Điều khiển kết cấu - Chương 6 pdf

421
Part II: Active Control
Chapter 6
Introduction to active structural motion
control
6.1 The nature of active structural control
Active versus passive control
The design methodologies presented in the previous chapters provide systematic
procedures for distributing passive motion control resources which, by definition,
have fixed properties and do not require an external source of energy. Once
installed, a passive system cannot be modified instantaneously, and therefore one
needs a reliable estimate of the design loading and an accurate numerical model
of the physical system for any passive control scheme to be effective. The inability
to change a passive control system dynamically to compensate for an unexpected
loading tends to result in an over-conservative design. When self-weight is an
important design constraint, one cannot afford to be too conservative. Also,
simulation studies on the example building structures show that passive control
is not very effective in fine tuning the response in a local region. Considering
these limitations, the potential for improving the performance by dynamically
modifying the loading and system properties exists. An active structural control
system is one which has the ability to determine the present state of the structure,
decide on a set of actions that will change this state to a more desirable one, and
carry out these actions in a controlled manner and in a short period of time. Such
control systems can theoretically accommodate unpredictable environmental
422 Chapter 6: Introduction to Active Structural Motion Control
changes, meet exacting performance requirements over a wide range of operating
conditions, and compensate for the failure of a limited number of structural
components. In addition, they may be able to offer more efficient solutions for a
wide range of applications, from both technical and financial points of view.
Active motion control is obtained by integrating within the structure a
control system consisting of three main components: a) monitor, a data

acquisition system, b) controller, a cognitive module which decides on a course of
action in an intelligent manner, and c) actuator, a set of physical devices which
execute the instructions from the controller. Fig 6.1 shows the interaction and
function of these components; the information processing elements for active
control are illustrated in Fig 6.2. This control strategy is now possible due to
significant recent advances in materials that react to external stimuli in a non-
conventional manner, sensor and actuator technologies, real-time information
processing, and intelligent decision systems.
Fig. 6.1: Components of an active control system
STRUCTURE
RESPONSE
EXCITATION
SENSORS
SENSORS
ACTUATOR
AGENTS TO CARRY
OUT INSTRUCTIONS
IDENTIFY THE STATE OF SYSTEM
DECIDE ON COURSE OF ACTION
DEVELOP THE ACTION PLAN
(SET OF INSTRUCTIONS
TO BE COMMUNICATED
TO THE ACTUATORS)
CONTROLLER
MONITOR
to measure
external
loading
MONITOR
to measure

response
6.1 The Nature of Active Structural Control 423
Fig. 6.2: Information Processing Elements for an Active Control System
Fig. 6.3: Passive and active feedback diagrams
Physical System
Sensor
Sensor
Sensor
Data
Processing
Transmission channel
Processing
Modeling &
Analysis
Decision
Making
Action
Visualization
Archival
and
Access
Fusion
hp()
pu
(a) Passive
h' p ∆p
e
∆p
f
++()

u
(b) Active
observe u
u ε
u
+
decide on ∆p
f
+
∆p
f
p
observe p
p ε
p
+
decide on ∆p
e
+
∆p
e
decide on changing
htoh'
424 Chapter 6: Introduction to Active Structural Motion Control
The simple system shown in Fig. 6.3 is useful for comparing active and
passive control. Figure 6.3(a) corresponds to passive control. The input, , is
transformed to an output, , by the operation
(6.1)
One can interpret this system as a structure with denoting the loading, the
displacement, and the flexibility of the structure. The strategy for passive

motion control is to determine such that the estimated output due to the
expected loading is contained within the design limits, and then design the
structure for this specific flexibility.
Active control involves monitoring the input and output, and adjusting the
input and possibly also the system itself, to bring the response closer to the
desired response. Figure 6.3(b) illustrates the full range of possible actions.
Assuming the input corrections and system modifications are introduced
instantaneously, the input-output relation for the actively controlled system is
given by
(6.2)
Monitoring the input, and adjusting the loading is referred to as open-loop control.
Observing the response, and using the information to apply a correction to the
loading is called feedback control. The terminology closed-loop control is
synonymous with feedback control.
In addition to applying a correction to the input, the control system may also
adjust certain properties of the actual system represented by the transformation
. For example, one can envision changing the geometry, the connectivity,
and the properties of structural elements in real time. One can also envision
modifying the decision system. A system that can adjust its properties and
cognitive processes is said to be “adaptive”. The distinguishing characteristic of
an adaptive system is the self-adjustment feature. Non-adaptive active structural
control involves monitoring and applying external forces using an invariant
decision system. The make up of the structure is not changed. Adaptive control is
the highest level of active control.
The role of feedback
Feedback is a key element of the active control process. The importance of
feedback can be easily demonstrated by considering a linear static system and
p
uhp()
uhp()=

