كتاب Stress, Strain, and Structural Dynamics
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 كتاب Stress, Strain, and Structural Dynamics

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أحضرت لكم كتاب
Stress, Strain, and Structural Dynamics
An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes
Bingen Yang
University of Southern California

كتاب Stress, Strain, and Structural Dynamics  S_s_a_11
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Contents
Preface xv
Acknowledgements xvii
Chapter 1 Introduction 1
1.1 The Making of This Book 1
1.2 How to Use This Book 4
Part I Strength of Materials 7
Chapter 2
Chapter 3
Static Analysis of Euler-Bernoulli Beams
Getting Started
Beam Theory
2.2.1 Static Problem
2.2.2 Solution Methods
2.2.3 Fundamentals of Euler-Bernoulli Beams
Static Analysis by the Toolbox
2.3.1 System Setup
2.3.2 Response to External Forces
2.3.3 Response to Boundary Disturbances
2.3.4 Total Response
2.3.5 Exact Analytical Solutions
2.3.6 Other Useful Functions
Moments of Inertia of Beam Cross-Section Area
Quick Solution Guide
References
Static Analysis of Bars, Shafts, and Strings
3.1
3.2
3.3
Getting Started
System Description
Static Analysis by the Toolbox
3.3.1 System Setup
9
9
11
11
16
23
25
26
28
33
34
36
40
42
46
51
53
53
55
59
59
VII 3.3.2 Response to External Forces 60
3.3.3 Response to Boundary Disturbances 63
3.3.4 Total Response 65
3.4 Stepped Bars and Shafts 67
3.4.1 System Description 67
3.4.2 Solution by the Toolbox 68
3.5 The Distributed Transfer Function Method 74
3.6 Quick Solution Guide 78
3.7 References 83
C h a p t e r 4 Buckling Analysis of Columns 85
4.1 Getting Started 85
4.2 Uniform Columns 87
4.2.1 Column Buckling Theory 87
4.2.2 Solution by the Toolbox 91
4.2.3 Eccentric Loading 96
4.2.4 Beam-Column Problem 100
4.2.5 Geometric Imperfection 104
4.3 Stepped and Nonuniform Columns 109
4.3.1 Constrained Stepped Columns 109
4.3.2 Nonuniform Columns 116
4.4 The Distributed Transfer Function Method 123
4.5 Quick Solution Guide 127
4.6 References 133
C h a p t e r 5 Stress Analysis in Two-Dimensional Problems 135
5.1 Getting Started 135
5.2 Plane Stress and Strain 136
5.2.1 In-Plane Stresses 136
5.2.2 In-Plane Strains 142
5.2.3 Strain Rosette 147
5.2.4 Hooke's Law 149
5.2.5 Criteria of Failure 151
5.3 Quick Solution Guide 155
5.4 References 156
Part II Structural Mechanics 157
Chapter 6 Static Analysis of Constrained Multispan Beams
6.1 Getting Started
6.2 System Description
6.3 Solution by the Toolbox
6.3.1 System Setup
6.3.2 Response to External Forces
6.3.3 Response to Boundary Disturbances
6.3.4 Response to Support Settlement
6.3.5 Total Response
6.3.6 Influence Lines
6.4 The Distributed Transfer Function Method
159
159
161
166
166
171
176
178
181
183
189
viii Contents 6.5 Quick Solution Guide
6.6 References 200
Chapter 7
Chapter 8
Static Analysis of Plane Trusses
7.1
7.2
7.3
7.4
7.5
7.6
Stati
8.1
8.2
8.3
8.4
8.5
8.6
Getting Started
System Description
Solution by the Toolbox
7.3.1 System Setup
7.3.2 Response to External Forces
7.3.3 Support Settlement
7.3.4 Fabrication Errors and Thermal Effects
7.3.5 Total Response
The Stiffness Method
Quick Solution Guide
References
c Analysis of Plane Frames
Getting Started
System Description
Solution by the Toolbox
8.3.1 System Setup
8.3.2 Response to External Forces
8.3.3 Response to Settlement of Supports
8.3.4 Total Response
8.3.5 Response of Frame Members
The Distributed Transfer Function Method
Quick Solution Guide
References
201
201
203
205
205
210
213
216
220
224
230
235
237
237
240
243
243
248
252
256
260
263
268
276
Part III Dynamics and Vibrations 277
C h a p t e r 9 Dynamics of Particles and Rigid Bodies 279
9.1 Getting Started 279
9.2 Dynamics of Particles 280
9.2.1 Preliminaries 280
9.2.2 Kinematics 284
9.2.3 Kinetics of Single Particle 286
9.2.4 Particle Motion via Numerical Integration 289
9.2.5 Systems of Particles 297
9.2.6 Central Force Motion 303
9.3 Dynamics of Rigid Bodies in Plane Motion 307
9.3.1 Plane Motion 307
9.3.2 Mass Moments of Inertia 310
9.3.3 Kinetics 314
9.3.4 Simulation of Dynamic Response 317
9.4 Rigid Body Dynamics in Three Dimensions 323
9.4.1 Kinematics 323
9.4.2 Inertia Properties 327
9.4.3 Energy and Momentum 335
Contents 9.4.4 Equations of Motion 336
9.4.5 Rotation of Axisymmetric Bodies 340
9.5 Quick Solution Guide 349
9.6 References 350
C h a p t e r 1 0 Vibration Analysis of One-Degree-of-Freedom Systems 351
10.1 Getting Started 351
10.2 System Description 353
10.3 Time Response 357
10.3.1 Free Response 357
10.3.2 Forced Response 361
10.3.3 Specific Forcing Functions 365
10.3.4 Total Response 374
10.3.5 Delayed Forcing Functions 376
10.3.6 Solution by Superposition 377
10.3.7 Mechanical Energy 379
10.3.8 Plotting Computed Response 382
10.4 Analytical Vibration Solutions 384
10.4.1 Analytical Methods 384
10.4.2 Solution by the Toolbox 389
10.5 Frequency Response 394
10.5.1 Harmonic Excitation 394
10.5.2 Base Excitation 400
10.5.3 Response under Rotating Unbalance 403
10.5.4 Vibration Isolation 406
10.6 Response to Periodic Excitation 410
10.6.1 Steady-State Solution in Fourier Series 410
10.6.2 Solution by the Toolbox 412
10.6.3 Fourier Coefficients of Certain Loads 419
10.7 Nonlinear Vibration 421
10.8 Quick Solution Guide 431
10.9 References 435
C h a p t e r 1 1 Vibration and Control of Multiple-Degree-of-Freedom
Systems 437
11.1 Getting Started 437
11.2 System Description 439
11.2.1 Introduction 439
11.2.2 Modes of Vibration 442
11.2.3 Solution by the Toolbox 444
11.3 Dynamic Response 450
11.3.1 Modal Analysis 450
11.3.2 Laplace Transform Method 458
11.3.3 Plotting Analytical Solutions 462
11.3.4 Runge-Kutta Algorithm 465
11.3.5 Solution by Superposition 468
11.3.6 Harmonic Excitation 472
11.4 Dynamic Vibration Absorption 475
