CONTROL FOR FORCED VIBRATIONS OF ROTATING MATHEMATICAL PENDULUM WITH DEPENDENT FROM TIME LENGTH AND MASS - Gristchak, D. D. (Computer Science and Applied Mathematics, 2016)
This paper deals with the proposed method for investigation of control problem for forced vibrations of rotating mathematical pendulum with dependent from time length and mass under the given external loading. Rotating pendulum with time dependent parameters as a mathematical model of a dynamical system is discussed. The model includes a vibrating ball with dependent from time mass, which is embedded in the inner surface of the solid thin walled two dimensional torus of radius equal to length of pendulum. The ball can slide without friction in the inner surface of the torus, and the torus may be rotated with a circumferential speed (in the general case dependent from time).Basic equation of problem is nonlinear non homogeneous differential equation with variable coefficients that in the general case can not be integrated analytically exactly. There are more popular direct numerical methods of integration this type of equations or approximate asymptotic approaches. For the solution of nonlinear problem on the first step of hybrid asymptotic approximation the solution is presented with respect to perturbation method on parameter nonlinearity. Introducing it into initial equation and acquainting the coefficients at equal order of parameter nonlinearity, the system of singular linear equations with variable coefficients is obtained. On the second step the phase integral (or WKB) method is explored. Finally an approximate analytical solution for the problem is formulated. Comparison of obtained analytical solution with direct numerical integration of basic equation for forced vibration pendulum with time dependent parameters and an approximate analytical solution for control of forced rotating pendulum with length and mass dependent from time are given.