pu
h
hp()
uh' p ∆p
e
∆p
f
++()=
hp()
6.1 The Nature of Active Structural Control 425
taking the input correction to be a linear function of the output. For this case,
(6.3)
(6.4)
where and are constants. Substituting in eqn (6.2) specialized for h’=h and
solving for results in
(6.5)
When is positive, the sensitivity of the system to loading is increased by
feedback, i.e. the response is amplified. Taking negative has the opposite effect
on the response. Specializing eqn (6.5) for negative feedback ( ), the response
becomes
(6.6)
Increasing decreases the effect of external loading. However, the influence of
, the noise in the response observation, increases with and, for sufficiently
large , is essentially independent of the feedback parameter. This result
indicates that the accuracy of the monitoring system employed to observe the
response is an important design issue for a control system.
Computational requirements and models for active control
The monitor component identified in Figs 6.1 and 6.2 employs sensors to
measure a combination of variables relevant to motion such as strain,
acceleration, velocity, displacement, and other physical quantities such as

pressure, temperature, and ground motion. This data is usually in the form of
analog signals which are converted to discrete time sequences, fused with other
data, and transmitted to the controller module. Data compression is an important
issue for large scale remote sensing systems. Wavelet based data compression
(Amaratunga, 1997) is a promising approach for solving the data processing
problem.
The functional requirements of the controller are to compare the observed
response with the desired response, establish the control action such as the level
of feedback force, and communicate the appropriate commands to the actuator
uhp=
∆p
f
k
f
u ε
u
+[]=
hk
f
u
u
h
1 hk
f
–

p ∆p
e
+[]
hk

f
1 hk
f
–

ε
u
+=
k
f
k
f
k
f
0<
u
h
1 hk
f
+

p ∆p
e
+[]
hk
f
1 hk
f
+

ε
u
+=
k
f
ε
u
k
f
k
f
426 Chapter 6: Introduction to Active Structural Motion Control
which then carries out the actual control actions such as apply force or modify a
structural property. The controller unit is composed of a digital computer and
software designed to evaluate the input and generate the instructions for the
actuators.
There are 2 information processing tasks: state identification and decision
making. Given a limited amount of data on the response, one needs to generate a
more complete description of the state of the system. Some form of model
characterizing the spatial distributions of the response and data analysis are
required. Once the state has been identified, the corrective actions which bring the
present state closer to the desired state can be established. In this phase, a model
which defines the input - output relationship for the structure is used together
with an optimization method to decide upon an appropriate set of actions.
For algorithmic non-adaptive systems, the decision process is based on a
numerical procedure that is invariant during the period when the structure is
being controlled. Time invariant linear feedback is a typical non-adaptive control
algorithm. An adaptive controller may have, in addition to a numerical control
algorithm, other symbolic computational models in the form of rule-based
systems and neural networks which provide the capability of modifying the

structure and control algorithm in an intelligent manner when there is a change in
the environmental conditions. Examples illustrating time invariant linear
feedback control algorithms are presented in the following sections; a detailed
treatment of the algorithms is contained in Chapters 7 and 8.
6.2 An introductory example of quasi-static feedback control
Consider the cantilever beam shown in Fig 6.4. Suppose the beam acts like
a bending beam, and the design objective is to control the deflected shape such
that it has constant curvature. The target displacement distribution corresponding
to this constraint has the form
(6.7)
where is the desired curvature. One option is to select the bending rigidity
according to
(6.8)
u
*
x()
1
2

χ
*
x
2
=
χ
*
D
B
x()
Mx()

