In automatic assembly, a vibratory bowl feeder plays a crucial role for reorienting the parts and feeding them into an assembly process. Taking the bowl as a platform and three prismatic flat-spring supports as spatial compliant legs and based on the continuum dynamics, this paper examines the bowl feeder from the point-of-view of a compliant platform device by applying von Mises' compliance study to each of the flat-spring legs and establishes a screw system of each leg. The compliance and Jacobian matrix of the bowl feeder are presented, the potential and kinetic energies are analyzed, and the dynamics models are established, leading to the characteristic equations of the compliant platform device. This generates for the first time the shape function integrated stiffness matrix and inertia matrix. This paper further analyzes the two characteristic equations of both a simplified system and a generalized system, implements the comparative study of the system natural frequencies between the two system models, and presents the stability analysis involving the system hysteresis damping. The effect of platform design parameters on the natural frequencies of the system under damping is identified and modal analysis of the system is carried out according to different forms of the excitation force. This paper presents a comprehensive study of the dynamics of this kind of compliant devices.