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29.3 Seismic Vibration Control

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29.3.1 Historical Development of Seismic Vibration Control

It is worth noting that a structural vibration

control system for reducing seismic responses was

developed several centuries ago in Japan, by an

unknown engineer. This was applied to the

construction of a Gojunoto (pagoda). The Gojunoto

was five stories tall and was constructed of

closely fitting mortised wooden beams and

columns (Figure 29.31). During an earthquake,

the vibrations of such a vertical cantilever structure

would produce bending moments that could not

be resisted by tension at the mortised joints. To

overcome this weakness, a long wooden pole was

suspended freely from the upper part of the

pagoda so that it could undergo pendular

vibrations if the pagoda was excited into motion

by an earthquake. The weight of the pole exerted a

compressive prestress on the pagoda, thus increasing

the bending resistance. The bottom of the pole

extended into a cylindrical hole in the ground that

was of a larger diameter than the pole. Thus, when

the pagoda was excited into vibrations by an

earthquake, some of the vibrational energy would

be transferred into oscillations of the pole and the

impact of the pole on the sides of the hole would

dissipate energy.

In 1969, Gupta and Chandrasekaran (1969)

studied the effectiveness of a number of tuned

mass dampers (TMDs) put in a single-DoF

system to reduce seismic responses. The TMD

system has both linear and elasto-plastic restoring-

force characteristics and viscous-type damping. The study was shown in the Fourth World

Conference on Earthquake Engineering (WCEE) in Santiago, Chile. In the Fifth WCEE in 1973,

Skinner et al. (1973) presented their studies on energy absorption devices for earthquake-resistant

structures. The devices were based on the plastic deformation of mild steel. In 1977, Ohno et al.

(1977) introduced optimum tuning of the dynamic damper for civil structures under earthquake

Suspended

pole

Wooden pagoda

FIGURE 29.31 Sketch of a wooden pagoda with

suspended pole that is free to undergo pendular

vibration. (Source: Tanabashi, R., Proc. II World Conf.

on Earthquake Engineering, 1960. With permission.)

Seismic Base Isolation and Vibration Control 29-33

excitations. Skinner et al. (1977) showed their

works on studying hysteretic dampers for

increasing structural earthquake resistance, and

Tyler (1978) demonstrated the usefulness of

flexural yielding of steel plate energy-dissipation

devices in New Zealand.

In the seventh WCEE in 1980, Sladek and

Klingner (1980) reported their studies on tunedmass

dampers for seismic response reduction in

civil structures, whereas Keightley (1980) presented

his work on a dry friction damper for multistory

structures. Figure 29.32 shows the schematic of a

framed structure with additional friction dampers

and the hysteretic loop of the damper.

In 1984, Pall (1984) numerically studied the

performance of a framed building equipped with friction devices. The performance was found to be

superior to that of a conventional building. Scholl (1984) introduced brace dampers. The stiffness and

damping of the damper can be set to alter the stiffness and damping in a structure. It was typically

installed at the cross points in braced-frame structure systems. Liu et al. (1984) investigated the

effectiveness of an active mass-damper control system for controlling coupled lateral – torsional

motions of tall buildings subjected to strong earthquakes.

A special themed session named “Seismic response control of structural systems” was held at the Ninth

WCEE in 1988. Some studies on passive systems were reported such as: a hysteretic damper that is put

between two adjacent structures (Kobori et al., 1988); a multiple passive TMD (Clark, 1988); viscoelastic

materials (Bergman and Hanson, 1988); viscous damping walls (Arima et al., 1988); and magnetic

damping (Shimosaka et al., 1988). Also, some studies on active control were reported, such as an

experimental study on active mass dampers (Aizawa et al., 1988) and air injection for suppressing

sloshing in a liquid tank (Hara and Saito, 1988; Sogabe et al., 1988). Since then, the active vibration

control of buildings and other civil structures has attracted a worldwide growing interest as an innovative

technology in earthquake engineering.

During the Tenth WCEE in 1992, more thorough studies on structural control were reported such

as: predictive control of structures with reduced number of sensors and actuators (Lopez-Almansa

et al., 1992); applicability of vibration control to nonlinear structure (Shimada et al., 1992); full-scale

implementation of active control (Soong et al., 1992); effects of soil – structure interaction (SSI) on

actively controlled structure (Wong and Luco, 1992); the time-delay problem (Hou and Iwan, 1992);

and fuzzy logic control (Tani et al., 1992). In this conference, the decision to form an international

association and to hold a world conference dedicated to structural control was made. The

International Association for Structural Control (IASC) was formed in 1993. The efforts

led to the successful First World Conference on Structural Control in 1994 in Pasadena, California

(IASC, 1994).

It was followed by the Second World Conference on Structural Control in 1998, which was held

in Kyoto, Japan (IASC, 1999). The Third World Conference on Structural Control was held in

April 2002, in Como, Italy (IASC, 2003). The conferences showed increasing interest among

researchers and an increase of innovations and applications of structural control technologies in

civil structures.

29.3.2 Basic Principles

Structural vibration control basically involves the regulation of structural properties in order to achieve

structural a desirable response to a given external load and modification of the excitation. The regulation

of structural properties includes the modification of mass, damping, and stiffness of the structure so that

Friction

damper

Column

Floor

(a)

Force

Deflection

(b)

FIGURE 29.32 (a) Friction dampers in a framed

structure; (b) hysteretic loop of the damper. (Source:

Keightley, W.O. Proc. VII World Conf. on Earthquake

Engineering, 1980. With permission.)

29-34 Vibration and Shock Handbook

it can respond more favorably to the external excitation. Moreover, it is also possible to reduce the level of

excitation transmitted to the structure. Structural control can be defined as a mechanical system that is

installed in a structure to reduce structural vibrations during loadings such as strong earthquakes.

29.3.2.1 Passive Control System

The basic function of a passive control system is to absorb or consume a portion of the input energy,

thereby reducing energy-dissipation demand on primary structural members and minimizing the

possible structural damage. In the passive system, controlling forces develop at the locations of

installation of the mechanism itself. The energy necessary for generation of these forces is provided

through the motion of the mechanism during the dynamic excitation. The relative motion of the

mechanism defines the amplitude and the direction of the controlling force. Passive control has four

main advantages: (1) it is usually relatively inexpensive; (2) it consumes no external energy (energy may

not be available during a major earthquake); (3) it is inherently stable; and (4) it works even during a

major earthquake.

In the passive system area, structural response control can be divided into two groups. The first group

uses supplemental damping to reduce structural responses by conversion of kinetic energy to heat. The

second group uses supplemental oscillators to reduce the structural response by transferring energy

among vibration modes. The first group includes devices that operate on principles such as frictional

sliding, the yielding of metals, phase transformation in metals, deformation of viscoelastic solids or

fluids, and fluid orificing. The second group includes dynamic vibration absorbers such as TMDs and

tuned liquid dampers.

The major difference between viscous/viscoelastic devices and friction/yielding devices is the maximum

force that each device will develop during an earthquake. The maximum earthquake forces

developed in viscous/viscoelastic devices are determined by the maximum displacements and velocities

across these devices. The maximum earthquake forces in a friction/yielding device equals the design

friction force/design yield force plus strain hardening. Thus, the maximum earthquake forces are more

easily controlled in the friction/yielding devices.

The effect of increasing damping in a structure can be appreciated by studying the earthquake response

spectrum. The earthquake response spectrum of a quantity is a plot of the peak value of the response

quantity as a function of the natural vibration period of the system. Each such plot is for single-DoF

system having a fixed damping ratio, and several such plots for different values of damping ratio are

included to cover the range of damping values. The most commonly used spectra in earthquake

engineering are the absolute acceleration response spectrum, the velocity response spectrum, and the

displacement response spectrum. For more information on how to develop earthquake response

spectrum, readers are referred to Chopra (1995).

Figure 29.33 shows the response spectra of measured ground acceleration during the Kobe, Japan,

earthquake on January 17, 1995. The ground acceleration north – south (NS) component was recorded at

the Kobe Marine Meteorological Observatory in Chuo-ku. As is shown in the figure, increasing the

damping ratio from 5 to 30% decreases the response spectra. Therefore, on a level of ordinary structure

that has 5% of damping, the response can be reduced further by increasing the structural damping. In the

area of passive systems, a variety of mechanical energy dissipaters to increase structural damping has been

developed and tested in the laboratory and, in some cases, in actual structural applications, such as

bracing systems, friction dampers, viscoelastic dampers, and other mechanical dampers.

The principle of the TMD dates back to the 1940s when Den Hartog (1947) reintroduced the dynamic

absorber invented by Frahm in 1909. Figure 29.34 shows the principle of the TMD. A large system under

alternating force P0 sin vt is represented by a K – M system where K and M are the stiffness and the mass

of the system, respectively. The vibration absorber consists of a comparatively small vibratory system

with stiffness k and mass m. The natural frequency,

ffiffiffiffiffi

k=m p ; of the attached absorber is chosen to be equal

to the frequency, v, of the disturbing force. It has been shown (Den Hartog, 1947) that the main mass, M,

does not vibrate at all and that the small system, k – m, vibrates in such a way that its spring force is at all

instants equal and opposite to P0 sin vt: Thus, there is no net force acting on M and therefore that mass

Seismic Base Isolation and Vibration Control 29-35

does not vibrate

vM ¼

ffiffiffiffiffi

; vm ¼

ffiffiffiffi

; m ¼

M ð29:22Þ

It is useful to introduce the new notations, above.

Assume that m ¼ 0:2; and vM ¼ vm: For this

condition, the resonance curve of the system is

illustrated in Figure 29.35. At v ¼ vm or v=vm ¼

1; the motion of the main mass ceases altogether

(Chopra, 1995). Since the system has two DoF, two resonant frequencies exist and the response is

unbounded at those frequencies.

If the exciting frequency, v, is close to the natural frequency, vm, of the attached absorber, and the

operating restrictions make it possible to vary either one, the vibration absorber can be used to reduce the

response amplitude of the main system to near zero. It is worth noting that the larger the attached

absorber mass, m, the smaller the absorber displacement, ðx2 2 x1Þ; relative to the displacement of main

mass, M. However, a large absorber mass presents a practical problem. At the same time, the smaller is the

mass, m, the narrower is the operating frequency range of the absorber (Chopra, 1995).

29.3.2.2 Active Control System

Active control systems are used to control the response of structures to internal or external excitation,

such as machinery or traffic noise, wind, or earthquakes, where the safety or comfort level of the

M k

P0 sinwt

FIGURE 29.34 Additional vibratory k – m system to a

larger K – M system to reduce vibration of the larger

system.

FIGURE 29.33 Effect of damping on earthquake response spectra.