χ
*
=
6.2 An Introductory Example of Quasi-static Feedback Conrol 427
where M(x) is the moment at location x due to the design loading. This strategy is
a stiffness based passive control approach. A second option is to select a
representative bending rigidity distribution, and apply a set of control forces
which produce a displacement distribution that, when combined with the
displacement due to the design loading, results is a displacement profile that is
close to the desired distribution. In what follows, the latter option is discussed.
Fig. 6.4: Cantilever beam with control force
Suppose the control force system consists of a single force applied at x=L.
Assuming linear elastic behavior, and using the linear technical theory of beams
as the model for the structure, the displacement due to F is estimated as
(6.9)
where is the bending rigidity, considered constant in this example. The
displacement due to the design loading is also determined with the technical
beam theory. This term is denoted as , and expressed as
(6.10)
Combining the 2 displacement patterns results in the total displacement, u(x).
(6.11)
The expanded form of eqn (6.11) corresponding to the particular choice of control
force location for this example is
(6.12)
The difference between and is defined as and interpreted
L
u(x)
x
F
u

c
x()
F
2D
B

Lx
2
x
3
3
–


F
D
B

hx()==
D
B
u
o
x()
u
o
x()
1
D
B

gx()=
ux() u
o
x() u
c
x()+=
ux()
1
D
B

gx() Fh x()+[]=
ux() u
*
x() ex()
428 Chapter 6: Introduction to Active Structural Motion Control
as the displacement error.
(6.13)
For this example, is considered to be fixed, and therefore is a function
only of the single control force magnitude F.
(6.14)
Ideally, one wants for . This goal cannot be achieved, and it is
necessary to work with a relaxed condition.
The simplest choice is collocation, which involves setting equal to zero at
a set of prescribed locations. For example, setting at x=L leads to
(6.15)
A more demanding condition is a least square requirement, which involves
first forming the sum of evaluated at a set of prescribed points, and then
selecting F such that the sum is a minimum. The continuous least square sum is

given by the following integral
(6.16)
Taking J(F) as the measure of the square error sum, F is determined with the
stationary condition
(6.17)
Differentiating the integral expression for J,
(6.18)
and using eqn (6.14), which defines for this particular example, results in
(6.19)
The value of F defined by eqn (6.19) produces the absolute minimum value of J. A
proof of this statement is presented in Section 7.2 which treats in more detail the
ex() u
o
u
c
u
*
–+=
D
B
ex()
ex()
1
D
B

gx() Fh x()+[]u
*
x()–=
e 0= 0 xL≤≤

e
e 0=
F
D
B
u
*
L() gL()–
hL()
=
e
2
J
1
2

e
2
xd
0
L
∫
JF()==
F∂
∂J
0=
F∂
∂J
e
F∂

∂e
xd
0
L
∫
0==
ex()
F
hx()D
B
u
*
x() gx()–[]xd
0
L
∫
hx()()
2
xd
0
L
∫
=
6.2 An Introductory Example of Quasi-static Feedback Conrol 429
least square procedure for quasi-static loading.
Example 6.1: Shape control for uniform loading
This example illustrates the application of the approach described above to
the case where the design loading is a uniform distributed load extending over
the entire length of the beam. The corresponding deflected shape is
(1)

Applying collocation at x=L leads to
(2)
The least square solution is
(3)
Both solutions are approximate since they do not satisfy . One can
improve the performance by taking additional control forces. Selecting the spatial
distribution of the control forces is a key decision for the design of a control
system.
The above discussion assumes that there is some initial loading, and one
can determine the corresponding displacement field with the physical model of
the structural system. This control strategy is similar to the concept of
prestressing. A more general scenario is the case where one is observing the
response at a set of “observation” points and the loading is being applied
gradually so that there is negligible dynamic amplification and sufficient time to
u
o
x()
1
D
B

gx()
1
D
B

w
24

x

4
4x
3
L– 6x
2
L
2
+()==
F
eL() 0=
3
2



D
B
χ
*
L



3
2



wL–=
F

ls
91
66



D
B
χ
*
L

2065
5280



wL–=
1.379()
D
B
χ
*
L

0.391()wL–=
ex() 0=
430 Chapter 6: Introduction to Active Structural Motion Control
adjust the control forces. Here, one needs to establish using the observed
displacement data.

