EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 9, 385-409 (1981) NON-PARAMETRIC IDENTIFICATION OF A CLASS OF NON-LINEAR CLOSE-COUPLED DYNAMIC SYSTEMS F. E. UDWADIA* University of Southern California, Los Angeles, California, U.S.A. AND CHIN-PO KUO? Jet Propulsion Laboratories, Pasedena, California, U.S.A. SUMMARY A non-parametric identification techniquefor the identificationofarbitrary memorylessnon-linearities has been presented for a class ofclose-coupled dynamic systems which are commonly met with in mechanicaland structural engineering. The method is essentially a regression technique and expresses the non-linearities as series expansions in terms of orthogonal functions.Whereas no limitation on the type of test signals is imposed, the method requires the monitoring of the response of each of the masses in the system. The computational efficiency of the method, its easy implementation on analogue and digital machines and its relative insensitivity to measurement noise make it an attractive approach to the non-parametric identification problem. I. INTRODUCTION The increased importance given to the accurate prediction of the response of structures in various loading environments has led, in recent years, to a growing interest in the improvement of methodologies for proper structural modelling. Several investigators have worked in the area of identification of structural systems so as to extract, from various types of response data, improved characterizations of the systems involved.’ -6 The identification problem can be looked at in terms of a class of inputs I, a class of models M and an error criterion, E. It usually takes the following form: Given the system response (at various locations in a structure) to the class of inputs I, identify a member of the class of models M, which minimizes some error criterion E. When sufficient a priori information about the mathematical structure of the class M to which a particular physical system belongs, is available, it is often possible to restrict the identification procedure to the determination of the various parameter values, which then characterize the system. Such a procedure is referred to as parametric identification. On the other hand, as often happens for complex structural systems, the a priori information on M may not be sufficient, thus requiring the identification procedure to be ‘expanded’to a search in function space. This constitutes non-parametric identification and leads to the ‘best’ functional representation of the system. The error criterion E, usually takes the form of a norm of the difference between the system performance and the model prediction. Though a large amount of research has been done in the areas of both parametric and non-parametric identifi~ation,~ - l6 present-day techniques for both are, however, deficient when dealing with large structural systems. Parametric methods usually either require the solution of matrix Ricatti equations or take recourse to non-linear programming techniques. Often, when the number of unknowns in the dynamic system exceeds fifty or so, these methods in addition to being very expensive computationally may also yield inaccurate estimates. Non-parametric methods which employ the Volterra series or the Wiener Kernel approach are also computationally expensive and often do not provide adequate characterization of the types of non-linearities * Associate Professor. t Member of Technical Staff. @ 1981 by John Wiley & Sons, Ltd. 0098-8847/8 1 /O40385-25%02.50 Received 23 June 1980 Revised 3 November 1980
Transcript
EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 9, 385-409 (1981)
NON-PARAMETRIC IDENTIFICATION OF A CLASS OF NON-LINEAR CLOSE-COUPLED DYNAMIC SYSTEMS
F. E. UDWADIA*
University of Southern California, Los Angeles, California, U.S.A.
A non-parametric identification technique for the identification ofarbitrary memoryless non-linearities has been presented for a class ofclose-coupled dynamic systems which are commonly met with in mechanical and structural engineering. The method is essentially a regression technique and expresses the non-linearities as series expansions in terms of orthogonal functions. Whereas no limitation on the type of test signals is imposed, the method requires the monitoring of the response of each of the masses in the system. The computational efficiency of the method, its easy implementation on analogue and digital machines and its relative insensitivity to measurement noise make it an attractive approach to the non-parametric identification problem.
I. INTRODUCTION
The increased importance given to the accurate prediction of the response of structures in various loading environments has led, in recent years, to a growing interest in the improvement of methodologies for proper structural modelling. Several investigators have worked in the area of identification of structural systems so as to extract, from various types of response data, improved characterizations of the systems involved.’ -6
The identification problem can be looked at in terms of a class of inputs I, a class of models M and an error criterion, E. It usually takes the following form: Given the system response (at various locations in a structure) to the class of inputs I, identify a member of the class of models M , which minimizes some error criterion E. When sufficient a priori information about the mathematical structure of the class M to which a particular physical system belongs, is available, it is often possible to restrict the identification procedure to the determination of the various parameter values, which then characterize the system. Such a procedure is referred to as parametric identification. On the other hand, as often happens for complex structural systems, the a priori information on M may not be sufficient, thus requiring the identification procedure to be ‘expanded’ to a search in function space. This constitutes non-parametric identification and leads to the ‘best’ functional representation of the system. The error criterion E, usually takes the form of a norm of the difference between the system performance and the model prediction.
Though a large amount of research has been done in the areas of both parametric and non-parametric identifi~ation,~ - l6 present-day techniques for both are, however, deficient when dealing with large structural systems. Parametric methods usually either require the solution of matrix Ricatti equations or take recourse to non-linear programming techniques. Often, when the number of unknowns in the dynamic system exceeds fifty or so, these methods in addition to being very expensive computationally may also yield inaccurate estimates. Non-parametric methods which employ the Volterra series or the Wiener Kernel approach are also computationally expensive and often do not provide adequate characterization of the types of non-linearities
* Associate Professor. t Member of Technical Staff.
@ 1981 by John Wiley & Sons, Ltd. 0098-8847/8 1 /O40385-25%02.50 Received 23 June 1980
Revised 3 November 1980
3 8 0 F. E. UDWADIA AND c‘. P. K L O
met with in mechanical and structural systems.?- For instance, a ‘cubic spring’ type non-linearity would require the determination of third-order kernels whose computation in practice becomes prohibitively expensive.20, In addition, the Wiener approach uses white noise inputs. I t is often extremely difficult. i f not impossible, to generate large enough inputs of this nature so as to drive large (and often massive) dynamic systems in their non-linear range of response. Applications of such techniques to large non-linear rnulti- degree-of-freedom systems are few, if any.
