Abstract
We develop a method for systematically constructing Lagrangian functions for dissipative mechanical, electrical, and electromechanical systems. We derive the equations of motion for some typical electromechanical systems using deterministic principles that are strictly variational. We do not use any ad hoc features that are added on after the analysis has been completed, such as the Rayleigh dissipation function. We generalise the concept of potential, and define generalised potentials for dissipative lumped system elements. Our innovation offers a unified approach to the analysis of electromechanical systems where there are energy and power terms in both the mechanical and electrical parts of the system. Using our novel technique, we can take advantage of the analytic approach from mechanics, and we can apply these powerful analytical methods to electrical and to electromechanical systems. We can analyse systems that include non-conservative forces. Our methodology is deterministic, and does does require any special intuition, and is thus suitable for automation via a computer-based algebra package.
Highlights
Introduction and MotivationIt is a widely believed that the Lagrangian approach to dynamical systems cannot be applied to dissipative systems that include non-conservative forces
If the macroscopic parameters of a mechanical system are completed by the addition of microscopic parameters, forces not derivable from a work function would in all probability not occur.’’ Lanczos [2], and writes ‘‘Frictional forces which originate from a transfer of macroscopic into microscopic motions demand an increase in the number of degrees of freedom and the application of statistical principles
We have extended the range of applications of Lagrangian analysis, to include non-conservative systems that include dissipative forces
Summary
It is a widely believed that the Lagrangian approach to dynamical systems cannot be applied to dissipative systems that include non-conservative forces. In 2004, Dreisigmeyer and Young[15] published another paper on nonconservative Lagrangian mechanics, in which they derived purely causal equations of motion They made use of left fractional derivatives. Suppose that the Lagrangian function includes references to a generalised coordinate, x(t), and to its first derivative x_ so L~Lðx,x_ Þ, the action is extremal when we choose x(t) in such a way that the Euler-Lagrange equation is satisfied:. This is the same as saying that all first order variation of the action is zero, d1⁄2I ~0. We are satisfied if our definitions give rise to correct ordinary differential equations of motion that are valid in a closed time-interval, 1⁄20,t
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