22.3 Structure – Equipment Systems with Elastomeric Bearings

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Owing to the stringent requirements for normal functioning of high-tech facilities, such as printed circuit

boards, semiconductor factories, and sensitive medical devices, the need to suppress excessive vibrations in

sensitive structure – equipment systems has become an issue of great concern to structural designers.

Besides, these high-tech facilities may suffer significant damages during a major earthquake. Using

elastomeric isolation systems to reduce the earthquake forces transmitted from the ground is one of the

most popular ways adopted by structural designers. In this section, the performance of elastomeric

bearings in protecting structure – equipment systems against horizontal ground motions will be

investigated.

22.3.1 Formulation of Base Isolation Systems with Elastic Bearing

By modeling the structure, internal equipment and the base of an isolated structure – equipment

system as a lumped mass system, one can construct the mathematical model for the structure –

equipment isolation system supported by an elastic bearing in Figure 22.9. The following is the

Structure and Equipment Isolation 22-9

© 2005 by Taylor & Francis Group, LLC

equation of motion for the base-isolated structure – equipment system when it is subjected to a

ground acceleration, x€g:

mex€e

msx€s

mbx€b

8>><

>>:

9>>=

>>;

þ

ce 2ce 0

2ce cs þ ce 2cs

0 2cs cs þ cb

2

664

3

775

x_e

x_s

x_b

8>><

>>:

9>>=

>>;

þ

ke 2ke 0

2ke ks þ ke 2ks

0 2ks ks þ kb

2

664

3

775

xe

xs

xb

8>><

>>:

9>>=

>>;

¼ 2

me

ms

mb

8>><

>>:

9>>=

>>;

x€g ð22:11Þ

where m represents the mass, c the damping coefficient, and k the stiffness of the system. Also, the

subscripts ‘e’, ‘s’, and ‘b’ are associated with the DoF of the equipment, structure, and base,

respectively. The notations employed in Figure 22.9 have been defined in Table 22.1. It should be

mentioned that the elastic bearing stiffness, kb; appearing in Equation 22.11, is a parameter relating

to the boundary conditions of the system considered here. A small value of kb relative to the

structural stiffness, ks; means that the system is isolated by a set of soft bearings. In contrast, a large

value of kb means that the structure is rigidly supported.

kb

ce

xe

xg

..

xb

mb me

ke

ms

c cs b

ks

xs

Equipment

Base

Structure

FIGURE 22.9 Model of a structure – equipment isolation system with elastic bearing.

TABLE 22.1 Definition of Symbols

Symbol Definition

ce; cs Damping coefficients of equipment and superstructure

ke ; ks Stiffness of equipment and superstructure

kb Stiffness of elastic bearing or resilient stiffness of isolation system

me ; ms ; mb Masses of equipment, superstructure and base mat

xe ðtÞ; xs ðtÞ; xb ðtÞ Relative-to-the-ground displacements of equipment,

superstructure and base mat

x€gðtÞ Ground acceleration

m Frictional coefficient of sliding isolation system

vb Frequency of isolation system

ve ; vs Frequencies of equipment and superstructure

vg Frequency of ground excitation

ze ; zs Damping ratios of equipment and structure

22-10 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

22.3.2 Free Vibration Analysis

By neglecting the damping and forcing terms in Equation 22.11, the equation of motion for free vibration

can be written as

mex€e

msx€s

mbx€b

8>><

>>:

9>>=

>>;

þ

ke 2ke 0

2ke ks þ ke 2ks

0 2ks ks þ kb

2

664

3

775

xe

xs

xb

8>><

>>:

9>>=

>>;

¼

0

0

0

8>><

>>:

9>>=

>>;