This paper presents a relatively simple non-parametric approach to the identification of a class of multi- degree-of-freedom (MDF) close-coupled non-linear systems (Figure 1 ) . The method, following Graupe.” is
H
‘\ INERTIAL REFERENCE
Figure 1
basically a rcgression technique. Masri and Caughey’” were the first to apply this technique to the identification of a single-degree-of-freedom oscillator, by expanding the restoring force in a series of Chebyshev polynomials.22 Herein, we extend the method to include a class of MDF systems, and further generalize it through the use of arbitrary orthogonal sets of functions. The technique has the advantage of being computationally efficient and simple to implement on analogue and digital machines. Unlike the Wiener Kernel approach, it is not restricted to ‘white noise’ type of inputs, and can be used with almost any type of test input. The choice of the class of models, M, has been governed by its wide usage in problems involving the dynamic response of: (i) full scale building structures, (ii) layered soil ma~ses , ’~ (iii) mechanical eq~iprnent.’~. and (iv) machine components and subsystems in, for instance, the nuclear industry.26% ”
shown that even under extremely noisy measurement conditions, the method yields good results. The technique has been illustrated through application to linear and non-linear systems, and it has been
11. SYSTEM MODEL
The model consists of a lumped mass system with masses M , , I = 1, ... , N , which are connected to one another by the unknown memoryless non-linear elements K, , I = 1.2 ,... , N , as shown. It is assumed that the acceleration of the base of the structure f(t) and the relative accelerations (with respect to the base) .YI(t) . 1 - I , 2. ... . N of the various masses are obtained from noisy measurements. The excitation forces , fi(t). 1 = 1.2, ... . N , are assumed to be available and the masses M , , I = I , 2. .._. N . to be either known or fairly well estimated from design drawings.
Further, for [he close coupled system shown (Figure 1) i t is reasonable to assume that the restoring force K 8 depends upon the relative displacement and the relative velocity between the masses M , and M , ,. Thus we have
! ! K , ( t ) = K,(y,(f)..i,,(i)L I = I . 2 .... . N
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS 387
where
and
x N + l(t) ' The dot indicates differentiation with respect to time.
required so that the weighted error norm defined by Using noisy measurements of the response xl( t ) , I = 1,2, ... , N an estimate R, of the restoring force K, is
& = p ( Y A - K ( Y > j 4 l l G (2)
is minimized with, G, a suitable weighting matrix and K = { K J . The equations of motion governing the system are then given by
MN X N + K N b N , LN] - K N - l[YN- l r L N - 1 1 = - Mh' qt)+fdt) Adding the top I equations (I = 1,2, ... ,N) together at a time, and rearranging, we have the N equations,
K,CY~, LJ = wl(t), 1 = 172, ... , N (4)
where 1
I = 1 wl(t) = .x { - M , [ ( q t ) +Xi(t)] +jXt)). 1 = 1,2, ... , N ( 5 )
Since w,(t) contains quantities which are either known or available from measurements, the unknown restoring forces KAY,, j , ) can be estimated.
111. IDENTIFICATION PROCEDURE
(a) General memoryless restoring forces
the measurements Assuming that the measurements x,(t), I = 1,2, ... , N, and ift) are corrupted by Gaussian white noise, we have
<(t) = Xl(t) + nht)
and (6) 4 t ) = qt) + m(t)
Noisy measurements of the various quantities are denoted by circumflexes above them. The noise processes may be assumed to have the following characteristics:
E[ni(tl)n,(t2)] = adtl)6K(i-j)SD(t1-t2), i,j = 1,2, ..., N
E[ndt)m(t , ) ] = 0, i = 1,2 ,... , N (7) and
~ C ~ ( f l ) m ( r , l l = ~ r ( t l ) ~ L ? ( t , -t2) The symbol E [ . ] stands for the expected value, 6, stands for the Dirac-delta function and 6, for the Kronecker delta.
388 1;. E. UDWADIA AND C. P. K L O
The relative displacements x,(t) and velocities k,(t) may be assumed to be either obtained from measurements o r from successive integrations of the acceleration signals. Thus
2 t ) = ;c,(t)+q,(t)
R/[$ l , j,] = kf(t) 4 " / ( t )+ r , ( t ) ,
where p l ( t ) and yl(f). 1 = 1,2. ... . N , are noise processes. The measurement, equation (4), then transforms t o
I 0 I = 1.2 ,... . N
where 1
q( t ) = c -Mi[rn(l)+ni(t)] I = 1
The function K , [ y r , j,] can now be expanded in a double series in terms of two sets of functions {4,J and Assuming that each set is orthonormal with respect to the weighting functions 9 , and g 2 . over suitably defined intervals C , and C,, we have
The coefficients uf j are to be determined so that the error norm
is minimized, say in the least-squares sense. This yields the estimate ,P ,P
. . i I .'
where the measurements f, and 3 are used to replace the exact values 1, and bI.
I ~ 1.2, The response quantities that need to be measured for estimating a specific k,[j,, j J , 1 6 16 N are then kl(fl
, I t 1
( h ) Scpar t ih l~ memor 1'1es.5 re.\toriny force I f i t is assumed that
K,rL,, c.,1 = K , [ ~ ~ , ] +u,[I',I.
