Modeling of free and forced longitudinal oscillations of a straight rod using the direct discretization method

Abstract

In the study of the dynamics of mechanical systems with distributed parameters, the discretization of partial differential equations by the finite element method, finite difference method, or boundary element method is widely used. This makes it possible to simplify the analysis of dynamic processes by reducing it to solving a system of ordinary differential equations. The aim of this paper is to mathematically substantiate the method of direct discretization of long-dimensional elastic links as applied to free and forced longitudinal oscillations of a straight rod as a mechanical system with distributed parameters, and to investigate the convergence of the method in determining the natural frequencies and amplitudes of forced oscillations of mechanical systems. The method of direct discretization is considered in the application to a rod with distributed parameters, which performs longitudinal oscillations. It is shown that by mathematical discretization of the wave equation, the boundary value problem can be reduced to the analysis of oscillatory phenomena in a chain mechanical system of material points. The formulas for determining the masses of the inertial elements of the computational model and the stiffness coefficients of the elastic connections are given. The convergence of the method is investigated by determining the natural frequencies of oscillations of a rod with free ends, as well as with one end clamped and the other end free, and comparing the approximate results of calculations with the exact results. The effect of the number of degrees of freedom, rod length, and material density on the eigenfrequencies is investigated. It is found that to ensure sufficient accuracy for engineering practice in determining the three or four lowest natural frequencies, computational models with 8–10 degrees of freedom can be used. It is emphasized that the cross-sectional area of the rod does not affect the characteristics of the frequency spectrum of the mechanical system. The zero value of the lowest natural frequency of an elastic rod with free ends can be explained by the fact that this value corresponds not to the oscillatory but to the translational motion of a mechanical system that is not connected to the base. Forced longitudinal oscillations of an elastic rod, one end of which is clamped at the base and the other end is free, under power and kinematic excitation are considered. The amplitude-frequency characteristics of mechanical systems are constructed on the examples of the dependences of the movement amplitude of one of the material points and the amplitude of the longitudinal force in one of the elastic links of the computational model on the frequency of forced oscillations. Since the dissipation of the mechanical energy of the system during its oscillations is not taken into account, as the frequency of forced oscillations approaches one of the natural frequencies of the system, the movement amplitudes, as well as the amplitudes of internal forces, tend to infinity. The actual values of resonant amplitudes are not of practical interest, since resonant modes of operation are not expected for the vast majority of machine units. The scientific novelty of the results obtained is that the mathematical substantiation of the method of direct discretization of the rod in its application to its longitudinal oscillations, the creation of computational algorithms and the study of the convergence of the results of determining the natural frequencies and amplitudes of forced oscillations can be considered as a development of the computer methodology for the dynamic analysis of mechanical systems with distributed parameters. The practical significance and scope of further research of the results of this work are based on the wide possibilities of practical application of the direct discretization method in the study of dynamic processes in rather complex mechanical systems of such technical objects as drilling rigs, cranes, mine hoisting machines, conveyors, rotary kilns, etc. This is facilitated by the simplicity and computational efficiency of the method in solving problems of dynamics of both linear and nonlinear mechanical systems. The method can be applied in the study of both stationary and transient dynamic processes in machine units.