Ào2o x2EI Do2wo x2þq A€w¼fðx;tÞð15ÞBy using the normal modes of the intact beam,the general solution of Eq.(15)can be assumed aswðx;tÞ¼X MmW mðxÞ q mðtÞð16ÞSubstituting Eq.(16)into Eq.(15)and then applying Eqs.(8)and(9)yields the modal equations for the damaged beam as follows:€ qmþX2mq mÀX Mnk mn q n¼f mðtÞðm¼1;2;...;MÞð17ÞThe third term in the left side of Eq.(17)reflects the influence of damage,which is characterized by the symmetric matrix k mn defined byk mn¼EIZ LdðxÞW00mW00nd x DIMð18ÞThe matrix k mn,which is called‘damage influence matrix (DIM)’herein,depends on the mode curvatures as well as the damage distribution function.Eq.(18)shows that the off-diagonal terms of DIM induce the coupling be-tween modal coordinates,which is called herein‘dam-age-induced modal coupling(DIMC)’.To the authors’knowledge,the DIMC has not been discussed in the existing literatures on SDIM.The natural frequencies of the damaged beam(X m) can be obtained fromdet X2mjÀX2md mnÀk mnk¼0ðno sumÞð19ÞFor the harmonic point force acting at x¼x F,the general solutions of Eq.(17)can be assumed asq mðtÞ¼q mðtÞþD q mðtÞð20Þwhere q mðtÞare the modal coordinates for the intact beam satisfying Eq.(10),and D q mðtÞare the damage-induced small perturbed solutions.Substituting Eq.(20) into Eq.(17)givesD€q mþX2mD q mÀX Mnk mn D q n¼X Mnk mn q nðm¼1;2;...;MÞð21ÞOn applying Eq.(14)into the right side of Eq.(21) and solving for D q mðtÞgivesD q mðtÞ¼X MnX MlX2mÀÂÀx2Ád mlÀk mlÃÀ1k mn Q n e i x tð22ÞThe third term in the left side of Eq.(21)is so small that it can be neglected.Then,Eq.(22)can be approximated in a simplified form asU.Lee,J.Shin/Computers and Structures80(2002)117–132119D q mðtÞ¼X Mnk mn Q nX2mÀx2e i x tð23ÞOn substituting Eqs.(14)and(23)into Eq.(20)and substituting the result into Eq.(16)may yield the forced vibration response of the damaged beam as follows:wðx;tÞ¼X Mm W mðxÞW mðx FÞXmÀx2"þX Mm X Mnk mnW mðxÞX2mÀx2W nðx FÞX2nÀx2#F e i x tWðxÞe i x tð24Þwhere M indicates the number of normal modes super-posed in the analysis.The structural damping can be taken into account in Eq.(24)by simply replacing the natural frequencies X m in Eq.(24)with X m(1þi g m)1=2, where g m is the m th modal loss factor.2.4.Damage influence matrixThe DIM depends on how the structural damage is distributed along the beam.Once the damage distribu-tion function dðxÞis given,the DIM can be readily computed from Eq.(18).As shown in Fig.1,consider a thickness-through damage of magnitude06D61, which is uniformly distributed over the small span2 x, with its midpoint at x¼x D.The‘piecewise uniform’thickness-through damage can be represented bydðxÞ¼D f H½xÀðx DÀ xÞ ÀH½xÀðx Dþ xÞ gð25Þwhere HðxÞis the Heviside’s unit function.Substituting Eq.(25)into Eq.(18)yields the DIM as follows:k mn¼EIZ x Dþ xx DÀ x W00mW00nd x!D k mn Dð26ÞIf there exist many damages,say N local damages,Eq.(26)can be further generalized as follows:k mn¼X Nj¼1EIZ x Djþ x jx DjÀ x jW00mW00nd x!D jX Nj¼1k jmnD jð27Þwhere N is the number of damage detection zones (DDZs),and D j,x Dj,and2 x j represent the magnitude, location,and size of the piecewise uniform damage over the j th DDZ,respectively.Here,the‘DDZs’indicate the finite beam segments that are suspected of damages. It can be observed from Eq.(27)that the damage-free zones in which D j¼0can be removed from the domain of integration without degrading the accuracy of DIM. This may drastically reduce the domain of problem or the number of DDZs for which damage identification analysis should be conducted.Based on this observa-tion,the reduced-domain method of damage identifica-tion is introduced in Section3.3.Development of damage identification methodIf the DIMC is negligible,Eq.(27)can be approxi-mated ask mnffiK m d mnð28ÞwhereK m¼X Nj¼1EIZ x Djþ x jx DjÀ x jW00m2d x!D jX Nj¼1kmjD jð29ÞApplying Eq.(28)into Eq.(19)may yield a set of linear algebraic equations for unknown D j as½ k mj f D j g¼X2mnÀX2moðm¼1;2;...;M and j¼1;2;...;NÞð30ÞOnce the modal data(i.e.,natural frequencies and mode shapes)for a beam in both intact and damaged states are provided by modal testing or theoretical vibration analysis,Eq.(30)can be solved for unknown D j to lo-cate and quantify many local damages at a time,which implies the structural damage identification.Thus,Eq.(30)can be used as a means of structural damage iden-tification.The SDIM derived from Eq.