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Analysis of Structures offers an original way of introducing engineering students to the subject of stress and deformation analysis of solid objects, and helps them become more familiar with how numerical methods such as the finite element method are used in industry. Eisley and Waas secure for the reader a thorough understanding of the basic numerical skills and insight into interpreting the results these methods can generate. Throughout the text, they include analytical development alongside the computational equivalent, providing the student with the understanding that is necessary to…mehr
Analysis of Structures offers an original way of introducing engineering students to the subject of stress and deformation analysis of solid objects, and helps them become more familiar with how numerical methods such as the finite element method are used in industry. Eisley and Waas secure for the reader a thorough understanding of the basic numerical skills and insight into interpreting the results these methods can generate. Throughout the text, they include analytical development alongside the computational equivalent, providing the student with the understanding that is necessary to interpret and use the solutions that are obtained using software based on the finite element method. They then extend these methods to the analysis of solid and structural components that are used in modern aerospace, mechanical and civil engineering applications. Analysis of Structures is accompanied by a book companion website href="http://www.wiley.com/go/waas">www.wiley.com/go/waas housing exercises and examples that use modern software which generates color contour plots of deformation and internal stress.It offers invaluable guidance and understanding to senior level and graduate students studying courses in stress and deformation analysis as part of aerospace, mechanical and civil engineering degrees as well as to practicing engineers who want to re-train or re-engineer their set of analysis tools for contemporary stress and deformation analysis of solids and structures. * Provides a fresh, practical perspective to the teaching of structural analysis using numerical methods for obtaining answers to real engineering applications * Proposes a new way of introducing students to the subject of stress and deformation analysis of solid objects that are used in a wide variety of contemporary engineering applications * Casts axial, torsional and bending deformations of thin walled objects in a framework that is closely amenable to the methods by which modern stress analysis software operates.
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Autorenporträt
Anthony M. Waas and Joe G. Eisley, University of Michigan, USA Anthony Waas is Professor of Aerospace Engineering and Professor of Mechanical Engineering, and Director, Composite Structures Laboratory at the University of Michigan. His current research interests are damage tolerance analysis of composite materials and components made of composite materials, nanocomposites, structural engineering, biomaterials and bioengineering, and structures and mechanical components operating under "hot" conditions. A recipient of many awards for teaching and research excellence, Professor Waas is a Fellow of ASME and the AAM, and an Associate Fellow of AIAA and has served as an Associate Editor of the AIAA Journal (1995-02) and on the Editorial Advisory Board of the AIAA Journal of Aircraft (1995-00). He is currently on the editorial board of the Journal Composites: B and serves as an Associate Editor of the RAeS Aeronautical Journal, IJ of Engineering Science and Journal of Applied Mechanics, and is on the Editorial Board of Computer Modeling in Engineering and Sciences, and the Journal of the Mechanical Behavior of Materials. He was the Technical Chair of the 49th AIAA SDM conference. Joe G Eisley is Professor Emeritus - Aerospace Engineering in the College of Engineering at the University of Michigan. He is author of Mechanics of Elastic Structures.
