Identification of Distributed-Parameter Vibrating Systems

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IFA C Id entification and Systcm Paramcter

Estimation 1982. W ashington D .e.. CSA 1982

IDENTIFICATIO N OF DISTRIBUTED-PARAMETER VIBRATI NG SYSTEMS L. Meirovitch and H. B aruh Departm ent of Engineering Science and M echanics, Virginia Polyt echnic Institut e and Stat e University, Blacksburg, Virginia , USA

Abst r act . This paper describes a method for the identification of the parameters entering into t he equations of motion of distributed-parameter systems . Because the motion of distributed systems is described in terms of partial differential equations , these parame t ers a r e in general continuous functions of the spatial variables . For v ibrati ng systems, these parameters ordinarily represent the mass , stiffness and damping distributions . In this paper , these distribu tions are expanded in terms of finite series of known functio ns of the spatial variables multiplied by undetermined coefficients . Assuming that th e nature of th e equations of motion is known and that a limited number of eigenvalues and eigenfunctions is given , use is made of the least-squares method , in conjunction with the eigenfunc tions' ortho go nal it y , t o compu t e t he undetermined coefficients , thus identifying the system distributed parame t ers . Keywords . Parameter identification; dist ribut ed-parameter systems; partial differential equations ; vibrating systems; mass and stiffness dis trib utions ; least- squares method . with it (Berman and Flannelly, 1977; Caravani, Watson and Thomson, 1977; Polis, Goodso n and Wozny, 1972; Sehitoglu a nd Klein, 1978). The The motion of a distributed-parameter system second approach is to consid er th e actual dis is generally described by partial differential tributed system and t o identify the parameters equations (Meirovitch, 1967) . The parameters contained in the equations directly from the cont ained in the equa tions of motion are in system response ( Spalding , 1976; Klein and Segeneral cont inuous functions of the spatial hitoglu, 197 8 ; Saha and Prasad, 1980). A third variables . For vibrating systems , these paraapproach is t o identify firs t the eigensolution meters ordinarily represent mass , st i ffness and damping dis tribut ions . Because these para- associated with the ac tual distributed system and then to identi fy th e parameters cont ained meters are continuous functions of the spa in the equations of motion by making use of the tial variables, distributed- parameter systems identified e i gensolu ti on . In this paper, th e possess an infinite number of degrees of free third app r oach is considered . To this end, dom . Consequently , the eigensolution associthe parame ters entering into the equa tions of ated with distributed-parameter sys t ems conmotion a re identified by making use of the sists of a denumerably infinite set of eigenvalues and associated eigenfunctions eigensolution. The identification of the ei(Mei r ovitch , 1967) . gens olution is considered in a companion paper (Baruh and Meirovitch, 1982) . It should be To model a distributed system , the parameters noted that identification of the mass and stiffcontained in the equations of motion must be ness distributions of the actuaJ distributed known accurately . In many practical cases , system is a more logical approach than identithis information is not available, so that one f ication of the entries of the mass and stiffmust find ways of obtaining it . A frequently ness matrices of an associated discretized used approach i s t o elicit the system response mode ~, because the entries of the mass and and use t his information to deduce t he system stiffness matrices are abstract parameters parame t ers . This is the essence of parameter depending on the discretization method and identif ication . mesh size, whereas the mass and stiffness distributions represent actual parameters . The parameter identi ficat i on me th ods pertinent t o th e vibra ti on of distributed systems can be For discrete systems, the idea of using the e i gens olution to identify the mass and stiffdivided int o three approaches. The first is ness parameters is considered by Berman and to consider a discre tized or discrete model Flannelly (1977). Spalding ( 1976) carries out and id en ti fy th e parame ter s cont ained in the discrete model or the eigensolution associated the identification of the parameters by exINTRODUCTION