Suppose there are observation points located at ( j= 1, 2, , s), and at
time t the monitoring system produces the data set . This data can be
used together with an interpolation scheme to generate an estimate of for the
region adjacent to the observation points. A typical spatial interpolation model
has the form
(6.20)
where are interpolation functions.
Given , one forms the displacement error,
(6.21)
and determines F(t) with either collocation or a least square method. The
continuous least square estimate for F(t) is given by
(6.22)
Example 6.2: Discrete displacement data
Suppose the displacement observation points are located at x=L/2 and
x=L. Given these 2 values of displacement, one needs to employ an interpolation
scheme in order to estimate . Taking a quadratic expansion,
(1)
u
o
x()
sx
j
u
o
x
j
t,()
u
o
u

o
xt,() u
o
x
j
t,()Ψ
j
x()
j 1=
s
∑
=
Ψ
j
x()
u
o
xt,()
ext,()u
o
xt,()u
*
x()–
1
D
B

hx()Ft()+=
Ft() D
B

hx()u
*
x() u
o
xt,()–()xd
0
L
∫
hx()()
2
xd
0
L
∫

=
u
o
x()
u
o
x() a
o
a
1
xa
2
x
2
++=

6.3 An Introductory Example of Dynamic Feedback Control 431
and specializing eqn(1) for points 1 and 2 results in the following approximation,
(2)
The expression for the control force is obtained using eqn (6.22) and eqn
(2). Evaluating the integrals leads to
(3)
as an estimate for F.
6.3 An introductory example of dynamic feedback control
To gain further insight on the nature of feedback control, the simple SDOF system
shown in Fig. 6.5 is considered. The system is assumed to be subjected to both an
external force and ground motion, and controlled with the force . Starting with
the governing equation,
(6.23)
and introducing the definitions for frequency and damping ratio leads to the
L/2
u
1
u
2
L/2
x
u
o
x() u
1
4
x
L




4
x
L



2
– u
2
x
L



– 2
x
L



2
++≈
F
91
66



D

B
χ
*
L

D
B
L
3

98
33

u
1
133
66

u
2
+


–≈
F
mu
˙˙
cu
˙
ku++ ma

g
– Fp++=
432 Chapter 6: Introduction to Active Structural Motion Control
Fig. 6.5: Single-degree-of-freedom system.
standardized form of the governing equation
(6.24)
The free vibration response of the uncontrolled system has the general
form
(6.25)
Substituting for in eqn (6.24), one obtains two possible solutions
(6.26)
(6.27)
Considering and to be complex conjugates,
(6.28)
where and are real numbers representing the real and imaginary parts of
, the solution takes the form
(6.29)
k
c
m
uu
g
+
F
u
g
p
u
˙˙
2ξωu

˙
ω
2
u++ a
g
–
F
m

p
m
++=
uAe
λt
=
u
uA
1
e
λ
1
t
A
2
e
λ
2
t
+=
λ

12,
ξω– iω 1 ξ
2
–±ξω– iω′±==
A
1
A
2
A
12,
1
2

A
R
iA
I
±[]=
A
R
A
I
A
u e
ξωt–
A
R
ωt 1 ξ
2
–



cos A
I
ωt 1 ξ
2
–


sin+=
6.3 An Introductory Example of Dynamic Feedback Control 433
One determines and with the initial conditions for and . The
resulting expressions are
(6.30)
Considering negative linear feedback, the control force is expressed as a
linear combination of velocity and displacement
(6.31)
where the subscripts ‘v’ and ‘d’ refer to the nature of the feedback, i.e. velocity or
displacement. Feedback is implemented in the actual physical system by:
• observing the response
• determining and
• calculating with eqn (6.31)
• applying with an actuator
Mathematically, one incorporates feedback by substituting for in eqn (6.24). The
result is
(6.32)
Equation (6.32) can be transformed to the standardized form by defining
equivalent damping and frequency parameters as follows:
(6.33)
(6.34)

