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Maths in Snowboarding
- 11 страниц
- 2016 год
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The above quotation is the decipherment of the famous anagram «6accdae13eff7i3l9n4o4qrr4s8t12ux» which Sir Isaac Newton sent in his letter to G.W.Leibniz (October, 1676). In this anagram Newton pointed out that differential equations are important because they express the laws of nature and this was an epoch-making discovery of the outstanding scientist.
Differential equations are extremely widely used.
It is known that the basic rules of classical mechanics may be stated in the form of differential equations and this direction was developed by many outstanding scientists like L. Euler, J. Lagrange, J. Maxwell. Electro-magnetism, quantum mechanics, fluid mechanism rules may be stated in the form of differential dependencies. Besides, many problems from biology, economy, engineering, astronomy and other sciences may be described by differential equations.
Nowadays the theory of differential equations is a very powerful mathematical tool for modelling and analysis of problems appearing in different fields of our life.
In my exploration I consider a real-life problem that may be described using differential equations with control force. Namely, I consider the mathematical model of gondola lift. I construct this model using the classical pendulum model. Motion of the pendulum is described by a second order differential equation and I derive the law of motion of the pendulum using the Newton’s second law.
Further I explore the model of gondola influenced by wind and the effects arising from this model. Here the problem of stabilization arises, that is, the problem of finding and realizing a control force, which would compensate the force of wind.
In this part of exploration I use some results of the mathematical control theory. This theory was intensively developed during the last sixty years, due to various engineering and technical problems, in which it was a need for control of different objects. An example of such problems is cosmology research: landing on the Moon, stabilization of telecommunication satellites orbits; airplane technologies, industry automatics, robotics and other. Fundamentals of the control theory, stability and stabilizability of systems described by differential equations, were stated in the mid of XX century by L. Pontriagin, R. Bellman, R. Kalman and other mathematicians.
Besides, I consider a model with several gondolas and derive a stabilizability condition. My exploration is concluded by solving a concrete problem (designing a control).
Stabilization of gondola lift 1
Introduction 1
Mathematical model of gondola 2
How to stabilize gondola 4
Application of control theory 6
How to stabilize several gondola simultaneously 7
Design of a control stabilizing gondola 9
Conclusion. 10
Bibliography 11
In my research I have considered the problem of stabilization of gondola lift subject to the action of external forces, as an example of the application of the theory of differential equations and mathematical control theory. I have deduced the equation of motion of the pendulum as a model of the gondola’s motion, as well as the models for one and several gondolas moving on a steel rope. Using methods of the control theory it has been proven, that it is possible to stabilize the gondola using only the movement of steel rope. The relevant calculations are given. Moreover, I have designed a control stabilizing the gondola with some fixed parameters.
Pendulum
1. https://en.wikipedia.org/wiki/Pendulum_(mathematics)
Controllability and Stabilization
1. https://en.wikipedia.org/wiki/Controllability
2. E.D. Sontag. Mathematical Control Theory, volume 6 of Texts in Applied Mathematics. Springer-Verlag, New York, second edition, 1998. Deterministic finitedimensional systems.
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