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E-raamat: Structural Damping: Applications in Seismic Response Modification

(State University of New York at Buffalo, USA), (State University of New York at Buffalo, USA), (State University of New York at Buffalo, USA), (State University of New York at Buffalo, USA)
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Structural Damping: Applications in Seismic Response ModificationSuurem pilt
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Rapid advances have been made during the past few decades in earthquake response modification technologies for structures, most notably in base isolation and energy dissipation systems. Many practical applications of various dampers can be found worldwide and, in the United States, damper design has been included in building codes. The current design process is simple and useful for adding supplemental damping up to a reasonable levelbut it is not as useful with higher levels of damping.

Taking a different approach, Structural Damping: Applications in Seismic Response Modification considers the dynamic responses of structures with added damping devices as systems governed by the combined effect of the static stiffness, period, and dampingor "dynamic stiffness"of the structure-device system. This formulation supplies additional information for higher-level supplemental damping design that current provisions may not adequately cover. The authors also propose a more comprehensive consideration of the core issues in structural damping, which provides a useful foundation for continued research and development in seismic response modification technologies for performance-based engineering.

The book includes design examples, based on the authors research and practical experience, to illustrate approaches that include higher-level supplemental damping to complement the use of the current NEHRP/ASCE-7 provisions. A self-contained resource on damping design principles, this book helps earthquake engineers select the most effective type of damper and determine the amount and configuration of damping under given working conditions.
Series Preface xv
Preface xvii
Series Editor xxiii
Authors xxv
Part I Vibration Systems
Chapter 1 Free and Harmonic Vibration of Single-Degree-of-Freedom Systems
3(62)
1.1 Model of Linear SDOF Vibration Systems
3(26)
1.1.1 Equation of Motion and Basic Dynamic Parameters
3(8)
1.1.1.1 Equilibrium of Vibration Forces
3(2)
1.1.1.2 Basic Parameters of the Physical Model
5(2)
1.1.1.3 Characteristic Equation and Modal Model
7(4)
1.1.2 Homogeneous Solution, Free-Decay Vibration, and the Response Model
11(5)
1.1.3 Forced Vibration with Harmonic Excitation
16(8)
1.1.3.1 Steady-State Response
16(2)
1.1.3.2 Method of Complex Response for Steady-State Displacement
18(1)
1.1.3.3 Response of Harmonic Excitation with Zero Initial Condition
19(4)
1.1.3.4 Responses with Nonzero Initial Conditions
23(1)
1.1.4 Ground Excitation
24(5)
1.1.4.1 Governing Equation
24(2)
1.1.4.2 Responses of Harmonic Ground Excitations
26(3)
1.2 Dynamic Magnification
29(21)
1.2.1 Dynamic Magnification Factor, General Excitation
30(10)
1.2.1.1 Dynamic Magnification Factor of Displacement
30(2)
1.2.1.2 Dynamic Magnification Factor of Acceleration
32(1)
1.2.1.3 Peak Values of Dynamic Magnification Factors
33(2)
1.2.1.4 Dynamic Magnification Factors Reaching Unity