With this notation, the solution for the free vibration response of the linear
feedback controlled case has the same general form as for no control; one just
replaces and with and respectively in eqn (6.29). It follows that the
effect of linear feedback is to change the fundamental frequency and damping
A
R
A
I
uu
˙
A
R
u 0()=
A
I
1
ω′

u
˙
0() ξωu 0()+()–=
Fk
v
u
˙
– k
d
u–=
uu
˙

F
F
F
u
˙˙
2ξω
k
v
m
+


u
˙
ω
2
k
d
m
+


u++ a
g
–
p
m
+=
ω
eq

2
ω
2
k
d
m
+=
2ξ
eq
ω
eq
2ξω
k
v
m
+=
ξω ξ
eq
ω
eq
434 Chapter 6: Introduction to Active Structural Motion Control
ratio. Solving eqns (6.33) and (6.34) for and , results in
(6.35)
(6.36)
where is the increment in damping ratio due to active control
(6.37)
Critical damping corresponds to
(6.38)
Equation (6.35) shows that negative displacement feedback increases the
frequency. According to eqn (6.37), the damping ratio is increased by velocity

feedback and decreased by displacement feedback. If the objective of including
active control is to limit the response amplitude, velocity feedback is the
appropriate mechanism. Displacement feedback is destabilizing in the sense that
it reduces the effect of damping. Stability and other issues associated with linear
feedback are discussed in Chapter 8.
Example 6.3: Illustrative example - influence of velocity feedback
This example demonstrates the influence of velocity feedback on the response of 2
SDOF systems subjected to seismic excitation. The properties of the systems are
System 1:
ω
eq
ξ
eq
ω
eq
ω 1
k
d
k
+=
ξ
eq
ξξ
a
+=
ξ
a
ξ
a
1

1
k
d
k
+
12⁄

k
v
2ωm
ξ 1
k
d
k
+
12⁄
1–



–=
ξ
eq
1=
k
v
2mω

ξ
eq

1=
1
k
d
k
+ ξ–=
m 10 000kg,= ω
1
6.32rd s⁄=
k 400 000N m⁄,= T
1
0.99s=
c 2 500Ns m⁄,=
6.3 An Introductory Example of Dynamic Feedback Control 435
System 2:
The models are excited with the El Centro and Taft accelerograms scaled to
. Figures 6.6 through 6.8 contain plots of the maximum relative
displacement, maximum control force magnitude, and maximum power
requirement. The power requirement is computed using the following expression
(1)
which assumes the control force is a set of self-equilibrating forces applied as
shown in Fig 6.5. Ground motion has no effect on the work done by F with this
force scheme.
m 10 000kg,= ω
1
2rd s⁄=
k 40 000N m⁄,= T
1
3.14s=
c 830Ns m⁄= ξ 0.0208=

a
max
0.5g=
Power force velocity× Fu
˙
==
436 Chapter 6: Introduction to Active Structural Motion Control
Fig. 6.6: Variation of maximum displacement with active damping.
0.05 0.1 0.15 0.2 0.25 0.3
0.04
0.06
0.08
0.1
0.12
0.14
0.16
ξ
a
Maximum displacement - m
Taft
El Centro
System 1
T
1
=0.99s
0.05 0.1 0.15 0.2 0.25 0.3
0.1
0.15
0.2
0.25

0.3
0.35
Maximum displacement - m
ξ
a
El Centro
Taft
System 2
T
1
=3.14s
6.3 An Introductory Example of Dynamic Feedback Control 437
Fig. 6.7: Variation of maximum control force level with active damping.
0.05 0.1 0.15 0.2 0.25 0.3
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
x 10
4
Maximum force - N
ξ
a

El Centro
Taft
System 1
T
1
=0.99s
0.05 0.1 0.15 0.2 0.25 0.3
1000
2000
3000
4000
5000
6000
7000
Maximum force - N
ξ
a
System 2
El Centro
Taft
T
1
=3.14s
438 Chapter 6: Introduction to Active Structural Motion Control
Fig. 6.8: Variation of maximum power requirement with active damping.
0.05 0.1 0.15 0.2 0.25 0.3
2000
4000
6000
8000

10000
12000
14000
16000
Maximum power - N.m/s
ξ
a
El Centro
Taft
System 1
T
1
=0.99s
0.05 0.1 0.15 0.2 0.25 0.3
500
1000
1500
2000
2500
3000
3500
4000
4500
Maximum power - N.m/s
ξ
a
System 2
El Centro
Taft
T

