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Structural Vibration Control

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National Chung Hsing UniversityAssistant Professor at National Chung Hsing University

Chopra, A.K. Dynamics of Structures – Theory and Applications to Earthquake Engineering. Prentice Hall, 2001 Chen, C. T. Linear System Theory and Design. Oxford University Press, 1999 Soong, T.T. Active Structural Control: Theory and Practice. Longman, 1990 Chapter 1 Review of Structural Dynamics Chapter 2 Mathematical Description of Structural Systems Chapter 3 State-space Realizations Chapter 4 Introduction of Passive Energy Dissipation Systems Chapter 5 Application of Passive Control Chapter 6 Controllability and Observability Chapter 7 State Feedback and State Estimators Chapter 8 Application of Passive and Active Control Chapter 9 Application of Semi-active and Hybrid Control

Structural Vibration Control

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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
Shieh-Kung
Huang
Copyright © 2016 by Pearson Education, Inc. All rights reserved.
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
Shieh-Kung
Huang
Copyright © 2016 by Pearson Education, Inc. All rights reserved.
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
Chapter 1 Review of Structural Dynamics
Shieh-Kung
Huang
Copyright © 2016 by Pearson Education, Inc. All rights reserved.
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.
Chapter 1 Review of Structural Dynamics
0
mu ku
+ =
Shieh-Kung
Huang
Copyright © 2016 by Pearson Education, Inc. All rights reserved.
1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS
16
• 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.
Chapter 1 Review of Structural Dynamics

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Structural Vibration Control

  • 2. 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
  • 3. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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
  • 4. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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 Chapter 1 Review of Structural Dynamics
  • 5. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 1 Review of Structural Dynamics 0 mu ku + =
  • 6. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 16 • 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. Chapter 1 Review of Structural Dynamics
  • 7. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 17 • 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. Chapter 1 Review of Structural Dynamics Courtesy of Elnashai and Sarno, 2015
  • 8. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 18 • 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. Chapter 1 Review of Structural Dynamics 0 mu cu ku + + =
  • 9. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 19 • 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. Chapter 1 Review of Structural Dynamics
  • 10. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 20 • Linear Elastic System: − Elastic material − First-order analysis • Inelastic System: − Plastic material − Higher-order analysis Chapter 1 Review of Structural Dynamics S f k u =  ( , ) S f f u u =
  • 11. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 21 • 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. Chapter 1 Review of Structural Dynamics 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 − − = + + =  = = =
  • 12. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 22 • 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. Chapter 1 Review of Structural Dynamics or and or ( , ) D S D S S mu cu ku p mu f f p f cu f ku f f u u + + = + + =  = = =
  • 13. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS Matlab Demonstration (Demo_1_2_A.m) 23 Chapter 1 Review of Structural Dynamics 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 − − = + + =  = = =
  • 14. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 24 • 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. Chapter 1 Review of Structural Dynamics ( ) ( ) ( ) ( ) 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      = − 
  • 15. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS 25 − 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. Chapter 1 Review of Structural Dynamics 1 ( ) ( ) ( ) 2 i t u t H P e d       − = 
  • 16. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS Matlab Demonstration (Demo_1_2_B.m) 26 Chapter 1 Review of Structural Dynamics 0 0 ( ) (1 cos ) when 0, (0) , and ( ) 0 n p u t t k c p p p t  = − = = = p(t) u(t)
  • 17. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS Matlab Demonstration 27 Chapter 1 Review of Structural Dynamics
  • 18. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS Matlab Demonstration 28 Chapter 1 Review of Structural Dynamics 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.
  • 19. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS Matlab Demonstration 29 Chapter 1 Review of Structural Dynamics https://www.mathworks.com/matlabcentral/fileex change/71007-newmark-beta-method-for- nonlinear-single-dof-systems
  • 20. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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: Chapter 1 Review of Structural Dynamics 2 n n T   = (0) u (0) u
  • 21. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION 31 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. Chapter 1 Review of Structural Dynamics 1 n n f T = 2 n n f   = n k m  = Tn n fn
  • 22. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION 32 • 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. Chapter 1 Review of Structural Dynamics 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     = = = = =
  • 23. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION 33 • Underdamped Free Vibration The time Chapter 1 Review of Structural Dynamics
  • 24. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION 34 • Underdamped Free Vibration The time Chapter 1 Review of Structural Dynamics • 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.
