Structural Vibration Control

STRUCTURAL
VIBRATION
CONTROL
Shieh-Kung Huang
黃謝恭
1

Shieh-Kung
Huang
Copyright © 2016 by Pearson Education, Inc. All rights reserved.
REVIEW OF STRUCTURAL DYNAMICS
Chapter Outline
12
CHAPTER 1
1.1 Introduction of Structural Dynamics
1.2 Single-degree-of-freedom systems
1.3 Response of Free Vibration and Harmonic Vibration
1.4 Earthquake Response of Linear Systems
1.5 Response Spectrum
1.6 Earthquake Response of Inelastic Systems
1.7 Energy Concepts in Earthquake Engineering
1.8 Muliti-degree-of-freedom systems
1.9 Free and Force Vibration of MDOF Systems

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1.1 INTRODUCTION OF STRUCTURAL DYNAMICS
13
• Structural Dynamics
Determination of responses of
structures under the effect of dynamic
loading
• Responses
Responses are usually included the
displacement, velocity, and acceleration.
• Dynamic Loading
Dynamic loading is a loading whose
magnitude, direction, sense and point of
application changes in time.
Chapter 1 Review of Structural Dynamics

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1.1 INTRODUCTION OF STRUCTURAL DYNAMICS
14
• (Modeling) Assumption
− Discrete vs. Continuous
− Lumped vs. Distributed
• Dimension
− Structural member
− Finite element
• (Analysis) Domain
− Time
− Frequency
− Time-frequency

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1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS
15
• Simple Structures
We begin our study of structural dynamics with simple
structures; these structures simple because they can be idealized
as a concentrated or lumped mass m supported by a massless
structure with stiffness k in the lateral direction.
• Degrees of Freedom
The number of independent displacements required to define
the displaced positions of all the masses relative to their original
position is called the number of degrees of freedom (DOFs) for
dynamic analysis. Thus we call this simple structure a single-
degree-of-freedom (SDOF) system.
0
mu ku
+ =

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• Damping
The process by which vibration steadily
diminishes in amplitude is called damping. It is
usually represented in a highly idealized manner.
This idealization is therefore called equivalent
viscous damping.
• Damping in Real Structures
− Opening and closing of microcracks
− Friction in connections
− Friction between structure and non-structure
elements
Mathematical description of these components
is almost impossible, so the modelling of damping
in real structures is usually assumed to be
equivalent viscous damping.

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• Sources of Damping Mechanisms
Damping is utilized to characterize the ability of structures to dissipate energy during dynamic
response. Unlike the mass and stiffness of a structure, damping does not relate to a unique physical
process but rather to a number of possible processes.
Courtesy of Elnashai and Sarno, 2015

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• SDOF system
The system considered is shown schematically and It consists of a mass m concentrated at the
roof level, a massless frame that provides stiffness to the system, and a viscous damper (also known
as a dashpot) that dissipates vibrational energy of the system. The beam and columns are assumed to
be inextensible axially.
where the constant c is the viscous damping coefficient, which is a measure of the energy dissipated
in a complete cycle.
0
mu cu ku
+ + =

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• Force–Displacement Relation
The internal force resisting the displacement u is equal and opposite to the external force fS. It is
desired to determine the relationship between the force fS and the relative displacement u associated
with deformations in the structure during oscillatory motion. This force–displacement relation would be
linear at small deformations but would become nonlinear at larger deformations.

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• Linear Elastic System:
− Elastic material
− First-order analysis
• Inelastic System:
− Plastic material
− Higher-order analysis
S
f k u
= 
( , )
S
f f u u
=

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• Equation of Motion
The following figure is the free-body diagram at time t with the mass replaced by its inertia force.
The forces acting on the mass at some instant of time are balanced according to D’Alember’s principle
of dynamic equilibrium. These include the external force p, the elastic (or inelastic) resisting force fS,
the damping resisting force fD, and the inertial force fI.
or
and or ( , )
S D D S
D S S
p f f mu mu f f p
f cu f ku f f u u
− − = + + =
 = = =

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• Mass–Spring–Damper System
We have introduced the SDOF system by idealizing a one-story structure, an approach that
should appeal to structural engineering students. However, the classic SDOF system is the mass–
spring–damper system of the following figure.
or
and or ( , )
D S
D S S
mu cu ku p mu f f p
f cu f ku f f u u
+ + = + + =
 = = =

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Matlab Demonstration (Demo_1_2_A.m)
23
or
and or ( , )
S D D S
D S S
p f f mu mu f f p
f cu f ku f f u u
− − = + + =
 = = =

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• Solution of A Linear SDOF System
The equation of motion for a linear SDF system subjected to external force is the second-order
differential equation derived earlier.
The initial displacement and initial velocity at time zero must be specified to define the
problem completely. Typically, the structure is at rest before the onset of dynamic excitation, so that
the initial velocity and displacement are zero. A brief review of four methods of solution is given in the
following.
− Classical Solution
Complete solution of the linear differential equation of motion consists of the sum of the
complementary solution and the particular solution.
− Duhamel’s Integral
Another well-known approach to the solution of linear differential equations, such as the
equation of motion of an SDOF system, is based on representing the applied force as a sequence
of infinitesimally short impulses.
Duhamel’s integral provides an alternative method to the classical solution if the applied force p(t)
is defined analytically by a simple function that permits analytical evaluation of the integral.
( ) ( ) ( ) ( )
mu t cu t ku t p t
+ + =
(0)
u
(0)
u
0
0
( ) (1 cos ) when 0, (0) , and ( ) 0
n
p
u t t c p p p t
k

= − = = =
 
0
1
( ) ( )sin ( )
t
n
n
u t p t d
m
   

= −


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− Frequency-Domain Method
The Laplace and Fourier transforms provide powerful tools for the solution of linear differential
equations, in particular the equation of motion for a linear SDOF system. Because the two
transform methods are similar in concept, here we mention only the use of Fourier transform,
which leads to the frequency-domain method of dynamic analysis.
− Other Numerical Methods
The preceding three dynamic analysis methods are restricted to linear systems and cannot
consider the inelastic behavior of structures anticipated during earthquakes if the ground shaking
is intense. The only practical approach for such systems involves numerical time-stepping
methods, for example, Newmark-beta method, Runge-Kutta method, or state-space method
(which are presented latter). These methods are also useful for evaluating the response of linear
systems to excitation—applied force p(t) or ground motion—which is too complicated to be defined
analytically and is described only numerically.
1
( ) ( ) ( )
2
i t
u t H P e d

  


−
= 

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Matlab Demonstration (Demo_1_2_B.m)
26
0
0
( ) (1 cos )
when 0, (0) , and ( ) 0
n
p
u t t
k
c p p p t

= −
= = =
p(t)
u(t)

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Matlab Demonstration
27

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28
ode23 is a three-stage, third-
order, Runge-Kutta method. ode45
is a six-stage, fifth-order, Runge-
Kutta method. ode45 does more
work per step than ode23, but can
take much larger steps. For
differential equations with smooth
solutions, ode45 is often more
accurate than ode23. In fact, it may
be so accurate that the interpolant is
required to provide the desired
resolution. That's a good thing.
ode45 is the anchor of the
differential equation suite. The
MATLAB documentation
recommends ode45 as the first
choice. And Simulink blocks set
ode45 as the default solver.

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https://www.mathworks.com/matlabcentral/fileex
change/71007-newmark-beta-method-for-
nonlinear-single-dof-systems

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1.3 RESPONSE OF FREE AND HARMONIC VIBRATION
30
• Undamped Free Vibration
Free vibration is initiated by disturbing the system from its static equilibrium (or undeformed, u(0)
=0) position by imparting the mass some displacement and velocity at time zero.
The time required for the undamped system to complete one cycle of free vibration is the natural
period of vibration of the system, which we denote as Tn, in units of seconds. It is related to the natural
circular frequency of vibration, ωn, in units of radians per second:
2
n
n
T


=
(0)
u
(0)
u

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A system executes several cycles in 1 sec. This natural cyclic frequency of vibration is denoted by
The units of fn are hertz (Hz) [cycles per second (cps)]; fn is obviously related to ωn through
The term natural frequency of vibration applies to both ωn and fn.
By solving the dynamic equilibrium, we can further find the natural circular frequency of vibration
is related to mass and stiffness.
1
n
n
f
T
=
2
n
n
f


=
n
k
m
 =
Tn
n
fn

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• Viscously Damped Free Vibration
Setting p(t)=0 in dynamic equilibrium gives the differential equation governing free vibration of
SDOF systems with damping:
where ζ is the damping ratio or fraction of critical damping as:
The damping coefficient ccr is called the critical damping coefficient because it is the smallest value of
c that inhibits oscillation completely.
2
( ) ( ) ( ) 0 ( ) ( ) ( ) 0
( ) 2 ( ) ( ) 0
n n
c k
mu t cu t ku t u t u t u t
m m
u t u t u t
 
+ + =  + + =
 + + =
cr
cr
2
and 2 2
2
n
n n
c c k
c m km
m c
 
 
= = = = =

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• Underdamped Free Vibration
The time

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• Underdamped Free Vibration
The time
• Typical Damping Ratios
Damping ratios tabulated here are only provided to illustrate that real structures do not possess
inherent damping >15%. From the given data, it should also be clear that the damping ratio depends
on the type of building construction.