1
=3.14s
6.4 Actuator Technologies 439
6.4 Actuator technologies
Introduction
The actuator component of the control system generates and applies the
control forces at specific locations on the structure according to instructions from
the controller. Over the past several decades, a number of force generation
devices have been developed for a broad range of motion control applications.
These devices can be described in terms of performance parameters such as
response time, peak force, and operating requirements such as peak power and
total energy demand. The ideal device is one that can deliver a large force in a
short period of time for a small energy input.
Civil structures generally require large control forces, on the order of a
meganewton and, for seismic excitation, response times on the order of milli-
seconds. The requirement on peak force coupled with the constraint on energy
demand is very difficult to achieve with a fully active force actuator system. There
are force actuator systems that are capable of delivering large force, but they also
have a high energy demand. Included in this group are hydraulic,
electromechanical, and electromagnetic devices. All these types are based on very
mature technology.
There is considerable on-going research and development of new force
actuators that have a low energy demand. One approach is based on modifying
the physical makeup of the device in such a way that only a small amount of
energy is required to produce a significant increase in force. Varying the orifice
area of a viscous type damper is an example of this strategy. A typical viscous
damper can deliver a force on the order of a meganewton. By adjusting the
damping parameter, the damper can be transformed into a large scale force
actuator.
The second approach employs adaptive materials as the force generating

mechanism. These materials respond to a low energy input by changing their
properties and their state in a nonconventional manner which results in a force.
Although these technologies are promising, the current devices can produce only
low forces, on the order of a kilonewton, and therefore their applicability for civil
structures is limited at this time.
There are 2 issues that need to be addressed: i) how the force generation
mechanism works, and ii) how the forces are applied to the structure. The first
440 Chapter 6: Introduction to Active Structural Motion Control
issue is related to the physical make up and underlying physics of the device. The
second question is concerned with how the device is attached to the structure so
as to produce the “desired” control force. In what follows, the attachment issue is
discussed first, then the state of the art for linear actuator technologies is
reviewed, and lastly some adaptive material based actuators are described.
Force application schemes
The schematic drawing contained in Fig 6.9 shows the typical makeup of
hydraulic, electro-mechanical, and electro-magnetic linear actuators. There are 2
basic elements, a piston and a mechanism that translates the piston linearly either
by applying a force to one end or by moving the end with a gear mechanism. The
interaction of the piston with an adjacent body produces a pair of contact forces F
at the contact point and a corresponding reaction force at the actuator support. If
the body moves under the action of F, the mechanism usually compensates for
this motion such that the force remains constant until instructed by the controller
to change the force magnitude.
Fig. 6.9: Linear actuator
Consider the structural frame shown in Fig 6.10a. Suppose the objective is
to apply a horizontal force at point A and there is no adjacent structure which
could support the actuator. One option (Fig 6.10b) is to fasten a tendon to point A,
pass it over a pulley attached to the base, and then connect it to a linear actuator
which can generate a tensile force in the tendon. In this scheme, the actuator
reaction force is transmitted directly to the base. A second option would be to

place the actuator directly on the structure. The actuator reaction force is now
F
u
Structure
Piston
Mechanism
Actuator Support
F
6.4 Actuator Technologies 441
transmitted to the structure; however, the other end of the piston needs to be
restrained in order to generate a control force. If the restraining body is rigidly
connected to the structure as shown in Fig 6.10c, the force system is self-
equilibrating and the structure “feels” no lateral force. Member AC is in tension.
In order to have a “non zero” lateral force acting on the structure, the restraining
body must be allowed to move laterally. This objective can be achieved by
attaching an auxiliary mass, m
a
, to the piston and supporting the mass on rollers.
The mass moves with respect to the structure with an absolute acceleration equal
to F/m
a
. One specifies the peak force and magnitude of the auxiliary mass, and
designs the actuator so that it can provide the required force at that level of
acceleration. Since the force is generated by driving the mass, this scheme is
referred to as an active mass driver.
Fig. 6.10: Control force schemes
The extension of these schemes to a multi-story structure is shown in Fig.
6.11. A linear actuator placed on a diagonal produces a set of self-equilibrating
forces which impose a shearing action on the particular story to which it is
attached. The other stories experience no deformation since the story shear due to

