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Adaptive time schemes for responses of non-linear multi-degree-of-freedom systems under random excitations
Pergamon
00457949-(94)EOO57-9
Cm~urers & Sfrurrura Vol. 52. No. 3. pp. 563-571. 1994 Copyright 1: 1994 El&r Sxnce Ltd Printed in Great Britain. All rights reserved 0045-7949194 57.00 + 0.00
ADAPTIVE TIME SCHEMES FOR RESPONSES OF NON-LINEAR MULTI-DEGREE-OF-FREEDOM 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)
Abstract-Three adaptive time schemes are incorporated into the stochastic central difference method with time co-ordinate transformation for the determination of the covariances of responses of multi-degree-offreedom non-linear systems. The results of a Dulling’s oscillator and a two-degree-of-freedom system with asymmetric non-linear 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 non-linearities, under stationary and non-stationary 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 multi-degree-of-freedom (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 non-linear 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 degrees-of-freedom (DOF). Secondly, there is no restriction on the types of damping present in the systems. Thirdly, they are applicable to systems whose non-linearities 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 [5-71, examples employing the SCD method in conjunction with statistical linearization (SL) have shown that the SCD-SL 7 Author to whom correspondence should be addressed.
technique is effective and accurate for the computation of covariances and time-dependent mean squares of the responses of non-linear systems. In a parametric study [7] of the SCD method for a non-linear single-degree-of-freedom (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 non-linear 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 up-dating 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 co-ordinate transformation (SCD-ATST), and the stochastic central difference and time co-ordinate transformation with adaptive time scheme (SCD-TATS). For simplicity, these three schemes shall be collectively called SCD-MATS. In the SCD-ATS the lowest natural frequency of the non-linear 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 SCD-ATST the non-linear or linearized system at every time step is first transformed into its corresponding dimensionless time co-ordinate 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 co-ordinate. This scheme may prove to be relatively much more expensive than the
564
M. LIU and C. W. S. To
SCD-TATS for non-linear systems with a large number of DOF as it requires time co-ordinate transformation (TCT) at every time step. The SCD-TATS 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 SCD-MATS. Section 3 is concerned with results of a series of numerical studies of the SCD-MATS for two non-linear systems. The first system is a Dulling oscillator under nonstationary random excitation, and the second one is a two DOF system with non-symmetric non-linearities excited by a non-stationary 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 SCD-ATS, SCD-ATST, and SCD-TATS or collectively called the SCD-MATS 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 non-linearities 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 non-linear 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 non-linear 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 SCD-MATS 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 = 1.0-0.053~
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 SCD-MATS. The parametric study in [7] provides a methodology for constructing such a relation and many pertinent data.
0.5 0.4
: y
0.3 ” 1o-1
2
9 4 60
100
2
945
Fig. 1. Relation for time step size and natural frequency.
565
Time schemes for responses of non-linear 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 non-linea~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 co-ordinate 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 non-linear 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 sub-section 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 co-ordinate t to the dimensionless time co-ordinate 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 zero-mean 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 non-stationary random excitation, and a two DOF non-linear 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 non-linearity 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 non-stationary 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 zero-mean Gaussian white noise. For this example, e(t) is defined as e(t) = 4.0(emoo5’- c-o.‘o’).
(8)
Applying the SCD-SL 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 SCD-MATS. Note that the SCD-TATS is identical to the SCD-ATS numerically in this particular example. In the SCD-ATS, eqn (10) is used to determine the time-dependent 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 SCD-ATST 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 non-linear 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 SCD-ATS and SCD-ATST, and compares them with those from [5] using SCIX4L. Note that the SCD-SL results have been found to compare very well with the digital simulation data [5]. Table 2 indicates that the results obtained by employing the SCD-ATS and
Table 2. Variance of response of the Duffing’s oscillatort Maximum variance response
i
c
SCD-SL
SCD-ATS
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
SCD-ATST 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
SCD-SL results are from, [5] for S = IO-“’ and [7] for s = 1.0.