11.4.1 Undamped Vibration Absorbers 476
11.4.2 Damped Vibration Absorbers 481
x Contents 11.5 Transfer Function Formulation
11.5.1 Transfer Function and Green's Function 487
11.5.2 Open-Loop Transfer Function 491
11.6 Feedback Control 497
11.6.1 Vibration Control System 497
11.6.2 Position Control System 501
11.6.3 Solution by the Toolbox 506
11.7 Quick Solution Guide 515
11.8 References 518
Part IV Structural Dynamics 519
521
521
523
523
524
528
534
535
537
537
539
540
544
552
554
555
559
559
567
581
581
582
588
594
601
608
615
C h a p t e r 1 3 Dynamic Analysis of Bars, Shafts, and Strings 617
13.1 Getting Started 617
13.2 System Description 619
13.2.1 Governing Equations 619
13.2.2 Eigenvalue Problem 622
13.2.3 System Setup by the Toolbox 624
13.2.4 Display of Modes of Vibration 627
13.3 Dynamic Response 630
13.3.1 Modal Expansion 631
13.3.2 Free Vibration 632
Chapter 12 Dynamics and Control of Euler-Bernoulli Beams
12.1
12.2
12.3
12.4
12.5
12.6
12.7
Getting Started
System Description
12.2.1
12.2.2
12.2.3
12.2.4
12.2.5
Governing Equation
Eigenvalue Problem
System Setup by the Toolbox
Animation of Modes of Vibration
Distributed Transfer Function Method
Dynamic Response
12.3.1
12.3.2
12.3.3
12.3.4
12.3.5
12.3.6
12.3.7
Modal Expansion
Specification of Damping
Free Vibration
Forced Vibration
Total Response
Animation of Time Response
Frequency Response
Feedback Control
Dynami
Control System Formulation
Solution by the Toolbox
cs and Control of Nonuniform Beams
Problem Statement
Rayleigh-Ritz Discretization
Modes of Vibration
Time Response
Control System Formulation
Quick Solution Guide
References
Contents xi 13.3.3 Forced Vibration 635
13.3.4 Total Response 639
13.3.5 Animation of Time Response 642
13.3.6 Frequency Response 643
13.4 Free Vibration of Stepped Systems 647
13.4.1 System Description 647
13.4.2 Free Vibration Analysis 651
13.4.3 Solution by the Toolbox 655
13.5 Quick Solution Guide 661
13.6 References 665
C h a p t e r 1 4 Dynamic Analysis of Constrained, Combined, and Stepped
Beams 667
14.1 Getting Started 667
14.2 Constrained Beams 670
14.2.1 System Description 670
14.2.2 System Setup and Eigensolutions 675
14.2.3 Eigenvalue Locus 683
14.2.4 Frequency Response 687
14.2.5 Multispan Beam Structures 689
14.2.6 Transient Response 691
14.3 Combined Beams 691
14.3.1 System Description 692
14.3.2 System Setup and Eigensolutions 696
14.3.3 Transient Response 698
14.3.4 Frequency Response 707
14.3.5 Oscillators for Vibration Absorption 712
14.4 Stepped Beams 715
14.4.1 Free Vibration Analysis 715
14.4.2 Solution by the Toolbox 721
14.5 Quick Solution Guide 729
14.6 References 736
Part V Two-Dimensional Elastic Continua 737
;er 15 Static Analysis of Linearly Elastic Bodies
Getting Started
Theory of Linear Elasticity
Stress and Strain
Basic Equations of Elasticity
Conversion of Elasticity Constants
Strain Energy
Principal of Minimum Potential Energy
Elasticity Problems in Two Dimensions
Plane Stress and Plane Strain
Governing Equations
Solution by Stress Function
Thermal Stresses
Elasticity Problems in Polar Coordinates
Stress Concentrations
769 15.4 Finite Element Method for 2-D Elasticity Problems 770
15.4.1 Finite Element Formulation 770
15.4.2 MATLAB Solutions in Rectangular Regions 777
15.4.3 MATLAB Solutions in Arbitrary-Shaped
Regions 792
15.5 Quick Solution Guide 802
15.6 References 805
C h a p t e r 1 6 Free Vibration of Membranes and Plates
16.1 Getting Started
16.2 Free Vibration of Membranes
16.2.1 Rectangular Membranes
16.2.2 Circular Membranes
16.3 Free Vibration of Rectangular Plates
16.3.1 Plate Theory
16.3.2 Eigenvalue Problem
16.3.3 Solution by the Toolbox
16.4 Free Vibration of Circular Plates
16.4.1 Equations in Polar Coordinates
16.4.2 Modes of Vibration
Appendix A
Appendix B
16.5
16.6
Quick Solution Guide
References
Commonly Used Mathematical Formulas
Algebraic Formulas
Areas and Volumes of Common Shapes
Trigonometry
Hyperbolic Functions
Derivatives and Integration
Series Expansion
Analytical Geometry
Vector Analysis
Matrix Theory
Complex Numbers and Complex Functions
Laplace Transforms
Inverse Laplace Transform via Partial Fraction
Expansion
MATLAB Basics
Getting Started
Matrix and Vector Manipulations
Graphics
M-Files
Control Flow
Solution of Algebraic and Differential Equations
Control System Toolbox
Appendix C The Distributed Transfer Function Method 913
C.l DTFM for One-Dimensional Continua 913
C.2 Transfer Function Synthesis of Multibody Structures 917
Contents xiii C.3 DTFM for Two- and Three-Dimensional Problems 919
C.4 References 920
Appendix D Conversion of Units 923
Appendix E Mechanical Properties of Engineering Materials 925
Index 929
Index
A
Absorbers, 475^81
absorber mass, 476
damped vibration absorbers, 481
design criteria, 478
for control of beam vibration, 712
forced frequency ratio, 476
frequency ratio, 477
mass ratio, 476
operating range, 477
optimal tuning of, 482
primary mass, 476
undamped vibration absorbers, 476
Acceleration, 280, 284, 307
angular, 308, 314, 324
Coriolis, 326
cylindrical components of, 285
of mass center, 299, 336
polar components of, 285
rectangular Cartesian components
of, 285
relative, 309
relative to a moving frame, 326
relative to a translating frame, 324
Action and reaction forces, 280
Active vibration control, 497, 559, 601
Actuator locations, 491, 559
Actuators, 497, 501,559
Adjoint eigenvalue problem, 452
Admissible displacement, 756
Admissible functions, 582, 585
Air damping, 421
Airy stress function, 761
Amplification factor, 394
Analytical geometry, 864
Analytical methods
boundary value approach, 19
convolution integral, 385
DTFM {see Distributed transfer function
method)
Green's function formula, 487
Laplace transform method, 385
method of singularity functions, 16
method of undetermined coefficients,
386
