152
Index
Frequency domain analysis, 32–39
Fourier series representation, 107–108
matrix exponential, 110–113
steady-state solution
non-proportional damping, 109–110
proportional damping, 108–109
vector-valued forcing functions, 107
G
General homogeneous solution, 12–17
Generalized eigenvectors, 147–149
H
Harmonic force
damped system, 28–29
undamped system
beating phenomenon, 27–28
resonance phenomenon, 26–27
I
Impulse-momentum (IM), 5–7
L
Linear algebraic equations, 55–56
Linear differential equations, 135–139
M
Mass matrix, 95
Matrix exponential, 110–113
Modal combination rules, 91
Modal coordinates, 102
Modal damping ratios, 76
Modal frequencies
discrete approximation, 74
eigenvalues, 70
experimental determination, 92–93
homogeneous system, 70
linear space, 72
n-DOF structure, 72
properties, 71–72
square roots, 70
Modal participation factor, 90
Moving loads, 47–49
Multi-degree of freedom (MDOF) systems
classical modal analysis
displacement vector representations,
76–77
homogeneous system, 70
modal frequencies/shapes, 70–72
proportional damping, 75–76
vibration solution, 69
continuous systems, 67
displacement vectors/functions, 67
equations of motion, 68–69
linear spaces, 67
physical system, 68
N
Natural frequency, 15
Natural period, 15
Newton’s second law, 5–8
Non-proportional damping
eigenvalues, 99–102
eigenvectors, 99–102
forced vibration, 102–103
free vibration, 103–106
state-space representation, 98
Non symmetric matrices, 63–65
Numerical methods
finite difference (FD), 39–42
Fourier series (FS) method, 42–44
P
Particular inhomogeneous solutions, 10
Physical system, 8–9
Proportional damping, 75–76, 108–109
R
Resonance phenomenon, 26–27
Response spectra, 45–47
Road roughness, 1
S
Seismic ground acceleration, 89–91
Shear beams
forced vibration, 128–129
free vibration, 129–130
modal shapes and frequencies, 125–128
physical system, 124–125
Simple forcing functions, 17–20
Single degree of freedom (SDOF), 2
coulomb damping model, 49–50
equation of motion, 9
frequency domain analysis, 32–39
moving loads, 47–49
newton’s second law, 5–8
numerical methods, 39–44
physical system, 8–9
response spectra, 45–47
Index
Frequency domain analysis, 32–39
Fourier series representation, 107–108
matrix exponential, 110–113
steady-state solution
non-proportional damping, 109–110
proportional damping, 108–109
vector-valued forcing functions, 107
G
General homogeneous solution, 12–17
Generalized eigenvectors, 147–149
H
Harmonic force
damped system, 28–29
undamped system
beating phenomenon, 27–28
resonance phenomenon, 26–27
I
Impulse-momentum (IM), 5–7
L
Linear algebraic equations, 55–56
Linear differential equations, 135–139
M
Mass matrix, 95
Matrix exponential, 110–113
Modal combination rules, 91
Modal coordinates, 102
Modal damping ratios, 76
Modal frequencies
discrete approximation, 74
eigenvalues, 70
experimental determination, 92–93
homogeneous system, 70
linear space, 72
n-DOF structure, 72
properties, 71–72
square roots, 70
Modal participation factor, 90
Moving loads, 47–49
Multi-degree of freedom (MDOF) systems
classical modal analysis
displacement vector representations,
76–77
homogeneous system, 70
modal frequencies/shapes, 70–72
proportional damping, 75–76
vibration solution, 69
continuous systems, 67
displacement vectors/functions, 67
equations of motion, 68–69
linear spaces, 67
physical system, 68
N
Natural frequency, 15
Natural period, 15
Newton’s second law, 5–8
Non-proportional damping
eigenvalues, 99–102
eigenvectors, 99–102
forced vibration, 102–103
free vibration, 103–106
state-space representation, 98
Non symmetric matrices, 63–65
Numerical methods
finite difference (FD), 39–42
Fourier series (FS) method, 42–44
P
Particular inhomogeneous solutions, 10
Physical system, 8–9
Proportional damping, 75–76, 108–109
R
Resonance phenomenon, 26–27
Response spectra, 45–47
Road roughness, 1
S
Seismic ground acceleration, 89–91
Shear beams
forced vibration, 128–129
free vibration, 129–130
modal shapes and frequencies, 125–128
physical system, 124–125
Simple forcing functions, 17–20
Single degree of freedom (SDOF), 2
coulomb damping model, 49–50
equation of motion, 9
frequency domain analysis, 32–39
moving loads, 47–49
newton’s second law, 5–8
numerical methods, 39–44
physical system, 8–9
response spectra, 45–47