1283

1284

L. Meirovitch and H. Baruh

panding these parameters in terms of finite series . The identification is implemented by a Green ' s function transformation . In this paper, the mass and stiffness distributions are expanded in a finite series of known functions of the spatial variable, multiplied by undetermined coefficients . Finite series expansions often imply truncation, but the assumption here is that the mass and stiff ness distributions are generally smooth func tions, so that they can be approximated reasonably well by a finite series of low-order polynomials . Then, use is made of the leastsquares method , in conjunction with the orthogonality of the eigenfunctions, to identify the undetermined coefficients . The eigenvalues and eigenfunctions are identified by the method described by Baruh and Meirovitch (1982) . EQUATIONS OF MOTION FOR SELFADJOINT DISTRIBUTED SYSTEMS The equation of motion for a distributed parameter system can be written in the form of the partial differential equation (Meirovitch, 1967, Sec . 5- 4) 2 2 Lu(P,t) + M(P)d U(P,t)/dt = f(P ,t) (1) which must be satisfied at every point P of the domain D, where u(P,t) is the displacement of an arbitrary point P, L is a linear differ ential self-adjoint operator of order 2p, M(P) is the distributed mass and f(P,t) is the ext ernal force . The displacement u(P,t) is subject to the boundary conditions B. u(P,t) = 0 (i=1,2, ... ,p) to be satisfied a t ~very point of the boundary S of the domain D, where Bi (i=1,2, ... ,p) are linear differential operators of order ranging from zero to 2p-l . Let us consider the associated eigenvalue problem consis ting of the differential equation (Meirovitch, 1967, Sec . 5- 4) (2)

and the boundary conditions

Bi~ r

= 0 (i=1, 2,

... ,p; r=1,2, ... ) . The solution of Eqs. (2) consists of a denumerably infinite set of eigenvalues Ar and associated eigenfunctions ~ r(r=1,2,

... ) .

Assuming that the operator

to the natural frequencies W of the undamped 2 r oscilla tion by Ar = wr (r=1,2, ... ) . Because L is self - adjoint, the eigenfunctions possess the or tho gonality property; they can be normalized so as to satisfy dD

=

Ar 6rs ,

r,s=1,2, . .. where 6

rs

is the Kronecker delta .

u (P , t)

(4)

Introducing Eq. (4) into Eq . (1), multiplying both sides of the result by ~ s a nd integrating over 0, we obtain 2

r=1,2, .. .

(5)

wherp. fret) are generalized forces . THE IDENTIFICATION PROCEDURE It is assumed here that the nature of the equations of motion is known , in the sense that the form of the mass and stiffness oper ators is known. However, explicit expressions for the mass and stiffness distributions contained in these operators are not known . Generally, the mass operator is merely a function representing the mass distribution itself. The stiffness operator is a differential operator of order 2p and it contains the stiffness at any point . For example , for the bending vibration of a beam the stiffness operator is 2 2 2 2 L = d [EI(x)d /dx ]/dx and the stiffness at a point is K(x) = EI(x), where E is the modulus of elasticity and I(x) is the mass moment of inertia , in Ivhich x is the spatial variable . Because the mass and stiffness distributions are continuous functions of the spatial vari ables , one can expand them in terms of series of known functions of the spatial variables multiplied by undetermined coefficients . This idea is very similar to that behind the expansion theorem for the response (Meirovitch , 1967), but the expansion functions here need not be orthogonal . We assume that an accurate representation of the mass and stiffness distributions is possible in terms of finite series as follows : ~l ~2 m(P) =

L

drMr(P), K(P) =

r=l

L

erKr(P)

(6)

r=l

where Mr(P) and Kr(P) are functions from a r ~l

efficients and

and er are undetermined coand

~2

are the number of

terms in the expansions of the mass and sti ffness distributions, respectively. The func tions Mr(P) and Kr(P) can be global or lo c al functions. The stiffness operator can also be expressed in terms of the finite series Q, l 1:. 2 L

(3)

f r (t) = JD~ r (P)f(P,t) dD,

ur(t) + wrur(t)

complete set , d

L is positive definite, all the eigenvalues are positive . They will be ordered so that The eigenvalues are related Al 2. A2 2. ..•

JD~ s L~ r

ent eigenfunctions ~ (P) multiplied by time - de pendent generalized ~oordinates u (t) of the form r

L

r=l

e L[K (P),P]

r

r

=

L

r=l

e L r

r

(7)

Ivhere we note that the stiffness operat o r is expressed in terms of expansion functions Kr(P) . As mentioned before , the mass oper -