modal expansion, 537
stiffness method, 224
Angular acceleration, 308, 314, 324
Angular displacement, 314
Angular momentum, 288, 316, 328
conservation of, 299,
of a particle, 288
of a rigid body, 316, 328
of a system of particles, 299
Angular velocity, 308, 323
Animation of
dynamic response of a particle, 290
dynamic response of a rigid body, 317, 343
modes of vibration, 446, 534, 627, 680,
814, 821
transient time response, 554, 642, 701
Approximate methods
assumed-modes method, 582
finite element method, 770, 829
Rayleigh-Ritz method, 582
Runge-Kutta numerical integration, 290, 423,
465, 501, 567, 701
truncated modal expansion, 671
929 Area
beam cross-section, 42
table of, 855
Average normal strain, 145
Average normal stress, 139
Axial force, 56, 204, 620
B
Balancing, of rotating machines, 404
Bars in static equilibrium, 53-83
axial force, 56
bar elements, 67
compatibility conditions, 68
external loads, 62, 71
boundary conditions, 56
distributed transfer function method, 74
governing equation, 56
longitudinal deformation, 56
longitudinal rigidity, 56
MATLAB functions,79
spring constraints, 66
stepped bars, 66
Bars, shafts and strings in vibration, 617-665
animation of modes of vibration, 627, 659
animation of time response, 642
eigenvalue problem, 622
equation of motion, 619
frequency response, 643
forced vibration, 635
free vibration, 632
MATLAB functions, 662
modal expansion, 631
mode shapes, 622
natural frequencies, 622
stepped bars in longitudinal vibration, 648
stepped shafts in torsional vibration, 647
Base excitation, 400
Beam-column, 100
Beam cross-section area, 42
Beam theory, 11
Beams in static equilibrium, 9-51
beam axis, 23
bending moment, 12
bending stiffness, 12
boundary conditions, 13
boundary disturbances, 13, 27
boundary value approach, 19
curvatures of beam axis, 24
distributed transfer function method, 21
equilibrium equations, 12, 25
Euler-Bernoulli, 11
Euler-Bernoulli hypothesis, 23
external loads, 12, 30
flexural rigidity, 24
MATLAB functions, 47
method of singular functions, 16
moment-curvature relation, 24
moment of inertia of area, 42
neutral axis, 23
normal stress and strain, 23
pure bending, 24
reactions at supports, 14
shear force, 14
sign convention, 14
singular functions of, 16
transverse displacement, 11
Beams in vibration, 521-615
animation of modes, 534
animation of time response, 554
boundary conditions, 525, 583
damping, 539
distributed transfer function method, 535
eigenvalue problem, 524
equation of motion, 523
Euler-Bernoulli, 521
external loads, 548
feedback control (see Control of beams)
forced vibration, 544
free vibration, 540
frequency response, 555
MATLAB functions, 609
modal coordinates, 538
modal expansion, 537
mode shapes, 524
natural frequencies, 524
nonuniform (see Nonuniform beams)
normalized eigenfunctions, 530
rigid-body mode, 526
steady-state vibration, 557
transverse displacement, 523
uniform beams, 523
Beating, 363, 372
frequency of beating, 373
period of beating, 373
Bending moment, 12, 88, 524, 823
Bending stiffness
beams, 12, 524
frame members, 240
plates, 823
Bessel functions, 818, 842
Biharmonic operator, 823, 840
Bi-orthogonality relations, 452
Body forces, 749
Boundary disturbances, 12, 58
Boundary influence functions, 22, 75, 125, 264
Brittle materials, 151
Buckling of columns, 85-133
axial compression load, 87
beam-column, 100
boundary conditions, 89
930 Index buckling load, 88
buckling mode shape, 88
characteristic equation, 90
constrained stepped columns, 109
distributed transfer function method, 123
eccentric loading, 96
eigenvalue problem, 88
elastic foundation, 109
Euler buckling load, 88
geometric imperfection, 104
MATLAB functions, 128
nonuniform columns, 117
uniform columns, 87
Bulk modulus of elasticity {see Elasticity
constants)
C
Calculus of variations, 756
operator of variation, 757
virtual displacement, 756
Center of mass, 299, 310, 327
Central force, 303
Central force motion, 303
conic section, 306
constant areal velocity, 304
Newton's law of gravitation, 304
space vehicle in free flight, 305
trajectories, 306
universal gravitational constant, 304
Characteristic equation
bars, shafts and strings, 623
beams, 526, 673, 718
columns, 90
membranes, 811, 819
multiple-degree-of-freedom systems, 440
one-degree-of-freedom systems, 355
plates, 827, 832, 843
Circular membranes {see Membranes)
Circular natural frequency, 353
Circular plates {also see Plates), 839
Circular shafts in static equilibrium, 53-83
boundary conditions, 57
compatibility conditions, 68
external loads, 62
governing equation, 57
MATLAB functions, 79
rotation (twist), 57
shaft elements, 68
spring constraints, 69
stepped shafts, 68
torque, 57
torsional rigidity, 57
Circular shafts in vibration {see Bars, shafts and
strings in vibration)
Closed-loop system, 499, 564
Closed-loop transfer function, 502
Coefficient of friction, 281, 421
Coefficient of linear thermal expansion, 216, 925
Coefficient of restitution, 302
Coefficient of viscous damping, 353, 524
Collision, 302
Colocated control system, 491, 563
Columns {see Buckling of columns)
Combined beams in vibration {see Constrained,
combined and stepped beams in
vibration)
Complex eigenvalues, 443, 444
Complex functions, 879
Complex harmonic factor, 473
Complex numbers, 878
Conic sections, 306, 866
Conservation of energy, 288, 316, 335, 380
Conservation of momentum, 299
Conservative force, 287
Conservative system, 287
Constants of Lame, {see Elasticity constants)
Constitutive law {see Hooke's law)
Constrained beams in vibration {see Constrained,
combined and stepped beams in
vibration)
Constrained, combined and stepped beams in
vibration, 667-736
animation of modes of vibration, 680, 724
animation of time response, 701
attached springs, masses and
dampers (SMD), 670
beam segments, 715
boundary conditions, 672