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• Comparison between Underdamped and Damped Free Vibration
The time required for the undamped system to complete one cycle of free vibration is changed
because the natural circular frequency of vibration, ωn, is affected by the damping.
This is the natural frequency of damped vibration. The natural period of damped vibration or the
natural frequency of damped vibration, is related to the one without damping by
2
1 where
D n n
k
m
   
= − =
2
2
2
or 1
2
1
n D
D D n
D
T
T f f
 

 

= = = = −
−

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• Attenuation of Motion
Ratio between displacement at an arbitrary time, t, and the one after a period, TD, is independent
of time
and
Hence, the natural logarithm of the above ratio is called logarithmic decrement.
(0) (0) ( )
( ) (0)cos sin
( )
n n D
t T
n
D D
D D
u u u t
u t e u t t e
u t T
 

 

−  
+
= +  =
 
+
 
2 2
2 2
1 1
2
1
( ) 2
where and
( ) 1
n D
T n i
n D
D n i
u t T u
e e T T e
u t T u
 
 
 
 
− −
+
= = = =  =
+ −
2
2
1
2
ln 2 where 1 1
1
i
i
u
u

   

+
= =  = − 
−

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2
( ) ( ) ( ) 0
(0) (0)
( ) (0)cos sin where 1
1
nt n
D D
D
D n
mu t cu t ku t
u u
u t e u t t
 
  

  
−
+ + = 
 
+
= + 
 
 
= −

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• Transient Response and Stead-state Response
The difference between the two is the free response, which decays exponentially with time at a
rate depending on  / n and  ; eventually, the free response becomes negligible, hence we call it
transient response; compare this with no decay for undamped systems. After awhile, essentially the
forced response remains, and we therefore call it steady-state response. It should be recognized,
however, that the largest deformation peak may occur before the system has reached steady state.

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• Transient Response and Stead-state Response
( )
( ) ( )
( )
( ) ( )
0
2
0 0
2 2
2 2
2 2
( ) ( ) ( ) sin
( ) ( cos sin ) cos sin
1 2
where
1 2 1 2
nt
D D
n n
n n n n
mu t cu t ku t p t
u t e A t B t C t D t
p p
C D
k k


   
    
         
−
+ + = 
= + + +
− −
= =
   
   
− + − +
   
   

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• Resonant Response of Viscously Damped System
As noted earlier for undamped systems, the motion becomes unbounded when ω approaches ωn,
as t goes to infinity. However, for damped cases, motion remains bounded to a maximum of 0.5ζ, as
shown in the following figure.

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1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS
41
• Earthquakes in Taiwan
Date
(UTC+8)
Area
Affected
ML Dead
Houses
Destroyed
1916/08/28 Central Taiwan 6.8 16 614
1920/06/05 Hualien 8.3 5 273
1927/08/25 Tainan 6.5 11 214
1935/04/21
Hsinchu,
Taichung
7.1 3,276 17,907
1935/07/17
Hsinchu,
Taichung
6.2 44 1,734
1941/12/17 Chiayi 7.1 360 4,520
1946/12/05 Tainan 6.1 74 1,954
1959/08/15 Pingtung 7.1 16 1,214
1964/01/18 Chiayi, Tainan 6.3 106 10,924
1999/09/21 Island-wide 7.3 2,415 51,711

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• Earthquakes in Taiwan
https://scweb.cwb.gov.tw/zh-tw/page/disaster/

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• Earthquake Excitation
For engineering purposes, the time
variation of ground acceleration is the
most useful way of defining the shaking of
the ground during an earthquake.
Actually, the ground acceleration
governs the response of structures to
earthquake excitation.
Courtesy of USGS https://pubs.usgs.gov/gip/dynamic/fire.html

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North–south component of horizontal ground acceleration recorded at the Imperial Valley
Irrigation District substation, El Centro, California, during the Imperial Valley earthquake of May 18,
1940. The ground velocity and ground displacement were computed by integrating the ground
acceleration.

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• Earthquake–induced Force
In earthquake-prone regions, the principal problem of structural dynamics that concerns structural
engineers is the response of structures subjected to earthquake-induced motion of the base of the
structure.
where ug(t) is the displacement of the ground
ut(t) is the total (or absolute) displacement (of the mass)
The concept of dynamic equilibrium is used. From the free-body diagram including the inertia
force fI, the equation of dynamic equilibrium is
( ) ( ) ( )
t
g
u t u t u t
= +
0 and ( ) ( ) ( )
( ) ( ) ( ) ( ) or ( ) ( ) ( ( ), ( )) ( )
t
I D S I g
g g
f f f f mu t mu t mu t
mu t cu t ku t mu t mu t cu t f u t u t mu t
+ + = = = +
 + + = − + + = −

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The ground motion can therefore be replaced by the effective earthquake force (indicated by the
subscript “eff”):
eff ( ) ( )
g
p t mu t
= −
Courtesy of Wikiwand https://www.wikiwand.com/en/Seismic_base_isolation

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The above equation governs the motion (or the response) of a linear SDOF system subjected to
ground acceleration.
Dividing this equation by mass m gives
When the responses are evaluated, please know the responses are:
− Absolute (or total) responses
− Relative responses (to ground)
− Relative responses (to other points)
eff
( ) ( ) ( ) ( ) ( )
g
mu t cu t ku t p t mu t
+ + = = −
2 2
( ) ( ) ( ) ( )
( ) 2 ( ) ( ) ( ) or ( ) 2 ( ) ( ) 0
g
t
n n g n n
c k
u t u t u t u t
m m
u t u t u t u t u t u t u t
   
+ + = −
 + + = − + + =

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• Response History
The following figure shows the deformation response of SODF systems to El Centro ground
motion.

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• Concept of Response Spectrum
A plot of the peak value of a response quantity as a function of the natural vibration period Tn of
the system, or a related parameter such as circular frequency ωn or cyclic frequency fn, is called the
response spectrum for that quantity.

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1.5 RESPONSE SPECTRUM
50
• Response Spectrum
A plot of the peak value of a response quantity as a function of the natural vibration period Tn of
the system, or a related parameter such as circular frequency ωn or cyclic frequency fn, is called the
response spectrum for that quantity.
A variety of response spectra can be defined depending on the response quantity that is plotted.
Consider the following peak responses:
The deformation response spectrum is a plot of deformation against Tn for fixed ζ . A similar plot for
velocity is the relative velocity response spectrum, and for total acceleration is the acceleration
response spectrum.
For engineering purposes, the relative velocity response spectrum is replaced by the pseudo-
velocity response spectrum and the acceleration response spectrum is replaced by the pseudo-
acceleration response spectrum.
0
0
0
( , ) max ( , , )
( , ) max ( , , )
( , ) max ( , , )
n n
t
n n
t
t t
n n
t
u T u t T
u T u t T
u T u t T
 
 
 




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The procedure to determine the deformation response spectrum.
0
0
0
( , ) max ( , , )
( , ) max ( , , )
( , ) max ( , , )
n n
t
n n
t
t t
n n
t
u T u t T
u T u t T
u T u t T
 
 
 




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Courtesy of Estrada and Lee, 2008

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The response spectrum for El Centro ground motion with various damping ratios.

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Courtesy of Chopra, 2020

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Courtesy of Chopra, 2020

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The mean spectra with probability distributions for the construction of elastic design spectrum.

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• Pseudo Response Spectrum
Considering the peak responses, the spectral displacement, Sd, spectral velocity, Sv, and spectral
acceleration, Sa, can be defined as
And, the relative velocity response spectrum is replaced by the pseudo response spectrums and the
can be defined as
if and only if ζ is small.
0
0
0
( , ) max ( , , )
( , ) max ( , , )
( , ) max ( , , )
d n n
t
v n n
t
t t
a n n
t
S u T u t T
S u T u t T
S u T u t T
 
 
 
 
 
 
2
max ( , , )
max ( , , ) named as
max ( , , ) named as
d n
t
v n n d
t
t
a n n d
t
S u t T
S u t T S PSV
S u t T S PSA

 
 

 
 

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Courtesy of Estrada and Lee, 2008

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https://www.mathworks.com/matlabcentral/fileexchange/78029-elastic-response-spectra

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https://www.mathworks.com/matlabcentral/fileexchange/50843-response-spectrum

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1.6 EARTHQUAKE RESPONSE OF INELASTIC SYSTEMS
61
• Elastoplastic Idealization
Consider the force–deformation relation for a structure during its initial loading shown in the
following figure. It is convenient to idealize this curve by an elastic–perfectly plastic (or elastoplastic for
brevity) force–deformation relation because this approximation permits the development of response
spectra in a manner similar to linearly elastic systems.
where fy is the yield strength
uy is the yield deformation
um is the maximum displacement
μ is the ductility
m
y
u
u
 =

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• Dissipated Energy
The input energy imparted to an inelastic system by an earthquake is dissipated by both viscous
damping and yielding. The various energy terms can be defined by integrating the equation of motion
of an inelastic system, as follows:
where EK(t) is the kinetic energy of the mass associated with its motion relative to the ground
ED(t) is the energy dissipated by viscous damping
ES(t) is the recoverable strain energy of the system (k is the initial stiffness)
EY(t) is the energy dissipated by yielding of the system
EI(t) is the energy input to the structure since the earthquake excitation
Concurrent with the earthquake response analysis of a system these energy quantities can be
computed conveniently by rewriting the integrals with respect to time. Thus
0 0 0 0
( ) ( ) ( ( ), ( )) ( )
( ) ( ) ( ( ), ( )) ( )
( ) ( ) ( ( ) ( )) ( )
g
u u u u
g
K D S Y I
mu t cu t f u t u t mu t
mu t du cu t du f u t u t du mu t du
E t E t E t E t E t
+ + = −
 + + = −
 + + + =
   
 
 
2
0 0
2
0 0
( )
( ) ( ) , ( ) ( ) ,
2
( )
( ) , ( ) ( ( ), ( )) ( ), and ( ) ( )
2
u u
K D
u u
S
S Y S I g
m u t
E t mu t du E t cu t du
f t
E t E t f u t u t du E t E t mu t du
k
= = =
= = − = −
 
 
 
2
0 0
( ) ( ) , ( ) ( ) ( ( ), ( )) ( )
t t
D Y S
E t c u t dt E t u t f u t u t dt E t
= = −
 

