steady state rocking response of rigid blocks part 1: analysis

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 18,89 106 (1989)

STEADY STATE ROCKING RESPONSE OF RIGID BLOCKS PART 1: ANALYSIS

W. K. TSO AND C. M. WONG Department of Civil Engineering and Engineering Mechanics, McMaster University, Hamilton, Ontario, Canada U S 4L7

SUMMARY The result of a theoretical study on the rocking response of rigid blocks subjected to sinusoidal base motion is presented. The study indicates that, for a given excitation amplitude and frequency, a rigid block can respond in several different ways. Based on analysis, the regions of different classes of steady state symmetric response solutions are mapped on the excitation amplitude-frequency parameter space. The steady state response solutions (both harmonic and subharmonic) are classified into two classes, out-of-phase and in-phase with respect to the excitation. Only out-of-phase solutions are found to be stable. A parametric study shows that steady rocking response amplitude is highly sensitive to the size of the block and the excitation frequency in the low frequency range. It is relatively insensitive to the excitation amplitude and the system's coefficient of restitution of impact. For two blocks of the same aspect ratio and coefficient of restitution subjected to the same excitation, the larger block always responds in smaller amplitude than the smaller block. Computer simulation is carried out to study the stability of the symmetric steady state response solutions obtained from analysis. It is found that as the excitation frequency is decreased beyond the boundary of stable symmetric response, the response becomes unsymmetric where the mean amplitude of oscillation is non-zero. Further decrease in excitation frequency beyond the stable unsymmetric response boundary causes instability in the form of overturning.

INTRODUCTION

The problem of rocking response of a rigid block subjected to base motion has been studied for a number of technical reasons. By evaluating the rocking response of free standing tombstones and monuments which led to their overturning, an estimate of the ground shaking intensity of past earthquakes which toppled these objects may be established.' Recently, concern for possible overturning of unanchored structure and equipment during earthquakes led to a renewed interest to this Apart from the technical interests, the problem of rigid block rocking is intrinsically of interest from a theoretical point of view. The problem is highly non-linear in nature. Since the non-linearity is not linearizable, the problem does not render itself to be studied by the traditional approaches for non-linear systems with small non-linearities. As a result, its dynamic behaviour is less understood than many other non-linear vibrational problems.

While the conditions which lead to overturning of a rigid block are the main focus of attention in most investigations, a significant advance in the understanding of the rocking phenomenon has been made through fundamental studies. A study by Housner7 has provided the basic understanding on the free rocking response of a rigid block. Some results on the forced rocking response of rigid blocks subjected to sinusoidal base excitations have been presented by Ogawa' and Spanos and K O ~ . ~ Ogawa analytically determined the conditions when steady state harmonic responses are possible, and predicted the steady state harmonic response amplitude as a function of excitation amplitudes and frequencies. Spanos and Koh showed by computation that, in addition to steady state harmonic response, a rigid block can be excited into steady state subharmonic rocking response. Also, a rigid block can execute steady state response with a zero mean amplitude under one set of conditions and steady state response with a non-zero mean amplitude under another set of conditions. For ease of identification, these are referred to as symmetric (zero mean amplitude) and unsymmetric (non-zero mean amplitude) modes of response. It is clear that, owing to the highly non- linear nature of the problem, a variety of rocking response modes can exist even when the input excitation is relatively simple, namely, sinusoidal excitation with constant amplitude.

0098-8847/89/010089-18$09.00 0 1989 by John Wiley & Sons, Ltd.

