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频响函数用于转子振动信号诊断

A frequency response function-based structural

damage identi?cation method

Usik Lee *,Jinho Shin

Department of Mechanical Engineering,Inha University,253Yonghyun-Dong,Nam-Ku,Incheon 402-751,South Korea

Received 9March 2001;accepted 9October 2001

Abstract

This paper introduces an frequency response function (FRF)-based structural damage identi?cation 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 e?ects 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 identi?cation problem.The ?rst 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 veri?ed through some numerically simulated damage identi?cation tests.ó2002Elsevier Science Ltd.All rights reserved.

Keywords:Structural damage;Damage identi?cation;Beams;Frequency response function;Damage-induced modal coupling;Reduced-domain method

1.Introduction

Existence 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 identi?cation methods (SDIM),and the extensive reviews on the subject can be found in Refs.[1–3].

The ?nite 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 sti?-ness di?usion that smears the damage-induced localized changes in sti?ness matrix into the entire sti?ness 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 ?exibility 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,the

modal

Corresponding author.Tel.:+82-32-860-7318;fax:+82-32-866-1434.

E-mail address:ulee@inha.ac.kr (U.Lee).

data 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 identi?cation.

How to minimize the experimental measurement errors,structure model errors,and the damage identi?-cation analysis errors has been an important issue in most structural damage identi?cation researches.To develop or to choose a proper reliable SDIM,one needs to well understand the degree of damage e?ects 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 e?ects 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 identi?cation viewpoint.

The purposes of the present paper are:to develop an FRF-based SDIM,in which an e?cient reduced-domain method of damage identi?cation can be used,to inves-tigate the e?ects of the damage-induced modal coupling and the omitted higher vibration modes on the accuracy of predicted vibration characteristics of the damaged beam,and?nally to verify the feasibility of the present SDIM through some numerically simulated damage identi?cation tests.

2.Dynamics of damaged beams

2.1.Dynamic equation of motion for damaged beams

Though 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]o2

o x2

o2w

o x2

tq A€w?fex;tTe1T

where wex;tTis the?exural de?ection,fex;tTthe external force,and EI is the bending sti?ness 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 di?cult to assign a de?nitive representation for the sti?ness 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 sti?ness at damage location by the degradation of elastic modulus as follows[16, 32,35]:

E dexT?E1?àdexT e2T

where E d is the e?ective Young’s modulus in the dam-aged state,and dexTis the damage distribution function which may characterize the state of damage.The case dexT?0indicates the intact state,while dexT?1indi-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 e?ective Young’s modulus

E d to derive the dynamic equation of motion for the beams in the damaged state as follows:

o x2

o2w

o x2

EI D

o2w

o x2

tq A€w?fex;tTe3T

where EI D is the e?ective reduction of bending sti?ness due to the presence of damages:

EI DexT?

EdexTy2d Ae4T

The 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 beam

The dynamic equation of motion for uniform intact beams(i.e.,EI?constant)is reduced from Eq.(1)as

118U.Lee,J.Shin/Computers and Structures80(2002)117–132

EI o4w

o x4

tq A€w?fex;tTe5TForced vibration response can be obtained by su-

perposing M normal modes as

wex;tT?

X M

W mexTq metTe6T

where q metTare the modal(or generalized)coordinates and W mexTare the normal modes satisfying the eigen-value problem

EIW0000

m àq A X2

W m?0em?1;2;...;MTe7T

and the orthogonality property

Z L

q AW m W n d x?d mne8T

Z L 0EIW00

W00

d x?X2

d mne9T

where 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 mtX2

q m?f metTem?1;2;...;MTe10Twhere f metTare the modal(or generalized)forces de?ned by

f metT?

Z L

fex;tTW m d xe11T

Assume that a harmonic point force is applied at x?x F as

fex;tT?F0d xeàx FTe i x te12Twhere F0is the amplitude of the harmonic point force and x is the excitation(circular)frequency.Substituting Eq.(12)into Eq.(11)gives

f metT?W mex FTF0e i x te13T

Solving Eq.(10)for q m yields

q metT?

W mex FT

àx2

F0e i x t Q m e i x te14T

The vibration response of the intact beam can be readily obtained by substituting Eq.(14)into Eq.(6).

2.3.Dynamic response of the damaged beam

The dynamic equation of motion for damaged uni-form beams can be reduced from Eq.(3)as EI

o4w

o x4

o x2

EI D

o2w

o x2

tq A€w?fex;tTe15T

By using the normal modes of the intact beam,the general solution of Eq.(15)can be assumed as

wex;tT?

X M

W mexT q metTe16T

Substituting Eq.(16)into Eq.(15)and then applying Eqs.(8)and(9)yields the modal equations for the damaged beam as follows:

€ q

tX2

q mà

X M

k mn q n?f metTem?1;2;...;MTe17T

The third term in the left side of Eq.(17)re?ects the in?uence of damage,which is characterized by the symmetric matrix k mn de?ned by

k mn?EI

Z L

dexTW00

W00

d x DIMe18T

The matrix k mn,which is called‘damage in?uence matrix (DIM)’herein,depends on the mode curvatures as well as the damage distribution function.Eq.(18)shows that the o?-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 from

det X2

àX2

d mnàk mn

?0eno sumTe19T

For the harmonic point force acting at x?x F,the general solutions of Eq.(17)can be assumed as

q metT?q metTtD q metTe20Twhere q metTare the modal coordinates for the intact beam satisfying Eq.(10),and D q metTare the damage-induced small perturbed solutions.Substituting Eq.(20) into Eq.(17)gives

D€q mtX2

D q mà

X M

k mn D q n?

X M

k mn q n

em?1;2;...;MTe21TOn applying Eq.(14)into the right side of Eq.(21) and solving for D q metTgives

D q metT?

X M

à?

àx2

d mlàk ml

?à1

k mn Q n e i x te22T

The 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 simpli?ed form as

U.Lee,J.Shin/Computers and Structures80(2002)117–132119

D q metT?

X M

k mn Q n

àx2

e i x te23T

On 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:

wex;tT?

X M

m W mexTW mex FTX

àx2

tX M

m X M

k mn

W mexT

àx2

W nex FT

àx2

F e i x t

WexTe i x te24Twhere 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(1ti g m)1=2, where g m is the m th modal loss factor.

2.4.Damage in?uence matrix

The DIM depends on how the structural damage is distributed along the beam.Once the damage distribu-tion function dexTis 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 x?x D.The‘piecewise uniform’thickness-through damage can be represented by

dexT?D f H?xàex Dà xT àH?xàex Dt xT ge25Twhere HexTis the Heviside’s unit function.Substituting Eq.(25)into Eq.(18)yields the DIM as follows:

k mn?EI

Z x Dt x

x Dà x W00

W00

d x

D k mn De26T

If there exist many damages,say N local damages,Eq.

(26)can be further generalized as follows:k mn?

X N

j?1

Z x Djt x j

x Djà x j

W00

d x

D j

X N

j?1

k j

D je27T

where 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 ?nite beam segments that are suspected of damages. It can be observed from Eq.(27)that the damage-free zones in which D j?0can 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 identi?cation analysis should be conducted.Based on this observa-tion,the reduced-domain method of damage identi?ca-tion is introduced in Section3.

3.Development of damage identi?cation method

If the DIMC is negligible,Eq.(27)can be approxi-mated as

k mn?K m d mne28Twhere

K m?

X N

j?1

Z x Djt x j

x Djà x j

W00

2d x

D j

X N

j?1

D je29T

Applying Eq.(28)into Eq.(19)may yield a set of linear algebraic equations for unknown D j as

? k mj f D j g?X2

àX2

em?1;2;...;M and j?1;2;...;NTe30TOnce 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 identi?cation.Thus,Eq.

