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22.4 Sliding Isolation Systems

Back

Sliding isolation can be an effective means for the seismic protection of structural systems. By

implementing sliding isolators under the base mat of a structure, the transmission of ground excitation

to the structure can be greatly reduced. Currently, applications of sliding isolation systems can be found

elsewhere (Naeim and Kelly, 1999). A sliding isolator usually consists of a slider with frictional surfaces.

For this reason, it is also referred to as a friction isolator. When subjected to an earthquake, the slider will

slide along the frictional contact surfaces whenever the horizontal seismic force exceeds the maximum

frictional force of the support, which, by Coulomb’s theory, is equal to the normal contact force

multiplied by the static (or dynamic) coefficient of friction of the sliding surfaces. Because of this, the

seismic force transmitted to the superstructure is generally less than the maximum frictional force of the

isolator. Obviously, the maximum frictional force is an important parameter for the design of a sliding

isolation system, because it decides when the system starts to slide and how large the shear force is to be

transmitted to the superstructure.

The motion of a sliding structure consists of two different states, namely, the sliding state and the stick

(or nonsliding) state. At any instant of motion, the structure can only belong to one of the two states.

Although in each state the sliding structure can be modeled as a linear system, the governing equations of

motion for the two states are different. As a result, the overall behavior of the sliding structure is

nonlinear. Such nonlinearity has resulted in the occurrence of subharmonic resonance in the frequency

response of a sliding structure (Mostaghel et al., 1983; Westermo and Udwadia, 1983), making the

dynamic response much more complicated. In some applications, a sliding isolation system has been

designed with an automatic recentering mechanism (or resilient mechanism), so that the structure can

slide back to its original position after the earthquake (Mokha et al., 1991). This type of sliding systems

has been called the resilient sliding isolation system, which will be investigated in Section 22.5. The

implementation of a recentering mechanism offers some advantages, but will inevitably introduce some

disadvantages, as will be discussed in Section 22.5.

The purpose of this section is to investigate the seismic behavior of a sliding isolated structure and

also the behavior of an equipment item mounted on the structure. No consideration will be made for

the recentering mechanism. The nonlinear dynamic equation for a structure with an underneath friction

element is first formulated. Next, two numerical approaches will be presented for solving the nonlinear

equation, the shear balance method and fictitious spring method. Finally, using some assumed data, the

harmonic response and seismic behavior of a sliding structure, together with the equipment mounted

on it, will be presented. In this section, the frictional coefficient of the sliding system is assumed to be of

the Coulomb type, that is, a time-invariant constant. For simplification, no distinction will be made

between the static and dynamic frictional coefficients, or between the dynamic and maximum static

frictional force.

Structure and Equipment Isolation 22-17

22.4.1 Mathematical Modeling and Formulation

22.4.1.1 Equation of Motion

A sliding isolated structure with an attached

equipment item, as schematically shown in

Figure 22.15, can be represented as a mass –

spring – dashpot system of three DoF, as shown in

Figure 22.16, for which the notations employed

have been defined in Table 22.1. When the

structural system is excited by an earthquake, the

equation of motion can be written as

Mx€ðtÞ þ Cx_ðtÞ þ KxðtÞ ¼ 2ML1x€gðtÞ þ L2f ðtÞ

ð22:27Þ

where the vector x denotes the dynamic responses

of the whole structural system

xðtÞ ¼

xeðtÞ

xsðtÞ

xbðtÞ

8>><

>>:

9>>=

>>;

ð22:28Þ

The mass, damping, and stiffness matrices in

Equation 22.27 are defined as

M ¼

me 0 0

0 ms 0

0 0 mb

664

775

;

C ¼

ce 2ce 0

2ce ce þ cs 2cs

0 2cs cs

664

775

;

K ¼

ke 2ke 0

2ke ke þ ks 2ks

0 2ks ks

664

775

ð22:29Þ

and the force distribution vectors as

L1 ¼

8>><

>>:

9>>=

>>;

; L2 ¼

8>><

>>:

9>>=

>>;

ð22:30Þ

Note that in Equation 22.27, the isolator frictional force, f ðtÞ; which is not a constant, is moved to the

right-hand side of the equation. This nonlinear force requires a special treatment in an analysis

procedure, as will be explained later on.

For a systematic treatment, the above equation of motion can be further written in a state space form

as shown below (Meirovitch, 1990):

zðtÞ ¼ AzðtÞ þ Ex€gðtÞ þ Bf ðtÞ ð22:31Þ

Ground

Sliding Bearing

Equipment

Superstructure

Base Mat

FIGURE 22.15 Schematic for an isolated structure –

equipment system with sliding bearing.

ks cs

ce xe

ke me

FIGURE 22.16 Model for an isolated structure –

equipment system with sliding support.