(30)is found to be the same as the modal-data-based SDIM introduced by Luo and Hanagud[16].However,as discussed in Section1,the modal-data-based SDIM may have some important limitations.Thus,this study aims to develop an FRF-based SDIM as an alternative to the modal-data-based SDIM derived Eq.(30).It might be relatively cheap and easy to use accel-erometers to measure vibration responses of a structure. The vibration signals measured by accelerometers can be readily processed to obtain FRFs.There are several different definitions of FRF[39].Though any ofthemcan be used to develop an FRF-based SDIM,the ‘in-ertance’FRF is adopted in this paper.The inertance FRF generated by the harmonic point force applied at a point x F can be measured at a point x as follows:A ðx ;x Þ¼€w ðx ;t Þf ðx F ;t Þ¼Àx 2W ðx ÞF 0ð31ÞSubstituting Eqs.(12)and (24)into Eq.(31)and applying Eq.(27)may yieldÀx 2X N j X M m X M n W m ðx ÞX 2m Àx 2k j mn W n ðx F ÞX 2nÀx 2D j ¼A ðx ;x Þþx2X M mW m ðx ÞW m ðx F ÞX m Àx2ð32ÞBecause Eq.(32)provides the relationship between un-known damage information (i.e.,damage location andmagnitude)and known vibration data,it can be used to develop an algorithm for structural damage identifica-tion.In Eq.(32),the mode shapes (W m )and natural frequencies (X m )of the intact beam are considered as known quantities because they are provided in advance by the modal testing or theoretical vibration analysis.The inertance FRF,A ðx ;x Þ,is also considered as known quantity because it is measured directly from the dam-aged beam.However,the damage magnitudes D j are the unknown quantities to be determined.In Eq.(32),the (response,FRF)measurement point x and the excitation frequency x can be chosen arbi-trary.For a specific set of x and x ,Eq.(32)may yield a linear algebraic equation for N unknown D j .Thus,choosing N different sets of excitation frequency and measurement point may yield N linear algebraic equa-tions for N unknown D j in the form of b X ij cf D j g ¼f Y i g ði ;j ¼1;2;...;N Þð33ÞwhereX ij ¼Àx 2qW m ðx p ÞX m Àx 2q()Tk j mnÂÃW n ðx F ÞX n Àx 2q()ð34ÞY i ¼A x p ;x q ÀÁþx 2qX m W m ðx p ÞW m ðx F ÞX m Àx 2qð35Þk jmn¼EIZx Dj þ x j x Dj À x jW 00m W 00n d xð36Þi ¼p þðq À1ÞPðp ¼1;2;...;P ;q ¼1;2;...;Q ;PQ P N Þð37Þwhere x p ðp ¼1;2;...;P Þdenote the measurement points and x q ðq ¼1;2;...;Q Þdenote the excitation frequencies.Solving Eq.(33)for N unknown D j simplyimplies the location and quantification of damages at atime.Thus,Eq.(33)provides a new algorithm for FRF-based SDIM.The present FRF-based SDIM requires the following data only:1.natural frequencies of intact beam,i.e.,X m ;2.modes shapes of intact beam,i.e.,W m ;3.FRF of damaged beam,i.e.,A ðx p ;x q Þ.The damage identification problem is a sort of in-verse problem.Thus,if the number of useful data (or equations)is not equal to the number of unknown quantities to be determined,a proper optimization solution technique is required.One of traditional approaches is to minimize a suitable norm of the dis-crepancy between measured and computed quantities,which is usually a quadratic form associated to the in-verse of the covariance matrix.The minimization pro-cedure may smear the damage over intact zones,which results in the incorrect damage identification.Thus,to avoid this sort of problem,how to setup a well-posed damage identification problem has been an important research issue in the subject of damage identification.To cope with this issue,two feasible strategies are intro-duced in the following.The first strategy is to obtain a sufficient number of equations from Eq.(32)by choosing as many sets of excitation frequency and (response)measurement point as needed.The use of FRFs may help realize this strategy.Because it is not always easy or practical to increase the number of measurement points over a cer-tain number,it seems to be much simple and easy first to fix the measurement points and then to vary the exci-tation frequency until a sufficient number of equations are derived.The second strategy is to reduce the (spatial)domain of problem.From Eqs.