29-36 Vibration and Shock Handbook

occupants is of concern. Active control makes use

of a wide variety of actuators, which may employ

hydraulic, pneumatic, electromagnetic, or motordriven

ball-screw actuation. An essential feature of

active control is that external power is used to

effect the control action. This makes the system

vulnerable to power failure, which is always a

possibility during a strong earthquake.

For the actuators properly to apply the

desired forces, sensors are placed in the structure

in order to measure structural response. The

sensors convey response information to a central

computer that then uses this information to

calculate the desired actuator forces. The advantage

of an active control system is that it attains

excellent control results. However, there are many

drawbacks to using the active control system. It is

relatively expensive to design and to operate due to

the large amount of power needed. It has the

potential to destabilize the structural system.

Furthermore, it tends to take up more space than

passive control devices.

A fully active structural control system has the

basic configuration as shown schematically in

Figure 29.36. The structural control system in the

figure basically consists of sensors, controllers,

and actuators. Sensors are used to measure

either external excitations or structural responses,

or both. Controllers process the measured information

and compute necessary control forces

needed based on a given control algorithm. Actuators

are used to produce the required forces and are

usually powered by external energy sources.

According to the characteristics of the controlling

effects, active control can be classified into two

categories: open-loop control and closed-loop control. A control system in which the control input is

applied without the knowledge of the plant output is called an open-loop control system. Figure 29.37

shows a block diagram of an open-loop control system, where the subsystems (controller and plant) are

shown as rectangular blocks. Open-loop control will be successful only if the controller has a reasonably

good prior knowledge of the behavior of the plant, which can be defined as the relationship between the

control input and the plant output. Mathematically, the relationship between the output of a linear plant

and the control input can be described by a transfer function. However, in actuality, the presence of plant

behavior uncertainty is unavoidable. Therefore, it is clear that an open-loop control system is unlikely to

be successful.

In the case that the controller adjusts the control input according to the actual observed output, the

system is called a closed-loop system. In this system, the control input is a function of the plant’s output.

Since in a closed-loop system the controller is constantly in touch with the actual output, it is likely to

succeed in achieving the desired output even in the presence of uncertainty in the linear plant’s behavior

(the transfer function). The mechanism by which the information about the actual output is conveyed to

the controller is called feedback. On a block diagram, the path from the plant output to the controller is

called a feedback loop. A block diagram of a possible closed-loop system is given in Figure 29.38.

01 0.8 1 1.25 ω/ωm

Po/K

FIGURE 29.35 Resonance curve for the system in

Figure 29.34 that shows response amplitude vs. exciting

frequency (the dashed curve indicates negative x2 or

phase opposite to excitation) for m ¼ 0:2 and vM ¼ vm.

(Source: Chopra, A.K. 1995. Dynamics of Structures,

Theory and Applications to Earthquake Engineering,

Prentice Hall, New York. With permission.)

Structure

Sensors

Excitation Response

Control

Sensors Controller

FIGURE 29.36 Schematic diagram of active control

system.

Controller Plant

Desired

output Output

Control

input

FIGURE 29.37 An open-loop control system; the

controller applies control input without knowing the

plant output.

Seismic Base Isolation and Vibration Control 29-37

When the desired output is a constant, the resulting controller is called a regulator. If the desired output

changes with time, the corresponding control system is called a tracking system. In any case, the principal

task of a closed-loop controller is to minimize the “error” as quickly as possible.

29.3.2.3 Semiactive Control System

Semiactive control falls between passive and active on the control spectrum. A semiactive control

system is similar to an active system in that the system uses sensors and controllers, and operates on

external power. However, the source of external energy is used only for adjustment of the mechanical

characteristics of the system (Rakicevic and Jurukovski, 2001). The inherent benefit of a semiactive

control device is that the mechanism used does not require large amounts of external power. Many

semiactive devices can be powered by batteries protecting them from sudden power loss during

earthquakes. Semiactive control systems are basically derived from passive systems; they are modified in

such a way that they enable adjustment or correction of their mechanical characteristics.

A typical strategy for a semiactive control system is that an “ideal” actively controlled device is first

assumed and appropriate primary controller designs for this device are designed. Then, a secondary

controller is used that clips the optimal control force so it is dissipative in a manner consistent with the

physical nature of the device. This strategy has been widely used (e.g., Dyke et al., 1996a, 1996b, Jung et al.,

2001).

Spencer and Sain (1997) found that many active control systems for civil engineering applications

operate primarily to modify structural damping. They claimed that preliminary studies indicate that

appropriately implemented semiactive systems perform significantly better than passive devices and have

the potential to achieve the majority of the performance of fully active systems, thus allowing for the

possibility of effective response reduction during a wide array of dynamic loading conditions. In other

words, semiactive control devices offer the adaptability of active control devices without requiring the

associated large power sources (Spencer and Sain, 1997).

Moreover, according to presently accepted definitions (Housner et al., 1997), a semiactive control

device is one that cannot inject mechanical energy into the controlled structural system (including the

structure and the control device), but has properties that can be controlled optimally to reduce the

responses of the system. Therefore, in contrast to active control devices, semiactive control devices do not

have the potential to destabilize (in the bounded input/bounded output sense) the structural system.

Semiactive control devices are often viewed as controllable passive devices.

Semiactive control technologies have recently been widely investigated in terms of the reduction of the

dynamic response of structures subjected to earthquake and wind excitations (Housner et al., 1997;

Spencer et al., 1997; Patten, 1998; Kurata et al., 1999; He et al., 2001; Iemura et al., 2001; Jung et al., 2001).

Various semiactive devices have been proposed that utilize forces generated by surface friction or viscous

fluids to dissipate vibratory energy in a structural system.

29.3.2.4 Hybrid Control System

The common usage of the term “hybrid control” implies the combined use of active and passive

control systems. For example, this may be a structure equipped with distributed viscoelastic damping

Controller Plant

Desired

output Output

Control

Error input

Feedback loop

+ −

FIGURE 29.38 Example of a closed-loop control system with feedback; the controller applies a control input based

on the plant output.

29-38 Vibration and Shock Handbook

supplemented with an active mass damper on the top of the structure, or a base-isolated structure with

actuators actively controlled to enhance performance. The method, which consists of using both passive

and active devices, should utilize the merits of both passive and active methods and avoid the demerits

of these methods. Thus, higher levels of performance should be achievable. Additionally, the

resulting hybrid control system can be more reliable than a fully active system because, should

the active control malfunction, the minimum seismic protection of the structure can be done by the

passive control.

29.3.2.5 Categorization of Basic Principles

From the above group of structural control systems, there are basically five principles concerning

earthquake-induced structural response control that are important to consider. They are listed

below:

1. Cutting off the input energy from the earthquake ground motion; examples:

(a) Floating structures

(b) Frictional structures

2. Isolating the natural frequencies of the structures from the predominant seismic power

components; examples:

(a) Base-isolated structures

(b) Long period structures

3. Providing nonlinear structural characteristics and establishing a nonstationary state nonresonant

system; examples:

(a) Inelastic structures

(b) Varying stiffness and damping structures

4. Utilizing energy absorption mechanism; examples:

(a) Viscous damper

(b) Viscoelastic damper

5. Supplying control force to suppress the structural response; examples:

(a) Active mass damper

(b) Active tendon

29.3.3 Important Issues in Vibration Control

29.3.3.1 Soil – Structure Interaction

Seismic vibration control of a civil structure deals with methods to suppress the response in a structure

subjected to earthquake excitation. To control the structural dynamic response, the structural system

model must be known. Modeling of a structure is relatively straightforward, using a finite number of

DoF, because the dimensions of the structure are finite. However, in general, civil structures are

supported on surrounding soil from which the tremor excites the structure. This makes the structure

interact with the surrounding soil. Therefore, it is very important to include SSI for controlling the

seismic response.

SSI will result in a structural response that may be quite different from the structural response

computed for a fixed-base building. The frequency of vibration of the structure may be lower because

of the interaction. The change in frequency may also affect the response of the overall structure or its

substructures or components. Moreover, soils are notoriously nonlinear when subjected to strong

ground motions at the level of engineering interest (Marshall, 2001). Damping of the final

system increases because of the radiation of energy of the propagating waves away from the structure

(Wolf, 1985).

Seismic Base Isolation and Vibration Control 29-39

29.3.3.2 Device – Structure Interaction and Time Delay

Other important effects come from the device – structure dynamic interaction. Dyke et al. (1995) studied

the role of control – structure interaction in mitigating dynamic response. Their findings show that

accounting for control – structure interaction and actuator dynamics in the design process can improve

the performance and robustness of a control system.

Time delay between control command and actual control force also causes problems. A study by

Hou and Iwan (1992) showed the problems of time delay in the vibration control of structures. Also,

Agrawal and Yang (2000) claimed that applications of unsynchronized control forces due to time delay

may result in a degradation of the control performance, and it may even render the controlled structure

unstable. Therefore, they provide a state-of-the-art review for available methods of time-delay

compensation.

29.3.3.3 Design Guidelines

Since structural control is a new concept in civil structures, design guidelines are now being developed to

provide the design engineers with tools for the safe design or seismic rehabilitation of structures. For a

passive control system, the American Society of Civil Engineers has prepared for the Federal Emergency

Management Agency a “Prestandard and Commentary for the Seismic Rehabilitation of Buildings”

(ASCE, 2000), in which requirements for the systematic rehabilitation of buildings using energydissipation

systems is set forth. In the standard, analysis and design criteria for passive energy dissipation

systems are provided.

In this prestandard, two sections deal with structural control technology. First, is “Passive Energy

Dissipation Systems,” which contains (1) General Requirements, (2) Implementation of Energy

Dissipation Devices, (2) Modeling of Energy Dissipation Devices, (3) Linear Procedures, (4) Nonlinear

Procedures, (5) Detailed Systems Requirements, (G) Design Review, and (H) Required Test of Energy

Dissipation Devices. Each subsection is explained in detail in the prestandard for the users.

However, in the second section, “Other Response Control Systems,” it is mentioned that analysis and

design of response control other than in the passive systems above shall be reviewed by an independent

engineering review panel. This is because the technology of active, semiactive, and hybrid control is not

sufficiently mature and the necessary hardware is not sufficiently robust to warrant the preparation of

general guidelines for the implementation of the technology.

29.3.4 Vibration-Control Devices

29.3.4.1 Passive Control Systems

29.3.4.1.1 Metallic Yielding Dampers

One of the effective mechanisms useful for the dissipation of energy input to a structure from an

earthquake is through the inelastic deformation of metals. The idea of utilizing added metallic energy

dissipaters within a structure to absorb a large portion of the seismic energy began with the conceptual

and experimental work of Kelly et al. (1972) and Skinner et al. (1975). Devices considered by them

included torsional beams, flexural beams, and U-strip energy dissipaters as shown schematically in

Figure 29.39.

During the following years, considerable progress has been made in the development of metallic

dampers. For example, many new designs have been proposed, including the hourglass or X-shaped and

triangular plate dampers as shown in Figure 29.40.

