characterization on the rocking vibration of rigid blocks under horizontal harmonic excitations

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 2, pp. 229-236 FEBRUARY 2012 / 229

DOI: 10.1007/s12541-012-0028-0

1. Introduction

Many structures are not firmly fixed to their supporting bases.

Some of them, often modeled as rigid objects (blocks,) in fact, can

be virtually considered to stand free. They may be occasionally set

in rocking or sliding motion due to base movements. At that,

structure overturning or falling down can be extremely damaging in

terms of both material assets and human health and lives. Thus,

serious damage to freestanding structures, neighboring objects, and

people (including loss of life,) has been reported during Kobe

earthquake on January 1995, because high excitation level,

particularly in the vertical direction. Therefore, understanding

rocking and sliding motions is essential for the development of safe

operational guidelines, safeguarding of the existing structures, and

improving structure analysis and design.

The authors thought that damages of freestanding objects or

structures was related sliding occurrence by the high excitation

level in the vertical direction and have investigated the chaotic

characteristics and the effects to rocking behavior by sliding motion

since Kobe earthquake when the block shape structure is subjected

to two-dimensional excitation. Previous analytical and experimental

studies on dynamic behavior of rocking objects have proved that

rocking motion is very complex and sensitive. Further, these studies

have also demonstrated that sliding motion greatly affects rocking

response, its distribution and stability.1-9

Therefore, in this study nonlinear rocking model has been used

for sliding assuming that the contact between the block and the base

is changed only slightly, namely, static and kinetic friction

coefficients are changing commonly, nonlinear characteristics is

examined in detail. In addition, using analysis results, we will

discuss why rocking response was not recurring in some harmonic

excitation tests which we had carried out. Chaotic characteristics

and the change of the response distribution are examined with

respect to sliding motion and minute changes of friction coefficients.

It is thought that this may bring on an initial establishment of

earthquake proofing stability standard for block structures.

2. Experimental

2.1 Nonlinear rocking model

A rocking and sliding rigid block system under horizontal

excitations of the rigid base, ah(t), is shown in Fig. 1, where x, y and

θ are the horizontal, vertical and angular displacements,

respectively, of the center of mass of the block with respect to the

rigid (or supporting) base. The block is of width B, height H, mass

Characterization on the Rocking Vibration of Rigid Blocks under Horizontal Harmonic Excitations

Man-Yong Jeong1 and In-Young Yang2,#

1 Department of Digital Engineering, Numazu College of Technology, 3600, Numazu, Shizuoka, Japan, 410-85012 Department of Mechanical Design Engineering, Chosun University, 375, Seosuk-dong, Dong-gu, Gwangju, Korea, 501-750

# Corresponding Author / E-mail: [email protected], TEL: +82-62-230-7840, FAX: +82-62-222-7705

KEYWORDS: Rocking vibration, Nonlinear, Poincaré section, Normal probability distribution, Bifurcation diagram

This investigation deals with characteristics of rocking vibration of rigid block subjected to the horizontal harmonic

excitations. Nonlinear rocking system may include minute changes of system parameters, such as friction and restitution

coefficients depending on contact conditions between the block and the base, and friction force that may vary as the block

moves. This paper is the first part of the examination how minute changes of the above system parameters are related with

minute changes of friction coefficients. A numerical analysis program has been developed to solve nonlinear motion

equations for rocking motion of a rigid block subjected to horizontal base excitation. Natural changes of friction coefficient

are simulated by random numbers with a normal probability distribution. Analytical results have shown that rocking

responses are very sensitive to small changes of friction conditions between the block and the base. Minute changes of the

friction coefficients make rocking response unstable and greatly change the response distribution itself. Because of the

dynamic nature of friction coefficients changes, experimental rocking responses are not recurring.

Manuscript received: March 22, 2011 / Accepted: August 22, 2011

© KSPE and Springer 2012

230 / FEBRUARY 2012 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 2

m, the distance between either base edge, O or O’, and the center of

mass R and the shape angle ψ.

Note that for the purpose of this study we assume there is never

a loss of contact between the object and the base. Accordingly, the

vertical motion of the center of mass, y, relative to the base is a

function of the rocking angle, θ, given by

cos( )y R ψ θ= − (1)

However, we will continue to use the variable y in some

equations for convenience. Reaction forces between the base of the

object and the base surface in horizontal and vertical directions, fx

and fy, respectively, are given by

( )x hf mx ma t= + (2)

yf my m= + g (3)

Rocking motion is initiated (at time t) when the following

condition is satisfied

( )h

Ba t

H> g (4)

where g is acceleration of gravity.

