Pergamon
00457949(94)EOO579
Cm~urers & Sfrurrura Vol. 52. No. 3. pp. 563571. 1994 Copyright 1: 1994 El&r Sxnce Ltd Printed in Great Britain. All rights reserved 00457949194 57.00 + 0.00
ADAPTIVE TIME SCHEMES FOR RESPONSES OF NONLINEAR MULTIDEGREEOFFREEDOM SYSTEMS UNDER RANDOM EXCITATIONS M. L. Lru and C. W. S. Tot Department of Mechanical Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 (Received 22 January 1993)
AbstractThree adaptive time schemes are incorporated into the stochastic central difference method with time coordinate transformation for the determination of the covariances of responses of multidegreeoffreedom nonlinear systems. The results of a Dulling’s oscillator and a twodegreeoffreedom system with asymmetric nonlinear stiffness indicate that the proposed procedures are very accurate and efficient to apply compared with digital simulation. They are free from computational instability. A relation for the computation of the time step size required in the implementation of the procedures has also been presented.
1. INTRODUCTION
The response analysis of complex structures, with material and geometrical nonlinearities, under stationary and nonstationary random excitations has been the focal point of investigation by the second author since 1982. The stochastic central difference (SCD) method for such analysis was introduced in 1984 [l], and further developed and presented in [2,3], for example. Over the last seven years various similar direct integration methods for
the random response of multidegreeoffreedom (MDOF) systems have also been presented. A recent example is the stochastic version of the Newmark family of algorithms [4]. These direct integration schemes have been developed for the computation of responses of linear and nonlinear MDOF systems. There are several important advantages of the aforementioned direct integration techniques over other existing procedures available in the literature. First, these direct integration techniques do not require modal transformation which can prove to be very expensive and inaccurate for systems with many degreesoffreedom (DOF). Secondly, there is no restriction on the types of damping present in the systems. Thirdly, they are applicable to systems whose nonlinearities are not explicitly defined. The latter are typical in analyses applying finite elements (FE). They are also applicable to systems with nonlinearities which are explicitly defined as functions of displacements and velocities. In [571, examples employing the SCD method in conjunction with statistical linearization (SL) have shown that the SCDSL 7 Author to whom correspondence should be addressed.
technique is effective and accurate for the computation of covariances and timedependent mean squares of the responses of nonlinear systems. In a parametric study [7] of the SCD method for a nonlinear singledegreeoffreedom (SDOF) system under stationary Gaussian white noise and in [5], it has been shown that the time step size used in the computation is reduced with increased natural frequency. This suggests that for nonlinear systems the time step size at every time step has to be changed. Consequently, many direct integration schemes may be derived by using a combination of the stochastic Newmark family of algorithms [4] and some time step size updating strategies. In what follows three representative schemes are proposed. These are the stochastic central difference and adaptive time scheme (SC&ATS), the stochastic central difference and adaptive time scheme with time coordinate transformation (SCDATST), and the stochastic central difference and time coordinate transformation with adaptive time scheme (SCDTATS). For simplicity, these three schemes shall be collectively called SCDMATS. In the SCDATS the lowest natural frequency of the nonlinear or linearized system at every time step is evaluated so that the time step size for the application of the SCD can be determined in accordance with the procedure given in [7]. In the SCDATST the nonlinear or linearized system at every time step is first transformed into its corresponding dimensionless time coordinate and then the adaptive time scheme (ATS) is performed. The response is evaluated in the dimensionless time coordinate and subsequently converted back to the original time coordinate. This scheme may prove to be relatively much more expensive than the
564
M. LIU and C. W. S. To
