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Nonlinear Stability and Bifurcation Theory : An Introduction for Engineers and Applied Scientists - Hans Troger

Nonlinear Stability and Bifurcation Theory

An Introduction for Engineers and Applied Scientists


Published: 1st September 1991
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This study aims to make progress in the field of stability theory available to scientists and engineers. A treatment of the different types of loss of stability of equilibrium positions of static and dynamic systems and of periodic solutions of dynamic systems is given by means of the methods of bifurcation and singuality theory. The reader needs only a background in mathematics. Among others, concepts such as centre manifold theory, the method of Ljapunov-Schmidt, normal form theory, unfolding theory, bifurcation diagrams, classifications and bifurcations in symmetric systems are discussed, as far as they are relevant in applications. Most important for the whole representation is a set of examples taken from mechanics and engineering showing the usefulness of the above mentioned concepts. These examples include buckling problems of rods, plates and shells, the loss of stability of the motion of road and rail vehicles, of a simple robot, and of fluid conveying elastic tubes. With these examples, questions like symmetry breaking, pattern formation, imperfection sensitivity, transition to chaos and correct modelling of systems are explored. Finally, a number of selected FORTRAN-routines should encourage readers to apply these theories to their own individual problems.

1 Introduction.- 2 Representation of systems.- 2.1 Dynamical systems.- 2.1.1 Time continuous system.- 2.1.2 Time discrete system.- 2.2 Statical systems.- 2.3 Definitions of stability.- 2.3.1 Stability in the sense of Ljapunov.- 2.3.2 Structural stability (robustness, coarseness).- 3 Reduction process, bifurcation equations.- 3.1 Finite-dimensional dynamical systems.- 3.1.1 Steady states.- 3.1.2 Periodic motions.- 3.2 Infinite-dimensional statical and dynamical systems..- 3.2.1 Statical systems.- 3.2.2 Dynamical systems.- 4 Application of the reduction process.- 4.1 Equilibria of finite-dimensional systems.- 4.1.1 Double pendulum with axially elastic rods and follower force loading.- 4.1.2 Double pendulum with elastic end support and follower force loading.- 4.1.3 Double pendulum under aerodynamic excitation..- 4.1.4 Loss of stability of the straight line motion of a tractor-semitrailer.- 4.1.5 Loss of stability of the straight line motion of a railway vehicle.- 4.1.6 Summary of Section 4.1.- 4.2 Periodic solutions of finite-dimensional systems.- 4.2.1 Mechanical model and equations of motion.- 4.2.2 Calculation of the power series expansion of the Poincare mapping.- 4.2.3 Stability boundary in parameter space.- 4.2.4 Center manifold reduction.- 4.3 Finite- and infinite-dimensional statical systems.- 4.3.1 Buckling of a rod: discrete model.- 4.3.2 Buckling of a rod: continuous model.- 4.3.3 Buckling of a circular ring.- 4.3.4 Buckling at a double eigenvalue: rectangular plate.- 4.3.5 The pattern formation problem: buckling of complete spherical shells.- 5 Bifurcations under symmetries.- 5.1 Introduction.- 5.2 Finite dimensional dynamical systems.- 5.2.1 Two zero roots.- 5.2.2 Two purely imaginary pairs.- 5.3 Infinite dimensional statical systems.- 5.4 Infinite dimensional dynamical systems.- 6 Discussion of the bifurcation equations.- 6.1 Transformation to normal form.- 6.1.1 Time-continuous dynamical systems.- 6.1.2 Time-discrete dynamical systems.- 6.1.3 Statical systems.- 6.2 Codimension.- 6.2.1 Static bifurcation.- 6.2.2 Dynamic bifurcation.- 6.3 Determinacy.- 6.4 Unfolding.- 6.5 Classification.- 6.5.1 Dynamic bifurcation.- 6.5.2 Static bifurcation: elementary catastrophe theory.- 6.5.3 The unfolding theory of Golubitsky and Schaeffer.- 6.5.4 Restricted generic bifurcation.- 6.6 Bifurcation diagrams.- 6.6.1 Statical systems.- 6.6.2 Time-continuous dynamical systems.- 6.6.3 Time-discrete dynamical systems.- 6.6.4 Symmetric dynamical systems.- 6.6.5 Symmetric statical systems.- A Linear spaces and linear operators.- A.1 Linear spaces.- A.2 Linear operators.- B Transformation of matrices to Jordan form.- C Adjoint and self-adjoint linear differential operators.- C.1 Calculation of the adjoint operator.- C.2 Self-adjoint differential operators.- D Projection operators.- D.1 General considerations.- D.2 Projection for non-self-adjoint operators.- D.3 Application to the Galerkin reduction.- E Spectral decomposition.- E.1 Derivation of an inversion formula.- E.2 Three examples.- F Shell equations on the complete sphere.- F.1 Tensor notations in curvilinear coordinates.- F.2 Spherical harmonics.- G Some properties of groups.- G.1 Naive definition of a group.- G.2 Symmetry groups.- G.3 Representation of groups by matrices.- G.4 Transformation of functions and operators.- G.5 Examples of invariant functions and operators.- G.6 Abstract definition of a group.- H Stability boundaries in parameter space.- I Differential equation of an elastic ring.- I.1 Equilibrium equations and bending.- I.2 Ring equations.- J Shallow shell and plate equations.- J.1 Deformation of the shell.- J.2 Constitutive law.- J.3 Equations of equilibrium.- J.4 Special cases.- J.4.1 Plate.- J.4.2 Sphere.- J.4.3 Cylinder.- K Shell equations for axisymmetric deformations.- K.1 Geometrical relations.- K.2 Stress resultants, couples and equilibrium equations.- K.3 Stress strain relations.- K.4 Spherical shell.- L Equations of motion of a fluid conveying tube.- L.1 Geometry of tube deformation.- L.2 Stress-strain relationship.- L.3 Linear and angular momentum.- L.4 Tube equations and boundary conditions.- M Various concepts of equivalences.- M.1 Right-equivalence.- M.2 Contact equivalence.- M.3 Vector field equivalence.- M.4 Bifurcation equivalence.- M.5 Recognition problem.- N Slowly varying parameter.- O Transformation of dynamical systems into standard form.- O.1 Power series expansion.- O.2 Recursive calculation.

ISBN: 9783211822920
ISBN-10: 3211822925
Audience: General
Format: Paperback
Language: English
Number Of Pages: 407
Published: 1st September 1991
Country of Publication: US
Dimensions (cm): 24.41 x 16.99  x 2.18
Weight (kg): 0.67