The X-shaped steel plates form devices called ADAS (Added Damping and Stiffness; see Perry et al.,

1993), because the devices essentially add stiffness as well as damping to the element where they are

installed. For example, if they are installed between two adjacent floors, then it will increase the stiffness

and damping between those two floors. Increasing the stiffness will generally attract more seismic forces.

However, since the devices have much lower yielding forces than the elements where they are installed,

then the postyield stiffness of the devices is dominant during a strong earthquake.

29-40 Vibration and Shock Handbook

The devices dissipate energy through the flexural yielding deformation of mild steel plates. Each

device consists of a series of steel plates arranged in parallel, with boundary elements at the top and

bottom to establish end fixity, which bend in double-curvature flexure when subjected to lateral loading.

The direction of the lateral loadings is shown in Figure 29.41. The plates are cut in an hour-glass shape

to match the moment diagram and thus maximize the uniformity of plastification over the height of

the steel plates.

Cyclic loading tests to determine the reliability of the ADAS were performed by Bergman and Goel

(1987). The tests demonstrated that the devices maintained stable hysteretic properties, dissipated

significant energy with no pinching or slip zones, and continued to have these properties until

plate fracture at high strain or high numbers of cycles. The schematic hysteretic loop of ADAS

Force

(a) (b)

FIGURE 29.39 Schematic of metallic dampers: (a) torsional beam and (b) U-strip, and the directions of alternating

forces. (Source: Skinner, R.I. et al. Earthquake Eng. Struct. Dyn., 3, 287 – 296, 1975. With permission.)

Force

Beam of

Upper Floor

Braced to Beam

of Lower Floor

(a) (b)

FIGURE 29.40 Schematic of (a) X-shaped (Source: Tsai, K.-C. et al. Earthquake Spectra, 9, 505 – 528, 1993. With

permission.) and (b) triangular plate dampers, and the directions of alternating forces. (Source: Perry, C.L. et al.

Earthquake Spectra, 9, 559 – 579, 1993. With permission.)

Seismic Base Isolation and Vibration Control 29-41

devices tested by Bergman and Goel (1987) is shown in Figure 29.42. It is worth noting that the apparent

stiffening of the device in the figure is due to finite deformation, not strain hardening of the material;

therefore, the effects of finite deformation should be included in the assessment of a design.

Tsai et al. (1993) have performed cyclic loading tests on TADAS (Triangular-plate ADAS; see Figure

29.40b). The hysteretic loops of the tested TADAS are similar to those of ADAS, such as the apparent

stiffening at large displacement, since the mechanism is the same. Although the slotted pin connection at

the apex reduces the axial tension force, some friction at the pin location contributes to the stiffening of

hysteretic loops (Soong and Dargush, 1997).

29.3.4.1.2 Friction Dampers

Another excellent mechanism for energy dissipation is friction. The mechanism has been used

for many years in automotive brakes to dissipate kinetic energy of motion. In friction dampers,

stick-slip phenomena must be minimized to avoid introducing high-frequency excitation.

Force

Moment

diagram

FIGURE 29.41 ADAS element: (a) lateral view; (b) longitudinal view with lateral forces; and (c) moment diagram

over the height of the steel plates.

Displacement (inch)

−8

−0.5 0.5 Displacement (inch)

Force

(kips)

−10

(a) (b) −1.1 1.1

Displacement (inch)

Force

(kips)

−12

FIGURE 29.42 Schematic hysteresis loop of the ADAS device with maximum displacements of (a) 0.42 in.

(10.7 mm); (b) 1.04 in. (26.4 mm); and (c) 1.56 in. (39.6 mm). (Source: Bergman, D. and Goel, S., Rpt. UMCE 87-10,

Univ. Michigan, 1987. With permission.)

29-42 Vibration and Shock Handbook

Furthermore, friction materials should have a

consistent coefficient of friction over the intended

life of the device (Housner et al., 1997).

The Pall device (Pall and Marsh, 1982) is one of

the friction dampers that can be installed in a

structure in an X-braced frame as illustrated in

Figure 29.43. The mechanism of the damper is that,

when there is axial tension force in the bracing

system, as shown in the figure by outward arrows,

then plates A and B are moving outward from each

other. This movement is resisted by some friction

force at the slip lap joint between plates A and B.

This movement also causes plates C and D to move

toward each other because of the existence of the links. This movement is also resisted by some frictional

force between plates C and D. Without the existence of the links, the bracing may buckle because of axial

compression force. The friction level between plates is designed so that the plates will not slip to each other

during wind storms or moderate earthquakes. Under severe loading conditions, the devices slip in order

to dissipate energy so that structural response can be reduced. The force – displacement relationship of

Pall dampers has been studied extensively. A plot of its typical cyclic response is illustrated in Figure 29.44

(Filiatrault and Cherry, 1987).

29.3.4.1.3 Viscoelastic Dampers

The application of viscoelastic dampers to civil

engineering structures seems to have begun in

1969, when approximately 10,000 viscoelastic

dampers were installed in each of the twin towers

of the late World Trade Center in New York to

reduce wind-induced vibrations. An example of a

viscoelastic damper is illustrated in Figure 29.45.

The damper consists of a viscoelastic material

bonded to steel plates. The viscoelastic materials

used in structural application are typically copolymers

or glassy substances, which dissipate energy when subjected to shear deformation (Soong and

Dargush, 1997). With induced structural vibration, the damper will absorb and dissipate the vibrational

column

brace

beam

damper

links

brace

slip lap joint

with friction pad

(a) (b)

FIGURE 29.43 Pall friction damper: (a) set up in a structure; (b) schematic of the damper.

Force

Displacement

−0.7

−17.78 17.78

0.7

−2000

2000

−8.9

8.9

(lbs) (kN)

(in)

(mm)

FIGURE 29.44 Schematic of force – displacement

hysteresis loop of the Pall friction damper. (Source:

Filiatrault, A. and Cherry, S. Earthquake Spectra, 3,

57 – 78, 1987.)

Force

Force/2

Center

plate

Viscoelastic

material

Steel flange

FIGURE 29.45 Typical viscoelastic-damper configuration.

Seismic Base Isolation and Vibration Control 29-43

energy by shearing deformation within the viscoelastic material. Heat will be generated in the viscoelastic

material and released through the steel members of the damper.

A typical hysteresis loop produced by viscoelastic dampers is shown in Figure 29.46. By using

viscoelastic dampers, although the structural response is elastic, hysteresis loops are formed because of

the viscoelastic material. The area enclosed in the hysteresis loops is the energy dissipated by the

viscoelastic dampers during one cycle of oscillation.

The behavior of viscoelastic materials under dynamic loading depends on vibrational frequency, strain,

and ambient temperature. In general, the relationship between shear strain, gðtÞ; and shear stress, t ðtÞ;

under harmonic shear strain with frequency, v; can be expressed as (Zhang et al., 1989)

tðtÞ ¼ G0ðvÞgðtÞ þ

G00ðvÞ

g_ðtÞ ð29:23Þ

G0ðvÞ and G00ðvÞ are shear storage modulus and shear loss modulus of the viscoelastic material,

respectively. The loss factor is defined by h ¼ G00ðvÞ=G0ðvÞ: In general, as the vibrational frequency

increases, the values of G0ðvÞ and G00ðvÞ become larger. However, if the ambient temperature increases,

those values become smaller. Test results of a typical viscoelastic damper averaged over the first 20 cycles

are shown in Table 29.4.

For a viscoelastic damper with shear area, A; and thickness, d; the corresponding force – displacement

relationship is

FðtÞ ¼ kdðvÞX þ cdðvÞ X_ ð29:24Þ

in which X and X_ are the relative displacement and velocity of the damper, respectively, and

kdðvÞ ¼

AG0ðvÞ

cdðvÞ ¼

AG00ðvÞ

vd ð29:25Þ

Force (kips)

1.00

−1.00

0.00

−0.05 Displacement (in) 0.05

Elastic

stiffness

Energy dissipated in

one cycle of oscillation

FIGURE 29.46 Typical schematic hysteresis loop of viscoelastic damper. (Source: Shen, K.L. and Soong, T.T. J. Eng.

Mech. ASCE, 121, 694 – 701, 1995. With permission.)

TABLE 29.4 Typical Viscoelastic Damper Properties

Temperature (8C) Frequency (Hz) Strain (%) kd

a (lb/in.) G 0 (psi) G 00 (psi) h

24 1 5 2124 142 193 1.36

24 1 20 2082 139 192 1.38

24 3 5 4084 272 324 1.19

24 3 20 3840 256 306 1.20

36 1 5 880 59 67 1.13

36 1 20 873 58 65 1.12

36 3 5 1626 108 119 1.10

36 3 20 1542 103 112 1.09

a The definition of kd is shown in the text.

Source: Data from Chang, K.C. et al. Earthquake Spectra, 9, 371 – 387, 1993.

29-44 Vibration and Shock Handbook

From the above equations, it is clear that a linear system with added viscoelastic dampers remains linear

with the dampers contributing to increased viscous damping as well as stiffness. However, it needs to be

pointed out that, for viscoelastic material at large strains, there is considerable self-heating due to the

large amount of energy dissipated. The heat generated changes the mechanical properties of the material,

and the overall behavior becomes nonlinear. This means that a linear analysis utilizing the above

equations can only be for approximation of the response.

29.3.4.1.4 Viscous Fluid Dampers

The most convenient and common functional output equation for a damper comes from classical system

theory, and is that of the so-called “linear” or “viscous” damping element

FðtÞ ¼ cd

X_ ðtÞ ð29:26Þ

in which FðtÞ; cd, and X_ ðtÞ are the damping force, damping coefficient, and relative velocity across the

damper, respectively.

In mechanical engineering, it is difficult to manufacture a useable fluid-filled component having a

purely viscous output, because even moderate pressure hydraulic flows through a simple orifice follow a

very different output equation, in which differential pressure varies with the fluid velocity squared.

Therefore, the output of the basic hydraulic damping element is

FðtÞ ¼ sgnð X_ ðtÞÞcdlX_ ðtÞl2 ð29:27Þ

As a result of long research in this area, which was started in the 1960s, the most useful dampers being

used in buildings today are the so-called “low exponent” type, with an output equation of the form:

FðtÞ ¼ sgnð X_ ðtÞÞcdlX_ ðtÞla ð29:28Þ

In most cases, a is an exponent having a specified value in the range of 0.3 to 1.0. Values of a that have

proven to be the most popular are in the range of 0.4 to 0.5 for building designs with seismic input

(Taylor, 2002).

The design elements of a fluid damper are relatively few. However, the detailing of these elements

varies greatly. Typical elements for a fluid damper are shown in Figure 29.47. Typical experimentally

measured force – displacement loops are shown in Figure 29.48.

