Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft

Abstract

Based on the general theory of fractional order derivatives and integrals, application of the Caputo–Fabrizio operator is analyzed to improve a mathematical model of a two-mass system with a long shaft and concentrated parameters. Thus, the real transmission of complex electric drives, which consist of long shafts with a sufficient degree of adequacy, is presented as a two-mass system. Such a system is described by ordinary fractional order differential equations. In addition, it is well known that an elastic mechanical wave, propagating along a drive transmission with a long stiff shaft, creates a retardation effect on distribution of the time–space angular velocity, the rotation angle of the shaft, and its elastic moment. The approach proposed in the current work helps to take in account the moving elastic wave along the shaft of electric drive mechanism. On this basis, it is demonstrated that the use of the fractional order integrator in the model for the elastic moment enables it to reproduce real transient processes in the joint coordinates of the system. It also provides an accuracy equivalent to the model with distributed parameters. The distance between the traditional model and the model in which the fractional integral is used for the elastic moment modelling in a two-mass system, with a long shaft, is analyzed.

Voltage Stabilisation of a Drive System Including a Power Transformer and Asynchronous and Synchronous Motors of Susceptible Motion Transmission

“…Mathematical models of the drive system elements will be presented on the basis of Figures 1 and 2 and of [9,10,21,[28][29][30][31][32]. A general interdisciplinary theory modifies the Hamilton-Ostrogradsky principle by expanding the Lagrangian with two components: Mathematical models of the drive system elements will be presented on the basis of Figures 1 and 2 and of [9,10,21,[28][29][30][31][32].…”

“…Mathematical models of the drive system elements will be presented on the basis of Figures 1 and 2 and of [9,10,21,[28][29][30][31][32]. A general interdisciplinary theory modifies the Hamilton-Ostrogradsky principle by expanding the Lagrangian with two components: Mathematical models of the drive system elements will be presented on the basis of Figures 1 and 2 and of [9,10,21,[28][29][30][31][32]. A general interdisciplinary theory modifies the Hamilton-Ostrogradsky principle by expanding the Lagrangian with two components: energy of internal and external dissipation and energy of external non-potential forces is applied to mathematical modelling of the drive system's electric elements.…”

Fuzzy Adaptive Type II Controller for Two-Mass System

“…The characteristic feature of the above-mentioned applications is a low frequency of torsional vibrations. Due to the progress of modern power electronics, microprocessor techniques which allows forcing the driving torque of the motor practically without delay (as compared to the time constant of the driving elements) the torsional vibrations are recognized nowadays in almost all modern drive systems, for instance micro-electromechanical systems, robot-arm drives, CNC-tools, deep space antenna systems, drives in electrical cars, windmills and others [3][4][5]. Up until the present moment, different control approaches developed in order to suppress torsional vibrations can be divided into two main frameworks.…”

Mathematical Modelling of Transient Processes in an Asynchronous Drive with a Long Shaft Including Cardan Joints

“…It concerns physical effects in motion transmission, where a drive motor's torque is transferred to a load system. When the distance between the drive motor and the load system is substantial, the torque is determined by describing a long shaft in a distributed mechanical parameters system [1][2][3][4][5][6]. Points of power receipt can obviously have varying practical configurations, which may in turn complicate kinematic and drive system equations.…”