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22.5 Sliding Isolation Systems with Resilient Mechanism

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In Section 22.4, the relevant equations of analysis have been presented for a sliding isolated structural

system, with no consideration made for the resilient (or recentering) mechanism. Because of this, rather

large residual displacements may occur on the sliding isolation system, as have been numerically

illustrated. If the concept of sliding isolators is to be applied to a real structure, it is important that the

residual base displacements be controlled within certain limits, since they are not tolerable for some

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 22.37 Maximum equipment acceleration vs. PGA ðve ¼ 5vsÞ:

10−1 100 101

10−1

100

101

102

Equipment Freq. (Hz)

Acceleration (m/s2)

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.38 Maximum equipment acceleration vs. equipment frequency ðPGA ¼ 0:5gÞ:

22-36 Vibration and Shock Handbook

engineering applications. For example, residual base displacements may distort the networking of water

and power lines, change the space between the isolated structure and adjacent buildings, and widen the

gaps at building entrances. Therefore, in practice, a sliding isolation system is usually enhanced through

inclusion of mechanisms that can provide some resilient force. However, for certain ground motions, the

added resilient force may also present some negative effects not readily transparent to structural

designers, as will be illustrated in the numerical studies later on.

As shown in Figure 22.39, there are at least two ways of implementing the resilient mechanism in a

sliding isolation system. Figure 22.39a shows an isolation system that combines the elastomeric bearings

with sliding bearings (Chalhoub and Kelly, 1990), in which the elastomeric bearings are used to provide

the resilient force, and the sliding bearings to uncouple the structural system from the ground motion.

On the other hand, the resilient mechanism can also be incorporated into each single sliding bearing, in a

way similar to that in the RFBI described in Section 22.2.3 (Mostaghel and Khodaverdian, 1987) or the

FPS shown in Figure 22.39b (Mokha et al., 1991). The FPS isolation system has been implemented in

many existing buildings and bridges. A typical FPS bearing consists a spherical sliding surface and a slider,

which usually has a smooth coating of very low friction. When an FPS device is implemented under a

structure, the slider will slide on the spherical surface during an earthquake, and the gravitational load of

the structure, together with the curved sliding surface, will provide the resilient force for the system to

return to its original position. The resilient stiffness of an FPS bearing depends on the radius of curvature

of the sliding surface and the structural weight carried by the bearing.

This section is aimed at investigating the behavior of a structure – equipment system isolated by a

sliding system with resilient device. For convenience of discussion, a system with resilient device will be

referred to as the resilient sliding isolation (RSI), and a sliding system without resilient device, as the one

studied previously, as the pure sliding isolation (PSI).

22.5.1 Mathematical Modeling and Formulation

Both the RSI systems shown in Figure 22.39 can be represented by the mathematical model given in

Figure 22.40, for which the symbols used have been defined in Table 22.1. The RSI model shown in

Figure 22.40 differs from the PSI model shown Figure 22.16 in that a linear spring of stiffness, kb; is added

to simulate the resilient force of the isolator. Obviously, an RSI model can be considered as the

composition of a friction element and a spring element in parallel. Owing to addition of resilient stiffness,

the number of vibration frequencies of the system is increased by one. The newly introduced frequency,

which depends on the resilient stiffness, is called the isolation frequency, which can be approximated by

vb ¼

ffiffiffiffiffiffiffiffiffiffiffi

kb=ðW =gÞ

ð22:61Þ

FIGURE 22.39 Schematic for a structure – equipment system isolated by a sliding bearing with resilience capability:

(a) combined isolation system; (b) friction pendulum system (FPS).

Elastomeric Bearing

Ground

Sliding Bearing

Equipment

Superstructure

Base Mat

(a)

Ground

FPS Sliding Bearing

Equipment

Superstructure

Base Mat

(b)

Structure and Equipment Isolation 22-37

where W is the total weight of the isolated

structure – equipment system. The isolation frequency

commonly used in design is between 0.33

and 0.5 Hz, which implies a period of 2 to 3 sec

(Naeim and Kelly, 1999).

For the type of combined sliding system shown

in Figure 22.39(a), the actual value of resilient

stiffness, kb; is decided by the total horizontal

stiffness of the elastomeric bearings implemented.

