In this paper, an extended Lagrangian formalism for general class of dynamical systems with dissipative, non-potential fields is formulated with the aim to obtain invariants of motion for such systems. A new concept of umbra-time has been introduced for this extension. D’Alembert basic idea of allowing displacement, when the real time is frozen is conveniently expressed in the terms of umbra-time. This leads to a peculiar form of equations, which is termed as umbra-Lagrange’s equations. A variational or least action doctrine leading to the proposed form of equation is introduced, which is based on recursive minimization of functionals. The concept of umbra-time extends the classical manifold over which the system evolves. The extension of Noether’s theorem in this extended space has been presented. The idea of umbra time is then used to propose the concept of umbra-Hamiltonian, which is used along with the extended Noether’s theorem to get into the dynamics of the dynamical systems with symmetries. In the mathematical models of dynamical system, the equations for the system can be formulated in a systematic way from the bondgraph representation as bondgraph representation of a system may be constructed in a total abstraction from the mathematical models of the dynamical system. In present paper, bond graphs are conveniently used to arrive at umbra-Lagrangian of the system. As a case study, we present a dynamic analysis of an electro-mechanical system through the proposed extended Lagrangian Formulation. The major objective of this paper is an analysis of symmetries of an electro-mechanical system comprising of an externally and internally damped, symmetric, elastic rotor driven by a three-phase induction motor, for which the umbra-Lagrangian remains unchanged under two families of transformations. The behaviour of limiting dynamics is obtained and validated through simulation studies.
Stability analysis and controller synthesis for hybrid dynamical systems