On an Asymptotic Method for Solving HomogeneousIntegro-Differential Equations Describing Vibrationsof Objects with Moving Boundaries

The problem of vibrations of objects with moving boundaries is presented as a differential equation with boundary and initial conditions and is a nonclassical generalization of a hyperbolic problem. In the present paper, equivalent integro-differential equations with symmetric and time-dependent kernels and time-varying integration limits are constructed. An expansion of the integro-differential equation of motion of variable-length objects into an infinite system of ordinary differential equations with variable coefficients is given. The concept of eigenfunctions and eigenvalues is defined for a boundary value problem in a domain bounded by time-varying integration limits. Solutions to homogeneous integro-differential equations describing vibrations of variable-length objects and systems of ordinary differential equations with changing parameters are constructed using asymptotic methods. Expressions for the amplitudes and phases of vibrations are obtained. This approach is especially useful in studying complex dynamical systems with lumped masses that oscillate under the influence of moving loads.