Некоторые аспекты динамики переходных процессов поворотного рабочего элемента под воздействием упругих сил

Abstract

Получена математическая модель динамики поворотного рабочего элемента, работающего на высоких скоростях и ускорении, выполняющего высокоточную координатную ориентацию. Математическая модель представлена дифференциальными уравнениями второго порядка, учитывает силы трения и упругости. Поворотный рабочий элемент рассмотрен как звено системы автоматического регулирования.

Отримано математичну модель динаміки поворотного робочого елемента, який працює на високих швидкостях та з прискоренням і виконує високоточну координатну орієнтацію. Математична модель представлена диференційними рівняннями другого порядку і враховує сили тертя та пружності. Поворотний робочий елемент розглянуто як ланку системи автоматичного регулювання.

Background. It is vitally necessary to create the mathematical model of the dynamic behavior of the rotary operating element, which is under the influence of inertial forces, the forces of friction and elastic forces in transient regimes of acceleration and deceleration for using the model in robotics, manipulators, etc. Statement. Due to the mathematical model results of the rotary operating element, working, as a rule, with high angular and linear velocities and accelerations, performing herewith high-precision coordinate orientation, there is the possibility to examine the dynamics of its behavior (movement) in acceleration and deceleration modes. This moving may be realized either by an exponential law, or under the law of damped harmonic oscillations. Except stated, the rotary working element is considered as a link of the automatic control system (ACS). In this case, the rotary working element can be represented either by the 2nd order link and this link can be replaced by two series-connected inertial links or as an oscillating link. Research methodology. These results are achieved due to the resulting inhomogeneous differential equation of 2nd order. Depending on the structure of the characteristic equation roots (real or complex roots), transients can be carried out either by an exponential law (real roots), or under the law of damped harmonic oscillations (complex roots). Moreover, the rotary operating element is considered as a link of ACS, that has been made possible by the resulting differential equation, which is presented in a symbolic way by which amplitude-frequency, phase-frequency and amplitude-phase characteristics of the rotary operating element are obtained as a link of the ACS. Conclusion. At the end of the article there are conclusions that enable the practical use of the results.