Received 23 December 1987

90 W. K. TSO AND C. M. WONG

In the present paper, and its companion paper," a detailed theoretical and experimental study on the steady state rocking response of rigid blocks subjected to sinusoidal base excitations is reported. Analytically, the approach adopted follows closely the one used by Ogawa. Three useful findings are presented. First, the regions where steady state harmonic and 1/3 subharmonic response are possible in the excitation amplitude-frequency parameter space are established analytically. Second, a parametric study is carried out to evaluate the effects in the variation of the system parameters on the steady state response amplitude. It is found that the steady state amplitude is highly sensitive to the system 'frequency' but relatively insensitive to both the excitation amplitude and the coefficient of restitution for impact. Third, by evaluating the solutions of the equations of motion numerically, the behaviour of a rigid block at transitional stages where the mode of steady state response changes and becomes unstable is investigated. It is found that, in some instances, the transition from steady state response results in the overturning of the block, while in other instances, the transition is from one type of steady state motion to another type of steady state motion (e.g. from 1/3 subharmonic to harmonic). The purpose of the present study is to provide additional understanding and information on the steady state rocking response of rigid blocks. Many of these analytical and numerical predictions on steady state rocking response are checked by an experimental investigation which is described in a companion paper."

STATEMENT OF THE PROBLEM

Consider the two-dimensional problem of a symmetrical rigid block resting on a rigid horizontal base and subjected to a base motion, U&C), as shown in Figure 1. It is assumed that the contact surface between the block and the base is sufficiently rough so that sliding cannot occur. In this case, the block either remains stationary relative to the base when the excitation is small, or rotates by pivoting about one of its edges. For sufficiently large oscillatory base motions, the block tends to respond by rotating about each of its edges in turn, leading to the phenomenon of rocking. For rocking response, the rigid block is characterized by its mass, M , and its mass moment of inertia about the pivoting points, I , . The location of the mass centre of the block with respect to the pivotal base points, 0 and 0', is defined by the distance, R, and the angle, OCr. Here 8,, is the maximum angle to which the block can be tilted without toppling over under gravity. The displaced position of the block from its equilibrium position is denoted by the angle, 0, measured with respect to the horizontal base. A positive 0 indicates the block is rotating about point 0, and a negative 0 indicates rotation about point 0'.

When subjected to a base excitation given by u&), the equations of motion when the block is pivoted about point 0 and 0' respectively are

(1)

(2)

I ,B'+ M g R sin(e,,-8)- M R U , COS(H, , -~) = o

I ,@- M g R sin (Ocr + 0) - MRU, cos (OCr + 0) = 0

where 0 > 0, and

e < o e = o e > o

c=====r "b Figure 1 . Rigid block

STEADY STATE ROCKING RESPONSE 1 91

where 8 < 0. The sinusoidal base excitation can be conveniently expressed as

ii, = M , , g sin (ot + 4) (3) Since B,,g is the base acceleration required just to uplift the block from the rest position, A becomes a convenient measure of peak base acceleration intensities. For slender blocks, 8,, is in general small and one may linearize the equations of motion by using the approximations sin (eCr f 8) = 8,, k 8 and cos (6,, f 8) = 1. Substituting equation (3) into equations ( I ) and (2), the linearized equations of motion can be expressed in a dimensionless form given by

Q 2 i j - u = A s in( t+d)-1 for u > O (4)

and Q 2 u - u = A sin(z+@)+l for u < O

where u = 8/0,,, T = wt and Q = o / p , and p can be considered as the system frequency given by

P = JMgR/I , (6) Equations (4) and (5) have the following solutions:

u + = a + sinh (t/Q) + b + cosh (z/Q) + 1 - f i sin (T + 4) (7)

u - = a - sinh(.r/Q)+b-cosh(t/fJ) - l - f i s in(z+#) fo ru<0 (8)

for u > 0

and

where f i = A/( 1 + Q'). The superscript indicates that the denoted quantities are valid only for positive angular displacement and

that they are valid only for negative angular displacement. The integration constants, a + , b', a - and b- , depend on the initial conditions. When u =0, the change in angular velocity due to impacts can be accounted for by the relation

b, = so, (9) where bb and b, are the angular velocities just before and after an impact respectively, and S is the coefficient of restitution during impact. Equations (7-9) will be used to arrive at a sufficient condition for steady state response.