SCDTATS for nonlinear systems with a large number of DOF as it requires time coordinate transformation (TCT) at every time step. The SCDTATS involves with the TCT being performed once at the beginning of the computation before the application of the ATS. It may be appropriate to note that the technique of employing the SCD with TCT, first proposed by the second author in [8], eliminates the computational instability inherent in the SCD method and therefore reduces very significantly the computational costs for stiff systems. This has clearly been demonstrated in [9] in which discretized beam and plate structures were considered. The following section includes an outline and implementation of the SCDMATS. Section 3 is concerned with results of a series of numerical studies of the SCDMATS for two nonlinear systems. The first system is a Dulling oscillator under nonstationary random excitation, and the second one is a two DOF system with nonsymmetric nonlinearities excited by a nonstationary random process representing an earthquake excitation. The first system has been studied in [5] while the second one has been investigated by Kimura and Sakata [lo], and the authors [1 I]. A summary and concluding remarks are given in Sec. 4. 2. STOCHASTIC
CENTRAL DIFFERENCE WITH ADAPTIVE TIME SCHEMES AND IMPLEMENTATION
The SCDATS, SCDATST, and SCDTATS or collectively called the SCDMATS reported in this paper are simple and easy to implement in a digital computer program. The crucial steps in their implementation are (a) the selection of the time step size, and (b) TCT when the systems have relatively high natural frequencies. These two steps will be considered in this section. For systems whose nonlinearities are explicitly defined and whose number of DOF are small, determination of their equivalent natural frequencies may be appropriate as application of the equivalent natural frequencies in the ATS reduces the computational time required for evaluation of the natural frequencies at every time step. For comparison and illustration, and as the two nonlinear systems in this paper have one and two DOF, the strategy of using the equivalent natural frequency of a system is also adopted in the following. Of course, for nonlinear systems with a large number of DOF derivation of the equivalent natural frequency can be formidable. if not impossible. For such systems the application of the equivalent natural frequency in the SCDMATS is not feasible.
Table 1. Time step size and natural frequency w
Al
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.0 1.2 1.4 1.6 1.8
0.9975 0.9902 0.9785 0.9630 0.9443 0.9232 0.9006 0.877 0.8525 0.83 0.78 0.74 0.69 0.65
w
Al
2.0
0.61 0.57 0.53 0.48 0.45 0.42 0.39 0.37 0.35 0.33 0.1812 0.04876 0.02816 0.0198
2.3 2.6 3.0 3.3 3.6 4.0 4.3 4.6 5.0 10.0 40.0 70.0 100.0
Results for the range of natural frequencies that are less than 1.0 were presented in [9]. Table 1 summarizes the data taken from [7] and [9]. The dependence of At on w is shown in Fig. 1. It is observed that as the natural frequency reduces to a small value the time step size At approaches 1.0. From Fig. 1 the relation between At and w may be written as At = 0.83  0.72 log,,, w, At = 1.00.053~
I .O < w < 5.0
O.l2w’,
(la)
w < 1.0,
(lb)
where w is in rad/sec. In arriving at eqn (1) the following pairs of (0, At) data have been used: (0.0, l.O), (0.5,0.9443), (1.0,0.83) and (5.0,0.33). Note that eqn (1) does not include a formula for w 2 5.0 rad/sec. This is because if a natural frequency is higher than the latter the time step size becomes too small to be computationally effective. When the natural frequency of the system is higher than 5.0 rad/sec the TCT of [8,9] should be used. If necessary, a formula for the relation between At and w that is not included in eqn (1) can be similarly constructed. Such a construction can be performed by applying the results from [7]. It should be noted that the results obtained by using the SCD method with TCT for MDOF linear systems reported in [9] were based on the dimensionless time step sizes associated with the dimensionless tstochaetlo centraldlfferenoe
method) 1
data from Table 1
2.1. Time step size and natural frequency Determining the relation between the time step size At and the natural frequency w of a SDOF system is the first step in implementing the SCDMATS. The parametric study in [7] provides a methodology for constructing such a relation and many pertinent data.
0.5 0.4
: y
0.3 ” 1o1
2
9 4 60
100
2
945
Fig. 1. Relation for time step size and natural frequency.
