Cmpurarr & Snu~rures Vol 56. No. I, pp I 13-I 20. 1995

0045-7949(94)00537-O CopyrIght (’ 1995 Elsevier Science Ltd

Printed m Great Brnain. All rights reserved cws-7949/95 $9.50 + 0.00

A HIGH PRECISION DIRECT INTEGRATION SCHEME FOR STRUCTURES SUBJECTED TO TRANSIENT

DYNAMIC LOADING

Jiahao Lin,t$ Weiping She@j and F. W. WilliamsT$

tResearch Institute of Engineering Mechanics, Dalian University of Technology, Dalian 116023. Peoples Republic of China

§Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, Peoples Republic of China

SDivision of Structural Engineering, School of Engineering, University of Wales, Cardiff CF2 IXH, U.K.

(Received 8 February 1994)

Abstract-A high precision direct (HPD) integration scheme is given. Non-linearly varying loadings are decomposed into Fourier components before direct integration is performed, so that accurate responses to each harmonic loading component can be obtained even for very large time-step sizes. Numerical examples include comparisons with the Newmark method and demonstrate the high precision and efficiency of this scheme.

INTRODUCTION

Dynamic responses of structures subjected to transient loading are often important, e.g. the response of rockets to reactions as the fuel burns, the response of buildings to impact waves caused by explosions and the response of ships to slamming forces caused by waves. Such responses are usually analysed by means of direct integration, or step-by-step, schemes among which there are implicit methods such as the Newmark, Wilson-8 or Houbolt schemes, and explicit methods such as the central difference scheme [l]. When using such methods, the time-step size must be carefully selected, relative to the natural periods of the structures and the variation of the loading, so as to ensure proper integration precision with reasonable computational effort. If explicit integration schemes [I] are adopted, the time-step size must also be very strictly constrained by the highest modal period of the discretized structure, in order to achieve integra- tion stability. Hence, such direct integration analyses need very small time steps and so are generally very time consuming and costly.

Recently, Zhong and Williams [2] proposed a very special explicit direct integration scheme, for which the time-step sizes are not constrained by any of the natural periods of the discretized structure. For this method, the applied loading is simulated by piece-wise linear loadings, so that the time-step size is restricted only by the form of the loading. This method is extended in this paper, in which it is called

IVisiting Professors at Cardiff when the paper was written.

the high precision direct-L (HPD-L) method, by an a priori decomposition of the applied loading into Fourier components, to give what will be called the high precision direct-F (HPD-F) method.

For each Fourier loading component, exact responses can be obtained by taking a large time-step size, which may cover an arbitrary segment of a period of this loading component, or even several periods. Hence the time-step size will be constrained by neither the natural periods of the discretized structure, nor the loading period of each harmonic loading component. In many cases, the adoption of a large time-step size enables considerable reduction of computational effort to be achieved. As a sort of explicit direct integration scheme, however, the HPD scheme is unconditionally stable. A proof of this is given in this paper, as a supplement to Ref. [2].

Four numerical examples are given to compare the precision and efficiency of the new HPD-F method with those of the original HPD-L method and with the Newmark method [ 11.

HIGH PRECISION DIRECT INTEGRATION SCHEME

The equations of motion of a descretized structural model can be written as [3]

[Wi-f) +1mi_) +iaf.x) = V(t)} (1) with initial conditions given as

j.40)) = {&I ), {i(O)) = ii-0 }. (2)

Here [Ml, [C] and [K] are time-invariant mass, damping and stiffness matrices, respectively, with

113

114 Jiahao Lin et al.

[M] assumed positively definite. Adopting the trans- where TI(=tk+,-tk, and the initial state vector formation proposed by Zhong and Williams [2], let {u(t,)} will be known from the previous step. Thus,

the problem which remains is to find the matrix [T(r)]

{P) =Wl{*_) +[Cl~x~/2 or and the particular solution {r,(t)).

{k} = [M]-‘(p) - [Ml-‘[C](x)/2. (3)

Equation (1) becomes

{@I = -WI - KIPf-‘W4)~x~

- Klw4-'bJI2-t if).

