Motion Equations of the Single­mass Vibratory Machine with a Rotary­oscillatory Motion of the Platform and a Vibration Exciter in the Form of a Passive Auto­balancer

This paper describes a mechanical model of the single-mass vibratory machine with a rotary-oscillatory motion of the platform and a vibration exciter in the form of a passive auto-balancer. The platform can oscillate around a fixed axis. The platform holds a multi-ball, a multi-roller, or a multi-pendulum auto-balancer. The auto-balancer's axis of rotation is parallel to the turning axis of the platform. The auto-balancer rotates relative to the platform at a constant angular velocity. The auto-balancer's casing hosts an unbalanced mass in order to excite rapid oscillations of the platform at rotation speed of the auto-balancer. It was assumed that the balls or rollers roll over rolling tracks inside the auto-balancer's casing without detachment or slip. The relative motion of loads is impeded by the Newtonian forces of viscous resistance. Under a normally operating auto-balancer, the loads (pendulums, balls, rollers) cannot catch up with the casing and get stuck at the resonance frequency of the platform's oscillations. This induces the slow resonant oscillations of the platform. Thus, the auto-balancer is applied to excite the dual-frequency vibrations.Employing the Lagrangian equations of the second kind, we have derived differential motion equations of the vibratory machine. It was established that for the case of a ball-type and a roller-type auto-balancer the differential motion equations of the vibratory machine are similar (with accuracy to signs) and for the case of a pendulum-type vibratory machine, they differ in their form.Differential equations of the vibratory machine motion are recorded for the case of identical loads.The models constructed are applicable both in order to study the dynamics of the respective vibratory machines analytically and in order to perform computational experiments.In analytical research, the models are designed to search for the steady-state motion modes of the vibratory machine, to determine the condition for their existence and stabilit