Singular surfaces and multiple eigenvalues in stability and optimization of non-conservative systems

Singular surfaces and multiple eigenvalues in stability and optimization of non-conservative systems

Kirillov, O.

it is common to consider that generically discrete eigenvalues of an operator are simple. However, in multiparameter operator families multiple eigenvalues are a robust phenomenon. Singular ruled surfaces were studied already by John Wallis, who in 1655 introduced his famous conical wedge known in the modern physical literature under the name of the “double coffee filter”. Later on, due to the efforts of Monge, Catalan, Plücker, Steiner and Cayley, the development of the theory of the ruled surfaces had lead to the formulation of the projective geometry. In XXth century the singular surfaces reappeared again in the form of a powerful singularity theory. In my presentation I will talk about multiple eigenvalues and associated singular surfaces in stability and optimization problems for non-conservative systems. First, I consider circulatory systems without damping that describe stability of columns under follower loads and torques, friction-induced instabilities in rotor dynamics as well as aeroelastic stability problems. I present an algorithm for classification of generic singularities on the stability boundary of a circulatory system and list all generic singularities up to codimension 10. I plot generic singularities on the stability boundary for one-, two-, and three-parameter families of circulatory systems and show how to approximate the boundary near the singularities studying perturbation of simple and multiple eigenvalues. I illustrate the general theory by the examples from robotics, rotor dynamics and structural optimization. In the latter case structural optimization of the m-link Ziegler pendulum will be considered and derivation of optimality conditions as well as the connection of the optimal solutions to singularities on the stability boundary will be discussed. Then, I will study the effect of small dissipation on the stability of circulatory systems. In 1952 Ziegler found that an infinitesimally small amount of damping leads to a finite change in the stability domain of a two-link pendulum loaded by the follower force. In 1956 Bottema resolved this destabilization paradox by means of the Whitney umbrella singularity that as he established exists on the stability boundary of the damped Ziegler’s pendulum. I will talk about extensions of this result to general finite dimensional and continuous circulatory systems as well as to the gyroscopic systems with small damping and non-conservative positional forces. Examples of similar paradoxal phenomena from rotor dynamics, soil mechanics and magnetohydrodynamics will be considered in detail.

Keywords: Multiparameter stability problems; stability boundary; singularities; parametric optimization; multiple eigenvalues; dissipation-induced instabilities

  • Lecture (others)
    Lecture at the seminar of Prof. Felix Darve, Laboratoire 3S-R: Sols, Solids, Structures, Risques, 24.02.2011, Grenoble, France


Publ.-Id: 16415