Abstract
We developed a technique to determine the non-chaotic oscillations of loads in the conservative pendulum systems by using the graphic technology of projection focusing. In this case, phase trajectories of the differential equations of oscillation are considered as projections of integral curves from the phase space onto the phase plane. The effect exerted by the value of one of the system's parameters on the image of phase trajectories was examined (at stable values of other parameters). By using projection focusing, the element that defines the critical value of variable parameter is selected among a family of phase trajectories, which, in a combination with other parameters, allows us to describe a non-chaotic trajectory in the load motion. The need for such studies is predetermined by the absence, in practice, of an engineering method for computing the non-chaotic trajectory of the load motion for a certain pendulum system. We proposed the notion of a focus-line of the parametric family of curves and the technique of projection focusing, which is based on it. We constructed integral curves in the phase space based on the numerical solution of second order Lagrange differential equations. A procedure is presented to determine the critical value of pendulum oscillation parameter by using the graphic notion of projection focusing of phase trajectories in the solutions of second order Lagrange differential equations. The examples are presented of determining the parameters of certain pendulums, which would provide for the non-chaotic trajectory of the load oscillations. The developed computerized projection technique for the simulation of oscillations in the pendulum mechanical systems makes it possible to choose the required values of parameters and initial conditions for initiating the oscillations, which provide for the non-chaotic technological character of oscillation trajectory of their elements, which is important for the practical implementation in the designs of pendulum systems.
Highlights
An experience in solving a number of applied problems demonstrates that a road to the result will be more efficient if one considers the “inner” projection nature of such problems
The problem on calculating the dynamics of pendulum system is a traditional subject of research in theoretical mechanics, where the simplicity of setting a task is combined with the complexity of its solution [2]
Mathematics and cybernetics – applied aspects choosing the combination of parameters and initial conditions for the initiation of oscillations, which would provide for the technological trajectory of load in the implemention of specific designs of pendulum systems
Summary
An experience in solving a number of applied problems demonstrates that a road to the result will be more efficient if one considers the “inner” projection nature of such problems. One manages to reduce a “parametric dimensionality” of problems by mapping a certain geometric object, stipulated by the problem statement, onto a coordinate hyper-plane in the same parameter space. In this case, computations, associated with the conversion of graphic information about the image representation, allow obtaining an effective solution for the original problem. Mathematics and cybernetics – applied aspects choosing the combination of parameters and initial conditions for the initiation of oscillations, which would provide for the technological trajectory of load in the implemention of specific designs of pendulum systems. In order to solve this problem, we propose in this work to apply geometric apparatus for mapping onto the phase plane the integral curves of second order Lagrange differential equations that describe pendulum mechanical oscillations. Here we tackle only those idealized systems in which the reserve of mechanical energy in the process of oscillations remains constant [5]
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