Inhaltsangabe
About the Authors xiii Preface xv 1 Forces and Moments 1 1.1 Introduction 1 1.2 Units 1 1.3 Forces in Mechanics of Materials 3 1.4 Concentrated Forces 4 1.5 Moment of a Concentrated Force 9 1.6 Distributed Forces-Force and Moment Resultants 19 1.7 Internal Forces and Stresses-Stress Resultants 27 1.8 Restraint Forces and Restraint Force Resultants 32 1.9 Summary and Conclusions 33 2 Static Equilibrium 35 2.1 Introduction 35 2.2 Free Body Diagrams 35 2.3 Equilibrium-Concentrated Forces 38 2.3.1 Two Force Members and Pin Jointed Trusses 38 2.3.2 Slender Rigid Bars 44 2.3.3 Pulleys and Cables 49 2.3.4 Springs 52 2.4 Equilibrium-Distributed Forces 55 2.5 Equilibrium in Three Dimensions 59 2.6 Equilibrium-Internal Forces and Stresses 62 2.6.1 Equilibrium of Internal Forces in Three Dimensions 65 2.6.2 Equilibrium in Two Dimensions-Plane Stress 69 2.6.3 Equilibrium in One Dimension-Uniaxial Stress 70 2.7 Summary and Conclusions 70 3 Displacement, Strain, and Material Properties 71 3.1 Introduction 71 3.2 Displacement and Strain 71 3.2.1 Displacement 72 3.2.2 Strain 72 3.3 Compatibility 76 3.4 Linear Material Properties 77 3.4.1 Hooke's Law in One Dimension-Tension 77 3.4.2 Poisson's Ratio 81 3.4.3 Hooke's Law in One Dimension-Shear in Isotropic Materials 82 3.4.4 Hooke's Law in Two Dimensions for Isotropic Materials 83 3.4.5 Generalized Hooke's Law for Isotropic Materials 84 3.5 Some Simple Solutions for Stress, Strain, and Displacement 85 3.6 Thermal Strain 89 3.7 Engineering Materials 90 3.8 Fiber Reinforced Composite Laminates 90 3.8.1 Hooke's Law in Two Dimensions for a FRP Lamina 91 3.8.2 Properties of Unidirectional Lamina 94 3.9 Plan for the Following Chapters 96 3.10 Summary and Conclusions 98 4 Classical Analysis of the Axially Loaded Slender Bar 99 4.1 Introduction 99 4.2 Solutions from the Theory of Elasticity 99 4.3 Derivation and Solution of the Governing Equations 109 4.4 The Statically Determinate Case 116 4.5 The Statically Indeterminate Case 129 4.6 Variable Cross Sections 136 4.7 Thermal Stress and Strain in an Axially Loaded Bar 142 4.8 Shearing Stress in an Axially Loaded Bar 143 4.9 Design of Axially Loaded Bars 145 4.10 Analysis and Design of Pin Jointed Trusses 149 4.11 Work and Energy-Castigliano's Second Theorem 153 4.12 Summary and Conclusions 162 5 A General Method for the Axially Loaded Slender Bar 165 5.1 Introduction 165 5.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix 165 5.3 The Assembled Global Equations and Their Solution 169 5.4 A General Method-Distributed Applied Loads 182 5.5 Variable Cross Sections 196 5.6 Analysis and Design of Pin-jointed Trusses 202 5.7 Summary and Conclusions 211 6 Torsion 213 6.1 Introduction 213 6.2 Torsional Displacement, Strain, and Stress 213 6.3 Derivation and Solution of the Governing Equations 216 6.4 Solutions from the Theory of Elasticity 225 6.5 Torsional Stress in Thin Walled Cross Sections 229 6.6 Work and Energy-Torsional Stiffness in a Thin Walled Tube 231 6.7 Torsional Stress and Stiffness in Multicell Sections 239 6.8 Torsional Stress and Displacement in Thin Walled Open Sections 242 6.9 A General (Finite Element) Method 245 6.10 Continuously Variable Cross Sections 254 6.11 Summary and Conclusions 255 7 Classical Analysis of the Bending of Beams 257 7.1 Introduction 257 7.2 Area Properties-Sign Conventions 257 7.2.1 Area Properties 257 7.2.2 Sign Conventions 259 7.3 Derivation and Solution of the Governing Equations 260 7.4 The Statically Determinate Case 271 7.5 Work and Energy-Castigliano's Second Theorem 278 7.6 The Statically Indeterminate Case 281 7.7 Solutions from the Theory of Elasticity 290 7.8 Variable Cross Sections 300 7.9 Shear Stress in Non Rectangular Cross Sections-Thin Walled Cross Sections 302 7.10 Design of Beams 309 7.11 Large Displacements 313 7.12 Summary and Conclusions 314 8 A General Method (FEM) for the Bending of Beams 315 8.1 Introduction 315 8.