Design of a Pulse-Width-Modulation Spacecraft Attitude Control System Via Digital Redesign

DESIGN OF A PULSE-WIDTH-MODULATION SPACECRAFT ATTT...

14th World Congress ofIFAC

Copyright (' 1999 IFAC 14th Triennial World Congress. Beijing, P.R. China

P-8a-04-2

DESIGN OF A PULSE-WIDTH.MODULATION SPACECRAFT ATTITUDE CONTROL SYSTEM VIA DIGITAL REDESIGN

Tohru Ieko·, Yoshimasa Ochit, Kimio Kanai*. Noriyuki Doli·· and Peter N. Nikifornk1f

., t.

*Department of Aerospace Engineering. National Defense Academy

Yokosuka, KaTlLlgawa 239-8686. Japan [email protected] .. Department of Mechanical Engineering, McGill University. USA it Department of Mechanical Engineering, University of Saskatchewan, CANADA

Abstract: This paper proposes a design method for pulse-width modulation (PWM) control systems. In the method, a discrete-time (DT) control system of thearnplitude modulation type is first designed, and then it is converted into a PWM control system. A pulse width and a delay time for each PWM input are determined based on the principle of equivalent area (PEA) so that the output error between the DT and PWM control systems can be as small as possible. In this paper, the DT controller is detennined by the digital redesign of a given continuous-time (eT) controller. The digital redesign method is also based on the PEA. Thus, in spite of the inherent nonlinearity of the PWM input, linear control theories can be employed in the controller design. The effectiveness of the proposed method is illustrated through computer simulation using a linear model of the Japanese HOPE spacecraft. Copyright © J999lFAC Keywords: Aerospace control, Space vehicles, Digital control, Pulse-width modulation, Digital redesign

1. INTRODUCRION Unlike a control input that has its amplitude modulated, an on/off-type control input such as a reaction jet and an electronic relay has a fixed amplitude and its pulsewidth is modulated. This way of input modu1ation is called pulse-width modulation or PWM. PWM control systems have been analyzed and synthesized as non linear systems (Anthony et al., 1990 and Shigehara, 1994). However, analysis and synthesis by nonlinear methods such as the describing function method are generally complicated and difficult especial1y in multi-input-multi-output systems. This paper proposes a linear state-space design method for PWM control systems. In the method, a discrete-time (DT) linear time invariant (LTI) control system of the amplitudemodulation type is first designed, and then it is convened into aPWM control system. Since there are many established design methods for DT-L TI control systems, the proposed method aUows us to choose an appropriate method for PWM control system design. In PWM control, a pulse width and a delay time of

fIring must be determined at each sampling time. This paper derives a decision method for their determination from the principle of equivalent area (PEA) (Andeen, 1960) so that the error between the DT and the PWM state equations are small. A similar methodology based on the PEA has been proposed by Bemelli-Zazzera et al. (1992)~ however, that described in this paper is better in providing smaller error, which is of the third order of the sampling period. No matter what approach is chosen fOT the DT contmller design, the decision method of PWM control input can be applied. In this paper, a digital-redesign approach proposed by the authors (leko et al., 1996) is employed. The method determines the DT control input, also based on the PEA, so as to approximate the mean value of the pre-designed continuous-time (CT) closed-loop input during each sampling period. The layout of this paper is as follows. In the next section DT models and the PEA in the time domain are described. Then the PWM control law is derived from the PEA in Section 3. Section 4 follows to represent

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ISBN: 008 0432484

DESIGN OF A PULSE-WIDTH-MODULATION SPACECRAFT ATTT...

the digital redesign method also based on the PEA. In S=tion 5, the proposed approach is applied to the attitude control system design of the Japanese HOPE spacecraft to illustrate its effectiveness. Finally, some conclusions are given.