22-18 Vibration and Shock Handbook

where the state vector zðtÞ and the system matrix A are defined as

zðtÞ ¼

x_ðtÞ

xðtÞ

" #

; A ¼

2M21C 2M21K

I 0

" #

ð22:32Þ

and the excitation and friction distribution vectors as

E ¼

2L1

" #

; B ¼

M21L2

" #

ð22:33Þ

22.4.1.2 Conditions for Stick and Sliding States

As mentioned above, the motion of a sliding structure at any instant has two possible states, namely, the

stick (or nonsliding) and sliding states. The following are the conditions that must be satisfied by the

sliding structure:

(1) In stick state

lf ðtÞl , fmax ¼ mW ð22:34aÞ

x_bðtÞ ¼ 0 ð22:34bÞ

where m is the coefficient of friction and W is the total weight of the structure. According

to the preceding equations, the frictional force in the stick state is an unknown with a

magnitude less than the maximum frictional force, fmax; which equals the product of m and

W ; while the sliding velocity of the structure is simply zero. Whenever the frictional force

satisfies Equation 22.34a, the sliding system remains in the stick state, otherwise it changes

into the sliding state.

(2) In sliding state

f ðtÞ ¼ 2sgnðx_bðtÞÞfmax ¼ 2sgnðx_bðtÞÞmW ð22:35aÞ

x_bðtÞ – 0 ð22:35bÞ

where the function sgnðxÞ denotes the sign of the variable x: According to Equation 22.35a

and Equation 22.35b, the frictional force in the sliding state has a magnitude equal to the

maximum frictional force, but directed in a sense opposite to that of the sliding velocity. On

the other hand, the sliding velocity of the isolator remains as an unknown.

22.4.2 Methods for Numerical Analysis

Two numerical methods commonly used for the analysis of sliding isolated structural systems, the shear

balance method and fictitious spring method, will be introduced in this section. By employing the

discrete-time state-space formula, both methods can be cast in an incremental form that is suitable for

the analysis of sliding systems with multiple DoF.

22.4.2.1 Shear Balance Method

Consider the state-space equation, Equation 22.31, and assume that both the ground acceleration and

frictional force vary linearly within each time interval, as shown in Figure 22.17. Equation 22.31 may be

written in the following incremental form (Meirovitch, 1990)

z½k þ 1􀀉 ¼ Adz½k􀀉 þ E0x€g½k􀀉 þ E1x€g½k þ 1􀀉 þ B0f ½k􀀉 þ B1f ½k þ 1􀀉 ð22:36Þ

where the symbol x½k􀀉 denotes that the variable x is evaluated at the kth time step. The other coefficient

Structure and Equipment Isolation 22-19

matrices in equation 22.36 are defined as

Ad ¼ eADt ¼

i¼0

Dti

Ai ð22:37Þ

B0 ¼ ðAÞ21Ad þ

Dt ðAÞ22ðI 2 AdÞ

􀀒 􀀓

B ð22:38aÞ

B1 ¼ 2ðAÞ21 þ

Dt ðAÞ22ðAd 2 IÞ

􀀒 􀀓

B ð22:38bÞ

where Dt denotes the size of the time step

considered for analysis. The matrices E0 and E1

can be computed in a way similar to B0 and B1;

except that the matrix B in Equation 22.38a and

Equation 22.38b should be replaced by the matrix

E. In some applications, the system matrix A

may not be invertible, that is, A 21 does not always exist. If this is the case, one may compute B0 and B1

(and similarly E0 and E1) by the following formulas:

B0 ¼ Ap

dB; B1 ¼ ðA^ d 2 Ap

dÞB ð22:39Þ

where

d ¼

i¼0

ðDtÞiþ1

i!ði þ 2Þ

Ai; A^ d ¼

i¼0

ðDtÞiþ1

ði þ 1Þ!

Ai ð22:40Þ

Note that, on the right-hand side of Equation 22.36, the only unknown at the kth time step is the

frictional force f ½k þ 1􀀉; therefore, f ½k þ 1􀀉 must be determined before the next time step response

z½k þ 1􀀉 is computed. Wang et al. (1998) proposed the shear balance method for computing the frictional

force f ½k þ 1􀀉: By this method, the sliding structure is first assumed to be in the stick state at the ðk þ 1Þth

step, for which the condition given in Equation 22.34b must be satisfied

x_b½k þ 1􀀉 ¼ Dz½k þ 1􀀉 ¼ 0 ð22:41Þ

where D is a relation matrix, equal to D ¼ ½0 0 1 0 0 0􀀉 for the model shown in Figure 22.16. Substituting

z½k þ 1􀀉 in Equation 22.36 into Equation 22.41, one may solve for the estimated frictional force at the

ðk þ 1Þth time step as

f􀀊½k þ 1􀀉 ¼ 2ðDB1Þ21DðAdz½k􀀉 þ B0f ½k􀀉 þ E0x€g½k􀀉 þ E1x€g½k þ 1􀀉Þ ð22:42Þ

where f􀀊½k þ 1􀀉 with an overbar signifies that the frictional force is an estimate obtained by assuming the

sliding structure to be at the stick state. Such a value may not be the actual one if the system is not in the

stick state. The physical meaning for f􀀊½k þ 1􀀉 is that it represents the balanced shear force required at

the ðk þ 1Þth time step for the structure to remain in the stick state. Therefore, the sign of f􀀊½k þ 1􀀉

indicates the direction of the resistant force provided by the isolation system. In spite of the fact that

f􀀊½k þ 1�� may not be the actual frictional force, it plays an important role for determining the actual state

(stick or sliding) and the actual frictional force of the sliding isolated structure, as will be described below

based on Equation 22.34a and Equation 22.35a.