29.3.4.1.5 Tuned Mass Dampers

Much of the early development of dynamic vibration absorbers, as mentioned in Section 29.3.2.1, has

been limited to the use of dynamic absorbers in mechanical engineering systems in which one operating

frequency is in resonance with a machine’s fundamental frequency. However, building structures

are subjected to earthquakes which posses many frequency components. The performance of a

Accumulator

housing

Compressible

silicon fluid

Rod make-up

Control accumulator

valve

Chamber 2

Chamber 1

Piston head

with orifices

Piston Cylinder

rod

Seal

retainer

High strength

acetal resin seal

FIGURE 29.47 Typical schematic of a fluid damper. (Source: Taylor, D.P. Passive Structural Control Symp., 2002.

With permission.)

Seismic Base Isolation and Vibration Control 29-45

dynamic vibration absorber, referred to as the

TMD herein, in complex multi-DoF systems is

expected to be different. Consider the resonance

curve in Figure 29.49 of a system shown in

Figure 29.34, but now with different parameters:

m ¼ 0:05; but still vM ¼ vm (see Equation 29.22

for the definition of these parameters). An

additional parameter, the damping ratio of TMD,

is shown in Equation 29.29. Where c is damping

coefficient of a linear viscous damper installed

between structural mass, M; and TMD mass, m;

6TMD ¼

2mvm ð29:29Þ

When the damping ratio in the TMD equals zero,

the response amplitude is infinite at the two

resonant frequencies. When the damping ratio

becomes infinite, the two masses are virtually stuck

to each other; the result is a single-DoF system with mass, 1.05 ðMÞ with the amplitude becoming infinite

again at a resonant frequency (see Figure 29.49). Therefore, somewhere between these extremes, there

must be a value of the TMD damping ratio for which the peak becomes a minimum. Therefore, the

objective in adding the TMD herein is to bring the resonant peak of the amplitude to its lowest possible

value so that smaller amplifications over a wider frequency bandwidth can be achieved.

There are many types of TMD for implementation and the following are some examples

(Figure 29.50). An innovative challenge is highly expected in this field.

There is another type of TMD that uses liquid as the mass; this damper is called the tuned liquid

damper (TLD). The TLD has been used in ships for controlling vibrations because of water waves.

The TLD uses water or other liquid as the moving mass and the restoring force is generated by gravity.

Energy absorption comes from boundaries between liquid and containers and turbulence in the liquid

flow. The basic principle of the TLD in absorbing kinetic energy of the main structure is the same as the

TMD. Figure 29.51 shows types of TLD. Favorable properties of TLD compared with TMD are as follows:

* Smooth movement in small vibration is possible because of no mechanical friction.

* It is reasonable in cost and maintenance because of no complex mechanism.

* It can be applied easily in two horizontal directions with a single TLD.

* It can be compact and portable if large numbers are used.

1000

−1000

−1.5 1.5

0.0

Force (lb)

Displacement (in)

f = 4 Hz

f = 1 Hz

FIGURE 29.48 Typical schematic of fluid damper

hysteretic loops. (Source: Constantinou, M.C. and

Symans, M.D. Struct. Des. Tall Build., 2, 93 – 132, 1993.

With permission.)

0.32

0.1

0.6 1 ω/ωm 1.3

Po/K

ζ TMD = 0

FIGURE 29.49 Resonance curve for the system in Figure 29.34 that shows response amplitude vs. exciting

frequency for m ¼ 0:05 and vM ¼ vm:

29-46 Vibration and Shock Handbook

The TLD can be divided into two categories. First, is the sloshing damper as shown in Figure 29.51a.

The vibration period is adjusted by the size of the container and the depth of the liquid. The damping

capacity is increased by placing meshes or rods in the liquid. The second category is the tuned liquid

column damper as shown in Figure 29.51b. The vibration period is adjusted by the shape of the column

or the air pressure in the column. The damping capacity is increased by adjusting the orifice in the

column, which generates a high turbulence.

29.3.4.2 Active Control System

29.3.4.2.1 Electromagnetic Actuator

An electromagnetic actuator is frequently employed where it is necessary to provide a mechanical force

depending on an electric current. Figure 29.52 shows the elements of such a device. A magnetic structure

supports a circuit of magnetic flux driven by the coil of N turns carrying current i, in ampere. Part of the

magnetic circuit is a movable armature that slides smoothly on the support member. An air gap (or

possibly a vacuum gap) of length d m is also in the magnetic circuit. Assume that d is small enough for

the magnetic flux density to be essentially constant at B Wb/m2 across the face of the armature. The crosssectional

area of the armature face is A m2. A tractive force, F, in Newtons, developed on the face of the

armature, is related to the area and field strength as

F ¼

AB2

2m0 ð29:30Þ

(a)

(d)

(b) (c)

(e) (f)

FIGURE 29.50 Examples of different types of TMD: (a) pendulum with damper; (b) inverted pendulum with

spring and damper; (c) pendulum of which hangers are winded to save space; (d) swinging mass on rotational

bearings; (e) sliding mass with spring and damper; (f) mass on rubber bearings. (Source: Iemura, H. Passive and Active

Vibration Control in Civil Engineering, Springer, New York, 1994a,b. With permission.)

(a) Vibration Vibration

(b)

FIGURE 29.51 Types of TLD: (a) tuned sloshing damper with meshes and rods; (b) tuned liquid column damper

with orifice. (Source: Iemura, H. Passive and Active Vibration Control in Civil Engineering, Springer, New York,

1994a,b. With permission.)

Seismic Base Isolation and Vibration Control 29-47

where m0 is the magnetic permeability of the air

gap having a value of 4p £ 1027 H/m (Henries/

meter). The magnetic field intensity is proportional

to the magnetomotive force N £ i

ampere turns. For an efficient magnetic structure,

the magnetic flux density is

B ¼

N £ i ð29:31Þ

which, combined with Equation 29.30, gives a

useful relationship:

F ¼

Am0N2

" #

􀁻 !

Newton ð29:32Þ

Note that F is positive when i is either positive or

negative. This means that the force, F, is always

tractive if the current, I, is either positive or

negative. The maximum value for magnetic flux

density, B, which can be realized using modern

magnetic materials in normal environments, is

approximately 1 Wb/m2 (Clark, 1996).

In most cases, it is necessary to apply force in

both the positive and negative directions. In this

case, the device is constructed with two air gaps

having the corresponding traction forces opposed

to one another, as shown in Figure 29.53.

It is interesting to observe how the net force, F,

on the armature depends on z and Di (differential

current ¼ iL 2 i0 ¼ iR 2 i0; see Figure 29.53 for

the parameters) when these are constants.

The schematic plot of F vs. z, with Di as a

parameter, is shown in Figure 29.54. It is worth

noting that the plot is nonlinear. It is also noted

from the figure that, if z is positive, F is also

positive. It means that the force developed on the

armature due to a small displacement from the

equilibrium position is in the direction to

increase that small displacement. This characterizes

a condition of static instability. Feedback

control can be used to stabilize this inherently

unstable device when such a device is used as an

actuator.

29.3.4.2.2 Hydraulic Actuators

Hydraulic actuators can produce large forces even at large displacements, which is useful for seismic

response-control applications. The actuator is usually driven using electrical signals. In other words, an

electro-hydraulic device is used to convert the low-powered signals into high-powered hydraulic fluid

flow. One example of such a device is an electro-hydraulic servo-valve. The schematic of such a device is

shown in Figure 29.55.

Sliding

armature

Magnet flux density

= B Webers/meter2

Armature A

i (Amperes)

N turns

z (t)

FIGURE 29.52 Elements of an electromagnet. (Source:

Clark, R.N. Control System Dynamics, Cambridge Univ.

Press, U.K., 1996. With permission.)

N turns

z(t)

Armature

N turns

2i0

FIGURE 29.53 Push – pull electromagnet. (Source:

Clark, R.N. Control System Dynamics, Cambridge Univ.

Press, U.K., 1996. With permission.)

−3 0 3

−30

i0 = 2 A

d0 = 5 mm

z (mm)

(Newton)

Di = 2A

Di = 1A

Di = −1A

Di = 0A

Di = −2A

FIGURE 29.54 Force vs. displacement of push – pull

electromagnet with differential current as a parameter.

(Source: Clark, R.N. Control System Dynamics, Cambridge

Univ. Press, U.K., 1996. With permission.)

29-48 Vibration and Shock Handbook

The valve shown in the figure is a single-stage spool valve. The simple design shown in the figure

exhibits the basic principles, inherent in most types of servo-valves, of conversion from electromagnetic

to hydraulic energy; in other words, actuating force. By changing the electric signals going to the spool

valve, the actuating force can be generated because the hydraulic pressure acting on the piston in one side

is greater than the other side (see Figure 29.55). The actuating force can also be altered in real time,

enabling real-time control of seismically excited structures.

29.3.4.3 Semiactive Control System

29.3.4.3.1 Variable-Orifice Hydraulic Dampers

A variable damping device used for the semiactive

control method can be achieved by using a

controllable, electro-mechanical, variable-orifice

valve to alter the resistance to flow of a conventional

hydraulic fluid damper. Figure 29.56 shows

a schematic of such a device. The effectiveness of

variable-orifice dampers in controlling seismically

excited buildings has been demonstrated through

both simulation and small scale experimental

studies (Hrovat et al., 1983; Mizuno et al., 1992;

Kurata et al., 1994; Patten et al., 1994; Sack et al.,

1994; Liang et al., 1995; Niwa et al., 2000).

29.3.4.3.2 Controllable Fluid Dampers

Another class of semiactive devices use controllable fluids. The advantage of controllable fluid devices

over controllable valve devices is that they contain no moving parts other than the piston. The essential

characteristic of controllable fluids is their ability to change reversibly from a free-flowing, linear viscous

fluid to a semisolid with a controllable yield strength in milliseconds, when exposed to an electric (for

electro-rheological fluid) or magnetic (for magneto-rheological [MR] fluid) field (Housner et al., 1997).

Hydraulic

Pressure

Hydraulic

Return

Actuator Force

To push-pull

current amplifier

To push-pull

current amplifier

Single-state

spool valve

High hydraulic

pressure

Low hydraulic

pressure

FIGURE 29.55 Schematic of servo-valve actuator. (Source: Clark, R.N. Control System Dynamics, Cambridge Univ.

Press, U.K., 1996. With permission.)

Controllable

Valve

Load

FIGURE 29.56 Schematic diagram of variable-orifice

valve oil damper.

Seismic Base Isolation and Vibration Control 29-49

MR fluids typically consist of micron-sized,

magnetically polarizable particles dispersed in a

carrier medium such as mineral or silicone oil.

When a magnetic field is applied to the fluid,

particle chains form and the fluid becomes a

semisolid and exhibits viscoplastic characteristics.

Transition to rheological equilibrium can be

achieved in a few milliseconds. A schematic

diagram of the controllable fluid damper is

shown in Figure 29.57.