35(1)
1.2.1.5 Half-Power Points and Resonant Region
36(2)
1.2.1.6 Response Reduction due to Increase of Damping
38(2)
1.2.2 Dynamic Magnification Factor, Ground Excitation
40(10)
1.2.2.1 Dynamic Magnification Factor of Relative Displacement
41(3)
1.2.2.2 Dynamic Magnification Factor of Absolute Acceleration
44(6)
1.3 Energy Dissipation and Effective Damping
50(13)
1.3.1 Energy Dissipated per Cycle
50(4)
1.3.1.1 Linear Viscous Damping
50(1)
1.3.1.2 Energy Dissipated by Damping Force
51(1)
1.3.1.3 Damping Coefficient and Damping Ratio of Linear Viscous Damping
52(1)
1.3.1.4 Linear System
53(1)
1.3.2 Damping and Seismic Force
54(4)
1.3.2.1 Parametric Equation
54(4)
1.3.3 Effective Damping
58(7)
1.3.3.1 Effective Damping Coefficient
59(1)
1.3.3.2 Effective Damping Ratio
60(1)
1.3.3.3 General Forced Vibration, Energy Dissipation
61(1)
1.3.3.4 Alternative Form of Damping Ratio
62(1)
1.4 Summary
63(1)
References
63(2)
Chapter 2 Linear Single-Degree-of-Freedom Systems with Arbitrary Excitations
65(78)
2.1 Periodic Excitations
65(26)
2.1.1 Periodic Signals
65(3)
2.1.1.1 Periodic Functions
65(3)
2.1.2 Fourier Series
68(4)
2.1.2.1 Fourier Coefficients
68(4)
2.1.3 Discrete Fourier Transform
72(4)
2.1.3.1 Discretization of Signals
72(2)
2.1.3.2 Discrete Fourier Series
74(2)
2.1.4 General Damping Force
76(13)
2.1.4.1 Linearity of Fourier Series
78(1)
2.1.4.2 Steady-State Structural Force
78(3)
2.1.4.3 Dry Friction Damping
81(4)
2.1.4.4 General Nonlinear Viscous Damping
85(4)
2.1.5 Response to Periodic Excitations
89(2)
2.1.5.1 General Response
89(1)
2.1.5.2 The nth Steady-State Response
90(1)
2.1.5.3 Transient Response
90(1)
2.2 Transient Excitations
91(12)
2.2.1 Transient Signals
91(1)
2.2.2 Fourier Transform
91(3)
2.2.2.1 Important Features of the Fourier Transform: A Summary
93(1)
2.2.3 Laplace Transform
94(2)
2.2.3.1 Important Features of the Laplace Transform: A Summary
95(1)
2.2.4 Impulse Response
96(2)
2.2.5 General Force and the Duhamel Integral
98(2)
2.2.5.1 Convolution Integral
98(2)
2.2.6 Transfer Function of Unit Impulse Response
100(2)
2.2.7 Integral Transform of Convolution
102(1)
2.3 Random Excitations
103(25)
2.3.1 Random Variables
103(6)
2.3.1.1 Mean Value, Mathematic Expectation
105(2)
2.3.1.2 Variance and Mean Square Value
107(1)
2.3.1.3 Standard Deviation and Root Mean Square Value
108(1)
2.3.2 Random Process
109(3)
2.3.2.1 Random Time Histories
109(1)
2.3.2.2 Statistical Averaging
109(3)
2.3.3 Correlation Functions and Power Spectral Density Functions
112(12)
2.3.3.1 Correlation Analysis
112(3)
2.3.3.2 Power Spectral Density Function
115(9)
2.3.4 Correlation between Forcing Function and Impulse Response Function
124(4)
2.3.5 Basic Approach to Dealing with Random Vibrations
128(1)
2.4 Earthquake Responses of SDOF Linear Systems
128(13)
2.4.1 Response Spectrum
128(3)
2.4.2 Design Spectra
131(6)
2.4.3 Control Factor for Damper Design: The Base Shear
137(6)
2.4.3.1 Spectral Accelerations and Displacement
141(1)
2.5 Summary
141(1)
References
141(2)
Chapter 3 Linear Proportionally Damped Multi-Degree-of-Freedom Systems
143(54)
3.1 Undamped MDOF Systems
143(16)
3.1.1 Eigen-Parameters of Linear Undamped Systems
143(5)
3.1.1.1 Governing Equations
143(2)
3.1.1.2 Modal Response of Free Vibrations
145(1)
3.1.1.3 General Eigen-Parameters
146(2)
3.1.2 Brief Discussion of Vectors and Matrices
148(6)
3.1.2.1 Vector
148(1)