(1) The system is in the “stick state” if lf􀀊½k þ 1􀀉l , fmax and the frictional force is

f ½k þ 1􀀉 ¼ f􀀊½k þ 1􀀉 ð22:43Þ

(2) The system is in the “sliding state” if lf􀀊½k þ 1􀀉l $ fmax and the frictional force is

f ½k þ 1􀀉 ¼ sgnð f􀀊½k þ 1􀀉Þfmax ð22:44Þ

k k + 1 Time step

Ground acceleration

xg[k + 1]

xg[k]

FIGURE 22.17 Force integration scheme with linear

interpolation.

22-20 Vibration and Shock Handbook

As can be seen, the term 2sgnðx_b½k þ 1􀀉Þ in Equation 22.35a is replaced by sgnðf􀀊½k þ 1􀀉Þ in Equation

22.44. Such a replacement is justified since the sign of f􀀊½k þ 1􀀉 indicates the direction of the resistant

force at the ðk þ 1Þth time step. Once the correct frictional force, f ½k þ 1􀀉; is determined by using

either Equation 22.43 or Equation 22.44, it can be substituted into Equation 22.36 to obtain the

response z½k þ 1􀀉 for the next time step. The computational flow-chart for the shear balance method

has been given in Figure 22.18.

22.4.2.2 Fictitious Spring Method

The fictitious spring method was first proposed by Yang et al. (1990) for the analysis of a sliding structure.

Later, Lu and Yang (1997) reformulated the method into a state-space form for the analysis of equipment

Yes

Compute response

End of

time steps

Stop

Assume stick state and compute

estimated friction force f [k + 1]

Start

k = 0

No Yes

Sliding State Stick State

|f [k + 1]| < fmax

f [k + 1] = sgn( f [k + 1]) fmax f [k+1] = |f [k+1]|

z[k + 1] = Ad z[k] + E0 xg [k] + E1 xg [k + 1] + B0 f [k] + B1 f [k + 1]

k = k + 1

FIGURE 22.18 Computational flow-chart for shear balance method.

Structure and Equipment Isolation 22-21

mounted on a sliding structure. By this method, a

fictitious spring, kf ; is introduced between the base

mat and the ground, as in Figure 22.19, to

represent the mechanism of sliding or friction.

The stiffness, kf ; of the fictitious spring is taken as

zero for the sliding state and as a very large value

for the stick state. With the introduction of the

fictitious spring, the stiffness matrix, K; in

Equation 22.29 should be modified as follows:

K ¼ Kðkf Þ ¼

ke 2ke 0

2ke ke þ ks 2ks

0 2ks ks þ kf

664

775

ð22:45Þ

Accordingly, the state-space dynamic equation, Equation 22.31, should be modified as

zðtÞ ¼ Aðkf ÞzðtÞ þ Ex€gðtÞ þ Bf~ðtÞ ð22:46Þ

where

A ¼ Aðkf Þ ¼

2M21C 2M21Kðkf Þ

I 0

" #

ð22:47Þ

Depending on the current state of the sliding system, the fictitious stiffness, kf ; and the modified friction

term, f~ðtÞ; in Equation 22.46 may take one of the following two sets of values:

(1) In the stick state

kf ¼ aks; f~ðtÞ ¼ kf xb0 ð22:48Þ

(2) In the sliding state

kf ¼ 0; f~ðtÞ ¼ 2sgnðx_bðtÞÞmW ð22:49Þ

In Equation 22.48, the symbol a represents a constant of very large value, and xb0 the initial elongation

of the fictitious spring in the current stick state (computation of xb0 will be explained later). Note that

the modified friction term, f~ðtÞ; may not be the actual frictional force. The actual frictional force can be

determined as follows:

(1) In the stick state

f ðtÞ ¼ kf ðxbðtÞ 2 xb0Þ ð22:50Þ

(2) In the sliding state

f ðtÞ ¼ f~ðtÞ ð22:51Þ

According to Equation 22.50 and Equation 22.51, the actual frictional force of the isolation system in the

stick state is equal to the internal force of the fictitious spring, while in the sliding state it is equal to the

modified frictional force, f~ðtÞ: The frictional force computed from the preceding two equations should

obey the conditions given in Equation 22.34a and Equation 22.35a as well.

With the conditions imposed for the stick and sliding states in Equation 22.48 and Equation 22.49,

respectively, the equation of motion in Equation 22.46 actually represents two different sets of equations.

ks cs

ce xe

ke me

xb0

FIGURE 22.19 Model for isolated structures with

fictitious spring.

