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Contents
3 Forced Vibration of Single Degree of Freedom System . . . . . . . . . . . . 59
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2
Response of Damped Systems to Harmonic Loading . . . . . . . . . 59
3.3
Rotating Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4
Reciprocating Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5
Whirling of Rotating Shafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6
Vibration Isolation and Transmissibility . . . . . . . . . . . . . . . . . . . . 73
3.6.1
Transmissibility Due to Support Motions . . . . . . . . . . . 75
3.7
Energy Dissipation by Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.8
Equivalent Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.9
Self-excited Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.10 Vibration Measuring Seismic Instruments . . . . . . . . . . . . . . . . . . . 86
3.10.1 Vibrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.10.2 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.10.3 Discussion About the Instruments . . . . . . . . . . . . . . . . . 89
3.11 Response of Structures Due to Transient Vibration . . . . . . . . . . . 90
3.11.1 Response of SDF System to an Ideal Step Input . . . . . 90
3.11.2 Response of SDF System to Gradually Applied
Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.12 Response to SDF Systems to a General Type of Forcing
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.13 Dynamic Load Factor and Response Spectrum . . . . . . . . . . . . . . 97
3.14 Response Due to Periodic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.14.1 Real Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.14.2 Response of SDF System to Periodic Forces
Represented by Real Fourier Series . . . . . . . . . . . . . . . 101
3.14.3 Complex Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.14.4 Response of SDF System to Periodic Forces
Represented by Complex Fourier Series . . . . . . . . . . . . 104
3.15 Response Due to Non-periodic Excitation . . . . . . . . . . . . . . . . . . . 106
3.16 Relationship Between Complex Frequency Response
Function and Unit Impulse Response Function . . . . . . . . . . . . . . 109
3.17 Support Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.17.1 Displacement Approach . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.17.2 Acceleration Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.18 Response of SDF Systems Related to Earthquakes . . . . . . . . . . . 113
3.19 Techniques for Analysing Earthquake Response . . . . . . . . . . . . . 115
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4 Numerical Methods in Structural Dynamics: Applied to SDF
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.2
Direct Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.2.1
Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 130
4.2.2
Linear Acceleration Method . . . . . . . . . . . . . . . . . . . . . . 135
Contents
3 Forced Vibration of Single Degree of Freedom System . . . . . . . . . . . . 59
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2
Response of Damped Systems to Harmonic Loading . . . . . . . . . 59
3.3
Rotating Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4
Reciprocating Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5
Whirling of Rotating Shafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6
Vibration Isolation and Transmissibility . . . . . . . . . . . . . . . . . . . . 73
3.6.1
Transmissibility Due to Support Motions . . . . . . . . . . . 75
3.7
Energy Dissipation by Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.8
Equivalent Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.9
Self-excited Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.10 Vibration Measuring Seismic Instruments . . . . . . . . . . . . . . . . . . . 86
3.10.1 Vibrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.10.2 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.10.3 Discussion About the Instruments . . . . . . . . . . . . . . . . . 89
3.11 Response of Structures Due to Transient Vibration . . . . . . . . . . . 90
3.11.1 Response of SDF System to an Ideal Step Input . . . . . 90
3.11.2 Response of SDF System to Gradually Applied
Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.12 Response to SDF Systems to a General Type of Forcing
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.13 Dynamic Load Factor and Response Spectrum . . . . . . . . . . . . . . 97
3.14 Response Due to Periodic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.14.1 Real Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.14.2 Response of SDF System to Periodic Forces
Represented by Real Fourier Series . . . . . . . . . . . . . . . 101
3.14.3 Complex Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.14.4 Response of SDF System to Periodic Forces
Represented by Complex Fourier Series . . . . . . . . . . . . 104
3.15 Response Due to Non-periodic Excitation . . . . . . . . . . . . . . . . . . . 106
3.16 Relationship Between Complex Frequency Response
Function and Unit Impulse Response Function . . . . . . . . . . . . . . 109
3.17 Support Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.17.1 Displacement Approach . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.17.2 Acceleration Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.18 Response of SDF Systems Related to Earthquakes . . . . . . . . . . . 113
3.19 Techniques for Analysing Earthquake Response . . . . . . . . . . . . . 115
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4 Numerical Methods in Structural Dynamics: Applied to SDF
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.2
Direct Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.2.1
Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 130
4.2.2
Linear Acceleration Method . . . . . . . . . . . . . . . . . . . . . . 135