characteristic equation, 673, 718
combined beams, 691-715
constrained beams, 670-691
constraints, 715
eigenvalue locus, 683
eigenvalue problem, 673, 695, 718
elastic foundation, 719
end mass, 717
external loads, 703
frequency response, 687, 707
governing equations, 670, 692
hinged disk, 717
MATLAB functions, 730
modal expansion, 671, 693
mode shape, 673, 696, 716
mounted rigid bodies, 691
natural frequencies, 674, 719
nodes, 715
nonproportionally damped system, 675
oscillators, 691
proportionally damped system, 674
Runge-Kutta method, 701
stepped beams, 715-729
state equation, 700
Index 931 Constrained, combined and stepped beams in
vibration, (continued)
transient response, 691, 698
undamped system, 674
vibration absorption, 712
Constrained multispan beams in static
equilibrium, 160-200
beam elements, 161
boundary conditions, 163
constraints, 164
distributed transfer function method,
189
external loads, 165, 173
hinge connection, 164
influence lines, 183
MATLAB functions, 195
nodes, 161
reactions at supports, 165
settlement of supports, 178
sign convention, 162
Constraints, 109, 164, 648, 715
Continuous beams (see Constrained multispan
beams static equilibrium)
Control laws (see Feedback control laws)
Control of beams, 559-581
actuators, 560
augmented state equation, 566
closed-loop, 564
discretized model, 561
feedback controllers (see Feedback control
laws)
open-loop, 562
MATLAB functions, 610
nonuniform beams, 601
pole-zero alternation, 563
poles and zeros, 570
root locus, 574
sensors, 561
state equation, 563
transfer function, 562
Control of mechanical systems (see Control of
multiple-degree-of-freedom systems)
Control of multiple-degree-of-freedom systems,
487-514
actuators, 491, 497, 501
augmented state equation, 500
closed-loop, 500
feedback controllers (see Feedback control
laws)
MATLAB functions, 506, 516
open-loop, 491
poles and zeros, 492
position control system, 501
sensors, 491, 497, 501
state representation, 498, 503
transfer function, 491, 500, 502
vibration control system, 497
Control system, 497, 559
Conversion of units, 136, 280
table of, 923
Convolution integral, 385, 881
Coordinate systems, 285
Coordinate transformation, 225, 265, 341
Coordinates
cylindrical, 285
generalized, 582, 594, 605, 671, 687
modal, 538, 631
polar, 285, 765
rectangular Cartesian, 285
Coriolis acceleration, 326
Coulomb friction, 281, 421
Criteria of failure, 151-155
Maximum distortion-energy theory, 152
Maximum normal-stress theory, 153
Maximum shear-stress theory, 151
Critical damping coefficient, 354
Critically damped system, 355
Cross or vector product, 281, 288, 298, 870
Curvature, 24, 824
D
D'Alembert's principle, 315
Damped frequency, 355
Dampers, 252,438, 670, 692
Damping, 353,439, 475
air, 421
Coulomb, 281, 421
energy dissipation per cycle, 422
forced response, 361
free response, 358
viscous, 353
Damping coefficient (see Coefficient of viscous
damping)
Damping form, 440
undamped systems, 355, 442
nonproportionally damped systems, 440, 675
proportionally damped systems, 440, 674
Damping matrix, 439, 674
Damping ratio, 353, 443, 673, 732
Damping status, 355
critically damped system, 355
over-damped system, 355
undamped system, 355
underdamped system, 355
Decoupled differential equations, 450
Deflection
lateral, 55, 87, 810, 823
of support, 205, 243
transverse, 12, 164, 240 Deformation
bending, 23
elastic, 213, 252, 743
longitudinal, 203, 240, 755
of structures, 178
torsional, 55, 755
Degree of freedom, 351, 437
Delta function, 12, 61, 165, 365, 487, 524, 588,
623, 671
Delayed forcing functions, 376, 466
Derivatives, 860
of vectors, 325, 871
table of, 861
Derivatives and integration, 860
Determinant, 874
Diagonal matrix, 874
Dilatation or unit volume change, 750
Dirac delta function {see delta function)
Direction cosines, 331, 744
Displacement
angular, 314
boundary, 12, 58
lateral, 621
longitudinal, 56, 203, 240, 620
modal, 124, 454, 533, 589, 658, 723, 815
nodal, 77, 192, 206, 224, 241,266, 771,
831,918
of elastic bodies, 289, 743
of multiple-degree-of-freedom systems, 439
of one-degree-of-freedom systems, 353
rotational, 68, 308, 649, 693
steady-state, 395, 413,473, 481, 557, 643,
687, 708
transverse, 11, 88, 161, 240, 523, 671,
810, 823
virtual, 587, 755
Displacement transmissibility, 401
Distortion energy per unit volume, 152
Distributed load or distributed force, 18, 61,
165, 700
Distributed transfer function method, 4, 913
for bars, shafts and strings, 74, 645
for beams, 21, 189,535
for columns, 123
for frames, 263
for multibody structures, 917
for one-dimensional continua, 913
for two- and three-dimensional continua, 919
Dry friction {the same as Coulomb friction)
Dot or scalar product, 287, 870
DTFM {see Distributed transfer function method)
Ductile materials, 151
Duhamel integral, 385
Dynamic balance in rotation, 403
Dynamic vibration absorbers {see Absorbers)
Dynamic vibration absorption, 475, 712
Dynamics
of bars, shafts and strings, 617
of beams, 521, 667
of particles, 280
of rigid bodies, 307, 323
structural, 519
E
Earth
mass of, 304
radius of, 304
Eigenfunction
bars, shafts and strings, 622, 651
beams, 524, 588, 622, 671, 719
columns, 88
membranes, 811, 818
multiple-degree-of-freedom systems, 450
plates, 827, 845
Eigenfunction expansion {see Modal expansion)
Eigenvalue
bars, shafts and strings, 622, 651
beams, 524, 588, 622, 673, 719
columns, 45, 88, 124
matrix, 876
matrix of inertia, 331
membranes, 811, 818
multiple-degree-of-freedom systems, 440, 444
plates, 827, 845
strain matrix, 747
stress matrix, 745
Eigenvalue locus, 683
Eigenvector
bars, shafts and strings
beams, 588, 673, 695
matrix, 876
matrix of inertia, 331
multiple-degree-of-freedom systems, 440, 444
strain matrix, 747
stress matrix, 745
Elastic body, 136, 724, 759
Elastic foundation, 109, 715
