Lecture Note for IISEE
                Fundamentals of
              Structural Dynamics
                  Dr. Izuru Okawa
               Building Research Institute
              Prof. Emeritus Yuji Ishiyama
                 Hokkaido University
                Dr. Makoto WATABE
                     Deceased
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
           This lecture note was originally written by Dr. Makoto Watabe and Dr. Yuji Ishiyama 
         for the participants of the International Institute of Seismology and Earthquake Engi-
         neering (IISEE), Building Research Institute (BRI). After that, Dr. Izuru Okawa revised 
         the note, adding descriptions, examples how to solve questions, etc. Then Prof. Yuji Ishi-
         yama who moved from BRI to Hokkaido University further revised the note for the 
         graduate students of English Graduate Program for Socio-Enviromental Engineering 
         (EGPSEE), Graduate School of Engineering, Hokkaido University. 
           The authors do not intend that the note be used for professional engineers or highly 
         educated researchers, but for those who have just started learning structural engineering. 
         Therefore the note only contains the fundamental concepts in structural dynamics. The 
         authors hope that readers will look into professional books to understand the background 
         in more detail. 
           If you have questions, suggestions, or comments on this lecture note, please write to 
         us. We thank you in advance for taking the time and interest to do so. 
                                         Dr. Izuru Okawa 
                                         Dr. Yuji Ishiyama 
                 Contents
                 1 Introduction                                                                                    5
                 2 Single Degree of Freedom (SDOF) Systems                                                         7
                     2.1   Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        8
                     2.2   Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      8
                           (1) Undamped Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . .           9
                           (2) Damped Free Vibration        . . . . . . . . . . . . . . . . . . . . . . . . . .   13
                     2.3   Response to Harmonic Loading . . . . . . . . . . . . . . . . . . . . . . . .           18
                           (1) Undamped Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         18
                           (2) Underdamped Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .          20
                           (3) Accelerometer and Displacement Meter . . . . . . . . . . . . . . . . . .           25
                           (4) Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      26
                     2.4   Response to Arbitrary Loading        . . . . . . . . . . . . . . . . . . . . . . . .   28
                           (1) Linear Acceleration Method . . . . . . . . . . . . . . . . . . . . . . . .         28
                           (2) Duhamel Integral - Convolution Integral . . . . . . . . . . . . . . . . .          32
                           (3) Response Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        33
                     2.5   Principle of Virtual Work - Generalized SDOF Systems . . . . . . . . . . .             37
                     2.6   Rayleigh’s Method - Vibration of Continuous Members . . . . . . . . . . .              39
                           (1) Basis for the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .       39
                           (2) Approximate Analysis of General Systems . . . . . . . . . . . . . . . .            40
                           (3) Selection of Shape Function . . . . . . . . . . . . . . . . . . . . . . . .        43
                           (4) Improved Rayleigh’s Method . . . . . . . . . . . . . . . . . . . . . . . .         44
                     2.7   Frequency Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .          47
                           (1) Complex Requency Response Function . . . . . . . . . . . . . . . . . .             47
                           (2) Responce to Arbitrary Excitation . . . . . . . . . . . . . . . . . . . . .         48
                           (3) Complex Frequency Response Function and Unit Impulse Function . . 49
                           (4) Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . .       49
                 3 Multi Degree of Freedom (MDOF) Systems                                                         51
                     3.1   Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        51
                     3.2   Undamped Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . .          55
                     3.3   Orthogonality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .       58
                     3.4   Concept of Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . .          60
                     3.5   Damping Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         62
                           (1) Voigt Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      63
                                                                  3
                 4                                                                                    CONTENTS
                           (2) Maxwell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        65
                           (3) Hysteretic Damping Model . . . . . . . . . . . . . . . . . . . . . . . . .         66
                           (4) Comparison of Damping Ratios . . . . . . . . . . . . . . . . . . . . . .           66
                     3.6   Stodola (Matrix Iteration) Method . . . . . . . . . . . . . . . . . . . . . .          67
                           (1) Procedure of Stodola method . . . . . . . . . . . . . . . . . . . . . . .          67
                           (2) Proof of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . .       68
                           (3) Analysis of Higher Modes       . . . . . . . . . . . . . . . . . . . . . . . . .   68
                     3.7   Holzer Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        71
                     3.8   Mode Superposition and Modal Analysis . . . . . . . . . . . . . . . . . . .            76
                           (1) Derivation of Equations      . . . . . . . . . . . . . . . . . . . . . . . . . .   76
                           (2) Mode Superposition       . . . . . . . . . . . . . . . . . . . . . . . . . . . .   81
                           (3) Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       82
                           (3) Mothods for the Estimation of Maximum Responses . . . . . . . . . . .              83
                     3.9   Solution by Step-by-step Integration Method . . . . . . . . . . . . . . . . .          89
                 4 Nonlinear Analysis                                                                             93
                     4.1   Outline of Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . . .        93
                     4.2   Nonlinear Response of SDOF Systems            . . . . . . . . . . . . . . . . . . . .  93
                     4.3   Nonlinear Response of MDOF Systems . . . . . . . . . . . . . . . . . . . .             95