(30)can be used as a means of structural damage iden-ti?cation.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 di?erent de?nitions of FRF[39].Though any of

them

can 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 ex ;x T?€w ex ;t Tf ex F ;t T?àx 2

W ex T

F 0

e31T

Substituting Eqs.(12)and (24)into Eq.(31)and applying Eq.(27)may yield

àx 2

X N j X M m X M n W m ex TX 2m àx 2k j mn W n ex F T

X 2n

àx 2D j ?A ex ;x Ttx

X M m

W m ex TW m ex F TX m àx

2e32T

Because Eq.(32)provides the relationship between un-known damage information (i.e.,damage location and

magnitude)and known vibration data,it can be used to develop an algorithm for structural damage identi?ca-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 ex ;x T,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 speci?c set of x and x ,Eq.(32)may yield a linear algebraic equation for N unknown D j .Thus,choosing N di?erent 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 ?f Y i g ei ;j ?1;2;...;N T

e33T

where

X ij ?àx 2q

W m ex p T

X m àx 2q

(

k j mn

W n ex F T

X n àx 2q

()

e34T

Y i ?A x p ;x q àátx 2

X m W m ex p TW m ex F TX m àx 2q

e35T

k j

?EI

x Dj t x j x Dj à x j

W 00

m W 00n d x

e36T

i ?p teq à1TP

ep ?1;2;...;P ;q ?1;2;...;Q ;PQ P N T

e37T

where x p ep ?1;2;...;P Tdenote the measurement points and x q eq ?1;2;...;Q Tdenote the excitation frequencies.Solving Eq.(33)for N unknown D j simply

implies the location and quanti?cation of damages at a

time.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 ex p ;x q T.

The damage identi?cation 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 identi?cation.Thus,to avoid this sort of problem,how to setup a well-posed damage identi?cation problem has been an important research issue in the subject of damage identi?cation.To cope with this issue,two feasible strategies are intro-duced in the following.

The ?rst strategy is to obtain a su?cient 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 ?rst to ?x the measurement points and then to vary the exci-tation frequency until a su?cient number of equations are derived.

The second strategy is to reduce the (spatial)domain of problem.From Eqs.(33)and (36),one may ?nd that the number of unknown quantities is equal to that of

DDZs and the matrix k j

mn requires de?nite 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 identi?cation 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 identi?cation analysis.In this paper,a three-steps method is introduced and its feasibility is numerically veri?ed in Section 4.

The ?rst step:Divide the domain of problem into N DDZs and use Eq.(33)to predict N unknown damages

U.Lee,J.Shin /Computers and Structures 80(2002)117–132121

D j for N DDZs.The?rst prediction results are repre-sented by D j(?rst step)ej?1;2;...;NT.

The second step:Divide each DDZ at the?rst step into M sub-DDZs to have total(M?N)sub-DDZs and use Eq.(33)to re-predict(M?N)unknown damages for (M?N)sub-DDZs.The second prediction results are

represented by D i

j (second step)(i?1;2;...;M and

j?1;2;...;N).

The third step:If D i

j esecond stepT

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 D?0for all removed damage-free zones and to conduct damage identi?cation only for the reduced domain,which is the reduced-domain method of damage identi?cation 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 quanti?ed from Eq.(33)every iteration.

In summary,an FRF-based SDIM is introduced based on the damage identi?cation 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 identi?cation 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 e?-ciently reduce the domain of problem and?nally to identify many local damages just within a few iterations.

4.Vibration characteristics of damaged beams

Many 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 identi?cation view-point,on the e?ects 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 L?1:2m is considered herein.The beam has the intact bending sti?ness EI?11:2N m2and the mass density per length q A?0:324kg/m.

4.1.E?ects of damage-induced model coupling

The DIM for the cantilevered beam with a piecewise uniform damage at the midpoint of beam,i.e.,x D?0:6 m,is shown in Table1.Similarly,the DIM for the cantilevered beam with three identical piecewise uniform damages at x D?0:3,0.6,and0.9m is given in Table2. The piecewise uniform damages considered for Tables1 Table1

Damage in?uence matrixek mn=k refTfor the cantilevered beam with one piecewise uniform damage:D?0:5;x D?0:6m;

2 x?0:133m;k ref?3:

Table2

Damage in?uence matrixek mn=k refTfor the cantilevered beam with three piecewise uniform damages:D1?D2?D3?0:5; x D1?0:3m,x D2?0:6m,x D3?0:9m;2 x1?2 x2?2 x3?0:133 m;k ref?3:

122U.Lee,J.Shin/Computers and Structures80(2002)117–132

and2have the same magnitude D?0:5and the same

size2 x?0:133m.

Tables1and2show that,as a general rule,the di-

agonal terms of DIM(i.e.,the direct e?ects of damage)

increase in magnitude as the mode number increases.

However,they decrease momentary at certain vibration

modes if a node of the modes is located in damaged

zones.For instance,k33and k55in Table1are smaller

than k22and k44,respectively,because a node of the third

and?fth modes is located in the damaged zone.Eq.(27)

shows that,in general,DIM becomes larger as the

damage magnitudes increase.The o?-diagonal terms of

DIM(i.e.,the indirect e?ects of damage or the DIMC)

are relatively smaller than the diagonal terms.The o?-

diagonal terms vanish completely when the damage is

uniformly distributed over the whole beam,regardless of

its magnitude,which can be readily proved from Eq.

(27)by using the orthogonality property for normal

modes.

Fig.2shows the e?ects of DIMC on the damage-

induced changes in natural frequencies of the cantile-

vered beam depending on the magnitude of a piecewise

uniform damage.Fig.3is for the simply supported

beam.Neglecting the DIMC tends to underestimate

the damage-induced changes in natural frequencies.In

general,the e?ects of DIMC on the changes in natural

frequencies are found to be negligible,especially when

the damage is very weak.However,it will be desirable to

include the DIMC in the damage identi?cation analysis

because damages are not known in prior for most

practical 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 the

higher modes,and vice versa for the absolute changes in

natural frequencies.Second,the percent changes in

natural frequencies highly depend on mode number and

damage location.If damages are located at or very near

the nodes of a mode,the percent change in the natural

frequency of the corresponding mode is very small.For

instance,the percent changes in natural frequencies are

very small for the odd(e.g.,third and?fth)modes of

cantilevered 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 D?0:5and about0.1%when D?0:05for the cantilevered beam.Similarly,about0.5%when D?0:5 and about0.05%when D?0:05for 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 e?ects of DIMC on the changes in inertance FRFs are found to be neg-ligible.One notes that the third and?fth resonance peaks are not appeared in Fig.4because the FRF

measurement point(x?0:6)coincides with a node of the third and?fth modes.

4.2.E?ects of the omitted higher modes

A su?ciently large number of normal modes and natural frequencies of the intact beam are required for accurate damage identi?cation.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 de?ned by the di?erence between the exact and approximate natural frequencies of the dam-aged beam.The approximate natural frequencies are calculated by using a?nite number of normal modes.On the other hand,the damage-induced change in natural frequency,denoted by D X(damage)in Figs.5and6,is de?ned by the di?erence 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 identi?cation.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 signi?cant for the natural frequency corresponding to the normal mode.For example,the damage considered herein is located at a node of the third and?fth modes of the cantilevered beam.Thus, when total?ve normal modes are used to calculate natural frequencies,for instance,the omitted

higher

modes-induced errors in the third and?fth natural fre-quencies are larger than16%of the damage-induced changes while those in the?rst,second and fourth nat-ural frequencies are about10%.A very similar obser-vation can be made for the simply supported beam.This means that,from a damage identi?cation viewpoint,it is desirable to use only the natural frequencies of the intact beam of which modes do not have nodes at or very near the damage locations.However,this is almost imprac-ticable because the damage locations are not known in advance.This limitation is certainly one of the short-comings for the modal-data-based SDIMs in which only modal parameters are used for damage identi?cation. Second,the omitted higher modes-induced errors be-come increasingly signi?cant for weak damages.For example,if total?ve normal modes are used to calculate the?rst natural frequency of the cantilevered beam,the omitted higher modes-induced error is about10%of the pure damage-induced changes when D?0:5,whereas about110%when D?0:05.A similar observation can be made for the simply supported beam.Because the omitted higher modes-induced errors become very sig-ni?cant especially for weak damages,a su?ciently large number of normal modes should be considered to compute su?ciently accurate modal data for the dam-aged beam.