Elasticity, 739-805
boundary conditions, 752
boundary tractions, 761, 783
conditions of compatibility, 752
equations of equilibrium, 749
finite element solution, 770
principle of minimum potential energy, 755
problems in polar coordinates, 765
problems of axial symmetry, 767
Saint-Venant's principle, 753
strain-displacement relations, 749
strain energy, 755
stress function, 761
Index 933 Elasticity, (continued)
stress-strain relations, 750
theory of, 742
Elasticity constants, 83, 753
bulk modulus of elasticity, 750
conversion of, 753
Lame coefficient or constants of Lame, 750
Poisson's ratio, 149, 750, 808, 823
shear modulus, 55, 149, 620, 750
Young's modulus, 83, 750
Elongation, 142,281,744
Energy, 287, 335
conservation of, 288, 316, 335, 380
kinetic, 287, 302, 315, 335, 379, 585, 826, 841
mechanical, 288, 379
potential, 287, 316, 335, 379, 585, 757, 773
strain, 585, 755, 826, 841
Energy dissipation, 303, 422, 475
Engineering materials, 925
Equation of motion
absorbers, 476, 481
axisymmetric rigid bodies, 341
bars, shafts and strings, 619
beams, 523, 587, 670, 692
membranes, 810, 816
multiple-degree-of-freedom systems, 439
one-degree-of-freedom systems, 353,421
plates, 823, 840
rigid bodies in two dimensions, 314
rigid bodies in three dimensions, 336
single particle, 286, 305
system of particles, 301
Equilibrium equation
bars, shafts and strings, 56
beams, 12, 161
columns, 88, 109, 117
elastic bodies, 749
frames, 240
global, 192, 227, 267, 775
trusses, 204
Equivalent viscous damping coefficient, 422
Euler angles, 340
Euler-Bernoulli beams, 11, 523
Euler buckling load, 88
Euler's equations, 337
Euler's laws, 298
Exponential matrix, 876
bars, shafts, and strings, 76, 646, 652
beams, 22, 190, 537
columns, 123
frame members, 264
F
Failure criteria (see Criteria of failure)
Feedback control laws
PD feedback, 503
PID feedback, 499, 504, 565
output feedback, 499, $64
state feedback, 499, 564
Finite element method, 770-801, 829-839
displacement interpolation, 771
element stiffness matrix, 772
for 2-D elasticity problems, 770
for vibration of plates, 829
global stiffness matrix, 775
mesh, 777, 799, 829
nodal displacements, 771, 831
nodes, 770, 829
rectangular elements, 829
shape functions, 771, 831
triangular elements, 771
Flexible beams (see Beams in static equilibrium
or Beams in vibration)
Flexible structures (see Structures and structural
components)
Flexural rigidity (the same as Bending stiffness)
Force
definition of, 281
conservative, 2, 87
external, 297
friction, 281
gravitational, 281
inertia, 315
internal, 297
nonconservative, 287
resultant, 282
spring, 281
Force transmissibility, 395,401, 404
Forced response or forced vibration
bars, shafts and strings, 635
beams, 544, 701
multiple-degree-of-freedom systems, 454, 469
one-degree-of-freedom systems, 361, 384
Forcing functions, 362, 455
exponential, 369
impulse, 365
ramp, 368
periodic, 410
pulse excitation, 373
sinusoidal, 371
step, 366
Fourier coefficient, 410, 419
Fourier series, 410
of periodic forcing function, 410
of steady-state response, 411
Frame of reference (see Reference frame)
Frames, 237-276
coordinate transformation, 265
distributed transfer function method, 263
external loads, 248, 250
global equilibrium equation, 267
934 Index internal forces, 241
MATLAB functions, 269
member stiffness matrix, 265
members, 240, 263
nodal displacements, 241
nodal forces, 241
nodes, 240, 267
settlement of supports, 252
static response, 248, 252
supports, 242
Free response or free vibration
bars, shafts and strings, 632
beams, 540, 715
membranes, 809
multiple-degree-of-freedom systems, 453,
459, 469
one-degree-of-freedom systems, 357
plates, 823, 839
Frequency
driving, 477
excitation, 371,472, 557
of beating, 373
Frequency, natural, {see Natural frequency)
Frequency ratio, 394, 477, 483
Frequency response
bars, shafts and strings, 643
beams, 555, 687, 707
multiple-degree-of-freedom systems, 472
one-degree-of-freedom systems, 394, 401, 405
Friction coefficient, 281
kinetic, 281, 421
static, 422
Functional, 757, 826
G
Generalized coordinates, 582, 594, 605, 671, 687
Generalized force, 587
Generalized Hooke's law {see Hooke's law)
Global equilibrium equation, 192, 227, 267, 775
Global stiffness matrix, 192, 267, 775
Gradient operator, 287, 871
Gravitation, 304
Newton's law of, 304
universal constant of, 304
Gravitational acceleration, 281
Gravity, 281
Green's function, 22, 75, 124, 190, 263, 487
Harmonic excitation
base excitation, 400
external excitation, 400
magnitude, 394, 473
phase angle, 400, 473
resonance, 372
rotating unbalance, 403
steady-state response, 394, 472
vibration absorber, 476, 481
vibration isolators, 476, 481
Hertz (unit), 354, 529, 636
Hooke's law, 149, 750
for plane strain, 150
for plane stress, 150
generalized, 149, 750
Hyperbolic functions, 860
I
Identity matrix, 874
Impact, 302
Impulse, 288, 316,335
Impulse force, 365, 455, 548, 637, 703
Impulse and momentum, principle of, 288,
315,335
Impulse response, 385, 487
Inertia
area moments of {see Moments of inertia of
area)
moments of, 329
polar moment of, 55, 620
principal axes of
principal moments of, 331
products of, 329
properties, 327
Inertia matrix, 439
Inertial force and moment, 315
Influence functions, 451
Influence lines, 183
Initial conditions of
bars, shafts and strings, 631
beams, 524
combined beams, 692, 699
constrained beams, 670
multiple-degree-of-freedom systems, 439
one-degree-of-freedom systems, 361
rigid bodies, 317, 343
single particles, 289
Instantaneous center of rotation, 309
Interactive computing, 2
Integration, 860
table of, 861
Internal force
bars, shafts and strings, 56, 622
beams, 14, 162, 716
columns, 124
frames, 241
plates, 823, 840
systems of particles, 297
trusses, 205, 226
Interpolation, 771,919
Inverse Laplace transforms, 880, 882
