A new dynamic stability theory for distributed structural systems with periodically varying parameters

A new method for determining the exact parametric instability regions of a class of second-order distributed structural (or hyperbolic) systems with periodically varying parameters is developed. The method is based on the non-dispersive wave behavior of a distributed system, and the conditions for the existence of such non-dispersive wave behavior are presented. Several example systems are considered, including a translating string with constant length and periodically varying velocity, a translating string with periodically varying length, a stationary string with one or two periodically moving boundaries, and a translating string with a periodically varying tension. The parametric instabilities of these systems are characterized by bounded displacements, exponentially growing vibratory energies, and formation of shock waves.;The period-1 instability regions in a parameter plane are obtained analytically using the new wave method and the concept of a fixed point. A general formulation for calculating the period-i (i>1) instability regions is given. The bifurcation diagram is introduced for calculating the period-i (i>1) instability regions numerically. The basins of attraction for period-1 attracting fixed points are obtained analytically and a physical explanation of shock wave instabilities is provided using wave propagation.;The dynamic response of a translating string with constant length and arbitrarily varying length is obtained using an exact wave method, real and complex modal analyses, and an implicit finite difference scheme. The least upper bound of the displacement of the freely vibrating string is obtained for an arbitrary movement profile. A numerical damping term is added to stabilize the finite difference scheme when the system has shock wave instability.;The Floquet-Liapunov theorem is used to calculate the eigenvalues of a transition matrix for dynamic systems with periodically varying parameters. A numerical method is provided for approximating the periodic transition matrix in one period using step functions. The instability regions of a translating string with constant length and periodically varying velocity are obtained using the numerical method. The effects of the number of spatial discretization terms on the prediction of the parametric instability regions are discussed. The effects of damping, tension variation, the constant part of the velocity, and the real and complex trial functions on the instability regions predicted are also discussed for the same system.