Вільні коливання осцилятора з синусоїдальною силовою характеристикою

Nonlinear oscillations caused by the initial deviation of the oscillator from the equilibrium position or the initial velocity given to it in this position are considered. It is assumed that the restoring force is proportional to the sine of the displacement of the oscillatory system. There are two variants of the sine: trigonometric and hyperbolic. In the first variant, the power characteristic of the oscillator is soft, and in the second, it is rigid. With a soft power characteristic, restrictions on the initial perturbations of the system are introduced. The exact analytic solutions of the nonlinear Cauchy problem are constructed in elliptic functions. Closed formulas for calculating the displacements of the oscillator and the period of cyclic motion are derived and tested by calculations. To simplify the calculations, in the absence of tables of elliptic Jacobi functions, approximate representations of them in elementary functions are proposed. Examples of calculations are given.