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix 315 8.3 The Global Equations and their Solution 320 8.4 Distributed Loads in FEM 327 8.5 Variable Cross Sections 341 8.6 Summary and Conclusions 345 9 More about Stress and Strain, and Material Properties 347 9.1 Introduction 347 9.2 Transformation of Stress in Two Dimensions 347 9.3 Principal Axes and Principal Stresses in Two Dimensions 350 9.4 Transformation of Strain in Two Dimensions 354 9.5 Strain Rosettes 356 9.6 Stress Transformation and Principal Stresses in Three Dimensions 358 9.7 Allowable and Ultimate Stress, and Factors of Safety 361 9.8 Fatigue 363 9.9 Creep 364 9.10 Orthotropic Materials-Composites 365 9.11 Summary and Conclusions 366 10 Combined Loadings on Slender Bars-ThinWalled Cross Sections 367 10.1 Introduction 367 10.2 Review and Summary of Slender Bar Equations 367 10.2.1 Axial Loading 367 10.2.2 Torsional Loading 369 10.2.3 Bending in One Plane 370 10.3 Axial and Torsional Loads 372 10.4 Axial and Bending Loads-2D Frames 375 10.5 Bending in Two Planes 384 10.5.1 When Iyz is Equal to Zero 384 10.5.2 When Iyz is Not Equal to Zero 386 10.6 Bending and Torsion in Thin Walled Open Sections-Shear Center 393 10.7 Bending and Torsion in Thin Walled Closed Sections-Shear Center 399 10.8 Stiffened Thin Walled Beams 405 10.9 Summary and Conclusions 416 11 Work and Energy Methods-Virtual Work 417 11.1 Introduction 417 11.2 Introduction to the Principle of Virtual Work 417 11.3 Static Analysis of Slender Bars by Virtual Work 421 11.3.1 Axially Loading 421 11.3.2 Torsional Loading 426 11.3.3 Beams in Bending 427 11.3.4 Combined Axial, Torsional, and Bending Behavior 430 11.4 Static Analysis of 3D and 2D Solids by Virtual Work 430 11.5 The Element Stiffness Matrix for Plane Stress 433 11.6 The Element Stiffness Matrix for 3D Solids 436 11.7 Summary and Conclusions 437 12 Structural Analysis in Two and Three Dimensions 439 12.1 Introduction 439 12.2 The Governing Equations in Two Dimensions-Plane Stress 440 12.3 Finite Elements and the Stiffness Matrix for Plane Stress 445 12.4 Thin Flat Plates-Classical Analysis 452 12.5 Thin Flat Plates-FEM Analysis 455 12.6 Shell Structures 459 12.7 Stiffened Shell Structures 466 12.8 Three Dimensional Structures-Classical and FEM Analysis 470 12.9 Summary and Conclusions 477 13 Analysis of Thin Laminated Composite Material Structures 479 13.1 Introduction to Classical Lamination Theory 479 13.2 Strain Displacement Equations for Laminates 480 13.3 Stress-Strain Relations for a Single Lamina 482 13.4 Stress Resultants for Laminates 486 13.5 CLT Constitutive Description 489 13.6 Determining Laminae Stress/Strains 492 13.7 Laminated Plates Subject to Transverse Loads 493 13.8 Summary and Conclusion 498 14 Buckling 499 14.1 Introduction 499 14.2 The Equations for a Beam with Combined Lateral and Axial Loading 499 14.3 Buckling of a Column 504 14.4 The Beam Column 512 14.5 The Finite Element Method for Bending and Buckling 515 14.6 Buckling of Frames 524 14.7 Buckling of Thin Plates and Other Structures 524 14.8 Summary and Conclusions 527 15 Structural Dynamics 529 15.1 Introduction 529 15.2 Dynamics of Mass/Spring Systems 529 15.2.1 Free Motion 529 15.2.2 Forced Motion-Resonance 540 15.2.3 Forced Motion-Response 547 15.3 Axial Vibration of a Slender Bar 548 15.3.1 Solutions Based on the Differential Equation 548 15.3.2 Solutions Based on FEM 560 15.4 Torsional Vibration 567 15.4.1 Torsional Mass/Spring Systems 567 15.4.2 Distributed Torsional Systems 