ð22:12Þ

By solving the preceding equation, one can obtain the natural frequencies and vibration modes of the

structure – equipment system with elastic bearings. As for the present problem, the horizontal stiffness of

the elastic bearing is designed to be quite low compared with that of the superstructure. It follows that the

superstructure in its entirety behaves essentially as a rigid body for the fundamental vibration mode shape

of the combined system, which implies that the displacement responses for the equipment, structure, and

base under free vibration can be approximately taken as the same, that is, xe ¼ xs ¼ xb ¼ x: By

introducing such a condition into Equation 22.12, the equation of motion for the equivalent single-DoF

base-isolated system can be written as

ðme þ ms þ mbÞx€ þ kbx ¼ 0 ð22:13Þ

Equation 22.13 indicates that the fundamental frequency, v1; of the base-isolated system can be

approximated by v1 <

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kb=ðme þ ms þ mbÞ

p

: Further, if the condition of fixed base is considered, that is,

by letting the responses of the base be equal to zero, xb ¼ x€b ¼ 0; the structure–equipment isolation

system will be reduced to the case of a fixed-base system, such that the equation of motion becomes

mex€e

msx€s

( )

þ

ke 2ke

2ke ks þ ke

" #

xe

xs

( )

¼

0

0

( )

ð22:14Þ

as is well known.

22.3.3 Dynamics of Structure – Equipment Isolation Systems

to Harmonic Excitations

The advantage of a closed-form solution is that it allows us to examine the key parameters involved in the

problem considered. This is what will be sought herein. For the case of a harmonic ground excitation, xg;

with amplitude, Xg; that is, with xg ¼ Xg eivt ; one may derive from Equation 22.11 the following:

mex€e

msx€s

mbx€b

8>><

>>:

9>>=

>>;

þ

ke 2ke 0

2ke ks þ ke 2ks

0 2ks ks þ kb

2

664

3

775

xe

xs

xb

8>><

>>:

9>>=

>>;

¼

me

ms

mb

8>><

>>:

9>>=

>>;

Xgv2 eivt ð22:15Þ

Correspondingly, the steady-state responses of the system can be expressed as

xe

xs

xb

8>><

>>:

9>>=

>>;

¼

Xe

Xs

Xb

8>><

>>:

9>>=

>>;

eivt ð22:16Þ

where ðXe; Xs; and XbÞ represent the amplitudes of the equipment, structure, and base, respectively.

Substituting Equation 22.16 into Equation 22.15 yields

ke 2 mev2 2ke 0

2ke ks þ ke 2 msv2 2ks

0 2ks ks þ kb 2 mbv2

2

664

3

775

Xe

Xs

Xb

8>><

>>:

9>>=

>>;

¼

me

ms

mb

8>><

>>:

9>>=

>>;

Xgv2 ð22:17Þ

Structure and Equipment Isolation 22-11

© 2005 by Taylor & Francis Group, LLC

from which the amplitudes ðXe; Xs; and XbÞ for the system can be solved as follows:

Xe ¼

Xs þ Xg f 2

e

1 2 f 2

e ð22:18aÞ

Xb ¼

Xs þ 1bf 2

s Xg

1 þ kb=ks 2 1bf 2

s ð22:18bÞ

Xs ¼

ms þ

me

1 2 f 2

e þ

mb

1 þ kb=ks 2 1bf 2

s

􀀒 􀀓

Xgv2

kb 2 mbv2

1 þ kb=ks 2 1bf 2

s

2 ms þ

me

1 2 f 2

e

􀀏 􀀐

v2 ð22:18cÞ

where the amplitudes of the equipment and base, that is, Xe and Xb; have been expressed in terms of the

amplitude of the base, Xs: The parameters in Equation 22.18 are defined as

fe ¼ v=ve ð22:19aÞ

fs ¼ v=vs ð22:19bÞ

1b ¼ mb=ms ð22:19cÞ

ve ¼

ffiffiffiffiffiffiffi

ke=me

p

ð22:19dÞ

vs ¼

ffiffiffiffiffiffiffi

ks=ms

p

ð22:19eÞ

Finally, the state-steady absolute acceleration responses of the structure, equipment, and base can be

