Conceptual Theories in Structural Dynamics

227 Pages·2020·8.22 MB·English

by  Farzad Hejazi, Tan Kar Chun

Advanced Structured Materials
Farzad Hejazi
Tan Kar Chun
Conceptual
Theories in
Structural
Dynamics
Advanced Structured Materials
Volume 135
Series Editors
Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of
Applied Sciences, Esslingen, Germany
Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of
Engineering, University of Porto, Porto, Portugal
Holm Altenbach , Faculty of Mechanical Engineering,
Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany
Common engineering materials reach in many applications their limits and new
developments are required to fulfil increasing demands on engineering materials.
The performance ofmaterials can beincreasedby combiningdifferent materials to
achieve better properties than a single constituent or by shaping the material or
constituents in a specific structure. The interaction between material and structure
mayariseondifferentlengthscales,suchasmicro-,meso-ormacroscale,andoffers
possible applications in quite diverse fields.
Thisbookseriesaddressesthefundamentalrelationshipbetweenmaterialsandtheir
structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic
etc.) and applications.
The topics of Advanced Structured Materials include but are not limited to
(cid:129) classical fibre-reinforced composites (e.g. glass, carbon or Aramid reinforced
plastics)
(cid:129) metal matrix composites (MMCs)
(cid:129) micro porous composites
(cid:129) micro channel materials
(cid:129) multilayered materials
(cid:129) cellular materials (e.g., metallic or polymer foams, sponges, hollow sphere
structures)
(cid:129) porous materials
(cid:129) truss structures
(cid:129) nanocomposite materials
(cid:129) biomaterials
(cid:129) nanoporous metals
(cid:129) concrete
(cid:129) coated materials
(cid:129) smart materials
Advanced Structured Materials is indexed in Google Scholar and Scopus.
More information about this series at http://www.springer.com/series/8611
Farzad Hejazi Tan Kar Chun
(cid:129)
Conceptual Theories
in Structural Dynamics
123
Farzad Hejazi TanKar Chun
Department ofCivil Engineering Department ofCivil Engineering
University PutraMalaysia University PutraMalaysia
Serdang,Selangor, Malaysia Serdang,Selangor, Malaysia
ISSN 1869-8433 ISSN 1869-8441 (electronic)
AdvancedStructured Materials
ISBN978-981-15-5439-1 ISBN978-981-15-5440-7 (eBook)
https://doi.org/10.1007/978-981-15-5440-7
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Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Static and Dynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Degree of Freedom (DOF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Components of Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Mass Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Stiffness Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Damping Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Idealized Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Free Vibration of Single Degree of Freedom System . . . . . . . . . . . . 29
3.1 Free Vibration of Undamped System . . . . . . . . . . . . . . . . . . . . . 29
3.2 Free Vibration of Damped System. . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1 Critical Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Overdamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.3 Underdamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Logarithmic Decremental. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Force Vibration of Single Degree of Freedom System. . . . . . . . . . . . 61
4.1 Force Vibration of Undamped System . . . . . . . . . . . . . . . . . . . . 62
4.2 Force Vibration of Damped System . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Dynamic Magnification Factor. . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 Frequency Ratio Nearly Equal to Zero ðbffi0Þ . . . . . . . . 83
4.3.2 Frequency Ratio Is Much Greater Than One ðb(cid:3)1Þ. . . . 84
4.3.3 Frequency Ratio Is Equal to One ðb¼1Þ . . . . . . . . . . . . 86
v
vi Contents
4.4 Dynamic Response Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Single Degree of Freedom Systems Subjected to Various
Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.1 Harmonic Excitation by Generator Machine. . . . . . . . . . . 95
5.1.2 Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.3 Harmonic Motion of Foundation (Motion Excitation) . . . . 100
5.1.4 Force Transmissibility (Force Excitation). . . . . . . . . . . . . 106
5.1.5 Vibration Measurement Instrument . . . . . . . . . . . . . . . . . 110
5.2 Impact Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2.1 Duhamel’s Integral for Undamped System. . . . . . . . . . . . 112