Fig.7shows the ratios between the omitted higher modes-induced errors in inertance FRFs and the dam-age-induced changes in inertance FRFs,depending on damage magnitude.A harmonic point force of x?30 rad/s is applied at the midpoint of beam and the iner-tance FRFs are measured also at the midpoint of beam. When D?0:5,about four normal modes are required for the cantilevered beam to lower the omitted higher modes-induced errors10%below the pure damage-induced changes,and about three normal modes for the simply supported beam.As the damage becomes weaker,a much larger number of normal modes are required to compute the FRFs to a required accuracy. Fig.7show that,when D?0:05,about eight and

?ve

normal modes are required for the cantilevered beam and simply supported beam,respectively.

4.3.E?ects of damage magnitude and location

Fig.8shows the percent changes in natural fre-quencies depending on damage magnitude.In general, as can be quickly observed from Eq.(30),the percent changes in natural frequencies increase in proportion to the magnitudes of damages.

Figs.9and10show the percent changes in natural frequencies depending on damage location for the can-tilevered and simply supported beams,respectively.The sensitivity of the natural frequency to the change of damage location at the lower modes is found to be higher than that at the higher modes.And,as discussed in Section4.1,the percent change in natural frequency tends to converge to a certain steady state value as the mode number increases.It can be also observed from Figs.9and10that the e?ects of damages on the change in a natural frequency become very weak when the damages are located at the nodes of the corresponding vibration mode.Because the?fth mode of cantilevered beam has nodes at x?0:33,0.66,and0.87m,for in-stance,Fig.9shows that the percent change in the?fth natural frequency is found to be very small for all damage locations considered in the?gure.Similarly,as the fourth mode of the simply supported beam has nodes at x D?0:33and0.60m,the percent change in the fourth natural frequency is also found to be very small when damage is located at x D?0:33and0.60m.

5.Numerical damage identi?cation tests and discussions

In this section,the feasibility of the present FRF-based SDIM is tested through some numerically simu-lated damage identi?cations.The feasibility tests are conducted by?rst placing some pre-de?ned piecewise uniform damages in a uniform beam and then inversely identifying them by use of the present SDIM.

Eqs.(33)–(37)show that the present SDIM requires the normal modes and natural frequencies for the intact beam and the experimentally measured FRFs for the damaged beam.For the intact beam,the normal modes and natural frequencies are analytically computed from Eqs.(7)–(9)in which the material and structural prop-erties are for a re?ned intact structure model.By‘re-?ned’,we mean that the measured and analytical modal parameters are in good agreement.

As the FRFs are measured experimentally,it is liable to be contaminated by various measurement noises. Thus,following the approach used by Thyagarajan et

al.

[31],an e %random noise is added to the FRFs analyt-ically simulated from Eq.(31)to represent the mea-surement noises in measured FRFs:

A x p ;x q àá?A x p ;x q àá1 te

randn e38T

where A is the FRFs simulated to include the measure-ment noises,and randn is the random noise generator function in MATLAB a.In this study,it is assumed that the random noise is uniformly distributed,with the mean ?0and variance ?1.

To measure the accuracy of the predicted damage state with respect to the true one,a root mean squared ‘damage identi?cation error (DIE)’is de?ned as

DIE ?

??????????????????????????????????????????????????????????

1L Z

L 0

d Pred

ex Tàd True ex T? 2d x s ????????????????????????????????????????????????1L X N

j 2 x j D Pred j àD True j

??2v u u t e39Twhere L is the length of beam and the superscripts ‘True’and ‘Pred’indicate the true and predicted damage states,respectively.The subscript j indicates the quantities for the j th DDZ.As the value of DIE is getting smaller,the predicted damage state is getting closer to the true one.

As an illustrative example,a simply supported uni-form beam is considered.The beam has the length L ?1:2m,the intact bending sti?ness EI ?11:2N m 2,and the mass density per length q A ?0:324kg/m.As shown in Fig.11,two damage problems are considered:the one piecewise uniform damage problem (i.e.,D ?0:6,x D ?1:0m,2 x ?0:044m)and the three piecewise uniform damages problem (i.e.,D 1?0:4,

x D 1?0:2m,2 x 1?0:015m;D 2?0:5,x D 2?0:6m,2 x 2?0:015m;D 3?0:6,x D 3?1:0m,2 x 3?0:015m).

https://www.wendangku.net/doc/1d13517237.html,parison between full-domain method and re-duced-domain method

The full-domain and reduced-domain methods of damage identi?cation are compared.In the full-domain method,damage identi?cation is iteratively conducted always over the whole span of beam with increasing the number of DDZs,for instance,by three times every it-eration,until the predicted damage converges to the true one.In the reduced-domain method,the domain of problem is reduced by searching out and removing damage-free zones by use of the three-steps process introduced in Section 3.The same procedure is repeated in the next iteration,but now only for the reduced domain.

Fig.12shows the detailed damage identi?cation processes for the full-domain and reduced-domain methods.No random noise in FRFs is considered for Fig.12.The reduced-domain method is found to pro-vide more accurate damage identi?cation when com-pared with the full-domain method.Fig.12(b)clearly shows that the damage magnitudes in the truly damaged zones are always predicted larger at the second step,whereas the damage magnitudes in the truly damage-free zones are always predicted smaller at the second https://www.wendangku.net/doc/1d13517237.html,paring the damage magnitudes predicted at the ?rst and second steps of iteration,the zones in which the predicted damage magnitudes become smaller at the second step are considered to be ‘damage-free’and they are all indicated by the minus (à)sign at the last step (i.e.,third step)of iteration,as shown in Fig.12(b).On the other hand,the other zones are kept suspecting of damage and indicated by the plus (t)sign.In Section 5.3,the three-steps process will be further tested with taking into account the noise in FRFs.

5.2.How to choose excitation frequencies

It might be important to well understand which ex-citation frequencies should be chosen in order to obtain reliable damage identi?cation from the measured FRFs which may depend on excitation frequency.Fig.13shows the DIEs obtained by varying the excitation fre-quency x of the harmonic point force applied at the midpoint of beam.It is found that the DIEs are strongly dependent of the excitation frequency.It is interesting to observe from Fig.13that the most reliable damage identi?cation (i.e.,small value of DIE)can be obtained when the excitation frequencies are chosen very near the natural frequencies.In theory,if structural damping is taken into account in the analysis,the

excitation

frequencies can be chosen to be equal to the natural frequencies.However,exciting the beam at even natural frequencies(e.g.,second and fourth natural frequencies) does not provide good damage identi?cation because the pre-speci?ed location of damage exactly coincides with a node of the even normal modes.In general,it is rec-ommended to choose the excitation frequencies very near to the natural frequencies of which normal modes do not have any node within the most candidate-dam-aged zones.5.3.Feasibility of the three-steps process for reduced-domain method

In Section5.1,the feasibility of the three-steps pro-cess used in conjunction with the reduced-domain method was tested for the beam with a single damage, shown in Fig.11(a),without taking into account the noise in FRFs.In this section,the feasibility tests are rerun with taking into account the random noise in FRFs.Fig.14shows the damage predictions at the?rst two steps of the?rst iteration of the reduced-domain method of damage identi?cation,depending on the level of random noise in FRFs.Fig.14shows that in general the damage magnitudes in the truly damaged zones are predicted larger at the second step,whereas those in the truly damage-free zones are predicted smaller.However, this rule is broken if the random noise in FRFs becomes larger than a certain value:for instance,about10%for the problem considered herein.Thus,for the successful applications of the present three-steps process to dam-age identi?cation,the‘true’noise in FRFs should be minimized as low as possible,probably less than about 5%,which can be met by most well-prepared vibration tests.Otherwise,the full-domain method would be rather recommended.