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Time variation of energy dissipated by viscous damping and yielding, and of kinetic plus strain
energy; (left) linear system, (right) elastoplastic system
0 0 0 0
( ) ( ) ( ( ), ( )) ( )
( ) ( ) ( ( ) ( )) ( )
u u u u
g
K D S Y I
mu t du cu t du f u t u t du mu t du
E t E t E t E t E t
+ + = −
+ + + =
   

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1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING
64
Seismic energy formulation natural way to understand effect of supplemental energy dissipation
device and seismic isolation systems.
Main advantages of energy formulation:
− replacement of vector quantities (displacements, velocities and accelerations) by scalar energy
quantities
− flow of energy quantities can be tracked during seismic response
• Rain Flow Analogy
During seismic shaking
Courtesy of Filiatrault, A., Christopoulos, C. 2006, ‘Principles of passive supplemental
damping and seismic isolation’

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At the end of seismic shaking
( ) ( ) ( )
I D Y
in d k
E t E t E t
V V V
= +
= +

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Using supplemental energy dissipation device

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Using seismic isolation systems

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68
• Supplemental Energy Dissipation Device
If part of this energy could be dissipated through supplemental devices that can easily be
replaced, as necessary, after an earthquake, the structural damage could be reduced. Such devices
may be cost-effective in the design of new structures and for seismic protection of existing structures.
Available devices can be classified into three main categories: fluid viscous and viscoelastic dampers,
metallic yielding dampers, friction dampers, and tuned mass dampers.
− Fluid Viscous Dampers
− Viscoelastic Dampers
Courtesy of G. Alotta, L. Cavaleri, M. Di Paola, and M.F. Ferrotto, 2016, ‘Solutions for the Design and Increasing of
Efficiency of Viscous Dampers’
Courtesy of Jiangsu ROAD Damping Technology CO., Ltd.

Shieh-Kung
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69
− Metallic Yielding Dampers
Buckling-Restrained Brace (BRB)
Courtesy of K. Ramadevi and A. Abinayaa, 2017,
‘Buckling Restrained Braces (BRB) in framed
structures as Structural Fuses in Seismic Regions – A
Review’
Courtesy of U.S. General Services Administration

Shieh-Kung
Huang
70
− Friction Dampers
Slotted Bolted Connection (SBC);
Courtesy of Rozlyn K. Bubela, Carlos Ventura, and
Helmut G.L. Prion, 2010, ‘Cyclic Testing of Steel
Chevron Braces with Vertically Slotted Beam Connection’

Shieh-Kung
Huang
71
Theoretical Behavior of Different Types of Dampers
Friction dampers Metallic yielding dampers
Viscoelastic dampers Fluid viscous dampers

Shieh-Kung
Huang
72
− Tuned Mass Dampers
Courtesy of Gebrail Bekdaş, Sinan MelihNigdeli, 2011, ‘Estimating optimum
parameters of tuned mass dampers using harmony search’
From Wikimedia Commons

Shieh-Kung
Huang
73
− Tuned Mass Dampers

Shieh-Kung
Huang
74
− Seismic Isolation Systems

Shieh-Kung
Huang
1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS
75
• Simple System: Two-story Shear Building
We first formulate the equations of motion for the simplest possible muliti-degree-of-freedom (MDOF)
system, a highly idealized two-story frame subjected to external forces p1(t) and p2(t). In this idealization
the beams and floor systems are rigid (infinitely stiff) in flexure, and several factors are neglected: axial
deformation of the beams and columns, and the effect of axial force on the stiffness of the columns. This
shear-frame or shear-building idealization, although unrealistic, is convenient for illustrating how the
equations of motion for an MDF system are developed.
Similar with Chapter 1.2, we can develop the dynamic equilibrium as:
1
1 1 1 1
2
2 2 2 2
or
0
0
j Sj Dj j j j j Dj Sj j
S
D
S
D
D S
p f f m u m u f f p
f
m u f p
f
m u f p
− − = + + =
 
       
 + + =
 
       
       
 
 + + =
mu f f p

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76
This matrix equation represents two ordinary differential equations governing the displacements
u1 and u2 of the two-story frame subjected to external dynamic forces p1(t) and p2(t). Each equation
contains both unknowns u1 and u2. The two equations are therefore coupled and in their present form
must be solved simultaneously.
1 1 2 2 1
2 2 2 2
1 1 2 2 1
2 2 2 2
or
or
S
S
S
D
D
D
f k k k u
f k k u
f c c c u
f c c u
+ −
     
= =
     
−
   
 
+ −
     
= =
     
−
     
 + + =
f ku
f cu
mu cu ku p

Shieh-Kung
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77
• Mass–Spring–Damper System
We have introduced the linear two-DOF system by idealizing a two-story frame—an approach
that should appeal to structural engineering students. However, the classic two-DOF system, shown in
the following figure, consists of two masses connected by linear springs and linear viscous dampers
subjected to external forces p1(t) and p2(t).
1 1 1 2 2 1 1 2 2 1 1
2 2 2 2 2 2 2 2 2
0
or
0
m u c c c u k k k u p
m u c c u k k u p
+ − + −
             
+ + = + + =
             
− −
             
mu cu ku p

Shieh-Kung
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78
• General Formulation of N-story Shear Building
Although the shear-frame or shear-building idealization is unrealistic in some manners, it is still
convenient and, most importantly, useful for studying the fundamental structural control of an MDOF
system. the dynamic equilibrium is the same as:
− Inertia Forces
− Damping Forces
I D S
+ + =
f f f p
1 1
2 2
3 3
0 0 0
0 0 0
0 0 0
0 0 0
I
n n
m u
m u
m u
m u
   
   
   
   
= =
   
   
   
   
f mu
m1
m2
m3
mn-1
mn
…
m1
m2
1 2 2 1
2 2 3 3 2
3 3 4 3
0 0
0
0 0
0 0 0
D
n n
c c c u
c c c c u
c c c u
c u
+ −
   
   
− + −
   
   
= = − +
   
   
   
   
f cu

Shieh-Kung
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79
− Elastic Forces
1 2 2 1
2 2 3 3 2
3 3 4 3
0 0
0
0 0
0 0 0
S
n n
k k k u
k k k k u
k k k u
k u
+ −
   
   
− + −
   
   
= = − +
   
   
   
   
f ku
m1
m2
m3
mn-1
mn
…

Shieh-Kung
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80
• Earthquake–induced Force
Similar with Chapter 1.4, we can develop the response of structures subjected to earthquake-
induced motion as:
where ug(t) is the displacement of the ground
ut(t) is the total (or absolute) displacement (of the mass)
l is n by 1 vector filled with 1
From the free-body diagram, the equation of dynamic equilibrium is
and
( ) ( ) ( )
t
g
t t u t
= +
u u l
0 and ( ) ( ) ( )
( ) ( ) ( ) ( ) or ( ) ( ) ( ( ), ( )) ( )
t
I D S I g
g g
t t u t
t t t u t t t f t t u t
+ + = = = +
 + + = − + + = −
f f f f mu mu ml
mu cu ku ml mu cu u u ml
eff ( ) ( )
g
t u t
= −
p ml

Shieh-Kung
Huang
81
• General Formulation of Structural System
For a generalized structural system, the dynamic equilibrium is described as:
− Inertia Forces
− Damping Forces
− Elastic Forces
I D S
+ + =
f f f p
11 12 13 1 1
21 22 23 2 2
31 32 33 3 3
1 2 3
and
n
n
T
I n
n n n nn n
m m m m u
m m m m u
m m m m u
m m m m u
   
   
   
   
= = =
   
   
   
   
f mu m m
11 12 13 1 1
21 22 23 2 2
31 32 33 3 3
1 2 3
and
n
n
T
D n
n n n nn n
c c c c u
c c c c u
c c c c u
c c c c u
   
   
   
   
= = =
   
   
   
   
f cu c c
11 12 13 1 1
21 22 23 2 2
31 32 33 3 3
1 2 3
and
n
n
T
S n
n n n nn n
k k k k u
k k k k u
k k k k u
k k k k u
   
   
   
   
= = =
   
   
   
   
f ku k k

Shieh-Kung
Huang
82
• General Formulation of Structural System
For a generalized structural system, the dynamic equilibrium is described as:
I D S
+ + =
f f f p

Shieh-Kung
Huang
83

Shieh-Kung
Huang
1.9 FREE AND FORCE VIBRATION OF MDOF SYSTEMS
84
• Natural Vibration Frequencies and Modes
In this section we introduce the eigenvalue problem whose solution gives the natural frequencies
and modes of a system. The free vibration of an undamped system can be described mathematically
by
where qn(t) is the displacement harmonic function,
fn is the deflected shape that does not vary with time
Substituting this form of u(t) in the equation of dynamic equilibrium gives
The result shows that the natural frequencies n and modes fn must satisfy the algebraic equation.
This algebraic equation is called the matrix eigenvalue problem. When necessary it is called the real
eigenvalue problem as
This equation is known as the characteristic equation or frequency equation.
Corresponding to the N natural vibration frequencies n of an N-DOF system, there are N
independent vectors fn, which are known as natural modes of vibration, or natural mode shapes of
vibration. These vectors are also known as eigenvectors, characteristic vectors, or normal modes. The
term natural is used to qualify each of these vibration properties to emphasize the fact that these are
natural properties of the structure in free vibration, and they depend only on its mass and stiffness
properties.
( ) ( )
n n
t q t f
=
u
2 2
( ) 0
n n n n n n n
q t
 f f  f f
 
