Parametrization of electromechanical systems must acknowledge Newton and Maxwell

Abstract

Electromechanical systems are an interesting type of coupled systems. They are composed by two subsystems with different nature: mechanical and electromagnetic. The subsystems interact. To represent the dynamics of a coupled system, it is necessary to properly characterize their interaction. The dynamics of an electromechanical system is given by an initial value problem (IVP) comprising a set of coupled differential equations involving, necessarily, mechanical and electromagnetic variables. Despite the ubiquity of electromechanical systems, a few authors do not parametrize them properly. Frequently, by some artifice, strange to the problem, the coupled system is uncoupled disregarding the electromagnetic subsystem. Hence, the uncoupled system has a different dynamics, resulting a reduced IVP with only a mechanical equation. This paper discusses this uncoupling using as example a galvanometer, a well-known measuring device. To analyze the effects of the decoupling, numerical simulations of the two IVP, complete and reduced, are performed.

Stability Analysis Applied to Mechanical, Electromagnetic and Electromechanical Systems With Parametric Excitation

“…Beginning now with (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19), y 0 = sin(τ ) for p = 1, the process leads to…”

“…Parametric resonance is thus a phenomenon that can result in large response amplitudes even for small excitation frequencies. For critical frequencies originated from (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), the effect is always destabilizing [28]. However, for some of the critical frequencies originated from the combination of natural frequencies in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21), a stabilizing effect can be obtained.…”

“…For critical frequencies originated from (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), the effect is always destabilizing [28]. However, for some of the critical frequencies originated from the combination of natural frequencies in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21), a stabilizing effect can be obtained. This is where parametric anti-resonance occurs, that, differently from the parametric resonance, does not result in instability.…”

“…By using equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) with the just obtained natural frequencies, it is possible to say where the "tongues" of instability start forming. For example, by taking δ = 4, the critical frequencies are: 4, 2, 1 and so on.…”

“…By making use of equations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21), the critical frequencies can be obtained, where the phenomenon of parametric resonance occurs. They are of interest so the frequency intervals where instability takes place are known.…”