568 15.5 Vibration of Beams in Bending 569 15.5.1 Solutions of the Differential Equation 569 15.5.2 Solutions Based on FEM 574 15.6 The Finite Element Method for all Elastic Structures 577 15.7 Addition of Damping 577 15.8 Summary and Conclusions 582 16 Evolution in the (Intelligent) Design and Analysis of Structural Members 583 16.1 Introduction 583 16.2 Evolution of a Truss Member 584 16.2.1 Step 1. Slender Bar Analysis 584 16.2.2 Step 2. Rectangular Bar-Plane Stress FEM 585 16.2.3 Step 3. Rectangular Bar with Pin Holes-Plane Stress Analysis 586 16.2.4 Step 4. Rectangular Bar with Pin Holes-Solid Body Analysis 587 16.2.5 Step 5. Add Material Around the Hole-Solid Element Analysis 588 16.2.6 Step 6. Bosses Added-Solid Element Analysis 590 16.2.7 Step 7. Reducing the Weight-Solid Element Analysis 591 16.2.8 Step 8. Buckling Analysis 592 16.3 Evolution of a Plate with a Hole-Plane Stress 592 16.4 Materials in Design 594 16.5 Summary and Conclusions 594 A Matrix Definitions and Operations 595 A.1 Introduction 595 A.2 Matrix Definitions 595 A.3 Matrix Algebra 597 A.4 Partitioned Matrices 598 A.5 Differentiating and Integrating a Matrix 598 A.6 Summary of Useful Matrix Relations 599 B Area Properties of Cross Sections 601 B.1 Introduction 601 B.2 Centroids of Cross Sections 601 B.3 Area Moments and Product of Inertia 603 B.4 Properties of Common Cross Sections 609 C Solving Sets of Linear Algebraic Equations with Mathematica 611 C.1 Introduction 611 C.2 Systems of Linear Algebraic Equations 611 C.3 Solving Numerical Equations in Mathematica 611 C.4 Solving Symbolic Equations in Mathematica 612 C.5 Matrix Multiplication 613 D Orthogonality of Normal Modes 615 D.1 Introduction 615 D.2 Proof of Orthogonality for Discrete Systems 615 D.3 Proof of Orthogonality for Continuous Systems 616 References 617 Index 619
About the Authors xiii Preface xv 1 Forces and Moments 1 1.1 Introduction 1 1.2 Units 1 1.3 Forces in Mechanics of Materials 3 1.4 Concentrated Forces 4 1.5 Moment of a Concentrated Force 9 1.6 Distributed Forces-Force and Moment Resultants 19 1.7 Internal Forces and Stresses-Stress Resultants 27 1.8 Restraint Forces and Restraint Force Resultants 32 1.9 Summary and Conclusions 33 2 Static Equilibrium 35 2.1 Introduction 35 2.2 Free Body Diagrams 35 2.3 Equilibrium-Concentrated Forces 38 2.3.1 Two Force Members and Pin Jointed Trusses 38 2.3.2 Slender Rigid Bars 44 2.3.3 Pulleys and Cables 49 2.3.4 Springs 52 2.4 Equilibrium-Distributed Forces 55 2.5 Equilibrium in Three Dimensions 59 2.6 Equilibrium-Internal Forces and Stresses 62 2.6.1 Equilibrium of Internal Forces in Three Dimensions 65 2.6.2 Equilibrium in Two Dimensions-Plane Stress 69 2.6.3 Equilibrium in One Dimension-Uniaxial Stress 70 2.7 Summary and Conclusions 70 3 Displacement, Strain, and Material Properties 71 3.1 Introduction 71 3.2 Displacement and Strain 71 3.2.1 Displacement 72 3.2.2 Strain 72 3.3 Compatibility 76 3.4 Linear Material Properties 77 3.4.1 Hooke's Law in One Dimension-Tension 77 3.4.2 Poisson's Ratio 81 3.4.3 Hooke's Law in One Dimension-Shear in Isotropic Materials 82 3.4.4 Hooke's Law in Two Dimensions for Isotropic Materials 83 3.4.5 Generalized Hooke's Law for Isotropic Materials 84 3.5 Some Simple Solutions for Stress, Strain, and Displacement 85 3.6 Thermal Strain 89 3.7 Engineering Materials 90 3.8 Fiber Reinforced Composite Laminates 90 3.8.1 Hooke's Law in Two Dimensions for a FRP Lamina 91 3.8.2 Properties of Unidirectional Lamina 94 3.9 Plan for the Following Chapters 96 3.10 Summary and Conclusions 98 4 Classical Analysis of the Axially Loaded Slender Bar 99 4.1 Introduction 99 4.2 Solutions from the Theory of Elasticity 99 4.3 Derivation and Solution of the Governing Equations 109 4.4 The Statically Determinate Case 116 4.5 The Statically Indeterminate Case 129 4.6 Variable Cross Sections 136 4.7 Thermal Stress and Strain in an Axially Loaded Bar 142 4.8 Shearing Stress in an Axially Loaded Bar 143 4.9 Design of Axially Loaded Bars 145 4.10 Analysis and Design of Pin Jointed Trusses 149 4.11 Work and Energy-Castigliano's Second Theorem 153 4.12 Summary and Conclusions 162 5 A General Method for the Axially Loaded Slender Bar 165 5.1 Introduction 165 5.