565
Time schemes for responses of nonlinear systems
fundamental natural frequencies. The rationale behind this was to better represent the response caused by the fundamental mode. This reasoning has also been adopted by Bathe and Wilson in [12] for deterministic responses. The pitfall with this approach in the deterministic response computation is that the time step size may well exceed the critical time step size, if the fundamental and the highest natural frequencies are far apart from each other. Indeed, the wide separation of the fundamental and highest natural frequencies occurs in many structural systems represented by the FE. The SCD with TCT scheme has eliminated this pitfall because the dimensionless time step size associated with the dimensionless fundamental natural frequency reaches 1.0 as the dimensionless fundamental natural frequency approaches zero. This dimensionless time step size of 1.O is well below the dimensionless critical time step, (As),, = 2”2 according to eqn (27) of [4], of the highest dimensionless natural frequency which is 1.
2.2. Equivalent natural frequency For systems whose nonlinea~ties are explicitly defined and whose numbers of DOF are small, the determination of their equivalent natural frequencies may be appropriate as the application of the equivatent natural frequency in the SCDMATS reduces the computational time required for the evaluation of the natural frequencies at every time step. Therefore, the strategy of using the equivalent natural frequencies is adopted in Sec. 3. 2.3. Time coordinate transformation Consider the following matrix equation of motion for a stiff MDOF MX+CX+KX=F,
(2)
where M, C and K are the assembled mass, damping and stiffness matrices, respectively, x, X and X are the stochastic acceleration, velocity and displacement vectors and F is the stochastic excitation vector. The matrices M, C and K are understood to be (a) time dependent for genera1 nonlinear MDOF systems, or (b) the assembled mass, damping and stiffness matrices, respectively, of the linearized MDOF system if the equivalent natural frequency approach mentioned in the last subsection is employed. It is assumed that the linear or linearized system of eqn (2) has its highest natural frequency SL Divide both sides of eqn (2) by the square of R and transform the resulting equation of motion from the time coordinate t to the dimensionless time coordinate r such that
where the prime and double prime designate the first and second derivatives with respect to the dimension
less time 7; the response vector x and the forcing vector F are functions of r which is chosen as z =Rt.
(4)
In eqn (3) the highest dimensionless natural frequency is 1. Thus, the dimensionless critical time step is 2”2 according to eqn (27) of [4]. The recursive covariance matrix expression for eqn (3) can be obtained by the procedure in [4], for example, as R(s + 1) = N,R(s)N: +(At)4N~~(~)N~+ where the superscript and
+ N, R(s  I)N; N,D(s)N;+
N,DT(s)N:,
T denotes the transpose
(5) of
D(s) = N,R(s  1) + N3DT(r  1). Equation (5) is understood to be in the dimensionless time domain such that R(s) =
, D(s) = @(r,)xT(r,
,)>,
in which the angular brackets denote ‘ensemble average’ of the enclosing quantity, while S, and +(r,) are the spectral density of the zeromean Gaussian white noise process and the discrete deterministic modulating function vector of the excitation vector in eqn (3), respectively. Once the variances and covariances of the responses in z are determined by using eqn (5) they are converted back to the t domain. They are related by the following formulae R,(s) = OR(s)
(64
GW where the subscript t denotes the recursive covariance matrix expression in the t domain. 3. NUMERICAL STUDIES
In this section of the Duffing’s oscillator under a nonstationary random excitation, and a two DOF nonlinear system excited by an earthquake are studied. These two systems have been studied and Monte Carlo simuIation results have been presented in 15, 111. Therefore, comparison can be made and the effectiveness of the presently proposed algorithms can be assessed.
