On the Euler–Lagrange Equation for Planar Systems of Rigid Bodies or Lumped Masses

Abstract

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.