  • 25. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION 35 • 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 Chapter 1 Review of Structural Dynamics 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       = = = = − −
  • 26. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 36 • 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. Chapter 1 Review of Structural Dynamics (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       + = =  = −  − 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION
  • 27. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION Matlab Demonstration (Demo_1_3_A.m) 37 Chapter 1 Review of Structural Dynamics 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          − + + =    + = +      = −
  • 28. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION 38 • 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. Chapter 1 Review of Structural Dynamics
  • 29. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION 39 • Transient Response and Stead-state Response Chapter 1 Review of Structural Dynamics ( ) ( ) ( ) ( ) ( ) ( ) 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                      − + + =  = + + + − − = =         − + − +        
  • 30. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.3 RESPONSE OF FREE AND HARMONIC VIBRATION 40 • 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. Chapter 1 Review of Structural Dynamics
  • 31. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 41 • Earthquakes in Taiwan Chapter 1 Review of Structural Dynamics 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
  • 32. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 42 • Earthquakes in Taiwan Chapter 1 Review of Structural Dynamics https://scweb.cwb.gov.tw/zh-tw/page/disaster/
  • 33. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 43 • 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. Chapter 1 Review of Structural Dynamics Courtesy of USGS https://pubs.usgs.gov/gip/dynamic/fire.html
  • 34. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 44 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. Chapter 1 Review of Structural Dynamics
  • 35. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 45 • 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 Chapter 1 Review of Structural Dynamics ( ) ( ) ( ) 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 + + = = = +  + + = − + + = −
  • 36. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 46 The ground motion can therefore be replaced by the effective earthquake force (indicated by the subscript “eff”): Chapter 1 Review of Structural Dynamics eff ( ) ( ) g p t mu t = − Courtesy of Wikiwand https://www.wikiwand.com/en/Seismic_base_isolation
  • 37. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 47 • Equation of Motion 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) Chapter 1 Review of Structural Dynamics 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     + + = −  + + = − + + =
  • 38. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 48 • Response History The following figure shows the deformation response of SODF systems to El Centro ground motion. Chapter 1 Review of Structural Dynamics
  • 39. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.4 EARTHQUAKE RESPONSE OF LINEAR SYSTEMS 49 • 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. Chapter 1 Review of Structural Dynamics
  • 40. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 1 Review of Structural Dynamics 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         
  • 41. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM 51 Chapter 1 Review of Structural Dynamics 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         
  • 42. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM 52 Chapter 1 Review of Structural Dynamics Courtesy of Estrada and Lee, 2008
  • 43. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM 53 Chapter 1 Review of Structural Dynamics The response spectrum for El Centro ground motion with various damping ratios.
  • 44. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM 54 Courtesy of Chopra, 2020 Chapter 1 Review of Structural Dynamics
  • 45. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM 55 Courtesy of Chopra, 2020 Chapter 1 Review of Structural Dynamics
  • 46. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM 56 Chapter 1 Review of Structural Dynamics The mean spectra with probability distributions for the construction of elastic design spectrum.
  • 47. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM 57 • 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. Chapter 1 Review of Structural Dynamics 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          
  • 48. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM 58 Chapter 1 Review of Structural Dynamics Courtesy of Estrada and Lee, 2008
  • 49. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM Matlab Demonstration 59 Chapter 1 Review of Structural Dynamics https://www.mathworks.com/matlabcentral/fileexchange/78029-elastic-response-spectra
  • 50. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.5 RESPONSE SPECTRUM Matlab Demonstration 60 Chapter 1 Review of Structural Dynamics https://www.mathworks.com/matlabcentral/fileexchange/50843-response-spectrum
  • 51. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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 Chapter 1 Review of Structural Dynamics m y u u  =
  • 52. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.6 EARTHQUAKE RESPONSE OF INELASTIC SYSTEMS 62 • 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 Chapter 1 Review of Structural Dynamics 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 = = −  
  • 53. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.6 EARTHQUAKE RESPONSE OF INELASTIC SYSTEMS 63 Time variation of energy dissipated by viscous damping and yielding, and of kinetic plus strain energy; (left) linear system, (right) elastoplastic system Chapter 1 Review of Structural Dynamics 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 + + = − + + + =    
  • 54. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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 Chapter 1 Review of Structural Dynamics Courtesy of Filiatrault, A., Christopoulos, C. 2006, ‘Principles of passive supplemental damping and seismic isolation’
  • 55. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 65 • Rain Flow Analogy At the end of seismic shaking Chapter 1 Review of Structural Dynamics ( ) ( ) ( ) I D Y in d k E t E t E t V V V = + = + Courtesy of Filiatrault, A., Christopoulos, C. 2006, ‘Principles of passive supplemental damping and seismic isolation’
  • 56. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 66 • Rain Flow Analogy Using supplemental energy dissipation device Chapter 1 Review of Structural Dynamics Courtesy of Filiatrault, A., Christopoulos, C. 2006, ‘Principles of passive supplemental damping and seismic isolation’
  • 57. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 67 • Rain Flow Analogy Using seismic isolation systems Chapter 1 Review of Structural Dynamics Courtesy of Filiatrault, A., Christopoulos, C. 2006, ‘Principles of passive supplemental damping and seismic isolation’
  • 58. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 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 Chapter 1 Review of Structural Dynamics 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.
