ДОСЛІДЖЕННЯ КОЛИВАНЬ МАЯТНИКА ЗМІННОЇ ДОВЖИНИ

Jazyk: ukrajinština
Rok vydání: 2021
Předmět:
Zdroj: Технічні науки та технології; № 3(25) (2021): Технічні науки та технології; 38-44
Technical sciences and technology; No. 3(25) (2021): Technical Sciences and Technologies ; 38-44
Технические науки и технологии; № 3(25) (2021): Технічні науки та технології; 38-44
ISSN: 2411-5363
2519-4569
Popis: The problem of movement of a pendulum of variable length is considered in the article. The study of the movements of pendulum systems is relevant, as they can demonstrate significantly nonlinear and quite diverse behavior and are often used as a source of model problems for the development and study of nonlinear control methods. First of all, this applies to various types of lifting mechanisms.Some lifting mechanisms use a bicylindro-conical drum to wind the rope, which significantly complicates the kinematics and dynamics of the oscillating load, which requires a separate study.Analysis of scientific publications has shown that research in the field of modeling of complex nonlinear pendulum systems continues and does not stop.The problem of studying the motion of a load that swings during the winding of a rope on a conical drum has not been considered before.The purpose of the work is to create a mathematical model, establish the basic laws and study the oscillating motion of a load with point mass, which performs flat oscillating motions on a weightless flexible cable wound on a rotating bicylindrical-conical drum.Using the Lagrange equations of the second kind, the second problem of dynamics is solved and a mathematical model of the considered mechanical system is obtained. The equation of motion and the relationship between angular and Cartesian coordinates are determined. A program was compiled and a numerical experiment was conducted. The model and the program make it possible to obtain the dependences of linear and angular displacements, as well as linear and angular velocities, and to construct appropriate graphs, phase portraits, and the trajectory of the load. The position of the load, its speed and acceleration make it possible to find the value of the lifting rope tension at any time. Tension acts on both the load and the drum, allowing you to determine the safe modes of lifting or lowering the load and the load on the drive mechanism. The study was performed on a nonlinear model without the use of asymptotic methods, which allowed to exclude the methodological error of the solution.Having a mathematical model and calculation programs, it is possible to conduct further research of the considered system. The obtained formulas make it possible to design such pendulum systems with the most rational characteristics and the optimal ratio of design parameters. The obtained results can be used for modeling of controlled pendulum motions of different mechanical systems. The methodology and program are recommended for solving applied problems of design and operation of various hoisting and transport systems and technical devices capable of demonstrating complex behavior. The methodologically proposed material is interesting for students and graduate students in terms of teaching the principles of construction and analysis of complex nonlinear dynamical systems.
Розглянуто задачу руху маятника змінної довжини, який являє собою вантаж із точковою масою, який здійснює 2D коливання на невагомому гнучкому канаті, що намотують на біциліндро-конічний барабан, який обертається на-вколо власної осі. Отримана математична модель системи, визначені рівняння руху і співвідношення між кутовими та декартовими координатами. Складена програма й виконаний числовий експеримент. Отримані залежності від часу лінійних та кутових переміщень та швидкостей, побудовані відповідні графіки, фазові портрети та траєкторія руху вантажу. Знайдено величину натягу підйомного канату.
Databáze: OpenAIRE
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