The Euler–Lagrange equation is frequently used to develop the governing dynamic equilibrium expressions for rigid-body or lumped-mass systems. In many cases, however, the rectangular coordinates are constrained, necessitating either the use of Lagrange multipliers or the introduction of generalized coordinates that are consistent with the kinematic constraints. For such cases, evaluating the derivatives needed to obtain the governing equations can become a very laborious process. Motivated by several relevant problems related to rigid-body structures under seismic motions, this paper focuses on extending the elegant equations of motion developed by Greenwood in the 1970s, for the special case of planar systems of rigid bodies, to include rigid-body rotations and accelerating reference frames. The derived form of the Euler–Lagrange equation is then demonstrated with two examples: the double pendulum and a rocking object on a double rolling isolation system. The work herein uses an approach that is used by many analysts to derive governing equations for planar systems in translating reference frames (in particular, ground motions), but effectively precalculates some of the important stages of the analysis. It is hoped that beyond re-emphasizing the work by Greenwood, the specific form developed herein may help researchers save a significant amount of time, reduce the potential for errors in the formulation of the equations of motion for dynamical systems, and help introduce more researchers to the Euler–Lagrange equation.
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September 2019
Technical Briefs
On the Euler–Lagrange Equation for Planar Systems of Rigid Bodies or Lumped Masses
R. Wiebe,
R. Wiebe
Department of Civil and Environmental Engineering,
University of Washington,
201 More Hall,
P.O. Box 352700,
Seattle, WA 98195-2700
e-mail: rwiebe@uw.edu
University of Washington,
201 More Hall,
P.O. Box 352700,
Seattle, WA 98195-2700
e-mail: rwiebe@uw.edu
1Corresponding author.
Search for other works by this author on:
P. S. Harvey, Jr.
P. S. Harvey, Jr.
School of Civil Engineering and Environmental Science,
University of Oklahoma,
202 W. Boyd St., Room 334,
Norman, OK 73019-2024
e-mail: harvey@ou.edu
University of Oklahoma,
202 W. Boyd St., Room 334,
Norman, OK 73019-2024
e-mail: harvey@ou.edu
Search for other works by this author on:
R. Wiebe
Department of Civil and Environmental Engineering,
University of Washington,
201 More Hall,
P.O. Box 352700,
Seattle, WA 98195-2700
e-mail: rwiebe@uw.edu
University of Washington,
201 More Hall,
P.O. Box 352700,
Seattle, WA 98195-2700
e-mail: rwiebe@uw.edu
P. S. Harvey, Jr.
School of Civil Engineering and Environmental Science,
University of Oklahoma,
202 W. Boyd St., Room 334,
Norman, OK 73019-2024
e-mail: harvey@ou.edu
University of Oklahoma,
202 W. Boyd St., Room 334,
Norman, OK 73019-2024
e-mail: harvey@ou.edu
1Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 21, 2018; final manuscript received June 21, 2019; published online July 15, 2019. Assoc. Editor: Zdravko Terze.
J. Comput. Nonlinear Dynam. Sep 2019, 14(9): 094502 (7 pages)
Published Online: July 15, 2019
Article history
Received:
October 21, 2018
Revised:
June 21, 2019
Citation
Wiebe, R., and Harvey, P. S., Jr. (July 15, 2019). "On the Euler–Lagrange Equation for Planar Systems of Rigid Bodies or Lumped Masses." ASME. J. Comput. Nonlinear Dynam. September 2019; 14(9): 094502. https://doi.org/10.1115/1.4044145
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