14th World Congress ofTFAC

;;(kT) =

ir

y{kT +m)dm

= C, E(A,T)x(kT)

(7)

+{D+ic! mE(A.m~}U(kT) :=Cx~T)+Du(kT)

2. PRELIMINARIES

2.1 Step Invariance DT Model In this paper, in order to express better the relation between the CT and DT systems, DT-LTI systems are described using the delta operator (Middleton, 1990). The operator is defined as £= (z-l)IT=(e'T-l)/T, where s, z and T denote the Laplace operator, the Z operator and a sampling period. respectively. Obviously the relation lim € ::: s holds. Since the delta and shift T-.
forms are theoretically equivalent, the content of this paper can be described using the shift operator. Consider a CT-LTI system described by

.:i(t) = Ax(t)+Bu(t), AER""" ,BE Ry(t}= Cx(t)+Du(t), C ER""" , DE R"'"

(1)

Note that the state equation of this model is the same as that of the step invariance model described by Eq. (4). Let us call the DT model defined by Eqs. (4) and (7) the step average model. Some characteristics of this DT model are as follows. (i) If both the CT system and its step invariance model are minimal realization, then this model is also minimal realization. (ii) The relationship of the poles between the system and this model satisfies c= (e'T-I)/T. (iii) If the er system is stable, then steady state gains of the eT system and this model are equivalent; namely.D - CA;IBS == D -CA -lB· (iv) The parameters of this model can easily be calculated by

er

ozr [D

where x(t)eRft, "(t)ER' and y(t)eRJ' are state. input, and output vectors, respectively, and A, B, C and Dare constant matrices. The solution of Eq. (1) for the initial condition x(to) is given by

x(t[)==e.t(·,-t.,) x(to)+

r:e.t(·,-P) Bu(p)dp

(3)

where to=kT, tl=kT+T (k=O, 1,2, ..... )andu(p)is

assumed to be constant during each sampling interval. Then the step invariance model is defined as fX(kT) = x(kT + T)- x (kT) T y(kT) =Cx(kT)+Du(kT) where As

= (eAT -I)/T

= Asx(kT) + B su(kT)(4)

and

0::,,] C

OMP

As

_I

(8)

T

2.3 Principle oj Equivalent Area Let us show that the PEA can provide a DT-control input that reproduces the time responses of the state to CT-control input with a good precision. FOT t[ to+.1, Eq. (3) becomes

=

X (10

+.1)==e.....d x(to)+B

r

TA

u(p):lp

(9)

Bs:o' e Ap dpB /T.

As = E{A,T)· A == A· E{A,T) Bs = E(A,T)·B, respectively. written

Bs

O",p 0pxp

(5)

The state and output of this model are equal to those of the eT system at each sampling time. Defining a function E(A, 1) as E(A , r)= '£"'n=::(j ~~ (AT)" , As and B. can (n+I)! be

J: ~ ~t

(2)

as

and

2.2 Step Average Model (/eko et aL, 1997)

From the mean value theorem there exists hj E [0

• . d~ f' p'u{to + d - p}1p == u(tf) +.1- h,. )-.-

Jo

1+1

XVf) + .1)= e~d x(to)+ B + 1:

Let us introduce a DT model whose output is the mean

;=1

rro ...A

J~

u(p)ctp (IJ)

.
('1+1.), u(to + d - h,)

Let the state at t J "" to +.1 for the same initial state x(to ) and another input u'(t) be denoted by x'(to +,6,). If the following equation holds (12)

Y(kT+m);:::ceAmX(kT)+( D+Cr e"P dp }O:T) (6) Then the mean value y(kT) of the CT system's output in kT:;;; t :;;; kT + T is obtained as

(10)

Then Eq. (9) becomes

~

value of the eT system's output during each sampling interval. This model is utilized in the digital redesign method in Section 5. Under the assumption of a constant input during the sampling interval, a inter-sample output at t = "T + m(O :s; m:S;; T) is described by

Lt]

which satisfies

then from Eq. (11) it follows that

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Copyright 1999 IF AC

ISBN: 008 0432484

DESIGN OF A PULSE-WIDTH-MODULATION SPACECRAFT ATTT...

14th World Congress ofTFAC

x'Vu + .1)-x(to +.1) ~

= 1: ,,,,J

Lii+'AiB

('1+1.)1

4. DJGITAL REDESIGN BASED ON PEA

{u'(to + .1-I{)- u(to + A - hi)}

(13)

