Trajectory-oriented synthesis of a planetary-type vibration exciter for vibratory technological equipment

The paper considers an oscillatory system that consists of an elastically supported working member equipped with a ring gear, a carrier, two planet gears, and an eccentric unbalanced mass. A kinematic model of the planetary mechanism is developed to describe the motion of the excitation point and to determine the conditions under which circular, elliptical, straight-line, and hypotrochoidal trajectories can be generated. On this basis, a dynamic model of the vibratory system is formulated in the form of coupled second-order differential equations for the translational motion of the working member. Particular attention is paid to the synthesis of the geometric and kinematic parameters of the exciter, including the gear ratio, eccentricity, carrier arm length, and initial phase angle. The governing equations are solved numerically for representative parameter sets, and the resulting steady-state trajectories of both the working member and the unbalanced mass are analyzed. The results substantiate the condition that represents a boundary case separating single-frequency trajectories from genuine hypotrochoidal motion. The specific parameters governing the transition from circular to elliptical and rectilinear trajectories and determining the number and shape of the lobes in the hypotrochoidal regime are theoretically justified.