On the other hand, for the FPS shown in

Figure 22.39(b), the resilient stiffness, kb; is

approximated by the following equation for

small isolator displacements:

kb ¼

R ð22:62Þ

where R denotes the radius of curvature of the sliding surface. By substituting Equation 22.62 into

Equation 22.61, one can verify that the isolation period of an FPS is equal to the oscillation period of a

pendulum; that is

Tb ¼

vb ¼

2p ffiffiffiffiffiffiffiffiffiffiffi

kb=ðW =gÞ

p ¼ 2p

ffiffiffiffiffi

R=g

ð22:63Þ

When the sliding isolated system of Figure 22.40 is excited by a ground motion, its equation of motion

may be written in exactly the same form as that of Equation 22.27; that is

Mx€ðtÞ þ Cx_ðtÞ þ KxðtÞ ¼ 2ML1x€gðtÞ þ L2f ðtÞ ð22:64Þ

All the variables used in the preceding equation are the same as those defined in Equation 22.28 to

Equation 22.30, except that the stiffness matrix, K; should be modified as

K ¼

ke 2ke 0

2ke ke þ ks 2ks

0 2ks ks þ kb

664

775

ð22:65Þ

Note that, in Equation 22.64, the frictional force, f ðtÞ; which does not remain constant, is placed on the

right-hand side, while the resilient stiffness, kb; which remains constant, is absorbed by the stiffness

matrix, K; as in Equation 22.65. However, if the total shear force, sðtÞ of the isolation system is of interest,

it should be computed as the summation of the frictional force and resilient force (see Figure 22.40);

that is

sðtÞ ¼ kbxbðtÞ þ f ðtÞ ð22:66Þ

The equation of motion as given in Equation 22.64 can be recast in the following form of the first-order

state-space equation:

zðtÞ ¼ AzðtÞ þ Ex€gðtÞ þ Bf ðtÞ ð22:67Þ

The definitions of the matrices zðtÞ; E; A; and B are the same as those defined in Equation 22.32 and

Equation 22.33, except that the system matrix, A; should be modified to account for the addition of the

resilient stiffness kb in the stiffness matrix, K:

22.5.2 Methods for Numerical Analysis

If one compares the equation of motion for the RSI system in Equation 22.67 with that for the PSI system

in Equation 22.31, one will conclude that the only source of nonlinearity in both equations comes from

ks cs

me ce xe

FIGURE 22.40 Model for a structure – equipment

system isolated by a sliding bearing with resilience

capability.

22-38 Vibration and Shock Handbook

the same term, namely, the frictional force, f ðtÞ: As a result, the two methods of solution mentioned in

Section 22.4.2, the shear balance method (Wang et al., 1998) and fictitious spring method (Yang et al.,

1990), remain valid for the analysis of the RSI systems, with no modification required. Moreover, owing

to inclusion of the resilient stiffness, kb; in the structural stiffness matrix, K; the system matrix, A;

becomes nonsingular and invertible. This introduces some advantage in computation of relevant

coefficient matrices, including the B0 and B1 matrices in Equation 22.38a and Equation 22.38b. In

Section 22.5.3, the shear balance method will be employed to simulate the response of an RSI system.

22.5.3 Simulation Results for Sliding Isolation with Resilient Mechanism

22.5.3.1 Numerical Model and Ground Excitations

In this section, the dynamic behavior of a sliding system represented by the model shown in Figure 22.40

will be investigated. The data adopted in the analysis for the equipment, structure, and the isolator have

been listed in Table 22.4. To facilitate comparison, some of the data are selected to be the same as those in

Table 22.3. In particular, the isolation frequency chosen is vb ¼ 0:4 Hz, falling in the common range of

0.33 to 0.5 Hz. Again, two types of ground excitations are considered, namely, the harmonic and

earthquake excitations. For the harmonic excitation, a waveform of ground acceleration identical to the

one given in Equation 22.60 is used. And for the earthquake excitation, the 1940 El Centro earthquake

with different levels of PGA will be used, of which the acceleration waveform has been given in Figure

22.10. The harmonic excitation is adopted mainly for studying the frequency response of the sliding

isolated system, while the earthquake excitation is for studying the effect of earthquake intensity. The

dynamic responses computed for the RSI system, including the structure and equipment, will be

presented, with emphasis placed on comparison with the PSI system of the same parameters. Similar to

what was done in Section 22.4.3, the symbol m will be used to denote the frictional coefficient of the

sliding isolation system in all figures, and the word “fixed” denotes the fixed-base structure.