29.3.4.3.3 Controllable Friction Dampers

Various controllable friction devices have been

proposed to dissipate vibratory energy in a

structural system. Akbay and Aktan (1990, 1991)

and Kannan et al. (1995) proposed a variablefriction

device in which the force at the frictional

interface was adjusted by allowing slippage in

controlled amounts. A similar device was also

studied by Cherry (1994) and Dowdell and Cherry

(1994a, 1994b).

A recent work by He et al. (2003) studied a

semiactive electromagnetic friction damper

(SAEMFD) for controlling seismic responses.

Figure 29.58 shows a schematic diagram of the

SAEMFD. The device consists of a friction pad

sandwiched between two steel plates. These three layers are slot-bolted together so that sliding takes place

between the steel plates and the friction pad. The friction force between steel plates and the friction pad

depends on the coefficient of friction ðmÞ and the normal force NðtÞ: Two insulated solenoids are installed

on the outer surfaces of the steel plates and the electric current in these solenoids is regulated such that an

electromagnetic attractive force exists between the two solenoids. Hence, the normal force NðtÞ between

the steel plates is directly proportional to the square of the current in the solenoids.

29.3.4.4 Hybrid Control System

The hybrid control methods which consist of both passive and active devices have been proposed and

implemented, utilizing the merits of both passive and active methods and avoiding the demerits of these

methods. Thus, higher levels of performance may be achievable. Additionally, the resulting hybrid

control system can be more reliable than a fully active system, although it is also often somewhat more

complicated.

One example of a hybrid system is a TMD with actuators that is put between the TMD mass and its

support so that the effectiveness of the TMD is increased by this technique. Figure 29.59 shows a

schematic diagram of a passive TMD, an active AMD (active mass damper), and a hybrid ATMD (active

TMD). Another example of hybrid control is a combination of base isolation with some form of active

control to limit excessive displacement (Fujii et al., 1992; Kageyama, M. and Yasui, 1992; Feng et al., 1993;

Reinhorn and Riley, 1994).

29.3.5 Control Algorithm

29.3.5.1 Active Control System

The most important part of active control is the algorithm, because the control forces are based on this.

Research efforts in active structural control have been focused on a variety of control algorithms based

Piston

Wires to

electromagnet

MR fluid

Magnetic

Choke Accumulator

Rod

Load

FIGURE 29.57 Schematic diagram of magnetorheological

damper. (Source: Spencer, B.F. et al. J. Eng.

Mech., 123, 1997. With permission.)

Friction Pad

Slotted Bolts

Upper Plate

Lower Plate

+ N(t) N(t) +

− −

FIGURE 29.58 Schematic drawing of semiactive

electro-magnetic friction damper. (Source: He, W.L.

et al. J. Struct. Eng., ASCE, 129, 941 – 950, 2003. With

permission.)

29-50 Vibration and Shock Handbook

on several control design criteria. An example of a famous control algorithm is linear optimal control

because all of the control design parameters can be determined for multiinput and multioutput systems.

Also, the control allows us to formulate directly the performance objectives of a control system. The

adjective optimal above means that a control system can be designed to meet the desired performance

objectives with the smallest control energy, which is the energy associated with generating the control

inputs. Such a control system that minimizes the cost associated with generating control inputs is called

an optimal control system. The optimal control system directly addresses the desired performance

objectives, while minimizing the control energy, by formulating an objective function that must be

minimized in the design process.

If the transient energy of a system is the total energy of the system when it is undergoing the transient

response, then the successful control system must have the capability to decay quickly the transient

energy to zero. By including the transient energy and the control energy in the objective function, both

parameters can be minimized. The objective function for the optimal control problem is a time integral

of the sum of transient energy and control energy expressed as a function of time.

The general, optimal control formulation for regulators can be explained as follows. Consider a

structure (Figure 29.60) under dynamic loading that is represented by Equation 29.33:

Mx€ðtÞ þ Cx_ðtÞ þ KxðtÞ ¼ DuðtÞ þ EfðtÞ ð29:33Þ

where M, C, K are, respectively, the n £ n mass, damping, and stiffness matrices, and xðtÞ is the ndimensional

displacement vector, f ðtÞis an r-vector representing applied load or external excitation, and

uðtÞ is the m-dimensional control force vector. The n £ m matrix, D, and n £ r matrix, E, are location

matrices that define locations of the control force and the excitation, respectively.

The equation can be rewritten using the state-space representation in the form

z_ðtÞ ¼ AzðtÞ þ BuðtÞ þ Hf ðtÞ; zð0Þ ¼ z0 ð29:34Þ

where

zðtÞ ¼

xðtÞ

x_ðtÞ

" #

ð29:35Þ

m2 m2

Spring Damper

Auxiliary Mass

Passive

Actuator

Auxiliary Mass Auxiliary Mass

Controller

Sensor

Active

Controller

Sensor

Hybrid

m1 m1

FIGURE 29.59 Schematic diagram of a passive, an active, and a hybrid mass damper system.

Seismic Base Isolation and Vibration Control 29-51

is the 2n-dimensional state vector

A ¼

0 I

2M21K 2M21C

" #

ð29:36Þ

is the 2n £ 2n system matrix, and

B ¼

M21D

" #

and H ¼

M21E

" #

ð29:37Þ

are 2n £ m and 2n £ r location matrices specifying, respectively, the locations of controllers and external

excitation in the state space. In Equation 29.36, 0 and I denote the null matrix and the identity matrix of

appropriate dimensions, respectively.

For simplicity, assume that we have a linear time invarying plant, as shown in Equation 29.34 above,

and suppose we would like to design a full-state feedback regulator for the plant such that the control

input vector is given by

uðtÞ ¼ 2GxðtÞ ð29:38Þ

where G is a feedback gain matrix. The control law given by the above equation is linear. Since the plant is

also linear, the closed-loop control system is linear. The control energy, CE, can be expressed as

CE ¼ uTðtÞRuðtÞ ð29:39Þ

where R is a square, symmetric matrix called the control cost matrix. Such an expression for control

energy is called a quadratic form, because the scalar function in Equation 29.39 contains quadratic

functions of the elements of uðtÞ: The transient energy, TE, can also be expressed in a quadratic form as

TE ¼ xTðtÞQxðtÞ ð29:40Þ

where Q is a square, symmetric matrix called the state weighting matrix. The objective function can then

be written as follows:

Jðt; tf Þ ¼

ðtf

t ½xTðtÞQxðtÞ þ uTRuðtÞ􀀉dt ð29:41Þ

where t and tf are the initial and final times, respectively; that is, the control begins at t ¼ t and ends at

t ¼ tf ; where t is the variable of integration.

cn kn

   





   





M =

...

   





   





−cn

−c2

c1 + c2

C =

... ...

   





   





−kn

−k2

k1 + k2

K =

... ...

FIGURE 29.60 Typical civil structure with its dynamic properties.

29-52 Vibration and Shock Handbook

The optimal control problem consists of solving for the feedback gain matrix, G, such that the

scalar objective function, Jðt; tf Þ; given by Equation 29.41, is minimized. Hence, the optimal control

problem solves a regulator gain matrix, G, which minimizes Jðt; tf Þ; subject to the constraint given by

Equation 29.34.

MATLABw Control System Toolbox (MathWorks, 1998) provides a function “lqr” for the solution of

the linear optimal control problem shown above (also see Appendix 32A). By using this command, the

gain matrix, G, can easily be obtained. For a numerical simulation, the matrix, G, is used for obtaining

optimal forces, uðtÞ; as shown in Equation 29.38. The MATLABw Control System Toolbox user manual is a

good reference (MathWorks, 1998) for applying this command. For a real application in a structure, the

optimal forces, uðtÞ; must be converted to real forces by actuating devices. Actuating devices are covered

in Section 29.3.4.

In the above explanation, the linear optimal controller has been derived with full-state feedback that

minimizes a quadratic objective function. The controller robustness to process and measurement noise

can only be indirectly ensured by iterative techniques. There are more advanced topics in modern control

that directly address the problem of robustness by deriving controllers that maintain system response and

error signals to within prescribed tolerances. One example is the H1 (pronounced H-infinity) optimal

control design technique. The reader may refer to control design textbooks, such as that written by

Tewari (2002).

29.3.5.2 Semiactive Control System

Because of the intrinsically nonlinear nature of semiactive control devices, development of control

strategies that are practically implementable and can fully utilize the capabilities of these unique

devices is an important and challenging task. Various nonlinear control strategies have been developed

to take advantage of the particular characteristics of semiactive devices, including bang-bang control

(Mukai et al., 1994; McClamroch and Gavin, 1995), clipped optimal control (Dyke et al., 1996a),

bistate control (Patten et al., 1994), fuzzy control methods (Sun and Goto, 1994), and adaptive

nonlinear control (Kamagata and Kobori, 1994). Caughey (1993) proposed a variable stiffness system

that employed a semiactive implementation of the Reid (1956) spring as a structural element. He et al.

(2001) proposed a resetting semiactive stiffness damper used for controlling seismically excited cablestayed

bridges. Iemura and Pradono (2003) introduced a pseudonegative stiffness control algorithm

used for producing artificially rigid – perfectly plastic force – deformation hysteretic loops by using

controllable damper.

29.3.5.2.1 Common Control Schemes for Controllable Dampers

An example will be given here for common control schemes for a controllable damper. Examples of a

controllable damper are variable-orifice damper and MR fluid dampers. Both types of dampers are

covered in Section 29.3.4. The strategy of a clipped-optimal control algorithm (Dyke et al., 1996a, 1996b)

for seismic protection using MR fluid dampers is as follows. First, an “ideal” active control device is

assumed, and an appropriate primary controller for this active device is designed. Then a secondary bangbang-

type controller causes the smart damper to generate the desired active control force, so long as this

force is dissipative. The primary controller can be one of active control algorithms shown above. For the

general smart damping device, the secondary control strategy is given by

fsa;i ¼

fa;i; fa;i £ x_dev , 0

0; otherwise

(

ð29:42Þ

where fsa,i is the control force of the ith MR fluid damper, fa,i is the desired control force for the ith device,

and x_dev is the velocity across the ith damper. Since the controllable damper is an energy-dissipative

device that cannot add mechanical energy to the structural system, special care must be taken in the

design of the primary controller so that the desired control force, fa,i, is dissipative during the majority of

the seismic event.

Seismic Base Isolation and Vibration Control 29-53

29.3.5.2.2 Special Control Schemes for Controllable Damper

The term “special” here refers to control schemes that have certain objectives for the hysteretic loops of

the controllable dampers. The control schemes are not centralized, so that each damper is controlled

separately in one structure or, in other words, one controller is for one damper. The advantage is that

should one controller malfunction, this controller will not affect the other controller.