− + =  =
 
m k m k
( ) cos sin
n n n n n
q t A t B t
 
= +
2 2
0 det 0
n n n
 f 
   
− =  − =
   
k m k m

Shieh-Kung
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85
• Modal and Spectral Matrices
The N eigenvalues and N natural modes can be assembled compactly into matrices. The N
eigenvectors can then be displayed in a single square matrix, each column of which is a natural mode:
where
• Modal and Spectral Matrices
The natural modes corresponding to different natural frequencies can be shown to satisfy the
following orthogonality conditions
where q isn’t equal to r.
The orthogonality of natural modes implies that the following square matrices are diagonal:
where the diagonal elements are
2 2
n n n
 f f
=  =
m k mΦΩ kΦ
0
T T
q r q r
f f f f
= =
m k
2
11 12 13 1
1
2
21 22 23 2
2
2 2
31 32 33 3
3
2
1 2 3
0 0 0
0 0 0
and
0 0 0
0 0 0
N
N
N
N N N NN
N
f f f f

f f f f

f f f f

f f f f

   
   
   
   
= =
   
   
   
 
 
Ω Φ
and
T T
= =
K Φ kΦ M Φ mΦ
2
, and
T T
n n n n n n n n n
K M K M
f f f f 
= = =
k m

Shieh-Kung
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86
• Modal Expansion of Displacements
In the following sections the natural modes are used as such a basis. Thus, a modal expansion of
any displacement vector u(t) has the form
where qr(t) are scalar multipliers called modal coordinates or normal coordinates. Because of the
orthogonality relation, all terms in the summation above vanish except the r = n term; thus the matrix
products on both sides of this equation are scalars.
This is the modal expansion of the displacement vector u(t).
1
( ) ( ) ( )
N
r r
r
t q t t
f
=
= =

u Φq
( ) ( )
( ) ( ) ( ) ( )
T T
T T n n
n n n n n T
n n n
t t
t q t q t
M
f f
f f f
f f
=  = =
mu mu
mu m
m

Shieh-Kung
Huang
87
• System with Damping
When damping is included, the free vibration response of the system is governed by
If the damping matrix of a linear system satisfies the identity
all the natural modes of vibration are real-valued and identical to those of the associated undamped
system; they were determined by solving the real eigenvalue problem. Such systems are said to
possess classical damping because they have classical natural modes. We have
For classically damped systems, the square matrix C is diagonal. Then, above equation represents N
uncoupled differential equations in modal coordinates qn(t), and classical modal analysis is applicable
to such systems. On the other hand, a linear system is said to possess nonclassical damping if its
damping matrix does not satisfy above equation.
( ) ( ) ( )
t t t
+ + =
mu cu ku 0
1 1
− −
=
cm k km c
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
T T T
t t t t t t
t t t
+ + =  + + =
 + + =
mΦq cΦq kΦq 0 Φ mΦq Φ cΦq Φ kΦq 0
Mq Cq Kq 0

Shieh-Kung
Huang
88
Classical damping is an appropriate idealization if similar damping mechanisms are distributed
throughout the structure (e.g., a multistory building with a similar structural system and structural materials
over its height). In this section we develop two procedures for constructing a classical damping matrix for
a structure.
• Rayleigh Damping
Consider first mass-proportional damping and stiffness-proportional damping:
For this damping matrices, the matrix C is diagonal by virtue of the modal orthogonality properties;
therefore, these are classical damping matrices.
0 1
a a
= +
c m k

Shieh-Kung
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89
• Caughey Damping
If we wish to specify values for damping ratios in more than two modes, we need to consider the
general form for a classical damping matrix, known as Caughey damping:
where N is the number of degrees of freedom in the system and al are constants. The first three terms
of the series are
1 0 1 1 1 2 1
0 0 1 1 2 2
( ) , ( ) , and ( )
a a a a a a
− − − −
= = =
m m k m m m k k m m k km k
1
1
0
N l
l
l
a
−
−
=
 
=  

c m m k

Shieh-Kung
Huang
90
• Modal Damping
An alternative procedure to determine a classical damping matrix from modal damping ratios can
be derived as
where C is a diagonal matrix with the nth diagonal element equal to the generalized modal damping:
Therefore, the first equation can be rewritten as
• Modal Damping and Rayleigh Damping Models
https://www.youtube.com/watch?v=4rgTdWGbmpQ
T
=
Φ cΦ C
(2 )
n n n n
C M
 
=
1 1
( ) ( )
T − −
=
=
c Φ C Φ
ΦCΦ

Shieh-Kung
Huang
MATHEMATICALDESCRIPTIONOFSTRUCTURALSYSTEMS
Chapter Outline
91
CHAPTER 2
2.1 Introduction of Systems
2.2 Introduction of Linear Systems
2.3 Introduction of Linear Time-Invariant Systems
2.4 Introduction of Discrete-Time Systems
2.5 Introduction of Structural Systems
2.6 Problems of Dynamics Systems

Shieh-Kung
Huang
2.1 INTRODUCTION OF SYSTEMS
92
Chapter 2 Mathematical Description of Structural Systems
• Systems
The systems studied here is assumed to have some input terminals and output terminals as
shown in the following figure. We assume that if an excitation or input is applied to the input terminals,
a unique response or output signal can be measured at the output terminals. This unique relationship
between the excitation and response, input and output, or cause and effect is essential in defining a
system.

Shieh-Kung
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93
A system with only one input terminal and only one output terminal is called a single-variable
system or a single-input single-output (SISO) system. A system with two or more input terminals
and/or two or more output terminals is called a multivariable system. More specifically, we can call a
system a multi-input multi-output (MIMO) system if it has two or more input terminals and output
terminals, a single-input multi-output (SIMO) system if it has one input terminal and two or more output
terminals.

Shieh-Kung
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94
A system is called a continuous-time system if it accepts continuous-time signals as its input and
generates continuous-time signals as its output. The input will be denoted by lowercase italic u(t) for
single input or by boldface u(t) for multiple inputs. Similarly, the output will be denoted by y(t) or y(t).
The time t is assumed to range from −∞ to +∞.
A system is called a discrete-time system if it accepts discrete-time signals as its input and
generates discrete-time signals as its output. All discrete-time signals in a system will be assumed to
have the same sampling period T. The input and output will be denoted by u[k] := u(kT) and y[k] :=
y(kT), where k denotes discrete-time instant and is an integer ranging from −∞ to +∞. They become
boldface for multiple inputs and multiple outputs.

Shieh-Kung
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95
• Comparison
Continuous-time System Discrete-time System
Representation u(t) and y(t)
u[k] and y[k]
uk and yk
Integral
Differential
Laplace Transform S Transform Z Transform
Fourier Transform Fourier Transform Discrete Fourier Transform
2
1
( )
t
t
u t dt

2
1
[ ]
k
k k
u k
=

( )
( ) or ( )
d
u t
dt
u t u t

[ ] [ 1]
[ 1]
u k u k
t
u k
− −

−

Shieh-Kung
Huang
2.2 INTRODUCTION OF LINEAR SYSTEMS
96
• Linear Systems
A system is called a linear system if the additivity property and the homogeneity property can be
applied for any time instant.
Additivity (or Super-position) Property:
Homogeneity Property:
The systems to be studied here are limited to linear systems. Using the concept of linearity, we
can develop that every linear system can be described by
where
Because G(t,τ) is the response excited by an impulse, it is called the impulse response matrix. This
equation describes the relationship between the input u(t) and output y(t) and is called the input–
output or external description.
0
( ) ( , ) ( )
t
t
t t d
  
= 
y G u
system system
( ) ( ) ( ) ( ) and
t t t t
  
⎯⎯⎯
→  ⎯⎯⎯
→ 
u y u y
system system system
1 1 2 2 1 2 1 2
( ) ( ) and ( ) ( ) ( ) ( ) ( ) ( )
t t t t t t t t
⎯⎯⎯
→ ⎯⎯⎯
→  + ⎯⎯⎯
→ +
u y u y u u y y
11 12 1
21 22 2
1 2
( , ) ( , ) ( , )
( , ) ( , ) ( , )
( , )
( , ) ( , ) ( , )
p
p
q q qp
g t g t g t
g t g t g t
t
g t g t g t
  
  

  
 
 
 
=
 
 
 
G

Shieh-Kung
Huang
2.3INTRODUCTIONOFLINEARTIME-INVARIANTSYSTEMS
97
• Linear Time-Invariant (LTI) Systems
A system is said to be time invariant if the time shifting property can be applied for any time
instant.
Time Shifting Property:
In other words, if the initial state and the input are the same, no matter at what time they are
applied, the output waveform will always be the same. Therefore, for time-invariant systems, we can
always assume, without loss of generality, that . If a system is not time invariant, it is said to be
time invariant (time-varying).
The input–output or external description for LTI systems can be described by
On the contrary, the input–output or external description for linear time-variant systems is still
described by
(recall the Duhamel’s integral from Chapter 1.2)
system system
( ) ( ) ( ) ( ) and
t t t T t T T
⎯⎯⎯
→  + ⎯⎯⎯
→ + 
u y u y
0 0
t =
0 0
( ) ( ) ( ) or ( ) ( ) ( )
t t
t
t t d t t d
     
= − = −
 
y G u y G u
0
( ) ( , ) ( )
t
t
t t d
  
= 
y G u
 
0
1
( ) ( )sin ( )
t
n
n
u t p t d
m
   

= −


Shieh-Kung
Huang
2.4 INTRODUCTION OF DISCRETE-TIME SYSTEMS
98
• Discrete-time Systems
This section develops the discrete counterpart of continuous-time systems. The input and output
of every discrete-time system will be assumed to have the same sampling period T and will be
denoted by u[k] := u(kT), y[k] := y(kT), where k is an integer ranging from −∞ to +∞.
Let u[k] be any input sequence. Then it can be expressed as
Thus the output y[k] excited by the input u[k] equals
The sequence G[k,m] is called the impulse response matrix sequence.
       