22.5.3.2 Harmonic Response of Structure

Time history. Consider an RSI system subjected to a harmonic excitation of vg ¼ 1 Hz. The base

displacement and structural acceleration of the RSI system have been plotted in Figure 22.41 and

Figure 22.42, respectively. Clearly, both the base displacement and structural acceleration of the RSI

system reach their steady-state harmonic responses in the first few cycles. Moreover, a smaller sliding

frictional coefficient ðm ¼ 0:1Þ is more effective for suppressing the structural acceleration, as indicated

by Figure 22.42. However, this is achieved only at the expense of a larger base displacement, as

indicated by Figure 22.41. By comparing the result for the RSI system in Figure 22.41 with those for the

PSI system in Figure 22.23, the effect of resilient mechanism in eliminating the permanent base

displacement for the case with a small frictional coefficient of m ¼ 0:1 can be clearly appreciated. In spite

of the large difference in base displacement, the structural accelerations for the RSI and PSI systems

shown in Figure 22.42 and Figure 22.24, respectively, appear to be quite similar, when interpreted in

terms of the waveform and response amplitude. This implies that, for the harmonic excitation

considered, the resilient mechanism in RSI has little influence on the isolation effectiveness.

Hysteretic behavior. In Figure 22.43, the hysteresis loops for RSI systems with m ¼ 0:1 and 0.25

subjected to a harmonic excitation of vg ¼ 1 Hz have been plotted, in which the vertical axis represents

TABLE 22.4 System Parameters Used in Simulation (Section 22.5.3)

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% Isolation frequency vb 0.4 Hz

Structure and Equipment Isolation 22-39

the total shear, sðtÞ; of the bearing computed from Equation 22.66. It is interesting to note that, for an RSI

system, the hysteresis loop is a parallelogram, of which the slope of the inclined upper and lower sides

exactly represents the total resilient stiffness, kb; and the height and width, respectively, are decided by the

maximum frictional force and maximum base displacement. As the frictional coefficient decreases, the

height of the parallelogram decreases, but the width increases. The total area of the hysteresis loop

represents the portion of energy dissipated by the RSI system. Noteworthy is the fact that, when the

resilient stiffness, kb; reduces to zero, the hysteresis parallelogram reduces to a square as well, identical to

the one shown in Figure 22.25 for the PSI system.

Frequency responses. The maximum accelerations of the steady-state response of the structure for four

different frictional coefficients, that is, m ¼ 0:05; 0.1, 0.25 and 1 (fixed-base), have been plotted in

Figure 22.44. From this figure, the following observations can be made: (1) Compared with the fixed-base

case, the resonant peak occurring around the structural frequency, vs; of 1.67 Hz was effectively

suppressed by the RSI system, but a resonance of higher amplitude was induced in the lower frequency

range (with frequencies lower than 0.6 Hz for the case studied). A further investigation reveals that the

newly induced resonance is associated with the isolation frequency, vb; of 0.4 Hz. Such an observation

remains valid for all values of frictional coefficients, m: (2) The use of a lower frictional coefficient, m; will

result in a smaller response in the high-frequency range for the RSI system, for example, with frequencies

higher than 0.6 Hz, but a larger response for the low-frequency range. (3) Although both the RSI and PSI

systems can effectively remove the resonant peak around the structural frequency of 1.67 Hz, the RSI

system has the side effect of creating a low-frequency resonant peak at the isolation frequency, vb: This

implies that the RSI system is more sensitive to the excitation frequency.

The frequency responses of the maximum base displacement for the RSI system with various frictional

coefficients have been plotted in Figure 22.45. When compared with the results for the PSI system in

Figure 22.27, it is clear that the resilient mechanism of the RSI system considerably reduces the base

displacement in the nonresonant excitation range, but it also amplifies the base displacement in the

region when the excitation frequency is close to the isolation frequency, vb: From Figure 22.44 and

Figure 22.45, we observe that both the structural acceleration and base displacement of an RSI system

may resonate at the isolation frequency, which is usually designed to be less than 0.5 Hz. This implies

that an RSI system may be ineffective or unsafe for a ground motion with enriched low-frequency

Time (s)

0 5 10 15

−0.4

−0.3

−0.2

−0.1

0.1

0.2

0.3

Displacement (m)

m = 0.1

m = 0.25

FIGURE 22.41 Comparison of base displacements ðvg ¼ 1 Hz, ve ¼ 5vsÞ:

22-40 Vibration and Shock Handbook

(long-period) vibrations, such as the case with a near-fault earthquake containing a long-period, pulselike

waveform (Jangid and Kelly, 2001; Lu et al., 2003). Structural designers should be aware of such a side

effect when designing an RSI system.