Equations (3) and (4) can be combined to give

{:>=[“R :](5}+{;] or simply

Ff = tffl{uj + {r)

COMPUTATION OF THE MATRIX Ir(r)j

In order to compute [T(T)] accurately, it is desirable [2] to transform eqn (10) into the form

[T(r)] = exp([H] x r) = [exp([H] x At)]“’ (12)

(4) in which

At = t/m (13)

(5) where m = 2N, and the use of N = 20 has been suggested [2], which leads to At = t/1048576. This extremely small time interval is usually much less than the highest modal period of any particular

(6) discretized structure. Using a Taylor expansion,

in which

exp([H] x At) = [I + T,,] (14)

in which

(7) [T,,] = [H] x At + ([N] x At)‘/2!

PI = -WI - Km4-W4) + ([HI x At)3/3! + ([HI x At)“/4!. (15)

[G] = -[C][M]-‘/2, [D] = [Ml-‘. J Substituting eqn (I 4) into eqn (I 2) gives

The general solution (u} to eqn (6) consists of two [T(r)] 2 ([I] + [T&]Y. (16) parts, i.e. the homogeneous solution {uh} and the particular solution (a,}. The homogeneous solution Note that must satisfy eqn (6) with the loading item (r} removed, and is simply [I + T&l2 = [I + 2 X T,, + r,, X T&l ’

{oh(f)1 = exp(F1 x r){e) (8) = [I + T”,]

in which 7 = t - tk for the current integration interval [I + T,,12 = [I + 2 X r,, + T,, X T,,] t E [tk, t, + ,], and {e} must be determined in terms of the initial conditions at time tk. E [I + T”,]

If the particular solution to eqn (6), {u,(t)}, is P. (17) . . . temporarily assumed to have been found, then the general solution of eqn (6) is

{W,) = [mMw,)l - {qtk)I) + {~,O)) (9)

in which

tV7)1= exp(tW x T). (10) So that clearly

In particular, at time t = t,, , , i.e. the end of the time-step,

{a(&+ 1)) = ]T(rk)l({U(&))

]I + Tozvl = [I + Tu..c , I* = [I + T<,.w 214

=...= [I + T<,o]“’ = [T(r)]. (18)

Equations (17) and (18) suggest the computing

-{~,(tk)~)+ {U&+,)1 (11) strategy. In order to avoid the loss of significant digits

High precision integration scheme 115

in the matrix [T(z)], it is necessary to compute [T,,] directly from [T,], [T,,] directly from [T,,], etc. by using