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix 165 5.3 The Assembled Global Equations and Their Solution 169 5.4 A General Method-Distributed Applied Loads 182 5.5 Variable Cross Sections 196 5.6 Analysis and Design of Pin-jointed Trusses 202 5.7 Summary and Conclusions 211 6 Torsion 213 6.1 Introduction 213 6.2 Torsional Displacement, Strain, and Stress 213 6.3 Derivation and Solution of the Governing Equations 216 6.4 Solutions from the Theory of Elasticity 225 6.5 Torsional Stress in Thin Walled Cross Sections 229 6.6 Work and Energy-Torsional Stiffness in a Thin Walled Tube 231 6.7 Torsional Stress and Stiffness in Multicell Sections 239 6.8 Torsional Stress and Displacement in Thin Walled Open Sections 242 6.9 A General (Finite Element) Method 245 6.10 Continuously Variable Cross Sections 254 6.11 Summary and Conclusions 255 7 Classical Analysis of the Bending of Beams 257 7.1 Introduction 257 7.2 Area Properties-Sign Conventions 257 7.2.1 Area Properties 257 7.2.2 Sign Conventions 259 7.3 Derivation and Solution of the Governing Equations 260 7.4 The Statically Determinate Case 271 7.5 Work and Energy-Castigliano's Second Theorem 278 7.6 The Statically Indeterminate Case 281 7.7 Solutions from the Theory of Elasticity 290 7.8 Variable Cross Sections 300 7.9 Shear Stress in Non Rectangular Cross Sections-Thin Walled Cross Sections 302 7.10 Design of Beams 309 7.11 Large Displacements 313 7.12 Summary and Conclusions 314 8 A General Method (FEM) for the Bending of Beams 315 8.1 Introduction 315 8.2 Nodes, Elements, Shape Functions, and the Element Stiffness Matrix 315 8.3 The Global Equations and their Solution 320 8.4 Distributed Loads in FEM 327 8.5 Variable Cross Sections 341 8.6 Summary and Conclusions 345 9 More about Stress and Strain, and Material Properties 347 9.1 Introduction 347 9.2 Transformation of Stress in Two Dimensions 347 9.3 Principal Axes and Principal Stresses in Two Dimensions 350 9.4 Transformation of Strain in Two Dimensions 354 9.5 Strain Rosettes 356 9.6 Stress Transformation and Principal Stresses in Three Dimensions 358 9.7 Allowable and Ultimate Stress, and Factors of Safety 361 9.8 Fatigue 363 9.9 Creep 364 9.10 Orthotropic Materials-Composites 365 9.11 Summary and Conclusions 366 10 Combined Loadings on Slender Bars-ThinWalled Cross Sections 367 10.1 Introduction 367 10.2 Review and Summary of Slender Bar Equations 367 10.2.1 Axial Loading 367 10.2.2 Torsional Loading 369 10.2.3 Bending in One Plane 370 10.3 Axial and Torsional Loads 372 10.4 Axial and Bending Loads-2D Frames 375 10.5 Bending in Two Planes 384 10.5.1 When Iyz is Equal to Zero 384 10.5.2 When Iyz is Not Equal to Zero 386 10.6 Bending and Torsion in Thin Walled Open Sections-Shear Center 393 10.7 Bending and Torsion in Thin Walled Closed Sections-Shear Center 399 10.8 Stiffened Thin Walled Beams 405 10.9 Summary and Conclusions 416 11 Work and Energy Methods-Virtual Work 417 11.1 Introduction 417 11.2 Introduction to the Principle of Virtual Work 417 11.3 Static Analysis of Slender Bars by Virtual Work 421 11.3.1 Axially Loading 421 11.3.2 Torsional Loading 426 11.3.3 Beams in Bending 427 11.3.4 Combined Axial, Torsional, and Bending Behavior 430 11.4 Static