Mathematical modelling of elliptic helicoid screw transmission systems

Abstract

This paper presents a systematic modelling procedure for producing high precision elliptic helicoid screw transmission systems. The modelling procedure includes both the derivation of the geometric model of the helicoid screw and the construction of pro®le models for the revolving conjugate elements. The pro®le models are derived by using several techniques including the theories of diVerential geometry, coordinate transformation and meshing principles. Since the meshing model of the screw transmission system is constructed systematically by rigorous diVerential geometry and meshing theories, the lead screw model thus developed can lead to good transmission quality and high system performance. In order to verify the eVectiveness of the proposed models and the modelling procedure, the performance of the proposed system is evaluated numerically by several diVerent indices including the transmissivity index, the manipulability index and the lubrication angle. The numerical results obtained by using the proposed modelling procedure are also veri®ed by experimental data taken directly from a sample product produced by the NC machine using the proposed theory. A comparative study is conducted. The results indicate that the proposed elliptic helicoid screw model proposed in this paper is accurate and e cient. The proposed theory and performance indices can be used to design for various screw systems of high performance. NOTATION a one-half length of the long axis of the basic generation ellipse b one-half length of the short axis of the basic generation ellipse C cos d c central distance of the basic generation ellipse and the meshing element D length of the normal vector N …1 † h unit rotating helical pitch of the screw M i, jtransformation matrix from coordinate system S j to coordinate system S i MI manipulability index n unit normal vector at the contact point between the surfaces of the swinging measurement ball and the elliptic helicoid screw N …1 † 1 normal vector N …1 † at the contact point on elliptic helicoid screw as expressed in the S 1 coordinate system N …2 † 1 normal vector N …2 † at the contact point on the revolving meshing element as expressed in the S 1 coordinate system Q lub full-cycle quality of lubrication angle Q m full-cycle quality of manipulability Q t full-cycle quality of transmissivity r a position vector of an arbitrary point on the basic ellipse as expressed in the S a coordinate system r 1 position vector of the elliptic helicoid surface of the screw as expressed in the S 1 coordinate system r …R † 2L 1 , r …R † 2L 2 surface families of the elliptic helicoid screw as expressed in the S 2L i (iˆ1, 2) coordinate systems r …R † 2R surface family of the elliptic helicoid screw as expressed in the S …R † 2R coordinate system r , r ¿ partial derivatives of helicoid surface r with respect to and ¿ respectively R e radius of helical curves on the screw cylinder 265

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