Using the expansion theorem (Meirovitch, 1967, a t or is the mass distribution itself . Sec . 5- 4) , the solution of Eq . (1) can be represented by an infinite series of space-d epend-

Identification of Distributed- Paramete r Vibrating Systems We assume that ID eigenvalues and eigenfunc tions are known . These eigenvalues and eigenfunctions are either given or they can be obtained by means of an identification procedure (Baruh and Meirovitch, 1982). The orthogonality relations, Eqs . (3), indicate that the eigenfunctions can be normalized with respect to the mass and stiffness distributions . Because the mass and stiffness distributions are not kno\Yn, the amplitudes of the identified eigenfunctions cannot be identified uniquely . They can be identified to within a multiplicative constant of the actual eigenfunctions . Hence, \.Je \.Jrite r=1,2, ... ,m where ~ ;

and

~r

~ ;L ~ ~dD

D

a a J ~ L ~ dD = a 2w2 6 r s D r s r r rs r,s = 1,2, ... , m

(9)

Considering Eqs . (6- 7) , we can write £.1

J

NCjl ;~ ~dD

D

I

j=l

d. J

J

~1. ~ ' ~ ' dD D J r s

~ ;L~ ~dD

D

.

rSJ

Next let us define the matrix U lll

U 112

U IH

U 12l

U 122

U ln

I

e. J ~ ' L. ~ ' dD r J s j=l J D r,s = 1 , 2 , ... , m

JDCP r' ~I.J ~ s' dD ,

r ,s

=

r,s

.

rSJ

=

1 1

U

(13)

U U m-l , m,l m-l , m,2

U m-I,m,£.

U mml

U mm£. 1

U mm2

Then, introducing the m(m + 1)/2 x m matrix T with all its entries equal to zero except the r-l entries T (i= (m- k+l) + 1; r=1,2, ..• ,m), ir k=l which are equal to -1, as well as the vectors T 2 2 ~ = [d d£.l) and a = [a l a 2 l d2

I

.

T [aT : d ) -

(10)

1,2, ... ,m; (Ha)

j = 1,2, ... , £.1 V

by using the remaining relations in Eqs . (12) . We assume that Eqs . (12a) , which contain the coeffic ients associated with the mass distribution , will be solved first . Note that, the choice of the set of coefficients to be deter mined first is arbitrary .

[T ; U)

where we note that the integrals on the right sides of Eqs . (10) contain given or identified functions . Denoting these int egrals by U

using one of Eqs . (12), and solve for the remaining £. 2 ( £.1) coefficien ts of the second set

write Eqs . (12) as

£. 2

J

(r = 1,2, ... , m) are the same in both sets r of undetermined coefficients, one can solve for the first set of m + £.1 (m + £'2) unknowns by

(r=1 ,2, ... ,m) are the identi-

are the r multiplicative constants . Introducing Eqs . (8) into the orthogonality conditions , Eqs . (3), \.Je obtain 2 J M~ ; ~ ~dD a r a s JD M~ r ~ s dD = a r 6rs D

J

a

(8)

fied and actual eigenfunctions and a

1285

1,2, ... ,m;

j = 1,2" " ' ~ 2

(lIb)

T

= 0

(14)

I

Note that, the right side of Eq . (14) is zero. Because of this, one cannot obtain a unique solution of Eq . (14) . To solve Eq . (14) uniqu e ly, we must have some addi t ional information about the distributed system . One such piece of additional information can be knowledge about the mass distribution ~ ~ certain point Po along the domain of the distributed parameter system . If the value of the mass distribution is known at point Po and if we consider Eq . (6), we can write £.1

and considering Eqs . (9 -11 ) , we obtain ,. 1

I

d.U rsj j=l J

j=l

£2

} 6

r rs'

I

} w2r 6rs'

I

e.V rsj j=l J

r

r,s = 1,2, ... , m (12a,b) 2 Equations (12) represent m relations for each of the three sets of the unknmYn coefficien ts d.(j = 1,2" " ' ;'1) ' e . (j = 1,2, ... , 9.? ) and J

J

(15)

dJ.MJ. (P O) = N(P O)

The availabi lity of Eq . (15) enab les us to add one row t o Eq . (14), so th a t th e equation becomes (16)

-

ar(r = 1,2, .. . , m) . Because the distributed sys 2

tern is self - adjoint, not all of the m relations are independent . This is so because U . rSJ U ., V . = V .' In vie\oJ of this, the number sr] rSJ srJ of independent relations for the unknown coef ficients is redu c ed to m(m+l)/2 , so that one is faced with the problem of solving two sets of m(m + 1)/2 equations for two sets of m + £.1 m + £. 2 unknowns. Because m of the unknowns

where ~(P 0) = [NI (PO)

M2 (PO) ...