  • 59. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 69 − Metallic Yielding Dampers Buckling-Restrained Brace (BRB) Chapter 1 Review of Structural Dynamics 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
  • 60. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 70 − Friction Dampers Slotted Bolted Connection (SBC); Chapter 1 Review of Structural Dynamics Courtesy of Rozlyn K. Bubela, Carlos Ventura, and Helmut G.L. Prion, 2010, ‘Cyclic Testing of Steel Chevron Braces with Vertically Slotted Beam Connection’
  • 61. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 71 Theoretical Behavior of Different Types of Dampers Chapter 1 Review of Structural Dynamics Friction dampers Metallic yielding dampers Viscoelastic dampers Fluid viscous dampers
  • 62. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 72 Chapter 1 Review of Structural Dynamics − Tuned Mass Dampers Courtesy of Gebrail Bekdaş, Sinan MelihNigdeli, 2011, ‘Estimating optimum parameters of tuned mass dampers using harmony search’ From Wikimedia Commons
  • 63. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 73 Chapter 1 Review of Structural Dynamics − Tuned Mass Dampers
  • 64. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.7 ENERGY CONCEPTS IN EARTHQUAKE ENGINEERING 74 Chapter 1 Review of Structural Dynamics − Seismic Isolation Systems
  • 65. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. • Equation of Motion Similar with Chapter 1.2, we can develop the dynamic equilibrium as: Chapter 1 Review of Structural Dynamics 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
  • 66. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS 76 • Equation of Motion 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. Chapter 1 Review of Structural Dynamics 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
  • 67. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS 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). Chapter 1 Review of Structural Dynamics 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
  • 68. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS 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 Chapter 1 Review of Structural Dynamics 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
  • 69. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS 79 − Elastic Forces Chapter 1 Review of Structural Dynamics 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 …
  • 70. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS 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 Chapter 1 Review of Structural Dynamics ( ) ( ) ( ) 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
  • 71. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS 81 • General Formulation of Structural System For a generalized structural system, the dynamic equilibrium is described as: − Inertia Forces − Damping Forces − Elastic Forces Chapter 1 Review of Structural Dynamics 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
  • 72. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS 82 • General Formulation of Structural System For a generalized structural system, the dynamic equilibrium is described as: Chapter 1 Review of Structural Dynamics I D S + + = f f f p
  • 73. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.8 MULITI-DEGREE-OF-FREEDOM SYSTEMS Matlab Demonstration (Demo_1_8_A.m) 83 Chapter 1 Review of Structural Dynamics
  • 74. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 1 Review of Structural Dynamics ( ) ( ) 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
  • 75. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.9 FREE AND FORCE VIBRATION OF MDOF SYSTEMS 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 Chapter 1 Review of Structural Dynamics 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
  • 76. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.9 FREE AND FORCE VIBRATION OF MDOF SYSTEMS 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). Chapter 1 Review of Structural Dynamics 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
  • 77. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.9 FREE AND FORCE VIBRATION OF MDOF SYSTEMS 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. Chapter 1 Review of Structural Dynamics ( ) ( ) ( ) 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
  • 78. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.9 FREE AND FORCE VIBRATION OF MDOF SYSTEMS 88 Chapter 1 Review of Structural Dynamics 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
  • 79. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.9 FREE AND FORCE VIBRATION OF MDOF SYSTEMS 89 Chapter 1 Review of Structural Dynamics • 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
  • 80. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 1.9 FREE AND FORCE VIBRATION OF MDOF SYSTEMS 90 Chapter 1 Review of Structural Dynamics • 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
  • 81. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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
  • 82. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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.
  • 83. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.1 INTRODUCTION OF SYSTEMS 93 Chapter 2 Mathematical Description of Structural Systems 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.
  • 84. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.1 INTRODUCTION OF SYSTEMS 94 Chapter 2 Mathematical Description of Structural Systems 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.