22-22 Vibration and Shock Handbook

Specifically, Equation 22.46 and Equation 22.48

collectively describe the motion of the structure in

the stick state, while Equation 22.46 and Equation

22.49 represent the motion of the structure in the

sliding state. Owing to the fact that a sliding

system may switch between the two states at

certain instants, the behavior of the entire system

should undoubtedly be regarded as a nonlinear

one. Nevertheless, within each particular state, the

behavior of the system as represented either by

Equation 22.46 and Equation 22.48 or Equation

22.46 and Equation 22.49 is a linear one.

In the following, a numerical solution scheme

based on the concept of fictitious spring will be

introduced. Let Dt denote a time increment, which is usually taken as a very small value, and assume that

the ground excitation and frictional force are constant within each time increment, Dt (see Figure 22.20).

Accordingly, the discrete-time solution of Equation 22.46 can be rewritten in an incremental form

(Meirovitch, 1990) as

z½k þ 1􀀉 ¼ Adz½k􀀉 þ Edx€g½k􀀉 þ Bd

f~½k􀀉 ð22:52Þ

where

Ad ¼ Adðkf Þ ¼ eAðkf ÞDt ¼

i¼0

Dti

Aðkf Þi ð22:53Þ

Ed ¼ Edðkf Þ ¼ Aðkf Þ21ðAdðkf Þ 2 IÞE ð22:54aÞ

Bd ¼ Bdðkf Þ ¼ Aðkf Þ21ðAdðkf Þ 2 IÞB ð22:54bÞ

For the case where the system matrix A is invertible, Bd and Ed may be computed instead using the

following formulas:

Ed ¼ Edðkf Þ ¼

i¼0

Dti

Aðkf Þi21

" #

E ð22:55Þ

Bd ¼ Bdðkf Þ ¼

i¼0

Dti

Aðkf Þi21

" #

B ð22:56Þ

Equation 22.52 is the solution of the sliding system given in incremental form, because the response,

z½k þ 1􀀉; can be computed from the solution of the previous step, z½k􀀉: Note that, in Equation 22.53

and Equation 22.54, the coefficient matrices Ad; Ed; and Bd have two possible sets of values, as the

fictitious spring constant, kf ; may take different values for the sliding and stick states. Nevertheless,

once the time step size, Dt; is chosen, the coefficient matrices Ad; Ed; and Bd; remain constant for

each state. As such, they need only be calculated once at the beginning of the incremental procedure.

The computational flow-chart for the fictitious spring method described above has been given in

Figure 22.21.

The dynamic equation and its discrete-time solution for the sliding structure in the two states

have been presented above. In the following, we shall describe how to determine the transition time

for the sliding structure to switch from one state to the other. Once the transition time is

determined, the original step size should be scaled down accordingly to reflect the transition point

(Yang et al., 1990).

xg [k]

k k + 1 Time step

Ground acceleration

FIGURE 22.20 Constant force integration scheme.

Structure and Equipment Isolation 22-23

Transition from stick to sliding state. As stated in Equation 22.34a, the condition for a sliding isolated

structure to remain in the stick state is that the static frictional force under the base mat be less than its

maximum value, fmax. Once the frictional force exceeds this maximum value, the system starts to slide.

With the fictitious spring method, the static frictional force is computed as the internal force of the

fictitious spring, as in Equation 22.50. Based on the above considerations and Equation 22.50, the

condition for the sliding system to transfer from the stick to the sliding state is

lf ðt0Þl ¼ lkf ðxbðt0Þ 2 xb0Þl ¼ fmax ¼ mW ð22:57Þ

where t0 denotes the transition time at which the structure starts to slide and kf is the spring constant

given in Equation 22.48 for the stick state. Because a very large value has been used for the fictitious

spring constant, kf ; in the stick state, the deviation in base displacement due to spring elongation is very

small in this state, which can just be neglected. In practice, the transition time, t0 may not occur precisely

yes

k = 0;

Assume stick state

Compute response:

z[k + 1] = Ad z[k] + Ed xg[k] + Bd f [k]

yes

kf = 0 and f [k] = -sgn(xb[k]) fmax

kf ≠ 0 and f [k] = kf xb0

End of time

steps?

Stop

k = k + 1

Check sliding

state?

Sliding state

Start

Stick state

Determine the state of

the next time step

FIGURE 22.21 Computational flow-chart for fictitious spring method.

22-24 Vibration and Shock Handbook

at the discrete points of time considered. It is likely that the spring force is less than fmax at the current

time step, say, the kth step, but exceeds fmax at the following time step. If this is the case, numerical

methods such as the bisection method should be employed to locate the transition time, t0 within the

time interval ðkDt; ðk þ 1ÞDtÞ considered based on Equation 22.57.