expressed in terms of the ground acceleration x€g as

as ¼ x€s þ x€g ¼ 2ðXs þ XgÞv2 eivt ¼

kbx€g

DðvÞ ð22:20aÞ

ae ¼ x€e þ x€g ¼ 2 ðXs þ XgÞv2 eivt

1 2 f 2

e ¼

kbx€g

ð1 2 f 2

e ÞDðvÞ ð22:20bÞ

ab ¼ x€b þ x€g ¼

2½Xs þ Xg þ ðkb=ksÞXg􀀉v2 eivt

1 þ kb=ks 2 1bf 2

s ¼ ðD21ðvÞ þ k21

s Þkbx€g

1 þ kb=ks 2 1bf 2

s ð22:20cÞ

DðvÞ ¼ ðkb 2 mbv2Þ 2 ð1 þ kb=ks 2 1bf 2

s Þ ms þ

me

1 2 f 2

e

􀀏 􀀐

v2 ð22:20dÞ

As can be seen, the acceleration response of each component in the structure – equipment system

depends mainly on the stiffness of the elastic bearing, kb: In particular, the use of a smaller bearing

stiffness, kb; can result in significant reduction of the shear forces transmitted to the superstructure, as

indicated by Equation 22.20a. This explains why an elastic bearing can be effectively used as an isolator

for reducing the base shear of the structure – equipment system. In contrast, if the bearing stiffness, kb; is

made to be infinitely large, that is, by letting kb ! 1; the acceleration responses in Equation 22.20,

reduce to

as ¼ ð1 2 f 2

e Þx€g

ð1 2 f 2

e Þð1 2 f 2

s Þ 2 1e f 2

s ð22:21aÞ

ae ¼

x€g

ð1 2 f 2

e Þð1 2 f 2

s Þ 2 1e f 2

s ð22:21bÞ

ab ¼ x€g ð22:21cÞ

with the use of L’Hospital’s Rule, where 1e ¼ me=ms: As can be seen from Equation 22.21c, the

acceleration of the structural base is equal to the ground acceleration. Clearly, the present problem has

been reduced to a two-DoF system with a rigid base, for which the solutions have been given in Equation

22.21a and Equation 22.21b.

22-12 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Some important high-tech facilities, such as semiconductor factories and medical devices, are very

sensitive to vibrations, especially to those caused by resonance. To consider the effect of resonance, we

shall let the ground excitation frequency, v; coincide with the equipment frequency, ve; that is, by letting

fe ¼ 1 or ve ¼ v: For this case, the acceleration responses of the system in Equation 22.20 reduce to

as ¼ 0 ð22:22aÞ

ae ¼

2kbx€g

ke½1 þ ðkb 2 mbv2

e Þ=ks􀀉 ð22:22bÞ

ab ¼

kbx€g

ks þ kb 2 mbv2

e ð22:22cÞ

Because of the coincidence of the ground excitation frequency with the equipment frequency, the

equipment behaves like a vibration absorber of the structure. For this reason, the response of the

equipment is greatly amplified, as implied by Equation 22.22b, while the response of structure is

completely suppressed, as indicated by Equation 22.22a. Moreover, if the frequency of the equipment is

eqffiuffiffiaffilffiffitffioffiffitffihffiffieffiffifffiuffiffinffidffiffiaffimffiffiental frequency of the structure–equipment isolation system, that is, veð< v1Þ ¼

kb=ðme þ ms þ mbÞ

p

; then the responses of the system in Equation 22.22 can further be expressed as

follows:

as ¼ 0 ð22:23aÞ

ae ¼

2kbx€g

ke½1 þ ðms þ meÞv2

e =ks􀀉 ð22:23bÞ

ab ¼

kbx€g

ks þ ðms þ meÞv2

e ð22:23cÞ

Since the equipment mass is generally much smaller than the structural mass, the preceding equation can

be further reduced to

as ¼ 0 ð22:24aÞ

ae ¼

2kbx€g

keð1 þ v2

e =v2s Þ ð22:24bÞ

ab ¼

kbx€g

ksð1 þ v2

e =v2s

Þ ð22:24cÞ

As indicated by Equation 22.24b, the acceleration response of the equipment depends on the stiffness

ratio, kb=ke; of the base to the equipment.