in which

{A 1 = (r@l + ]H12/o)-‘({r2j - Wl{r, )/WI

~~~,l=~~~~~.,~,l+~~~.,-,l~~~,,i-,l {BJ = (WI + W12h-I(--{r, 1 - Wl{r21/~).

(i = I, 2,. . , N).

Then [T(z)] should be computed from

(19) Substituting eqn (25) into eqn (1 I), and letting t = tk+ , gives the general solution of eqn (6), i.e. the direct integration formula, as

ux~)l z VI + [T”NI. (20)

In eqn (20), the approximation is caused by the truncation of the Taylor expansion of eqn (IS). It is generally negligibly small because when N = 20, the first term ignored by the truncation is of the order O(At’) = 10-300(zs), which is of the order of the round-off errors of ordinary computers. Indeed, so far as general dynamic analyses are concerned the exact [T(z)] is obtained, as the examples given later in this paper demonstrate.

{Gkt ,)1 = [~(~dl(4h) - {A 1 sin w

-{B}cosor,)+{A}sinwt,+,

+{Bjcoswt,+,. (27)

The time interval tt can cover an arbitrary seg- ment, or even many periods, of a sinusoidal wave because, no matter how large the step size may be, exact responses will be obtained provided the matrix [T(7k)] has been generated accurately.

PARTICULAR SOLUTIONS FOR VARIOUS FORMS OF LOADING

3. Fourier expansion of an arbitrary loading form (HPD-F FORM)

1. Linear loading form (HPD-L FORM)

Zhong and Williams [2] assumed that the loading varies linearly within the time step (t,, t, + I)r i.e.

An arbitrary loading curve (r(t)} can be expanded, by Fourier transformation within the time interval

]tk, tk + ,I, to give

ir) = ir01-t {rlj x (t -b) (21)

in which {ro} and {r, } are time-invariant vectors. The particular solution of eqn (6) denoted as {up}, is then [2]

{r(t)} z {b,} + i$, ({a,}sin(iwt) + {b,}cos(iwt)) (28)

where q is an integer which should be selected to ensure that the truncation error is negligible. Sub- stituting into eqn (6) gives the particular solution as

{a,(t)) = ([HI-’ + VI x t)(-[Hl-‘{r, 1)

- W-‘({ro) - {rl 1 x ~1. (22)

Substituting eqn (22) into eqn (11) gives the integration formula

{a,(t)1 = {Bo) + ,$, ({A,)sin(iwr) + (&)cos(iat))

(29) in which

Voj = -[HI-‘{bo)

{A,} = (i20z[Il + [H12)-I(-[Hl{ai} +iw(b,}) {a(fk+r)) = V(7k)l x b(tk) -tW-‘({rol

+ [ff-‘{r, ))I- W-‘({r~I {B,} = (i2w2[1] + [HJ2)-I(-im{ai} - [H]{b,)) (31)

+[Hl-‘{r,J+(r,) XG). (23)

2. Sinusoidal loading form (HPD-S FORM)

If the applied loading varies sinusoidally within the time region t E [tk, t, + ,I, then

(i=l,2 ,..., q). :I

These values are then substituted into eqn (11).

{r(f)} = {r,} sin wt + {r2} cos wt (24)

in which {r,) and {rz} are time-invariant vectors. Substituting eqn (24) into eqn (6) enables the particu- lar solution to be obtained, by routine mathematical derivations, as

Here {u(fk)} is the known initial state vector, and {u,(t,)} and (a,(fk+,)} can be computed by eqn (29). Clearly, no matter how large the step size 7k may be, exact solutions will be produced provided the matrix [T(Q)] has been accurately generated, because eqn (11) is the exact solution to the initial valued problem of the ordinary differential eqn (6) with non-homogeneous RHS, see eqn (24), in the time interval [tk, t, + ,I.

(a,} = {A} sinwt + {B} cos~t (25)

The above HPD-S and HPD-F extensions of the earlier HPD-L method [2] are good alternatives to it and sometimes give outstandingly good results.

(26)

(30)

116 Jiahao Lin el a/

STABILITY OF THE EXPLICIT HPD INTEGRATION SCHEME

In order to investigate the stability of a direct integration scheme, it is only necessary to prove that the spectral radius p of the integration approximation operator for a SDOF system is less than or equal to unity [l]. For the present HPD method, this operator is the matrix [T(T)] [2]. Because of eqn (lo), it is convenient to find the eigenvalues of the matrix [H] first, i.e. to solve the eigenproblem