=R(.•f)+R{A')+ ... where R(Ai ) denotes the error of the order of .1'. If L1 is sufficiently small, the error becomes negligible. Equation (12), which suggests the 'Equivalent Area', means that there exists a constant input u'(t) for to:::;,g. that satisfies Eq. (13). As shown 1ater, in the PWM control system design and digital redesign, we determine the control inputu' SO that it can satisfy Eq. (12). 3. PWM CONTROL LAW BASED ON PEA Consider the input shown in Fig. 1, where ii p , a(k1) and t(k1) denote a fixed magnitude, a pulse width and a delay time, respectively. It is assumed that three control inputs, - ii po 0, and ii p (>0) are available. Let the PWM control input be ii p and the state equation is described as

x(kT + T)= eAT X{kT)+ e"(T-Hr)

r

eAPd pBii (l4)

Suppose a DT-control input system is given and let us consider determining ii, cr, and 1: from the DT-control input. The error between Eqs. (4) and (14) is written as

x(kT+T)--X(kT +T)

=E(A ,T):rBu (kT)

(15)

-eA{r-T('T}-
:= R(Tl )+ R(T2)+ R{r 3 )+ ...

In open-loop digital redesign methods. such as Tustin's, a very fast sampling eaU: is often required to guarantee stability and good control performance. For the purpose of ensuring a larget N, however, digital redesign methods that allow a larger sampling period are desired. In the following, such a closed-loop digital redesign method (Ieko et al .• 1996) proposed by the authors is presented. Suppose that for a er plant described by Eqs. (1) and (2) a CT state-feedback control law. u(t)

= Fx(t) + Gr(t)

(19)

is given, where F and G are constant matrices and r{t) is external input. The eT plant-input transfer function (PITF) H(s), which is a closed-loop transfer function from the external input r(t) to the control input of the plant u(t), can be written as H(s):

,Bk [A+BF F IJ

(20)

Let us assume that a zero-order hold is used in the application of the DT pJant-input uj,kTJ. Then, regardjng u' as ucl,.k1), fa as kT, and L1 as T, apply the PEA or Eq. (12) to u(t) and u.l..k1); that is, (21)

In Fig. 1 the DT control input u.tkT) is determined by Eq. (21). From Eqs. (7) and (21) it can be seen that u.l..kT) is the output of a step average model of H(s). Hence, by replacing A, B, C and Din Eqs. (4) and (7) by A+BF, BG, F and G in Eq. (20), respectively, the step average model is obtained as (22)

where

R(T)::;; B{Tu(kT)-cr(kTp} R{T2)= AB[{Tu{kT)-cr(kT)i:tr +{-T +2'f(kT)+cr(kT)}:r(kT)iV2

(15-1) (15-2)

Now for the j-th (j = 1,2,. ",r) control input, select the amplitude, pulse width and delay time of the control input signal, respectively, as

uj =sign{uj(kT)}.uPj

(16)

crikT)~ Tu j(kT)juj

(17)

l)kT)=V-erikT)'v2

(l8)

Then, R(D ::;; R(r) = 0 is achieved, and the resulting third-order term in Eq. (15) becomes R(:f!)

= {Ji!-
Fd and G d are also calculated in the same way as Eq. (8). Since the DT control law is derived from the PEA, it reproduces the time response of the eT state well, as shown by Eq. (13). This digital redesign method has the fol1owing features. a) When T -70 the obtained gains become the predesigned eT ones. b) Although the redesigned closed-loop system matrix A.+B.Fd is different from the desired one ~(".+B,I')T _ I}lT, the first-order approximation of the sampling period is achieved in the de1ta form. In fact, from the Taylor series expansion around T = 0 it follows that {e("p+B.I':rr

Equation (17) corresponds to the PEA, and the most suitable delay time obtained from Eq. (18) means that the center of the pulse width should be kT+TI2. This delay time is different from the result in BernelliZazzera et al. (1992). Since the minimum pulse width a mlD is finite in practical PWM control systems, we

-1}/T -{As + BsFd}

~-ApBpF(Ap +BpF)r2/12+R{T3)

(15-3)

(23)

Please note that this is equivalent to the secondorder approximation in the shift form. This feature provides stability for relatively large T. c) If the redesjgned digital control system is stable, then the steady DT state is equivalent to steady eT one; that is,

suppose that T = N er... holds for a natural number N and assume that the number is large enough to guarantee the small state-error ofEq. (13).

xAoo)::::-(As + BsF"tBsG" '=-(Ap +BpFdt BpG

= x(oo)

(24)

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DESIGN OF A PULSE-WIDTH-MODULATION SPACECRAFT ATTT...