22.5.3.3 Harmonic Response of Equipment

Time history. Consider an equipment item with a frequency of ve ¼ 5vs (¼ 8.34 Hz), attached to the RSI

system. The harmonic acceleration responses of the equipment for the case with m ¼ 0:1 and 0.25 have

been plotted in Figure 22.46a and b, respectively, together with those for the fixed-base case. As can be

seen, the equipment acceleration has been effectively suppressed by the RSI with the smaller frictional

coefficient ðm ¼ 0:1Þ: Moreover, the acceleration waveforms shown in Figure 22.46 are similar to those of

FIGURE 22.42 Comparison of structural accelerations ðvg ¼ 1 Hz, ve ¼ 5vs Þ:

Time (s)

0 5 10 15

−15

−10

−5

Acceleration (m/s2)

m = 0.1

Fixed

Time (s)

0 5 10 15

−15

−10

−5

Acceleration (m/s2)

m = 0.25

Fixed

Structure and Equipment Isolation 22-41

the PSI system shown in Figure 22.28. This implies that for the given excitation, the behavior of the

equipment was not altered by introduction of the resilient mechanism in the RSI system.

Frequency responses. Figure 22.47 shows the acceleration frequency response curve of the attached

equipment with a frequency of ve ¼ 5vs (8.34 Hz). A comparison of Figure 22.47 with Figure 22.44

indicates that the frequency responses of the equipment and primary structure are generally similar,

except that a resonant peak associated with the equipment frequency around 8.34 Hz appears in

Figure 22.47. Owing to such a similarity, the observations made previously for Figure 22.44 are applicable

to Figure 22.47 for the attached equipment.

Effect of equipment tuning. Figure 22.48 shows the frequency response of the attached equipment for the

case when the equipment frequency is tuned to the structural frequency, that is, with ve ¼ vs: Similar to

−0.4 −0.2 0.2 0.4

Displacement (m)

−1500

−1000

−500

500

1000

1500

Total base shear (kN)

m = 0.1

m = 0.25

FIGURE 22.43 Hysteresis loops of a sliding bearing with resilience capability ðvg ¼ 1 Hz, ve ¼ 5vs Þ:

Excitation Freq. (Hz)

100 101 10−1

100

101

102

103

104

10−1

Acceleration (m/s2)

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.44 Maximum structural acceleration vs. ground excitation frequency ðve ¼ 5vs Þ:

22-42 Vibration and Shock Handbook

the structural frequency response shown in Figure 22.44, the equipment attached to the RSI system also

resonates at the isolation frequency vb of 0.4 Hz. Such a resonance does not occur for the equipment

attached to the PSI system (see Figure 22.30). Through comparison of the tuned case in Figure 22.48 with

the detuned case in Figure 22.47, the following observations can be made: (1) Even when the equipment

tuning occurs, an RSI system mitigates the equipment’s resonant peak associated with the structural

frequency at 1.67 Hz, although the effectiveness of isolation has been reduced. (2) The tuning effect has no

influence on the resonant response associated with the isolation frequency of 0.4 Hz.

22.5.3.4 Earthquake Response of Structure

Time history. For an RSI system subjected to the El Centro earthquake with PGA ¼ 0:5g; the structural

acceleration and base displacement have been shown in Figure 22.49 and Equation 22.50, respectively. By

comparing Figure 22.49 with Figure 22.31 for the corresponding PSI system, one observes that the

structural accelerations of the RSI and PSI systems are generally similar, in terms of the response

waveform and the response magnitude. Both systems reduce the maximum structural acceleration quite

effectively, for example, by about 80% for m ¼ 0:1: However, significant difference does exist between the

Excitation Freq. (Hz)

10−1 100 10−8

10−6

10−4

10−2

100

102

104

101

Displacement (m)

m = 0.05

m = 0.1

m = 0.25

FIGURE 22.45 Maximum base displacement vs. ground excitation frequency ðve ¼ 5vs Þ:

FIGURE 22.46 Comparison of equipment accelerations ðvg ¼ 1 Hz, ve ¼ 5vs Þ:

Time (s)

0 5 10 15

−15

−10

−5

Acceleration (m/s2)

m = 0.1

Fixed

Time (s)

0 5 10 15

−15

−10

−5

Acceleration (m/s2)

m = 0.25

Fixed

Structure and Equipment Isolation 22-43

base displacements for the RSI system in Figure 22.50 and those for the PSI system in Figure 22.32. For

example, for m ¼ 0:1; the maximum base displacement experienced by the RSI system has been reduced

by about 30%, while the residual base displacement has been reduced by about 70%, as can be seen by

comparing Figure 22.50 with Figure 22.32. This implies that the resilient mechanism of the RSI system

plays an important role in reducing the maximum and residual base displacements, especially the latter.