(33)and (36),one may find that the number of unknown quantities is equal to that ofDDZs and the matrix k jmn requires definite integrals only over the zones with damages.Thus,instead of examin-ing whole domain of problem to search out damages (i.e.,full-domain method),one can reduce the domain of problem in advance by removing the zones that are found out to be damage-free to examine only the reduced domain of problem (i.e.,reduced-domain method).The reduced-domain method will not degrade the accuracy of damage identification results at all.To realize the reduced-domain method,however one should know the locations and sizes of damage-free zones in advance.Unfortunately,this is impracticable for most cases.Thus,one needs a method to search out damage-free zones in the process of damage identification analysis.In this paper,a three-steps method is introduced and its feasibility is numerically verified in Section 4.The first step:Divide the domain of problem into N DDZs and use Eq.(33)to predict N unknown damagesU.Lee,J.Shin /Computers and Structures 80(2002)117–132121D j for N DDZs.Thefirst prediction results are repre-sented by D j(first step)ðj¼1;2;...;NÞ.The second step:Divide each DDZ at thefirst step into M sub-DDZs to have total(MÂN)sub-DDZs and use Eq.(33)to re-predict(MÂN)unknown damages for (MÂN)sub-DDZs.The second prediction results arerepresented by D ij (second step)(i¼1;2;...;M andj¼1;2;...;N).The third step:If D ij ðsecond stepÞ<D jðfirst stepÞ,conclude that the i th sub-DDZ within the j th DDZ is damage-free.Otherwise,the sub-DDZ is suspected of damage.Once damage-free zones are searched out and re-moved from the domain of problem by using the present three-steps method,it is possible to put D¼0for all removed damage-free zones and to conduct damage identification only for the reduced domain,which is the reduced-domain method of damage identification in-troduced in the present study.By iteratively using the reduced-domain method,all damage-free zones can be removed from the original domain of problem to leave damaged zones only,which simply implies the location of damages.The damage magnitudes are quantified from Eq.(33)every iteration.In summary,an FRF-based SDIM is introduced based on the damage identification algorithm of Eq.(33).In the present SDIM,the reduced-domain method can be iteratively used to reduce the domain of problem. The present SDIM can locate and quantify many local damages at a time by using the FRFs experimentally measured from the damaged beam.The appealing fea-tures of the present SDIM may include the followings: (1)the modal data of damaged beam are not required in the analysis;(2)as many equations as required to setup a well-posed damage identification problem can be gen-erated from the measured FRFs by varying the excita-tion frequency as well as the measurement point;(3)the reduced-domain method based on the three-steps pro-cess of domain reduction can be iteratively used to effi-ciently reduce the domain of problem andfinally to identify many local damages just within a few iterations.4.Vibration characteristics of damaged beamsMany researchers[13,16,32–37]have investigated the damage-induced changes in natural frequencies,mode shapes,and curvature mode shapes varying the location and severity of damage.However,there have been very few investigations,from a damage identification view-point,on the effects of the DIMC as well as the higher vibration modes omitted in the analysis(simply the omitted higher modes)on the accuracy of predicted vi-bration characteristics of the damaged beam.Thus,in this section,some numerical investigations are given to the DIMC and the omitted higher modes.As a repre-sentative problem,a uniform beam of length L¼1:2m is considered herein.The beam has the intact bending stiffness EI¼11:2N m2and the mass density per length q A¼0:324kg/m.4.1.Effects of damage-induced model couplingThe DIM for the cantilevered beam with a piecewise uniform damage at the midpoint of beam,i.e.,x D¼0:6 m,is shown in Table1.Similarly,the DIM for the cantilevered beam with three identical piecewise uniform damages at x D¼0:3,0.6,and0.9m is given in Table2. The piecewise uniform damages considered for Tables1 Table1Damage influence matrixðk mn=k refÞfor the cantilevered beam with one piecewise uniform damage:D¼0:5;x D¼0:6m;2 x¼0:133m;k ref¼3:87Table2Damage influence matrixðk mn=k refÞfor the cantilevered beam with three piecewise uniform damages:D1¼D2¼D3¼0:5; x D1¼0:3m,x D2¼0:6m,x D3¼0:9m;2 x1¼2 x2¼2 x3¼0:133 m;k ref¼3:87122U.Lee,J.Shin/Computers and Structures80(2002)117–132and2have the same magnitude D¼0:5and the samesize2 x¼0:133m.Tables1and2show that,as a general rule,the di-agonal terms of DIM(i.e.,the direct effects of damage)increase in magnitude as the mode number increases.However,they decrease momentary at certain vibrationmodes if a node of the modes is located in damagedzones.For instance,k33and k55in Table1are smallerthan k22and k44,respectively,because a node of the thirdandfifth modes is located in the damaged zone.Eq.