561
Time schemes for responses of non-linear systems
SCD-ATST agree very well with those using the SCD-SL. This is particularly true for the cases with damping ratios higher than 0.05. 3.2. A two degree-of-freedom
p=M2, WC?,
system with non-sym -
metric non-linear stlflness
This two DOF non-linear system, shown in Fig. 2, has previously been investigated by Kimura and Sakata [lo], and the authors [ 111.It may be applied to study soil-structure interaction or a primary structure and secondary equipment under an earthquake excitation treated as a non-stationary 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 SCD-ATST 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 non-linear restoring forces are defined explicitly. In [ 111it was found that the SCD-SL 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 over-dot and double over-dot 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 non-symmetric 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 SCD-MATS in the 7 domain and transformed back to the original dimensionless time co-ordinate. 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
SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS
(20)
- 2(X2(r,))‘}
1
(16)
(17)
and parameters w =/l = 1.0,
z =Ri
e-o.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(e-O 12% _
i, = 12= 0.1,
So = 0.0012.
(21)
The ranges of non-linearities 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 non-linearities. 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 SCD-SL [ll]. On the other hand. the presently proposed SCD-ATS, SCD-ATST and SCD-TATS 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 SCD-MATS are presented in Table 3 and compared with those evaluated by the SCD-SL and Monte Carlo simulation (DS) in Figs 3-8. Note that the fundamental frequency after the TCT described in Sub-section 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 non-linear
systems
569
25
+
SCD-ATS
+
SW-TATS
+r
SCD-ATST
-
SCD-SL (Az=O.945)
+
SCD-ATS
z! 20
+
SW-TATS
15
+
SCD-ATST
-
SCD-SL (A~=O.995)
B
>
10
5
"0
IO
20
30
46
50
60
70
60
90
100
DIMENSKYLESSTME
Fig. 3. Non-stationary
responses
DMENSIONLESSTIM
of the two DOF system.
quency. From Table 3 and Figs 3-8 it can be observed that the computed maximum variances are slightly smaller than those obtained by the SCD-SL. Of course, the discrepancy increases with increasing intensities of non-linearities. For small intensities of non-linearities the three SCD-MATS give very close results. As the intensities of the non-linearities are increased the SCD-TATS results have the best agreement with those using the SCD-SL 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 non-linearities.
Fig. 6. Non-stationary responses of the two DOF system.
The results obtained by the SCD-MATS 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 9-l 5. It can be seen that results by SCD-MATS, SCD-SL and DS have better agreement, in term of the maximum values of variances, over a wide range of non-linearities than using eqn (lb). In particular, the SCD-TATS 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 non-linearities the system experiences stiffness softening and hardening. Stiffness softening
(II- - 02,e - 0.3,so - 0.0012) 30
40
I
a5
25
so
-4
SCD-ATS
-Et
SCD-TATS
+
SCD-ATST
-
SCD-SL (AmO.950)
8 P
25
%
20
I
15
+
SCD-ATS
+
SCD-TATS
+
SCD-ATST
-
SCD-SL (Az=l .I 00)
IO 5 0 0
10
20
80
46
50
60
70
60
90
100
DMENSIONESSTIME
Fig. 4. Non-stationary
responses
of the two DOF system.
(q - - 0.6.e- O.Q,.So - 0.0012) 1.I ., ., ., .,
30,
Fig. 7. Non-stationary
.,
!
25
SCD-ATS SCD-TATS
responses
of the two DOF system.
60
35 8 5
+
80
SCD-ATS
+
SCD-TATS
+r
SCD-ATST
-
SIMULATION
26
SCD-ATST I '
SCD-SL (AT-0.975)
20 15 10 6
0
10
20
30
40
50
60
J 70
60
90
100
0
t 0
10
20
IWENStOMESSliM
Fig. 5. Non-stationary
responses
of the two DOF system.
30
46
50
60
70
50
90
100
ciMENslct&E99TlME
Fig. 8. Non-stationary
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 SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCDPATS SCDPATST SCD-TATS SCD-SL SCD-ATS SCDPATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS
(-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 SCD-TATS 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)
+$+ SCD-ATS(a) -
-3
SCD-ATST (a)
25
SCD-SL(A~0.975)
20
(As-1 .I 00)
t
SCD-SL
-
SIMULATION
15
10
20
30
40
50
60
70
30
80
1
&A 100
100
MMENSi+ZWESSTlME
DMENSIONLESSTIME
Fig. 9. Non-stationary
responses
of the two
DOF system.
Fig. 11. Non-stationary
responses of the two DOF system.
(l)- - 1.0,e- l.d.So - 0.0012)
f
0
10
20
30
40
SCD-ATS(a)
50
DMENMNLESS
Fig. 10. Non-stationary
60
70
80
90
-+
SCD-ATST (a)
-
SIMULATION
100
UMENS0NLESS71ME
TIME
responses of the two DOF system.