STEADY STATE SYMMETRIC RESPONSE

A sufficient condition for steady state response is that the response period is an integer multiple (n) of the excitation period. This condition is shown in Figure 2 for symmetric harmonic response with n = 1. The sinusoidal base motion at time zero is shown to have a phase shift of 4. For symmetric steady state responses, the angular displacements at the beginning of the positive cycle (z=O) and at the beginning of the negative cycle (T = n) must be equal to zero, and the angular velocities must be equal in magnitude and opposite in sense. Since an impact occurs at the end of the positive half cycle, the angular velocity at the beginning of the negative half cycle is reduced by a factor of 6. To maintain steady state response, this implies that the magnitude of the angular velocity at the end of the positive half cycle must be equal to the magnitude of the angular velocity at the beginning of the positive half cycle divided by 6.

Based on the above considerations the following equations can be established to describe the sufficient conditions for a symmetric steady state response mode with a half period of n7c ( n = 1,2,3, . . . ): when z = 0,

u + ( O ) = 0 (10) and


.. Excitation

Figure 2. Sufficient conditions for symmetric harmonic steady state response

and when t = nz,

u+(nn)= 0

and

d + (nz) = - 1j,,/6 (131

With equations (10) and (11) as the initial conditions the constants a+ and b+ in equation (7) can be

(14)

where 6, is the angular velocity at the beginning of a half cycle, as defined in equation (1 1).

determined as the following:

a+ = R(C, + f l cos 4) and

h+ = f l sin 4 - 1

Equations (12) and ( I 3) lead to the equations

0 = a'sinh (nz/R) + b' cosh (nz/R) + 1 + f l sin 4 and

b+ ' a+ _ - us - 6 R R (17)

Equations (16) and ( 1 7) are the conditions for determining b,, the angular velocity at the beginning of a half cycle, and the phase angle, 4, of a steady state response. Arranging equations ( 1 6) and ( 1 7), ti, can be expressed explicitly by the following two equations:

-- cosh (nz/R) + -sinh (nz/R) + flcos 4

v = - ( ' R h F s i n @ + h ) - f l c o s +

and

b = - ' (Ps in4- -h) - f lcosQ R h

where h = tanh (nz/2R).


Eliminating the unknown fis in equations (18) and (19) results in a single equation for 4, namely

(20) P cos 4 + sin 4 = Q

(1 -4 (1 +a) ’ where P = DhR, Q = Dh2/P and D = -

From equation (20), the following equations can be written to determine 4 explicitly:

and

Q f P,/- 1+P2 sin 4=

cos 4 = QPTJ- 1 +P2

In order for 4 to be real, the radicals in equations (21) and (22) must be positive, namely, P2 + 1 > Q2. Expressed in terms of the non-dimensional acceleration amplitude, A, this condition can be written as

(1 + R2)Dh2 A’J@i@jTi

TYPES OF STEADY STATE SYMMETRIC RESPONSE

In addition to the excitation amplitude inequality (23), two other inequalities can be derived to express the boundaries of different types of steady state response solution.

The signs in front of the radicals in equations (21) and (22) cannot be chosen arbitrarily. If sin 4 takes on the positive solution of equation (21), cos 4 has to take on the negative solution of equation (22) and vice versa. To determine 4 uniquely, both equations (21) and (22) are required because there are two possible values of 4 for a single value of sin 4 or cos 6. Denoting as the value of 4 obtained from equations (21-22) by taking the positive radical in equation (21) and the negative radical in equation (22), one arrives at the conclusions that

0 < 4 1 < n / 2 if Q > 1

and ~ / 2 < 4 ~ < n i f Q < 1

Similarly, denoting d2 as the value of 4 determined based on taking the negative radical in equation (21) and the positive radical in equation (22), the ranges of phase angle 42 are given by

0 < 4 1 ~ < n / 2 i fP<Q

- ~ / 2 < 4 ~ < 0 i f P > Q and

Denoting the class of steady state solutions according to the range of 4 such that - 7112 < 4 < n/2 as in- phase solutions and n / 2 < 4 <3n/2 as out-of-phase solutions, one can conclude that the steady state symmetric response of a rigid block has either a pair of in-phase solutions (when Q > I), or a pair of solutions with one out-of-phase and one in-phase (when Q < 1).