K,[O] = D/[O] = 0. i = 1.2, . ,,\7
R,[y,-j + D/[.jj] = M'/(t). I = I , 2, .. , N
= 1.2 .... . N
M l t h
then bq (4) we have
Again expanding R,[\x,] and D,[j,] in orthonormal sets (cbn) and {@,,}, we get
Rf[yfl 5 hf 4A 1 ,) \ - 0
'Inti
Estimates o f hf. and d!; can be similarly obtained by minimwing c l in the least-squares sense
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS 3 89
In the case where equation (13) is applicable, a simpler alternative approach may be followed. As ni t ) , 1 = 1,2, ... , N, is measured for t E (0, T), the quantities j,(t) and $t) can be obtained through integration, if xit) and are not actually measured. Thus times i,, ~ ( 0 , T) can be found such that
- A
j ( t k , , ) = 0 , k = 1 , 2 ,..., k,, 1=1 ,2 ,..., N (16)
For each time c, , which satisfies equation (16) the value of jl(c, I) can be obtained. As the times i,, will, because of measurement noise, be slightly different from those at which j, equals zero, D,[jl(t,)] though close to zero may not exactly equal it. In fact if c,, = tk,, + z ~ , , where rk,, is such that j,(t,, ,) = 0 then using (13).
Thus
R f [Yl(<, 111 = wl(G, 1 ) - ED,(c, I )
The coefficients bf, s = 0,1,2, ... , N,, can now be estimated by minimizing
3' ki N,
' I = kgl gl(Yl(tk,I)) $ , ( $ , I ) - 1 bf, @s(yl(tk,l)) 7 1 = 1, 2,... [ s = o
Estimates of bi then require the solution of the normal equations:
where
' 7 ' N
and the quantities y,(C,,), k = 1,2, ... , k,, have been replaced by their estimates j,(i,,) jk . By a proper choice of {4,,} (e.g. Chebyshev polynomials) and by a proper selection of the points y t (actually achieved in practice by interpolation), the matrix Ss, may often be made a diagonal matrix, so that
4 r C j Ti, (224
where C j can be thought of as a normalized constant. As k, becomes large, and the measurement noise tends to zero, arbitrarily precise approximations to RI[y,]
will be obtained by considering a variety of excitation inputs. Similarly, the set ofpoints tp, , E (0,T) can be found so that j,(t^,, ,) = 0, p = 1,2, ... , p,; I = ,1,2, ... , N, yielding a
normal set of equations similar to (20) and (21), with g1 replaced by g2, replaced by ji and the functions {q5,,(j,)} replaced by {@"( j )} . Again, by a proper choice of {$,,} and a proper selection of points y;, the estimate of dfi can be expressed in the form
d,. J = E . I T! J' (22b) where E j is a normalization constant.
The method outlined above is schematically illustrated (for noise free data) in Figure 2. We begin with the time histories wit), j , ( t ) and y,(t), 1 = 1,2, ... , N. The various times tk,,, k = 1,2, ... , k,, and t,,,, p = 1,2, ... , p l , at which j+ ( f ) and yl(t) are respectively zero are determined. The values of y?(tk,,) and j?(t,,,) corresponding to these times are obtained [Figure 2(a)]. The corresponding restoring forces R,[Y*(tt,l)] and D,[j*(t,,,)] are found as wI[tk,,] and w,[t,,,] respectively. The values of R, and D, are then plotted versus y, and j, respectively [Figure 2(b)].
390 F. E. UDWADIA AND C. P. KUO
I I
I t
R’ T
I (a)
Figure 2.
(c) Forced vibration testing of systems
explicitly knowing (or measuring) the forces off;.(t) if we specifiy that In the absence of a base motion z(t), the identification of K,[y,, j,] can be performed without the need of
16i<l, and = {:rbitrary i > I (2:;;
For noise free data, various arbitrary functions f;.(t), i > / , can be used so that arbitrarily accurate approximations of K , can be found.
TV. ERROR ANALYSIS
The estimates 6ij, 6; and 2: obtained by the simple regression analysis technique outlined in the previous sections differ from the exact values primarily because of the presence of measurement noise. The influence of noise may be thought of as affecting: (1) the measured value of wl(t) and (2) the estimates of yl( t ) and j l(t).
To acquire an appreciation of the effect of measurement noise on the estimates, we shall consider here the case where the restoring force is separable. Error analysis of equation (1 2) for the general restoring force case. besides being more complex, will not, it appears, yield any additional physical insight into the effect of measurement noise on the estimates.
Let the discrete time points Lp,, be utilized where the times fP,, are chosen so that, say, jl(fp,I) = 0.
If we assume that the noise in measuring, i,(t^,, J, p = 1,2, ... , p I ; 1 = 1,2, ... , N , has zero mean, is uncorrelated p = 1,2 ...., p,: 1 = 1,2 ...., N .
and has a constant variance, then by equation (lo),
E[v,(t^,, ,)]=O,VttE(O,T), I = 1.2 ,..., N
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS 39 1
Furthermore, if xdt) and i1 ( t ) , 1 = 1,2, ... , N + 1, are measured then we have the following relations:
and
with pN+ = qN+ I ( t ) = The random variables aI and fl, are assumed uncorrelated, such that for any two times and Lq, E (0, T),
and
The variances of the random errors in measuring x,(CpJ and iitp,,), I = 1,2, ... , N, are assumed to be 0: and u:respectively. The measurements of z(cp,,) and i(fPJ are also corrupted with noise whose variances are uf and 0:. The various variances could be functions of time.