5.2.2 Constant Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2.3 Rectangular Load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.4 Triangular Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3 Earthquake Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3.1 Spectra Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.2 Base Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.4 Numerical Evaluation of Dynamic Response . . . . . . . . . . . . . . . 133
5.4.1 Central Difference Method (CDM) . . . . . . . . . . . . . . . . . 133
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6 Multi-degree of Freedom System . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.1 Equation of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2 Multi-degree of Freedom for Undamped System
Under Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.3 Normalization of Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.4 Orthogonality of Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.5 Multi-degree of Freedom System Subjected to Impact Load . . . . 164
6.6 Multi-degree of Freedom System Subjected to Earthquake
Excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.8 Rayleigh’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.9 Holzer’s Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Appendix. .... .... .... .... ..... .... .... .... .... .... ..... .... 213
References.... .... .... .... ..... .... .... .... .... .... ..... .... 219
List of Figures
Fig. 1.1 An at rest pendulum.. ..... .... .... .... .... .... ..... .. 2
Fig. 1.2 A falling pendulum... ..... .... .... .... .... .... ..... .. 2
Fig. 1.3 Change of dead load with respect to time... .... .... ..... .. 3
Fig. 1.4 Degree offreedom of a body .... .... .... .... .... ..... .. 4
Fig. 1.5 Spring–mass system in simple harmonic motion.. .... ..... .. 5
Fig. 2.1 Concept of point mass ..... .... .... .... .... .... ..... .. 8
Fig. 2.2 Reinforce concrete as a heterogeneous material .. .... ..... .. 8
Fig. 2.3 Spring under tension.. ..... .... .... .... .... .... ..... .. 10
Fig. 2.4 Beam under bending.. ..... .... .... .... .... .... ..... .. 11
Fig. 2.5 Deformation of two member in different length .. .... ..... .. 12
Fig. 2.6 Beams deflection. .... ..... .... .... .... .... .... ..... .. 12
Fig. 2.7 Damping of spring-mass system.. .... .... .... .... ..... .. 13
Fig. 2.8 Rotation at joint in frame ... .... .... .... .... .... ..... .. 14
Fig. 2.9 Single concrete member response to movement .. .... ..... .. 15
Fig. 2.10 A functioning viscous damper.... .... .... .... .... ..... .. 16
Fig. 2.11 Structural motion as response to external force... .... ..... .. 17
Fig. 2.12 Symbol of dynamic system components.... .... .... ..... .. 18
Fig. 2.13 Series system ... .... ..... .... .... .... .... .... ..... .. 19
Fig. 2.14 Parallel system .. .... ..... .... .... .... .... .... ..... .. 20
Fig. 2.15 Example 2.1 .... .... ..... .... .... .... .... .... ..... .. 22
Fig. 2.16 Example 2.2 .... .... ..... .... .... .... .... .... ..... .. 22
Fig. 2.17 Example 2.3 .... .... ..... .... .... .... .... .... ..... .. 23
Fig. 2.18 Example 2.4 .... .... ..... .... .... .... .... .... ..... .. 24
Fig. 2.19 Solution for Example 2.4 ... .... .... .... .... .... ..... .. 26
Fig. 2.20 Exercise 2.1 .... .... ..... .... .... .... .... .... ..... .. 26
Fig. 2.21 Exercise 2.2 .... .... ..... .... .... .... .... .... ..... .. 27
Fig. 2.22 Exercise 2.3 .... .... ..... .... .... .... .... .... ..... .. 27
Fig. 2.23 Exercise 2.4 .... .... ..... .... .... .... .... .... ..... .. 28
Fig. 3.1 Analytical models for SDOF system... .... .... .... ..... .. 30
Fig. 3.2 Idealization of structure with a base isolation system .. ..... .. 30
vii
viii ListofFigures
Fig. 3.3 Analytical models for undamped SDOF system under free
vibration... .... .... ..... .... .... .... .... .... ..... .. 31
Fig. 3.4 Displacement response of undamped system under free
vibration... .... .... ..... .... .... .... .... .... ..... .. 34
Fig. 3.5 Example 3.1 .... .... ..... .... .... .... .... .... ..... .. 35
Fig. 3.6 Displacement, velocity and acceleration response
for the frame in Example 3.1 .... .... .... .... .... ..... .. 37
Fig. 3.7 Example 3.2 .... .... ..... .... .... .... .... .... ..... .. 38
Fig. 3.8 Example 3.3 .... .... ..... .... .... .... .... .... ..... .. 39
Fig. 3.9 Analytical models for damped SDOF system under free
vibration... .... .... ..... .... .... .... .... .... ..... .. 41
Fig. 3.10 Displacementresponseofdampedsystem under free vibration
(critical damping condition) . .... .... .... .... .... ..... .. 44
Fig. 3.11 Displacementresponseofdampedsystem under free vibration
(critical and overdamping condition)... .... .... .... ..... .. 45
Fig. 3.12 Displacement response of damped system under
free vibration.... .... ..... .... .... .... .... .... ..... .. 49
Fig. 3.13 Example 3.4 .... .... ..... .... .... .... .... .... ..... .. 50
Fig. 3.14 Dynamic response for Example 3.4 ... .... .... .... ..... .. 52
Fig. 3.15 Undamped system under free vibration. .... .... .... ..... .. 53
