Explicit generalization of Lagrange's equations for hybrid coordinate dynamical systems

Abstract

An explicit generalization of the classical Lagrange's equations (for discrete coordinate dynamical systems) to cover a large family of multibody hybrid discrete/distributed parameter systems is presented. The coupled system of ordinary and partial differential equations follows directly from spatial and time differentiation of various Lagrangian functional, whereas the boundary conditions are directly established from another explicit set of symbolic variational equations. Five illustrative examples are presented. E consider a family of multibody hybrid discrete/dis- tributed parameter systems that can be regarded as consisting of a collection of interconnecte d rigid and elastic bodies. Such models are useful for dynamics and control anal- ysis of flexible spacecraft. The equations of motion are hybrid, in the sense that the rigid-body motions are described by dis- crete time-varying coordinates, and the elastic motions are described by time- and space-varying coordinates; the resulting hybrid system of ordinary and partial integro-differential equations embodies significant coupling between the rigid- body and elastic motions.1 Meirovitch2 extended the classical Lagrange's equations for hybrid systems using the extended Hamilton's principle. Al- though Meirovitch found the correct forms for the hybrid system, his equations embodied a differential operator that must be developed through integration by parts for each specific application. Also, the boundary condition operator in Meirovitch's developments must be found by integration by parts for each specific application. Berbyuk and Demid- yuk3 formulated the dynamic equations and boundary condi- tions for a specific mechanical system (two-link manipulator with one rigid link and one flexible link) by means of the extended Hamilton's principle. In deriving the kinetic energy and boundary conditions, they included the effects of end pay load. Low and Vidyasagar4 presented a procedure for deriving dynamic equations for manipulators containing both rigid and flexible links. They proposed a method for producing a compact symbolic expression for the equation of flexible manipulator systems. As they mentioned, their boundary con- ditions do not make allowance for ends that involve discrete elements, such as lumped masses and springs. Of course, Hamilton's principle can produce the appropriate boundary conditions in such cases, but the procedure is system-specific and tedious, especially when dealing with multiple-connected flexible bodies. We were motivated by Meirovitch's developments to estab- lish, at least for significant classes of systems, explicit La- grange differential equations and boundary conditions that make allowance for lumped masses, springs, and similar forces at the boundaries. In essence, we seek to symbolically carry out the integration by parts once and for all for a large class of systems. In the present paper, explicit Lagrange's equations