The inequalities, P 2 Q and Q 2 1 , can be expressed respectively as the following in terms of the excitation parameters, A and R:

and

A 3 Dh2 (R2 + 1)


The above two inequalities together with inequality (23) basically divide the A-R excitation parameter space into four regions. Such divisions for harmonic responses are presented in Figures 3(a) and 3(b) for two values of coefficient of restitution, 8. A point in regions 1 and 2 of the A-R space denotes an excitation condition that can leadto both in-phase and out-of-phase solutions. The in-phase solution in region 1 has a positive phase angle but a negative phase angle in region 2. Region 3 represents the excitation conditions which can lead to two in-phase solutions: one has a positive phase angle and the other has a negative phase angle. No symmetric steady state solution is possible in region 4.

6 Region Solution 6 = 5.5

5 - 1 1

1, 2 One out-of-phase and one in-phase

3 t w o in-phase

4 no solution

0 2 4 6 8 10 1 2 1 4

Excit. Froq. Ln)

(a)

6 Region Solution 6 = 0.95

1. 2 One out-of-phase and one in-phase

3 two in-phase

4 no solution

0 , , , I I I I , I I I I I I

0 2 4 6 e 10 ? 2 1 4

Excit. Freq. (n) (b )

--@

--a Figure 3. Regions for different types of symmetric harmonic steady state solution

95 STEADY STATE ROCKING RESPONSE 1

6

5 -

4 - A a Y

a E 3 -

.- @ :: 2 -

w

1 -

Region 3 at which both solutions are in-phase is small in general and it disappears entirely for large (close to unity) values of 6. Region 4 also decreases in size as 6 increases.

The divisions of the excitation parameter space for 1/3 subharmonic steady state response (n = 3) are shown in Figure 4. Again, as 6 increases (decreasing damping in the system), region 2 expands at the expense of regions 3 and 4. Region 1 is insensitive to the value of 6.

d = 0 . 9 5

Region Solution

I , 2 One out-of-phase and one in-phase

3 two in-phase

P no solution

n E <

6 = 0 . 5

6 -

4 -

Region Solution

1, 2 One out-of-phase and one in-phase

3 . 5 two in-phase

4 no solution

0 ! , , , , , , , , , , , , , , 1 0 2 4 6 8 10 12 14

Excit. Froq. (n) (a)


PARAMETRIC STUDY FOR SYMMETRIC STEADY STATE SOLUTIONS

A study is made to investigate the effect on the steady state symmetric rocking solutions of a rigid block if its system parameters 6 and p and the excitation parameters A and w are altered. Results for both harmonic and 1/3 subharmonic responses are presented. Only the out-of-phase steady state solutions are considered since the in-phase solutions are always unstable, as will be shown in the next section. Frequency response curves are used to present the results of this parametric study.

1

0.9

0.8

0 . 2

0.1

0 0 2 4 6 8 10 12 14 16

Excit. Frsq. (n) (a)

1

0 9

0 8

u O 7 '73

0.6

*.a 0 5 " \

0 . 4

0 3

0 2

0 1

n

0

A = I A = 2 A = 3 A = 4 A = 5 A = 6

\ \

I 1 1 1 1 1 1 1 1 . . .

0 2 4 6 a 10 12 14 16

Ereit. Freq. (n) (b )

Figure 5. Frequency response curves for different excitation levels, A: (a) symmetric harmonic response; (b) symmetric 1/3 subharmonic response

STEADY STATE ROCKING RESPONSE: 1 97

The steady state response amplitude is calculated first by determining the phase angle, 4, using equations (21) and (22). The angular velocity, O,, is then computed from equation (19). Once fiS and 4 are known, the constants, a + and b + , are determined from equations (14) and (15), and the solution, u + (T), as given in equation (7) is completely defined. The steady state response amplitude, which corresponds to the value of u + ( . r ) when the angular velocity, t j + ( ~ ) , becomes zero, is then determined.

In Figures 5(a) and 5(b) are the frequency response curves for harmonic and 1/3 subharmonic responses. They are calculated based on different excitation amplitudes, A, and a coefficient of restitution, 6, of 0.8.