5.4.Multi-excitation frequency/multi-measurement point approach

Eq.(23)shows that the number of DDZs is equal to the number of unknown damage magnitudes,D j.Once the number of DDZs is?xed,one can choose a proper number of excitation frequencies and measurement points to derive as many linear algebraic equations as the number of unknown D j.Thus,depending on the number of excitation frequencies and measurement points,various approaches can be considered:the single-excitation frequency multi-measurement point(SFMP) approach,the multi-excitation frequency single-mea-surement point(MFSP)approach,and the multi-exci-tation frequency multi-measurement point(MFMP) approach.For a damaged beam divided into total 27DDZs,for instance,one may choose either‘one ex-citation frequency twenty-seven measurement points (1F27P)’,or‘three excitation frequencies nine measure-ment point(3F9P)’,or‘nine excitation frequencies three measurement point(9F3P)’,or‘27excitation frequencies one measurement points(27F1P)’.The SFMP approach seems to be impractical for most problems because it is not so easy to increase the number of measurement points over a certain limit number.Because the FRFs are dependent of measurement points,the MFSP ap-proach can provide poor damage identi?cation if the measurement points are not properly chosen.Thus,to cope with these problems met when SFMP or

MFSP

approach is applied,it is recommended in general to use the MFMP approach for most reliable damage identi-?cation.Table 3compares the damage identi?cations by SFMP and MFMP approaches,in terms of DIE with varying the random noise in FRFs up to 9%.

The

MFMP approach is shown to provide more accurate results for both the full-domain and reduced-domain methods.This is probably because the abundant damage information contained in the measured FRFs can be re?ected in the damage identi?cation analysis more ef-?ciently by the use of MFMP approach.Table 3shows that in general the reduced-domain method used in combination with MFMP approach provides the most reliable damage identi?cation.

5.5.Application to the three damages problem

Both the full-domain and reduced-domain methods of damage identi?cation (all in combination with MFMP approach)are applied to the beam with three damages (Fig.11b)and their results are given in Figs.15and 16,respectively,for di?erent levels of random noise in FRFs.It is shown that both methods fairly well locate

and quantify the pre-de?ned three damages up to about 9%random noise in FRFs,but putting a small level of incorrect damages at damage-free zones.

When compared with the full-domain method,the reduced-domain method seems to provide more re-liable results as far as the noise in FRFs is so small that the three-steps process works correctly.Fig.17shows the details of a reduced-domain method based damage identi?cation,in which a 5%random noise in FRFs is considered so that the three-steps process works correctly.However,it is numerically investigated that the three-steps process fails to discriminate damage-free zones from damaged zones if the random noise in FRFs becomes larger than about 9%for the present example problem.Thus,when a 10%random noise in FRFs is considered,the three-steps process fails to work and Fig.16(d)certainly shows that the reduced-domain method provides a misleading damage identi?cation by putting non-zero damages at damage-free zones.

Table 3

Comparison of the DIEs depending on the application method of the present SDIM and the random noise in FRFs Random noise in FRFs (%)SFMP approach MFMP approach Full-domain method Reduced-domain method Full-domain method Reduced-domain method 09:56?10à128:21?10à125:32?10à123:76?10à1217:99?10à33:25?10à33:16?10à32:57?10à332:33?10à21:47?10à21:12?10à29:56?10à353:61?10à22:16?10à22:04?10à21:16?10à274:24?10à22:61?10à22:52?10à22:18?10à29

5:01?10à2

3:62?10à2

3:33?10à2

2:84?10à

130U.Lee,J.Shin /Computers and Structures 80(2002)117–132

6.Conclusions

In this paper,an FRF-based SDIM is derived from dynamic equation of motion for damaged beams.The appealing features of the present SDIM include the followings.First,the modal data of damaged structure are not required in the analysis.Second,a large number of equations can be readily derived by varying the ex-citation frequency as well as the response measurement https://www.wendangku.net/doc/1d13517237.html,stly,the domain or size of problem can be drastically reduced by iteratively using the reduced-domain method introduced in this paper.

Numerical investigations on the dynamics of dam-aged beam may conclude that:(1)neglecting the DIMC may underestimate the damage-induced changes in natural frequencies,(2)the e?ects of DIMC on the changes in natural frequencies and FRFs are in general negligible,and (3)the damage-induced changes in nat-ural frequencies are relatively large at the lower modes,but highly dependent of modes.

The feasibility of the present SDIM is veri?ed through some numerically simulated damage identi?ca-tion tests.It is shown that the three-steps process in-troduced for the reduced-domain method of damage identi?cation is valid as far as the noise in FRFs is smaller than a certain limit value:for instance,9%for the example problems considered in the present study.In general,the reduced-domain method is found to provide the most reliable damage identi?cation when it is used in combination with the MFMP

approach.

U.Lee,J.Shin /Computers and Structures 80(2002)117–132131

Acknowledgements

This work was supported by Korean Research Foundation Grant(KRF-2001-041-E00034). References

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旋转机械振动的基本特性

旋转机械振动的基本特性概述绝大多数机械都有旋转件，所谓旋转机械是指主要功能由旋转运动来完成的机械，尤其是指主要部件作旋转运动的、转速较高的机械。旋转机械种类繁多，有汽轮机、燃气轮机、离心式压缩机、发电机、水泵、水轮机、通风机以及电动机等。这类设备的主要部件有转子、轴承系统、定子和机组壳体、联轴器等组成，转速从每分钟几十到几万、几十万转。故障是指机器的功能失效，即其动态性能劣化，不符合技术要求。例如，机器运行失稳，产生异常振动和噪声，工作转速、输出功率发生变化，以及介质的温度、压力、流量异常等。机器发生故障的原因不同，所反映出的信息也不一样，根据这些特有的信息，可以对故障进行诊断。但是，机器发生故障的原因往往不是单一的因素，一般都是多种因素共同作用的结果，所以对设备进行故障诊断时，必须进行全面的综合分析研究。由于旋转机械的结构及零部件设计加工、安装调试、维护检修等方面的原因和运行操作方面的失误，使得机器在运行过程中会引起振动，其振动类型可分为径向振动、轴向振动和扭转振动三类，其中过大的径向振动往往是造成机器损坏的主要原因，也是状态监测的主要参数和进行故障诊断的主要依据。从仿生学的角度来看，诊断设备的故障类似于确定人的病因：医生需要向患者询问病情、病史、切脉（听诊）以及量体温、验血相、测心电图等，根据获得的多种数据，进行综合分析才能得出诊断结果，提出治疗方案。同样，对旋转机械的故障诊断，也应在获取机器的稳态数据、瞬态数据以及过程参数和运行状态等信息的基础上，通过信号分析和数据处理提取机器特有的故障症兆及故障敏感参数等，经过综合分析判断，才能确定故障原因，做出符合实际的诊断结论，提出治理措施。 ^WWWWWVWWWIWWVWWWVWWWWWWWWWIHWMVWWWVWWWMWWWWWWIWWhVWWWWWWWWBWWVWWMWWWHIWW^'.a'tn'.- 根据故障原因和造成故障原因的不同阶段，可以将旋转机械的故障原因分为几个方面，见表1。