Similarly, sliding motion occurs when the following condition

is satisfied

( )

h

s

x a t

yµ

+<

+

g (5)

Where ,s

µ is static friction coefficient between the object and

the base. When sliding or sliding and rocking (sliding/rocking)

occurs, the relationship between the horizontal and vertical reaction

forces and the kinetic friction coefficient, ,s

µ between the object

and the base is as follows

( ) S( ) h o k

m x a t x m yµ+ = − + g (6)

The governing equation of the motion about rotation centers O

and O’ is presented by

S( )sin( ) S( ) cos( )o o kI mR y xθ θ ψ θ µ ψ θ= + − + − g (7)

where Io is the inertia moment of the block about its edge, O or

O’ given by

2

4

3o

mRI = (8)

and alternative equations are defined by signum function, ( ),S a

where

( ) 1( 0)S a a= > (9)

( ) 1( 0)S a a= − < (10)

Introducing the following normalization / ,θ ψΘ = / ,p

Θ Θ p=

,ptτ = / pωΩ = and / ,o o

X x R= where

2 3

4o

m Rp

I R= =

g g (11)

the (non-dimensional) rocking and sliding equations can be

expressed as

2

1( , , ) 0

pp f XΘ+ Θ Θ = (12)

2

2

( )( , , ) h

p

aX p f X

R

τ

+ Θ Θ = − (13)

where 1( , , )

pf Θ Θ X and

2( , , )

pf Θ Θ X are given as follows

1( , , )

pf XΘ Θ =

2

2

S( )sin (1 ) S( ) ( ) cos (1 )1 cos (1 )

1 sin (1 ) S( ) ( ) cos (1 )sin (1 )

o k p

o k

S x

S X

ψ µ ψ γ ψ

γ ψ γ µ ψ ψ

Θ − Θ + Θ − Θ − − Θ Θ

+ − Θ + Θ − Θ − Θ

(14)

2

2

2

( , , )

S( ) ( ) cos (1 )1 cos (1 )

1 sin (1 ) S( ) ( ) cos (1 )sin (1 )

p

o k p

o k

f X

S x

S X

µ ψ γ ψ

γ γ ψ γ µ ψ ψ

Θ Θ =

Θ − Θ − − Θ Θ

+ − Θ + Θ − Θ − Θ

(15)

Where

2

o

mR

Iγ = (16)

The horizontal base excitation is accordingly assumed as

( ) sin( )h h

a Aτ ψ τ= Ω +Φg (17)

The non-dimensional horizontal displacement o

X and o

X of

the object edge are expressed as follows

S( ) sin (1 )o

X X R ψ= − Θ − Θ (18)

cos (1 )o

X X R ψ= + − Θ (19)

2.2 Impact of block against base

Impact of the block against the base is nonlinear and needs to

be careful modeled. Energy dissipation at impact without sliding

motion is constant and depends on object shape (i.e., slenderness

ratio). In the literature, energy dissipation rate is often characterized

by the restitution coefficient, e, as described by Yim et al..1,2

However, when sliding motion occurs, energy dissipation is a

function of the magnitude of horizontal sliding motion. Using the

principle of impulse and momentum, Shenton has suggested the

x

y

θ

ah(t)

B

H ψ−θ

ψ

θ

R

O

O’

ykox fxf µ)(S −=

yf

Fig. 1 Rocking of the rigid block

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 13, No. 2 FEBRUARY 2012 / 231

corresponding impact model.3 The effect of sliding motion at

impact moment had been discussed in our previous studies.5,6

When the object goes towards to impact from rocking or

rocking and sliding (rock/slide) motion, the normalized post-impact

velocities of the mass center, scalar components 2,X

2Y and

2,Θ

are expressed through pre-impact velocities 1,X

1Y and

1Θ as

follows:

2 1

2

iH

Xδ

= − Θ (20)

2 1 1

S( ) ( 2 )2

i

BY eδ= − Θ + Θ (21)

2 1i

δΘ = Θ (22)

where energy dissipation rate δi is defined as

2 23 31 (1 )cos (1 )sin

4 2i x

eδ λ ψ ψ= − + − + (23)

and the velocity rate x

λ and restitution coefficient e, respectively,

are given by

1

1

2

x

X

Hλ =

Θ

(24)

231 sin

2e ψ= − (25)