3.1.2.2 Vector Norm
149(1)
3.1.2.3 Orthonormal Vectors
150(1)
3.1.2.4 Unit Vector
150(2)
3.1.2.5 Vector Space
152(1)
3.1.2.6 Linear Independence
152(2)
3.1.3 Symmetric Matrix and Rayleigh Quotient
154(5)
3.1.3.1 Eigen-Parameters of Symmetric Matrices
154(2)
3.1.3.2 Rayleigh Quotient
156(3)
3.2 Proportionally Damped MDOF Systems
159(10)
3.2.1 Modal Analysis and Decoupling Procedure
159(5)
3.2.1.1 Governing Equation of Damped MDOF Systems
159(1)
3.2.1.2 Decoupling by Means of Rayleigh Quotient
160(2)
3.2.1.3 Decoupling by Means of Linear Transformation
162(2)
3.2.2 Free-Decay Vibration
164(5)
3.3 Modal Participation and Truncation
169(13)
3.3.1 Modal Participation Factor
169(1)
3.3.2 Modal Contribution Indicator
170(3)
3.3.2.1 Theory of the Indicator
170(2)
3.3.2.2 Realization of Indicators
172(1)
3.3.3 Response Computation of Truncated Modal Superposition
173(8)
3.3.3.1 Computation Procedure
173(8)
3.3.4 Peak Responses
181(1)
3.4 Base Shear and Lateral Force
182(3)
3.5 Natural Frequency and Mode Shape Estimation
185(5)
3.5.1 Natural Frequency
185(3)
3.5.2 Mode Shape Estimation
188(2)
3.6 Coefficient Matrix for Proportional Damping
190(6)
3.6.1 Rayleigh Damping
190(1)
3.6.2 Caughey Damping
191(1)
3.6.3 Modification of Caughey Damping
192(1)
3.6.4 Overdamped Modes
193(1)
3.6.5 Alternative Expression of Proportional Damping
194(3)
3.6.5.1 Generalized Symmetric Damping Matrix
195(1)
3.7 Summary
196(1)
References
196(1)
Chapter 4 Multi-Degree-of-Freedom Systems with General Damping
197(82)
4.1 State Equations and Conventional Treatment
197(28)
4.1.1 State Matrix and Eigen-Decomposition
197(7)
4.1.1.1 State Equations
197(7)
4.1.2 Accompanist Matrix of Mode Shapes
204(3)
4.1.3 Linear Independency and Orthogonality Conditions
207(5)
4.1.4 Approach of Complex Damping
212(6)
4.1.5 Solutions in 2n Modal Space
218(7)
4.1.5.1 Mode Decoupling
218(1)
4.1.5.2 Alternative Computation Method
219(6)
4.2 Damper Design for Nonproportionally Damped Systems
225(5)
4.2.1 Issues with General Damper Design
225(1)
4.2.2 Essence of the Solution of Nonproportionally Damped System
226(4)
4.2.3 Modal Truncations for Nonproportionally Damped Systems
230(1)
4.3 Overdamped Subsystems
230(10)
4.3.1 Concept of Overdamped System
231(5)
4.3.2 Design Response Spectra for Overdamped Subsystem
236(4)
4.3.2.1 Spectral Value
236(1)
4.3.2.2 Overdamping Constant
237(3)
4.4 Responses of Generally Damped Systems and the Design Spectra
240(9)
4.4.1 Approach in 2n- and n-Space, Design Codes
241(3)
4.4.2 Modal Solution in n-Dimensional Space
244(4)
4.4.3 Modal Truncations for a Generally Damped System
248(1)
4.5 Modal Participation and Modal Criteria
249(26)
4.5.1 Criteria on Complex Mode
249(4)
4.5.1.1 Modal Energy Index
249(2)
4.5.1.2 Complex Modal Factor
251(2)
4.5.2 Modal Participation Factors
253(3)
4.5.2.1 Nonproportionally Damped Modes
254(1)
4.5.2.2 Proportionally Damped Modes
254(2)
4.5.2.3 Overdamped Subsystems
256(1)
4.5.3 Modal Contribution Indicators
256(6)
4.5.3.1 Modal Mass Ratio
256(4)
4.5.3.2 Static Modal Energy Ratio
260(2)
4.5.4 Modal Reconstruction of Generally Damped System
262(17)
4.5.4.1 Modal Reconstruction for Damper Design
264(4)
4.5.4.2 Damper Design without Stiffness Matrix
268(7)
4.6 Summary
275(1)
References
275(4)
Part II Principles and Guidelines for Damping Control
Chapter 5 Principles of Damper Design
279(64)
5.1 Modeling of Damping