5. HOPE Attitude Control In this section the proposed methodology is applied to the design of an attitude control system of the H-II Orbiting Plane (HOPE}, which is Japanese small-size unmanned space shuttle. Assuming that the HOPE is symmetrical about the XZ plane, the equations of motion in the body axes are as follows. Translational motions: (25) X axis: mC(u B + qBWB - rs"II)== Fx (26) Yaxis: mc(ve + rB"1j - p"wB )= Fy Z axis: mc(wg + PS"B -qguB )= Fz

(27)

Rotational motions: X axis: I}{xPB -lxzrB +(/22 -lyy")qBrB -lxzPBqlJ = Mx (28) Yaxis: IyyqlJ + (In - Izz )r/lPlJ + Ixz{Pi -r; )=M y (29) z axis: -lxzPB + IzzrB + (lyy -Ixx )PBqB + IxzqorB == M z (30) where Fx , F y, Fz and U B ' "0' W B are forces and velocities along the X, Yand Z axis respectively; Mx. My, M z and Ps' Q8' r8 are torques and angular velocities about the respective axes; and the mass me== 7,900 (kg), the moment of inertia [xx == 13,000 (kgm2), Iyy == 59,000 (lcgm2), Izz.== 66,000 (kgm2), and the product of inertia Ixz= 2,000 (kgm2). The thruster system of the HOPE is shown by Fig. 2 (M utota, 1997). The force or torque with respect to a certain axis is generated by a combination of some thruster units with appropriate thrust ratios. Murota derived the optimum combinations for translational and rotational motions with respect to each axis by the LP method (Murota, 1997). Table 1 shows basic combinations of the thruster units. For example, the force +Fx is produced using thruster T3, T4, TJ 1, T12 and T15 with the output ratio of 0.546 : 0.546 : 0.386 : 0.386 ; 1.0. The resulting force along the X-axis is then +800 (N). Meanwhile, the equations of motion can be decoupled and linearized by openloop decoupling as fonows. Solving Eqs. (28) and (30) for Pll and ill' we obtain PB=

1

2

InIzz-lxz

[(Iyylzz-/;"-I~hBrB

+ (lxx1xz + lzzlxz -IyylKZ )PSqB

(31)

+ IzzM x + l."zM z ]

rB ==

1 2 Ixx/zz -Ixz

[(-

I xx/yy + Jix + l~ )PBQe

+ (-l/
(32)

In the case of roU motion without yaw motion, from Eq. (32) and Table 1, if qjl =; 0 and M z = C~x' where Cp~-I06.4Id(706/xx)' are satisfied, then 78 =0 is achieved. Similarly from Eq. (31) yaw motion without roll motion requires qB = 0 and Mx = C,Mz, where C r=-706lxz1(I06.4Iz0. Thus, we can realize six decoupled motions described by the following linear differential equations.

14th World Congress ofTFAC

Translational motions without rotations: YB = Fy/mc (34), ZB = Fz/mc (35) B Rotational motions about a single axis: ~B = Mx (lzz +Cp/xz)j(/xx/zz -liz) (36)

x = Fx/mc (33), iiB=My/lyy

ViB

=Mz{lxx +Cr/xz)j(lxx1zz -I~)

(37)

(38)

where urdxsfdt, vo==dyr/dt, w/I=tizsfdl, pu=dt/J,/dt, Qg=:d9s1dt and rrdlfl,/dt. To generate the forces or moments required for the decoupling control, the thruster units must produce thrusts that achieve the thrust ratios in Table I. However, since the thrust amplitude of each thruster unit is fixed, the ratio must be realized by the modulating timing and the duration of the emitting jets. PWM control is therefore required for the attitude control of the HOPE. The proposed method is applied to design of the attitude control system and its performance is evaluated through computer simulation.