or

][H] - ;[I]] = 0 (32)

f(t) 4

1.00 - :n 0.00 L_--_ ._\_ t

0.00 1.00

Fig. I. Half-sine wave loading.

Using eqn (7), eqn (33) can be reduced into

Mi’f Ci + K =O. (34)

This is no more than the characteristic equation of the SDOF equation of motion

Let

M.;i + C.2 + Kx = 0. (35)

p’= KIM, < = CI’(2pM). (36)

The eigenvalues of [HI, or the roots of eqns (33) or (34) can be expressed as

[-< k i(l - (‘)“5]p when i < 1 E.,,2 =

{

(37) [-[ _+ (<‘- l)“‘]p when 5 > 1

where i = J-1 and for the sign +, “ +” gives i., whilst “ - ” gives AZ.

Using eqn (IO), the eigenvalues of [r(t)], denoted

as IA and pL?, must be exp(i,t) and exp(&t). Obviously,when< < 1,1~,I=J~L2(=lexp(-ip7)16 1; when [ > 1, i, and AZ are all minus real numbers,

and so ]p,] = ]exp(i,r)] and ]p2] = ]exp(&r)] must be both less than unity. Hence, for any damping ratio [, the spectral radius of [T(7)] cannot be greater than unity. i.e.

This proves the unconditional stability of the HPD integration scheme.

NUMERICAL EXAMPLES

E.umple I

The equation of motion of an SDOF system and its boundary conditions are

(39)

x(0) = 0, i(0) = 0 (40)

in which p = 1 .O and [ = 0.05, where the units are omitted for convenience. The applied loading is f‘(f) = sin(wt) with w = rc and 0 d t < 1.0, see

Fig. 1. The displacement responses at t = 0.2, 0.4, 0.6,

0.8 and 1 .O were computed and are listed in Table I. The first row of results are exact and are given for

Table I. Displacement responses of the SDOF system due to a half-sine loading

Scheme 5 0.2 0.4 I

0.6 0.8 I.0 Max err.

Eqn (41)

HPD-S 0.2 1.0

HPD-L 0.001 0.01 0.05 0.20

Newmark 0.001 0.01 0.05 0.20

0.00407805 0.0303926 0.0913166 0.183735

0.00407805 0.0303926 0.0913166 0.183735

0.00407805 0.0303926 0.0913166 0.183735 0.289484 0.000% 0.00407772 0.0303901 0.09 I309 1 0.183720 0.289461 0.008% 0.00406947 0.0303298 0.0911285 0.183357 0.288889 0.210% 0.00389127 0.0293407 0.0882162 0.177613 0.279920 4.580%

0.00407810 0.0303926 0.0913166 0.183735 0.289484 0.001% 0.00408242 0.0303969 0.0913142 0.183720 0.289453 0.107% 0.00418686 0.0305004 0.0912551 0.183349 0.288701 2.668% 0.00576260 0.0320355 0.0902718 0.177561 0.277025 41.307%

0.289484 0.000%

0.289484 0.000% 0.289484 0.000%

High precision integration scheme 117

0.00 0.50 1.00

Fig. 2. Bi-linear loading.

comparative purpose. They were obtained from the analytical solution to eqns (39) and (40), i.e.

x(t) = 7~ exp( -it) [

i cos qt + - sm ift 1 -n cos nt (41)

V

in which n = 3.14159265 and 1 =(I -<‘)“.5= 0.998749217. The remaining results were obtained from: the HPD-S method (i.e. a sinusoidal loading simulation) with w = 7~ and t = 0.2 and 1.0; the HPD-L method (i.e. a linear loading simulation) with z = 0.001, 0.01, 0.05 and 0.20; and the Newmark method [l] with s( = 0.5, 6 = 0.25 and 7 = 0.001, 0.01, 0.05 and 0.20.

It is seen from Table 1 that, for the type of sinu- soidal loading used, the HPD-S form is of course the best of the three step-by-step methods. This is because no matter how large the step size t may be, exact solutions can always be obtained for this problem by using the HPD-S method. In addition, the Newmark method produces about 10 times bigger errors than the HPD-L one does when the same step

size is used for both methods. This can be seen from the final column of the table, which gives the maxi- mum error for any of the five times t = 0.2, 0.4, 0.6, 0.8 and 1.0.

Example 2

Letf(t) in example 1 be replaced by the following bi-linear loading (see Fig. 2):