T

M£. (PO») ' 1

The matrix on the left side of Eq. (16) is an [m(m + 1)/2 + 1) x (m + £'1) matrix . We shall denote the matrix by U* . For most cases , U* is not square . If U* is rectangular, then we can use a least - squares or a minimum nqrm type approach to determine a and d . For cases when an ove rspe c ifi ed model-is encountered, one can convert U* into a square matrix by ignoring

1286

L. Meirovitch and H. Baruh

some of the orth ogona lit y relations . It turns out that such an approach is not necessary . Indeed, because the orthogonality conditions are consistent , having more rows than columns does not imply that the mass distribution is identified as a least-squares fit . The actual mass distribution satisfies all of the orthogonality relations, so that having more rows than columns in the matrix U* merely implies reduncancy . Re dundancy occurs if the ~ l terms in the expansion (6) can describe the mass distribution exactly . \~en the mass distribution cannot be represented exactly be a series of ~ l terms we need as many orthogonality relations as possible to obtain an accurate description of the mass distribution. Note that we cannot determine the number of terms necessary to describe the mass distrib4tion in advance . In view of the above, it is advisable to retain all of the available orthogonality relations in Eq . (14) . Using more equations than unknowns is equivalent to identifying a larger number of coefficients in the expansion of the mass distribution than ~ l' in which the surplus coeff i cients are identified as zero . If U* is changed into a square matrix, the terms in th e expansion of the mass distribution higher than ~ l are totally ignored . Upon obta ining the solution for the mass distribution and the normalization factors for the eigenfunctions, we can obtain the solution for the stiffness distribution by using Eq . C12b), where we are faced with only ~ 2 unknowns , namely, the coefficients of the terms in the expansion of the stiffness distribution . We can represent Eq . C12b) in matrix form as Ve

z

respond to overspecified models . Both the pseudo - inverse and least- squares approaches require the inversion of a symmetric positive definite matrix . With the exception of very ill- conditioned matrices, the inversion of a positive definite symmetric matrix is a very stable operation, so that dealing with rectangular matrices does not represent a computational problem . It was assumed that the value of the mass distribution was known at least at one point along the domain of the distributed - parameter system . For cases when the mass distribution is known at more than one point, one can include this information in the problem formulation as well . In fact, it is recommended that one include as much information as possible in the least-squares identification scheme . Note that, instead of the mass distribution, if the value of the stiffness distribution is known at a point, one can solve for the coefficients of the functions in the expansion of the stiffness distribu tion first . The coefficients of the func tions in the expansion of the mass distribu tion can be identified subsequently . In the abov e analysis, no prior knowledge of the mass and stiffness distributions was as summed. In most cases , however, the analyst has a reasonably good initial estimate of the mass and stiffness distributions . Inclusion of the initial estimate in the identification procedure is likely to increase the accuracy of the identification. Equation (6) can be augmented by the addition of the illitial estimates as follows:

~l

Cl7) MCP)

where

HOCP) +

L

drHrCP)

r=l V 1l2

V 1H

V 12l

V 122

Vl 2£ • 2

V

2

V V m- l,m,l m- l,m,2

V

V mml

V mm ~ 2

KCP) = KOCP) + (18)

is an [mCm + 1)/2] x ~ 2 matrix ,

~

T

and z

2 2

[alw l

0

erKrCP)

where HOCP) and KOCP) represent the initial estimates of the mass and stiffness distri In view of the above, Eq . (7) bebutions. comes

~2

L e~ ]

L

r=l

m- l,m' ~ 2

V mm2

(19)

~2

V lll

e

0

inverting Eq . (17), we can determine the coef ficients of the functions in the expansion of , the stiffness distribution , thus identifying the stiffness distribution . The discussion presented earlier in connection with rectangular matrices is equally valid for the case of identification of the stiffness distribution .