Fig. 12. Non-stationary
responses of the two DOF system.
571
Time schemes for responses of non-linear systems
random excitation, and a two DOF non-linear system under a non-stationary 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 non-linear systems have insignificant differences. However, it is expected that the SCD-ATST is more expensive to employ for non-linear systems with many DOF. The three pro-
as a0
SCD-TATS(a)
+
25
-SIMULATION
La 11 1 '
1c ! i 10
w
w
40
w
a0
70
a0
w
IW
cwENaK2uws~
Fig. 13. Non-stationary
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 non-linear systems. Acknowledgemenr-The results presented were obtained in the course of research sup~rted by the Natural Sciences and Engineering Research Council of Canada.
4-
SCO-TATS(a)
-
SIMULATION
REFERENCES
10 S 0
0
12
20
22
Fig. 14. Non-stationary
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. Non-stationary
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 non-linear systems. The resulting procedures are called the SCDATS, SCD-ATST and SCD-TATS. A SDOF Duffing’s oscillator disturbed by a non-stationary
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 multi-degree-offreedom systems by the stochastic central difference method. In Proc. ht. Co@ Computational Mechanics, Tokyo, Japan, 25-29 May, X1.181-186 (1986). 4. C. W. S. To, A stochastic version of tbe Newmark family of algorithms for discretired dynamic systems. Compt. Strucf. 44(3), 667-673 (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), 427-433 (1988). 7. C. W. S. To, Parametric. effects on the time step of the stochastic central difference method. J. Sound V&r. 137(3), 509-515 (1990). 8. C. W. S. To, Random response of multidegree-offreedom 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 co-ordinate transformation. Submitted to Compur. Srrucf. (1992). 10. K. Kimura and M. Sakata, Nonstationary response analysis of a nonsymmetric nonlinear multi-degree-offreedom system to nonwhite random excitation. JSME hr. J. 31(4), 690-697 (1988).
Il. C. W. S. To and M. L. Liu, Recursive expressions for the time-dependent means and mean square response of multi-degree-of-freedom 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, 283-291 (1973).
00457949-(94)EOO57-9
Cm~urers & Sfrurrura Vol. 52. No. 3. pp. 563-571. 1994 Copyright 1: 1994 El&r Sxnce Ltd Printed in Great Britain. All rights reserved 0045-7949194 57.00 + 0.00
ADAPTIVE TIME SCHEMES FOR RESPONSES OF NON-LINEAR MULTI-DEGREE-OF-FREEDOM 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)
Abstract-Three adaptive time schemes are incorporated into the stochastic central difference method with time co-ordinate transformation for the determination of the covariances of responses of multi-degree-offreedom non-linear systems. The results of a Dulling’s oscillator and a two-degree-of-freedom system with asymmetric non-linear 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 non-linearities, under stationary and non-stationary 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 multi-degree-of-freedom (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 non-linear 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 degrees-of-freedom (DOF). Secondly, there is no restriction on the types of damping present in the systems. Thirdly, they are applicable to systems whose non-linearities 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 [5-71, examples employing the SCD method in conjunction with statistical linearization (SL) have shown that the SCD-SL 7 Author to whom correspondence should be addressed.
technique is effective and accurate for the computation of covariances and time-dependent mean squares of the responses of non-linear systems. In a parametric study [7] of the SCD method for a non-linear single-degree-of-freedom (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 non-linear 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 up-dating 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 co-ordinate transformation (SCD-ATST), and the stochastic central difference and time co-ordinate transformation with adaptive time scheme (SCD-TATS). For simplicity, these three schemes shall be collectively called SCD-MATS. In the SCD-ATS the lowest natural frequency of the non-linear 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 SCD-ATST the non-linear or linearized system at every time step is first transformed into its corresponding dimensionless time co-ordinate 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 co-ordinate. This scheme may prove to be relatively much more expensive than the
564
M. LIU and C. W. S. To
SCD-TATS for non-linear systems with a large number of DOF as it requires time co-ordinate transformation (TCT) at every time step. The SCD-TATS 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 SCD-MATS. Section 3 is concerned with results of a series of numerical studies of the SCD-MATS for two non-linear systems. The first system is a Dulling oscillator under nonstationary random excitation, and the second one is a two DOF system with non-symmetric non-linearities excited by a non-stationary 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 SCD-ATS, SCD-ATST, and SCD-TATS or collectively called the SCD-MATS 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 non-linearities 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 non-linear 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 non-linear 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 SCD-MATS 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 = 1.0-0.053~
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 SCD-MATS. The parametric study in [7] provides a methodology for constructing such a relation and many pertinent data.
