Modeling the motion of an oscillator with a soft elastic characteristic

Abstract

The free oscillations of a system with one degree of freedom are considered under the assumption that the elasticity of a spring is proportional to the cubic root of its deformation. Two forms of the analytical solution of the nonlinear differential equation of motion of the oscillator are obtained. In the first displacement of the oscillator in time is expressed in terms of incomplete elliptic integrals of the first and second kind. In the second form, the solution is expressed in terms of periodic Ateb-functions. The tables of the involved functions are made, which simplify the calculation. Formulas are also derived for calculating the oscillation periods when the oscillator is signaled or the initial deviation from the equilibrium position or the initial velocity (instantaneous pulse) in this position. The dependence of the oscillation period on the parameters of the oscillator and the initial conditions is established. Examples of calculations of oscillations are presented with the use of compiled tables of special functions and using the proposed approximations of the Ateb-functions. Comparison of numerical results obtained by different methods is made.

Adaptive grid implementation for parallel continuum mechanics methods in particle simulations

“…Alternative approaches for extending the finite difference method (FDM) to adaptive grids have been proposed in [193][194][195][196][197].…”

Construction of a physical-mathematical model of oscillations of the unbalanced vibrator of the pneumatic sorting table

“…At the same time, the amplitude of the deflection of the plate, after its separation from the base, is greater than the amplitude of the deflection of the supported plate in the direction of its impulse. This is the fundamental difference between the dynamic behavior of a plate with a one-sided bond and the behavior of a plate on a two-sided basis [19,20]. In the case of two-sided contact of the plate with the elastic base, which is traditionally considered in the scientific literature, the mentioned effect cannot be manifested.…”

“…The analysis of the latest publications reveals that they assume a two-sided connection of the plates with the base. Such an assumption simplifies the statement of the problem and its solution but does not correspond to the research task, where the connections are one-sided [19,20]. Dynamic problems of deformation of a plate with a one-sided bond with the base belong to nonlinear problems since in the general case they have unknown areas of separation and contact that change during movement.…”