M. LIU and C. W. S. To
566 3.1. Dujjing’s oscillator
The Duffing’s oscillator considered here has the following equation of motion [5]
where x, i and jt are the displacement, velocity and acceleration of the oscillator, [ is the damping ratio, E the strength of the nonlinearity and w the natural frequency of the corresponding linear oscillator (that is, when 6 = 0). If the excitation r(t) is stationary random, u is defined as a* = 7rS/(2403). If the excitation is nonstationary random, a2 is the maximum value of the variance response of the corresponding linear system. The excitation is defined as r(t) = e(t)w(t), where e(t) is a deterministic envelope function and w(r) the zeromean Gaussian white noise. For this example, e(t) is defined as e(t) = 4.0(emoo5’ co.‘o’).
(8)
Applying the SCDSL technique, the recursive expression for eqn (7) is given as [5] R(s + 1) = a*R(s) + b*R(s  l)+ZabD(s)
+ z2B(s),
D(s) = aR(s  1) + bD(s  1),
(9)
where 12 (o,At)*l b = _(I 
Without ambiguity the responses and excitation in eqn (9) are understood to be in the t domain. The equivalent natural frequency at time instant t, is[5]
Note that the above equations also hold true in the t domain provided that one substitutes Ar for At, and recognizes that all the other quantities are with respect to the t domain. The variance of response for the system in eqn (7) is evaluated here using the proposed SCDMATS. Note that the SCDTATS is identical to the SCDATS numerically in this particular example. In the SCDATS, eqn (10) is used to determine the timedependent natural frequency at time I, which is substituted into eqn (1) to obtain At,, the time step size at time instant tS. At, is then applied to determine the terms a, 6 and Z, and the variance of response at the next time instant R(s + 1). Therefore, in the entire computational process the time step sizes are computed automatically. In the SCDATST the w,(s) of eqn (10) is applied to perform the TCT. Consequently, in this particular example the dimensionless natural frequency of the transformed oscillator in the T domain is always 1.0 and the dimensionless time step size in the r domain is always 0.83. Of course, for MDOF nonlinear systems, the time step size in the 5 domain, in general, varies from one time step to another. Table 2 lists the results applying the SCDATS and SCDATST, and compares them with those from [5] using SCIX4L. Note that the SCDSL results have been found to compare very well with the digital simulation data [5]. Table 2 indicates that the results obtained by employing the SCDATS and
Table 2. Variance of response of the Duffing’s oscillatort Maximum variance response
i
c
SCDSL
SCDATS
10‘O
0.01
1.0
0.05
I.0
0.5
0.0 0.1 0.3 0.5 1.0 5.0 0.0 0.1 0.3 0.5 1.0 5.0 0.0 0.1 0.3 0.5 1.0 5.0
51.659 x lo” 43.358 36.358 31.918 26.242 14.861 23.53 19.97 16.02 14.20 Il.15 6.35 3.1 2.48 2.0 1.76 1.41 0.71
51.659 x IO” 44.756 37.209 32.782 26.575 14.344 23.533 19.634 15.789 13.716 10.933 5.783 3.137 2.507 I.969 1.695 I.340 0.707
SO
t The
SCDATST 51.659 x lo“’ 47.969 42.259 38.206 31.813 17.784 23.533 20.366 16.738 14.610 11.712 6.227 3.137 2.533 2.001 I .726 I.366 0.714
SCDSL results are from, [5] for S = IO“’ and [7] for s = 1.0.
561
Time schemes for responses of nonlinear systems
SCDATST agree very well with those using the SCDSL. This is particularly true for the cases with damping ratios higher than 0.05. 3.2. A two degreeoffreedom
p=M2, WC?,
system with nonsym 
metric nonlinear stlflness
This two DOF nonlinear system, shown in Fig. 2, has previously been investigated by Kimura and Sakata [lo], and the authors [ 111.It may be applied to study soilstructure interaction or a primary structure and secondary equipment under an earthquake excitation treated as a nonstationary random process. The equation of motion for this system expressed in relative displacements X, = y, y,, and XZ= yz y, (where the lower cases denote absolute displacements) is
and w(i) is the Gaussian white noise with a zero mean and a spectral intensity S,. At this stage two routes can be followed in applying the SCDMATS. The first route is to apply the SCDATST introduced in Sec. 2 directly to eqn (1 1), while the second route is to linearize eqn (11) so that the equivalent natural frequencies of the system may be obtained. The latter route is followed here in this study as the present system has only two DOF and its nonlinear restoring forces are defined explicitly. In [ 111it was found that the SCDSL scheme IV gave excellent results compared with those by digital simulation. Therefore, the SCDSL scheme IV is adopted here for the linearization of eqn (11). Let the linearized equation be Mx + CB + KJ
= h
{ 0
=
I{
i=w,t
“2
ml
=f(i).