In spite of the observations made above, one should not forget that the frequency content of one

earthquake may be different from an other. As was demonstrated in Figure 22.44 and Figure 22.45, an RSI

system is generally sensitive to low-frequency excitations and may resonate at the isolation frequency.

10–1 100 101

10–1

100

101

102

103

104

Excitation Freq. (Hz)

Acceleration (m/s)

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.47 Maximum equipment acceleration vs. ground excitation frequency ðve ¼ 5vs Þ:

Excitation Freq. (Hz)

10–1 100 101

10–1

100

101

102

103

104

Acceleration (m/s)

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.48 Maximum equipment acceleration vs. ground excitation frequency under tuning condition

ðve ¼ 5vs Þ:

22-44 Vibration and Shock Handbook

Therefore, if the RSI system is subjected to an earthquake containing more low-frequency components,

unlike the El Centro earthquake, it is likely that the maximum structural responses induced exceed those

of the PSI system.

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

RSI system have been plotted with respect to the PGA in Figure 22.51 and Figure 22.52, respectively.

These figures indicate that as the earthquake intensity increases from 0.1 to 1g; the structural acceleration

is reduced by an increasing amount by the RSI system, while the maximum base displacement also

increases. By comparing Figure 22.51 and Figure 22.52 with Figure 22.33 and Figure 22.34 for the PSI

system, one observes that both the RSI and PSI systems perform equally well for the El Centro

earthquake, although the PSI system induces a slightly larger base displacement. On the other hand,

unlike the response for the PSI system, the use of a smaller frictional coefficient for the RSI system does

not always lead to a lower structural acceleration, as can be verified by comparing the responses for

m ¼ 0:1 and 0.05 with a PGA greater than 0:8g in Figure 22.33 and Figure 22.51. This can be attributed to

the large resilient force induced by the large base displacement under higher PGA levels.

FIGURE 22.49 Comparison of structural 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

Time (s)

0 10 20 30 40 50 60

–15

–10

–5

Acceleration (m/s2)

m = 0.25

Fixed

Structure and Equipment Isolation 22-45

Residual base displacement. Figure 22.53 shows the residual base displacement of the RSI system vs. the

PGA of the earthquake. For a given m; it is difficult to establish a relation between the earthquake

intensity and residual displacement, because a larger PGA may result in a smaller residual base

displacement in some cases. However, if one takes the average of residual displacements over the PGA

range from 0.1 to 1g; the following can be computed: xres ¼ 0:0065; 0.011, and 0.014 m for m ¼ 0:05; 0.1,

and 0.25, respectively. These values indicate that a smaller frictional coefficient leads to a smaller residual

base displacement, which can be attributed to the fact that for a SRI system with a smaller coefficient of

friction, it is easier for the resilient mechanism to return the structure to its initial position after an

earthquake. On the other hand, a comparison of Figure 22.53 with Figure 22.35 for the PSI system

indicates that for the same value of m; the residual displacement was reduced substantially by the RSI

system. This is certainly an advantage offered by the resilient mechanism of the RSI system.

Time (s)

Displacement (m)

0 10 20 30 40 50 60

–0.06

–0.04

–0.02

0.02

0.04

0.06

0.08

0.1

m = 0.1

m = 0.25

FIGURE 22.50 Comparison of base displacements ðve ¼ 5vs; PGA ¼ 0:5gÞ:

0.2 0.4 0.6 0.8

Peak Ground Acceleration (g)

Acceleration (m/sec)

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.51 Maximum structural acceleration vs. PGA ðve ¼ 5vsÞ:

22-46 Vibration and Shock Handbook

22.5.3.5 Earthquake Response of Equipment

Time history. Let us consider an equipment item of natural frequency equal to five times the structural

frequency, that is, ve ¼ 5vs (¼ 8.34 Hz). The acceleration responses of the equipment mounted on the

RSI system that were subjected to the El Centro earthquake with a PGA of 0:5g for m ¼ 0:1 and 0.25 have

been plotted in Figure 22.54a and b, respectively, along with those for the fixed-base cases. As can be seen,

the main-shock response of the equipment appearing during the first 10 sec for the fixed-base system was

effectively suppressed by the RSI system with m ¼ 0:1 or 0.25. The level of reduction is more pronounced

for the case with a smaller frictional coefficient, that is, with m ¼ 0:1: By comparing Figure 22.54 with

Figure 22.36 for the PSI system, one concludes that the effect of the resilient mechanism of the RSI system

on the equipment response is insignificant for the earthquake and equipment frequency considered.

