Dynamic Stability and Bifurcation in Nonconservative Mechanics (CISM International Centre for Mechanical Sciences Book 586)

The book offers a unified view on classical results and recent advances in the dynamics of nonconservative systems. The theoretical fundamentals are presented systematically and include: Lagrangian and Hamiltonian formalism, non-holonomic constraints, Lyapunov stability theory, Krein theory of spectra of Hamiltonian systems and modes of negative and positive energy, anomalous Doppler effect, reversible systems, sensitivity analysis of non-self-adjoint operators, dissipation-induced instabilities, local and global instabilities. They are applied to engineering situations such as the coupled mode flutter of wings, flags and pipes, flutter in granular materials, piezoelectric mechanical metamaterials, wave dynamics of infinitely long structures, radiative damping, stability of high-speed trains, experimental realization of follower forces, soft-robot locomotion, wave energy converters, friction-induced instabilities, brake squeal, non-holonomic sailing, dynamics of moving continua, and stability of bicycles and walking robots. The book responds to a demand in the modern theory of nonconservative systems coming from the growing number of scientific and engineering disciplines including physics, fluid and solids mechanics, fluid-structure interactions, and modern multidisciplinary research areas such as biomechanics, micro- and nanomechanics, optomechanics, robotics, and material science. It is targeted at both young and experienced researchers and engineers working in fields associated with the dynamics of structures and materials. The book will help to get a comprehensive and systematic knowledge on the stability, bifurcations and dynamics of nonconservative systems and establish links between approaches and methods developed in different areas of mechanics and physics and modern applied mathematics. Preface......Page 6 Contents......Page 8 1 Introduction......Page 9 2.1 The Ziegler Double Pendulum......Page 12 2.2 The Standard Case as a Reference: The Dead Load H on the Double Pendulum......Page 17 2.3 The Follower Force P on the Double Pendulum......Page 20 2.4 Surprising Effects Related to the Viscosity: The Ziegler Paradox......Page 25 2.5 The Deformation of an Elastic Rod: The Euler's Elastica......Page 27 2.6 Constitutive Equation and Dynamics......Page 33 2.7 The Beck and Pflüger Rods......Page 36 2.8 Self-adjointness, an Exclusion Condition for Flutter......Page 42 2.9 Beyond the Linearized Solution: Limit Cycle Behaviour......Page 43 2.10 Follower Forces from Coulomb Friction......Page 45 2.11 Self-oscillating Systems......Page 47 3 Flutter in Frictional Solids......Page 48 3.1 Contact with Coulomb Friction Versus Nonassociative Elastoplasticity......Page 51 3.2 The Rate Equations of Nonassociative Elastoplasticity for Frictional Solids......Page 54 3.3 The Propagation of Incremental Plane Waves......Page 56 3.4 Strain Localization into Planar Bands......Page 58 3.5 The Analysis of the Acoustic Tensor and Flutter Instability......Page 61 4 Concluding Remarks on Flutter Instability in Structures and Solids......Page 64 References......Page 67 Dissipation Induced Instabilities of Structures Coupled to a Flow......Page 70 1.1 Cross-Flow Instabilities......Page 71 1.2 Dynamic Instability by Negative Flow-Induced Damping......Page 75 1.3 Wing Instabilities Due to Mode Coupling......Page 77 1.4 Axial Flow Problems......Page 80 2 Damping Induced Instabilities of Structures Coupled to a Flow......Page 89 2.1 Damping Induced Instabilities of Wings......Page 90 2.2 The Fluid-Conveying Pipe Model System......Page 91 2.3 Conclusion......Page 99 3.1 Energy Converters......Page 100 3.2 Models of Energy Harvesting Systems Based on Flow-Induced Vibrations......Page 101 3.3 Conclusion......Page 107 References......Page 109 1 Introduction......Page 110 2 Background on Euler Angles and Bases......Page 112 2.1 The Euler and Dual Euler Bases......Page 114 2.2 Vector Representations......Page 115 3 Lagrange's Equations of Motion and the Newton–Euler Equations of Motion......Page 116 3.1 A Force FA Acting at a Material Point XA......Page 118 3.2 Ideal Integrable Constraints......Page 119 3.4 A Canonical Form, Equilibria, and Linearization......Page 120 4 Simple Conservative Moments......Page 122 4.2 Ziegler's Example Revisited......Page 123 5 The Case of a Fixed Axis of Rotation......Page 124 6.1 Kinematical Considerations......Page 125 6.2 Constraints and Constraint Forces......Page 126 6.4 The Equations of Motion......Page 127 6.5 Equilibria and Linearized Equations of Motion......Page 129 6.6 Solving for the Reaction Force......Page 130 7 The Satellite Dynamics Problem......Page 131 References......Page 133 1.1 ``It was Greenhill who Started the Trouble.........Page 135 1.2 Greenhill's Shaft as a Non-self-adjoint Problem......Page 137 1.3 From Follower Torques to Follower Forces......Page 142 2 Reversible and Circulatory Systems......Page 145 2.1 Zubov-Zhuravlev Decomposition of Non-potential Force Fields......Page 146 2.2 Circulatory Forces in Rotor Dynamics......Page 150 2.3 Stability Criteria for Circulatory Systems......Page 152 2.4 Geometrical Interpretation for m=2 Degrees of Freedom......Page 154 2.5 Approximating Flutter Cone by Perturbation of Eigenvalues......Page 155 3.1 Shieh–Masur Shaft with Dissipative Forces......Page 158 3.2 A Circulatory System Perturbed by Dissipative Forces......Page 161 4 Krein Signature and Stability of Hamiltonian Systems......Page 166 4.1 Canonical and Hamiltonian Equations......Page 170 4.2 Krein Signature of Eigenvalues......Page 171 4.3 Krein Collision or Linear Hamiltonian-Hopf Bifurcation......Page 172 5.2 Secular Instability of the Maclaurin Spheroids......Page 174 6.1 Rotating Shaft by SM1968......Page 182 6.2 Two-Mass-Skate (TMS) Model of a Bicycle......Page 187 References......Page 193