2.0t when 0 ,< t < 0.5

f(t) = (42) 2.0( 1 .O - t) when 0.5 < t ,< 1 .O

with the other parameters remaining unchanged. Displacement responses x(t) at t = 0.25, 0.5, 0.75

and 1 .O were computed and are listed in Table 2. The first row of results is exact and was obtained from the analytical solution which follows. When t E [O.O, 0.51

x(t) = 0.2S(t) - 2Q(r) - 0.2 + 21

(43) i(t) = -0.2&(r) - 2R(t) + 2

in which

S(t) = exp( --CT) [

cos ~5 + i sin qz

1 1

R(z)=exp(-CT) cosqr -ksinqr 1 04 i Q(t)=exp(-j7)-sinty~

rl

z = t, q = (1 - [*)05 = 0.998749217

and when t E [O.S, 1.01

x(t) = S(T)[X(O.5) - 1.21

+ Q(z)[a(O.5) + 2]+ 1.2 - 27 (45)

in which 7 = t -0.5, x(0.5) and i(O.5) can be calculated from eqn (43), and other quantities are again given by eqn (44). The remaining results were obtained from: the HPD-L method (i.e. a linear loading simulation) with time-step T = 0.125 and 0.5; the Newmark method; and the HPD-F method

Table 2. Displacement responses of the SDOF system due to a bi-linear loading

Scheme 5 0.25 0.50 -0.75 I.0 Max err.

Eqn (43)

HPD-L

HPD-F

;I:

q = 10 q = 50

Newmark

0.125 0.5

0.00515983

0.00515983

0.04064 I8

0.0406418 0.0406418

0.123901

0.12390 I

0.228135 0.000%

0.228135 0.000% 0.228 135 0.000%

0.25 0.00471467 0.040 108 1 0.122818 0.226892 8.627% 0.25 0.005 18503 0.0407124 0.123977 0.228232 0.488% 0.25 0.00515588 0.0406361 0.123891 0.228 I22 0.077% 0.25 0.005 15980 0.0406417 0.123901 0.228 134 0.000%

l/8 0.00577355 0.04 I724 I 0.123985 0.227206 1 1.894% I/32 0.00519826 0.0407096 0.123906 0.228076 0.745% l/l28 0.005 16223 0.0406460 0.123901 0.228131 0.047% l/512 0.005 15998 0.040642 1 0.123901 0.228 134 0.003%

118 Jiahao Lin et nl.

f(t) +

1.00 -, ,/---Y\ / \

/ \ /’ \ \

0.50

0.00 Il.7 _-__B t

0.00 1.00

Fig. 3. Off-peak half-sine wave loading

(i.e. a Fourier decomposition simulation) with q = 2, 5, 10 and 50 Fourier components used, for which

with

f(t) = i; a, sin 0, t (46) ,=I

wi = (2i - 1)~ and a, = 8( - I)‘-‘/[n2(2i - I)*]. (47)

Table 2 shows that, for the piece-wise linear loading used, the HPD-L form is the best of the three direct integration methods used. However, the table also shows that the HPD-F method deals flexibly and efficiently with arbitrarily varied loadings and rather large time-steps. The additional comput- ational effort needed by the Fourier decomposition and superposition only takes a very small percentage of the solution time. (Example 4 gives some compar- isons of CPU times required by the methods for a 3 DOF system.)

Example 3

Let the loading formf(t) of example I be replaced by the off-peak half-sine form of eqn (48) (see also

Fig. 3) with other parameters remaining unchanged.

(

sin(rct) when t E [0, l/6] or [5/6, I.01

f(t) = (48) 0.5 when t E (l/6, 5/6).

The responses of the displacement .X at some speci- fied times were computed and are given in Table 3. They were obtained as follows. Firstly, the HPD-S method was used within the intervals I E [0, l/6] and [5/6, 1.01, while the HPD-L method was used within the interval t E (l/6, 5/6). Within the whole region t f [0, I], T = l/12 and T = l/6 were both tried. Obvi- ously, the solutions thus obtained (denoted as HPD- S&L in Table 3) must be exact. This approach was used because it was necessary to have an exact result to test the convergence of the remaining methods, since an analytical solution was not easy to obtain. The predicted exactness is confirmed by the two rows in the table (for T = l/l2 and z = l/6) being equal. Secondly, the HPD-L method was used on its own, with 7 = l/12, l/24, l/96 and l/384 tried. Thirdly, the HPD-F method was tried with q = 5, 20, 50 or 100. with the Fourier coefficients determined from