Fig. 3.16 Exact and approximated solution of logarithmic decrement .. .. 55
Fig. 3.17 Example 3.5 .... .... ..... .... .... .... .... .... ..... .. 57
Fig. 3.18 Exercise 3.1 .... .... ..... .... .... .... .... .... ..... .. 58
Fig. 3.19 Exercise 3.2 .... .... ..... .... .... .... .... .... ..... .. 58
Fig. 3.20 Exercise 3.3 .... .... ..... .... .... .... .... .... ..... .. 59
Fig. 3.21 Exercise 3.4 .... .... ..... .... .... .... .... .... ..... .. 59
Fig. 4.1 Analytical models for undamped SDOF system under
force vibration... .... ..... .... .... .... .... .... ..... .. 62
Fig. 4.2 Example 4.1 .... .... ..... .... .... .... .... .... ..... .. 70
Fig. 4.3 Analytical models for damped SDOF system under force
vibration... .... .... ..... .... .... .... .... .... ..... .. 72
Fig. 4.4 Trigonometry relationship for 2nb and 1(cid:4)b2
in sine function.. .... ..... .... .... .... .... .... ..... .. 76
Fig. 4.5 Trigonometry relationship for 2nb and 1(cid:4)b2 in cosine
function.... .... .... ..... .... .... .... .... .... ..... .. 81
Fig. 4.6 Damping magnification factor for various frequency
ratio and damping ratio..... .... .... .... .... .... ..... .. 83
Fig. 4.7 Corresponding response for B nearly equal to 0.. .... ..... .. 84
Fig. 4.8 Corresponding response for B greater than 1 .... .... ..... .. 86
Fig. 4.9 Correlation between frequency ratio and phase angle.. ..... .. 87
Fig. 4.10 Example 4.2 .... .... ..... .... .... .... .... .... ..... .. 90
Fig. 4.11 Example 4.3 .... .... ..... .... .... .... .... .... ..... .. 91
Fig. 4.12 Exercise 4.1 .... .... ..... .... .... .... .... .... ..... .. 93
Fig. 4.13 Example 4.2 .... .... ..... .... .... .... .... .... ..... .. 93
ListofFigures ix
Fig. 5.1 Vibration generated by a rotating machine .. .... .... ..... .. 96
Fig. 5.2 Example 5.1 .... .... ..... .... .... .... .... .... ..... .. 98
Fig. 5.3 Vibration isolation for a force and b motion excitations..... .. 99
Fig. 5.4 Analytical model of ground harmonic motion.... .... ..... .. 100
Fig. 5.5 Trigonometry relationship for cxf and k under motion
excitation .. .... .... ..... .... .... .... .... .... ..... .. 101
Fig. 5.6 Transmissibility ratio of structure system ... .... .... ..... .. 103
Fig. 5.7 Example 5.2 .... .... ..... .... .... .... .... .... ..... .. 104
Fig. 5.8 Analytical model offorce transmissibility... .... .... ..... .. 106
Fig. 5.9 Trigonometry relationship for cxf and k under
force excitation.. .... ..... .... .... .... .... .... ..... .. 107
Fig. 5.10 Example 5.3 .... .... ..... .... .... .... .... .... ..... .. 108
Fig. 5.11 Analytical model for vibration measurement. .... .... ..... .. 110
Fig. 5.12 General load function as impulsive loading . .... .... ..... .. 112
Fig. 5.13 Mass–spring system acted upon by constant load. .... ..... .. 114
Fig. 5.14 Dynamic response of undamped system due to sudden
applied constant force. ..... .... .... .... .... .... ..... .. 115
Fig. 5.15 Rectangular constant force acting on mass–spring system.... .. 116
Fig. 5.16 Triangular dynamic force acted upon undamped oscillator... .. 118
Fig. 5.17 Maximumdynamicfactorforrectangularandtriangularforces
(Paz and Leigh 2004). ..... .... .... .... .... .... ..... .. 123
Fig. 5.18 Example 5.4 .... .... ..... .... .... .... .... .... ..... .. 124
Fig. 5.19 Analytical model of ground motion caused by acceleration
excitation .. .... .... ..... .... .... .... .... .... ..... .. 126
Fig. 5.20 Response spectra for elastic system based on El Centro
earthquake recorded in year 1940 (Paz and Leigh 2004) .... .. 130
Fig. 5.21 Example 5.5 .... .... ..... .... .... .... .... .... ..... .. 132
Fig. 5.22 Central difference method (CDM). .... .... .... .... ..... .. 134
Fig. 5.23 Example 5.6 .... .... ..... .... .... .... .... .... ..... .. 137
Fig. 5.24 Exercise 5.1 .... .... ..... .... .... .... .... .... ..... .. 140
Fig. 5.25 Exercise 5.2 .... .... ..... .... .... .... .... .... ..... .. 140
Fig. 5.26 Exercise 5.3 .... .... ..... .... .... .... .... .... ..... .. 141
Fig. 5.27 Exercise 5.4 .... .... ..... .... .... .... .... .... ..... .. 141
Fig. 6.1 Analytical models for MDOF system .. .... .... .... ..... .. 144
Fig. 6.2 Three-storey building and its MDOF analytical model . ..... .. 144
Fig. 6.3 Undamped 2-DOF system under free vibration... .... ..... .. 146
Fig. 6.4 Normal modes of vibration.. .... .... .... .... .... ..... .. 150
Fig. 6.5 Two-storey building as undamped 2-DOF system under
free vibration.... .... ..... .... .... .... .... .... ..... .. 151
Fig. 6.6 Example 6.1 .... .... ..... .... .... .... .... .... ..... .. 156
Fig. 6.7 Idealization of structure for Example 6.1 ... .... .... ..... .. 157
Fig. 6.8 Normal modes for structure in Example 6.1. .... .... ..... .. 164
Fig. 6.9 2-DOF system subjected to impact load .... .... .... ..... .. 164
Fig. 6.10 Example 6.2 .... .... ..... .... .... .... .... .... ..... .. 168