The identification method described above requires the inversion of two matrices . These matrices are in general rectangular and cor-

L

r=l

2

2 2 2 T o 0 am_ l wm_ l 0 0 amw ] are vectors of m dimensions ~ 2 and mCm + 1)/2 respectively. By

GENERAL CONSIDERATIONS

LO +

e L r r

(20)

where LO = LO[KOCP), P] represents the stiff ness operator associated with the initial es timate of the stiffness distribution . Sub stituting Eqs . (19) and (20) into the orthogonality relations Eqs . (3), and considering Eqs . C9 - l2), we obtain

\

L

d ,U , J rSJ

j=l

- U

rsO

~2

L

e ,V , J rSJ

j=l

- V

rsO

r = 1,2, ... ,m, s=r,r+l, ... ,m where U rSO

J' D

M t ' t'dD V = 0 r s ' rsO

(21)

fDt r'L 0 t s' dD .

Identification of Distributed-Parameter Vibrating Systems In addition, knowledge of the value of the mass distribution at point Po can be expressed as ,(,1

HO(PO) +

L

j=l

d.H.(P ) J J O

(22)

We note from the above that in the presence of initial conditions the left sides of Eqs. (16) and (17) remain unchanged . Only the vectors on the right sides of Eqs . (16) and (17) change , which implies that the iden ti fication procedure can be car ried out for different initial estimates without added computational effort . NUMERICAL EXAMPLE As an illustration of the method, let us consider the problem of identification of the parameters of a tapered bar in axial vibra tion . The mass and stiffness distributions are given by M(x) = 2(1 - x) , EA(x) = 2(1 - x) , 0 < x < 1, where the beam length is chosen as 1 . The stiffness opera t or is L = - d[EA(x)d/dx]/dx . The boundary operators are Bl(O) = 1, B (1) = EA d/dx. 2 It can be shown that the actual eigenvalues must solve the transcendental equa ti ons J (w) = 0 (r = 1 , 2, ... ) , where J is the O o r zeroth - order Bessel function of the first kind and-w~ is the rth frequency of undamped r oscillation . The solution of t he transcendental equation can be found in Abramow itz and Stegun (1972 , p . 469 , Table 9 . 5). It can also be shown that th e normalized eigenfunctions have the form

where J

is the first - order Bessel function l of the first kind . The eigensolution of the bar was identified using a time-domain approach , as described in Baruh and Meirovitch (1982) . It was assumed that only the first 20 modes contribut ed to the overall motion. The identification procedure was carried out by using 12 sensors, and the response was obtained for a unit impulse, applied at x = 0 . 63 . The eigenfuncti ons were identified by interpola ting the sensors data using c ubic splines . The r eason for this will become evident shortly . Let us now identify the mass and stiffness distributions . We assume that they can be expanded in the form £.1 r-l \' M(x) = L d x , r =l r

K(x)

(24)

We also assume that the value of the mass dis tributi on is known at the point x = 0 . 5 . Then, using Eqs . (10) , (14) and (17) , the coeffic ients d and e were determined. r r Tables 1 and 2 compare the identified coefficien ts d and e for different numbers

r

r

1287

£.1 and .Q. of terms . It is observed that, by 2 using a lower number of terms in the expansions (24), the coefficients d and e are r

r

identified with greater accuracy . Figure 1 compares the exact and identified stiffness distributions for 4 terms in the expansion . We note that, even when the coef ficients of er do not agree with their actu al values exactly, the identified stiffness distribution is very close to the actual one . It should be noted that the V

