Finite Element Model for Motion and Contact Analysis in a Trochoidal- Type Machine

In a trochoidal-type machine, the rotor and chamber will be in direct contact when apex seals arc removed. The reliability and efficiency of the machine will depend on the magnitude of these contact forces. In this paper, a finite element model that simulates the planetary rotation of a rotor inside a chamber is developed. The model is then used to analyze contact forces between rotor and chamber for a range of shaft angle. The results from this model are compared with the results from a simplified analytical model. Good agreements are obtained. This model will be fi.niher developed in the future to do dynamic contact analysis in a trochoidal-typc machine without apex seals. INTRODUCTION Trochoidal-type machines belong to the category of planctm}' rotation machines. Because of simplicity, compactness, possibility ofhigher speeds, high power-to-weight ratio, and low operating noise, they offer significant advantages over other types of machines. They provide a wide range of applications, such as engines, pumps, compressors, and blowers. The contact problem in a trochoidal-type machine has been studied by several researchers. Hall analyzed the contact forces and stresses in a gerotor-type pump where chamber is fanned by circular-arc teeth. Hall [1968] derived an impmtant equation for the minimum radius of cwvature of rotor. Colboume [I 97 6] applied the equation to modify the gerotor parameters and reduced the contact stresses. Later, Hoffmann [1975] designed, manufactured and tested a trochoidal-type gas compressor. He discovered that apex seals contJ.ibuted approximately 25% of all friction drive due to seal, and suggested eliminating apex seals by keeping tight running clearance in order to improve the life and efficiency of the machine. Shung and Pennock [ 1994a] derived closed form equation of type 2(bii) inner conjugate envelope for peritrochoid. Since there is no cusp on this type of envelope, it is a good candidate for trochoidal-type machine without apex seal. Reducing the contact force in a trochoidal-type machine without apex seals is important. Shung and Pennock [ 1994b] developed an analytical model to predict the normal contact forces between rotor and chamber. They also developed a combined model (analytical model and finite element model) to include friction and deformation at the contact regions. Kannan and Shung [1995] applied finite element method to study the variation in running clearance in a trochoidal-type pump when design parameters were changed but the rotor position remained the same. In this trochoidal-type pump, the chamber is a peritrochoid, and the rotor is a type 2(bii) inner conjugate envelope. The pinion gear is attached to the rotor, and the ring gear is attached to the chamber. The pinion gear rolls inside the ring gear through an eccenttic shaft, and the rotor does planetary rotation inside the chamber. The pockets between rotor and chamber provide the necessary positive displacements for this type of machine. In this paper, this model is developed fmther so that the model simulates the planetary rotation of a rotor inside a chamber and contact forces due to shaft deflection can be studied for a range of shaft angle. ANALYTICAL MODEL First a simplified analytical model is developed. Later the contact forces from this model are compared with the contact forces from Finite Element Analysis (FEA) and the shift of the rotor center due to body force is applied as shaft deflection to the finite element model developed in the next section. A model for the analysis of the contact forces is shown in Figure 1. Ocis the center of the chamber, OR is the center of the rotor, Pis the pitch point, and ~ is the jth contact regioxt The body force acting on the rotor is relocated from OR. to P with a corresponding moment Mb and can be expressed as