Subscripts 1 and 2 refer to pre-impact and post-impact values,

respectively, and subscript i refers to the impact edge. For pure

rocking motion (no sliding) at pre-impact, the rate x

λ is -1. During

impact, the condition of sliding occurrence is given by

( )

(1 2 )

i x

s

i

H

B e

δ λµ

δ

+>

+ +

(26)

where s

µ is static friction coefficient between the object and

the base during impact. For the case of rotating and sliding object,

normalized post-impact velocities are expressed by

2 1 1 2 1

S( )S( ) (1 2 )2

i k i

BX X X e

R

ψµ δ= + Θ + + Θ

(27)

2 1 1

S( ) ( 2 )2

i

BY eδ= − Θ + Θ (28)

2 1i

δΘ = Θ (29)

where k

µ is kinetic friction coefficient at impact and

normalized energy dissipation i

δ in case of sliding is given by

2

1 2

2

1 2

1 3 1 S( )S( ) (1 2 )sin

1 3 1 S( )S( ) sin

i k

i

i k

HX e

B

HX

B

µ ψ

δ

µ ψ

− − Θ +

=

+ − Θ

(30)

In the previous expressions, 1

S( )Θ denotes the pre-impact sign

of ,Θ2

S( )i

X the sign of post-impact velocity in the horizontal

direction.

Fig. 2 Histogram of normally distributed random number (µ = 0.4,

σ = 0.03)

3. Rocking with minute changes of friction coefficients

3.1 Minute change of normal probability distribution and

numerical analysis

In order to identify natural minute changes of rocking system

parameters, Box-Muller method for substituting homogenous

random numbers with normally distributed random numbers has

been used. With Box-Muller transformation we can convert two

homogenous random numbers u1 and u2 into random numbers, z1

and z2 with normal probability distribution.

1 1 2

2ln cos2z u uπ= − (31)

2 1 2

2ln sin 2z u uπ= − (32)

Figure 2 shows the histogram of a series of normally distributed

random number for 20,000 random values, the mean value of 0.4,

and the standard deviation of 0.03. As shown in this figure, the

generated random number can be considered to have normal

distribution. The kurtosis of the generated random number is almost

the same as the target value, and the skewness is no more than 0.01

that is small enough. In this study, we assume that the kinetic

friction coefficient always changes and the static friction coefficient

changes at the moment of sliding stopping.

Using a variable time-step Runge-Kutta method, the rocking

and sliding responses of the rigid block system are simulated

numerically to investigate nonlinear characteristics. The standard

normalized sampling time of the numerical integration time-step is

iteratively reduced by a factor of 10 each time until normalized

angular displacement becomes sufficiently small (10-6 or less).

When contact is reached (rotation angle equals to zero,) the

transition of the rocking equations is carried out by adopting

Equations (22) and (30) according to kinetic energy dissipation at

impact. An accurate rocking response analysis procedure has been

developed to obtain accurate numerical results. The algorithm

includes two major phases, namely rotation and impact. During the

rotation phase based on formula (5) it is determined if sliding

occurs and simultaneous equations (12) and (13) are solved using

Runge-Kutta method. During the impact phase the point of


normalized zero angular displacement is determined, the post-

impact velocity is calculated by formula (22) or (29) and then

exchanges the signum function S(Θ) in the rocking equation are

determined. In this study, unless otherwise stated, representative

block width of B = 1m and height H = 4m and the associated

restitution coefficient e based on Equation (25) are considered. The

following four methods of chaotic analysis are used to examine

rocking and sliding responses. First, bifurcation diagrams are

created by plotting normalized angular displacement sampled at

periodic points in the rocking response for 50,000 points (after

steady state has been achieved). The diagram shows variations in

periodicity of the response as a function of amplitude excitation.

Bifurcation diagrams are constructed with sampling amplitude

increments of 0.05 for horizontal excitation. Blanks (of gaps)

between amplitude increments represent overturning of the block.

Second, time values of selected individual responses are plotted to

form response characteristic. Third, Poincaré sections are

constructed using strobe points in the phase space for rocking

response sampled at periodic points of horizontal excitations. The

diagram forms a strange attractor in case of chaotic response and

reduces to a few fixed points in case of periodic response.

3.2 Influence of sliding in case of fixed friction coefficients

For practical freestanding rocking blocks sliding may occur

depending on friction coefficient between the block and the

supporting base. The probability of sliding increases with decreased

slenderness ratio (which is the case for a wide range of rigid blocks).

