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