1 if
where
0 if
m
k m
k m k m k m
k m
 

=−
=

= − − = 



u u
   
[ , ]
m
k k m m

=−
= 
y G u
From http://signalsworld.blogspot.com/2009/11/continuoustime-and-discrete-time.html

Shieh-Kung
Huang
2.5 INTRODUCTION OF STRUCTURAL SYSTEMS
99
• Structural Systems
Recalling the dynamic equilibrium of an MDOF system, the Laplace transform is an important tool
in analysis. Applying the Laplace transform to the external description yields
This equation describes the relationship between the input (earthquake-induced acceleration) and the
output (displacement of the structure) in the Laplace domain (also called s-domain). It can be further
simplified
where a variable with (s) denotes the Laplace transform of the variable. The function is called the
transfer matrix.
1 1
2 1 1
2 1 1
2 1 1
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
( )
g
g
g
g
g
t t t u t
t t t u t
s s s s s u s
s s s u s
s
u s s s
− −
− −
− −
− −
+ + = −
 + + = −
 + + = −
 + + = −
 = −
+ +
mu cu ku ml
u m cu m ku l
Iu m cu m ku l
I m c m k u l
u l
I m c m k
( )
s
G
2 1 1
output
( )
input
s
s s
− −
= = −
+ +
l
G
I m c m k

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Huang
2.6 PROBLEMS OF DYNAMICS SYSTEMS
100
• Dynamic Analysis
Inputs and systems are known and only outputs are unknown.
• System Identification
Inputs and outputs are known and only systems are unknown.
• Inverse Problem (or Input Force Identification)
Outputs and systems are known and only inputs are unknown.
• Control
An additional force (either internal or external) is used to drive the desired (specified) outputs.

Shieh-Kung
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101
• Dynamic System
• Passive Control
• Active and Semi-active Control
Structure/
System/
Plant
u (t) y(t)
Structure/
System/
Plant
u (t) y(t)
Control Device
Structure/
System/
Plant
u (t) y(t)
Control Device
(Controller)

Shieh-Kung
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102
• Open-loop Control (or Feedforward Control)
• Close-loop Control (or Feedback Control)
Structure/
System/
Plant
u (t) y(t)
Control Device
(Controller)
Sensor
uc (t)
Structure/
System/
Plant
u (t) y(t)
Control Device
(Controller)
uc (t)

Shieh-Kung
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103
• State Feedback Control
Structure/
System/
Plant
u (t) y(t)
Control Device
(Controller)
Sensor
uc (t)
Estimator
x(t)

Shieh-Kung
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104
• General Assumptions of Our Course
Continuous-time Systems v.s. Discrete-time Systems
Linear Time-Invariant (LTI) Systems v.s. Linear Time-Variant (LVI) Systems (or Nonlinear Systems)
Time Domain v.s. Frequency Domain
Active Control v.s. Semi-active Control v.s. Passive Control

Shieh-Kung
Huang
STATE-SPACE REALIZATIONS
105
CHAPTER 3
3.1 External and Internal Description
3.2 Solution of State-Space Equations
3.3 Equivalent State-Space Equations
3.4 Realizations
3.5 Characteristics Analysis
3.6 Solution of Linear Time-Variant (LTV) Equations
3.7 Stability

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3.1 EXTERNAL AND INTERNAL DESCRIPTION
106
• External Description
The systems to be studied in this course are limited to linear systems. Using the concept of
linearity, we develop in Chapter 2 that every linear system can be described by
This equation describes the relationship between the input u(t) and output y(t) and is called the input–
output or external description.
• Internal Description
If a linear system is lumped as well, then it can also be described by
The first equation (called state equation) is a set of first-order differential equations and the second
equation (called observation equation) is a set of algebraic equations. They are called the internal
description of linear systems. Because the vector x(t) is called the state, the set of two equations is
called the state-space or, simply, the state equations.
If a linear system has, in addition, the property of time invariance, then equations reduce to
and
Chapter 3 State-space Realizations
0
( ) ( , ) ( )
t
t
t t d
  
= 
y G u
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
t t t t t
t t t t t
= +
= +
x A x B u
y C x D u
0 0
( ) ( ) ( ) or ( ) ( ) ( )
t t
t
t t d t t d
     
= − = −
 
y G u y G u
( ) ( ) ( )
( ) ( ) ( )
t t t
t t t
= +
= +
x Ax Bu
y Cx Du

Shieh-Kung
Huang
107
• Relationship between External and Internal Description
Applying the Laplace transform to the external description yields
where a variable with (s) denotes the Laplace transform of the variable. The function is called the
transfer matrix.
Similarly, the internal description of linear systems can be analyzed by the Laplace transform as:
The equations also reveal the fact that the response of a linear system can be decomposed as the
zero-state response and the zero-input response. If the initial state is zero, then equation reduces to
This relates the input–output (or transfer matrix) and state-space descriptions.
The functions tf2ss and ss2tf in MATLAB compute one description from the other. They
compute only the SISO and SIMO cases. For example, ss2tf computes the transfer matrix from the
first input to all outputs or, equivalently, the first column of . If the last argument 1 in ss2tf is
replaced by 3, then the function generates the third column of .
( ) ( ) ( )
s s s
=
y G u
( )
s
G
1 1
1 1
( ) (0) ( ) ( )
( ) ( ) ( )
( ) ( ) (0) ( ) ( )
( ) ( ) (0) ( ) ( ) ( )
s s s s
s s s
s s s s
s s s s s
− −
− −
− = +
= +
= − + −

= − + − +
x x Ax Bu
y Cx Du
x I A x I A Bu
y C I A x C I A Bu Du
1
( ) ( ) ( ) and ( ) ( )
s s s s s −
= = − +
y G u G C I A B D
( )
t
G
( )
t
G

Shieh-Kung
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108

Shieh-Kung
Huang
109
• State-space Equations for Structural System
Considering a linear n DOF structural system subjected to an earthquake excitation. The equation
of motion for this can be expressed as
The equation of motion can be written as the continuous-time state-space equations:
where x(t) is state vector
y(t) is observation (or output) vector
A is state (or linear elastic system) matrix
B is input (or excitation influence) matrix
C is observation (or output) matrix.
D is feedthrough (or excitation influence) matrix
The matrices in the state equation can be derived by
The observation vector has various forms and can be derived accordingly. For example, if the absolute
acceleration is measured, the observation equation can be derived as:
( ) ( ) ( ) ( )
g
t t t u t
+ + = −
mu cu ku ml
( ) ( ) ( )
( ) ( ) ( )
g
g
t t u t
t t u t
= +
= +
x Ax B
y Cx D
1 1
( )
( ) , , and
( )
t
t
t − −
     
= = =
     
− − −
     
u 0 I 0
x A B
u m k m c l
1 1
( ) ( ) and ( ) ( ) ( )
and
t t
g
t t t t u t
− −
= = +
 
 = − − =
 
y u u u l
C m k m c D 0

Shieh-Kung
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110
The transfer matrix from the external description of the structural system can be derived as:
For another point of view, we can also derive the transfer matrix from Chapter 2.5 as:
1
1
1 1
1 1
1
1 1
1 1
1
1 1
2 1 1 1
1
2 1 1
( ) ( )
1
1
s s
s
s
s
s
s s s
s s
−
−
− −
− −
−
− −
− −
−
− −
− − −
−
− −
= − +
 
   
 
= − − − +
 
   
  − − −
   
 
−
   
 
= − −    
  + −
   
 
+  
 
= − −    
  −
+ + −  
 
= −
+ +
G C I A B D
0 I 0
m k m c I 0
m k m c l
I I 0
m k m c
m k I m c l
0
I m c I
m k m c
l
I m c m k m k I
m k
I m c m k
1
1 1
2 1 1
s
s
s s
−
− −
− −
−
 
 
−  
  −
 
+
=
+ +
l
m c
l
m cl m kl
I m c m k
2
2 1 1
1 1
2 1 1
( ) ( )
( )
( )
( ) ( )
t
g
g g
s u s
s
s
u s u s
s
s s
s
s s
− −
− −
− −
+
= =
= − +
+ +
+
=
+ +
u l
u
G
l
l
I m c m k
m cl m kl
I m c m k

Shieh-Kung
Huang
111
• State-space Equations for Structural System with Control Devices
Considering a linear n DOF structural system with control device subjected to an earthquake
excitation. The equation of motion for this can be expressed as
where uc(t) is the control force from control devices and h is the location vector for the devices. The
equation of motion can be written as the continuous-time state-space equations:
( ) ( ) ( ) ( ) ( )
c g
t t t t u t
+ + = −
mu cu ku hu ml
( ) ( ) ( )
( ) ( ) ( )
t t t
t t t
= +
= +
x Ax Bu
y Cx Du
1 1 1
( )
( )
( ) , , , and
( )
( )
c
g
t
t
t
u t
t − − −
 
     
= = = =  
     
− − −
       
u
u 0 I 0 0
x A B u
u m k m c m h l
1 1 1
( ) ( ) and ( ) ( ) ( )
, and
t t
g
t t t t u t
− − −
= = +
   
 = − − =
   
y u u u l
C m k m c D m h 0

Shieh-Kung
Huang
112
Considering the same structural system. The alternative equation of motion can be written as the
continuous-time state-space equations:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
c g
c g
t t t u t
t t t u t
= + +
= + +
x Ax Bu E
y Cx Du F
1 1 1
( )
( ) , , , and
( )
t
t
t − − −
       
= = = =
       
− − −
       
u 0 I 0 0
x A B E
u m k m c m h l
1 1
1
( ) ( ) and ( ) ( ) ( )
, , and
t t
g
t t t t u t
− −
−
= = +
 
 
 = − − = =
 
 
 
y u u u l
0
C m k m c D F 0
m h

Shieh-Kung
Huang
113
The function ss in MATLAB is quite useful when you construct the numerical model. For example, a
SDOF oscillates as
 
2
2
0 1 0
0 1 0
, and
, and
2 1
/ / 1
/ / , and 0 2 , and 0
n n
n n
k m c m
k m c m
 
 
   
   
= =
= =    
    − − −
− − −   
 
   
 
= − − = = − − =
 
A B
A B
C D C D

Shieh-Kung
Huang
114
The function ss in MATLAB is quite useful when you construct the numerical model. There are other
functions in MATLAB helping us construct and convert models in discrete-time, we will discuss them later.
Taking the slide 76 as an example
1 1
1 1
, and
, and
− −
− −
   
= =
   
− − −
   
 
= − − =
 
0 I 0
A B
m k m c l
C m k m c D 0

Shieh-Kung
Huang
Matlab Demonstration (Demo_3_1_C.m)
115
Now we can use the function lsim shown in slide 24 &
25 to calculate the structural responses.