(12)
Following the procedure in (1 I] it can be shown that the dimensionless time dependent mean (from here on it shall simply be called mean unless ambiguity arises)
e(f)W)\ 0 I
m(s + 1) = N2m(s) + iV,m(s  1)
or in a more compact form
A42 + CA?+ KX + g(X) = f(S), where the overdot and double overdot denote, respectively, differentiation with respect to dimensionless time once and twice. The other symbols in eqn (11) are
(13)
 At’N, (g(WQ)> and the mean squares of responses R(s + 1) = (AQ4N, B(s)Nf+ N2(s)R(s)N2(s)T +N,R(s
 l)N;+N,(s)D(s)N;
+ N, D (s)~N~(s)~ D(s) = N,(s)R(s
 1) + N,D(s
 l)T
(14)
Y2
where l_
(4 (is))
m(s) =
i
(X2(fs)>
(Xl
i
w2>
R(s) =[ (X2(f,)Xl (i.,)>
(xlKJx~(~~l)) (x,(i,)x2(f,,))
KI
Cl
Y0
D(s)= [
$
BAaE
Fig. 2. Two
DOF
N, = (M + O.SArC)’
system with nonsymmetric nonlinearities.
N2(s) = N, [2M  (Ai)‘K,(s)]
(x2(f,)x2(iy,))
1
568
M. LIU and C. W. S. To
m, = N, [2M  (Ar)%,,]
In this study the system described by eqn (11) has the following modulating function
NJ = N, (OSAX  M)
(15)
(g(x(‘y))’ =
Pl<(x2(fs))2)

(1  ~)tl((&(fs))‘) Kc=&+
0 0
&rlGf2(fs)) 2(1  p)ttV2(fs))
{I) 0 0 W)=W)= o [ o1> 0 0 NO)= o o [I 0 o
(18)
such that the recursion is from s = 1. Note that K, is included in N2(s) of eqn (15). The precedure in Sec. 2.3 can now be applied to the preceding recursive relations, (13) and (14), provided that eqn (4) is replaced by the following (19)
and eqns (13) and (14) are understood to be in the 7 domain after the TCT. In turn, eqn (12) can now be evaluated by the SCDMATS in the 7 domain and transformed back to the original dimensionless time coordinate. Of course, in this particular case Q in eqn (19) is also dimensionless. Table 3. Maximum (V, 6) (0.02,0.03)
(0.2.0.3)
(0.6.0.9)
( 1.o, 1.5)
(1.5,2.25)
(2.0,3.0)
variance
+ 3~<(X2tf~))~) + 3U + ~k((~2(fJ)?)
responses Scheme
SCDSL SCDATS SCDATST SCDTATS SCDSL SCDATS SCDATST SCDTATS SCDSL SCDATS SCDATST SCDTATS SCDSL SCDATS SCDATST SCDTATS SCDSL SCDATS SCDATST SCDTATS SCDSL SCDATS SCDATST SCDTATS
(20)
 2(X2(r,))‘}
1
(16)
(17)
and parameters w =/l = 1.0,
z =Ri
eo.25i)
~~(3(x2(~~))((X2(i.~:,))2)  2(X2(C))‘j
+ (1 +~)~{3(X,(i,))((X,(i,))2)
and the starting conditions for eqns (13) and (14) are m(O)=m(l)=
e(f) = (j(eO 12% _
i, = 12= 0.1,
So = 0.0012.