Excit. Freq. (Hz)

(a)

0 2 4 6 8 10 12 14 16

Excit . Freq. (Hz)

(b)

Figure 6. Frequency response curves for different values of p : (a) symmetric response; (b) symmetric 1/3 subharmonic response


Unlike the characteristics of a single degree of freedom system with linearizable non-linear spring, no response peak can be observed from the rocking response curves. For both types (harmonic and 1/3 subharmonic) of response, the steady state amplitude increases as excitation frequency, R, decreases. Also, the response amplitude does not increase in proportion to the increase in the excitation amplitude. The 1/3 subharmonic response curves for excitation amplitudes of A = 1 and A = 2 terminate at certain frequencies because there is no 1/3 subharmonic steady state solution beyond these excitation frequencies for the given excitation amplitude and restitution factor.

Shown in Figures 6(a) and 6(b) are the frequency response curves for different p values for both harmonic and 1/3 subharmonic responses. The response amplitudes are calculated based on an excitation amplitude of A = 3 , and a restitution factor of 6=0% The response amplitude is sensitive to the parameter, p , and it increases with an increase in p. As defined by equation (6), one can express p as

I , + M R 2 R; + R 2 P =

where R,= a= radius of gyration about the mass centre. Equation (26) implies that the steady response amplitude is independent of the mass of the block, but

depends on the size, R, and mass distribution, R,, of the block. For the particular case of a homogeneous rectangular block, R,= R d , and p is given by

P=&

From the above expression, it can be seen that for blocks with the same critical angle, O,,, the smaller ones (i. e. those with smaller R) lead to larger p values, which in turn lead to larger steady state response amplitudes according to Figures 6(a) and 6(b). This trend of size effect on the dynamic.behaviour of rocking blocks was first noted by Housner' in association with the overturning potential of such blocks.

The effect of the restitution factor, 6, on the steady state response amplitudes for harmonic and 1/3 subharmonic response is shown in Figures 7(a) and 7(b) respectively. The range of 6 values considered is from 0-7 to 1-0. It can be seen that the steady state amplitude is insensitive to the restitution factor, 6, in general, and particularly insensitive in the case of harmonic responses. As the restitution factor increases (ie. decrease in damping in the system), the response amplitude increases for harmonic responses. However, for 1/3 subharmonic responses, the response amplitude increases as the system damping increases (i.e. 6 decreases). This kind of behaviour is contradictory to common intuition.

In summary, the steady state rocking response is sensitive to.the system frequency, p, and for a given system, it is sensitive to the excitation frequency, a. It is relatively insensitive to the restitution factor, 6, of the system and the excitation amplitude, A.

STABILITY ANALYSIS USING TIME HISTORY SOLUTION COMPUTATION

Instead of using the classical approaches of stability analysis," a time history computational approach is used to evaluate the stability of the steady state solutions.

In this approach, a steady state response time history is generated by first calculating the phase angle, 4, and angular velocity, b,, using equations (19), (21) and (22). Together with a zero angular displacement, these parameters are used as the initial conditions for the computation. Equations (7), (8) and (9) are used to compute the time history. If the solution is stable, the computation will give a steady state response time history, and the resulting steady state amplitude will be the same as the steady state amplitude determined in the previous section. If the contemplated solution is unstable, the instability will manifest itself since any computational round off error is equivalent to a small disturbance applied to the contemplated steady state solution. One advantage of the current approach, as compared to the classical approach of stability analysis, is that it not only provides information on stability, it also describes the dynamic behaviour of the system during the transition away from stable steady state response.