One example is given by He et al. (2001). A resetting semiactive stiffness damper (RSASD) is used to

control the peak seismic response of a structure. The RSASD consists of a cylinder-piston system with an

on – off valve in the bypass pipes connecting two sides of the cylinder. Basically, the damper is similar to a

variable-orifice oil damper as shown in Figure 29.58; however, the orifice valve is replaced by an on – off

valve. This damper serves as a stiffness element in which the stiffness is provided by the bulk modulus of

the fluid in the cylinder when the valve is closed. When the valve is open, the piston is free to move, and

the hydraulic damper provides only a small damping, without stiffness.

Such a stiffness damper can be operated in the resetting mode. During the operation in this mode, the

valve is always closed. The energy is then stored in the hydraulic oil of the damper in the form of potential

energy. At an appropriate time, the valve is pulsed to open and close quickly. At that moment, the piston

is at the resetting position, and the energy stored in the hydraulic damper is released and converted into

the head loss across the damper. Hence, by pulsing the valve at appropriate times, structural response can

be reduced by drawing energy from the system (He et al., 2001).

Another example is that of Kurino et al. (2003). They presented a device developed for an actual

application whose system employs a decentralized control algorithm that uses information only from

built-in sensors.

The pseudonegative stiffness control algorithm (Iemura and Pradono, 2003) is also intended for

controlling a variable damper based only on the displacement and velocity sensors located within the

damper. The purpose is to control the variable damper’s hysteresis loop. An example of the application of

this control algorithm to a bridge model is shown in Section 29.3.7.2.

29.3.6 Experimental Performance Verification

29.3.6.1 Shaking Table Tests of a Flexible Structural Model with TMD, AMD,

and ATMD

For practical implementation of TMD, AMD, and

ATMD for structures, it is important to find the

efficiency of each control system for random

excitations. The author has made an analytical

and numerical studies on the efficiency of different

control methods. To verify the results, a three-DoF

structural frame model (Figure 29.61) with and

without control devices is tested a on shaking table

at Kyoto University, Japan (Iemura et al., 1992).

Natural periods for modes 1, 2, and 3 are 0.6578,

0.2580, and 0.1568 sec, respectively. The relevant

participation factors for each mode are 1.2204,

0.3493, and 20.1341, respectively. The moving

mass and other mass of the TMD are 3.5 and

5.5 kg, respectively. The spring constant is

0.581 kgf/cm. The damping ratio is 25.06%.

These properties are used for the experiment.

The masses of the TMD consist of the AC

servo-motor, moving mass, driving guides, and

velocity meter. At the time of the experiment of the

TMD, the moving mass was fixed and the TMD

Shaking Table

Velocity Meter

Velocity

Meter

Velocity Meter

Moving

Mass

Motor

FIGURE 29.61 Three-DoF experimental model with

TMD/ATMD.

29-54 Vibration and Shock Handbook

was hung from the third floor. In order to work as the hybrid type ATMD, the moving mass was driven by

the motor. For the pure active control experiment, the motor and the moving mass are set directly on the

third floor.

It was clearly found that the second mode response is not reduced by the TMD but is effectively

reduced by the AMD and ATMD. The TMD is effective only in the first mode frequency range, while

active control force can cover a wide frequency range.

It was also found that the control force of the ATMD is much lower than that of AMD, especially in the

first mode frequency range. The first mode response is reduced mainly by the TMD and the second mode

response is reduced by the AMD. This verifies the energy efficiency of ATMD. This is the reason that the

ATMD concept is now popularly used for practical application.

29.3.6.2 Substructure Hybrid Test

Before introducing additional dampers to a structure for seismic-response reduction, the precise

frequency-dependent properties of the damper should be obtained from the damper. These properties will

be used for numerical simulation on the effectiveness of the damper in reducing seismic responses. It is

quite difficult to work out the appropriate model for a specified device because of the existence of strong

nonlinearity. For a more reliable experimental test of structures with structural control devices, so-called

“substructure hybrid experiment” techniques have been developed.

The term “substructure hybrid” implies a technique that combines the device loading experiment and

numerical simulation of structural response. Why should it be separated? It is because civil structures are

relatively large and expensive to construct in a laboratory. Therefore, only the inelastic part is tested

experimentally. The elastic part, which is easier to model, is numerically simulated on a computer. Both

results are combined in real time at every time step of the simulation. Up until now, various kinds of test

methods have been proposed. Most of them can be classified into three categories from the viewpoint of

the loading equipment.

29.3.6.2.1 Hybrid Tests Using Hydraulic Actuator

Hydraulic actuators are commonly used for loading experiments. They are advantageous for testing

specimens that needs large excitation force and displacements (Tanzo et al., 1992; Igarashi et al., 1993;

Igarashi, 1994; Williams and Blakeborough, 2001). Various algorithms and techniques have been

proposed in order to conduct precise, real-time experiments such as the “operator splitting numerical

integration scheme,” which is suitable for online controlled experiments (Nakashima, 1993). A

compensation method based on extrapolation is proposed by Horiuchi and Konno (2001) for the

response delay of the actuator. Similar feedforward-based compensation methods are widely used for the

numerical algorithm’s development and real-time testing (Nakashima and Masaoka, 1999; Nakashima

et al., 1999; Blakeborough et al., 2001).

29.3.6.2.2 Hybrid Tests Using Shaking Table

Substructure hybrid loading test systems have been developed for shaking table equipment. Since most

shaking tables are driven by hydraulic actuators, algorithms as well as technologies for the hydraulic

actuators system are directly applicable to the shaking table test systems. Iemura et al. (2002) introduced

the inverted digital filter of the shaking table for compensating its dynamics, and conducted a real-time

hybrid experiment using the electromagnetic mass damper installed in the nonlinear structure. The

shaking table test is applicable to a test specimen such as the TMD.

29.3.6.2.3 Hybrid Tests Using Inertia-Force-Driven Loading System

This is a newly developed method for hybrid loading test (Toyooka, 2002; Iemura et al., 2003). The

system consists of a large size mass, rubber and roller supports, and active mass driver. The schematic

figure of the system is shown in Figure 29.62 and the property of the system is in Table 29.5. The active

mass driver is attached to the mass. The test specimen is attached to the mass and the ground. The mass

Seismic Base Isolation and Vibration Control 29-55

(i.e., the concrete slab) can be excited with large displacement, velocity, and acceleration by making use of

the shaker (Iemura et al., 2003). The Inertia-Force-Driven Loading (IFDL) system was developed to allow

an economical and accurate loading environment for energy dissipation devices to characterize the

dynamic properties and to comprehend the performances of these devices under the realistic loading

conditions.

29.3.7 Implementations

29.3.7.1 Semiactive Control of Full-Scale Structures Using Variable Joint

Damper System

29.3.7.1.1 Background of Study

In order to verify the effectiveness of the application of the semiactive control technique to the joint

damper system (JDS), seismic response control tests using full-scale multistory steel-frame structures,

excitation devices, and a variable damper are performed at the Disaster Prevention Research Institute,

Kyoto University, Japan. The variable damper allows external control of damping force by the electric

servo-valve that regulates the oil flow through the cylinder/piston mechanism. The test results show that

the variable damper was successfully controlled with high accuracy, as well as having the advantage of JDS

application of the semiactive control in reducing the dynamic response of structures over the

FIGURE 29.62 Test set up of the IFDL system.

TABLE 29.5 Properties of the IFDL System

Weight of slab mass 23.853 tonf

Total stiffness 344.43 kN/m

Total damping 6.32 kN/m/sec

Natural frequency 0.55 Hz

Equivalent damping 3.86%

Stroke limit ^ 10 cm

29-56 Vibration and Shock Handbook

conventional passive control. A comparison of the semiactive control algorithms to JDSs in terms of

the feasibility and the advantage in the engineering application is also based on the test results.

Extensive research on the semiactive control approach has been conducted in order to reduce the

seismic response of structures, induced especially by strong earthquake ground motions. The JDS, which

aims to achieve the dynamic response reduction of adjacent structures through the use of connection

devices with energy-absorbing capability, has been considered a promising approach to establish effective

semiactive structural systems for earthquakes.

The purpose of this study is to verify experimentally the effectiveness of the application of semiactive

control to the JDS. Seismic response control tests using full-scale multistory frame structures, excitation

devices, and a variable damper are performed at the Disaster Prevention Research Institute, Kyoto

University. The variable damper allows for external control of the damping force using the electric servovalve

that regulates the oil flow through the cylinder/piston mechanism.

Two types of semiactive control algorithms were employed, namely the linear quadratic regulator

(LQR) control theory and the newly proposed pseudonegative stiffness control for JDSs. Parametersetting

strategies for the algorithms are studied prior to the tests through numerical simulations based on

the modeling of the full-scale steel frame structures and the control device used in the tests.

29.3.7.1.2 Test System Set Up

As shown in Figure 29.63, the test structure of the JDS consists of two full-scale structural steel frames: a

five-story frame (1 £ 2 span) and a three-story frame (1 £ 1 span). The dimensions of both frames are

shown in Table 29.6. Natural frequencies are also shown in the table.

In this test system, mass-driver devices are used to reproduce the vibration conditions under both

sinusoidal and real earthquake inputs. One-directional horizontal earthquake excitation is applied.

Although three mass-driver systems can be seen in the figure, two of them are used at the same time. The

accuracy of the simulated response using the mass driver devices has been verified by a series of research

Damper

Control PC

Shaker

Control PC

Shaker

Velocity Sensors Variable Damper

Data

Acq. PC

FIGURE 29.63 Schematic diagram of joint damper system.

Seismic Base Isolation and Vibration Control 29-57

conducted prior to this study. Velocities, relative displacements, and absolute accelerations of all floors

are measured through instrumentation. All measured responses are sent to a digital signal processingbased

system for the feedback control.

As the control device in the JDS, a variable damping device (variable damper) is used in the test system.

The variable damper is installed at the third story of the five-story frame so as to connect the two frames.

The mechanism of the variable damper is similar to that shown in Figure 29.56. It is a semiactive

hydraulic damper consisting of a cylinder/piston mechanism filled with oil, double rods that connect the

frames, a by-pass pipe that contains a flow control valve, and an accumulator that keeps the by-pass line

pressure constant. The opening ratio of the flow control valve can be changed by a servo-controller using

an external signal. The flow volume through the valve can be regulated to control the pressure loss. The

delay time for the opening ratio control is sufficiently short to allow real-time control.

29.3.7.1.3 Control Algorithm

In this study, three types of control algorithms are used: linear viscous damper control, LQR control

(discussed in Section 29.3.5), and pseudonegative stiffness (PNS) control. The linear viscous damper

control algorithm is intended to be the reference response in the case of passive control, and the

effectiveness of the semiactive control is demonstrated by comparing the other two cases with the linear

viscous damper control case.

