Simultaneous Differential Equations and Multi-Dimensional Vibrations Mathematics and Physics for Science and Technology Series Editor: L.M.B.C. Campos Director of the Center for Aeronautical and Space Science and Technology Lisbon University Volumes in the series: Topic A – Theory of Functions and Potential Problems Volume I (Book 1) – Complex Analysis with Applications to Flows and Fields L.M.B.C. Campos Volume II (Book 2) – Elementary Transcendentals with Applications to Solids and Fluids L.M.B.C. Campos Volume III (Book 3) – Generalized Calculus with Applications to Matter and Forces L.M.B.C. Campos Topic B – Boundary and Initial-Value Problems Volume IV – Ordinary Differential Equations with Applications to Trajectories and Oscillations L.M.B.C. Campos Book 4 – Linear Differential Equations and Oscillators L.M.B.C. Campos Book 5 – Non-Linear Differential Equations and Dynamical Systems L.M.B.C. Campos Book 6 – Higher-Order Differential Equations and Elasticity L.M.B.C. Campos Book 7 – Simultaneous Differential Equations and Multi-Dimensional Vibrations L.M.B.C. Campos Book 8 – Singular Differential Equations and Special Functions L.M.B.C. Campos Book 9 – Classification and Examples of Differential Equations and their Applications L.M.B.C. Campos For more information about this series, please visit: https://www.crcpress. com/Mathematics-and-Physics-for-Science-and-Technology/book-series/ CRCMATPHYSCI Mathematics and Physics for Science and Technology Volume IV Ordinary Differential Equations with Applications to Trajectories and Oscillations Book 7 Simultaneous Differential Equations and Multi- Dimensional Vibrations By L.M.B.C. Campos Director of the Center for Aeronautical and Space Science and Technology Lisbon University CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 334 87-2742 © 2020 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-0-367-13721-2 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Campos, Luis Manuel Braga da Costa, author. Title: Simultaneous differential equations and multi-dimensional vibrations/ Luis Manuel Braga da Campos. Description: First edition. | Boca Raton, FL : CRC Press/Taylor & Francis Group, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2018049440| ISBN 9780367137212 (hardback : acid-free paper) | ISBN 9780429030253 (ebook) Subjects: LCSH: Vibration--Mathematical models. | Oscillations–Mathematical models. | Differential equations. | Equations, Simultaneous. | Engineering mathematics. Classification: LCC TA355 .C28 2018 | DDC 531/.320151535–dc23 LC record available at https://lccn.loc.gov/2018049440 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com to Leonor Campos Contents Contents .................................................................................................................vii Diagrams, Notes, and Tables .............................................................................xiii Preface ..................................................................................................................xvii Acknowledgments ..............................................................................................xix About the Author ................................................................................................xxi Physical Quantities ...........................................................................................xxiii 7. Simultaneous Differential Equations ........................................................1 7.1 Reduction of General to Autonomous Systems ................................2 7.1.1 Autonomous System of Differential Equations ...................2 7.1.2 General System of Simultaneous Differential Equations ...3 7.2 Tangents, Trajectories, and Paths in N-Dimensions .........................4 7.2.1 N-Dimensional Hypercurve Specified by Tangent Vectors .......................................................................................5 7.2.2 Families of Curves in the Plane or in Space .........................6 7.2.3 N-Dimensional Curve Lying on the Intersection of M Hypersurfaces ......................................................................7 7.2.4 Space Curves as the Intersection of Two Surfaces ..............8 7.2.5 Hypersurfaces Orthogonal to a Vector Field .....................10 7.3 Order of a Simultaneous System of Differential Equations ..........12 7.3.1 Definition of Order for Simultaneous Differential Equations .................................................................................12 7.3.2 Transformation from a Simultaneous to a Decoupled System ......................................................................................14 7.3.3 Constants of Integration and Depression of the Order ........................................................................................15 7.4 Linear Simultaneous System with Constant Coefficients .............16 7.4.1 Linear Simultaneous System with Variable Coefficients .............................................................................17 7.4.2 Linear Forced System with Constant Coefficients ............18 7.4.3 Characteristic Polynomial of a Simultaneous System ......19 7.4.4 Non-Degenerate and Degenerate Differential Systems ....................................................................................21 7.4.5 Distinct Roots of the Characteristic Polynomial ...............21 7.4.6 Multiple Roots of the Characteristic Polynomial ..............22 7.4.7 General Integral and Linearly Independent Particular Integrals ................................................................23 7.4.8 General Integral for Distinct Roots .....................................25 7.4.9 Arbitrary Constants and Boundary Conditions ...............25 vii viii Contents 7.4.10 General Integral with Multiple Roots .................................27 7.4.11 Natural Integrals and Diagonal or Banded System ..........28 7.4.12 Block-Banded Diagonal System ...........................................29 7.4.13 Diagonalization of a Square System ...................................30 7.4.14 Transformation from a Non-Diagonal to a Banded System ......................................................................................32 7.5 Integrals of Forced and Unforced Systems......................................33 7.5.1 Forcing of a Simultaneous System by an Exponential .....33 7.5.2 Single and Multiple Resonant Forcing................................34 7.5.3 Non-Resonant and Resonant Forcing by an Exponential .............................................................................35 7.5.4 Forcing by the Product of an Exponential by a Sine or Cosine .................................................................................37 7.5.5 Forcing by Hyperbolic or Circular Cosines or Sines ........38 7.5.6 Inverse Matrix of Polynomials of Derivatives ...................39 7.5.7 Power Series Expansion of Inverse Polynomial Operator ..................................................................................39 7.5.8 Principle of Superposition and Addition of Particular Integrals ................................................................41 7.6 Natural Integrals for Simultaneous Homogeneous Systems ........42 7.6.1 Linear System of Homogeneous Derivatives .....................42 7.6.2 Matrix of Polynomials of Homogeneous Derivatives ..............................................................................44 7.6.3 Unforced System and Characteristic Polynomial .............45 7.6.4 Distinct and Multiple Roots of the Characteristic Polynomial ..............................................................................46 7.6.5 Natural Integrals and the General Integral .......................47 7.6.6 Compatibility Conditions for the Dependent Variables ..................................................................................48 7.6.7 Arbitrary Constants and Boundary Conditions ...............48 7.6.8 Decoupled or Minimally-Coupled Natural Differential System ................................................................49 7.6.9 Block Diagonal-Banded System ...........................................50 7.7 Forced and Unforced Homogeneous Systems ................................51 7.7.1 Analogy of Constant and Homogeneous Coefficients .............................................................................52 7.7.2 Forcing of a Homogeneous System by a Power .................53 7.7.3 Non-Resonant and Multiply Resonant Particular Integrals ...................................................................................55 7.7.4 Power Forcing and Single Resonance .................................55 7.7.5 Double Root and Double Resonance ...................................56 7.7.6 Cosine and Sine of Multiples of Logarithms .....................57 7.7.7 Forcing by a Power Multiplied by a Double Product .......59 7.7.8 Inverse Matrix of Polynomials of Homogeneous Derivatives ..............................................................................60 Contents ix 7.7.9 Homogeneous Forcing by a Polynomial of Logarithms ..............................................................................61 7.7.10 Complete Integral of the Forced Homogeneous Derivatives ..............................................................................62 7.8 Simultaneous Finite Difference Equations ......................................64 7.8.1 Non-Linear and Linear Finite Difference Equations .................................................................................64 7.8.2 Operator Forward Finite Difference ...................................65 7.8.3 Matrix of Polynomials of Finite Differences ......................66 7.8.4 Simple and Multiple Roots of the Characteristic Polynomial ..............................................................................68 7.8.5 General Solution of an Unforced System............................68 7.8.6 Compatibility Conditions for the Dependent Variables ..................................................................................69 7.8.7 Arbitrary Constants and Starting Conditions ...................70 7.8.8 Diagonal or Lower Triangular System ...............................70 7.8.9 Block-Diagonal Lower Triangular System .........................71 7.8.10 Diagonalization of a Finite Difference System ..................72 7.9 Unforced and Forced Finite Difference ...........................................73 7.9.1 Forward, Backward, and Central Differences ...................74 7.9.2 Forcing by a Power with Integer Exponent ........................75 7.9.3 Non-Resonant Forcing by Integral Powers ........................77 7.9.4 Three Cases of Simple Resonance .......................................78 7.9.5 Product of Power by Circular and Hyperbolic Functions .................................................................................80 7.9.6 Products of Powers by Cosines of Multiple Angles ..........82 7.9.7 Complete Integral of Forced Finite Differences ................82 7.9.8 Comparison of Three Matrix Polynomial Systems ...........83 Conclusion 7 .................................................................................................156 8. Oscillations with Several Degrees-of-Freedom ...................................159 8.1 Balance of Forces, Energy, and Dissipation ...................................160 8.1.1 Restoring, Friction, Inertia, and Applied Forces .............161 8.1.2 Linear Restoring Force and Quadratic Potential .............161 8.1.3 Friction Force and Dissipation Function ..........................162 8.1.4 Coupled and Decoupled Equations of Motion ................163 8.1.5 Activity/Power and Work of the Applied Forces ............164 8.1.6 Kinetic, Potential, and Total Energies ...............................166 8.2 Modal Frequencies, Damping, Coordinates, and Forces ............167 8.2.1 Mass, Damping, and Oscillation Matrices .......................167 8.2.2 Friction, Oscillation, and Dispersion Matrices ................168 8.2.3 Free Undamped Decoupled Oscillations .........................169 8.2.4 Modal Frequencies of Undamped Oscillations ...............170 8.2.5 Modal Dampings of Decaying Oscillations .....................171 8.2.6 Modal Coordinates and Oscillation Frequencies ............173