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| موضوع: كتاب Stress, Strain, and Structural Dynamics الإثنين 04 أبريل 2022, 2:28 pm | |
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أخواني في الله أحضرت لكم كتاب Stress, Strain, and Structural Dynamics An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes Bingen Yang University of Southern California
و المحتوى كما يلي :
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 #ماتلاب,#متلاب,#Matlab,
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