For the resonance condition of ve ¼ v; mentioned previously, let us consider the case when the

structural frequency is equal to the equipment frequency, that is, ve ¼ vs: For this case, the responses of

the system in Equation 22.22 reduce to

as < 0 ð22:25aÞ

ae <

2kbx€g

1e½ksð1 2 1bÞ þ kb􀀉 ð22:25bÞ

ab ¼

kbx€g

ksð1 2 1bÞ þ kb ð22:25cÞ

which indicates that the acceleration response of the equipment may be greatly amplified, as implied by

the relatively small mass ratio 1e ð¼ me=msÞ and large stiffness ratio, kb=ks; in Equation 22.25b. Such a

phenomenon has been referred to as the tuning of equipment.

On the other hand, when the excitation frequency, v; coincides with the fundamental frequency, v1; of

the isolated system, that is, vð< v1Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kb=ðme þ ms þ mbÞ

p

; resonant response may be induced on the

structure – equipment isolation system. Considering that the first priority in design of high-tech

Structure and Equipment Isolation 22-13

© 2005 by Taylor & Francis Group, LLC

equipment is to reduce the vibrations of the equipment, rather than the structure, by comparing the

denominators in Equation 22.20a and Equation 22.20b, one may assume that the condition l1 2 f 2

e l $ 1

or fe ¼ v=ve $

ffiffi

2 p remains satisfied for a good design, which is equivalent to

ve

vs

#

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kb=ks

2½1 þ ðmb þ meÞ=ms􀀉

s

ð22:26Þ

Since the fundamental frequency, vs; of a base-isolated structure is generally low in practice, the

horizontal stiffness of the equipment attached to the structure must be designed to be soft enough such

that Equation 22.26 can be satisfied. Certainly, this is one of the guidelines to be obeyed in the design of

equipment for the sake of vibration reduction.

22.3.4 Illustrative Example

The forgoing formulations have been made by neglecting the damping of the structural system and by

assuming the ground motion to be of the harmonic type. In practice, there is always some damping with

the structural system, while the ground motion may be random in nature. To deal with such problems,

the only recourse is to use numerical methods that are readily available. In this section, the Newmark b

method, proposed by Newmark (1959), with g ¼ 1=2 and b ¼ 1=4; will be adopted for solving the

second-order differential equation presented in Equation 22.11, which has the advantage of being

numerically stable.

The example considered is the structure – equipment system isolated by elastomeric bearings, shown in

Figure 22.9, with the data given in Table 22.2. As can be seen, the equipment has a frequency equal to five

times the structural frequency, that is, ve ¼ 5vs (¼ 8.34 Hz). The 1940 El Centro earthquake (NS

component) with a peak ground acceleration (PGA) of 341.55 gal is adopted as the ground excitation, as

given in Figure 22.10. By an eigenvalue analysis, the natural frequencies solved for the base-isolated

system are 2.46, 21.41, and 52.74 rad/sec. Because of the installation of elastic bearings on the structure –

equipment system, the fundamental frequency of the system decreases significantly and is approximately

equal to v1 <

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kbðme þ ms þ mbÞ

p

¼ 2:51 rad=sec; according to Section 22.3.2. From this example, one

observes that the use of a single-DoF system to model a base-isolated system can give a generally good

result for the first frequency of vibration.

As can be seen from Figure 22.11, for the structural acceleration of the system, the main-shock

response of the fixed-base structure has been effectively eliminated due to installation of the elastic

bearings. However, as indicated by Figure 22.12, because of the installation of soft bearings, the base

displacement response of the isolated system is much larger and decays much slowly, even after the main

shocks.

