A study of a bi-symmetric electro-mechanical system through umbra-Lagrangian generated by bondgraphs, and Noether's theorem
This paper presents a dynamic analysis of an electro-mechanical system through the umbra-Lagrange's equation proposed in a previous paper, which includes dissipative and non-potential fields in a compact Lagrangian form. An extended Noether's theorem along with an umbra-Hamiltonian is employed to get invariants of motion, or possible invariant trajectories. The umbra-Lagrangian of this electro-mechanical system is obtained through an extended Karnopp's algorithm. The major contribution of this paper is the dynamic analysis exploiting the symmetries of an electro-mechanical system comprising an externally and internally damped, symmetric, elastic rotor driven by a two-phase induction motor. For such a system the umbra-Lagrangian remains unchanged under two families of continuous transformations. The behavior of the limiting dynamics is obtained and validated through simulations. The stability issue of this system is also examined. Similar analysis is also extended to an identical rotor driven by a three-phase induction motor. As SO (2) symmetries are not obvious in this case due to the presence of three stator inertances, the three-phase induction motor is reduced to an equivalent SO (2) symmetric two-phase induction motor, to which the analysis is applied.
electro-mechanical system umbra-lagrangian bondgraphs noether's theorem umbra-hamiltonian