# 2019 ASEE Annual Conference & Exposition

## An Alternative Form of Euler’s Equation for the Rotational Dynamics of a Rigid Body Confined to Planar (2-D) Motion

#### Presented at Mechanics Division Technical Session 1

An Alternative Form of Euler’s Equation for the Rotational
Dynamics of a Rigid Body Confined to Planar (2-D) Motion
Any instructor of engineering mechanics who has ever taught statics and dynamics courses for a sustained period of time should be already familiar with the practices listed below, which refer to the evaluation of the appropriate moments-of-forces/couples equation that governs the rotational behavior of a rigid body:
• Statics: Moments may be evaluated about axes through any selected point in space, which is typically on, in, or otherwise near the rigid body of interest.
• Dynamics: Moments should be evaluated about either (a) axes through the mass center of the body, or (b) a fixed axis about which the body is constrained to rotate (if applicable).
This article presents another option for evaluation of the moments-of-forces/couples equation for the specific case of dynamics. The scope of application of the method advocated is restricted to planar (2-D) motion of rigid bodies, though it is possible to extend this method to spatial (3-D) motion as well. However, it is considerably more involved in this context, and it likely would be less suited for engineering students at the undergraduate level.
In this method, the moments-of-forces/couples equation may be evaluated at any point on or in the rigid body, but it should be a convenient point at which the kinematics of the body motion is either already provided or readily assessed. However, as will be demonstrated and discussed in this article, the equation associated with this method lends itself especially well to problems that involve a composite rigid body (i.e., a set of rigid elements which are rigidly joined together).
Two illustrative examples are considered in this article to both introduce and apply the method advocated. These examples reveal the advantage of moment evaluation about a point that differs from the mass center. Incidentally, this method is employed by the author in several courses that rely upon dynamics topics, which he regularly teaches at his academic institution.
Several alternative forms of Euler’s equation for the rotational dynamics of a rigid body may be found in [1–3]. The author has examined standard textbooks and other technical references, and it appears that the specific form of the equation presented in this article is novel and useful.
References
[1] F. P. Beer, E. R. Johnston, Vector Mechanics for Engineers: Statics and Dynamics, 6th Edition, McGraw-Hill, New York, 1997, pp. 1126–1130.
[2] I. H. Shames, Engineering Mechanics: Statics and Dynamics, 4th Edition, Prentice-Hall, Upper Saddle River, NJ, 1997, pp. 914–916.
[3] R. C. Hibbeler, Engineering Mechanics: Statics and Dynamics, 12th Edition, Pearson/Prentice-Hall, Upper Saddle River, NJ, 2010, pp. 600–604.

Authors