Transition from sliding state to stick state. The structure in the sliding state may return to the stick state

whenever the following two conditions are satisfied. (1) The relative velocity of the base mat to the

ground reaches zero, that is, _j bðt0Þ ¼ 0 where t0 is the transition time; (2) the estimated static frictional

force, denoted by f􀀊ðt0Þ; is less than the maximum static frictional force, that is, f􀀊ðt0Þ , fmax: Here, the

estimated static frictional force, f􀀊ðt0Þ; is defined as the shear force required to balance the motion of the

superstructure if the system is assumed to be in the stick state, similar to the one given in Equation 22.42

for f􀀊½k þ 1􀀉: By letting the relative velocity and relative acceleration between the base mat and the ground

be equal to zero, that is, j€bðt0Þ ¼ _j bðt0Þ ¼ 0; the estimated frictional force can be calculated from the freebody

diagram of the base mat as

f􀀊ðt0Þ ¼ mbx€gðt0Þ 2 ksðxsðt0Þ 2 xbðt0ÞÞ 2 csx_sðt0Þ ð22:58Þ

For the sliding structure to transfer from the sliding to the stick state, both the aforementioned

conditions must be satisfied simultaneously. Once the structure enters the stick state, the term f ðt0Þ

should be set equal to f􀀊ðt0Þ and used as the initial frictional force. For the sake of equilibrium, the initial

base mat displacement, xb0; should be computed as

xb0 ¼ xbðt0Þ 2 ð f􀀊ðt0Þ=kfÞ ð22:59Þ

where the value of kf is the one given for the stick state in Equation 22.48.

Concerning the two conditions mentioned above, it may happen that only the first condition,

b ¼ 0 is satisfied, while the computed f􀀊ðt0Þ is still larger than fmax: If this is the case, the sliding

system should not be regarded as a transition to the stick state. Rather, the situation should be

regarded as an indication for reversing the direction of sliding in the next time step. Correspondingly,

the frictional force, f ðt0Þ; should be set equal to the sliding frictional force, rather than the estimated

one, f􀀊ðt0Þ:

22.4.3 Simulation Results for Sliding Isolated Systems

22.4.3.1 Numerical Model and Ground Excitations

In this section, the dynamic behavior of the sliding isolated structure – equipment system shown in

Figure 22.16 will be analyzed using the shear balance method. Although the sliding structure and

equipment considered are both of single-DoF, there exists no difficulty for use of the method to solve

problems with multi-DoF systems. In Table 22.3, the material properties adopted for the present model

have been listed, which are intended to simulate a small, five-story, reinforced concrete frame. For the

present purposes, two types of ground excitation are considered, namely, harmonic and earthquake

excitations. The harmonic excitation is considered primarily for studying the frequency response of the

sliding system, while the earthquake excitation is considered for the effect of earthquake intensity. For the

TABLE 22.3 System Parameters Used in Simulation (Section 22.4.3)

Equipment Superstructure Isolation System

Parameter Value Parameter Value Parameter Value

Mass me 3 t ð¼ ms =100Þ Mass ms 300t Mass mb 100 t ð¼ ms =3Þ

Frequency ve 5vs or a variable Frequency vs 1.67 Hz Frictional coefficient m 0.05, 0.1, 0.25

Damping ratio ze 5% Damping ratio zs 5% — —

Structure and Equipment Isolation 22-25

harmonic excitation, a sinusoidal ground acceleration of the following form is adopted:

x€gðtÞ ¼ 0:5g sin vgt ð22:60Þ

where vg denotes the excitation frequency and g is the gravitational acceleration. For the earthquake

excitation, the 1940 El Centro earthquake (NS component) is considered, for which the waveform has

been given in Figure 22.10. The PGA level of the earthquake will be adjusted for reasons of research.

Concerning the effectiveness of response reduction, three quantities are chosen as the indices, namely, the

base displacement (base drift), structural acceleration, and equipment acceleration. For all the figures

shown in Section 22.4.3 below, the symbol m in the legend is used to denote the frictional coefficient of

the sliding isolation system and the word “fixed” denotes the response of the corresponding fixed-base

structure.

22.4.3.2 Harmonic Response of Structure

Time history. For the isolated system subjected to a

harmonic excitation of vg ¼ 1 Hz, the base

displacements computed for different coefficients

of friction were plotted in Figure 22.22 and

Figure 22.23. As can be seen, the base displacements

quickly reach the steady-state response

within the first few cycles. Meanwhile, the use of

a smaller frictional coefficient results in a larger

permanent displacement before the steady state is

reached. From the structural accelerations plotted

in Figure 22.24, one observes that, for a sliding

structure with a smaller coefficient of friction, the

steady-state response is achieved in a faster way,

accompanied by a larger reduction on structural

acceleration. Of interest in Figure 22.24 is that the

response of the fixed-base case shows a clear period

of 1 sec, while in the sliding case, the response is

contaminated by high-frequency signals caused by

the sliding-stick transitions.

