An Approximate Method for Solving Boundary Value Problems with Moving Boundaries by Reduction to Integro-Differential Equations

The problem of vibrations of objects with moving boundaries formulated as a differential equation with boundary and initial conditions is a nonclassical generalization of a hyperbolic problem. To facilitate the construction of the solution to this problem and to justify the choice of the form of the solution, equivalent integro-differential equations with symmetric and time-dependent kernels and time-varying integration limits are constructed. The advantages of the method of integro-differential equations are revealed in the transition to more complex dynamic systems carrying concentrated masses vibrating under the action of moving loads. The method is extended to a wider class of model boundary value problems that take into account bending stiffness, the resistance of the external environment, and the stiffness of the base of a vibrating object. The solution is given in dimensionless variables and is accurate up to second-order values with respect to small parameters characterizing the velocity of the boundary. An approximate solution is found for the problem of transverse vibrations of a hoisting rope having bending stiffness, one end of which is wound on a drum and a load is fixed on the other.