279(16)
5.1.1 General Classifications of Damping
280(8)
5.1.1.1 Damping Ratios of Systems
280(1)
5.1.1.2 Damping Force of Systems
281(2)
5.1.1.3 Bilinear Damping
283(2)
5.1.1.4 Sublinear Damping
285(3)
5.1.2 Effective Damping Ratios for MDOF Systems
288(7)
5.1.2.1 Timoshenko Damping
288(6)
5.1.2.2 Force-Based Effective Damping
294(1)
5.2 Rectangular Law, Maximum Energy Dissipation per Device
295(8)
5.2.1 Maximum Energy Dissipation, Rectangular Law of Damping
296(1)
5.2.2 Smallest Maximum Dissipation, Rectangular Law of Seismic Work
297(4)
5.2.2.1 Minimum Work Done by Maximum Seismic Force
297(3)
5.2.2.2 Linearity of Nonlinear Responses
300(1)
5.2.3 Quality Factor
301(2)
5.2.4 Issues of Multiple Dynamic Equilibrium Positions
303(1)
5.3 Damping Adaptability
303(5)
5.3.1 Concept of Damping Adaptability
304(3)
5.3.2 Deformation Shape Function
307(1)
5.4 Design and Control Parameters
308(8)
5.4.1 Low Damping and High Damping Structures
308(6)
5.4.1.1 Control Parameters and Design Parameters of Linear Systems
308(3)
5.4.1.2 Necessity of Additional Design Parameters
311(3)
5.4.2 Issues of Damping Ratios
314(2)
5.5 Damping Force–Related Issues
316(23)
5.5.1 Recoverable Damping and Self-Centering Systems
316(5)
5.5.1.1 Recoverable Damping
317(2)
5.5.1.2 Self-Centering Structures
319(1)
5.5.1.3 Relation between Stiffness and Damping
319(2)
5.5.2 Working Frequency and Temperature of Dampers
321(2)
5.5.2.1 Effect on Self-Centering
321(1)
5.5.2.2 Frequency-Dependent and Temperature-Dependent Damping
321(2)
5.5.3 Damper Design Considering Supporting Stiffness
323(14)
5.5.3.1 Modeling
323(3)
5.5.3.2 Approximation
326(4)
5.5.3.3 Generalized Supporting Stiffness
330(1)
5.5.3.4 Design Considerations
331(6)
5.5.4 Damper Installation
337(2)
5.6 Summary
339(1)
References
339(4)
Chapter 6 System Nonlinearity and Damping of Irregular Structures
343(68)
6.1 Nonlinear Systems
343(25)
6.1.1 Classifications of Nonlinear Damping
343(4)
6.1.1.1 Control Parameters and Design Parameters of Nonlinear Systems
344(2)
6.1.1.2 Nonlinear Damper Classification
346(1)
6.1.2 Conventional Preliminary Estimation
347(17)
6.1.2.1 Nonlinear Dynamics
348(2)
6.1.2.2 Nonlinear Statics
350(5)
6.1.2.3 Engineering Issues
355(6)
6.1.2.4 Equivalent Linear SDOF Systems
361(3)
6.1.3 Consideration of Nonlinear Damping for MDOF Systems
364(2)
6.1.3.1 Generic Nonlinear Damping
364(1)
6.1.3.2 General Idea of Nonlinear Damping Design
365(1)
6.1.4 Yield Structure with Supplemental Damping: Simplified Approach
366(2)
6.1.4.1 Damping Ratio Summability
366(1)
6.1.4.2 Period Determination
367(1)
6.2 Irregular MDOF System
368(11)
6.2.1 Irregular Structures
368(2)
6.2.1.1 Effect of Structural Configuration
368(2)
6.2.1.2 Conventional Definitions
370(1)
6.2.2 Vertical Irregularity
370(1)
6.2.3 Plane Irregularity
371(3)
6.2.3.1 Principal Axes of Structures
371(2)
6.2.3.2 Cross Effect
373(1)
6.2.4 Irregular Damping
374(3)
6.2.4.1 Regular Mass-Stiffness, Irregular Damping
374(3)
6.2.5 Design Considerations for Nonproportionality Damping
377(1)
6.2.6 Response Estimation Using Response Spectra
378(1)
6.3 Minimizing Damping Nonproportionality
379(3)
6.3.1 Further Discussion on Conventional Energy Equation
379(3)
6.3.1.1 Minimization of Conservative Energy
379(3)
6.3.2 Minimization of Damping Nonproportionality
382(1)
6.4 Role of Damping in Nonlinear Systems
382(24)
6.4.1 Linear Systems
382(11)