Hysteretic behavior. In order to understand the

mechanical characteristics of a nonlinear device

used for vibration control, it is common to present

a diagram showing the force – deformation relation

of the device, also referred to as the hysteretic

diagram (Soong and Dargush, 1997). Figure 22.25

shows the hysteresis loops of the sliding isolation

system (the sliding layer) for m ¼ 0:1 and 0.25,

when the system is subjected to a harmonic

excitation of vg ¼ 1 Hz. In the figure, the

horizontal and vertical axes, respectively, represent

the base displacement and shear force, that is, the

frictional force, under the mat. Just like many

other frictional elements or devices, the shape

of the hysteresis loop of a sliding bearing is rectangular. The height of the rectangle is equal to the

maximum frictional force that depends on the coefficient of friction, while the width of the hysteresis

loop is determined by the base-sliding displacement. As the coefficient of friction decreases, the height of

the loop decreases, while the width increases. The total area of the hysteresis loop is equivalent to the

portion of the energy dissipated by the sliding bearing.

0 5 10 15

–2

−1.5

−1

−0.5

0.5

Time (s)

Displacement (m)

m = 0.1

m = 0.05

FIGURE 22.22 Comparison of base displacements

ðvg ¼ 1 Hz; ve ¼ 5vs Þ:

10 15

Time (s)

0 5

−0.8

−0.6

−0.4

−0.2

0.2

Displacement (m)

m = 0.1

m = 0.25

FIGURE 22.23 Comparison of base displacements

ðvg ¼ 1 Hz; ve ¼ 5vs Þ:

22-26 Vibration and Shock Handbook

Frequency response. Figure 22.26 shows the maximum structural acceleration with respect to the

excitation frequency for four different frictional coefficients, m ¼ 0:05; 0.1, 0.25 and 1 (for the fixed-base

case). Here, the maximum acceleration means the steady-state acceleration response. The following

observations can be made from Figure 22.26: (1) Compared with the fixed-base case, the use of a smaller

frictional coefficient can reduce the structural acceleration for the frequency range considered. (2) The

sliding mechanism can effectively suppress the main resonant response, associated with the natural

frequency of 1.67 Hz of the superstructure system. (3) As the coefficient of friction, m; decreases from 1

to 0.05, the main resonant frequency associated with the structural natural frequency drifts from the

fixed-base frequency of 1.67 Hz toward a higher value. (4) For the sliding cases of m ¼ 0:05; 0.1, 0.25,

there exist some minor peaks in the range of lower excitation frequencies, besides the main resonant

peak. Such a phenomenon is called the subharmonic resonance. For a large frictional coefficient, say, with

m ¼ 0:25; the subharmonic resonant response may be even larger than that of the main resonance.

FIGURE 22.24 Comparison of structural accelerations ðvg ¼ 1 Hz; ve ¼ 5vs Þ:

0 5 10 15

−15

−10

−5

Time (s)

Acceleration (m/s2)

m = 0.1

Fixed

0 5 10 15

−15

−10

−5

Time (s)

Acceleration (m/s2)

m = 0.25

Fixed

Structure and Equipment Isolation 22-27

Figure 22.27 shows the frequency responses of the maximum base displacement for various

frictional coefficients. The following are observed: (1) The larger the frictional coefficient, the smaller the

base drift is. (2) The base displacement has very large magnitudes in the lower excitation frequencies and

decreases monotonically as the excitation frequency increases. (3) The extremely large base drift

exhibited in the lower excitation frequency range is due to the initial permanent displacement observed

in Figure 22.22.

22.4.3.3 Harmonic Response of Equipment

Time history. Consider an equipment item of a natural frequency equal to five times of the structural

frequency, that is, ve ¼ 5vs: For the case of a harmonic excitation of vg ¼ 1 Hz, the accelerations solved

for the equipment mounted on the structure with m ¼ 0:1 and 0.25, along with the fixed-base case, have

−0.8 −0.6 −0.4 −0.2 0 0.2

−1000

−500

500

1000

Displacement (m)

Total base shear (kN)

m = 0.1

m = 0.25

FIGURE 22.25 Hysteresis loops of a sliding bearing ðvg ¼ 1 Hz; ve ¼ 5vs Þ:

Excitation Freq. (Hz)

10−1 100 101

10−1

100

101

102

m = 0.05

m = 0.1

m = 0.25

Fixed

Acceleration (m/s2)

FIGURE 22.26 Maximum structural acceleration vs. ground excitation frequency ðve ¼ 5vs Þ:

22-28 Vibration and Shock Handbook

been plotted in Figure 22.28. As can be seen, the equipment quickly reaches the steady state within a few

cycles of oscillation. The equipment response is effectively suppressed for the case with a smaller

frictional coefficient. Additionally, the waveforms shown in Figure 22.28 for the equipment appear to be

marginally higher than those of the primary structure shown in Figure 22.24, which can be attributed to

the use of a relatively stiff equipment, that is, with ve ¼ 5vs:

Frequency responses. For an equipment item of the frequency ve ¼ 5vs (¼ 8.34 Hz), the maximum

acceleration response has been plotted as a function of the excitation frequency in Figure 22.29.