1

0 9

0 8

0 .7

- 0.6

m 0.5

iw 0 4

V 0.3

0 - .- e t S'

0.2

0.1

0 I 0 2 4 6 8 10 12 14

I I I I 1 5 I

Excit. Froq. (n) (a)

. . 0 2 4 6 8 10 12 14

Excit. Freq. (n) (b)

Figure 7. Frequency response curves for different values of 6: (a) symmetric harmonic response; (b) symmetric 1/3 subharmonic response

(a) Stability of in-phase solutions It is found that none of the in-phase steady state solutions is stable. Shown in Figures 8(a) to 8(c) are typical

response time histories for in-phase harmonic responses. The system parameters are p = 211 = 6.283, and 6 = 0 8 . The excitation amplitude is A = 1. Even though the initial conditions are set for in-phase steady state response, the system leaves the in-phase response after a few cycles and eventually reaches to some other stable (out-of-phase) harmonic response as shown in Figures 8(b) and 8(c), or leads to toppling over as shown

W. K. TSO AND C. M. WONG

1

0.5

0

-0.5

- 1

I p I 6.283 d . 0 . 8

I I I I 1 I I I I I i 0 2 4 6 8 10

Time in n u m b r of axcitation poriad

(a)

n L L

. \ 6 2

I m m

v v " y v v v v u v u v v u u v Stab I e out-of-phase harmonlc response

V " -0.5 -

- 1 I I I I 8 I I I I

0 4 8 12 16 20 24

Time in nurnbor of axcitation poriod

(b )

S t a b l e out of phase harmonic response

-0.04 - -0.06 - -0.08 l l l l l l l l l l l l l l l l l , l

0 2 4 6 8 1 0 12 14 16 18 20

Time in nurnbor of excitation poriod

(C)

Figure 8. In-phase harmonic response traces for an excitation with A = 1: (a) frequency = 1 Hz; (b) frequency = 2 Hz; (c) frequency = 8 Hz

in Figure 8(a). The in-phase 1/3 subharmonic responses are also unstable. After a few cycles of in-phase 1/3 subharmonic response, the system may topple over as shown in Figure 9(a), it may settle to a stable harmonic out-of-phase response as in Figure 9(b), or to a stable 1/3 subharmonic out-of-phase response as in Figure 9(c).

(b) Stability of out-of-phase solutions A typical stable harmonic out-of-phase response trace is shown in Figure 10. Two stable 1/3 subharmonic

out-of-phase response traces are shown in Figures 1 l(a) and 1 l(b). A 1/3 subharmonic response has a period that is three times that of the excitation. The trace in Figure l l(a) has two peaks in each half cycle. The


TOPP I I ng p I 6.263 d I 0 . 6 c i

.D L

-li -2

0 2 4 8 a 10 12 14 1 6 i n 20

Tim. in numkr of oxcitation poriod

(a)

2

1

$ L s,u 62 s

0

rmonlc response -1

-2 0 I0 20 30 40

Tim. in numbor of oxcitotion poriod

( b )

1

0.5 n m i 8 . \ 0

-0.5

m m 5

-1 0 10 20 50 40

Tim. in numkr of oxcitation poriod

( C )

Figure 9. In-phase 1/3 subharmonic response traces for an excitation with A = 2 (a) frequency=35Hz; (b) frequency=4.0 Hz; (c) frequency = 5.0 Hz

distinctiveness of the second peak diminishes as the excitation amplitude, A, decreases from 4 to 2, as shown in Figure 1 l(b).

The behaviour of the system as it transforms from stable steady state harmonic response to toppling over when the excitation frequency is reduced is illustrated by a series of time history plots in Figure 12. The values of the system parameters, p and 6, are 2n and 0 8 respectively, and the excitation amplitude is kept constant at A =2, while the excitation frequency varies from 2.0 to 1.7 Hz (0 varies from 2.0 to 1.7). At 2.0 Hz, the steady state response is stable as shown in Figure 12(a). Decreasing the excitation frequency to 1.9 Hz, the symmetric


1

0.5

* L ","

0

-0.9

- 1 I I I I I I I I I I I 0 2 4 6 8 10

Tim. in nurnbor of oxcitation poriod

Figure 10. Typical harmonic response trace for sinusoidal excitation (frequency=4 Hz and A = 6)

- 1 I I I I I I I I I I I I I 1 I

0 + 8 1 2 16 20 2+ 28 32

Time in number of excitation period

(a)

1

0.5

4 ' i i o - 0 6 2 z

-0.5

- 1 0 4 a 12 16 20 24 20

Time in number of oxcitation period

(b)