(21)
The ranges of nonlinearities studied were: 0.02 to  1.5 for q, and 0.03 to 2.25 for 6. Digital simulation results were also obtained for smaller q and larger 6 in addition to the above two ranges of nonlinearities. The smaller q and larger 6 were q =  2.0 and c = 3.0. However, it was found that numerical instability could occur for certain combinations of q and t when using the SCDSL [ll]. On the other hand. the presently proposed SCDATS, SCDATST and SCDTATS do not have numerical instability. Note that when q and t are both equal to zero it is a linear system. In this special case, the two natural frequencies are 0.61804 and 1.61804. The results obtained by employing the three SCDMATS are presented in Table 3 and compared with those evaluated by the SCDSL and Monte Carlo simulation (DS) in Figs 38. Note that the fundamental frequency after the TCT described in Subsection 2.3 is less than 1.O. and therefore the time step size A7 is determined by eqn (lb) in which At is replaced by A7 and o is understood to be the dimensionless fundamental fre
of the two DOF system [AT based on eqn I(b)] (XI>
(X,X* >
25.6397 24.8217 24.6697 24.6744 25.9606 24.9091 24.8343 24.9065 26.8154 25.3070 25.2358 25.6460 28.3533 25.7276 25.7343 26.8538 33.7253 26.4225 26.5014 29.5495
15.1325 14.7068 14.7959 14.7810 15.1362 14.6228 14.7372 14.6663 15.2945 14.4733 14.6338 14.5125 IS.5543 14.3781 14.5750 14.6206 16.7801 14.3400 14.5749 IS.3829
10.3236 9.9657 9.6327 9.6202 10.2092 9.7952 9.4858 9.3600 9.9782 9.4364 9.1733 8.8307 9.9993 9.1117 8.8888 8.491 I 1 I .4832 8.7908 8.6116 8.4428
19.3983 14.7002 17.0326
8.5630 8.411 I 8.7753
Overflow 27.2930 27.4225 34.5062
Time schemes
t:: s
IO
for responses
of nonlinear
systems
569
25
+
SCDATS
+
SWTATS
+r
SCDATST

SCDSL (Az=O.945)
+
SCDATS
z! 20
+
SWTATS
15
+
SCDATST

SCDSL (A~=O.995)
B
>
10
5
"0
IO
20
30
46
50
60
70
60
90
100
DIMENSKYLESSTME
Fig. 3. Nonstationary
responses
DMENSIONLESSTIM
of the two DOF system.
quency. From Table 3 and Figs 38 it can be observed that the computed maximum variances are slightly smaller than those obtained by the SCDSL. Of course, the discrepancy increases with increasing intensities of nonlinearities. For small intensities of nonlinearities the three SCDMATS give very close results. As the intensities of the nonlinearities are increased the SCDTATS results have the best agreement with those using the SCDSL and DS. Therefore, in this study the SCDTATS seems to be the best algorithm to employ in terms of accuracy and computing cost over wide ranges of nonlinearities.
Fig. 6. Nonstationary responses of the two DOF system.
The results obtained by the SCDMATS based on the dimensionless time step size evaluated from the straight line in Fig. 1, designated as ‘extended eqn (la)‘, are included in Table 4 and Figs 9l 5. It can be seen that results by SCDMATS, SCDSL and DS have better agreement, in term of the maximum values of variances, over a wide range of nonlinearities than using eqn (lb). In particular, the SCDTATS and DS results for the case of q = 2.0 and t = 3.0 presented in Fig. 15 are in much better agreement than those in Fig. 8. Note that in this particular set of nonlinearities the system experiences stiffness softening and hardening. Stiffness softening
(II  02,e  0.3,so  0.0012) 30
40
I
a5
25
so
4
SCDATS
Et
SCDTATS
+
SCDATST

SCDSL (AmO.950)
8 P
25
%
20
I
15
+
SCDATS
+
SCDTATS
+
SCDATST

SCDSL (Az=l .I 00)
IO 5 0 0
10
20
80
46
50
60
70
60
90
100
DMENSIONESSTIME
Fig. 4. Nonstationary
responses
of the two DOF system.