By comparing Figure 22.29 with Figure 22.26, one observes that the frequency response curves of the

equipment and primary structure are generally similar, except that a secondary resonant peak

occurs around the equipment natural frequency of 8.34 Hz in Figure 22.29. The other observations from

Figure 22.29 are as follows: (1) In comparison with the fixed-base case, the sliding isolation alleviates

both the structural and equipment resonant peaks around the frequencies of 1.67 and 8.34 Hz,

respectively. However, the level of alleviation is more apparent for the former than for the latter. (2) The

equipment also exhibits the same subharmonic resonance behavior as that of the primary structure, in

terms of the resonance peaks and frequencies. (3) As the frictional coefficient m decreases from 1 (for the

fixed-base case) to 0.05, the main resonant frequency associated with the structure drifts toward a higher

value. However, the resonant frequency associated with the equipment remains the same.

Effect of equipment tuning. The effect of equipment tuning refers to the case when the equipment

frequency is tuned to the structural frequency, that is, ve ¼ vs: Figure 22.30 shows the frequency

response of the equipment when the equipment tuning occurs. Compared with Figure 22.29, this figure

shows the following: (1) When equipment tuning occurs, the sliding isolation system can still mitigate

the main resonant peak of the equipment, but the effectiveness of mitigation is drastically reduced.

(2) Although the subharmonic resonance can still be observed, the relevant frequencies of the equipment

are different from those of the primary structure. (3) The frequency of the maximum resonant response

remains equal to the tuned equipment’s natural frequency of 1.67 Hz, regardless of the change in the

frictional coefficient, m; from 1 to 0.05.

22.4.3.4 Earthquake Response of Structure

Time history. For the isolated system subjected to the El Centro earthquake with a PGA of 0:5g; the

structural acceleration and base displacement of the sliding system have been plotted in Figure 22.31 and

Figure 22.32, respectively, together with the response for the fixed-base case in Figure 22.31. As can be

10−1 100 101

10−8

10−6

10−4

10−2

100

102

104

Excitation Freq. (Hz)

Displacement (m)

m = 0.05

m = 0.1

m = 0.25

FIGURE 22.27 Maximum base displacement vs. ground excitation frequency ðve ¼ 5vs Þ:

Structure and Equipment Isolation 22-29

seen from Figure 22.31, the main-shock response occurring between 0 and 10 sec for the fixed-base

structure has been effectively suppressed by the sliding isolators with m ¼ 0:1 and 0.25. About 80 and

60% of the maximum structural acceleration have been suppressed by the isolators with m ¼ 0:1 and

0.25, respectively. On the other hand, Figure 22.32 demonstrates that the better suppression effect for the

case with m ¼ 0:1 is achieved at the expense of a larger base displacement. It is interesting to note that the

horizontal segments in the curves of Figure 22.32 actually represent the stick state of the sliding system,

which is useful for unveiling the sliding-stick mechanism involved.

Effect of earthquake intensity. The maximum structural acceleration and base displacement vs. the PGA

have been plotted in Figure 22.33 and Figure 22.34, respectively. As can be seen from Figure 22.33, the

maximum structural acceleration for the fixed-base case is proportional to the earthquake intensity,

while in all the sliding cases it remains essentially as a constant after the PGA reaches a certain level. In

other words, the reduction in structural maximum response and the efficiency of isolation have increased

FIGURE 22.28 Comparison of equipment accelerations ðvg ¼ 1 Hz; ve ¼ 5vs Þ:

0 5 10 15

−15

−10

−5

Time (s)

Acceleration (m/s2)

m = 0.1

Fixed

0 5 10 15

−15

−10

−5

Time (s)

Acceleration (m/s2)

m = 0.25

Fixed

22-30 Vibration and Shock Handbook

with the increase in earthquake intensity. For example, for the case of m ¼ 0:1; Figure 22.33 shows that

the structural acceleration is reduced by around 60% at PGA ¼ 0:2g; while it is reduced by more than

90% at PGA ¼ 1:0g: However, as indicated by Figure 22.34, the above reduction in structural response

has been achieved at the expense of increased base displacements. For the same PGA level, a sliding

system with a smaller frictional coefficient has a better effect of vibration reduction, but this is

accompanied by a larger base displacement.

Residual base displacement. The residual base displacement is defined as the permanent base

displacement of the structure after it stops vibrating. This quantity is important in the study of sliding

structures. Figure 22.35 shows the residual base displacement as a function of the earthquake PGA level.

A first look at the figure reveals that no clear relation exists between the earthquake intensity and residual

Acceleration (m/s2)

Excitation Freq. (Hz)

10−1 100 101

10−1

100

101

102

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.29 Maximum equipment acceleration vs. ground excitation frequency ðve ¼ 5vs Þ:

10−1 100 101

10−1

100

101

102

103

Excitation Freq. (Hz)

Acceleration (m/s2)

Fixed

m = 0.05

m = 0.1

m = 0.25

FIGURE 22.30 Maximum equipment acceleration vs. ground excitation frequency for equipment tuning ðve ¼ vs Þ:

Structure and Equipment Isolation 22-31

displacement, because a larger PGA may lead to a smaller residual base displacement. Nevertheless, after

taking the average of the residual displacements over the PGA range of 0.1 to 1g; we obtain xres ¼ 0:083;

0.084, 0:084 m for the case of m ¼ 0:05; 0.1, 0.25, respectively. These values indicate that a smaller

frictional coefficient leads to a larger residual base displacement in general. However, when the frictional

coefficient, m; approaches zero, the residual displacement approaches a constant equal to the permanent

ground displacement.