Thus, it is essential to take sliding into account when analyzing

rocking response. First, in order to investigate the effect of sliding

on rocking behavior, rocking responses are simulated for three

cases, without sliding, with static friction coefficient 0.4s s

µ µ= =

(overline means friction coefficient at impact) and kinetic

coefficient 0.35,k k

µ µ= = and with static friction coefficient

0.35s s

µ µ= = and kinetic coefficient 0.3.k k

µ µ= = The

corresponding bifurcation diagrams are shown in Figure 3. The

above friction coefficients are not identical with real rocking system

used in our previous works, which has non-recurrence for repeated

tests under the identical experimental condition.5,6 However, we can

estimate the rocking behavior qualitatively by examining simulated

responses.

Figure 3 shows bifurcation diagrams of rocking responses for a

rigid block subjected to a fixed horizontal excitation of Ωh = 10 and

Ah = 1-10 for next three cases: with sufficiently large friction

coefficients (Figure 3(a)), with static friction coefficient s s

µ µ= =

0.4 and kinetic friction coefficient 0.35k k

µ µ= = (Figure 3(b)), and

0.35s s

µ µ= = and 0.3k k

µ µ= = (Figure 3(c)). The restitution

coefficient e is set to 0.912. As shown in Figure 3(a), the rocking

response becomes periodic (harmonics of (1,1) mode, subharmonics

of (1,3) mode). The bifurcation diagram shown in 3b show periodic

responses of (1,1) mode, transition to (1,3) mode, back to periodic

responses of (1,1) mode and transition to chaotic responses as

excitation amplitude increases. As shown in Figure 3(c), the region

of (1,3) mode reduces with decrease of friction force between the

block and the base, so there are just periodic responses of (1,1)

mode and chaotic responses when 0.35s s

µ µ= = and k k

µ µ= = 0.3.

When compared to no-sliding case (Fig. 3(a)), one can see that

the initial part of the bifurcation remains unchanged as shown in

Figure 3(b) and 3(c). As shown in Figure 3(c), periodic responses of

(1,3)s mode are not suppressed by sliding effects when friction

coefficients are rather large, namely 0.4,s s

µ µ= = 0.35.k k

µ µ= =

If sliding occurs continuously, periodic rocking responses of (1,1)

or (1,3) mode turns into chaotic response, regardless of the

magnitude of sliding motion. On the contrary, if sliding is transient,

the periodicity of rocking response is maintained, but the sliding

motion makes it possible to change the mode of the rocking

response.

Figure 4 shows two examples of chaotic response changed by

sliding occurrence from periodic response of (1,3) mode. Here,

Θ shows time values of angular displacement and X indicates the

horizontal displacement by sliding. As illustrated in the figure, the

(a) Without sliding

(b) µs = 0.4, µk = 0.35

(c) µs = 0.35, µk = 0.3

Fig. 3 Change of bifurcation diagram by sliding (Ωh = 10)


response becomes random since sliding motion is continuous and

horizontal displacement by sliding is not homogenous. When

sliding occurs continuously, the kinetic energy of rocking motion is

dissipated non-homogenously by sliding occurrence and the

response becomes aperiodic. In addition, these examples indicate

that large rocking displacement causes big sliding motion. Note that

the magnitude of sliding increases with the magnitude of the

angular displacement by the rocking motion in Figure 4(a).

3.3 Influence of minute changes of friction coefficients

Effects of minute changes of friction coefficients are examined

by using bifurcation diagram, time values of rocking response and

Poincaré section. As stated in section 3.1, minute changes of static

and kinetic friction coefficients form normal probability distribution

around the mean value. Figure 5 with standard deviation of

fluctuation of friction coefficients for Figures 5(a), (b) and (c) are

0.003, 0.016, 0.03 respectively, shows bifurcation diagrams of the

identical system and excitation conditions, but with fluctuation of

friction coefficients with Figure 3(b). As shown in Figure 5(a),

rocking response remains almost unchanged, and response

distribution is nearly identical to that on Figure 3(b), when

fluctuation range of the friction coefficients are small enough. From

comparison of Figure 3(b) and Figure 5 it can be observed that the

region of (1,3) mode is enlarged and the region of chaotic response

is slightly reduced by increase of the fluctuation of friction

coefficients. Even though the change on rocking response is not so

large, the impact on rocking response distribution is large, and the

change of rocking response distribution also increases with

fluctuation range of friction coefficients. Analytical results for a

case when static and kinetic friction coefficients are 0.35 and 0.3

respectively, are not provided herein. However, similar results and

examinations are obtained by numerical study for various

fluctuation conditions of the friction coefficients.