  • 85. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.1 INTRODUCTION OF SYSTEMS 95 Chapter 2 Mathematical Description of Structural Systems • 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 − −  −
  • 86. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.2 INTRODUCTION OF LINEAR SYSTEMS 96 Chapter 2 Mathematical Description of Structural Systems • 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
  • 87. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.3INTRODUCTIONOFLINEARTIME-INVARIANTSYSTEMS 97 Chapter 2 Mathematical Description of Structural Systems • 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      = − 
  • 88. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.4 INTRODUCTION OF DISCRETE-TIME SYSTEMS 98 Chapter 2 Mathematical Description of Structural Systems • 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
  • 89. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.5 INTRODUCTION OF STRUCTURAL SYSTEMS 99 Chapter 2 Mathematical Description of Structural Systems • 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
  • 90. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.6 PROBLEMS OF DYNAMICS SYSTEMS 100 Chapter 2 Mathematical Description of Structural Systems • 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.
  • 91. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.6 PROBLEMS OF DYNAMICS SYSTEMS 101 Chapter 2 Mathematical Description of Structural Systems • 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)
  • 92. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.6 PROBLEMS OF DYNAMICS SYSTEMS 102 Chapter 2 Mathematical Description of Structural Systems • 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)
  • 93. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.6 PROBLEMS OF DYNAMICS SYSTEMS 103 Chapter 2 Mathematical Description of Structural Systems • State Feedback Control Structure/ System/ Plant u (t) y(t) Control Device (Controller) Sensor uc (t) Estimator x(t)
  • 94. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 2.6 PROBLEMS OF DYNAMICS SYSTEMS 104 Chapter 2 Mathematical Description of Structural Systems • 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
  • 95. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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
  • 96. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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
  • 97. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION 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 . Chapter 3 State-space Realizations ( ) ( ) ( ) 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
  • 98. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION Matlab Demonstration (Demo_3_1_A.m) 108 Chapter 3 State-space Realizations
  • 99. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION 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: Chapter 3 State-space Realizations ( ) ( ) ( ) ( ) 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
  • 100. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION 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: Chapter 3 State-space Realizations 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
  • 101. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION 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: 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: Chapter 3 State-space Realizations ( ) ( ) ( ) ( ) ( ) 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
  • 102. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION 112 Considering the same structural system. The alternative equation of motion can be written as the continuous-time state-space equations: 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: Chapter 3 State-space Realizations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 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
  • 103. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION Matlab Demonstration 113 Chapter 3 State-space Realizations 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
  • 104. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION Matlab Demonstration (Demo_3_1_B.m) 114 Chapter 3 State-space Realizations 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
  • 105. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.1 EXTERNAL AND INTERNAL DESCRIPTION Matlab Demonstration (Demo_3_1_C.m) 115 Chapter 3 State-space Realizations Now we can use the function lsim shown in slide 24 & 25 to calculate the structural responses.
  • 106. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 3 State-space Realizations ( ) 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
  • 107. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.2 SOLUTION OF STATE-SPACE EQUATIONS 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. Chapter 3 State-space Realizations ( ) 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
  • 108. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.2 SOLUTION OF STATE-SPACE EQUATIONS 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 Chapter 3 State-space Realizations 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
  • 109. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.2 SOLUTION OF STATE-SPACE EQUATIONS 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. Chapter 3 State-space Realizations             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
  • 110. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.2 SOLUTION OF STATE-SPACE EQUATIONS 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. Chapter 3 State-space Realizations 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
  • 111. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.2 SOLUTION OF STATE-SPACE EQUATIONS Matlab Demonstration (Demo_3_2_A.m) 121 Chapter 3 State-space Realizations The functions d2c, c2d, and d2d in MATLAB are quite useful when you convert the numerical model.
  • 112. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.2 SOLUTION OF STATE-SPACE EQUATIONS 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? Chapter 3 State-space Realizations             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
  • 113. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.2 SOLUTION OF STATE-SPACE EQUATIONS Matlab Demonstration (Demo_3_2_B.m) 123 Chapter 3 State-space Realizations 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.