6.4.1.1 Response Spectra of Seismic Work and Energy
382(6)
6.4.1.2 Brief Summary of Damping Effect for Linear Design Spectra
388(2)
6.4.1.3 Energy Equation in Linear Systems
390(2)
6.4.1.4 Limitation of Damping Control
392(1)
6.4.2 Nonlinear Damping and Nonlinear Systems
393(8)
6.4.2.1 Energy Dissipation in Nonlinear Systems
393(4)
6.4.2.2 Notes on Nonlinear Response Spectra
397(1)
6.4.2.3 Rule 0.65 and the Penzien Constant
397(3)
6.4.2.4 Nonlinear Spectra
400(1)
6.4.3 Inelastic Structure with Large Ductility
401(10)
6.4.3.1 Inelastic Structure with Supplemental Damping
401(1)
6.4.3.2 Criterion of Nonlinear Overdamping
402(1)
6.4.3.3 Energy Dissipations by Viscous and Bilinear Damping
402(1)
6.4.3.4 Seismic Responses of Inelastic Structures with Viscous Damping
403(1)
6.4.3.5 Biased Deformation of Inelastic Structures
404(1)
6.4.3.6 Force–Displacement Loop of Inelastic Deformation
405(1)
6.5 Summary
406(1)
References
407(4)
Part III Design of Supplemental Damping
Chapter 7 Linear Damping Design
411(64)
7.1 Overview of Design Approaches
411(4)
7.1.1 Design Philosophy
411(1)
7.1.2 Design Methods for Linear Systems
412(3)
7.2 MSSP Systems Simplified Approach
415(16)
7.2.1 General Description
415(1)
7.2.2 Feasibility of Damping Control
416(1)
7.2.3 SDOF and MSSP Systems
417(1)
7.2.4 Basic Design Procedure
418(13)
7.2.4.1 Estimation of Seismic Response of Original Structure
418(5)
7.2.4.2 Determination of Damping Ratio and Damping Coefficient
423(1)
7.2.4.3 Specifications of Dampers
424(7)
7.3 Proportionally Damped MDOF Systems Approach
431(10)
7.3.1 General Description
431(1)
7.3.2 Criterion for Modal Selection
431(1)
7.3.3 Basic Design Procedure
431(10)
7.3.3.1 Estimation of Seismic Response of Original Structure
431(5)
7.3.3.2 Determination of Damping Ratio and Damping Coefficient
436(1)
7.3.3.3 Selection of Damper
437(4)
7.4 Design of Generally Damped Systems
441(8)
7.4.1 Criteria for Generally Damped Systems
441(1)
7.4.2 Basic Design Procedure
441(8)
7.4.2.1 Estimation of Seismic Response of Structure with Dampers
441(6)
7.4.2.4 Redesign of Damping Devices
447(1)
7.4.2.5 Selection of Dampers
447(2)
7.5 Damper Design Issues
449(19)
7.5.1 Supporting Stiffness
449(4)
7.5.1.1 General Requirement
449(4)
7.5.2 Modification of Non-Timoshenko Damping
453(1)
7.5.3 Safety, Reliability, and Maintenance Issues
454(5)
7.5.3.1 Fail-Safe Concept
454(1)
7.5.3.2 Maximum Force in Dampers
454(1)
7.5.3.3 Stability of Damper System
455(2)
7.5.3.4 Combinations of Different Devices
457(1)
7.5.3.5 Safety Factors
457(1)
7.5.3.6 Reliability and Maintenance
458(1)
7.5.4 Numerical Damping Coefficient
459(6)
7.5.4.1 Concept of Numerical Damping Coefficient B
459(3)
7.5.4.2 Modification of Design Spectrum Based on Period Range
462(3)
7.5.5 Modified SRSS
465(3)
7.5.5.1 Absolute Acceleration
465(2)
7.5.5.2 Incomplete Modes
467(1)
7.6 Damper Design Codes
468(1)
7.7 Brief Summary of Damping Design of Linear Systems
468(5)
7.7.1 Major Step (1) Decision Making
468(2)
7.7.2 Major Step (2) Modal Analysis
470(1)
7.7.3 Major Step (3) Spectral Values
471(1)
7.7.4 Major Step (4) Model Responses
471(1)
7.7.5 Major Step (5) Criteria of Supplemental Damping
471(1)
7.7.6 Major Step (6) Design Damping Ratio, Specifications of Damping Devices, Reevaluation of Total Design
472(1)
References
473(2)
Chapter 8 Nonlinear Damping
475(74)
8.1 Overview of Design Approaches
475(5)