Effect of earthquake intensity. Figure 22.55 shows the maximum equipment acceleration vs. the PGA of

the earthquake. This figure illustrates that for all values of the frictional coefficient, m; considered, an

0.2 0.4 0.6 0.8

Peak Ground Acceleration (g)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

m = 0.05

m = 0.1

m = 0.25

Displacement (m)

FIGURE 22.52 Maximum base displacement vs. PGA ðve ¼ 5vs Þ:

0.2 0.4 0.6 0.8

Peak Ground Acceleration (g)

0.005

0.01

0.015

0.02

0.025

0.03

Displacement (m)

m = 0.05

m = 0.1

m = 0.25

FIGURE 22.53 Residual base displacement vs. PGA ðve ¼ 5vs Þ:

Structure and Equipment Isolation 22-47

increasing amount of reduction can be achieved by the RSI system as the earthquake intensity increases

from 0.1 to 1g: Because relatively stiff equipment (i.e., with ve ¼ 5vs) was assumed in the simulation, the

curves shown in Figure 22.55 are similar to those of Figure 22.51 for the primary structure. Therefore, the

observations made previously for Figure 22.51 apply here. The maximum response of equipment items

with other natural frequencies will be discussed below. Moreover, a comparison of Figure 22.55 with

Figure 22.37 (for the PSI system) reveals that the resilient mechanism can have some minor effect on the

equipment response, but only when a smaller frictional coefficient (i.e., m ¼ 0:05 or 0.1) is used and

when the PGA level is high.

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

of the equipment has been plotted in Figure 22.56 for equipment frequencies ranging from 0.1 to 10 Hz.

FIGURE 22.54 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

Time (s)

0 10 20 30 40 50 60

–15

–10

–5

Acceleration (m/s2)

m = 0.25

Fixed

22-48 Vibration and Shock Handbook

As can be seen, for all the values of m considered, the equipment response is amplified when the

equipment frequency is close to the structural frequency, vs; of 1.67 Hz, which means that the tuning

effect tends to enlarge the equipment response. However, the use of a smaller m can help in reducing the

amplification of the equipment response resulting from the tuning effect. Finally, a comparison between

Figure 22.56 and Figure 22.38 (for the PSI system) shows that the two diagrams are quite similar for an

equipment item with a frequency higher than 1 Hz, but are different for that with a lower frequency. This

implies that for the earthquake considered, the resilient mechanism of the RSI system has little effect on

the response of the equipment with a higher stiffness.

0.2 0.4 0.6 0.8

Peak Ground Acceleration (g)

Acceleration (m/s2)

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.55 Maximum equipment acceleration vs. PGA ðve ¼ 5vs Þ:

Equipment Freq. (Hz)

10–1 100 101

10–1

100

101

102

Acceleration (m/s2)

m = 0.05

m = 0.1

m = 0.25

Fixed

FIGURE 22.56 Maximum equipment acceleration vs. equipment frequency ðPGA ¼ 0:5gÞ:

Structure and Equipment Isolation 22-49

22.5.4 Concluding Remarks

In this section, the behavior of a structure – equipment system isolated by an RSI system under both the

harmonic and earthquake excitations has been investigated. Both the responses of the structure and

equipment were studied, with special attention given to the effect of the resilient mechanism that

characterizes an RSI system. The numerical results demonstrated that when subjected to a harmonic

excitation, an RSI system is able to effectively suppress the resonant peaks associated with the structural

frequency for both the structure and equipment, but it may also induce some resonant response near the

isolation frequency due to the presence of resilient stiffness. Therefore, an RSI system is more sensitive to

the frequency content of the ground excitation than a PSI system, especially to excitations of lowfrequency

components. As for the earthquake responses, the numerical results showed that the resilient

mechanism of an RSI system can considerably reduce the residual base displacement. The resilient

mechanism has a minor effect on the acceleration response of the structure and equipment, as long as no

resonance is induced by the RSI system at the isolation frequency. By and large, both the RSI and PSI

systems can be used as effective devices for reducing the acceleration responses of a structure and

equipment.

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22.5 Sliding Isolation Systems with Resilient Mechanism

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