22.4.3.5 Earthquake Response of Equipment

Time history. Consider an equipment item with a natural frequency equal to five times the structural

frequency, that is, ve ¼ 5vs (¼ 8.34 Hz). For the isolated system subjected to the El Centro earthquake

FIGURE 22.31 Comparison of structural accelerations ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

0 10 20 30 40 50 60

−15

−10

−5

Time (s)

Acceleration (m/s2)

m = 0.1

Fixed

0 10 20 30 40 50 60

−15

−10

−5

Time (s)

Acceleration (m/s2)

m = 0.25

Fixed

22-32 Vibration and Shock Handbook

with PGA ¼ 0:5g; the time histories computed for the equipment acceleration for the cases with m ¼ 0:1

and 0.25, along with the fixed-base case, have been plotted in Figure 22.36a and b. As can be seen, the

main-shock response of the fixed-base structure occurring for the first 10 sec has been effectively

suppressed through installation of the sliding isolator with m ¼ 0:1 and 0.25. A higher level of reduction

can be achieved if a smaller frictional coefficient is chosen.

Effect of earthquake intensity. Figure 22.37 shows the maximum equipment acceleration as a function

of the PGA level. Because of the use of a relatively stiff equipment ðve ¼ 5vsÞ; the curves shown in

Figure 22.37 are similar to those for the primary structure in Figure 22.33, but with slightly higher

values. Therefore, the observations made previously for Figure 22.33 are applicable to Figure 22.37.

The maximum response of equipment items with other frequencies will be discussed below.

Time (s)

0 10 20 30 40 50 60

−0.05

0.05

0.1

0.15

Displacement (m)

m = 0.1

m = 0.25

FIGURE 22.32 Comparison of base displacements ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

0.2 0.4 0.6 0.8 1

Peak Ground Acceleration (g)

Acceleration (m/s2)

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.33 Maximum structural acceleration vs. PGA ðve ¼ 5vs Þ:

Structure and Equipment Isolation 22-33

Effect of equipment tuning. In order to study the equipment tuning effect, the maximum equipment

acceleration has been plotted as a function of the equipment frequency in Figure 22.38. As can be seen,

for all the values of m considered, the equipment response is amplified when the equipment frequency

moves close to the structural frequency of 1.67 Hz for the fixed-base case. Note that, since the

resonant frequency of a sliding structure shifts to a higher value as the frictional coefficient decreases

(see Figure 22.26), the frequency for which the most severe tuning effect occurs in Figure 22.38 also shifts

from 1.67 Hz to a higher value as m decreases. Nevertheless, it is concluded that, by choosing a smaller m;

the amplification of the equipment response due to tuning effect can be effectively suppressed.

22.4.4 Concluding Remarks

The dynamic behavior of a sliding isolated structural system with an attached equipment item was

investigated in this section. A sliding isolated structure is classified as a nonlinear dynamic system, as the

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

Peak Ground Acceleration (g)

Displacement (m)

m = 0.05

m = 0.1

m = 0.25

FIGURE 22.34 Maximum base displacement vs. PGA ðve ¼ 5vsÞ:

Displacement (m)

0.2 0.4 0.6 0.8

Peak Ground Acceleration (g)

0.05

0.1

0.15

0.2

0.25

m = 0.05

m = 0.1

m = 0.25

FIGURE 22.35 Residual base displacement vs. PGA ðve ¼ 5vsÞ:

22-34 Vibration and Shock Handbook

frictional forces induced on the sliding surface do not remain constant. To deal with such nonlinear

systems, two analysis methods were formulated, the shear balance method and the fictitious spring

method, both of which were presented in an incremental form that is suitable for direct implementation.

Through the selection of a sliding isolated structure – equipment model, the responses of the structure

and equipment subjected to both harmonic and earthquake excitations were analyzed. For the case of

harmonic excitation, the results showed that the resonant responses of both the structure and attached

equipment can be effectively suppressed, which remains good even when the equipment frequency is

tuned to the structural frequency. For the case of seismic excitation, the results indicated that the level of

reduction on the structural and equipment responses increases as the PGA level of the earthquake

increases. Moreover, a sliding system with a smaller frictional coefficient has a higher isolation efficiency,

at the expense of a larger base displacement.

Time (s)

0 10 20 30 40 50 60

−15

−10

−5

Acceleration (m/s2)

m = 0.25

Fixed

FIGURE 22.36 Comparison of equipment accelerations ðve ¼ 5vs ; PGA ¼ 0:5gÞ:

Time (s)

0 10 20 30 40 50 60

−15

−10

−5

Acceleration (m/s2)

m = 0.1

Fixed

Structure and Equipment Isolation 22-35

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