Inverse matrix, 875
Isotropic material, 764
Index 935 Isolation (see Vibration isolation)
Isolator, 406
K
Kinematics, 284, 323
Kinetic energy
beams, 585
one-degree-of-freedom systems, 379
particle, 287
plates, 826, 841
rigid bodies, 315, 335
Kinetics
of particles, 286, 298
of rigid bodies, 314
Kirchhoff's assumptions, 824
Kronecker delta (see delta function)
L
Lagrange's equations, 587
Lame coefficient (see Elasticity constants)
Laplace transforms, 879
definition of, 879
for solutions, 385, 458
inverse Laplace transform, 880,
table of, 881
Laws of sines and cosines, 859
Laplacian operator, 810, 816
Linear density, 524, 590, 648, 670, 692, 715, 915
Linear independence of vectors, 875
Linear momentum, 298, 316, 327
conservation of, 299
Logarithms, 854
Longitudinal stiffness or rigidity, 56, 204, 240,
620, 648
Longitudinal vibration of bars (see Bars, shafts
and strings in vibration)
Lumped parameter systems (see
Multiple-degree-of-freedom systems)
M
Magnitude of
force, 281, 405
frequency response or steady-state response,
394, 401, 422, 473, 481, 557, 644, 687
vector, 868
velocity, 287, 308, 335, 395
Mass, 280
Mass center, 299, 310, 327
Mass density, 810, 823, 840
Mass matrix, 439, 694
Mass moments of inertia (see Moments of inertia
of mass)
Material failure, 151-155
Material properties, table of, 925
Mathematical formulas, 853-888
algebraic formulas, 853
analytical geometry, 864
areas and volumes, 855
complex numbers, 878
derivatives and integration, 860
hyperbolic functions, 860
Laplace transforms, 879, 882
matrix theory, 872
series expansion, 862
trigonometry, 857
vector analysis, 868
MATLAB, 5, 889-912
algebraic and differential equations, 907
control flow, 903
Control System Toolbox, 910
graphics, 898
mathematical functions, 892
matrix and vector manipulations, 893
M-files, 901
MATLAB toolboxes of this book, 3, 5
bars, shafts and strings in equilibrium, 79
bars, shafts and strings in vibration, 662
beams in equilibrium, 47
beams in vibration, 609
columns, 128
constrained, combined and stepped beams in
vibration, 730
constrained multispan beams in static
equilibrium, 195
elastic bodies in equilibrium, 802
inverse Laplace transforms, 884
membranes in vibration, 849
multiple-degree-of-freedom systems, 515
one-degree-of-freedom systems, 432
particles and rigid bodies, 349
plane frames, 269
plane trusses, 231
plates in vibration, 849
stress analysis in two dimensions, 155
Matrix, 872
determinant of, 874
eigenvalues and eigenvectors, 876
exponential of, 877
identify, 874
inversion, 875
rank of, 876
square, 874
Maximum and minimum normal strains, 144
Maximum distortion-energy theory, 152
Maximum normal-stress theory, 153
Maximum overshoot, 367
Maximum principal strain, 146,
Maximum principal stress, 153, 745
Maximum shear strain, 145
Maximum shear stress, 139, 746
Maximum shear-stress theory, 151
936 Index Mechanical energy, 288, 379
Mechanical properties, 925
table of, 925
Membranes, 809-823
boundary conditions, 810, 816
circular, 816
eigenvalue problems, 810, 817
equation of motion, 810, 816
free vibration, 809, 816
MATLAB functions, 849
mode shapes, 812, 818
natural frequencies, 811,819
nodal circles and nodal diameters, 819
rectangular, 810
Mechanical system, 367, 487, 497, 910
Method of singularity functions, 16
Method of undetermined coefficients, 386
Metric system (see Standard International
System)
Modal analysis 450,469, 538, 632
Modal damping, 443, 539, 671, 692
Modal expansion, 450, 491, 537, 560, 631,
671, 693
Modes of vibration, 442, 527, 588, 625, 842
Mode shapes
buckling, 88, 119, 124
vibration, 442,491, 524, 588, 622, 651, 673,
696,716,811,819,827,842
Modulus of elasticity (see Elasticity constants)
Mohr's circle for plane strain, 145
Mohr's circle for plane stress, 140
Moment, 281
bending, 12, 88, 524, 823
resultant 282, 314, 336
twisting, 823, 840
Moment equation of motion
in a body frame, 337
in a rotating frame, 336
rotation about a fixed axis, 337
Moment of a force, 281
Moment of inertia of area, 44
Moment of momentum (the same as Angular
momentum)
Moments of inertia of mass, 310
central axis, 311
matrix of inertia, 330
parallel-axis theorems, 311
principal, 331
principal axes, 331
table of, 312
transformation of, 331
Momentum
angular, 288, 298, 316, 328
conservation of, 299
linear, 288, 298, 316
Motion, 280
central force, 303
of mass center, 301
plane, 308
relative, 309, 324
rotation, 307, 323, 337, 340
rotation about a point, 307
translation, 307
Multiple-degree-of-freedom systems, 437-518
animation of modes of vibration, 446
bi-orthogonality relations, 452
closed-loop, 499, 501,509
complex eigensolutions, 444
dynamic vibration absorption (see Absorbers)
eigenvalue problem, 440
feedback control (see Control of
multiple-degree-of-freedom systems)
free response, 453, 458, 459
frequency response, 472
forced response, 454, 458, 459
harmonic excitation, 472
influence function, 451
MATLAB functions, 515
modal damping, 443
natural frequencies, 442
nonproportionally damped systems, 440
open-loop, 491
orthonormal relations, 442
proportionally damped systems, 440
Rayleigh damping, 443
rigid-body modes, 439, 452
state eigenvectors, 452
transfer function, 487
undamped systems, 442
Multispan beams (see Constrained multispan
beams)
N
Natural frequency, 354
a primary system combined with an
absorber, 476
bars, shafts and strings, 622, 651
beams, 524, 588, 674, 719
membranes, 811,819
multiple-degree-of-freedom systems, 442
one-degree-of-freedom systems, 354
plates, 827, 843
Natural period, 354
Neutral axis, 23
Neutral surface, 823
Newton's law of gravitation, 304
Newton's laws of motion, 280
Nodal displacement, 77, 192, 206, 224, 241,266,
771,831,918
Nodal line, 340, 819,843
Index 937 Nodal points, 491, 533, 563, 589, 628, 723
Nodes, 66, 109, 161, 203, 240, 648, 715, 770,