0.5 0.4
: y
0.3 ” 1o-1
2
9 4 60
100
2
945
Fig. 1. Relation for time step size and natural frequency.
565
Time schemes for responses of non-linear 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 non-linea~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 co-ordinate 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 non-linear 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 sub-section 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 co-ordinate t to the dimensionless time co-ordinate 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) =
,)>,
in which the angular brackets denote ‘ensemble average’ of the enclosing quantity, while S, and +(r,) are the spectral density of the zero-mean 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 non-stationary random excitation, and a two DOF non-linear 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 non-linearity 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 non-stationary 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 zero-mean Gaussian white noise. For this example, e(t) is defined as e(t) = 4.0(emoo5’- c-o.‘o’).
(8)
Applying the SCD-SL 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 SCD-MATS. Note that the SCD-TATS is identical to the SCD-ATS numerically in this particular example. In the SCD-ATS, eqn (10) is used to determine the time-dependent 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 SCD-ATST 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 non-linear 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 SCD-ATS and SCD-ATST, and compares them with those from [5] using SCIX4L. Note that the SCD-SL results have been found to compare very well with the digital simulation data [5]. Table 2 indicates that the results obtained by employing the SCD-ATS and
Table 2. Variance of response of the Duffing’s oscillatort Maximum variance response
i
c
SCD-SL
SCD-ATS
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
SCD-ATST 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
SCD-SL results are from, [5] for S = IO-“’ and [7] for s = 1.0.
561
Time schemes for responses of non-linear systems
SCD-ATST agree very well with those using the SCD-SL. This is particularly true for the cases with damping ratios higher than 0.05. 3.2. A two degree-of-freedom
p=M2, WC?,
system with non-sym -
metric non-linear stlflness
This two DOF non-linear system, shown in Fig. 2, has previously been investigated by Kimura and Sakata [lo], and the authors [ 111.It may be applied to study soil-structure interaction or a primary structure and secondary equipment under an earthquake excitation treated as a non-stationary 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 SCD-ATST 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 non-linear restoring forces are defined explicitly. In [ 111it was found that the SCD-SL 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 over-dot and double over-dot 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 non-symmetric 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 SCD-MATS in the 7 domain and transformed back to the original dimensionless time co-ordinate. 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
SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS
(20)
- 2(X2(r,))‘}
1
(16)
(17)
and parameters w =/l = 1.0,
z =Ri
e-o.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(e-O 12% _
i, = 12= 0.1,
So = 0.0012.
(21)
The ranges of non-linearities 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 non-linearities. 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 SCD-SL [ll]. On the other hand. the presently proposed SCD-ATS, SCD-ATST and SCD-TATS 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 SCD-MATS are presented in Table 3 and compared with those evaluated by the SCD-SL and Monte Carlo simulation (DS) in Figs 3-8. Note that the fundamental frequency after the TCT described in Sub-section 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 non-linear
systems
569
25
+
SCD-ATS
+
SW-TATS
+r
SCD-ATST
-
SCD-SL (Az=O.945)
+
SCD-ATS
z! 20
+
SW-TATS
15
+
SCD-ATST
-
SCD-SL (A~=O.995)
B
>
10
5
"0
IO
20
30
46
50
60
70
60
90
100
DIMENSKYLESSTME
Fig. 3. Non-stationary
responses
DMENSIONLESSTIM
of the two DOF system.
quency. From Table 3 and Figs 3-8 it can be observed that the computed maximum variances are slightly smaller than those obtained by the SCD-SL. Of course, the discrepancy increases with increasing intensities of non-linearities. For small intensities of non-linearities the three SCD-MATS give very close results. As the intensities of the non-linearities are increased the SCD-TATS results have the best agreement with those using the SCD-SL 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 non-linearities.
Fig. 6. Non-stationary responses of the two DOF system.