Shieh-Kung
Huang
3.2 SOLUTION OF STATE-SPACE EQUATIONS
116
Consider the LTI state-space equations,
the solution hinges on the exponential function of A studied in the state-space equations. In particular, we
need
to develop the solution. Premultiplying the exponential function on both sides of the state-space equations
yields
This is the solution of the state equation.
( )
t t t
d
e e e
dt
− − −
= − = −
A A A
A A
0 0
0
( )
0
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
( ) (0) ( )
( ) (0) ( )
t t t
t t t
t t
t
t
t
t
t
t
t
t t
e t e t e t
e t e t e t
d
e t e t
dt
e t e d
e t e d
t e e d



 
 
 
− − −
− − −
− −
− −
=
− −
−
= +
 − =
 
 =
 
 =
 − =
 = +



A A A
A A A
A A
A A
A A
A A
x Ax Bu
x Ax Bu
x Bu
x Bu
x x Bu
x x Bu
( ) ( ) ( )
t t t
= +
x Ax Bu

Shieh-Kung
Huang
117
Differentiating the solution, we obtain
Similarly, substituting the solution into the observation equation yields
Thus, the solution can be computed directly in the time domain.
( )
0
( ) ( )
0
( )
0
( ) (0) ( )
(0) ( ) ( )
(0) ( ) ( )
( ) ( )
t
t t
t
t t t
t
t
t t
d
t e e d
dt
e e d e
e e d t
t t

 


 
  
 
−
− −
=
−
 
= +
 
 
= + +
 
= + +
 
 
= +



A A
A A A
A A
x x Bu
A x A Bu Bu
A x Bu Bu
Ax Bu
( )
0
( ) ( ) ( ) ( ) ( )
( ) (0) ( ) ( )
t
t t
t t t t t
t e e d t

 
−
= +
= + +

A A
y C x D u
y C x C Bu Du

Shieh-Kung
Huang
118
• Discretization
As shown in Chapter 2.4, the state-space equations can also be discretized. If the set of
equations is to be computed on a digital computer, it must be discretized as
If and only if T is close to 0, it isn’t practicable.
From another point of view, because of discretization, the input changes values only at discrete-time
instants. For this input, the solution yields
and
0
0
( ) ( )
( ) lim
(( 1) ) ( )
( ) lim
(( 1) ) ( )
( ) ( )
(( 1) ) ( ) ( ) ( )
(( 1) ) ( ) ( ) ( )
T
T
x T t
t
T
k T kT
kT
T
k T kT
kT kT
T
k T kT T kT T kT
k T T kT T kT
→
→
+ −
=
+ −
 =
+ −
 + =
 + = + +
 + = + +
x x
x
x x
x
x x
Ax Bu
x x Ax Bu
x I A x Bu
  ( )
0
: ( ) (0) ( )
kt
kT kT
k kT e e d

 
−
= = + 
A A
x x x Bu
 
   
( 1)
( 1) (( 1) )
0
( 1)
( ) (( 1) )
0
0
1 : (( 1) ) (0) ( )
(0) ( ) ( )
where
k T
k T k T
kT k T
T kT kT k T
kT
T
T
k k T e e d
e e e d e d
e k e d k kT T

 

 
   
  
+
+ + −
+
− + −
+ = + = +
 
= + +
 
 
= + = + −

 

A A
A A A A
A A
x x x Bu
x Bu Bu
x Bu

Shieh-Kung
Huang
119
Thus, if an input changes value only at discrete-time instants kT and if we compute only the
responses at t = kT , then state-space equations become
with
This is a discrete-time state-space equations. Note that there is no approximation involved in this
derivation and the solution yields the exact solution of the continuous-time state-space equations at
t = kT, if the input is piecewise constant.
We can further discuss the computation of Bd.
     
     
1 d d
d d
k k k
k k k
+ = +
= +
x A x B u
y C x D u
0
, , , and
T
T
d d d d
e e d


= = = =

A A
A B B C C D D
2
2
0 0
2 3
2
2 3
1 2 3
2 3
1 2 3
1
2!
2! 3!
2! 3!
2! 3!
( )
T T
T
e d d
T T
T
T T
T
T T
T
e
 
  
−
−
−
 
= + + +
 
 
= + + +
 
= + + +
 
 
 
= + + + + −
 
 
= −
 
A
A
I A A
I A A
A A A A
A I A A A I
A I

Shieh-Kung
Huang
120
Thus we have
if and only if A is nonsingular. Using this formula, we can avoid computing an infinite series.
For conclusion, we have
with
Fortunately, the MATLAB function c2d transforms the continuous-time state-space equations into
the discrete-time state-space equations. Talking about the solution, it can be obtained by using the
MATLAB function lsim, an acronym for linear simulation.
1
( ) ,
T
d e
−
= −
A
B A I B
1
, ( ) , , and
T T
d d d d
e e
−
= = − = =
A A
A B A I B C C D D
     
     
1
( ) ( ) ( )
( ) ( ) ( )
d d
d d
k k k
t t t
t t t k k k
+ = +
= +

= + = +
x A x B u
x Ax Bu
y Cx Du y C x D u

Shieh-Kung
Huang
121
The functions d2c, c2d, and d2d in MATLAB are quite useful when you convert the numerical model.

Shieh-Kung
Huang
122
• Solution of Discrete-Time State-Space Equations
Consider the discrete-time state-space equation
In order to discuss the general behavior of discrete-time state equations, we will develop a general
form of solutions. We compute
and the observation equation
They are the discrete counterparts of the continuous-time solution. Their derivations are considerably
simpler than the continuous-time case.
Importantly, we can also solve the discrete-time state-space recursively. How we can do it?
     
     
1 d d
d d
k k k
k k k
+ = +
= +
x A x B u
y C x D u
     
     
     
     
2
1
1
0
1 0 0
2 1 1
0 0 1
0
d d
d d
d d d d
k
k k m
d d d
m
k m
−
− −
=
= +
= +
= + +
= + 
x A x B u
x A x B u
A x A B u B u
x A x A B u
     
1
1
0
0 [ ]
k
k k m
d d d
m
k m k
−
− −
=
= + +

y CA x CA B u Du
     
1 d d
k k k
+ = +
x A x B u

Shieh-Kung
Huang
123
Talking about the solution, it can be obtained by using the MATLAB function lsim, an acronym for
linear simulation. Let us try Example 3.1.C again.

Shieh-Kung
Huang
3.3 EQUIVALENT STATE-SPACE EQUATIONS
124
Consider the state-space equations
Let T be an n by n real nonsingular matrix and let
Then the state-space equations,
where
is said to be (algebraically) equivalent and is called an equivalence transformation.
Moreover, the transfer matrix (mentioned in Chapter 3.1) is the same
Two state equations are said to be zero-state equivalent if they have the same transfer matrix. The
MATLAB function ss2ss carries out equivalence transformations.
( ) ( ) ( )
( ) ( ) ( )
t t t
t t t
= +
= +
x Ax Bu
y Cx Du
( ) ( )
t t
=
x Tx
( ) ( ) ( )
( ) ( ) ( )
t t t
t t t
= +
= +
x Ax Bu
y Cx Du
1 1
, , , and
− −
= = = =
A TAT B TB C CT D D
1
1 1 1
1
1 1
1 1 1 1
1
( ) ( )
( )
( )
( )
( )
s s
s
s
s
s
−
− − −
−
− −
− − − −
−
= − +
= − +
 
= − +
 
= − +
= − +
G C I A B D
CT I TAT TB D
C T I TAT T B D
C T IT T TAT T B D
C I A B D
( ) ( )
t t
=
x Tx
x1
x2
x3
x3
x2
x1

Shieh-Kung
Huang
125
• Commonly Used Forms
− Canonical Forms
MATLAB contains the function canon. If last argument,
type=companion, the function will generate an equivalent
state equation with in the companion form as
Similar variations are controllable canonical form,
controllability canonical form, observable canonical form,
and observability canonical form.
− Jordan Forms
If last argument, type=modal, the function will
generate an equivalent state equation with in the Jordan
form diagonized as
Suppose A has some real eigenvalues and some complex
eigenvalues. Because A has only real coefficients, the two
complex eigenvalues must be complex conjugate. The
transformation matrix T now is the same with the
eigenvector matrix Q of the system matrix A.
1
2
3
1
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
n
n
n
n
a
a
a
a
a
−
−
−
−
 
 
−
 
 
−
=  
−
 
 
 
−
 
A
1
2
3 3 1
3 3
0 0 0 0
0 0 0 0
0 0 0
:
0 0 0
0 0 0 0 n


 
 

−
 
 
 
 
= =
 
−
 
 
 
 
J TAT

Shieh-Kung
Huang
126
• Absolute and Relative Responses
− The absolute acceleration, velocity, and displacement
− The relative (to the ground) acceleration, velocity, and displacement
− The relative (to the vicinity) acceleration, velocity, and displacement
( )
t
u
( )
a t
u
( )
t
u
( )
g
u t