829,917
Noncolocated control system, 491, 571
Nonlinear damping
air damping, 421
Coulomb damping or dry friction, 421
equivalent viscous damping coefficient, 422
Nonlinear spring, 421
Nonlinear vibration, 421
Nonproportionally damped system, 440, 675
Nonuniform beams, 581-608
admissible functions, 582, 585
boundary conditions, 583
discretized model, 561
equation of motion, 581
kinetic and strain energy, 585
modes of vibration, 588
Normal strain, 24, 142, 744
Normal stress, 23, 136, 742
Normalized amplitude, 477
Normalized eigenfunctions, 530, 625, 657
Numerical integration, 289, 317, 337,411,470,
541,632,699,831
Nutation, 340
O
One-degree-of-freedom systems, 351
characteristic equation, 355
characteristic roots, 355
convolution integral, 385
critically-damped, 355
damped frequency, 355
damping ratio, 353
forced response, 361
free response, 357
frequency response, 394
harmonic excitation, 394
Laplace transform, 385
MATLAB functions, 432
maximum overshoot, 367
mechanical energy, 379
method of undetermined coefficients, 386
natural frequency, 367
nonlinear vibration, 421
overdamped, 355
response, 365
to delayed forcing functions, 376
to exponential force, 369
to impulse force, 365
to ramp force, 368
to periodic excitation, 410
to pulse excitation, 373
to sinusoidal excitation, 371
to step force, 366
rise time, 367
Runge-Kutta algorithm, 423
settling time, 368
solution by superposition, 377
undamped, 355
underdamped, 355
Orthogonal axes, 330
Orthogonal eigenvectors, 444, 675
Orthogonal matrix, 745
Orthonormal relations, 444, 674
Open-loop transfer function, 491,463
Optimal tuning of absorbers, 482
Oscillators, 691, 712
Overdamped systems, 355
P
Parallel-axis theorems, 311
Partial fraction expansion, 882
Particle dynamics, 280-307
animation of motion, 290, 317, 343
dynamic response, 289
impulse and momentum, 288
kinematics, 284
kinetics, 286
MATLAB functions
particle system (see System of particles)
work and energy, 287
Particular solution, 21, 386
Passive vibration control, 475
Periodic excitation, 410
Periodic force, 410
period T of, 410
Phase angle, 394,473, 893
Plane strain and plane stress, 136, 142, 759
Plates, 823-848
bending stiffness, 823
boundary conditions, 825, 841
circular, 839
curvatures, 825
eigenfunctions, 827, 842
eigenvalue problem, 827
finite element solution, 829
free vibration, 823
kinetic energy, 826
Kirchhoff's assumptions, 824
Levy solutions, 828
MATLAB functions, 850
mode shape, 827, 843
modes of vibration, 842
moments, 823, 841
Navier solutions, 827
natural frequencies, 827, 842
nodal circles and nodal diameters, 843
rectangular, 823
shear forces, 823, 841
938 Index strain energy, 826
stress components, 825
Pointwise load or pointwise force, 18, 62, 102,
173, 250, 545, 636, 699, 740
Poisson's ratio {see Elasticity constants)
Polar moment of inertia, 55, 620
Poles and zeros, 492, 568, 879
Pole-zero alternation, 492, 563
Position vector, 284, 324, 743
Potential energy, 287, 755
Power, 380, 924
Precession, 340
Pressure, 924
Principal moments of inertia, 331
Principal strains, 144, 746
Principal stresses, 138, 745
Principle of impulse and momentum, 288,
315,335
Principle of minimum potential energy, 757
Principle of work and energy, 287, 302, 316
Products of inertia, 329
Proportionally damped system, 440, 674
Pulse excitation, 362, 373
Q
Quadratic equations, 854
Quick Solution Guide, 5
buckling of columns, 127
control of beams, 608
control of multiple-degree-of-freedom
systems, 515
dynamic analysis of beams, 608,
dynamics of bars, shafts and strings, 661
dynamics of constrained, combined and
stepped beams, 729
membranes in free vibration, 802
particle and rigid-body dynamics, 349
plates in free vibration, 802
static analysis of bars, shafts and strings, 78
statics of beams, 46
statics of elastic bodies, 729
statics of multispan beams, 194
statics of plane frames, 268
statics of plane trusses, 230
stress analysis in two dimensions, 155
vibration of multiple-degree-of-freedom
systems, 515
vibration of one-degree-of-freedom
systems, 431
R
Rayleigh damping, 440
Rayleigh-Ritz method, 582
Reactions at supports, 14, 165, 204, 243, 338
Rectangular membranes {see Membranes)
Rectangular plates {see Plates)
Reference frame
fixed, 325
moving, 325
rotating, 336
translating, 324
Resonance, 363, 372, 477, 713
Resonant frequency, 712
Resonant vibration, 372, 387,411, 547,
636, 712
Resultant force and moment, 282
Right-hand rule or screw, 282, 780
Rigid body dynamics, 307, 323
animation of motion, 317, 343
axisymmetric, 340
definition of, 307
Euler angles, 340
Kinematics, 314, 323
linear and angular momentum, 316, 335
MATLAB functions, 349
work and energy, 315, 335
Rigid-body mode, 439, 452, 526
Rise time, 367
Rotation
about a fixed axis, 323
about a point, 307
finite, 324
instantaneous center of, 309
of axisymmetric bodies, 340
of beam cross section, 12
of rigid bodies, 307, 323
Rotating unbalance, 403
Rotational displacement, 308
Runge-Kutta method, 290,423, 465, 501,
567, 701
S
Saint-Venant's principle, 753
Scalar product {see Dot product)
Sensors, 491, 497, 559, 601
Separation of variables, 811, 818, 827
Series expansion
eigenfunction {see Modal expansion)
Fourier, 863
Maclaurin, 862
Taylor, 862
Settling time, 368
Shafts in equilibrium {see Circular shafts in static
equilibrium)
Shafts in vibration {see Bars, shafts and strings in
vibration)
Shear forces, 14, 88, 162, 241, 524, 582, 671,
823, 840
Shear modulus {see Elasticity constants)
SI system {see Standard International System)
SI unit prefixes, 923
Index 939 Sign convention
axial force, 56
bending moment and shear force, 12, 47, 162
external forces, 30, 48, 173
internal forces, 241
reactions at supports, 14
stress components, 136
torque, 57
Signum function, 421, 893
Single-degree-of-freedom systems {see
