In this paper, the dynamic behavior of the damping system is analyzed with a two-mass pendulum absorber, the equations of motion of non-linear mechanical systems are built accordingly. AFC equation systems have been identified in the non-linear formulation. To obtain the frequency response, the Ritz averaging method is used. A new numerical method of determining the parameters of optimal tuning two-mass pendulum absorber in the non-linear formulation has been Proposed and implemented.

Dynamic behavior of a heavy homogeneous sphere in a spherical cavity of a supporting body that performs specified translational movements in space has been studied. Using the Appel formalism, the equations of ball motion in a moving spherical cavity without slip are constructed and a numerical analysis of the evolution of the ball motion is carried out.

The forced oscillations of the damping mechanical system of solids "Ball Vibration Absorber (BVA) with linearly viscous resistance – a movable carrier body" under the influence of external harmonic excitation are considered. Based on Appell's formalism, the dynamic equations for the joint motion of a heavy ball without sliding into a spherical cavity of a carrier body are formulated and numerically studied. The amplitude-frequency characteristic of the damping mechanical system and the curves of the dependences of the maximum amplitude of the oscillations of the carrier body on the values of the radius of the spherical cavity and the coefficient of viscous resistance of the BVA are obtained. The conditions and restrictions on the rolling of a heavy ball in the spherical recess of the absorber without sliding are determined.