Shieh-Kung
Huang
127
• Equivalence transformation for Structural System
Using the transformation matrix, T, and let
where
The state equation can be derived by
The equivalence transformation is the same with using the following transformation matrix
( ) ( ) and ( ) ( ) ( ) ( )
g
t t t t t u t
= + + = −
u Tu mu c u ku ml
1 1 1
, , , and
− − −
= = = =
m mT c cT k kT l Tl
1 1
( ) ( ) ( )
( )
( ) , and
( )
g
t t u t
t
t
t − −
= +
     
= = =
     
− − −
     
x Ax B
u 0 I 0
x A B
u m k m c l
( ) ( )
t t
 
=  
 
T 0
x x
0 T

Shieh-Kung
Huang
128
• Example of A 3-story Structural System
Considering a 3-story shear-frame
The inverse of the transformation matrix
We can compute the new stiffness matrix as
m1
m2
m3
1 1
2 2 1
3 3 2
( ) 1 0 0 ( )
( ) ( ) and 1 1 0 ( ) ( ) ( )
( ) 0 1 1 ( ) ( )
u t u t
t u t t u t u t
u t u t u t
   
 
   
 
= = −  = −
   
 
   
 
− −
 
   
u T u
1
1 0 0
1 1 0
1 1 1
−
 
 
=
 
 
 
T
1
2 1 2
2 3 2 3
3 3
1 2
2 3
3
0 1 0 0
1 1 0
0 1 1 1
0
0
0 0
k k k
k k k k
k k
k k
k k
k
−
=
+ −
   
   
= − + −
   
   
−  
 
−
 
 
= −
 
 
 
k kT

Shieh-Kung
Huang
129
The new damping matrix is derived as
and the mass matrix are
m1
m2
m3
1
1 2 2
2 3 2 3
3 3
1 2
2 3
3
0 1 0 0
1 1 0
0 1 1 1
0
0
0 0
c c c
c c c c
c c
c c
c c
c
−
=
+ −
   
   
= − + −
   
   
−  
 
−
 
 
= −
 
 
 
c cT
1
1
2
3
1
2 2
3 3 3
0 0 1 0 0
0 0 1 1 0
0 0 1 1 1
0 0
0
m
m
m
m
m m
m m m
−
=
   
   
=
   
   
 
 
 
 
=
 
 
 
m mT

Shieh-Kung
Huang
130
The MATLAB function ss2ss carries out equivalence transformations. This example transforms the
states (relative to ground) to the new states relative to the vicinal floor. Consider the model from slides 94
and 95
1 1 1
( ) ( )
, ,
( ) ( )
t t
t t
− − −
=
= = =
 
=  
 
u Tu
m mT c cT k kT
T 0
x x
0 T

Shieh-Kung
Huang
131
By the way, the MATLAB function canon performs canonical state-space realization. This example
modifies from the help

Shieh-Kung
Huang
3.4 REALIZATIONS
132
Every linear time-invariant (LTI) system can be described by the external description (with transfer
matrix) and internal description (with state-space equations). The computed transfer matrix is unique. So,
the converse problem can be introduced, that is, to find a state-space equation from a given transfer
matrix. This is called the realization problem.
A transfer matrix is said to be realizable if there exists a finite-dimensional state equation such that
This refers that a transfer matrix is realizable if and only if it is a proper rational matrix.
where Adj is to form adjugate matrix and det is to form determine of the matrix. If A is n by n, then
det(sI – A) has degree n and the transfer matrix is realizable.
Every pole of the transfer matrix is an eigenvalue of A; on the other hand, from the definition, every
solution of (sI – A) is a eigenvalue.
We can use the MATLAB function eig to generate eigenvalues and eigenvectors, so it can be used to
check if the transfer matrix is realizable and if the Jordan forms shown in Chapter 3.3 can be constructed.
If A cannot be diagonized, A is said to be defective and eig will yield an incorrect solution. Moreover, A is
nonsingular if and only if it has no zero eigenvalue.
Note that any (algebraic) equivalence transformation has no change on eigenvalue.
1
( ) ( )
s s −
= − +
G C I A B D
 
1 1
( ) ( ) Adj( )
det( )
s s s
s
−
= − + = − +
−
G C I A B D C I A B D
I A
( ) ( )
( ): det
p
   
=  − =  = −
AΨ Ψ A I Ψ 0 A I
1 1 1
   
− − −
=  =  =  =
AΨ Ψ TAT Ψ Ψ AT Ψ T Ψ AΨ Ψ

Shieh-Kung
Huang
3.4 REALIZATIONS
133
Every linear time-invariant (LTI) system can be converted to the internal description (with state-space
equations).
It’s actually canonical forms shown in slide 125.
2
1 2 3
3 2
1 2 3
( )
( )
( )
y s b s b s b
G s
u s s a s a s a
+ +
= =
+ + +
1 2 3 1 2 3
1 2 3
3 2 1
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) 0 1 0 ( ) 0
( ) 0 0 1 ( ) 0 ( ) where ( ) ( ) ( ) ( )
( ) ( ) 1
y t a y t a y t a y t bu t b u t b u t
y t y t
y t y t u t u t bu t b u t b u t
y t a a a y t
 + + + = + +
 
     
 
     
 = + = + +
 
     
 
     
− − −
     
 
From: https://eng.libretexts.org/

Shieh-Kung
Huang
3.4 REALIZATIONS
134
External Description
(Input–output Model)
Internal Description
(State-space Modal)
Continuous-time
System
Discrete-time System

Shieh-Kung
Huang
3.4 REALIZATIONS
table list of matlab conversion between models
135
Transfer
Function
State-Space
Zero-pole-
gain Form
Partial
Fraction
Expansion
Lattice
Filter Form
Second-
order
Sections
Form
Convolution
Matrix
Transfer
Function
tf2ss
tf2zp
roots
residuez
residue
tf2latc tf2sos convmtx
State-Space ss2tf ss2zp ss2sos
Zero-pole-
gain Form
zp2tf
poly
zp2ss zp2sos
Partial
Fraction
Expansion
residuez
residue
Lattice
Filter Form
latc2tf
Second-
order
Sections
Form
sos2tf sos2ss sos2zp
Convolution
Matrix

Shieh-Kung
Huang
3.5 CHARACTERISTICS ANALYSIS
136
• Characteristics Equation
After realizations, we always care about the dynamic characteristics (natural frequencies and
damping ratios described in Chapter 1.3) of the structural system. These can be found by performing
the eigen-analysis to the internal or external description of a system.
For the external description. The poles of the transfer matrix provide the information about the
eigenvalues; and for the internal description, the eigenvalues and eigenvectors can be obtained by
decomposing A as:
where eig is the eigenvalue decomposition operator that outputs eigenvector matrix Ψ of and
eigenvalues matrix Λ
It leads
Again, the MATLAB function eig performs the eigenvalue decomposition for an arbitrary matrix.
0 k k k
 
− =  =
A I Aψ ψ
1 1
− −
=  =
A ΨΛΨ Λ Ψ AΨ
1
2
0 0
0 0
0 0 n



 
 
 
=
 
 
 
Λ

Shieh-Kung
Huang
137
• Structural Responses
As shown in Chapter 3.2, the discretized solution (or responses) is
( )
0
2 3
2 3
2 3
1 1 2 1 3 1
2 3
2 3 1
1 1
( ) (0) ( )
2! 3!
2! 3!
2! 3!
diag( )
k
t
t t
t
t
t
t e e d
t t
e t
t t
t
t t
t
e e


 
−
− − − −
−
− −
= +
= + + + +
= + + + +
 
= + + + +
 
 
= =

A A
A
Λ
x x Bu
I A A A
ΨIΨ ΨΛΨ ΨΛ Ψ ΨΛ Ψ
Ψ I Λ Λ Λ Ψ
Ψ Ψ Ψ Ψ

Shieh-Kung
Huang
138
• Modal Coordinate
As shown in Chapter 3.3, the equivalence transformation of the state-space equations is
where
So, the state-space equations can be transformed to modal coordinate
where
Hence, each mode in the system matrix, Λ, is now decomposed.
1 1
, , , and
− −
= = = =
A TAT B TB C CT D D
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
t t
t t t t t t
t t t t t t
=

= + = +

⎯⎯⎯
→
 
= + = +
 
x Tx
x Ax Bu x Ax Bu
y Cx Du y Cx Du
1
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
t t
t t t t t t
t t t t t t
−
=

= + = +

⎯⎯⎯⎯
→
 
= + = +
 
x Ψ x
x Ax Bu x Ax Bu
y Cx Du y Cx Du
1 1
, , , and
− −
= = = = =
A Ψ AΨ Λ B Ψ B C CΨ D D

Shieh-Kung
Huang
139
• Dynamic Characteristics of Structural System
After realizations, we always care about the dynamic characteristics (natural frequencies and
damping ratios described in Chapter 1.3) of the structural system. These can be found by performing
the eigen-analysis to the internal or external description of a system. It should be noticed that the
eigenvalues and eigenvectors appear in complex conjugated pairs, and a pair of conjugated
eigenvalues is associated with a single natural frequency and damping ratio:
Thus, the natural frequencies and damping ratios can be computed as:
For a discrete-time system, system matrix can be transformed between continuous-time system
Hence, the eigenvalues have similar relationship
Fortunately, the MATLAB function damp computes the natural frequency and damping of system poles,
no matter it’s a continuous-time system or a discrete-time system.
2
1
, 1
k k k k k k
i
     
+ = −  −
2 2
and where Re( ) and Im( )
k
k k k k k k k k
k
a
a b a b
   

−
= + = = =
1
ln( )
T
d d
e
T
=  =
A
A A A
1
ln( )
kT
dk k dk
e
T

  
=  =

Shieh-Kung
Huang
140
The MATLAB function eig and damp carries out eigenvalue decomposition and modal parameters
extraction, respectively. Let us consider the example from Demo_3_1_B.m again
2 2
and
k k k
k
k k k k
k
a ib
a
a b