Let r p , l be such that
YI(t,l) = 0, p = 192, *..,PI. (27)
tp .1 = $J,f + T P , I (28)
Define Fp,[ by the relation A
where T ~ , , is the error in finding tp,I. We next expand y & , ) in a Taylor series giving
2 dYl + k d 2 Y, 2 K Y&, I ) = Yl(t, 1 ) + T p , I q
+ higher order terms
Using equation (24) and noting relations (26a) and (23) we get
where the higher order terms in zp , l have been neglected. Thus if j+( tp . l )#O then
For the oscillating system considered, it will generally be unlikely that j ~ l ( t ~ , ~ ) and yl(t,J be both zero, except when the oscillator is executing very small amplitude motions, preparatory to coming to rest. If, however, jdt,J = 0, then the next higher term in (29) can be used to estimate zp,I.
392 F. E. UDWADIA AND C. P. KUO
Then Gif((FPJ can be expanded as
+ higher order terms + ~ ~ ( f , , I )
and .$ ( L p I ) as
u2 I a:(; J _.. h (Lp. = .w,, 1 ) + up, cdp. j1pP, I ) + p , - pi- yr ( tp .
+higher order terms + [)
Neglecting the higher order terms, equation (33) gives
i(t^,,,) = L&, I ) + Y l ( F p , 1 )
where ui,l a:(t [) ..
Yf&. I ) = Q p , 1 @l(t^,, I ) Y,( tp, 1 ) + 2 p , Y f ( ~ p , l ) + P f ( ~ p . f )
Using equation (22b), we have
', % EJ T: p = 5 1 EJ 'l('p, f ) ' ('p, 1 ) )
where tJJ = g 2 ~cl,. Taking the expected value on both sides of equation (36) we get
E C ~ I * 5 Ej EIGX~ ,̂. I ) i ( t p , J)I p = 1
The function $j can be expanded about jI ( tpJ to yield, after some manipulation,
E C ~ I 2 Ej W / ( t p , 1) ~ j ~ t p , 1 ) ) p = 1
where
and
(391
! 40 1
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS 393
with
and
where
bk , l f i t k , l ) - for j j ( l k , l ) # o , and qj = 91 4, Had the signals x l and A, been obtained by integrating x,, the errors al and P I would have been correlated at various times leading to additional terms in equation (40) involving the expected value of the products of a i l , ,)
Equations (38H43) indicate the effect of the measurement noise on the expected value of the coefficients b; and di. We observe that the estimates are biased, the bias being independent of the noise in the measurement of xl(t) and zl(t). The bias is however dependent on the noise in the measurement of xl(t) , i l ( t ) , x,+ l(t) and il+ l(t). If the noise in these measurements goes to zero (i.e. a;s and pis equal zero), all the terms except the first on the right-hand side ofequations(38) and (41) go to zero so that the estimates become unbiased. Furthermore, ifsay, we use the orthonormal polynomials { $ j } or {4j}, greater biases would, in general, be obtained for the coefficient estimates dfi and 6 with increasingj. This is because the higher order polynomials oscillate more rapidly thus leading to larger values of the d$j/djj, dZ $j/dj:, etc., which in turn, by equations (39) and (41), increase the bias.
Further, to illustrate the effect of noise in the measurement of the acceleration terms, let us assume that air), PAL), 1 = 1,2 ,... ,N, are zero. We have then
and BXtp.1).
which for uncorrelated noise gives
But we have by equation (25)
which indicates that the variance of the estimate increases with 1 as well as with increasing magnitudes of the masses Mi, i = 1,2, ... , ,1. Since M I > 0, V 1, it follows that
(+y+ M?
Thus from equation (43), the variance is more sensitive to noise in the measurement of the base motion qr), than it is to noise in X,(t), 1 = 1,2,. . . , N .
V. APPLICATIONS
In this section we present a few select applications of the identification technique discussed earlier. We restrict ourselves to the separable restoring force case (Section IIIb). A much larger list of applications, including identification for the general restoring force case can be found in Reference 28.
394 F. E. UDWADIA AND C. P. K U O
7
I
Motivated by the simplicity of the method, we attempted to investigate its worthwhileness in a simulated real time environment using a small, 16-bit minicomputer with a maximum core storage of 64K bytes. The computations were all performed in single precision and only forty seconds ofdata in each case were analysed. The digitization rate for the data was taken to be 0.04 s, a rate which would allow the multiplexing of several channels using standardly available analogue to digital converters.
M1 = 1
R1
3 ’ 1
(a ) Linear system Consider the system in Figure 3 with the restoring forces given by
Figure 3
I f the system is linear, then
1 = 1.2.3.4 8 -!
1 4
The various parameters of the system are shown in Table I. The system is subjected to the swept-sine wave test excitation.
NL’I = b: I
D,[y] = d: j l
,inti
f,(t) = a, sin [o( t ) ] t , I = 1.2,3,4,
where the time dependent frequency w(t) changes linearly in the time interval, (0, T ) according to the relation.
w(t) = a, + r , f (491
NON-PARAMmRIC CLOSE-COUPLED DYNAMIC SYSTEMS
i
1
2
3
4
395
Ki[y,i] biy + d i i fi(t) = ai Sin[al+u2tlt mi/m*
" 1 n l n2 wo b i / b * di /d* a.