8.1.1 General Description
475(2)
8.1.1.1 Condition of Using Supplemental Damping
476(1)
8.1.1.2 Amount of Damping
476(1)
8.1.1.3 Type of Damping Devices
476(1)
8.1.1.4 Models for Reevaluation of Structural Responses
476(1)
8.1.1.5 Structural Ductility
477(1)
8.1.2 Types of Damping
477(2)
8.1.2.1 Bilinear Damping
477(1)
8.1.2.2 Sublinear Damping
478(1)
8.1.2.3 Basic Differences between Bilinear and Sublinear Damping
478(1)
8.1.2.4 Other Types of Dampers
479(1)
8.1.3 Design Procedures
479(1)
8.1.3.1 Equivalent Linear Systems
479(1)
8.1.3.2 Nonlinear Response Spectra
479(1)
8.2 Equivalent Linear Systems Approach with Bilinear Dampers
480(20)
8.2.1 General Description
480(8)
8.2.1.1 Selection of Design Models
480(1)
8.2.1.2 Response Estimation and First Round of Damper Design
480(8)
8.2.2 Response Estimation
488(4)
8.2.2.1 Equation of Motion
488(2)
8.2.2.2 Response Reevaluation
490(2)
8.2.3 Design Issues
492(5)
8.2.3.1 SDOF Systems
492(4)
8.2.3.2 MDOF Systems
496(1)
8.2.3.3 Brief Summary
496(1)
8.2.4 Damper Specification
497(3)
8.2.4.1 Basic Parameters
497(3)
8.2.4.2 Selection of Bilinear Dampers
500(1)
8.3 Equivalent Linear Systems Approach with Sublinear Dampers
500(15)
8.3.1 General Description
500(2)
8.3.1.1 Response Estimation and First Round of Damper Design
501(1)
8.3.2 Response Estimation
502(5)
8.3.2.1 Mode Shape Computations
502(1)
8.3.2.2 Response Reevaluation
503(3)
8.3.2.3 Summary of Simplified Design
506(1)
8.3.3 Design Issues
507(8)
8.3.3.1 SDOF Systems: Effective Mode
507(6)
8.3.3.2 MDOF Systems
513(2)
8.3.4 Damper Specifications
515(1)
8.4 Nonlinear Response Spectra Approach with Sublinear Dampers
515(10)
8.4.1 General Description
515(1)
8.4.1.1 Control Parameters and Response Spectra
515(1)
8.4.2 Response Estimation
515(5)
8.4.2.1 Linear Interpolations
517(3)
8.4.2.2 MDOF Systems
520(1)
8.4.3 Design Issues
520(5)
8.5 Nonlinear Response Spectra Approach with Bilinear Dampers
525(19)
8.5.1 General Description
525(3)
8.5.1.1 Supplemental Damping and Inelastic Structures
525(1)
8.5.1.2 Displacement-Based Design
526(2)
8.5.2 Response Estimation
528(15)
8.5.2.1 Overview
528(2)
8.5.2.2 Response Estimation Based on Bilinear Spectra
530(1)
8.5.2.3 Bilinear Design Spectra
530(13)
8.5.3 Design Issues
543(1)
8.6 Summary
544(3)
8.6.1 Preliminary Decision Making
544(1)
8.6.2 Initial Damping Design
545(1)
8.6.3 Response Estimation with Proper Model of Damping
545(1)
8.6.3.1 Parameters of an SDOF System
545(1)
8.6.3.2 Shape Function and Modal Participation Factor
546(1)
8.6.3.3 Trial-and-Error Iteration
546(1)
8.6.4 Selection of Damping Devices
546(1)
References
547(2)
Index 549
Dr. Zach Liang is a research professor in the Department of Mechanical and Aerospace Engineering at the State University of New York at Buffalo.

Dr. George C. Lee is a SUNY Distinguished Professor in the Department of Civil, Structural and Environmental Engineering at the State University of New York at Buffalo.

Dr. Gary F. Dargush is Professor and Chair of the Department of Mechanical and Aerospace Engineering (MAE) at the State University of New York at Buffalo.

Dr. Jianwei Song is a senior research scientist in the Multidisciplinary Center for Earthquake Engineering Research in the Department of Civil, Structural and Environmental Engineering at the State University of New York at Buffalo.
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