this actuator is zero. It follows that one needs to incorporate active braces
F
A
F
Tendon
Pulley
A
(a) Active force
(b) Active tendon
(c) Self-equilibrating forces
m
a
F
a
(d) Active mass driver
F
A
C
442 Chapter 6: Introduction to Active Structural Motion Control
throughout the structure in order to achieve global displacement control. Forces
generated with active mass drivers are not self-equilibrating and consequently
have more influence on the global displacement response.
Fig. 6.11: Control force schemes for a multi-story structure
The previous examples relate to shear beam structures which require forces
that act in the transverse direction. For bending beam problems, control force
systems that produce bending moments are required. This action can be obtained
with linear actuators placed on the upper and lower surfaces, as illustrated in Fig
6.12. The region between A and B is subjected to a constant moment equal to F d.
Another scheme is shown in Fig 6.13. The actuator is attached to the beam with
rods that provide the resistance to the piston motion, resulting in the self-

equilibrating system that produces a triangular moment distribution over the
region A-B-C. By combing a number of these actuator-rod configurations, one can
generate a piecewise linear bending moment distribution.
F
3
F
2
F
1
T
1
T
2
T
3
F
3
F
3
F
2
F
2
F
1
F
1
(b) Active mass driver(a) Active brace
6.4 Actuator Technologies 443
Fig. 6.12: Constant moment field

Fig. 6.13: Triangular moment field
F
F
d
a)
AB
F d
F d
b)
A
B
F d
c)
(-)
L
F
a)
F
F/2
F/2
b)
( - )
FL/4
A
B
C
c)
444 Chapter 6: Introduction to Active Structural Motion Control
Linear actuators generate control force systems composed of concentrated
forces. For discrete structures such as frames, this type of force distribution is

appropriate. However, for continuous structures such as beams and plates, a
continuous force distribution is more desirable. One strategy that has been
examined is based on using a adaptive material in the form of a thin plate. Fig
6.14 illustrates this approach for a continuous beam. Plates are attached by epoxy
to the upper and lower surfaces. Applying a voltage to the plate generates a
longitudinal strain. Since the plate is attached to the surface, motion of the plate is
restrained and an interfacial shear stress is generated. This stress produces a
distributed control moment equal to
(6.39)
where is the width of the plate. Spatial and temporal variation of the control
force system is achieved by varying the voltage applied to the plate.
Fig. 6.14: Moment generated by strain actuators
Large scale linear actuators
Referring back to Fig 6.9, a linear actuator can be considered to consist of a
piston and a mechanism which applies a force to the piston and also controls the
motion of the piston. This actuator type is the most widely available and
extensively used, particularly for applications requiring a large force and short
response time. The descriptors hydraulic, electro-mechanical, and electro-
magnetic refer to the nature of the force generation mechanisms. These devices
generally have a high energy demand.
τ t()
m
c
xt,()
m
c
xt,()τb
f
d=
b

f
d
+
m
o
dx
beam
film
dx
τ
τ
6.4 Actuator Technologies 445
Hydraulic systems generate the force by applying a pressure on the face of
a piston head contained within a cylinder. Fig 6.15 illustrates this concept. Fluid is
forced in or out of the cylinder through the orifice at C
1
to compensate for the
piston displacement and maintain a certain pressure. These systems have the
highest force capacity of the linear actuator group, on the order of meganewtons
(Dorey et al., 1996). Precise control movement and force can be achieved with a
suitable control system. Protection against overload is provided by a pressure
relief value. The disadvantages of this type of system are: the requirements for
fluid storage systems, complex valves and pumps to regulate the flow and
pressure, seals, and continuous maintenance. Durability of the seals and the
potential for fluid spills are critical issues.
Fig. 6.15: Schematic cross section view - a hydraulic cylinder (Dorey et al, 1996)
Electromechanical linear actuators generate the force by moving the piston
with a gear mechanism that is driven by an electric motor. The motion, and
therefore the force, is controlled by adjusting the power input to the motor. These
devices are compact in size, environmentally safe, and economical. Figure 6.16

illustrates the various components of a linear electric actuator system (Raco,
www.raco.de). The largest electric actuator that can be ordered “off the shelf” is
rated for 600kN force.