Figure 1 I. Typical 1/3 subharmonic response traces for sinusoidal excitation (a) frequency = 6 Hz and A =4; (b) frequency = 6 Hz and A = 2

STEADY STATE ROCKING RESPONSE: 1

Stab18 symnctrlc rcaponse

2 I

3 - 2 -

103

n 8 1.7

n 2 . 0 1

2 , 1

Stable unsymnetrlc harmonlc response

1 I 1 5 I 0 10 20 30 40

Tim. in number of oxcitation poriod (b)

2

1

P L a,. 0 @2 <

- 1

-2 0 4 8 12 16 20 24 2% 32

Tim0 in numbor of orcitotion poriod (C)

Figure 12. Harmonic response time histories: (a) R = 2 Q (b) R=1.9; (c) R= 1.8; and (d) R = 1.7

104

2 -

1 -

W. K. TSO AND C. M. WONG

n = 3 . 0

0 10 20 30 40

Time in nurnbor of excitation poriod (a)

2

S L s o 1

w; s 0

a

- 1 0 20 40 60 80

Time in number of excitation period (b)

n - 3.6

- 2 1 I I I I I I , ! , I I I I I r I 0 4 8 12 16 20 24 28 3 2

Time in number of excitation period ( C )

3 1

Figure 13. 1/3 subharmonic response time histories: (a) R = 4.4; (b) R = 4.2; (c) R = 3.6; and (d) R = 3.0


harmonic response becomes unstable and the system shifts gradually to an unsymmetric harmonic ste< dy state response mode. This mode of steady state response was first reported by Spanos and Koh.' The asymmetry of the response increases with further decrease in exciting frequency as shown in Figures 12(b) and 12(c). Further decrease in excitation frequency results in transition from steady state unsymmetric response to overturning of the block as shown in Figure 12(d).

The behaviour of the same system as it transforms from stable steady state 1/3 subharmonic response towards toppling over as a function of the excitation frequency is illustrated in Figure 13. At a frequency of 4.4 Hz (Q=4.4), the 1/3 subharmonic steady state symmetric response is stable as shown in Figure 13(a). With a decrease of excitation frequency to 4.2 Hz (Q=4*2), the symmetric form of response becomes unslable and the system responds in a 1/3 subharmonic unsymmetric response mode as shown in Figure 13(b). With further decrease of excitation frequency to 3.6 Hz, no form of 1/3 subharmonic response is stable and the system eventually responds with a stable symmetric harmonic response as illustrated in Figure 13(c). Finally, further decrease of frequency to 3.0 Hz results in overturning of the block after a few cycles of subharmonic response as can be seen in Figure 13(d).

It should be pointed out that the frequency range where the unsymmetric steady state responses can exist is very small when compared to the frequency range of symmetric steady state responses, for both harmonic and 1/3 subharmonic responses. Also, the peak to peak response amplitude for unsymmetric response is always close to the predicted peak to peak symmetric response amplitude from analysis. Additional information on the computed responses can be found in Reference 12.

CONCLUSIONS

A detailed study on the steady state rocking response of rigid blocks on a rigid base when subjected to sinusoidal base excitation is presented. Because of the highly non-linear nature of the problem, a variety of steady state solutions appear to be possible. In the present study, emphasis is placed on steady state harmonic responses and steady state 1/3 subharmonic responses.

The possibility of having steady state response is best presented in the excitation amplitude-frequency parameter space. Analysis shows that, for a given forcing frequency, no steady state solution is possible if the base excitation is either too large or too small. For intermediate values of excitation amplitude, steady state solutions are possible only when the response is out of phase with the base motion. All in-phase steady state solutions are unstable, and as a result they cannot be realized in reality.

A parametric study shows that the out-of-phase steady state response amplitude is sensitive to the system frequency parameter, p , and also to the excitation frequency ratio, a. The larger the value of p , the higher is the steady state amplitude. For a homogeneous rectangular block of a given aspect ratio, p is inversely proportional to the square root of the size of the block. Therefore, smaller blocks would respond with larger steady state amplitudes, when compared with larger blocks with the same aspect ratio. On the other hand, the steady state response amplitude is relatively insensitive to the restitution factor, 6, of the system. These observations are applicable to both harmonic and 1/3 subharmonic responses.

