Theory of parametric resonance: modern results

Linear dynamical systems with many degrees of freedom with periodic coefficients also depending on constant parameters are considered. Stability of the trivial solution is studied with the use of the Floquet theory. First and second order derivatives of the Floquet matrix with respect to parameters are derived in terms of matriciants of the main and adjoint problems and derivatives of the systems matrix. This allows finding the derivatives of simple multipliers, responsible for the stability of the system, with respect to parameters and predicting their behavior with a change of parameters. It is shown how to use this information in gradient procedures for stabilization or destabilization of the system. As a numerical example, the system described by Carsson-Cambi equation is considered. Then, strong and weak interactions of multipliers on the complex plane are studied, and geometric interpretation of these interactions is given. As application of the developed theory the resonance domains for Hill's equation with damping are studied. It is shown that they represent halves of cones in the three-parameter space. Then, parametric resonance of a pendulum with damping and vibrating suspension point following arbitrary periodic law is considered, and the parametric resonance domains are found. Another important application of damped Hill's equation is connected with the study of stability of periodic motions in non-linear dynamical systems. it is shown how to find stable and unstable regimes for harmonically excited Duffing's equation. Then, linear vibrational systems with periodic coefficients depending on three independent parameters: frequency and amplitude of periodic excitation, and damping parameter are considered with the assumption that the last two quantities are small. Instability of the trivial solution of the system (parametric resonance) is studied. For arbitrary matrix of periodic excitation and positive definite damping matrix general expressions for domains of the main (simple) and combination resonances are derived. Two important specific cases of excitation matrix are studied: a symmetric matrix and a stationary matrix multiplied by a scalar periodic function. It is shown that in both cases the resonance domains is halves of cones ion the three-dimensional space with the boundary surface coefficients depending only on the eigenfrequencies, eigenmodes and system matrices. The obtained relations allow to analyze influence of growing eigenfrequencies and resonance number on resonance domains. Two mechanical problems are considered and solved: Bolotin's problem of dynamic stability of a beam loaded by periodic bending moments, and parametric resonance of a non-uniform column loaded by periodic longitudinal force. The lecture is a review of the recent results on parametric resonance obtained by the author with Frederick Solem, Pauli Pedersen (Denmark), and Alexei A. Mailybaev (Russia).