Open-loop decoupling control: First, we consider performing tile six decoupJing controls both in the positive and negative directions of each axis. The proposed method is compared to the conventional one that modulates the pulse width only. Figure 3 shows the simulation results of positive rotational motion about the X-axis. The solid and broken lines indicate the proposed and conventional PWM methods. respectively. Although both PWM methods incur errors so as to excite coupled motions, the proposed method provides more precise control than the conventional one, which is observed in simulation results of other decoupling controls as well. Closed-loop comrnand-following control: The above simulation results indicate that the PWM control using combinations of the thruster units cannot achieve completely-decoupled motions. However, closed-loop control is robust against such an accumulation of errors. In this study we employ a proportional-plus-integral (PI)-type optimal regulator to design a command following control system. The position and attitude commands are generated for the HOPE to achieve the following series of motions: approaching a target spacecraft from 400 m away, yawing by 70 deg. and then rolling by 30 deg. The command of the translational motion along the X axis in the positive direction is generated by the solution of Hill's equation (Kaplan, 1976) for 05~OO s. The yaw command and the roll command are given as ramp functions of time for 530:5;t:G60 s and 62():::;~50 s, respectively. Hill's equation (KapJan. 1976 and Imado. 1989) is described in the target axes as Eq. (39). In this equation co denotes angular velocity of the target; the target axes are defined by Fig. 4; [xn YP ZT]T indicates the chaser's or HOPE's position relative to the target; and hp /'r> and hr are the control forces applied to the chaser. Figure 5 shows a position control system where the PI-type regulator is designed for the system represented by Eq. (39), and then the CT controller is digital-redesigned. The control input I/"p /yn hr]T given by the DT con-

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DESIGN OF A PULSE-WTDTH-MODULATION SPACECRAFT ATTI...

troller is transformed into the control forces along the body axes, (Fx , F r , Fz1T. Then the body-axis forces are realized by the PWM control along with the combinations of the thruster units and their thrust ratios. Figure 6 represents an angle control system. which is designed by the same methodology as the position control system. Figures 7(a) and (b) show time histories oftransla.tional and rotational motions, respectively. In the figures the solid and broken lines indicate the proposed and conventional PWM methods, respectively, and the dotted lines indicate commanded positiOns or angles. The results show that the proposed PWM method works well and gives better conlTol performance than the conventional one. ~

0

}

0

0

0

0

XT

.iT

0

0

0

2u>

0

0

.iT

0

0

1

0 0

0 0

YT

0 0 0

0

1

_ru 2

0

ZT (39) iT

ZT

0 0 0

iT

0

d YT dt YT

::::

-2m )w2 0

0

0

0

0 1

+me

0

0

Id"

0

0 0

0

/ yr

0

0

0 0 0 0

0

0

f.T

YT

6. Conclusions

In this paper a new design method for PWM control systems was proposed. The method consists of two steps: the redesign of a pre-designed eT control law into a DT one and then the conversion of the DT contro) input into a PWM input. The redesigned DT con1T011aw is derived by applying the PEA to the cJosedloop control input. A decision method for the pulse width and delay time of the PWM control input is also derived from the PEA. It is not dependent upon plant

14th World Congress ofIFAC

parameters. The pulse width and delay time guarantee that the state error between the DT and the PWM systerns is of third order of the sampling period in openloop systems. The proposed method is applied to the attitude control of the HOPE spacecraft and it is confrrmed through computer simulation that the proposed method gives better control performance than a conventional method. References Anthony, T. C. and B. Wie (1990). Pulse-Modulated Contra] Synthesis for Flexible Spacecraft. AIAA JOU11UJl of Guidance, Control and Dynamics, 13, 1014-1022. Shigehara, M. (1994). Introduction to Space Engineering - Navigation and Control of Artificial Satellite and Rocket (in Japanese). Baihuukan. Bemelli-Zazzera F. and P. Mantegazza (1992). PulseWidth Equivalent to Pulse-Amplitude Discrete Control of Linear Systems. AIAA Journal of Guidance, Control and Dynamics, 15,461-467. Andeen, R. E. (1960). The Principle of Equivalent Area. AIEE Applications and Industry, 79, 332-336. Middleton, R. H. and G. C . Goodwin (1990). Digital Control and EstilTUltion - A Unified Approach. Prentice Hall. Ieko, T., Y . Ochi, K. Kanai and N. Hori (1996). Digital Redesign Methods Based on Plant lnput Mapping and a New Discrete-Time Model. Proc. 0/ IEEE 35th CDC, 1569-1574. Ieko, T., Y. Ochi and K. Kanai (1997). A New Digital Redesign Method for Pulse-Width Modulation Control Systems. Proc. of the AIAA GNC Cont, 1730-1737. Mucota, A. (1997). Optimal Thruster Selection Algorithm for HOPE's Orbit Control by LP Method (in Japanese). Master's Thesis, Shinsyu University. Kaptan, M. H. (1976). Modern Spacecraft Dynamics & Control. Imado, F. (1989). Guidance and Navigation for Aerospace, Lecture Note of SICE 4th Basic Course on Guidance and Control/or Aerospace (in Japanese), 53/86.

u(t)

Fig. 2 Thruster system of HOPE

Fig . .1 DT and PWM control inputs by PEA

8037

Copyright 1999 IF AC

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DESIGN OF A PULSE-WIDTH-MODULATION SPACECRAFT ATTT...