In linear viscous control algorithm, the damping force demanded of the variable damper, FdðtÞ; is

FdðtÞ ¼ CcvrðtÞ ð29:43Þ

where Cc is the connecting damping coefficient and vr is the relative velocity of the damper position. Realtime

control of the valve opening ratio is required to generate the demanded control force, Fd, with the

variable damper, even for this simplest control algorithm.

The LQR control theory is used as a semiactive control algorithm in this study, as extensively used in

past studies. In the LQR control algorithm, the optimal control force, Fd, is regarded as the demanded

force and the variable damper is controlled to track the demanded force as close as possible within the

constraint, depending on the piston velocity. The control gain parameters are chosen on the basis of

numerical simulation in consideration of the capacity of the variable damper. The control force is

calculated in the following manner:

FdðtÞ ¼ 2GxðtÞ ð29:44Þ

where G is the optimal gain matrix given by the LQR theory and xðtÞ is the state vector for the structural

frames.

If the main purpose of a JDS is the response reduction of the upper floors in the adjacent structures,

the most interesting feature of the system is obtained by connecting them at lower stories with a negative

stiffness element. Although this characteristic has been reported theoretically and analytically in many

studies, there are many problems left in applying the active control device at the present time, mainly

owing to the difficulty in realizing the negative stiffness with a passive control device. However,

semiactive devices such as the variable damper can generate an apparent negative stiffness by controlling

the damping. Therefore, taking into account JDS and negative stiffness, a new, simple control algorithm

to realize pseudonegative stiffness with a damping element is proposed in this study. To generate the

TABLE 29.6 Test Frames and Mass-Drivers

Five-Story Three-Story

Height (m) 17.22 10.65

Weight (tonf) 163.1 62.1

First mode natural frequency (Hz) 1.78 2.41

Mass driver 5 ton mass at the fourth floor 2 ton mass at third floor

29-58 Vibration and Shock Handbook

negative stiffness, the demanded force of variable damper Fd is defined as follows:

FdðtÞ ¼ KcxrðtÞ þ CcvrðtÞ ð29:45Þ

where Kc and Cc are the connecting stiffness (negative value) and the damping coefficient, respectively.

The relative displacement and velocity of the damper piston, respectively, are xr and vr. It follows that this

algorithm simulates the state in which the two frames are connected by a negative stiffness and a positive

damping elements. The eigenvalue analysis is used to determine the value of Kc and Cc.

This pseudonegative stiffness algorithm has a great advantage in practical application. Most of the

previously proposed control algorithms require direct measurement of the structural system to produce

feedback and to calculate the demanded control force. That means that a considerable number of sensors

should be installed in the object structure. Considering practical application, it is difficult to install many

sensors because of the economical disadvantage. On the other hand, for the pseudonegative stiffness

algorithm, only the relative displacement and velocity are used for feedback and a sensor is required only

at the damper. In addition to the simplicity in installation, the required parameters are limited to the

connecting stiffness and damping.

29.3.7.1.4 Test Results

The response of variable damper to sinusoidal input with 1.8 Hz (max 10 gal), which is approximately

the first resonance frequency in the connected state, is shown in Figure 29.64. The five-story top-floor

velocity response in the LQR control theory and pseudonegative stiffness are smaller than that in the

viscous damper; especially; the peak value in pseudo negative stiffness is improved at 25% compared with

that in the viscous damper, though the velocities at the top story of the three-story frame are almost equal

for all control algorithms. On the other hand, the LQR control theory can reduce the peak response of

both frames as compared with the viscous damper. When the object is to moderate the response of total

system, the LQR theory is the most effective algorithm of the three being compared. Based on the test,

semiactive controls based on the LQR control theory and pseudonegative stiffness can reduce the peak

responses of the total system more effectively than viscous damper-type passive control.

For the earthquake excitation (El-Centro 1940 N – S and Kobe 1995 N – S, scaled to 20 gal max), the

influence of the friction of the variable damper appears in the dynamic characteristics of the variable

damper in every control algorithm because of the relatively small responses. The relative velocity and

displacement responses of the variable damper in both semiactive controls are larger than that in the

viscous damper. Judging from the test result, it is confirmed that the variable damper is controlled

effectively in the different control algorithm for real earthquake inputs. With respect to the velocity

response of the top story of the three-story frame, the responses are not very different when different

control algorithms are used. On the other hand, for the five-story top-floor response, both

semiactive controls can reduce the response more effectively than the passive control to both real

FIGURE 29.64 Variable damper responses in the sinusoidal excitation test: (a) at the top story of five-story fame;

(b) at the top story of three-story fame.

Seismic Base Isolation and Vibration Control 29-59

earthquake excitations. Semiactive controls are more effective than passive control to reduce the response

of the top story of the five-story frame in the earthquake excitation cases.

Figure 29.65 shows the hysteretic loops resulting from viscous-type, LQR-type, and pseudonegative

stiffness-type controls under El-Centro 1940 N – S, scaled to 20 gal max. It is obvious that the

pseudonegative hysteretic loop can be achieved experimentally by using a variable damper.

29.3.7.2 Application of Structural Control Technologies to Seismic Retrofit

of a Cable-Stayed Bridge

29.3.7.2.1 Background of Study

Owing to severe damage to bridges caused by the Hyogo-ken Nanbu earthquake in 1995, very high

ground accelerations (level II design) are now required in the new bridge design specification set in 1996,

in addition to the relatively frequent earthquake motion (level I design) by which old structures were

designed and constructed. Hence, the seismic safety of cable-stayed bridges that were built before the

present specification has to be reviewed and seismic retrofit has to be done, if it is found necessary.

In order to study the effectiveness of passive and semiactive control on the seismic retrofit of a

cable-stayed bridge, numerical analyses on a model is carried out. An existing cable-stayed bridge that has

fixed-hinge connections between the deck and towers is modeled and its connections are replaced with

isolation bearings and dampers. The isolation bearings are assumed to be elastic or hysteretic type. The

dampers are linear and variable type. The objective is to increase the damping ratio of the bridge by using

passive and semiactive control technologies. The calculation of the structural damping ratio at the main

mode is feasible, as the passive or semiactive control method produces certain hysteretic loops under

FIGURE 29.65 (a) Viscous-type control, (b) LQR-type control, and (c) pseudonegative stiffness-type control under

El-Centro 1940 N – S scaled to 20 gal max.

29-60 Vibration and Shock Handbook

harmonic motion, and the main mode has an effective modal mass that is larger than 90% of the total

mass. SSI effects on the structural damping ratio are also studied.

The Tempozan Bridge (Hanshin Highway Public Corporation, 1992), built in 1988, is a three-span

continuous steel cable-stayed bridge that is situated on reclaimed land and crosses the mouth of the Aji

River, Osaka, Japan. The total length of the bridge is 640 m with a center span of 350 m, and the lengths

of the side spans are 170 and 120 m (Figure 29.66). The main towers are A-shaped to

improve the torsional rigidity. The cable in the superstructure is a two-plane fan pattern multicable

system with nine stay cables each plane. The bridge is supported on a 35 m thick soft layer and the

foundation consists of cast-in-place RC piles of 2 m in diameter. The main deck is fixed at both towers to

resist horizontal seismic forces. The bridge is relatively flexible, with a predominant period of 3.7 sec. As

to the seismic design in the transverse direction, the main deck is fixed at the towers and the end piers.

Figure 29.67 shows the original design spectrum used for designing the bridge and the new

design spectrum specified in the bridge design specification set in 1996 for level I and level II

earthquakes (Japan Roadway Association, 1996). A level II earthquake has type I (interplate type)

FIGURE 29.66 Side view of the Tempozan Bridge.

100

1000

10000

0.1 1 10

Natural Period (sec.)

Absolute Acceleration (gal)

New Design Spectrum Level II (Type I)

New Design Spectrum Level II (Type II)

New Design Spectrum Level I

Original Design Spectrum

FIGURE 29.67 Design spectra for bridges.

Seismic Base Isolation and Vibration Control 29-61

and type II (intraplate type). As can be seen in the figure, the new design spectrum shows higher

acceleration response in all period ranges than the original one.

29.3.7.2.2 Basic Concept of Seismic Retrofit

If the deck is connected with very flexible bearings to the towers, the induced seismic forces will be kept to

minimum values, but the deck may have a large displacement response. On the other hand, a very stiff

connection between the deck and the towers will result in a lower deck displacement response but will

attract much higher seismic forces during an earthquake. This is the case in the original bridge structure,

the Tempozan Bridge. Therefore, it is important to replace the existing fixed-hinge bearings with special

bearings or devices at the deck-tower connection both to reduce seismic forces and to absorb large

seismic energy and reduce the response amplitudes. Additionally, energy-absorbing devices may also be

put between the deck-ends and piers; however, this will attract relatively large lateral force of the piers,

and therefore this kind of method has been avoided for this bridge at this time.

The bridge model that represents the existing Tempozan Bridge is termed the “original bridge model.”

The bridge model with the spring and damper (viscous, hysteretic, and semiactive) between the deck

and the towers is termed the “retrofitted bridge model.” The original and retrofitted bridge models are

shown in Figure 29.68. The original structure system has fixed-hinge connections between the towers and

the deck and rollers connection between the deck-ends and piers, so that the deck longitudinal

movement is constrained by the towers (Figure 29.68a). For the retrofitted bridge, the isolation bearings

and dampers connect the deck to the towers (Figure 29.68b).

The cables are modeled by truss elements. The towers and deck are modeled by beam elements, and the

isolation bearings are modeled by spring elements. The models were analyzed by a commercial finite

element program (Prakash and Powell, 1993). The moment – curvature relationship of the members is

calculated based on the sectional properties of members and material used.

29.3.7.2.3 Modal Shape Analysis

The first modes of the structures are interesting here because these modes have the largest contribution to

the longitudinal movement of the bridge (also see Chapters 3 and Chapter 4). The mode shapes of the

original bridge and the retrofitted bridge are shown in Figure 29.69. The first mode shape of the original

structure is shown in Figure 29.69a. The natural period ðTÞ of this mode is 3.75 sec (frequency ¼ 0.266

Hz), which is close to the design value for the bridge (3.7 sec, frequency ¼ 0.270 Hz; Hanshin Highway

Public Corporation, 1992). This first mode shape has the effective modal mass as a percentage of the total

mass of 84%.

For the retrofitted structure, the stiffness of the bearings is an important issue, as large stiffness produces

a large bearing force. However, very flexible connections produce a large displacement response.

Therefore, based on a study on a simplified model of the bridge under seismic motion, the bearing stiffness

(a) Fixed Hinge

Isolation Bearings + Passive or

(b) Semi Active Dampers

FIGURE 29.68 Cable-stayed bridge models: (a) original structure system; (b) retrofitted structure system.

29-62 Vibration and Shock Handbook

that produces retrofitted main period ðT0Þ 1.7 times the original main period ðTÞ was chosen. This bearing

stiffness makes the energy-absorbing devices work well in reducing seismic-induced force and

displacement. The main natural period ðT0Þ of the retrofitted bridge then becomes 6.31 sec and the

effective modal mass as a percentage of total mass is 92%.