When excitation amplitude is relatively small, it doesn’t affect

bifurcation diagrams even though friction coefficients are

continuously changed, because there is no sliding in that case.

However, when excitation amplitude is relatively large, sliding

continuously occurs, the effect by sliding motion occurrence

increases and the distribution of rocking responses changes

materially. When minute changes to friction coefficients, periodic

rocking responses of (1,1) mode change into periodic responses of

(1,3) mode and chaotic responses also change into periodic

responses of (1,1) mode around the boundary region between the

periodic mode of (1,1) and chaotic. Namely, fluctuation of friction

coefficients makes rocking responses change to almost no-sliding

state as shown in Figure 3(a).

State of rocking response depends on fluctuation range of

friction coefficients and brakes down into two types nearby the

boundary of (1,3) mode periodic and chaotic responses as shown in

Figure 6. The sliding motion continues first for a while, during the

transient state of rocking motion and then stops at steady state

response as shown in Figures 6(a) and (c). On the other hand, the

(a) µs = 0.4, µk = 0.35

(b) µs = 0.35, µk = 0.3

Fig. 4 Effect of sliding motion (Ah = 6.6, Ωh = 10)

(a) σ = 0.003

(b) σ = 0.016

(c) σ = 0.03

Fig. 5 Change of bifurcation diagram by minute fluctuation of fric-

tion coefficients (Ω = 10, µs = 0.4, µk = 0.35)


sliding motion is continuous and the response becomes chaotic as

shown in Figures 6(b) and (d).

Next, to investigate the effects of friction coefficient change on

the chaotic attractor, rocking responses are simulated within three

fluctuation ranges of the friction coefficients, the standard deviation

of the fluctuation of friction coefficients = 0.003, 0.016 and 0.03,

two typical examples of rocking responses are shown in Figures 7

and 8. They show Poincaré sections corresponding to rocking

responses for two excitation conditions, Ah = 8.0, Ω h = 10 and Ah =

9.0, Ω h = 10, in the bifurcation diagrams of Figure 5. The shapes of

the attractors of chaotic response are not different one from the

other under both conditions: fixed friction coefficients and minutely

changed friction coefficients. The shapes of the attractors are almost

not changed with small increase of the fluctuation range, unless

fluctuation parameters become extremely large. However, the

attractors become large with increase of fluctuation range. The

shapes of Poincaré sections change by increase of the excitation

amplitude as shown in Figure 6(d), 7 and 8.

3.4 Permutation of the fluctuations

The fluctuation of friction coefficients is calculated by creating

a random number with normal probability distribution using

application software to solve nonlinear equations of rocking motion.

The sequences of the finite series for fluctuation are not exactly

identical in all simulations. Instead, all simulation results are

different, even though the finite sequence of numbers makes up a

normal probability distribution. If the number of sliding instances is

m and the number of samples is n, the number of the fluctuation

instances is nPm. If sliding always occurs, the number of

permutations becomes n!. In this study the number fluctuation

instances was 500,000!. Here, n is the number of samples for

numerical analysis. The sampling number of the normalized

excitation amplitude is 90 from 1 to 10 with the interval of 0.02.

The probability of bifurcation is 1/(200,000!×90). However, since

there is a possibility that fluctuations of friction coefficients have

identical number, the expectation for the same bifurcation becomes

lower than 1/(200,000!×90) in practice.

(a) σ = 0.003 (b) σ = 0.016

(c) σ = 0.03

Fig. 7 Effect on Poincaré sections for minute fluctuation of friction

coefficients (Ah = 8.0, Ωh = 10, µs = 0.4, µk = 0.35)

(a) σ = 0.003 (b) σ = 0.016

(c) σ = 0.03

Fig. 8 Effect on Poincaré sections for minute fluctuations of

friction coefficients (Ah = 9.0, Ωh = 10, µs = 0.4, µk = 0.35)

(a) σ = 0.03

(b) σ = 0.003

(c) σ = 0.03 (d) σ = 0.003

Fig. 6 Minute fluctuations of friction coefficients (Ah = 6.6, Ωh =

10, µs = 0.4, µk = 0.35)


(a) (b)

(c) (d)

(e) (f)

Fig. 9 Bifurcation diagrams under the same friction coefficients

fluctuation conditions (Ωh = 10, µs = 0.35, µk = 0.3, σ = 0.03)

Accordingly, we may not simply examine those bifurcation

diagrams and we can only catch general trend based on those

diagrams and roughly discuss about their distribution shape. These

bifurcation diagrams can be an evidence of the non-recurrence of

rocking responses observed in our previous experimental works.