. terms defin-

rSJ

ed by Eq . (11) require integrations involving the eigenfunctions and the stiffness op erator . Because the stiffness operator is a differential operator , the identified eigenfunctions must have continuous derivatives throu gh order 2p ( p = 1 for our case) . This is one reas on why cubic splines are used as interpolation functions . Indeed, cubic splines interpolate the data points such that the estimate of the displacement of the beam has continuous first and second derivatives. Because one can design (2p + I) - order splines (p = 1, 2, ... ) and the design can be in one , two or three dimensions (Ahlberg, Nilson and Walsh, 1967), the use of splines as inter polation functions is fully justified . CONCLUSIONS A method is presented for the identification of the parameters entering into the equation of motion of distributed - parameter systems . For distributed systems , these parameters are in general continuous functions of the spatial variables . For vibrating systems, these parameters ordinarily represent the mass , stiffness and damping distributions. In this paper , thes e distributions are expanded in terms of finite series of known functions of the spatial variables , multi plied by undetermined coefficients . These functions can be global or local functions . Assuming that the nature of the equations of motion is known, and that a limited number of eigenvalues and eigenfunctions are given, use is made of the least- squares method , in conjunction with the eigenfunctions ' ortho gonality , to compute the undetermined coef ficients , thus identifying the actual distri buted system, without recourse to discreti zation . REFERENCES Meirovitch , L . (1967) . Analytical Hethods in Vibrations . The Macmillan Co . , New York, NY . Berman , A . , W. G. Flannelly . (1977) . Theory of Incomplete Models of Dynamic Structures . AlAA Journal, ~, pp . 1481- 1467 . Caravani , P , M. L . Watson, and W. T . Thomson (1977) . Recursive Least - Squares Time Domain Identifi ca tion of Structural Parameters, Journal of Applied Mechan ics, pp . 135-140 .

1288

L. Meirovitch and H. Baruh

Polis, M. P., R. E. Goodson, and M. J. Wozny (1972). Parameter Identification for the Beam Equation Using Galerkin's Criterion, Proceedings of the IEEE 1972 Conference on Decision and Control, Paper No. TP4-3, pp. 391- 395. Sehitoglu, H. and R. E. Klein (1978). A Finite Element and Gradient Method for Identification of Parameters in a Class of Distributed Parameter Systems, ASME Winter Annual Meeting, San Francisco, California, Paper No. 78- WA/DSC- 29 . Spalding, G. R. (1976). Distributed System Identification: A Green's Function Approach, Journal of Dynamic Systems, Measurement and Control, 98, pp. 146-151. Klein, Rand H. Sehitoglu (1978). Some Result on the Identification of Parameters in Distributed Processes Described by the One-Dimensional Wave Equation, yroTABLE 1 - Identified Mass and Stiffness Co efficients for the Case £1 = £2 = 3, m = 4

ceeding of the Joint Automatic Control Conference, pp. 133-143. Saha, D. C. and R. G. Prasad (1980). Identification of Distributed Parameter Systems Via Multidimensional Distributions, Proceedings of the IEEE, ~, Pt. D. pp. 45-50. Baruh, H. and L. Meirovitch (1982). Identification of the Eigensolution of Distributed-Parameter Systems. AIAA/ASME/ ASCE/AHS 23rd SDM Conference, New Orleans, Louisiana. Abramowitz, M and I. A. Stegun, editors. (1972). Handbook of Mathematical Functions, Dover Publications, Inc., New York, NY. Ahlberg, J. H., E. N. Nilson, and J. L. Walsh (1967) . ~he Theo~of Splines and Their Applications, Academic Press, Inc., New York, NY. TABLE - 2 Identified Mass and Stiffness Co efficients for the Case £1 = £2 = 4, m = 4 Mass Coefficients

Mass Coefficients

d d d

l 2 3

Identified

Identified

Exact

2.0000

2.0000

d

-1.9997

-2.0000

d

-0.0004

0.0000

d d

l 2 3 4

Exact

2.0062

2.0000

2.0337

-2.0000

0.0587

0.0000

-0.0320

0.0000

Stiffness Coefficients

e e e

l 2 3

Identified

Exact

2.0033

2.0000

-2.0104

- 2.0000

0.0028

0.0000

e e e e

8 N

8

x~

Cf.~00~--~0~.ZS~----0~.-50------0~.7S~--~I.OO X

Figure la - Exact Stiffness Distribution

I

Stiffness Coefficients

l 2 3 4

Identified

Exact

1.9964

2 . 0000

-1.9 490

-2.0000

-0.1115

0.0000

0.0596

0.0000

8 N

8

x~~.+-

____~____~____~____~

~.oo

o.zs

0.50

X

O.7S

1.00

Figure lb - Identified Stiffness Distribution