Analysis of 3D and 2D Solids by Virtual Work 430 11.5 The Element Stiffness Matrix for Plane Stress 433 11.6 The Element Stiffness Matrix for 3D Solids 436 11.7 Summary and Conclusions 437 12 Structural Analysis in Two and Three Dimensions 439 12.1 Introduction 439 12.2 The Governing Equations in Two Dimensions-Plane Stress 440 12.3 Finite Elements and the Stiffness Matrix for Plane Stress 445 12.4 Thin Flat Plates-Classical Analysis 452 12.5 Thin Flat Plates-FEM Analysis 455 12.6 Shell Structures 459 12.7 Stiffened Shell Structures 466 12.8 Three Dimensional Structures-Classical and FEM Analysis 470 12.9 Summary and Conclusions 477 13 Analysis of Thin Laminated Composite Material Structures 479 13.1 Introduction to Classical Lamination Theory 479 13.2 Strain Displacement Equations for Laminates 480 13.3 Stress-Strain Relations for a Single Lamina 482 13.4 Stress Resultants for Laminates 486 13.5 CLT Constitutive Description 489 13.6 Determining Laminae Stress/Strains 492 13.7 Laminated Plates Subject to Transverse Loads 493 13.8 Summary and Conclusion 498 14 Buckling 499 14.1 Introduction 499 14.2 The Equations for a Beam with Combined Lateral and Axial Loading 499 14.3 Buckling of a Column 504 14.4 The Beam Column 512 14.5 The Finite Element Method for Bending and Buckling 515 14.6 Buckling of Frames 524 14.7 Buckling of Thin Plates and Other Structures 524 14.8 Summary and Conclusions 527 15 Structural Dynamics 529 15.1 Introduction 529 15.2 Dynamics of Mass/Spring Systems 529 15.2.1 Free Motion 529 15.2.2 Forced Motion-Resonance 540 15.2.3 Forced Motion-Response 547 15.3 Axial Vibration of a Slender Bar 548 15.3.1 Solutions Based on the Differential Equation 548 15.3.2 Solutions Based on FEM 560 15.4 Torsional Vibration 567 15.4.1 Torsional Mass/Spring Systems 567 15.4.2 Distributed Torsional Systems 568 15.5 Vibration of Beams in Bending 569 15.5.1 Solutions of the Differential Equation 569 15.5.2 Solutions Based on FEM 574 15.6 The Finite Element Method for all Elastic Structures 577 15.7 Addition of Damping 577 15.8 Summary and Conclusions 582 16 Evolution in the (Intelligent) Design and Analysis of Structural Members 583 16.1 Introduction 583 16.2 Evolution of a Truss Member 584 16.2.1 Step 1. Slender Bar Analysis 584 16.2.2 Step 2. Rectangular Bar-Plane Stress FEM 585 16.2.3 Step 3. Rectangular Bar with Pin Holes-Plane Stress Analysis 586 16.2.4 Step 4. Rectangular Bar with Pin Holes-Solid Body Analysis 587 16.2.5 Step 5. Add Material Around the Hole-Solid Element Analysis 588 16.2.6 Step 6. Bosses Added-Solid Element Analysis 590 16.2.7 Step 7. Reducing the Weight-Solid Element Analysis 591 16.2.8 Step 8. Buckling Analysis 592 16.3 Evolution of a Plate with a Hole-Plane Stress 592 16.4 Materials in Design 594 16.5 Summary and Conclusions 594 A Matrix Definitions and Operations 595 A.1 Introduction 595 A.2 Matrix Definitions 595 A.3 Matrix Algebra 597 A.4 Partitioned Matrices 598 A.5 Differentiating and Integrating a Matrix 598 A.6 Summary of Useful Matrix Relations 599 B Area Properties of Cross Sections 601 B.1 Introduction 601 B.2 Centroids of Cross Sections 601 B.3 Area Moments and Product of Inertia 603 B.4 Properties of Common Cross Sections 609 C Solving Sets of Linear Algebraic Equations with Mathematica 611 C.1 Introduction 611 C.2 Systems of Linear Algebraic Equations 611 C.3 Solving Numerical Equations in Mathematica 611 C.4 Solving Symbolic Equations in Mathematica 612 C.5 Matrix Multiplication 613 D Orthogonality of Normal Modes 615 D.1 Introduction 615 D.2 Proof of Orthogonality for Discrete Systems 615 D.3 Proof of Orthogonality for Continuous Systems 616 References 617 Index 619
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