Because not only regions of (1,3) mode and chaotic responses are

different one from another, but also the details of rocking responses

are different for every numerical analysis. However, the cause of

non-recurrence may be related with another factor such as minute

fluctuations of the excitation acceleration, fluctuations of the

restitution coefficient or the combination thereof. Thus, additional

simulation may be necessary about these factors and their

combination such as simultaneous fluctuation of friction

coefficients and excitation amplitude.

3.5 Non-recurrence of rocking response

Previous experimental studies on dynamic behavior of rocking

objects with sliding motion have demonstrated that rocking

responses cannot be recur in repeated tests with identical excitation

conditions if sliding occurs. However, horizontal displacement by

sliding motion could not be measured in the previous experimental

works because of the rough sampling of laser displacement sensor.

Thus, there was no clue for the cause of non-recurrence that we

could find experimentally.5,6

Figure 10 shows the rocking response obtained from repeated

simulations at identical excitation conditions: Ah = 9.0, Ωh = 10, µs =

0.35, µk = 0.3, σ = 0.03. As shown in Figure 10, there are two types

of rocking responses of (1,1) mode and (1,3) mode when excitation

amplitude ranges from 3.2 to 4.6. From the results of this analysis,

we can partially make causes of various shapes of rocking

responses in repeated experiments under identical excitations be

obvious.

As possible causes of the non-recurrence of experimental

rocking responses we can also consider minute fluctuations of the

excitation force, minute dynamic change of the restitution

coefficient between the block and the base and rolling motion

around vertical axis. However, these causes, except for the

fluctuation of friction coefficients, should not greatly affect the

distribution of rocking responses. Sliding motion and its direction at

pre-impact yield the difference on the energy dissipation at impact

and it also effect massively to the rocking response and its

periodicity post-impact.

The minute fluctuation of the excitation force and minute

changes of the restitution coefficient can also be a cause of non-

recurrence of rocking responses. The contribution of these factors

needs, however, further investigation. In the previous experimental

works, we could not measure horizontal displacement by sliding

motion because of the low sensitivity of displacement sensors.

(a)

(b)

(c)

(d)

Fig. 10 Non-recurrence of rocking responses under same friction

coefficients fluctuation condition (Ah = 9.0, Ωh = 10, µs = 0.35, µk =

0.3, σ = 0.03)


Recently, the sensitivity and measurement accuracy of laser

displacement sensors have been improved. Accordingly, additional

experiments are required using new measurement instruments to

further investigate non-recurrence of experimental rocking

responses.

In this study, our attempt could not reach to obvious

clarification of non-recurrence cause on the experimental rocking

responses of (1,3) mode. However, the possibility of cause

clarification of the rocking response non-recurrence increases with

this study. As mentioned above, it is necessary to carry out

additional simulations to study the combination of such factors as

simultaneous fluctuations of friction coefficients and excitation

amplitude.

4. Conclusions

In this study, response characteristics of a rocking rigid block

subjected to horizontal harmonic excitations have been examined

for both cases: with and without fluctuation of friction coefficients.

Response analysis and examination for the results have been carried

out, and the conclusions can be outlined as follows.

(1) Sliding sets up chaotic response in the region which

otherwise would demonstrate periodic response. That is, a chaotic

response section appears when sliding starts. More specifically,

rocking responses are materially changed by sliding and are set to

into chaotic responses from periodic responses of (1,1) and (1,3)

mode when excitation is sufficiently large.

(2) The attractor shapes remain almost the same for a variety of

fluctuation ranges, provided fluctuation parameters are rather small.

However, the region of (1,3) mode is enlarged and the region of

chaotic response is slightly reduced with the increase of fluctuation

of friction coefficients.

(3) The occurrence of sliding motion makes periodic response

of (1,3) mode change into the periodic response of (1,1) mode. The

periodic rocking responses of (1,1) mode, which changes from

periodic response of (1,1) mode because of sliding, changes into

periodic responses of (1,3) mode because of fluctuations of friction

coefficients.

(4) The sequence of the finite series for fluctuations of friction

coefficients is different in detail every time. Various shape forms of

rocking responses in repeated experiments with identical excitation

conditions can be partially explained by simulations.

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