(27)shows that,in general,DIM becomes larger as thedamage magnitudes increase.The off-diagonal terms ofDIM(i.e.,the indirect effects of damage or the DIMC)are relatively smaller than the diagonal terms.The off-diagonal terms vanish completely when the damage isuniformly distributed over the whole beam,regardless ofits magnitude,which can be readily proved from Eq.(27)by using the orthogonality property for normalmodes.Fig.2shows the effects of DIMC on the damage-induced changes in natural frequencies of the cantile-vered beam depending on the magnitude of a piecewiseuniform damage.Fig.3is for the simply supportedbeam.Neglecting the DIMC tends to underestimatethe damage-induced changes in natural frequencies.Ingeneral,the effects of DIMC on the changes in naturalfrequencies are found to be negligible,especially whenthe damage is very weak.However,it will be desirable toinclude the DIMC in the damage identification analysisbecause damages are not known in prior for mostpractical cases.From Figs.2and3,one may observe the followings.First,in general,the percent changes in natural fre-quencies at the lower modes are larger than those at thehigher modes,and vice versa for the absolute changes innatural frequencies.Second,the percent changes innatural frequencies highly depend on mode number anddamage location.If damages are located at or very nearthe nodes of a mode,the percent change in the naturalfrequency of the corresponding mode is very small.Forinstance,the percent changes in natural frequencies arevery small for the odd(e.g.,third andfifth)modes ofcantilevered beam and for the even(e.g.,second and fourth)modes of simply supported beam.Very similar results have been experimentally observed by Capecchi and Vestroni[40].Third,the percent changes in natural frequencies converge to a certain steady state value as the mode number increases.For instance,about1% when D¼0:5and about0.1%when D¼0:05for the cantilevered beam.Similarly,about0.5%when D¼0:5 and about0.05%when D¼0:05for the simply sup-ported beam.Fig.4compares the inertance FRFs of damaged beam,calculated with and without including the DIMC, with that of intact beam.In general,the effects of DIMC on the changes in inertance FRFs are found to be neg-ligible.One notes that the third andfifth resonance peaks are not appeared in Fig.4because the FRFmeasurement point(x¼0:6)coincides with a node of the third andfifth modes.4.2.Effects of the omitted higher modesA sufficiently large number of normal modes and natural frequencies of the intact beam are required for accurate damage identification.However,in practice, only a limited number of the lower normal modes and natural frequencies can be provided by modal testing or theoretical modal analysis.Thus,the errors due to the omission of the higher normal modes are inevitable.Fig.5shows the ratios between the omitted higher modes-induced errors in natural frequencies and the damage-induced changes in natural frequencies for the cantilevered beam with a piecewise uniform damage. Similarly,Fig.6shows the results for the simply sup-ported beam.The omitted higher modes-induced error in natural frequency,denoted by D X(omitted modes)in Figs.5and6,is defined by the difference between the exact and approximate natural frequencies of the dam-aged beam.The approximate natural frequencies are calculated by using afinite number of normal modes.On the other hand,the damage-induced change in natural frequency,denoted by D X(damage)in Figs.5and6,is defined by the difference between the exact natural fre-quency of the intact beam and that of the damaged beam.The important thing here is that the omitted higher modes-induced errors should be much smaller than the damage-induced changes for very reliable damage identification.From Figs.5and6,one may observe the followings.First,if damages are located at or very near the nodes of a normal mode,the omitted higher modes-induced errors become very significant for the natural frequency corresponding to the normal mode.For example,the damage considered herein is located at a node of the third andfifth modes of the cantilevered beam.Thus, when totalfive normal modes are used to calculate natural frequencies,for instance,the omittedhigher。