Generalized Vector Derivatives for Systems With Multiple Relative Motion

[+] Author and Article Information
T. A. Sherby

Applied Research and Consulting Services, National Cash Register Company, Dayton, Ohio

J. F. Chmielewski

Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, Pa.

J. Appl. Mech 35(1), 20-24 (Mar 01, 1968) (5 pages) doi:10.1115/1.3601168 History: Received January 05, 1967; Revised July 11, 1967; Online September 14, 2011


For the analysis of relative motion, classical vector mathematics is limited to the use of one moving reference frame when taking vector derivatives. However, many dynamical systems consist of a number of rigid bodies in motion relative to one another. The classical procedure requires the specification of the position of each body relative to a single “main body.” The use of relative coordinates allows a natural specification of the position of one moving body relative to another moving body in network fashion. To use relative coordinates in dynamic and kinematic analyses, it is necessary to use relative vector derivatives involving more than one moving reference frame. This paper presents general expressions for the kth-order derivative of a relative position and angular velocity vector measured in any moving reference frame in a system of m reference frames with multiple relative motion. These expressions are used to develop a procedure which generates the differential equations of motion for the system by routine substitution of the relative coordinates and their scalar derivatives. This procedure offers promise as an algorithm for machine generation of the system equations and eliminates the possibility of neglecting subtle accelerations due to relative motion. The use of the procedure is demonstrated by generating the equations of motion of an offset unsymmetrical gyroscope.

Copyright © 1968 by ASME
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