(q   0.6.e O.Q,.So  0.0012) 1.I ., ., ., .,
30,
Fig. 7. Nonstationary
.,
!
25
SCDATS SCDTATS
responses
of the two DOF system.
60
35 8 5
+
80
SCDATS
+
SCDTATS
+r
SCDATST

SIMULATION
26
SCDATST I '
SCDSL (AT0.975)
20 15 10 6
0
10
20
30
40
50
60
J 70
60
90
100
0
t 0
10
20
IWENStOMESSliM
Fig. 5. Nonstationary
responses
of the two DOF system.
30
46
50
60
70
50
90
100
ciMENslct&E99TlME
Fig. 8. Nonstationary
responses
of the two DOF
system.
570
M. LIU and C. W. S. To
Table 4. Maximum variance responses of the two DOF system (ATbased on the straight line in Fig. I) Scheme
(n. c) (0.02.0.3)
tx:>
SCDPSL SCDATS SCDATST SCDTATS SCDSL SCDATS SCDATST SCDTATS SCDSL SCDATS SCDATST SCDTATS SCDSL SCDPATS SCDPATST SCDTATS SCDSL SCDATS SCDPATST SCDTATS SCDSL SCDATS SCDATST SCDTATS
(0.2,0.3)
(0.6,0.9)
(1.0,1.5)
( l&2.25)
(2.0,3.0)
25.6397 26.7718 29.3169 29.3735 25.9606 26.9342 29.5666 29.7079 26.8154 27.3736 30.2215 30.8914 28.3533 27.9245 3 1.0409 32.8527 33.7253 28.7892 32.2736 37.0238 Overtlow 29.8890 33.8461 45.4173
is due to the term &fz in eqn (1 1), while stiffness hardening arises from the term (1 + p)&. More specifically, it has been observed during the numerical experiments that in this case the dimensionless fundamental natural frequency is reduced while the
IS.1325 15.7640 17.5615 17.5742 15.1362 15.6627 17.4897 17.4055 15.2945 15.4885 17.3832 17.2573 15.5543 15.3842 17.3541 17.4843 16.7801 15.3524 17.4331 18.6338
10.3236 10.8222 11.5001 10.5036 10.2092 10.6353 1 I .2990 11.1378 9.9782 10.2273 10.8856 10.4525 9.9993 9.8893 10.5315 10.0332 1 I .4832 9.5663 10.1948 9.9645
15.4354 17.6816 21.2028
9.3328 9.9741 10.4130
dimensionless higher natural frequency is increased. In Figs 14 and 15 the DS results have two distinct peaks in each variance plot while those by the SCDTATS have only one maximum value in each curve. (n   1 s, e  2.25,So 0.0012)
hj . 0.6,e O.Q,So  0.0012)
+$+ SCDATS(a) 
3
SCDATST (a)
25
SCDSL(A~0.975)
20
(As1 .I 00)
t
SCDSL

SIMULATION
15
10
20
30
40
50
60
70
30
80
1
&A 100
100
MMENSi+ZWESSTlME
DMENSIONLESSTIME
Fig. 9. Nonstationary
responses
of the two
DOF system.
Fig. 11. Nonstationary
responses of the two DOF system.
(l)  1.0,e l.d.So  0.0012)
f
0
10
20
30
40
SCDATS(a)
50
DMENMNLESS
Fig. 10. Nonstationary
60
70
80
90
+
SCDATST (a)

SIMULATION
100
UMENS0NLESS71ME
TIME
responses of the two DOF system.
