Mechanical Vibration

Ivana Kovačić, University of Novi Sad, Serbia Ivana Kovačić graduated in Mechanical Engineering from the Faculty of Technical Sciences (FTN), University of Novi Sad, Serbia. She obtained her MSc and PhD in the Theory of Nonlinear Vibrations at the FTN. She is currently a Full Professor of Mechanics at the FTN and the head of the Centre of Excellence for Vibro-Acoustic Systems and Signal Processing CEVAS at the same faculty. Kovačić is the Subject Editor of three academic journals: the Journal of Sound and Vibration, the European Journal of Mechanics A/Solids and Meccanica. Her research involves the use of quantitative and qualitative methods to study differential equations arising from nonlinear dynamics problems mainly in mechanical engineering, and recently also in biomechanics and tree vibrations. Dragi Radomirović, University of Novi Sad, Serbia Dragi Radomirović graduated in Mechanical Engineering from the Faculty of Technical Sciences (FTN), University of Novi Sad (UNS), Serbia. He obtained his MSc and PhD in Analytical Mechanics at the FTN. He is a Full Professor of Mechanics at the Faculty of Agriculture, UNS. His research interests are directed towards Mechanical Vibrations and Analytical Mechanics.

• About the Authors ixPreface xi1 Preliminaries 1Chapter Outline 1Chapter Objectives 11.1 From Statics 11.1.1 Mechanical Systems and Equilibrium Equations 11.1.2 Constraints and Free-Body Diagrams 11.1.3 Equilibrium Condition Via Virtual Work 21.2 From Kinematics 41.2.1 Kinematics of Particles 41.2.2 Kinematics of Rigid Bodies 51.2.3 Kinematics of Particles in Compound Motion 71.3 From Kinetics 81.3.1 Kinetics of Particles 81.3.2 Kinetics of Rigid Bodies 91.4 From Strength of Materials 131.4.1 Axial Loading 131.4.2 Torsion 141.4.3 Bending 142 Lagrange’s Equation for Mechanical Oscillatory Systems 17Chapter Outline 17Chapter Objectives 172.1 About Lagrange’s Equation of the Second Kind 172.2 Kinetic Energy in Mechanical Oscillatory Systems 192.3 Potential Energy in Mechanical Oscillatory Systems 212.3.1 Gravitational Potential Energy 222.3.2 Potential Energy of a Spring (Elastic Potential Energy) 242.4 Generalised Forces in Mechanical Oscillatory Systems 272.5 Dissipative Function in Mechanical Oscillatory Systems 28References 303 Free Undamped Vibration of Single-Degree-of-Freedom Systems 31Chapter Outline 31Chapter Objectives 31Theoretical Introduction 314 Free Damped Vibration of Single-Degree-of-Freedom Systems 67Chapter Outline 67Chapter Objectives 67Theoretical Introduction 675 Forced Vibration of Single-Degree-of-Freedom Systems 101Chapter Outline 101Chapter Objectives 101Theoretical Introduction 1016 Free Undamped Vibration of Two-Degree-of-Freedom Systems 127Chapter Outline 127Chapter Objectives 127Theoretical Introduction 1277 Forced Vibration of Two-Degree-of-Freedom Systems 153Chapter Outline 153Chapter Objectives 153Theoretical Introduction 1538 Vibration of Systems with Infinite Number of Degrees of Freedom 183Chapter Outline 183Chapter Objectives 1838.1 Theoretical Introduction: Longitudinal Vibration of Bars 1838.2 Theoretical Introduction: Torsional Vibration of Shafts 1978.3 Theoretical Introduction: Transversal Vibration of Beams 2079 Additional Topics 225Chapter Outline 225Chapter Objectives 2259.1 Theoretical Introduction 2259.2 Equivalent Two-Element System for Concurrent Springs and Dampers 2269.2.1 Concurrent Springs 2279.2.2 Concurrent Dampers 2319.3 Nonlinear Springs in Series 2389.3.1 Purely Nonlinear Springs in Series 2399.3.2 Equal Duffing Springs in Series 2399.3.3 Two Different Nonlinear Springs 2409.4 On the Deflection and Potential Energy of Nonlinear Springs: Approximate Expressions 2429.4.1 Duffing-Type Spring Deformed in the Static Equilibrium Position 2429.4.2 Duffing-Type Spring Undeformed in the Static Equilibrium Position 2429.5 Corrections of Stiffness Properties of Certain Oscillatory Systems 2449.5.1 One-Degree-of-Freedom Systems 2459.5.2 Two-Degree-of-Freedom Systems 248Appendix: Mathematical Topics 255A.1 Geometry 255A.2 Trigonometry 257A.3 Algebra 258A.4 Vectors 258A.5 Derivatives 259A.6 Variation (Virtual Displacements) 260A.7 Series 260Index 261