Solving Problems of Dynamics of Systems with Elastic Elements and Variable Masses

Most mechanical systems exhibit nonlinear properties under certain parameters of external disturbance. Nonlinearity is a general property of mechanical systems, and their linear behavior is an exception. Typically, the linearity of the corresponding equations of motion follows from the assumption that the vibration amplitudes of mechanical systems are small. Usually, in science we are talking about linear oscillations, i.e., about oscillations described by linear differential equations, and the smallness of the oscillations is not a defining feature. In some cases, small oscillations, no matter how small, are described by nonlinear differential equations. There are also mechanical systems in which the nonlinearity of the dynamic characteristics manifests itself the more, the smaller the deviation of the system from the equilibrium position, for example, systems with tension in the elastic characteristic and systems with dry friction, where the influence of nonlinearity is especially large for small vibration amplitudes. The equivalent circuits of many machines with elastic links contain variable masses. The differential equations of the transition process then contain variable coefficients that are given functions of time. In this regard, great difficulties arise in analyzing the dynamics of such machines, since general methods for solving differential equations with variable coefficients have not yet been developed. The work deals with a method of bringing equivalent circuits of machines containing variable masses to equivalent circuits with constant masses by adding external forces that form the same transient process.