1 .o 0.50 1.00 10.0 2.0 5 40 10.0
1 .o 0.75 0.80 -20.0 2.0 5 40 10.0
1 . o 1 .00 0.60 15.0 2.0 5 40 10.0
2.0 0.50 2.00 -25.0 2.0 5 40 10.0
Table I. Description of linear system
1 SYSTEM1 (Stiffness = linear) (Darnoina = linear) TEST SIGNAL
I . , - I
where
n , and n2 are scaling constants, and To is a normalizing time constant. Figure 4 shows a segment of the excitation signal (described in Table I) at each of the four mass levels for ctl = 2 rad/s,n, = 5, n2 = 40, oo (2z/T0) = ,/(b:/m,) = 10, and T= 40 s. The time scale is shown normalized with - respect to To. A short
ACTING AT MASS M, ACTING AT M A S S M2 - 30r I - 1
24 24 g 18 " 12 z 6 2 0
z -12 z -12 -6
2 0 0 -6
-18 -18
2 - 2 4 t 1 , I , I I , I 2 - 2 4 t 1 I I I I , I , I I -30 -30
ACTING AT M A S S U j ACTING AT M A S S M4 1 I
24 24 g 18 i= 18 y 12 z" 12 2 6 Y
z -12 = -12 2 -18 s -18
3 6 0 y o
(3 -6 -6
2 - 2 4 1 , , I l I , l 1 t! -24 0 4 8 12 16 20 -=o 4 8 12 16 10 -30
TI ME/ To nwT, Figure 4. Swept-sine test signal for identification
portion of the system response to this excitation is indicated in Figure 5. Forty seconds of data (approximately 15 times the fundamental period of the system) are used for the identification. By digitizing these data at equispaced time increments At = 0.04, the digitized time histories x,(t), id t ) and xdt) are obtained.
To study the effect of measurement noise on the identification results, these digitized results are corrupted by the addition ofzero mean uncorrelated Gaussian noise. The same noise-to-signal ratio (N/S) is used for each of
F. E. UDWADIA AND C. P. KUO
RESPONSE OF M A S S M.
2
Y O
-4 50
U @
-50
u Q
R E S P O N S E OF M A S S M,
R E S P O N S E OF M 4 S S M, 1J I 3
..J
0 : P Y O
^" - I . s -LU
250
u 0 u Q
R E S P O N S E OF MASS M4 *I- " "
2 0
5 0 O E -3.6 -6
80
U u o Q
-80 20 24 28 32 36 40
TIML"To
Figure 5. Response of linear system to swept sine forcing
the measurements x,, i,, x,, 1 = 1,2,3,4. The identification results are obtained for three different values of the N/S ratio, namely, 0 ~ 0 0 1 , O . O l and 0.02. Whereas the first number represents data of exceptionally good quality, the second typically represents the situation pertinent to data available from accelerographs, and the third to what may be referred to as 'poor' quality data.
From these 'noisy' measurements, the corresponding time histories i(t), j l ( t ) and R(t) are calculated for f = ] A t , i = 0, 1,2 ...., 1,OOO.
The functions R , and D, are expanded in a series of Chebyshev polynomials { T,} so that Nx Q H
s = o *-0 R,CL',I= x bf xv,) = z hf LS
and No N"
q = O Y - 0 D"LI3 = z 4: T,(j.,) = c df, Pf
The values of N , and N , are chosen to be 4 and 3 respectively.
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS 397
The coefficient estimates bi and 3 are obtained (by performing a least-squares fit) by solving the normal equations [equations (20) and (21)].29 To improve the quality of the fit,z9 the &fk,,) and 2tp,,) arrays are normalized so that they lie in the interval ( - 1,l). Using the weighting functions g l (q ) = g2(q) = (1 -qz)-*, the coefficients 6; and 2: are found. For ease of comparison with the exact R,'s and D,'s, these coefficients are converted to 6 and 4 corresponding to the polynomial expansions [equation (5 l)]. These monomial base coefficients lend ease to the physical interpretation of the various terms in the expansion of R, and D,.
Figure 6 shows the results of the identification giving the estimates of the intermass stiffness (R,) and the intermass damping (0,) as functions of relative displacement and velocity respectively. The least-squares polynomial fits are calculated at the various points At,,,) and kt,,,) for various noise-to-signal ratios. The exact stiffness and damping are also plotted at the same values of 3 and 3 for comparison. As seen from the figure, the estimates gradually worsen with increasing values of N/S. The estimated coefficients of the polynomials are shown in Tables II(a) and II(b) for each R, and D,, I = 1,2,3,4. We observe that, in each case, the estimated coefficients for all except the linear term are small.
A measure of the accuracy of the identified stiffness and damping can be obtained by defining the root-mean- square errors (rms) as
and (52)
111 = [I [Dl - D,12 ,. dL, ;r.:dAl;i where the integrations are carried out over the complete response range of y, and j l respectively. The rms errors are indicated for each R, and D, and each N/S ratio in Tables H(a) and II(b).
It is interesting to note that the rms error does not change substantially when the N/S ratio changes from 0001 to 001. This is because of the fact that for such low values of the N/S ratio the digitization process as well as the single precision accuracy of the computations (which leads to round off) actually dominate the accuracy of results. Computations done in double precision" show lower rms errors for N/S = 0.001. We note from the tables that, in accordance with our discussion in Section IV, the rms error increases with increasing i values.
A comparison of the predicted response using I?, and b, and the exact response for an excitation different from the test excitation, and comprising of a base acceleration, qf), is indicated in Figure 7(a). This base acceleration is actually a sample ofzero mean Gaussian White Noise (ZMGWN) with a standard deviation (a) of unity. The stiffness and damping estimates corresponding to the N/S = 0.02 case are used. We observe that the predicted responses, using the identification results obtained even under very noisy test conditions (N/S = 0-02), and the exact responses are reasonably close to each other.
The solid lines in Figure 7(b) show the response of the system when mass M , is subjected to an impulsive (delta-function) force of ten units. The predicted response of the system, using the identification results obtained for N/S = 0.02 (Tables II(a) and II(b)), is indicated by the dashed lines. Again, the predicted response matches well with the exact response.