Finally, the stability of the out-of-phase steady state solution is studied by means of a computer simulation. It is shown that, as the system is excited near the boundary of stability, the steady state symmetric response shifts to another form of steady state response, namely, unsymmetric steady state response. In this form of response, the system no longer oscillates with a zero mean amplitude. Further changes of excitation frequency below unsymmetric steady state responses result in overturning of the block.

Many of these forms of response are actually observed experimentally. The detailed description of the experiment and the comparison of the experimental results with the theory are given in an accompanying paper."

ACKNOWLEDGEMENT The writers wish to acknowledge the support from the Natural Science and Engineering Research Council of Canada (NSERC) for the work presented herein.


APPENDIX

Notation a + , b’, a- , b - A D 9 h I , 10 M P P

R

t

Q

R ,

UP

i, U

fi,, fib

4 P 6 e,, e B 8’

4 5

0

R

constants of integration dimensionless sinusoidal base excitation amplitude damping parameter, (1 - 6)/( 1 + 6) gravitational acceleration tanh (nn/2R) mass moment of inertia of a rigid block about the centre of mass mass moment of inertia of a rigid block about the pivotal point mass of rigid block rocking frequency parameter, Jm D h R

distance between the centre of mass and the pivotal point of a rigid block radius of gyration about the mass centre time in seconds horizontal base excitation normalized angular displacement (d/e,,) normalized angular velocity (8/8,,) angular velocity just after and before impact normalized angular velocity at the beginning of a half cycle of steady state motion A / ( 1 + R2) restitution factor maximum angle which a block can tilt without toppling under gravity alone angular displacement angular velocity angular acceleration dimensionless time (ut) phase angle frequency of the sinusoidal base excitation normalized excitation frequency (alp)

Dh*/B

REFERENCES

1. Y. Ishiyama, ‘Review and discussion on overturning of bodies by earthquake motion’, BRI Research Paper No. 85, Building Research

2. M. Aslam, W. G. Godden and D. T. Scalise, ‘Earthquake rocking response of rigid bodies’, J. struct. din ASCE 106,377-392 (1980). 3. K. Muto, H. Umemura and Y. Sonobe. ‘Study 01 overturning vibration of slender structures’, Proc. 2nd world conf. earthquake eng.

4. Y. Ishiyama. ‘Motions of rigid bodies and criteria tor overturning by earthquake excitations’, Earthquake eng. struct. dyn. 10,635650

5. I. N. Psycharis and P. C. Jennings, ‘Rocking of slender rigid bodies allowed to uplift’, Earthquake eng. struct. dyn. 11.57-76 (1983). 6. C. S. Yim, A. K. Chopra and J. Penzien, ‘Rocking response of rigid blocks to earthquakes’, Earthquake eng. struct. dyn. 8, 565-587

7. G. W. Housner. ‘The behaviour of inverted pendulum structures during earthquakes’, Bull seism. soc. Am. 53.403417 (1963). 8. N. Ogawa, ‘A study on overturning vibration of rigid structure’, Proc. 7th world conf. earthquake eng. Istanbul, Turkey 7,205 (1980). 9. P. D. Spanos and A. S . Koh, ‘Rocking of rigid blocks due to harmonic shaking’, J. eng. mech. dio. ASCE 110, 1627-1642 (1984).

10. C. M. Wong and W. K., Tso, ‘Steady state rocking response of rigid blocks, Part 2 Experiment’, Earthquake eng. strucf. dyn. 18,

11. C. Hayashi. Nonlinear Oscillation in Physical Systems. McGraw-Hill, New York, 1964. 12. C. M. Wong, ‘Rocking response of rigid blocks subject to harmonic base excitation, M. Eng. Thesis, McMaster University, Hamilton,

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steady state rocking response of rigid blocks part 1: analysis

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