汽轮机振动大的原因分析及其解决方法[1]

汽轮机振动大的原因分析及其解决方法摘要：为了保障城市经济的发展与居民用电的稳定，加强汽轮机组日常保养与维护，保障城市供电已经成为了火力发电厂维护部门的重要任务。文章就汽轮机异常振动的原因进行了分析与故障的排除，在振动监测方面应做的工作进行了简要的论述。关键词：汽轮机；异常振动；故障排除；振动监测；汽流激振现象对转动机械来说，微小的振动是不可避免的，振动幅度不超过规定标准的属于正常振动。这里所说的振动，系指机组转动中振幅比原有水平增大，特别是增大到超过允许标准的振动，也就是异常振动。任何一种异常振动都潜伏着设备损坏的危险。比如轴系质量失去平衡（掉叶片、大轴弯曲、轴系中心变化、发电机转子内冷水路局部堵塞等）、动静磨擦、膨胀受阻、轴承磨损或轴承座松动，以及电磁力不平衡等等都会表面在振动增大，甚至强烈振动。而强烈振又会导致机组其他零部件松动甚至损坏，加剧动静部分摩擦，形成恶性循环，加剧设备损坏程度。异常振动是汽轮发电机运转中缺陷，隐患的综合反映，是发生故障的信号。因此，新安装或检修后的机组，必须经过试运行，测试各轴承振动及各轴承处轴振在合格标准以下，方可将机组投入运行。振动超标的则必须查找原因，采取措施将振动降到合格范围内，才能移交生产或投入正常运行。一、汽轮机异常振动原因分析汽轮机组担负着火力发电企业发电任务的重点。由于其运行时间长、关键部位长期磨损等原因，汽轮机组故障时常出现，这严重影响了发电机组的正常运行。汽轮机组异常振动是汽轮机常见故障中较为复杂的一种故障。由于机组的振动往往受多方面的影响，只要跟机本体有关的任何一个设备或介质都会是机组振动的原因，比如进汽参数、疏水、油温、油质、等等。因此，针对汽轮机异常震动原因的分析就显得尤为重要，只有查明原因才能对症维修。针对导致汽轮机异常振动的各个原因分析是维修汽轮机异常振动的关键。二、汽轮机组常见异常震动的分析与排除引起汽轮机组异常振动的主要原因有以下几个方面，汽流激振、转子热变形、摩擦振动等。（一）汽流激振现象与故障排除汽流激振有两个主要特征：一是应该出现较大量值的低频分量；二是振动的增大受运行参数的影响明显，且增大应该呈突发性，如负荷。其原因主要是由于叶片受不均衡的气体来流冲击就会发生汽流激振；对于大型机组，由于末级较长，气体在叶片膨胀末端产生流道紊乱也可能发生汽流激振现象；轴封也可能发生汽流激振现象。针对汽轮机组汽流激振的特征，其故障分析要通过长时间的记录每次机组振动的数据，连同机组满负荷时的数据记录，做出成组曲线，观察曲线的变化趋势和范围。通过改变升降负荷速率，从5T/h到50T/h的给水量逐一变化的过程，观察曲线变化情况。通过改变汽轮机不同负荷时高压调速汽门重调特性，消除气流激振。简单的说就是确定机组产生汽流激振的工作状态，采用减低负荷变化率和避开产生汽流激振的负荷范围的方式来避免汽流激振的产生。（二）转子热变形导致的机组异常振动特征、原因及排除转子热变形引发的振动特征是一倍频振幅的增加与转子温度和蒸汽参数有密切关系，大都发生在机组冷态启机定速后带负荷阶段，此时转子温度逐渐升高，材质内应力释放引起转子热变形，一倍频振动增大，同时可能伴随相位变化。由于引起了转子弯曲变形而导致机组异常振动。转子永久性弯曲和临时性弯曲是

转子不平衡的故障机理与诊断

转子不平衡的故障机理与诊断(1) 转子不平衡是由于转子部件质量偏心或转子部件出现缺损造成的故障，它是旋转机械最常见的故障。据统计，旋转机械约有一半以上的故障与转子不平衡有关。因此，对不平衡故障的研究与诊断也最有实际意义。一、不平衡的种类造成转子不平衡的具体原因很多，按发生不平衡的过程可分为原始不平衡、渐发性不平衡和突发性不平衡等几种情况。原始不平衡是由于转子制造误差、装配误差以及材质不均匀等原因造成的，如出厂时动平衡没有达到平衡精度要求，在投用之初，便会产生较大的振动。渐发性不平衡是由于转子上不均匀结垢，介质中粉尘的不均匀沉积，介质中颗粒对叶片及叶轮的不均匀磨损以及工作介质对转子的磨蚀等因素造成的。其表现为振值随运行时间的延长而逐渐增大。突发性不平衡是由于转子上零部件脱落或叶轮流道有异物附着、卡塞造成，机组振值突然显著增大后稳定在一定水平上。不平衡按其机理又可分为静失衡、力偶失衡、准静失衡、动失衡等四类。二、不平衡故障机理设转子的质量为M，偏心质量为m，偏心距为e，如果转子的质心到两轴承连心线的垂直距离不为零，具有挠度为a，如图1-1所示。

图1-1 转子力学模型由于有偏心质量m和偏心距e的存在，当转子转动时将产生离心力、离心力矩或两兼而有之。离心力的大小与偏心质量m、偏心距e及旋转角速度ω有关，即F＝meω2。众所周知，交变的力（方向、大小均周期性变化）会引起振动，这就是不平衡引起振动的原因。转子转动一周，离心力方向改变一次，因此不平衡振动的频率与转速相一致，振动的幅频特性及相频特性。三、不平衡故障的特征实际工程中，由于轴的各个方向上刚度有差别，特别是由于支承刚度各向不同，因而转子对平衡质量的响应在x、y方向不仅振幅不同，而且相位差也不是90°,因此转子的轴心轨迹不是圆而是椭圆，如图1-2所示。由上述分析知，转子不平衡故障的主要振动特征如下。 (1) 振动的时域波形近似为正弦波（图1-2）。 (2)频谱图中，谐波能量集中于基频。并且会出现较小的高次谐波，使整个频谱呈所谓的“枞树形”，如图1-3所示。

