Stability of Axially Moving Materials

Preface.- 1.Prototype Problems: Bifurcations of Different Kinds.- 1.1 Rigid column with elastic clamping.- 1.2 Elastic column and its optimization.- 1.3 Elastic rod under torsion.- 1.4 Divergence and optimization of wings.- 1.5 Stability of tensioned cantilever beam.- 1.6 Accelerating motion of rod (rocket, missile) under follower force.- 2 Bifurcation Analysis for Polynomial Equations.- 2.1 Bifurcation and parametric representations.- 2.2 Analysis of a cubic equation.- 2.3 Analysis of a quartic (fourth-order) polynomial equation.- 3 Nonconservative Systems with a Finite Number of Degrees of Freedom.- 3.1 Critical parameters and destabilizing perturbations.- 3.2 Characteristic polynomial and series expansions.- 3.3 Ideal perturbations and sufficient conditions for stability (n = 2).- 3.4 Matrices and examples of ideal perturbations.- 3.5 Stability of systems subjected to deficient perturbations and determination of the deficiency index.- 3.6 On the stability and trajectories of the double pendulum with linear springs and dampers.- 4 Some General Methods.- 4.1 Criteria of elastic stability.- 4.2 Bifurcations and multiplicity of critical loads.- 4.3 Decomposition method for bimodal solutions.- 4.4 Bifurcation and analysis of implicitly given functionals.- 4.5 Variational principle and bifurcation analysis.- 5 Modeling and Stability Analysis of Axially Moving Materials.- 5.1 General dynamics and geometric considerations.- 5.2 Kinematic relations of small deformations.- 5.3 Constitutive linear elastic and visco-elastic relations.- 5.4 Modeling of beams and panels.- 5.5 Modeling of axially moving materials.- 5.6 Transformation to weak form.- 5.7 Boundary conditions.- 5.8 Numerical examples in stability of axially moving elastic and viscoelastic panels.- 6 Stability of Axially Moving Plates.- 6.1 Isotropic plates.- 6.2 Orthotropic plates.- 6.3 Plates with a nonuniform axial tension distribution.- 7 Stability of Axially Moving Strings, Beams and Panels.- 7.1 Unified model and exact eigensolutions for torsional, longitudinal and transverse vibration types.- 7.2 Exact eigensolutions of the traveling string with damping.- 7.3 Exact eigensolutions of axially moving beams and panels.- 7.4 Long axially moving beam with periodic elastic supports.- 7.5 Stability of a traveling beam in a gravitational field.- 8 Stability in Fluid-Structure Interaction of Axially Moving Materials.- 8.1 Basic concepts.- 8.2 Analytical solution of two-dimensional potential flow.- 8.3 Added-mass approximation.- 8.4 Numerical examples.- 8.5 Recommendations for further reading.- 9 Optimization of Elastic Bodies Subjected to Thermal Loads.- 9.1 Optimal distribution of thickness in a thermoelastic beam.- 9.2 Optimal distribution of materials in a thermoelastic beam.- 9.3 A guaranteed double-sided estimate for energy dissipation in heat conduction of locally orthotropic solid bodies.- 9.4 Conclusion.- Appendix A Cartesian tensors.- A.1 Tensor algebra.- A.2 Tensor calculus.- A.3 Integration by parts in multiple dimensions.- Appendix B Numerical integration of ODEs and semidiscrete PDEs.- B.1 Explicit Runge-Kutta methods.- B.2 Classical implicit methods.- B.3 The theoretical basis of iterative implicit methods.- B.4 Time-discontinuous Galerkin (dG).- C Finite elements of the Hermite type.- Appendix C: Finite elements of the Hermite type.- C.1 Coordinate mapping and the derivative degrees of freedom.