  • 114. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 3 State-space Realizations ( ) ( ) ( ) ( ) ( ) ( ) 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
  • 115. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.3 EQUIVALENT STATE-SPACE EQUATIONS 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. Chapter 3 State-space Realizations 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
  • 116. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.3 EQUIVALENT STATE-SPACE EQUATIONS 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 Chapter 3 State-space Realizations ( ) t u ( ) a t u ( ) t u ( ) g u t
  • 117. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.3 EQUIVALENT STATE-SPACE EQUATIONS 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 Chapter 3 State-space Realizations ( ) ( ) 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
  • 118. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.3 EQUIVALENT STATE-SPACE EQUATIONS 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 Chapter 3 State-space Realizations 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
  • 119. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.3 EQUIVALENT STATE-SPACE EQUATIONS 129 The new damping matrix is derived as and the mass matrix are Chapter 3 State-space Realizations 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
  • 120. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.3 EQUIVALENT STATE-SPACE EQUATIONS Matlab Demonstration (Demo_3_3_A.m) 130 Chapter 3 State-space Realizations 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
  • 121. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.3 EQUIVALENT STATE-SPACE EQUATIONS Matlab Demonstration (Demo_3_3_B.m) 131 Chapter 3 State-space Realizations By the way, the MATLAB function canon performs canonical state-space realization. This example modifies from the help
  • 122. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 3 State-space Realizations 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
  • 123. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 3 State-space Realizations 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/
  • 124. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.4 REALIZATIONS 134 Chapter 3 State-space Realizations External Description (Input–output Model) Internal Description (State-space Modal) Continuous-time System Discrete-time System
  • 125. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.4 REALIZATIONS table list of matlab conversion between models 135 Chapter 3 State-space Realizations 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
  • 126. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 3 State-space Realizations 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
  • 127. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.5 CHARACTERISTICS ANALYSIS 137 • Structural Responses As shown in Chapter 3.2, the discretized solution (or responses) is Chapter 3 State-space Realizations ( ) 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
  • 128. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.5 CHARACTERISTICS ANALYSIS 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. Chapter 3 State-space Realizations 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
  • 129. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.5 CHARACTERISTICS ANALYSIS 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. Chapter 3 State-space Realizations 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     =  =
  • 130. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.5 CHARACTERISTICS ANALYSIS Matlab Demonstration (Demo_3_5_A.m) 140 Chapter 3 State-space Realizations 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     = + − = + =
  • 131. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.5 CHARACTERISTICS ANALYSIS Matlab Demonstration (Demo_3_5_B.m) 141 Chapter 3 State-space Realizations 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     =  =
  • 132. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 3 State-space Realizations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 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
  • 133. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.6SOLUTIONOFLINEARTIME-VARIANT(LTV)EQUATIONS Matlab Demonstration 143 Chapter 3 State-space Realizations 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).
  • 134. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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. Chapter 3 State-space Realizations ( ) 0 ( ) (0) ( ) t t t t e e d    − = +  A A x x Bu 0 ( ) Constant t dt      g ( ) s G
  • 135. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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 Chapter 3 State-space Realizations 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
  • 136. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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 Chapter 3 State-space Realizations 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
  • 137. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.7 STABILITY 147 Chapter 3 State-space Realizations 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’
  • 138. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.7 STABILITY Matlab Demonstration (Demo_3_7_A.m) 148 Chapter 3 State-space Realizations 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.
  • 139. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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 Chapter 3 State-space Realizations 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
  • 140. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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 Chapter 3 State-space Realizations ( ) 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
  • 141. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 3.7 STABILITY 151 Chapter 3 State-space Realizations
  • 142. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. INTRODUCTIONOFP ASSIVEENERGYDISSIP ATIONSYSTEMS 152 CHAPTER 4
  • 143. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. 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 of bilinear hysteretic isolation systems. Earthquakes and Structures, 19(5), 395-409. Wang, S. J., Lin, W. C., Chiang, Y. S., & Hwang, J. S. (2019). Mechanical behavior of lead rubber bearings under and after nonproportional plane loading. Earthquake Engineering & Structural Dynamics, 48(13), 1508-1531. Wang, S. J., Yu, C. H., Lin, W. C., Hwang, J. S., & Chang, K. C. (2017). A generalized analytical model for sloped rolling-type seismic isolators. Engineering Structures, 138, 434-446. Wang, S. J., Hwang, J. S., Chang, K. C., Shiau, C. Y., Lin, W. C., Tsai, M. S., ... & Yang, Y. H. (2014). Sloped multi‐roller isolation devices for seismic protection of equipment and facilities. Earthquake Engineering & Structural Dynamics, 43(10), 1443-1461. Chapter 4 Introduction of Passive Energy Dissipation Systems
  • 144. Shieh-Kung Huang Copyright © 2016 by Pearson Education, Inc. All rights reserved. APPLICATION OF PASSIVE CONTROL 154 CHAPTER 5