(完整word版)汽轮机异常振动分析及处理

汽轮机异常振动分析及处理一、汽轮机设备概述国华宝电汽轮机为上海汽轮机有限公司制造的超临界、一次中间再热、两缸两排汽、单轴、直接空冷凝汽式汽轮机，型号为NZK600-24.2/566/566。具有较高的效率和变负荷适应性，采用数字式电液调节（DEH）系统，可以采用定压和定—滑—定任何一种运行方式。定—滑—定运行时，滑压运行范围40～90%BMCR。本机设有7段非调整式抽汽向三台高压加热器、除氧器、三台低压加热器组成的回热系统及辅助蒸汽系统供汽。高中压转子、低压转子为无中心孔合金钢整锻转子，高中压转子和低压转子之间装有刚性法兰联轴器，低压转子和发电机转子通过联轴器刚性联接。整个轴系轴向位置是靠高压转子前端的推力盘来定位的，由此构成了机组动静之间的相对死点。整个轴系由 7个支持轴承支撑，高中压缸、低压缸和碳刷共五个支持轴承为四瓦块可倾瓦，发电机两个轴承为可倾瓦端盖式轴承，推力轴承安装在前轴承箱内。推力轴承采用LEG轴承，工作瓦块和定位瓦块各八块。盘车装置安装在发电机与低压缸之间，为链条、蜗轮蜗杆、齿轮复合减速摆动啮合低速盘车装置，盘车转速为2.38r/min。运行中为提高机组真空严密性，将机组轴封密封蒸汽压力由设计28kp提高至 40kp—60kp（以轴封漏汽量而定）。虽然提高了运行经济性但也增大了轴封漏汽量，可能会使润滑油带水并影响到机组胀差和振动，现为试验中，无法得出准确结论。#1机组大修后启机发生过因转子质量不平衡引起多瓦振动，经调整平衡块后得以改善。正常停机时出现过因胀差控制不当造成多瓦振动，也可能和滑销系统卡涩有一定关系。#2机组正常运行中（无负荷变化）偶尔会出现单各瓦振动上升现象，不做运行调整，振动达到高点之后迅速回落，一段时间后又会恢复正常，至今未查明原因。机组采用顺序阀运行时，在高低负荷变换时会发生#1瓦振动短时增大现象，暂定为高压调阀开关时汽流激振引起的振动。机组异常振动是经常发生又十分复杂的故障，要迅速做出判断处理，才能将危害降到最低。二、机组异常振动原因 1、机组运行中心不正引起振动（1）汽轮机启动时，如暖机时间不够，升速或加负荷过快，将引起汽缸受热膨胀不均匀，或滑销系统有卡涩，使汽缸不能自由膨胀，均会造成汽缸对转子发生相对偏斜，机组出现不正常的位移，产生振动。（2）机组运行中，若真空下降，将使低压缸排汽温度升高，后轴承座受热上抬，因而破坏机组的中心，引起振动。

转子故障振动机理分析

转子故障振动机理分析转子故障引起振动有许多形式, 现对其中的几个典型振动故障产生的原因及其对应的振动机理进行如下分析: 1.转子不平衡故障及振动机理分析转子不平衡包括转子系统的质量偏心及转子部件出现缺陷;转子质量偏心是由于转子的制造误差、装配误差、材料不均匀等原因造成的，称为初始不平衡。转子部件缺损是指转子在运行中由于腐蚀、磨损、介质结垢以及转子受疲劳力的作用，使转子的零部件（如叶轮、叶片等）局部损坏、脱落、碎片飞出等，造成的新的转子不平衡。转子质量偏心及转子部件缺损是两种不同的故障，但其不平衡振动机理却有共同之处。振动机理分析:旋转过程中,转子产生不平衡离心力与力矩通过支承点作用在轴及轴承上,引起振动.设转子质量为M（包括偏心质量m），偏心距e，旋转角频率w=2 f（v f为 v 转动频率），在t瞬时位移在直角坐标系分量x，y，如图6-3所示，则可得转子中心运动微分方程为图6-3 转子力学模型

则有以上几式中的K可以近似简化为机器的安装总刚度，M为机器的总质量，为K和M构成的振动体的无阻尼固有频率。为无量纲阻尼因子，它的取值不同，会影响到系统的响应，是激励频率与固有频率之比，也是无量纲因子。根据上式，按不同的频率比和阻尼系数的变化，作出幅频响应图及相频响应图，如下图所示：图6-4 幅频响应图及相频响应图转子不平衡所引起振动有下列特点:振动方向为径向,振动的特征频率等于转频;转子的轴承均发生较大的振动;在转子通过临界转速时振幅有特别显著的增大;在高速下随转轴转速上升振动很快增大;振动频率与转速相等且为正弦波;在没有带负荷时振动就达到最大值. 2.转子不对中故障振动机理分析机组各转子之间由联轴器联接构成轴系，传递运动和转动。由于机器的安装误差、承载后的变形以及机器基础的沉降不均等，造成机器工作状态时各转子轴线之间产生轴线平

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汽轮机转子振动分析与处理发表时间：2018-09-12T11:22:41.650Z 来源：《基层建设》2018年第21期作者：马玉清[导读] 摘要：在工业生产中，汽轮机作为重要的旋转设备，是必不可少的机械设备。哈尔滨汽轮机厂有限责任公司黑龙江哈尔滨 150046 摘要：在工业生产中，汽轮机作为重要的旋转设备，是必不可少的机械设备。其中汽轮机转子是汽轮机的主要零部件，其安全性、可靠性、适用性以及可维修性特点受到人们的关注。在汽轮机转子运行过程中，发生的振动信号是判断汽轮机工作状态的重要指标，更是影响机械设备运行安全与操作人员人身安全的因素，因此对汽轮机转子运行故障分析及诊断的研究工作迫在眉睫。关键词：汽轮机；转子；运行故障；分析与诊断汽轮机运行过程中，转子在高温工质中高速运转，不但要承受叶片、叶轮等带来的巨大离心力，同时受到蒸汽轴向推力、轴系振动力、扭转力矩等多重应力影响，在这样复杂的工况下，发生转子振动故障的概率相当高，因此加强汽轮机转子振动故障的分析及处理，对保障汽轮机安全稳定运转具有重要现实意义。一、汽轮机转子运行故障类型在汽轮机转子运行过程中，振动信号发生是转子发生故障的前提表现，对此应在汽轮机转子运行过程中，对其振动信号进行准确测量，为了更好地判断汽轮机转子运行故障类型，对其进行分类阐述。振动频率：基频振动、倍频振动、整分数基频振动、比例基频振动、超低基频振动以及超高基频振动；振幅方位：横向振动（水平振动和垂直振动）、轴向振动与扭转振动；振动原因：转子平衡度较差、轴系不对称和零件松动、摩擦（密封件摩擦、转子和定子之间产生的摩擦）、轴承损坏、轴承内部油膜涡动与油膜振动、动力和水力的影响、轴承刚度较差、电气等；振动部位：转子和轴系振动（轴颈、轴纹叶片）、轴承（油膜滑动和波动）、壳体振动与轴承座振动、基础振动（基座、工作台、支架）、其他结构振动（阀门、阀杆、管道等）。二、出现故障的原因分析 1.设计制造因素由于在汽轮机中，转子一直是处于高速运转的过程中，如果是在生产制造的过程中出现问题，就会使得转子在运行的过程中，其质心和几何中心没有重合在一起，并且由于转子在运行的过程中处于高速运转的过程中，这样就会产生一个离心力，离心力主要是通过旋转中心线的静止平面上进行投影，这是一个周期性的简谐外力，如果在这个时候进行强迫振动，这就会使得汽轮机转子的振动出现加剧现象。并且由于在运行的过程中，由于现代汽轮机的制造为了提高汽轮机工作的效率，因此汽轮机动静之间的间隙十分小，所以这就使得汽轮机在高温高速运行的过程中，很容易使得转子产生振动现象，并且由于汽缸在运行的过程中出现受热不均匀的现象，这样就会使得汽缸出现变形，加剧了转子振动，严重的时候就不断的产生循环，最终就导致故障的产生。 2.安装及检修因素汽轮机转子通过联轴器相互连接起来，转子两头均有轴承提供支撑，共同构成转子轴系。若安装时两转轴中心未精确调整到同一直线上，则汽轮机运行时会因转子不对中而发生振动。转子之间如果通过刚性联轴器相连，在对轮结合面处会形成很大的张口，此时如果用连接螺栓将转子强行连到一起，会发生静止绕曲变形，在转子上生成附加连接约束力，导致转子振动。此外，滑销系统对汽轮机组膨胀具有重要的引导功能，若因各种因素导致滑销系统卡涩，就会影响机组的正常膨胀，严重时会使机组发生强烈振动，甚至出现无法启动的情况。 3.运行因素（1）转子弯曲。汽轮机转子如果存在材质不均匀的情况，在高速运转受热后会发生弹性热弯曲，导致不可逆形变；汽轮机启动时，如果盘车或暖机不充分，上下缸温度差异大，转子横截面内温度场分布不均，也会因弹性热弯曲而出现不可逆形变。（2）机组启动。汽轮机从启动到正常运转的这段时间内，各金属构件及管道导热均处于一个不稳定的状态，期间容易受到各种机械应力、热应力的作用而产生振动、形变以及复杂的热膨胀效应。此外，机组冷态启动与热态启动的操作步骤存在差异，若人为混淆可能导致机组强烈振动。例如，某厂一660MW汽轮机组在冷启动时，操作人员未待缸体充分膨胀便过早结束低速冲转，导致机组在通过临界转速时发生剧烈振动，最后突破阀值而发生跳机。（3）润滑油温。油温与轴瓦间油膜的形成息息相关，而油膜对转子稳定性具有至关重要的影响。油温过高会导致润滑油粘度下降，不利于轴瓦内油膜的形成，进而引起转子系统干摩擦。温度过低会导致润滑油粘度超标，引起油压下降，进而引起转子振动的加剧。三、解决故障的措施 1.提高安装精度（1）轴系连接要尽量做到平直、同心。转子水平放置时，会由于自重作用而产生微弱的静挠曲，故转子安装完之后，应确保各转子轴线构成一平滑的曲线，否则会导致轴承本身负载的不一致，降低转子运行的平稳性。在实际安装过程中，应根据轴承的具体方位来确定曲线的实际方位，务必使整个转子呈一连续的光滑曲线。（2）精确安装轴承。汽轮机组中使用了很多的可倾瓦轴承，这类轴承的特点是稳定性极强，并且可以有效地缓解油膜振动。在安装过程中，应确保轴承盖与轴瓦之间的紧力满足设计要求。（3）提高轴承座安装精度。轴承座安装应当结合图纸要求及相关规范进行严格把关，根据实际需要，安装时可予以多次测量，求得加权平均值。同时应注意，轴承座几何中心应与轴颈承力中心保持重合。（4）精确安装滑销系统。正常情况下，机组运行时会由于高温、高压作用而发生缸体膨胀，通过正确安装花销系统，合理调整系统的间隙，能够将缸体膨胀控制在一定范围之内，降低对机组造成的影响。 2.减少摩擦力的产生汽轮机在运行的过程中，想要使得转子在运行的过程中减少摩擦力，因此这就需要我们要使用压力以及湿度符合相关要求的润滑油，并且在使用的过程中还需要降低润滑油的粘度。这主要是由于润滑油的粘度不断增加的话，那么就会使得油膜的承载力不断的增大，但是如果我们一直增加润滑油的粘度话，就会使得其均匀分布受到了破坏，这样反而就极大的增加了摩擦力。 3.对转子进行动平衡检查