(b) Non-linear systems Two non-linear systems have been considered. They represent non-linearities which are often encountered
in structural and mechanical systems. The first system has non-linear stiffness and linear damping of the form
and
The system description is given in Table 111. We note that whereas R,, R,, R, represent 'hardening' non- linearities, R, repesents a 'softening' nonlinearity. The test signal used is identical to that used for the linear system described in Table I. Using N , = 4 and N , = 3 ,& and & were obtained. The estimated functions R, and D, are shown, as before, in Figure 8. Tables IV(a) and IV(b) give the estimates for the coefficients of the
All 30 13A
8 PO P- 8-
I
oz 01 0 01- oz-
/
oz 01 0 01- oz-
8-
P-
O Pa
P
8
P-
O
P
8
'a
01-
O 'a
01
51 01 s 0 5- 01- Sl-
orin
P-
O 'a
P
8
lN3W33VldSIa
1 O L-
Z 1 0 1-
00 1-
0 %
001
00 1-
0 z8
00 1
-
OP-
OP
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS
Table II(a). The coefficients of the identified stiffness of System 1
i
1
2
3
4
399
i R . = blY
b;
50
7 5
100
50
0.07721
0.44445
0.36444 0.06970 0.00442
0.36817 0.27012 0.00607
0.001
1 0.5 0.010
0.00033 0.51512 -0.00034 0.00008 0.04534
-0.07826 0.53241 0.00085 -0.00006 0.05553
-i *i 2 *i 3 -i 4 * ' R . = b' + bly + b2y t b3y t b4y 1 0
N / S
- 0.001 0.59691 2.7956 -1.0551 0.01552
0.35053 2.5490 -0.57827 0.01527 0.010
0.020 0.15696 I 2.4581 1-0.19538 I 0.01617 I 0.001 -0.01 824 74.250 0.23820 I 0.39391 1-0.13063 1 0.00409 I 0.010 0.06529 74.325
0.020 0.17733 74.335
0.001 -0.00021 99.436 0.09831 I 0.479771-0.10275 I 0.00232 I 0.010 0.061 44 99.631 0.015141 0.149041 0.11477 I 0.00291 I 0.020
0.001
0.010
0.020
-
- -
0.09254
-0.06270
-0.04072
-0.01943
99.732
48.997
49.149
0.684841 0.442781-0.56636 I 0.01198 I 49.240
Table II(b). The coefficients of the identified damping of System 1
0.020 5 0.001
II II I I -0.07995 1 1 0.54058 1 1 o.00111 I -0.00012 I 0.06109
0.03075 1 1 0.41320 11 0.00008 I 0.00000 I 0.03410 I -0.06653 1 1 0.43327 11 0.00005 I -0.00002 I 0.06786 I
1 1 0.020 1-0.09249 1 1 0.45110 1 1 0.00005 I -0.00003 I 0.10025 I
300 F. E. UDWADIA AND C. P. KUO
I- Z
3 4 w U
L lA - n
I- Z 2 4 u1 U
a vr - n
+ Z ? 4 w U
a m - a
I-
t
4
25 w U
a
0 VI -
z 0
3 f 3 V 4
RESPONSE TO RANDOM BASE MOTION
RESPONSE OF MASS M1 0 . 3 r i 1 1 I 1 1 1 0.2
0.1 0
-0.1 -0.2 -0.3
RESPONSE TO I MPULS I VE FORCE ON MASS M I RESPONSE OF M A S S M,
2.4 I I I I I I l l ' ] 1.8 -4 I
+ z 1.2 y 0.6 ; f ; o 2 -0.6 z -1.2 n
- 1 a
I
-0.3 I l l l l l RESPONSE OF M A S S M-,
0.18 0.12 0.06
0 -0.06 -0.12
.J
0.12 0.06
0 -0.06 -0.12
- 2 . o y 1
-3.00 5.0 10.0 15.0 20.0 TIME/To
(a)
i_i ..-
-2.41 I ' I 1 I ' ' I
RESPONSE OF M A S S 4
RESPONSE OF MASS M3 1.61 I I l I I 1 I 1 1.2 0.8 0.4
0 -0.4 -0.8
RESPONSE OF M A S S M4 1.2 0.9 0.6 0.3
0 -0.3 -0.6 -0.9 -1.2
0 6.0 12.0 18.0 24.0 30.0 TIME/Tca
(b) f igure 7 A comparison ofthe response of the actual system( ~ ) wlth that of the identified system ( ) (a) under base e w t d t i r v
(b) under dn impulse force of the ten units applied dt M , for System 1
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS
i m. /m*
1 1 .o
2 1 .o
40 1
' i 3 ' Ki[y,);1 = b;y + b3Y + d i i
b; / b * b i / b * d; Id*
0.50 0.10 1 .oo 0.75 0.25 0.80
I
Table 111. Description of the system with non-linear stiffness and linear damping
1 m*=l , b*=100, d*=0.5 I
polynomial series representation of R, and D,. The coefficients are obtained via the Chebyshev polynomial expansion as mentioned earlier. The rms values for different N/S ratios are also indicated. It is seen that the identification procedure leads to fairly good estimates even when using noisy (N/S = 1/50) test data.
Figure 9(a) shows a comparison between the predicted response of the system [using the identification results of Tables IV(a) and IV(b)] and the exact response of the system when the system is subjected to twice the amplitude of the ZMGWN base acceleration used before [Figure 7(a)]. Identification results corresponding to the N/S ratio of 0.02 were used. Figure 9(b) shows the predicted and actual system response to an impulsive force of ten units applied to mass MI.
Lastly, a system with non-linear damping and non-linear stiffness is chosen with
and 1 = 1,2,3,4 RICYll = 6: Yl+ b: Y:
Table V shows the actual parameters of the system. Identification of the coefficients 8 and b is done using N , = N , = 3 with the test signal defined in Table I. The estimated functions R, and D, for different N/S ratios as well as the exact functions are plotted in Figure 10. The results together with the rms errors are shown in Tables VI(a) and VI(b). Figures 1 l(a) and 1 l(b) show the responses of the actual system and the identified model to the base acceleration of Figure 7(a), and the same impulsive loading used before.