.f(t) = i: a, sin w, t (49) /=I

in which

w, = x, a, = (n. I3 + 4/2)/?r,

w, = (2i - l)n, I

a, = n[(2i -21,2_ ,][m($+) > (50)

-I,,,(zi,i.)] 2i - I

(i = 2, 3, . ) J

Table 3. Displacement responses of the SDOF system due to off-peak half-sine loading

Scheme

HPD-S&L

HPD-L

HPD-F q=5

q = 20 q = 50 q = 100

Newmark

r

l/12 t/6

l/l2 l/24 l/96 l/384

11’3

J/6 116 l/6

l/12 l/24 l/96 l/384 11768

l/6 216 0.00237776 0.0161956 0.00237776 0.0161956

0.0820817 0.131923 0.188841 0.000% 0.08208 17 0.131923 0.188841 0.000”/0

0.00236285 0.0161412 0.0819556 0.131767 0.188633 0.627% 0.00237428 0.0161832 0.0820504 0.131884 0.188789 0.146% 0.00237754 0.0161948 0.0820798 0.131920 0.188837 0.009% 0.00237774 0.0161956 0.0820816 0.131923 0.188841 0.001%

0.00238195 0.0161943 0.0820755 0.131917 0.188826 0.00237823 0.0161963 0.0820829 0.131924 0.188842 0.00237779 0.0161957 0.08208 18 0.131923 0.188841 0.00237776 0.0161956 0.08208 17 0.131923 0.188841

0.00264298 0.0163885 0.0820777 0.131803 0.188294 0.00244435 0.0162441 0.0820809 0.131893 0.188704 0.00238192 0.0161987 0.08208 17 0.131921 0.188832 0.00237801 0.0161958 0.08208 I7 0.131923 0.188840 0.00237782 0.0161957 0.0820817 0.131923 0.188841

0.177% 0.021% 0.002% 0.000%

11.154% 2.801% 0.175% 0.01 1% 0.003%

416 516 1.0 Max err.

High precision integration scheme 119

M

K c r y3

M +--+ y2

KC

M

i

T K c

Fig. 4. A three DOF system.

Finally, the Newmark method was used with steps of r = l/12, l/24, l/96, l/384 and l/768.

Table 3 shows, for the loading used, which consists of linear and sinusoidal segments, that the HPD-F form gives very good results even when only five Fourier components are used, while the Newmark method produces about IO-20 times bigger errors than the HPD-L method when both methods share the same time step r. An advantage of all forms of the HPD method is that [r(t)] need only be generated once if identical step sizes tk are used within the whole time domain, because this matrix does not depend on the form of the loading.

Example 4

The equations of motion of the three DOF system shown in Fig. 4 are

Table 4. Displacement responses y, of the top mass of the three DOF system of Fig. 4

Scheme t 116 216 I

416 516 1.0 Max err.

HPD-S&L

HPD-L

HPD-F q=5 q = 20 q =50 q = 100

Newmark

l/12 0.00230184 116 0.00230184

l/l2 0.00228743 l/24 0.00229847 l/96 0.00230163 I/384 0.00230183

116 0.00230615 l/6 0.00230229 116 0.00230187 116 0.00230185

l/l2 0.00253527 l/24 0.00236037 l/96 0.00230550 l/384 0.00230207 l/768 0.00230190

0.0151731 0.0721257 0.112874 0.157740 0.000% 0.0151731 0.072 1257 0.112874 0.157740 0.000%

0.0151230 0.072020 1 0.112747 0.157573 0.626% 0.0151608 0.0720995 0.112842 0.157698 0.146% 0.0151723 0.0721241 0. I 12872 0.157738 0.009% 0.0151730 0.072 1256 0.112874 0.157740 0.000%

0.0151724 0.0721215 0.112871 0.157729 0.187% 0.0151736 0.0721267 0.112874 0.157740 0.020% 0.0151731 0.072 1258 0.112874 0.157740 0.001% 0.0151731 0.0721257 0.112874 0.157740 0.000%

0.0152946 0.0720710 0.112757 0.157302 10.141% 0.0 152036 0.0721121 0. I 12845 0.157631 2.543% 0.0151750 0.072 1249 0.112872 0.157734 0.159% 0.0151732 0.0721257 0.112874 0.157740 0.010% 0.0151731 0.0721257 0. I 12874 0.157740 0.003%

Table 5. Velocity responses u, of the top mass of the three DOF system of Fig. 4

Scheme

HPD-S&L

T l/6

l/12 0.0405455 l/6 0.0405455

216 416 516 I.0 Max err.

0.112019 0.223013 0.264574 0.259650 0.000% 0.112019 0.223013 0.264574 0.259650 0.000%

HPD-L

HPD-F q=5 q = 20 q = 50 q = 100

l/12 l/24 l/96 l/384

116 116 116 116