频响函数用于转子振动信号诊断

A frequency response function-based structural damage identi?cation method Usik Lee *,Jinho Shin Department of Mechanical Engineering,Inha University,253Yonghyun-Dong,Nam-Ku,Incheon 402-751,South Korea Received 9March 2001;accepted 9October 2001 Abstract This paper introduces an frequency response function (FRF)-based structural damage identi?cation 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 e?ects 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 identi?cation problem.The ?rst 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 veri?ed through some numerically simulated damage identi?cation tests.ó2002Elsevier Science Ltd.All rights reserved. Keywords:Structural damage;Damage identi?cation;Beams;Frequency response function;Damage-induced modal coupling;Reduced-domain method 1.Introduction Existence 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 identi?cation methods (SDIM),and the extensive reviews on the subject can be found in Refs.[1–3]. The ?nite 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 sti?-ness di?usion that smears the damage-induced localized changes in sti?ness matrix into the entire sti?ness 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 ?exibility 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,the modal * 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)00170-5

汽轮机振动分析与故障排除

成人高等教育毕业设计题目：汽轮机振动分析与故障排除学院(函授站）：机械工程学院年级专业：热能与动力工程层次：本科学号：姓名：张华指导教师：起止时间：年月日～月日

内容摘要我国经济的快速发展对我国电力供应提出了更高的要求。为了保障城市经济的发展与居民用电的稳定，加强汽轮机组日常保养与维护，保障城市供电已经成为了火力发电厂维护部门的重要任务。汽轮机组作为发电厂重要组成部分其异常振动对于整个发电系统都有着重要的影响，汽轮机组异常振动是汽轮机常见故障中较为复杂的一种故障。由于机组的振动往往受多方面的影响，只要跟机本体有关的任何一个设备或介质都会是机组振动的原因。因此，针对汽轮机异常震动原因的分析就显得尤为重要，只有查明原因才能对症维修。针对导致汽轮机异常振动的各个原因分析是维修汽轮机异常振动的关键。文章就汽轮机异常振动的原因进行了分析与故障的排除，在振动监测方面应做的工作进行了简要的论述。关键词:汽轮机；异常振动；分析；排除

内容摘要 0 前言 (3) 第一章振动原因查找和分析 (4) 第2章汽轮机组常见异常震动的分析与排除 (4) 2.1汽流激振现象与故障排除 (5) 2.2转子热变形导致的机组异常振动特征、原因及排除 (5) 2.3摩擦振动的特征、原因与排除 (6) 第三章运行方面 (6) 3.1 机组膨胀 (6) 3.2 润滑油温 (6) 3.3轴封进汽温度 (7) 3.4机组真空和排汽缸温度 (7) 3.5 发电机转子电流 (7) 3.6断叶片 (7) 第四章关于汽轮机异常振动故障原因查询步骤的分析 (7) 第五章在振动监测方面应做好的工作 (8) 结论 (10)

