An Approximate Method for Solving Boundary Value Problems with Moving Boundaries in the Developed Software Package TB-Analisys
Abstract
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 the bending stiffness, the resistance of the external environment, and the stiffness of the base of an oscillating object. Particular attention is paid to the consideration of the most common case in practice, when external disturbances act at the borders. The solution is made in dimensionless variables accurate to second-order values of relatively small parameters characterizing the velocity of the boundary. An approximate solution is found to the problem of transverse vibrations of a rope of a lifting device, which has bending rigidity, one end of which is wound on a drum, and a load is fixed on the other. The results obtained for the amplitude of oscillations corresponding to the nth dynamic mode are presented. The phenomenon of steady-state resonance and passage through resonance is investigated using numerical methods. The solution is made in dimensionless variables using the TB-Analisys software package developed in the Matlab environment, which allows using the results obtained for calculating a wide range of technical objects.
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