A numerical-analytical method for investigating parametric oscillations

The natural frequencies and modes of the parametric oscillations of mechanical systems are investigated using the Lyapunov–Poincaré theory and the numerical-analytical sagittary function and accelerated convergence methods. An example of a physical pendulum with movable internal masses is studied. The small parametric oscillations of a simple pendulum of variable length are investigated by the numerical-analytical accelerated convergence method for an arbitrary value of the modulation coefficient. Periodic solutions are constructed using a parameter continuation procedure, and the boundaries of the stability regions (in the linear approximation) and of the instability regions (of parametric resonance) are determined. Ince–Strutt diagrams are constructed for arbitrary permissible values of the modulation coefficient of the lowest oscillation modes. Several qualitative effects, which are, in principle, inaccessible when using routine calculations by perturbations methods, are established.