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

Jeffrey C. Hayen

Download Paper | Permalink

Conference: 2019 ASEE Annual Conference & Exposition
Location: Tampa, Florida
Publication Date: June 15, 2019
Start Date: June 15, 2019
End Date: June 19, 2019
Conference Session: Mechanics Division Technical Session 1
Tagged Division: Mechanics
Page Count: 23
DOI: 10.18260/1-2--32054
Permanent URL: https://peer.asee.org/32054
Download Count: 1164

Paper Authors

biography

Jeffrey C. Hayen Oregon Institute of Technology orcid.org/0000-0003-3771-339X

visit author page

Jeffrey Hayen joined the faculty in the MMET Department at the Oregon Institute of Technology (OIT) in 2011. Before arriving at OIT, Jeffrey served as a Professor of Engineering, Mathematics, and Physics at Southwestern Oregon Community College for 16 years. Prior to that experience, he worked in the aerospace industry as a thermodynamicist and propellant analyst for high-performance upper-stage rockets at the Space Systems Division of the General Dynamics Corporation. He also has conducted research concerning structural dynamics and control for the Kajima Corporation of Japan, and he currently provides technical analyses and performs computational simulations for the United Launch Alliance in Denver. Jeffrey earned his B.S. and M.S. degrees in Mechanical Engineering from San Diego State University, and his Ph.D. degree in Applied Mechanics and Physics from the California Institute of Technology.

visit author page

Download Paper | Permalink

Abstract

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.

Citation
Format

Hayen, J. C. (2019, June), An Alternative Form of Euler’s Equation for the Rotational Dynamics of a Rigid Body Confined to Planar (2-D) Motion Paper presented at 2019 ASEE Annual Conference & Exposition , Tampa, Florida. 10.18260/1-2--32054

TY - CPAPER
AB - 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.

AU - Jeffrey C. Hayen
CY - Tampa, Florida
DA - 2019/06/15
PB - ASEE Conferences
TI - An Alternative Form of Euler’s Equation for the Rotational Dynamics of a Rigid Body Confined to Planar (2-D) Motion
UR - https://peer.asee.org/32054
DO - 10.18260/1-2--32054
ER -