汽轮机转子运行故障分析及诊断

汽轮机转子运行故障分析及诊断发表时间：2017-05-12T09:03:43.900Z 来源：《防护工程》2017年第1期作者：李钢 [导读] 在目前工业生产中，汽轮机作为重要的旋转设备，是工业生产中必不可少的机械设备。辽宁大唐国际阜新煤制天然气有限责任公司辽宁阜新 123000 摘要：在目前工业生产中，汽轮机作为重要的旋转设备，是工业生产中必不可少的机械设备。其中汽轮机转子是汽轮机的主要零部件，使得汽轮机转子安全性、可靠性、适用性以及可维修性特点受到人们的关注，促使关于汽轮机转子运行故障机理与诊断技术也在飞速发展。在汽轮机转子运行过程中，发生的振动信号是判断汽轮机工作状态的重要指标，更是影响机械设备运行安全与操作人员人身安全的因素，因此对汽轮机转子运行故障分析及诊断的研究工作迫在眉睫。关键词：汽轮机转子；运行故障；诊断 1概述汽轮机组的振动是机组运行必须要监测的一个非常重要的参数，因为当机组振动超过规定的范围时，将会引起设备的损坏，甚至造成严重后果：(1)使转动部件损坏。当机组振动过大时，会使叶片、围带、叶轮等各部件的应力增加，从而产生很大的交变应力，导致疲劳而损坏；(2)使机组动、静部分发生磨损；(3)使各链接部件松动；(4)直接造成运行事故。当机组振动过大，同时又发生在高压缸端侧时，有可能危及保安器误动作而发生停机事故。因此，机组运行中要严格检测其振动值。近几年来，大庆油田宏伟热机组频繁出现振动大引起的停机事件，这就使得我们不得不引起对汽轮机组振动故障的重视。 2汽轮机转子运行故障类型在汽轮机转子运行过程中，振动信号发生是转子发生故障的前提表现，对此应在汽轮机转子运行过程中，对其振动信号进行准确测量，为了更好地判断汽轮机转子运行故障类型，对其进行分类阐述。振动频率：基频振动、倍频振动、整分数基频振动、比例基频振动、超低基频振动以及超高基频振动；振幅方位：横向振动（水平振动和垂直振动）、轴向振动与扭转振动；振动原因：转子平衡度较差、轴系不对称和零件松动、摩擦（密封件摩擦、转子和定子之间产生的摩擦）、轴承损坏、轴承内部油膜涡动与油膜振动、动力和水力的影响、轴承刚度较差、电气等；振动部位：转子和轴系振动（轴颈、轴纹叶片）、轴承（油膜滑动和波动）、壳体振动与轴承座振动、基础振动（基座、工作台、支架）、其他结构振动（阀门、阀杆、管道等）。 3结合实际案例对汽轮机转子运行故障及诊断进行分析某市炼油厂，利用延迟焦化装置中采用汽轮机，其具体的汽轮机厂商为杭州汽轮机厂，类型为凝气反动式汽轮机，现采用ENTEK振动检测系统对汽轮机运行状态进行诊断与监测。其详细的汽轮机转子运行故障诊流程为：对汽轮机转子振动信号信息进行检测和采集、分析与处理、传输、推理以及控制等。因为振动信号检测是判断汽轮机转子运行故障的主要依据，振动信号分析与处理工作是判断汽轮机转子故障的关键环节，传输与推理是整体运行故障判断的核心，控制是汽轮机转子运行故障诊断的最终目标。同时在汽轮机转子内部安装电涡流传感器，将线缆与控制箱相连，控制箱自带的振动监测模块可完成高速度数字振动信号的传输与处理工作，再使用以太网将信号处理结果上传至上位机中，从而完成汽轮机转子运行故障的诊断工作。 3.1对ENTEK振动检测系统的利用在该炼油厂使用的ENTEK振动检测系统性能参数如下所示：型号：NK25/NK28/NK12.5；额定功率：1178KW、常规功率：1071KW；额定转速：12176RPM、常规转速：9132RPM-12785RPM；最大进汽压力：1.2MPa（a）、常规进汽压力：1MPa（a）；常规排汽压力：0.012MPa（a）；最大进汽温度300摄氏度、常规进汽温度230摄氏度。在ENTEK振动检测系统中，对于汽轮机转子运行故障的诊断，产生的信号数据直接送至XM模块中，经过以太网的传输，将信号传输至emonitor系统软件内部，在该软件界面中，实现传感器与信号数据的相接，使其成为振幅型数据，从而可知由emonitor系统软件连接的采集器、监测模块以及保护监测表共同组成具有共享能力的数据库，其共享数据库内自主携带故障诊断工作，能够依据实际需求，对汽轮机转子的运行故障类别进行准确定位，对此，操作人员以手动输送的方式，完成故障诊断报告的生成工作。在此系统故障诊断环节中，由汽轮机转子振动值超出限定值而产生的故障，则需对汽轮机进行停机检修，同时加大对转子运行状态的监测工作，并对转子的转速进行妥善控制。汽轮机转子在初始运行期间，振动值均以达到限定值范围，但是由于难以在生产中对汽轮机进行检修。因此，采用转子减速与状态控制的方式，实现对汽轮机转子运行故障的诊断工作。 3.2报警和故障诊断在对汽轮机转子振动信号数据分析过程中，应利用事先采集的信号设置与之相对应的报警界定，进而才能在振动值高出正常限定值时，及时对汽轮机转子的运行故障类型进行识别和分类，其详细的振动值高超报警流程为：输定报警值界限——输入采集数据限号——汽轮机转子运行——发生警报。首先，对转子平衡度较差故障诊断：水平与垂直倍频不平衡值均大于等于1、单倍频振动效果较为明显；其次，转子摩擦故障诊断：4倍频占据1倍频20%以上、5倍频与0.5倍频占据1倍频10%以上、2倍频占据1倍频50%以上、3倍频占据1倍频20%以上以及1倍频在界定值以上；最后，油膜涡动与油膜振动故障诊断：0.5倍频、1倍频其幅值均在2.0以上。 3.3摩擦振动故障排查措施分析通常情况下，汽轮机转子运行的环境比较复杂，它在运行过程中不仅会受到高速旋转和气流冲击作用力，同时高温、潮湿以及高压的工作环境会对转子造成一定的破坏，影响机组转子的安全稳定运行。因此，应当对转子日常的保养和检查工作给予高度的重视，一旦检查过程中发现故障，维修技术人员应当立即采取解决措施，对产生摩擦振动的部件进行必要维修，而如果机组部件维修价值不高应当进行更换，以消除摩擦振动对汽轮机运行造成的不利影响。 3.4汽轮机积盐原因及处理措施对于正常运行的汽轮机，其饱和蒸汽实际含盐量会与过热蒸汽含盐量相同或饱和蒸汽含盐量略高。若汽轮机的过热蒸汽含盐量比过饱和蒸汽含盐量高时，则说明汽轮机内部积盐现象已很严重，此时应及时停机，全面清洗汽轮机。在清洗时我们常用到两种处理方法手工除垢与喷砂除垢。如果用这两种除垢法不能完全去除汽轮机内部污垢，可用柠檬酸溶液配合软水来进一步清洗汽轮机。

振动分析常见图谱

振动分析常见图谱一、跟踪轴心轨迹轴心轨迹是轴心相对于轴承座的运动轨迹，它反映了转子瞬时的涡动状况。对轴心轨迹的观察有利于了解和掌握转子的运动状况。跟踪轴心轨迹是在一组瞬态信号中，相隔一定的时间间隔（实际上是相隔一定的转速）对转子的轴心轨迹进行观察的一种方法。这种方法是近年来随着在线监测技术的普及而逐步被认可的，它具有简单、直观，判断故障简便等优点。图4-20是某压缩机高压缸轴承处轴心轨迹随转速升高的变化情况，在能过临界转速及升速结束之后，轨迹在轮廓上接近椭圆，说明这时基频为主要振动成分，如果振幅值不高，应该说机组是稳定的。如果达到正运行工况时机组振幅值仍比较高，应重点怀疑不平衡，转子弯曲一类的故障。二、波德（Bode）图波德图是描述某一频带下振幅和相位随过程的变化而变化的两组曲线。频带可以是1×、2×或其他谐波；这些谐波的幅、相位既可以用FFT法计算，也可以用滤波法得到。当过程的变化参数为转速时，例如启、停机期间，波德图实际上又是机组随激振频率（转速）不同而幅值和相位变化的幅频响应和相频响应曲线。当过程参数为速度时，比较关心的是转子接近和通过临界转速时的幅值响应和相位响应情况，从中可以辨识系统的临界转速以及系统

的阻尼状况。图4-21 某压缩机高压缸波德图图4-21是某转子在升速过程中的波德图。从图中可以看出，系统在通过临界转速时幅值响应有明显的共振峰，而相位在临界前后转了近180。。除了随转速变化的响应外，波德图实际上还可以做机组随其他参数变化时的响应曲线，比如时间，不过这时的横坐标应是时间，这对诊断转子缺损故障非常有效。也可以针对工况，当工况条件改变时做波德图，这时的幅频响应和相频响应如果不是两条直线，说明工况变化对振动的大小和相位有影响，利用这一特点可以甄别或确认其他症兆相近的故障。三、极坐标图极坐标图实质上就是振动向量图，和波德图一样，振动向量可以是1×、2 ×或其他谐波的振动分量。极坐标图有时也被称为振型圆和奈奎特图（Nyquist图），但严格说来，二者是有差别的，因为极坐标图是按实际响应的幅值相位来绘制的，而Nyquist图一般理解为是按机械导纳来绘制的。极坐标图可以看成是波德图在极坐标上的综合曲线，它对于说明不平衡质量的部位，判断临界转速以及进行故障分析是十分有用的。和波德图相比，极坐标图在表现旋转机械的动态特征性方面更为清楚和方便，所以其应用也越来越广。