In Figure 22.13 the acceleration response of the equipment for the isolated and fixed-base cases are

compared. As can be seen, the main-shock response of the fixed-base structure has been effectively

suppressed through the installation of the elastic bearings. Furthermore, the equipment response appears

to be almost identical to the structure response shown in Figure 22.11, due to the fact that the equipment

TABLE 22.2 System Parameters Used in Simulation (Section 22.3.4)

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

Horizontal

stiffness ke

8258 kN/m Horizontal

stiffness ks

33,030 kN/m Horizontal

stiffness kb

2546 kN/m

Damping 15.74 kN m/s Damping 314.79 kN m/s Damping 50.46 kN m/s

Frequency

ve ¼

ffiffiffiffiffiffiffi

ke =me

p 52.47 rad/sec Frequency

vs ¼

ffiffiffiffiffiffiffi

ks =ms

p 10.46 rad/sec Frequency

vb ¼

ffiffiffiffiffiffiffiffi

kb=mb

p 5.05 rad/sec

22-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

0 10 20 30 40 50 60

Time (s)

−400

−200

0

200

400

PGA (gal)

1940 El Centro NS, PGA = 341.55 gal

FIGURE 22.10 Waveform of 1940 El Centro earthquake (NS component).

Time (s)

−0.4

−0.2

0.0

0.2

0.4

Displacement of base (m)

Fixed base

Base isolation

0.0 10.0 20.0 30.0 40.0 50.0

FIGURE 22.12 Comparison of base displacements.

0.0 10.0 20.0 30.0 40.0 50.0

Time (s)

−10.0

−5.0

0.0

5.0

10.0

Acceleration of structure (m/s2)

Fixed base

Base isolation

FIGURE 22.11 Comparison of structural accelerations.

Structure and Equipment Isolation 22-15

© 2005 by Taylor & Francis Group, LLC

is rigidly attached to the structure, as implied by the relatively higher frequency of the equipment. As for

the present example, the effectiveness of the elastic bearings in reducing the equipment response is

ascertained.

For high-tech equipment, engineers may be concerned about the effect on equipment tuning induced

by external excitations, such as earthquakes and traffic-induced vibrations. To investigate this effect on

the structure – equipment isolation system considered, the maximum equipment acceleration was plotted

as a function of the frequency ratio, ve=vs; in Figure 22.14. As can be seen, the response of the equipment

is greatly amplified when it has a frequency close to the fundamental frequency of the structure –

equipment isolation system, that is, when ve ¼ 2:51 rad=sec or ve=vs ¼ 0:24: To avoid such a situation, it

is suggested that isolators be mounted at both the structure base and equipment base. From Figure 22.14,

one observes that the use of a small horizontal stiffness for the equipment will generally lead to greater

equipment response due to tuning effect. However, as the stiffness of the equipment is further reduced,

the equipment will reach another isolation state, in which the equipment response will be substantially

suppressed, as indicated by the region with relatively small values of ve=vs: The margin for such

a frequency ratio of ve=vs can be obtained by substituting the parameters in Table 22.2 into

0.0 10.0 20.0 30.0 40.0 50.0

Time (s)

−10.0

−5.0

0.0

5.0

10.0

Acceleration of equipment (m/s2)

Fixed base

Base isolation

FIGURE 22.13 Comparison of equipment accelerations.

0.1 1.0 10.0

Frequency ratio we /ws

0.0

2.0

4.0

6.0

8.0

Maximum Acceleration (m/s2)

Equipment

Structure

Base

0.16

FIGURE 22.14 Maximum accelerations of structure – equipment isolation systems.

22-16 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Equation 22.26, which yields a critical ratio of ve=vs ¼ 0:168; very close to the value of 0.16 marked in

Figure 22.14.

22.3.5 Concluding Remarks

This section investigates the dynamic response of a mathematical model of a structure – equipment

system isolated by elastomeric bearings and subjected to ground excitations. Based on the closed-form

solution of a structure – equipment isolation system subjected to harmonic support motions, one

observes that the coincidence of the ground excitation frequency with the equipment frequency will make

the equipment behave like a vibration absorber of the structure, of which the acceleration response will be

greatly amplified. For the case that the first priority in design is to reduce the vibration of the equipment

rather than that of the structure, Equation 22.26 provides a guideline for the design of equipment, which

has been verified in the numerical example.

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