14th World Congress ofTFAC

Table 1 Basic combinations of thruster units

-- ..

r""",

x

y

z

""'x "

y

Z

IIIJO

"

0

0

0

0

0

0

0

U

0

0

I !¥.2

0

0

0

&

0

~JJ9_2

0

0

0

0

2::'i3.
0

0

R

+F. F. ~r,

-F,

+F. p.

+M,

M, +M. M,

'M, -Ut-

n

T.

TlI

0.,..

o,~

0,_

1'5

T6 0.20'

I.'

n

T' 0.><11

OJ!I1l T:I 0_ TlI 0.168 T4

....., 'H

T6 1,0

TB

0102 1'9

'

T.O 1.0

TJ.

0'''. n.

U,2Ol

TII

O.JI7

TIS \,0

no

A... ,

TI2

'I13

0.3l1

OM'

TI2

O.7C\08 'Ill

T) Tt4 1,0 III 0·7211 ~.72i 1':! n TtO TI2 TB O,tl'1 0.023 0.0]1 I .• T4 TU r14 'N o.ng I .• 0.023 0..,.1 T4 T12 'Ill n 1.0 0.7<17 0,107 1.0 T •• TI3 1'0 1.0 O:/(f/ 0, TU 719 TIl Tl TO 1.1> "'076 (J.?j7 ...OI T4 T6 TlI r •• 0.616 0.151 O.08l ••()10 l.0

......

.....

T'

,..

""'"

•

• • •

0

0

.:n....

0

0

0

0

0

106.•

0

0

0

0

0

0

0

" "

• •

-u)O." 0

L361.2

0

0

-1021.1

0

0

•

0

0

0

0

0

0

0 0

0

0

.?
~~. l..

e---

+

~

comma.d gefleralor

r--' rH

lift

x

+~

~btlJ:tiilc;r

Mr

CD-Rirof

P,

.11

e.

HOPE dyuamicl

r~'"

pnin

'I'~.

~B

~;:: rHo

r---Jo

Q.

",

T.

~::::

+

I

Fig. 4 Target axes

Fig. 6 Angle control system

"...1 +

~[llIImaDd

~

lYe, ..

glmenHor

~

~.c",",

8-. IDffi-~

f. r

+.,....

target areS"

t

f,r

~"

/()

r--r--

!

~

I,r

+

~

10

body

t-,..

I--

from

UCJ

~::: dal\lste:r CODlr{)!

uait

~~

~:;:

+

from

HOPE dy nal»ic5

aDd Hill', equation

body

~

f--./ '0

uco

•

lU!tl.U:.t:S

K,

Fig. 5 Position control system

~y ~ ! ~k?~1 !.~Ir------------JJ !r:~d o

200

400

eoo.lQO

0

fi~~"'---r----.-----..------------

I

I

11

'r

...,

-5

0

!

2W

eo<>

400

.,..~

...

toO

100

r'

g

.,,".;.1 , .. ' -"::1 l.r

0

200

O!!OO

~

I/JOO

..

~

0

~ ' , I L -

I

'""

'"'"

~

~

'

~

j

--

,,,

... 0.1

1 " -0..•

cA.

0

~

~

'fIM. (MC'

~

~

0

nn. (..0)

~

-400

r ,

600

800

,,

-0.1

'"'"

rg

~rsr=j ~V\---------!... I~Ir--------'1 o

o.roo-

300

o.~,...-~--~-_----,

o

~

~

r ... (..::)

~

~

~I

!

0

~

~

-an. (MC)

D ~

~

Fig. 7(b) Time histories (rotational motions)

Fig. 7(a) Time histories (translational motions)

8038

Copyright 1999 IF AC

ISBN: 008 0432484