0.0403148 0.111822 0.222875 0.264461 0.259337 0.569% 0.0404877 0.111970 0.222978 0.264546 0.259572 0.143% 0.0405419 0.112016 0.223010 0.264572 0.259645 0.009% 0.0405453 0.112019 0.223012 0.264574 0.259650 0.000%

0.0404440 0.111894 0.223 108 0.26464 I 0.259626 0.250% 0.0405460 0.112023 0.223011 0.264576 0.259653 0.001% 0.0405456 0.112019 0.223012 0.264574 0.259650 0.000% 0.0405455 0.112019 0.223012 0.264574 0.259650 0.000%

Newmark l/12 0.0400782 0.111619 0.222773 0.264409 0.259562 1.166% l/24 0.0404285 0.111919 0.222953 0.264533 0.259628 0.289% 1196 0.0405382 0.112013 0.223009 0.264572 0.259649 0.018% l/384 0.0405450 0.112019 0.223012 0.264574 0.259650 0.001% I/768 0.0405454 0.112019 0.2230 13 0.264574 0.259650 0.000%

CAS 56/l-,

120 Jiahao Lin et al.

with

I

100.2 0 0

[Ml= 0 100.2 0

0 100.2

:c]=[:!: :;: -*51

I’

0

0 -85 85 z (52)

[

280 -140 0

[Kl= -140 280 -140

0 -140 140 I

{r,} = :

i i 100 J

The form off(t) was the same as that used in example 3, i.e. the off-peak half-sine wave form of eqn (48). The displacement y, and velocity uj of the top mass were computed by each of the four methods used for example 3, see Tables 4 and 5.

These two tables again use a proper combination of the HPD-S and HPD-L methods to obtain exact results for comparative purposes. They show that this type of problem can also be dealt with very efficiently by either of the HPD-F or HPD-L methods alone. In addition, for a given step size, the Newmark method generated about lo-20 times bigger errors than the HPD-L method for displacement responses, and about twice as large errors for the velocity responses.

The above computations were executed on the IBM/ 386SX notebook computer without a co-processor. When using the mixed HPD-S and HPD-L methods, i.e. the HPD-S&L method, a total of only 3 s CPU of computer time was used when ‘I = l/12. Using the HPD-L method with T = l/384 resulted in a maxi- mum error of about O.OOl%, but took a total of 43 s CPU. The HPD-F method with q = 50 Fourier

components gave similar accuracy too, and took a total computer time of 26 s. For each of these

schemes, the computation of [T(T)] took 2.4 s. For comparison, the Newmark method with step size

T = l/768 required 64 s CPU.

CONCLUSIONS

The earlier HPD-L integration method ]2] has been extended to form the new HPD-S and HPD-F

methods and all three methods have been applied

to various structures subjected to transient dynamic loading. It has been shown that all three methods are

efficient and have high precision. In particular, it has

been shown, for several loading cases, that a proper selection of, or flexible combination of, the HPD-L,

HPD-S and HPD-F direct integration methods can

be used to excellent effect. As explicit integration methods, all three are unconditionally stable and a

formal proof of this stability has been presented.

The HPD-S and HPD-F methods, proposed in this

paper as extensions of the HPD-L method [2], pro- vide effective alternative forms of the earlier HPD-L

method, making it even more competitive with the

Newmark method and other available good methods.

Acknowledgements-The authors arc grateful for the sup- port of the National Natural Science Foundation of China and of the British Royal Society. Many thanks are also due to Professor Wanxie Zhong for beneficial discussions.

REFERENCES

1. K. J. Bathe and E. L. Wilson, Numerical Med~ods in Finite Element Analysis, pp. 308-362. Prentice-Hall, Englewood Cliffs, NJ (1976).

2. W. X. Zhong and F. W. Williams, A precise time step integration method. J. med. Engng Sci., Proc. Insl. Mech Engrs, Part C (in press).

3. R. W. Clough and J. Penzien, Dynamics qf Structures. McGraw-Hill. New York (1975).