Research Papers

New Approach to the Modeling of Complex Multibody Dynamical Systems

[+] Author and Article Information
Aaron Schutte

Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453schutte@usc.edu

Firdaus Udwadia

Departments of Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, Systems Architecture Engineering, and Information and Operations Management, University of Southern California, 430K Olin Hall, Los Angeles, CA 90089-1453fudwadia@usc.edu

In general, we could take R=R(r1,r2,,rv,t)R(r,t) in Eq. 5 and S=S(u1,u2,,uw,t)S(u,t) in Eq. 7, but for the sake of simplicity and clarity of exposition we do not explicitly include the time t in these equations.

J. Appl. Mech 78(2), 021018 (Dec 20, 2010) (11 pages) doi:10.1115/1.4002329 History: Received July 28, 2009; Revised July 23, 2010; Posted August 09, 2010; Published December 20, 2010; Online December 20, 2010

In this paper, a general method for modeling complex multibody systems is presented. The method utilizes recent results in analytical dynamics adapted to general complex multibody systems. The term complex is employed to denote those multibody systems whose equations of motion are highly nonlinear, nonautonomous, and possibly yield motions at multiple time and distance scales. These types of problems can easily become difficult to analyze because of the complexity of the equations of motion, which may grow rapidly as the number of component bodies in the multibody system increases. The approach considered herein simplifies the effort required in modeling general multibody systems by explicitly developing closed form expressions in terms of any desirable number of generalized coordinates that may appropriately describe the configuration of the multibody system. Furthermore, the approach is simple in implementation because it poses no restrictions on the total number and nature of modeling constraints used to construct the equations of motion of the multibody system. Conceptually, the method relies on a simple three-step procedure. It utilizes the Udwadia–Phohomsiri equation, which describes the explicit equations of motion for constrained mechanical systems with singular mass matrices. The simplicity of the method and its accuracy is illustrated by modeling a multibody spacecraft system.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

A rigid body in an inertial frame of reference

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Figure 2

A multibody spacecraft system consisting of two interconnected rigid bodies (N=2) in a uniform gravitational field. The connection between the two bodies at points P1 and P2 is modeled by two springs and a damper. The spring constants kl and knl refer to the linear and cubically nonlinear restoring forces exerted by the springs, and the linear damping coefficient is denoted by c. The two bodies are free to rotate about and move along the line P1P2, which is fixed relative to each body.

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Figure 3

A view illustrating the initial configuration of the system

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Figure 5

The radial distance to bodies 1 and 2 over three orbits

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Figure 6

(a) Unit quaternion of body 1 and (b) unit quaternion of body 2; unit quaternions ui, i=1,2, over three orbits

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Figure 7

(a) Angular velocity of body 1 found by Eq. 40 and (b) angular velocity of body 2 found by Eq. 40; body-fixed angular velocities ωi, i=1,2, over three orbits

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Figure 8

(a) Vibrational motion between bodies 1 and 2 showing stretching and contracting and (b) vibrational motion over a fractional orbit period showing higher frequency oscillations; vibrational motion between the spring and damper connections P1 and P2 over three orbits

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Figure 9

(a) Error in the first component of e1, (b) error in the second component of e1, (c) error in the third component of e1, and (d) error in the first component of e2; errors e1 and e2 versus orbit number

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Figure 4

(a) The coordinate modeling constraints ϕ1i, i=1,2, over three orbits and (b) the physical modeling constraints Φk, k=1,2,…,6, over three orbits; error in the eight modeling constraints showing their satisfaction throughout the integration



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