 

= +
−
= + =

Shieh-Kung
Huang
141
The eigenvalue from a discrete-time system or a continuous-time system can be easily transform
using
1
ln( )
kT
dk k dk
e
T

  
=  =

Shieh-Kung
Huang
3.6SOLUTIONOFLINEARTIME-VARIANT(LTV)EQUATIONS
142
Consider the linear time-variant (LTV) state-space equations
It is assumed that, for every initial state x(t0) and any input u(t), the state equation has a unique solution. A
sufficient condition for such an assumption is that every entry of A(t) is a continuous function of t.
In conclusion, we cannot extend the solution of time-variant equations to the matrix case and must
use a different approach to develop the solution. Fortunately, for most of the cases in structural
engineering, the response is oscillation and the structural system is slow time-variant, so we can divide
the time step to a very small interval and assume the structural system is time-invariant in that interval.
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
t t t t t
t t t t t
= +
= +
x A x B u
y C x D u
         
         
1 d d
d d
k k k k k
k k k k k
+ = +
= +
x A x B u
y C x D u

Shieh-Kung
Huang
3.6SOLUTIONOFLINEARTIME-VARIANT(LTV)EQUATIONS
143
The responses of a LTV system can still be calculated without using the MATLAB function lsim;
alternatively, they can be acquired by recursive computation using discrete-time equations (similar to slide
122).

Shieh-Kung
Huang
3.7 STABILITY
144
Recall the solution of the state equation in Chapter 3.2,
where the first term in the right is the initial state response and the second term is the forced response.
The asymptotic stability (for initial state response) and input-output stability (for forced response) need to
be checked before the system is really controlled.
• Input–Output Stability (BIBO Stability)
A system is said to be BIBO stable (bounded-input bounded-output stable) if every bounded input
excites a bounded output. This stability is defined for the zero-state response and is applicable only if
the system is initially relaxed.
A system is BIBO stable if and only if g(t) is absolutely integrable in [0,∞). A system with proper rational
transfer function is BIBO stable if and only if every pole of the transfer function has a negative real
part or, equivalently, lies inside the left-half s-plane.
( )
0
( ) (0) ( )
t
t t
t e e d

 
−
= + 
A A
x x Bu
0
( ) Constant
t dt

  
 g
( )
s
G

Shieh-Kung
Huang
3.7 STABILITY
145
• Internal Stability (Asymptotic Stability)
As discussed earlier, every pole of the transfer matrix
is an eigenvalue of A. The LTI system is said to be asymptotically stable if every finite initial state x(0)
excites a bounded response ( or all eigenvalues of A have zero or negative real
parts). The system is said to be marginally stable or stable in the sense of Lyapunov if every finite
initial state x(0) excites a bounded response (all eigenvalues of A have negative real parts).
Thus asymptotic stability implies BIBO stability. Note that
asymptotic stability is defined for the zero-input response,
whereas BIBO stability is defined for the zero-state response.
The system in the following example has eigenvalue 1 and
is not asymptotically stable; however, it is BIBO stable.
Thus BIBO stability, in general, does not imply asymptotic
stability. We mention that marginal stability is useful only in the design of oscillators. Other than
oscillators, every physical system is designed to be asymptotically stable or BIBO stable with some
additional conditions.
As in the continuous-time case, any (algebraic) equivalence transformation will not alter the
stability of a state equation
1
( ) ( )
s s −
= − +
G C I A B D
0 0 0
( ) 0 0 0 ( )
0 0 1
t t
 
 
=
 
 
−
 
x x
( ) 0 when
t t
→ → 
x
t
x
0 when
x t
→ → 
Asymptotically Stable
Marginally Stable

Shieh-Kung
Huang
3.7 STABILITY
146
• Stability Analysis
Recall the solution of the solution in Chapter 3.5,
The solution in the modal coordinate is
So, the asymptotic stability (as well as BIBO stability) is achieve if and only if
the marginally stability is achieve if and only if
the mode oscillation is controlled by
• Connection with Structural Dynamics
− Attenuation Rate
− Damped Natural Frequency
− Mode Shape
1
diag( )
kt
t
e e −
=
A
Ψ Ψ
( )
(cos sin )
k k k
k
t a ib t
a t
k k
e e
e b t i b t
 +
=
= +
0
k
a 
0
k
a 
k
b
k k k
a  
= −
2
1
k k k
b  
= −
k
ψ

Shieh-Kung
Huang
3.7 STABILITY
147
Continuous-time System Discrete-time System
Marginally Stability Every eigenvalue of A has zero or
negative real parts
Eigenvalues of Ad have
magnitudes less than or equal to 1
Asymptotic Stability Every eigenvalue of A has negative
real parts
Eigenvalues of Ad have
magnitudes less than 1
BIBO Stability Every eigenvalue of A has negative
real part
Every eigenvalue of Ad has a
magnitude less than 1
Courtesy of Cheng Chen and James M. Ricles, 2008, ‘Development of Direct
Integration Algorithms for Structural Dynamics Using Discrete Control Theory’

Shieh-Kung
Huang
3.7 STABILITY
148
The MATLAB function pzmap and pzplot carriy out the pole-zero plot of a dynamic system. This
example shows the poles and zeros of the five DOFs shear-type structure.

Shieh-Kung
Huang
3.7 STABILITY
149
• Lyapunov Theorem
The Lyapunov theorem introduces a different method of checking asymptotic stability. For
convenience, we call A stable if every eigenvalue of A has a negative real part. The theorem states
that all eigenvalues of A have negative real parts if and only if for any given positive definite symmetric
matrix Q, the Lyapunov equation
has a unique symmetric solution P and P is positive definite. The solution can be expressed as
The Lyapunov theorem are valid for any given Q; therefore we shall use the simplest possible Q.
Even so, using them to check stability of A is not simple. It is much simpler to compute, using
MATLAB, the eigenvalues of A and then check their real parts. Thus the importance of this theorem is
not in checking the stability of A but rather in studying the stability of nonlinear systems. They are
essential in using the so-called second method of Lyapunov.
In the discrete-time system, All eigenvalues of Ad have magnitudes less than 1 if and only if for
any given positive definite symmetric matrix Q, the discrete Lyapunov equation
has a unique symmetric solution P and P is positive definite. The solution can be expressed as
T T
+ = − = −
A P PA Q Q Q
0
T
t t
e e dt

= 
A A
P Q
T T
d d
− = − = −
P A PA Q Q Q
0
( )
T m m
d
m

=
= 
P A QA

Shieh-Kung
Huang
3.7 STABILITY
150
The second method of Lyapunov is an energy function. The system is stable if we can define an
“energy-like” function for a system and prove that the “energy” is decreasing. For example,
The LaSalle’s theorem states that if the scalar function V(x) is and except at the origin,
then the system is asymptotically stable.
For both the first and second methods of Lyapunov, what is the relationship between P and Q?
Recall the initial state response of the state equation
by comparing two equations, we have
( ) T
V =
x x Px
( ) 0
V 
x ( ) 0
V 
x
( ) ( )
( ) ( )
( )
0
T T T
T T
T T
T
V V
=  = +
= +
= +
= − 
x x Px x x Px x Px
Ax Px x P Ax
x A P PA x
x Qx
0 0
0
0
( ) ( ) ( )
(0) (0)
(0)( ) (0)
T
T
T
T t t
T t t
V dt t t dt
e e dt
e e dt
 


− =
=
=
 


A A
A A
x x Qx
x Q x
x Q x
0
( ) ( ) (0)
(0) if system is stable, ( ) 0
(0) (0)
T
V dt V V
V V

− = −  +
=  →
=
 x
x Px
0
T
t t
e e dt

= 
A A
P Q

Shieh-Kung
Huang
3.7 STABILITY
151

Shieh-Kung
Huang
INTRODUCTIONOFP
ASSIVEENERGYDISSIP
ATIONSYSTEMS
152
CHAPTER 4

Shieh-Kung
Huang
153
• 林旺春（Wang-Chuen Lin）副研究員
Lin, W. C., Yu, C. H., Yang, C. Y., Hwang, J. S., & Wang, S. J. (2022). Seismic Retrofit of Coupled Hospitals
with Viscous Dampers. Journal of Innovative Technology, 4(2), 53-64.
Lin, W. C., Wang, S. J., & Hwang, J. S. (2022). Seismic Retrofit of Existing Critical Structures Using
Externally Connected Viscous Dampers. International Journal of Structural Stability and Dynamics, 22(13),
2250144.
Lin, W. C., Yu, C. H., Tsai, M. A., Chang, Y. W., Peng, S. K., & Wang, S. J. (2022). Hysteretic behavior of
viscoelastic dampers subjected to damage during seismic loading. Journal of Building Engineering, 53,
104538.
Wang, S. J., Sung, Y. L., Yang, C. Y., Lin, W. C., & Yu, C. H. (2020). Control Performances of Friction
Pendulum and Sloped Rolling-Type Bearings Designed with Single Parameters. Applied Sciences, 10(20),
7200.
Wang, S. J., Lin, W. C., Chiang, Y. S., & Hwang, J. S. (2020). Coupled Bilateral Hysteretic Behavior of High-
damping Rubber Bearings under Non-proportional Plane Loading. Journal of Earthquake Engineering, 1-28.
Wang, S. J., Lee, H. W., Yu, C. H., Yang, C. Y., & Lin, W. C. (2020). Equivalent linear and bounding analyses
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Chapter 4 Introduction of Passive Energy Dissipation Systems

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APPLICATION OF PASSIVE CONTROL
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CHAPTER 5

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