It is clear from the figures that smaller curvatures are found at the towers and the decks of the

retrofitted structure than in the original structure. This shows that the retrofitted structure is expected to

produce smaller moments at the towers and the decks than the original structure during a seismic attack.

29.3.7.2.4 Time-History Analysis

The models were analyzed by a commercial finite element program (Prakash and Powell, 1993), which

produces a piece-wise dynamic time history using Newmark’s constant average acceleration ðb ¼ 1=4Þ

integration of the equations of motion, governing the response of a nonlinear structure to a chosen base

excitation. The input earthquake motions were type I-III-3, I-III-2, and I-III-1 earthquakes, which are

artificial acceleration data used for design in Japan for soft soil condition. Those data are intended to be

type I (interplate type). With numerical comparison (Figure 29.67), type I earthquake motion gives

higher effect to the bridge than type II motion, in longer period range.

Table 29.7 shows the seismic response effects because of different kinds of bearings and dampers: fixedhinge

bearings for the original bridge model; elastic bearings, elastic bearings plus viscous dampers, and

hysteretic bearings for the retrofitted bridge models. The input earthquake was type I-III-3 earthquake

data and was in the longitudinal direction.

From the table, it is clearly seen that if only elastic bearings are used for seismic retrofit, then the

sectional forces are reduced to about 40% of the original ones. However, the displacement response is

increased to 176% of the original one. By adding viscous dampers to the elastic bearings, the sectional

forces can be reduced to about 25% of the original ones, and the displacement response is reduced to 63%

of the original. Thus, the viscous dampers together with bearings work to reduce the seismic response of

retrofitted bridge. The structural damping ratio is calculated as 35%.

If hysteretic bearings are used for seismic retrofit, the sectional forces are reduced to about 29% of the

original ones and the displacement response is reduced to 67% of the original one. The equivalent

structural damping ratio is calculated as 13.1% by using pushover analysis to obtain a hysteretic loop at

the main mode. The hysteretic bearings are modeled by a bilinear model, and the second stiffness of the

hysteretic bearings is 0.03 times the initial stiffness and produces a first mode natural period of 6.31 sec.

1st mode shape,

frequency = 0.266 Hz

1st mode shape,

frequency = 0.158 Hz

(a)

(b)

Flexible deck-tower connections

Fixed deck-tower connections

FIGURE 29.69 First mode shapes of (a) original structure and (b) retrofitted structure.

Seismic Base Isolation and Vibration Control 29-63

29.3.7.2.5 Soil – Structure Interaction Effect

One method to study the SSI effects is to take into

account the effects of flexible foundations and the

radiation of energy from foundations. In this

method, the cable-stayed bridge is idealized as in

Figure 29.70 (Kawashima and Unjoh, 1991). The

subsoil supporting the foundation was assumed to

be an elastic half space. The subsoil was assumed to

be elastic with no energy dissipation. The foundation

was idealized as a rigid massless circular

plate. The radius of the rigid circular plate was

simply assumed so that it gives the same surface area as the foundation. Dynamic stiffness of the

foundation was assumed in a frequency independent form:

Kx ¼

8Gsa

2 2 y

Cx ¼

pGsa2

Vs ð29:46Þ

Kr ¼

8Gsa3

3ð1 2 y Þ

Cr ¼

0:25p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p2ð1 2y Þ=ð1 2 2y ÞGsa4

Vs ð29:47Þ

in which Kx and Cx represent the spring and damping coefficient for sway motion, and Kr and Cr

represent those for rocking motion. Vs and a represent shear wave velocity of subsoils and the radius of

foundation, respectively.

The result shows that SSI increases the natural period and the damping ratio of the original structure.

However, the damping ratio of the retrofitted structure is reduced and the effectiveness of the bearings

and dampers in reducing seismic responses is also reduced (Table 29.8). This is mainly because the SSI

model introduces flexibility at the base. A flexible base will reduce the frequency of the structure.

A smaller frequency will reduce the effectiveness of viscous damping devices in absorbing earthquake

energy. Moreover, a flexible base will increase the elastic strain energy of the structure that reduces the

damping ratio. If the SSI model possesses an elemental damping ratio, as is a usual case for the soil, the

structural damping ratio will also be influenced by the SSI-model damping characteristics.

29.3.7.2.6 Semiactive Control

The semiactive control herein uses the pseudonegative stiffness control algorithm (Iemura and Pradono,

2003) so that the sum of the damper force and bearing force (plus other connecting stiffness forces)

are expected to produce a hysteresis loop that is as close as possible to that of rigid – perfectly plastic

TABLE 29.7 Maximum Earthquake Responses and Damping Ratios in Longitudinal Direction

Items Original

Structure

Retrofitted Structure

Elastic Bearings Elastic Bearings þ Viscous Damping Hysteretic Bearings

Deck displacement (m) 2.37 4.17 1.50 1.58

Tower momenta (MN m) 3,100 2,000 900 900

Tower axial forcea (kN) 48,000 15,000 15,000 21,000

Cable force (kN) 24,000 3,440 4,000 5,000

Bearing forceb (kN) 94,000 44,000 17,000 25,000

Deck moment (MN m) 370 95 75 95

Deck axial force (kN) 56,000 21,000 11,000 15,000

Damping ratio (%) 2 2 35 13.1

Natural period (sec) 3.75 6.31 6.73 3.86 and 6.31c

a Base of tower AP3.

b At connection between deck and tower AP3.

c Initial and postyield stiffness.

a elastic half space a

K Cx r

FIGURE 29.70 Cable-stayed bridge model with flexible

foundation and energy radiating from foundation.

29-64 Vibration and Shock Handbook

force – deformation characteristics (Figure 29.71a).

Moreover, no residual displacement is expected

at the bearings after an earthquake attack,

because the hysteresis loop is velocity dependent.

Figure 29.71 shows ideal and realistic force –

deformation characteristics of the variable

damper that can produce artificial rigid – perfectly

plastic force – deformation characteristics by using

variable damper.

One algorithm that can approach the hysteretic

loop in Figure 29.71b requires the following

variable-damper force (Iemura et al., 2001):

Fd;t ¼ Kdut þ Cdu_ t ð29:48Þ

where Kd is connecting stiffness (negative value) and Cd is damping coefficient (positive value). The

algorithm is practical because only displacement and velocity sensors are placed in the dampers.

Therefore, each damper can have its own controller. Should a malfunction happen in one damper or

controller, it will not affect the other dampers or controllers.

This algorithm produces the hysteretic loop shown in Figure 29.72b under harmonic motion. It is clear

from the figure that the variable damper is superior to the linear viscous damper, because the

maximum variable damper plus the connecting-stiffness force can be set to be equal to the maximum

connecting-stiffness force (Figure 29.72b). One can calculate that the damping ratio of the hysteresis loop

in Figure 29.72b is 53.4%. For the same damping ratio, the hysteresis loop in Figure 29.72a produces a

total force 1.46 times larger than the connecting-stiffness force (Iemura and Pradono, 2003). The

connecting stiffness between the deck and the tower of the retrofitted cable-stayed bridge comes from the

contribution of cable stiffness, upper tower stiffness, and bearing stiffness.

The cable-stayed bridge model with isolation bearings and variable dampers controlled with the

pseudonegative stiffness algorithm was analyzed by a program developed by the authors under the

MATLAB (MathWorks, 2000) and SIMULINK (MathWorks, 1999) environments. The program

produces a piece-wise dynamic time-history, using Newmark’s constant average acceleration ðb ¼ 1=4Þ

integration of the equations of motion, governing the response of a nonlinear structure to a chosen base

excitation. The input motions were type I-III-1, I-III-2, and I-III-3 earthquakes, which are artificial

acceleration data used for design in Japan (Japan Roadway Association, 1996).

The results show that the application of the pseudonegative stiffness control algorithm is effective in

reducing seismic response of the bridge model. Figure 29.73 shows the base shear-deck displacement

TABLE 29.8 Maximum Earthquake Responses and Damping Ratios (SSI Included)

Items Original Structure Retrofitted Structure

Elastic Bearings þ Viscous Damping Hysteretic Bearings

Deck displacement (m) 2.78 2.57 2.77

Tower momenta (MN m) 1,500 800 882

Tower axial forcea (kN) 36,200 12,500 19,800

Cable force (kN) 12,300 3,010 4,470

Deck moment (MN m) 228 58 86

Deck axial force (kN) 31,900 9,100 12,300

Foundation displacement (m) 0.171 0.079 0.093

Damping ratio (%) 3.1 23 9.3

Natural period (sec) 5.04 7.66 5.13 and 7.46b

a Base of tower AP3.

b Initial and postyielding stiffness.

(b)

Connecting

Stiffness

Variable

Damper

Total

(a)

FIGURE 29.71 (a) Ideal and (b) realistic hysteretic

loops produced by variable damper.

Seismic Base Isolation and Vibration Control 29-65

Connecting

stiffness

Linear damping F

(a)

Variable damping

(b)

Variable damping +

connecting stiffness

Linear damping +

connecting stiffness

FIGURE 29.72 Hysteresis loops for (a) linear viscous damping and (b) pseudonegative stiffness damping.

−80000

− 40000

40000

80000

−3.0 −2 .0 −1.0 0.0 1.0 2 .0 3.0

Deck Displacement (m)

Base Shear (kN)

−80000

−40000

40000

80000

−3.0 −2 .0 −1.0 0.0 1.0 2.0 3 .0

Deck Displacement (m)

Base Shear (kN)

(a) (b)

FIGURE 29.73 Base shear vs. deck displacement relationship of a cable-stayed bridge model with (a) linear dampers

(b) pseudonegative stiffness dampers (type I-III-1 earthquake).

−40000

−20000

20000

40000

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

Bearing Displacement (m)

Damping Force (kN)

−40000

−20000

20000

40000

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

Bearing Displacement (m)

Damping Force (kN)

(a) (b)

FIGURE 29.74 Damping force vs. damping displacement relationship of a cable-stayed bridge model with (a) linear

dampers, (b) pseudonegative stiffness dampers (type I-III-1 earthquake).

29-66 Vibration and Shock Handbook

relationship for both bridges, with a linear damper and pseudonegative stiffness damper, respectively,

under type I-III-1 earthquake input. The bridge model with pseudonegative stiffness dampers shows

lower base shear than that of the bridge with linear damper.

Figure 29.74 shows the hysteretic loops produced by both linear dampers and pseudonegative stiffness

dampers (at tower AP3). The damping force produced by the pseudonegative stiffness damper is larger

than that of the linear damper. However, the total force of damping plus the isolation bearing is lower for

the pseudonegative stiffness damper (Figure 29.75). Therefore, the base shear of the cable-stayed bridge

model is lower for the pseudonegative stiffness dampers.

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