VI. CONCLUSIONS AND DISCUSSION
A relatively simple non-parametric method for the identification of a class of close-coupled non-linear multi- degree-of-freedom systems has been developed. The class of systems is one which is often encountered in the fields of mechanical and structural engineering. Identification of arbitrary memoryless non-linearities is possible through knowledge of the accelerations, velocities and displacements of the various masses. These quantities are then used to obtain by regression techniques the surfaces of the restoring forces as functions of the intermass displacements and velocities.
A particularly simple technique is illustrated when the restoring force is linearly separable into two functions, one of intermass velocity and the other of intermass displacement.
An assessment has been made of the effect of measurement noise on the estimates of the coefficients that are obtained from the regression analysis. It is found that whereas the biases in the estimated coefficients are primarily dependent on the noise in the displacement and velocity measurements, their variances are controlled to a good extent by noise in the acceleration measurements.
402
N/S = 0.001-
N/S = 0.02
t-N/S = 0.01 -
F. E. UDWADIA AND C. P. KUO
/
N,S = OiOOl N/S = 0.01
Figure 8. A
,
80
40
0 R1
-40
/ /i- N/S=0.02 1
200
100
R2 0
-100
-200
100
R3 0
-100
40
20
R4 0
-20
-40
8
4
D l 0
-4
-8 -008 -0.4 0 0.4 0.8 -10 0 10
-1 0 1 2
I I I I
-1 0 1
- 1 0 1
D IS PL AC EM E N T
comparison of the actual stiffness and damping denoted by ) and noise (------), with 1 per cent noise ( - -
10
D2 0
-10
D3
-20 0 20
8
4
D4 O
-4
-8
-8 -4 0 4 8
V EL OC ITY
solid ( with 2 per cent noise ( -- 4 for System 2
) lines and the identification results ,with 0.1 percenf
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS
Table IV(a). The coefficients of the identified stiffness of System 2
0.001
0.010
0.020
403
-0.00499 39.677 -0.02314 20.525 0.04931 0.00147
0.10660 100.160 1.0550 19.993 -1.4486 0.00350
0.17982 100.56 2.2282 19.527 -3.0004 0.00701
Ri =b;y+b,y ’ i 3
i =
bl’
1 50
2 7 5
0.010
0.020
0.001 0.00902
0.010 0.02193
0.020 0.03476
-0.02244 49.146 0.28733 -9.4405 -0.30165 0.01273
-0.05730 48.607 0.60696 -9.1306 -0.71023 0.02433
0.001 1-0.02578 11 74.441 11 0.02301 11 25.556 11 -0.01337 I 0.00332 I
1
2
3
4
0.010 I 0.06611 11 74.486 11-0.1827011 25.566 1 O.O3323/ 0.00365 I
0.001 1-0.0202811 49.627 11 0.02107~~-9.7352 11 0.08131 1 0.00458 I
0.020
0.001
0.010
0.01377 0.42822 -0.00091 -0.00002 0.06562
0.00748 0.30273 0.00001 0.00003 0.03412
-0.02072 0.30118 0.00049 0.00004 0.03714
Table IV(b). The coefficients of the identified damping of System 2
0.001
0.010
0.020
0.08478 1.18900 -0.01297 -0.00428 0.07780
-0.30175 1.09520 -0.00124 -0.00217 0.07152
-0.64619 1.00790 0.00849 -0.00028 0.10322
0.010 1-0.01952 1 1 0.42118 11 -0.00070 1-0.00001 I 0.05485 I
0.020 I 0.08003 1 1 0.29381 11 0.00043 I 0.00006 I 0.05204 I
0. c 0. z 0. 2 0. 5 0. < -I
& -0. 0 -0.
-0. -0. -0.
F. E. UDWADIA AND
RESPONSE TO RANDOM B A S E MOTION
R E S P O N S E OF MASS M, I
0.5 + 0.4
2 0.2 5 0.1 4 0 & -0.1 0 -0.2
z 0.3
-0.3 -0.4 -0.5
5 4 3 2 1 0 1 2 3 4 5
I
0.4 3 0.3 2 0.2 w 0.1 v o 4 a- -0.1 - -0.2
-0.3
Vl
0
-()..“II I I I i I I I { -0.5
R E S P O N S E OF M A S S M, 0. 0.
w 0. 5 0.
2 0.
2-0. v
2 - 0 . 0-0.
-0. -0.
30 24 18 12 06
0 06 12 18 24 30
J
R E S P O N S E OF MASS M. 0.30 0.24
w 0.12 5 0.06 u o 2-0.06 --0.12 n -0.18
4
5 0.18
vl
‘ ‘ ‘ I I ‘ I i 0 4.0 8 .0 12.0 16.0 20.0
T I ME/T
(a 1
C. P. K U O
RESPONSE TO IMPULSE FORCE ON MASS M i
R E S P O N S E OF M A S S M.