The results obtained by the SCD-MATS 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 9-l 5. It can be seen that results by SCD-MATS, SCD-SL and DS have better agreement, in term of the maximum values of variances, over a wide range of non-linearities than using eqn (lb). In particular, the SCD-TATS 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 non-linearities the system experiences stiffness softening and hardening. Stiffness softening
(II- - 02,e - 0.3,so - 0.0012) 30
40
I
a5
25
so
-4
SCD-ATS
-Et
SCD-TATS
+
SCD-ATST
-
SCD-SL (AmO.950)
8 P
25
%
20
I
15
+
SCD-ATS
+
SCD-TATS
+
SCD-ATST
-
SCD-SL (Az=l .I 00)
IO 5 0 0
10
20
80
46
50
60
70
60
90
100
DMENSIONESSTIME
Fig. 4. Non-stationary
responses
of the two DOF system.
(q - - 0.6.e- O.Q,.So - 0.0012) 1.I ., ., ., .,
30,
Fig. 7. Non-stationary
.,
!
25
SCD-ATS SCD-TATS
responses
of the two DOF system.
60
35 8 5
+
80
SCD-ATS
+
SCD-TATS
+r
SCD-ATST
-
SIMULATION
26
SCD-ATST I '
SCD-SL (AT-0.975)
20 15 10 6
0
10
20
30
40
50
60
J 70
60
90
100
0
t 0
10
20
IWENStOMESSliM
Fig. 5. Non-stationary
responses
of the two DOF system.
30
46
50
60
70
50
90
100
ciMENslct&E99TlME
Fig. 8. Non-stationary
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 SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS SCD-SL SCDPATS SCDPATST SCD-TATS SCD-SL SCD-ATS SCDPATST SCD-TATS SCD-SL SCD-ATS SCD-ATST SCD-TATS
(-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 SCD-TATS 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)
+$+ SCD-ATS(a) -
-3
SCD-ATST (a)
25
SCD-SL(A~0.975)
20
(As-1 .I 00)
t
SCD-SL
-
SIMULATION
15
10
20
30
40
50
60
70
30
80
1
&A 100
100
MMENSi+ZWESSTlME
DMENSIONLESSTIME
Fig. 9. Non-stationary
responses
of the two
DOF system.
Fig. 11. Non-stationary
responses of the two DOF system.
(l)- - 1.0,e- l.d.So - 0.0012)
f
0
10
20
30
40
SCD-ATS(a)
50
DMENMNLESS
Fig. 10. Non-stationary
60
70
80
90
-+
SCD-ATST (a)
-
SIMULATION
100
UMENS0NLESS71ME
TIME
responses of the two DOF system.
Fig. 12. Non-stationary
responses of the two DOF system.
571
Time schemes for responses of non-linear systems
random excitation, and a two DOF non-linear system under a non-stationary 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 non-linear systems have insignificant differences. However, it is expected that the SCD-ATST is more expensive to employ for non-linear systems with many DOF. The three pro-
as a0
SCD-TATS(a)
+
25
-SIMULATION
La 11 1 '
1c ! i 10
w
w
40
w
a0
70
a0
w
IW
cwENaK2uws~
Fig. 13. Non-stationary
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 non-linear systems. Acknowledgemenr-The results presented were obtained in the course of research sup~rted by the Natural Sciences and Engineering Research Council of Canada.
4-
SCO-TATS(a)
-
SIMULATION
REFERENCES
10 S 0
0
12
20
22
Fig. 14. Non-stationary
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. Non-stationary
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 non-linear systems. The resulting procedures are called the SCDATS, SCD-ATST and SCD-TATS. A SDOF Duffing’s oscillator disturbed by a non-stationary
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 multi-degree-offreedom systems by the stochastic central difference method. In Proc. ht. Co@ Computational Mechanics, Tokyo, Japan, 25-29 May, X1.181-186 (1986). 4. C. W. S. To, A stochastic version of tbe Newmark family of algorithms for discretired dynamic systems. Compt. Strucf. 44(3), 667-673 (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), 427-433 (1988). 7. C. W. S. To, Parametric. effects on the time step of the stochastic central difference method. J. Sound V&r. 137(3), 509-515 (1990). 8. C. W. S. To, Random response of multidegree-offreedom 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 co-ordinate transformation. Submitted to Compur. Srrucf. (1992). 10. K. Kimura and M. Sakata, Nonstationary response analysis of a nonsymmetric nonlinear multi-degree-offreedom system to nonwhite random excitation. JSME hr. J. 31(4), 690-697 (1988).
Il. C. W. S. To and M. L. Liu, Recursive expressions for the time-dependent means and mean square response of multi-degree-of-freedom 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, 283-291 (1973).