Fig. 12. Nonstationary
responses of the two DOF system.
571
Time schemes for responses of nonlinear systems
random excitation, and a two DOF nonlinear system under a nonstationary random process have been studied. A relation, eqn (1) has also been presented for the evaluation of the time step size required in the implementation of the aforementioned procedures. It has been found that the computational times required by the above three procedures for obtaining the results of the two nonlinear systems have insignificant differences. However, it is expected that the SCDATST is more expensive to employ for nonlinear systems with many DOF. The three pro
as a0
SCDTATS(a)
+
25
SIMULATION
La 11 1 '
1c ! i 10
w
w
40
w
a0
70
a0
w
IW
cwENaK2uws~
Fig. 13. Nonstationary
responses of the two DOF system.
cedures are very efficient to apply compared with digital simulation. Furthermore, they are free from computational instability and give very accurate results for highly nonlinear systems. AcknowledgemenrThe results presented were obtained in the course of research sup~rted by the Natural Sciences and Engineering Research Council of Canada.
4
SCOTATS(a)

SIMULATION
REFERENCES
10 S 0
0
12
20
22
Fig. 14. Nonstationary
40
w
w
70
w
w
IW
responses of the two DOF system.
4s 42
as
+
SC&TATS (a)
a0

SIWLATION
25 20 IS 10 5 n 0
to
20
w
40
so
SO
70
w
20
IW
GwEummEsswIE
Fig. 15. Nonstationary
responses of the two DOF system.
4. SUMMARY AND
CONCLUDING REMARKS
To recapitulate, in the investigation reported here three adaptive time schemes are incorporated into the stochastic central difference method with time coordinate transformation for the determination of covariances of responses of MDOF nonlinear systems. The resulting procedures are called the SCDATS, SCDATST and SCDTATS. A SDOF Duffing’s oscillator disturbed by a nonstationary
1. C. W. S. To, The stochastic central difference method in structural dynamics. Report No. 310, Department of Mechanical Engineering, University of Calgary (1984). 2. C. W. S. To, The stochastic central difference method in structural dynamics. Compur. Srruet. 23, 813816 (1986). 3. C. W. S. To, Random responses of multidegreeoffreedom systems by the stochastic central difference method. In Proc. ht. [email protected] Computational Mechanics, Tokyo, Japan, 2529 May, X1.181186 (1986). 4. C. W. S. To, A stochastic version of tbe Newmark family of algorithms for discretired dynamic systems. Compt. Strucf. 44(3), 667673 (1992). 5. C. W. S. To, Recursive expressions for random response of nonlinear systems. Comput. Strucl. 29(3), 451457 (1988). 6. C. W. S. To, Random response of a Duffing oscillator by the stochastic central difference method. J. Sound V&r. 124(3), 427433 (1988). 7. C. W. S. To, Parametric. effects on the time step of the stochastic central difference method. J. Sound V&r. 137(3), 509515 (1990). 8. C. W. S. To, Random response of multidegreeoffreedom systems with geometrical nonlinearities by the stochastic central difference method. Submitted to J. coffin.
Dynam.
(1992).
9. C. W. S. To and M. L. Liu, Random responses of discretized beams and plates by the stochastic central difference method and time coordinate transformation. Submitted to Compur. Srrucf. (1992). 10. K. Kimura and M. Sakata, Nonstationary response analysis of a nonsymmetric nonlinear multidegreeoffreedom system to nonwhite random excitation. JSME hr. J. 31(4), 690697 (1988).
Il. C. W. S. To and M. L. Liu, Recursive expressions for the timedependent means and mean square response of multidegreeoffreedom system. Compur. Strucr., to be published. 12. K. J. Bathe and E. L. Wilson, Stability and accuracy analysis of direct integration methods. Ear&q. Engng Strucf. Dynam. 1, 283291 (1973).