1.6 0.8
0 -0.8 -1.6 -2.4 C -4 -3.2 -4.0 ’ I I I I ’
R E S P O N S E OF WSS M2 I , , , , , , , ,
01 12 0.06
0 -0.06 -0.12 -0.18- -0.24 - - -0.30 I ’ I I I ’ I ~
R E S P O N S E OF MASS M3
1.2 0.8 0.4
0 -0.4 -0.8
-1.6 -2.0
2.0 1 . 6 c
R E S P O N S E OF MASS M4 1 ’ 1 , 1 1 , , 1
1.2 0.8 0.4
0 -0.4 -0.8 -1.2 1 -4 -1.6 1
0 6.0 12.0 18.0 24.0 30.0 TIMEA
(b)
Figriie 9 A comparison of the response of the dctudl system ( -1 with thdt of the identified system( J (‘i) rintici b>iw t ’xc i t , i t !o
(b) under an impulse force of the ten units applied at M , for Svskm 1
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS
i
1
2
3
4
405
m. /m*
b i / b * b;/b* d;/d* d i / d *
1 .o 0.50 0.10 0.60 0.04
1 . o 0 .75 0.20 0.40 0.04
1 .o 1 .oo 0.25 0.40 0.04
2.0 0.50 -0 .10 0.20 0.04
Table V. Description of the system with both non-linear stiffness and damping
SYSTEM 3 (Stiffness = nonlinear) (Damping = nonlinear)
1 m*=l , b*=100, d*=0.5 I
The technique is implemented on a small 16-bit minicomputer and the calculations conducted in single precision. Even under very noisy measurement conditions (N/S ratio of 1/50), with only a few terms in the series expansion, the identification results yield low rms errors. The capability of predicting the response of the system to excitation other than the test excitation, by using the results from identification, has been illustrated. As has been observed in other studies’ accurate estimates of the damping are generally more difficult to obtain than estimates of the stiffness.
A drawback of the method is that it can only be used for identifying memoryless intermass non-linear restoring forces. This is so, because expansions of the type given by equations (1 1) and (13, where the y,’s and j,’s are treated as independent variables, are only valid if the restoring forces are single valued functions of the independent variables. Thus, for example, in a bilinear hysteretic system in which the restoring force is a multivalued function of y , and j , , the technique would fail. Alternatively speaking, for such systems, one could find a class of inputs which would yield incorrect identification. A simple example of such a class of inputs for the bilinear hysteretic case is the class of impulsive excitations which cause permanent displacements of the system.
The main advantages of the method are: 1. 2.
3.
4.
5.
The method is applicable to general memoryless intermass non-linear restoring forces. There is no limitation on the nature of the test excitation that can be used for the identification. This is a major advantage over some of the other non-parametric methods”. which often require Gaussian White Noise (GWN) excitations. Su&h GWN excitations are difficult to produce in high enough magnitudes in order to drive multi-degree-of-freedom systems, which are often large, in their non-linear ranges of response. The computational requirements, both in terms of CPU time as well as storage, are very small in comparison with the Wiener method making the method attractive for real time identification.” In fact, the results reported herein were computed using a 16-bit minicomputer with a 64KB memory. The duration of time over which the data are required to be taken is comparatively small compared to other non-parametric techniques.” The identification results obtained are relatively insensitive to measurement noise. The rms errors in the
determination of the restoring forces, increase in general, as we move towards the point of fixity of the system.
ACKNOWLEDGMENTS The research presented in this paper was sponsored by the National Aeronautics and Space Administration as well as a grant from the National Science Foundation. The NASA part of the effort was performed at the Jet Propulsion Laboratory, California Institute of Technology under contract No. NAS7-100 sponsored by Dr. A. Amos, Materials and Resources Division, Office of Aeronautics and Space Technology.
-1.5-1.0-0.5 0 0.5 1.0 1.5 -10-8 -6 -4 -2 0 2 4 6 8 D IS PIAC EME NT VELOCITY
Figure 10 A comparison of the actual stiffness and damping denoted by solid ( ) lines and the identification results uith 0 1 per cent noise ( - -). with 1 per cent noise ( - - - -) and with 2 per cent noise ( ~ ~ ) for System 3
0.020 -0.08937 49.501 0.04032 -9.6386 0.00586
i '
1
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS
Table VI(a). The coefficients of the identified stiffness of System 3
"i 2 ^i 3 D . = di + dry + d2y + dy 1 0 Di =d Y+d 4Y
Table VI(b). The coefficients of the identified damping of System 3
I I I I
407
408 F. E. UDWADIA AND C. P. KLTO
RESPONSE TO RANDOM BASE MOTION
RESPONSE OF MASS M1 1 0.4 r--
I- $ 1.2
8 0
K? -1.2
Z 0.6
2 -0.6
Q
RESPONSE TO IMPULSIVE FORCE ON MASS MI RESPONSE OF MASS M,
2.4 1
1.8 ~
-Oe3 -0.4 L ---Id RESPONSE OF MASS M2
1 0.4 [-
-1.8 -2.4 ! 1 . - -I.- . 8
RESPONSE OF M A S S M2
o 6 I----- 0.3 1 1.2
0.8 2 0.4
0
0 -0.8
c z 0.2- W
2 a -0.4 v)
RESPONSE OF MASS M3 0.24 1 0.18 I-
Z b-
Y 5 0.06 W
4 3 0
n - -0.12 Q
z 0.12
U
CL v) v) - 2 -0.06
-0.181 -0.24 , , , 1 Y
R E S P O N S E OF W S S M4 0.24
1 __ -L-- . - -1.6 RESPONSE OF M A S S Mg
2-4r----- 1.8 -- - - 1.2 0.6
0 -0.6
1.2
0 -0.12 -~
-0.18 .
-0.24 ' 1 I 0 4.0 8.0 12.0 16.0 20.0
TI M€/T, ia )
- 1 . 2 1 , I , 1 -1.6 d
0 6.0 12.0 18.0 24.0 30.0 TI ME/T,
(b 1 Figure 1 1 A comparison of the response of the actual system ( ) with thdt of the identified system ( 1 (a ) under b'rse sxcitnLio6-
(b) under an impulse force of the ten units applied at M , for System 1
NON-PARAMETRIC CLOSE-COUPLED DYNAMIC SYSTEMS 409
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