For citation:
Gladkov S. O., Bogdanova S. B. On the question accounting of the resistance force at the hinge point of setting physical pendulum and its influence on the dynamics of movement. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 1, pp. 53-62. DOI: 10.18500/0869-6632-2019-27-1-53-62
On the question accounting of the resistance force at the hinge point of setting physical pendulum and its influence on the dynamics of movement
Topic. The paper is devoted to the analysis of the dynamics of a complex system, i.e. a hinge mechanism plus a compound pendulum, in which where a differential equation is found, describing its nonlinear behavior. Aim. The paper is in the analysis of nonlinear oscillations of a complex dynamical system, which is a hinge, a rod and a ball, setting together in the one way. It is assumed to obtain differential equation of motion of the pendulum with regard to the gimbal friction and the resistance of the continuum. Method. Problem-solving procedure is based on the law of conservation of energy, accounting energy dissipation both in the hinge and in when the setting rod and ball move in a viscous medium. In this case, it is assumed to use the definition of dissipative functions in a viscous medium, which making allowance for inhomogeneous distribution of the velocity near the surface of the rod and ball. Results. In this paper, it is strictly analytically shown that energy losses in the hinge have impacts on the dynamics of the studied system (i.e. a hinge plus a rod plus a ball). These energy losses lead to a strong reduce amount of damping time at fluctuating motion, which has a highly nonlinear character that is described in the paper in details. The numerical solution of the nonlinear dynamic equation found, illustrated in the figures, points to strongly inhomogeneous oscillations of generalized coordinate, for which the angle of deviation of a pendulum from y-axis has chosen. Discussion. Thanks to the proposed method of differential equations of the movement of complex dynamical systems in the paper, which is the summation the expressions for the dissipative function and for derivative with time of a total energy of system, it is obtained the studied equation. Such an approach allows us to derive any differential equations (systems of equations) with regard to damping. Using the example of our studied dynamical system, it is shown how this method «works». Such an algorithm simplifies the analysis of the derivation of equations and keeps to a minimum making analytical errors.
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