Extension of Lagrangian–Hamiltonian mechanics for continuous systems—investigation of dynamics of a one-dimensional internally damped rotor driven through a dissipative coupling

Abstract

In this paper, the extended Lagrangian formulation for a one-dimensional continuous system with gyroscopic coupling and non-conservative fields has been developed. Using this formulation, the dynamics of an internally and externally damped rotor driven through a dissipative coupling has been studied. The invariance of the extended or so-called umbraLagrangian density is obtained through an extension of Noether's theorem. The rotor shaft is modeled as a Rayleigh beam. The dynamic behavior of the rotor shaft is obtained and validated through simulation studies. Results show an interesting phenomenon of limiting behavior of the rotor shaft with internal damping beyond certain threshold speeds which are obtained theoretically and affirmed by simulations. It is further observed that there is entrainment of whirling speeds at natural frequencies of the rotor shaft primarily depending on the damping ratio.

A review on extension of Lagrangian-Hamiltonian mechanics

“…The umbra Lagrangian theory has been used successfully to study invariants of motion for non-conservative mechanical and thermo-mechanical systems [48]. In another paper, the authors applied umbra Lagrangian to study dynamics of an electromechanical system comprising of an induction motor driving an elastic rotor (Mukherjee et al, 2009). This system was symmetric in two sets of coordinates, one set was mechanical or geometrical, and the other symmetry was in electrical domain.…”

“…This system was symmetric in two sets of coordinates, one set was mechanical or geometrical, and the other symmetry was in electrical domain. Recently, Mukherjee et al (2009) presented the extension for Lagrangian-Hamiltonian Mechanics for continuous systems and investigated the dynamics of an internally damped rotor through dissipative coupling. Some basic concepts of umbra-Hamiltonian theory may be given in Appendix A for ready reference.…”

“…The broad principle, on which the creation of umbra-Lagrangian and other relevant energies (Mukherjee, 2001;Rastogi, 2005;Mukherjee et al, 2009) are based, can be summarized as follows: (a) All temporal fluctuations of parameters are in real time.…”

“…The extended Noether's theorem, as discussed in paper (Mukherjee et al, 2009) may lead to a constant of motion, or trajectories, on which some dynamical quantity remains conserved.…”

“…In recent years, the authors have attempted to develop alternative method to the construction of the first integrals of dynamical systems by means of extended Noether's theorem. Typical contributions in this area are given in reference (Rastogi, 2005;Mukherjee et al, 2006;Mukherjee et al, 2007;Mukherjee et al, 2009). Investigating such alternatives has been applied to analyze the dynamics through invariance of the action integral for some engineering applications, which are rarely applied.…”