One-degree-of-freedom systems)
Singularity function, 17
Solution methods
boundary value approach, 19
distributed transfer function method, 21, 74,
263, 913
finite element method, 770, 829
Laplace transform method, 385,458
method of singularity functions, 16
Runge-Kutta numerical integration {see
Runge-Kutta method)
method of separation of variables, 811,
818,827
principle of superposition {see Superposition)
Spin, 340
Spring coefficient, 28, 92, 169, 281, 353, 525,
622,672,718
Spring-mass system, 475
Spring-mass-damper system, 353, 441,491
Stability, 509, 573, 607
Standard International System, 280, 923, 926
State equation, 290, 343, 465, 500, 563, 700, 877
State representation, 498, 503, 572, 605
State space formulation, 500
State variables, 289,422, 877
State vector, 343,498, 563, 606
Steady precession, 342
Steady-state response, 376, 394, 410, 422,476,
557,916
Step function, 17, 374, 434, 467, 545, 595,
636, 700
Stepped beams in free vibration {see Constrained,
combined and stepped beams in
vibration)
Stiffness
bending, 12, 240, 524, 823
longitudinal, 56, 204, 240, 620, 648
torsional (rigidity), 57, 621
Stiffness matrix, 190, 224, 264,439, 772, 831
Stiffness method for trusses, 224
Strain, 142, 744
average normal, 145
components, 142, 744
invariants, 747
maximum shear stress, 144, 746
Mohr's circle, 145
normal, 142, 744
principal, 144, 745
shear, 142, 744
transformation, 143
tensor, 745
Strain-displacement relations, 749
Strain energy, 755
density, 755
of elastic members, 756
of nonuniform beams, 585
of plates, 826
Strain gauges, 147
Strain invariants, 747
Stain rosette, 147
Strain transformation, 142, 745
Strength of materials, 7
Stress, 136, 742
average normal, 139
components, 136
invariants, 745
maximum shear, 139
Mohr's circle, 140
normal, 136,742
principal, 138, 745
shear, 136, 742
tensor, 744
thermal, 764
transformation, 137
units of, 136
Stress analysis, 135, 739
Stress concentration, 769
factor, 769
Stress function, 761
Stress invariants, 745
Stress transformation, 744
Stress-strain relations {see Hooke's law)
Strings in static equilibrium, 58-83
external loads, 62
boundary conditions, 58
governing equation, 58
lateral deformation, 58
MATLAB functions, 79
Strings in vibration {see Bars, shafts and strings in
vibration)
Structural control, 487,497, 559, 601
Structural dynamics, 519
Structural mechanics, 157
Structures and structural components
bars, shafts and strings, 53, 617
beams, 9, 521
columns, 85
constrained and combined beams, 667
elastic bodies, 135, 739
frames, 237
940 Index membranes, 809
multispan beams, 159
plates, 823, 839
trusses, 201
Superposition, 65, 101, 256, 377, 416, 468, 640
System of particles, 297-303
collision, 302
Euler's laws, 298
impulse and momentum, 298
mass center, 299, 301
work and energy, 302
T
Thermal stresses, 764
Thick cylinder, 768
Time derivatives of vectors in a moving
reference, 325
Time response
bars, shafts and strings, 630
beams, 537
constrained and combined beams, 698
multiple-degree-of-freedom systems, 450
one-degree-of-freedom systems, 357
particles, 289
rigid bodies, 317, 343
Torque-free motion, 342
Torsion of circular shafts (see Circular shafts in
static equilibrium)
Torsional rigidity, 57, 620,
Torsional vibration of shafts (see Bars, shafts and
strings in vibration)
Transfer function, 385, 487, 500, 563, 602
Transfer function formulation, 487, 601
Transformation equation
for matrix of inertia, 331
for strain, 137, 745
for stress, 143, 744
for stiffness matrix, 226, 266
Transient response (see Time response)
Transmissibility
displacement, 401
equation for isolator design, 408
force, 395,401, 404
Transmitted force, 395, 404
Transmitted nodal force, 191, 265, 918
Transverse displacement, 11, 523, 823
Trigonometric functions, 857
Trigonometric identities, 858
Trusses, 201-235
axial (internal) force, 204
coordinate transformation, 225
fabrication errors, 216
external forces, 210
MATLAB functions, 231
member, 203
member stiffness matrix, 224
nodal displacements, 206
nodes, 203
static response, 210
stiffness method, 224
support settlement, 213
supports, 204
thermal effects, 216
total response, 220
U
Ultimate normal stress, 153
Undamped system, 355, 442
Underdamped system, 355
Unit conversion (see Conversion of units)
Unit volume change or dilatation, 750
U.S. customary units, 136, 280, 923
V
Variational calculus (see Calculus of
variations)
Vectors, 868
addition and subtraction of, 869
components of, 869
cross or vector product, 870
curl, 872
divergence, 871
dot or scalar product, 870
gradient, 872
magnitude
scalar triple product of, 971
Velocity, 284
absolute, 328
angular, 308, 323, 328
cylindrical components of
instantaneous center of
polar components of
rectangular components of
relative, 328
relative to a fixed frame, 325
relative to a moving frame, 325
Vibration, 351, 437
absorption, 475
analysis, 384
control, 497, 559
damped, 355, 443
forced, 361, 544
free, 357, 540
isolation, 406
nonlinear, 421
undamped, 358, 366, 450
Vibration absorbers (see Absorbers)
Vibration absorption, 475, 712
Vibration isolation, 406
base excitation, 407
isolators, 406 Vibration isolation, (continued)
rotating unbalance, 407
transmissibility formulas, 407
Vibration isolators (see Vibration isolation)
Vibration of multiple-degree-of-freedom systems
(see Multiple-degree-of-freedom
systems)
Vibration of single-degree-of-freedom systems
(see One-degree-of-freedom systems)
Virtual displacement, 587,756,
Virtual work, 586, 757
Viscous damping, 353,421,481, 524, 539, 671,
692
Volume of three-dimensional shapes, 856
W
Weight, 281, 305
Windows, 5
Work, 287
Work and energy, principle of, 287, 302, 316
Y
Yield stress, 152
Young's modulus (see Elasticity constants)
Z
Zeros of a transfer function, 492, 563, 603

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