Anton O. Belyakov

Dynamics of the pendulum with periodically varying length

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of a childs swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with a numerical study. Chaotic motions of the pendulum depending on problem parameters are investigated numerically.

The Hula! Hoop Problem

A hula! hoop is sports equipment, which became popular in the 1960s, and is a thin! walled hoop that goes around the athlete’s waist. For spinning a hula! hoop, the athlete’s waist makes periodic motions in the horizontal plane resulting in stable rotations. In [1] the periodic motion of the athlete’s waist along one axis was considered, and the hula! hoop problem was reduced to the problem of a pendulum with a vibrating suspension point in the absence of gravity. The stable mode of pendulum rotation with an average angular velocity equal to the excitation frequency was found approximately, and the conditions of stability of this mode were obtained. In [2] the same mode of rotation was found for this pendulum by the method of averag! ing in the second approximation, and its stability con! ditions were investigated. Stable hula! hoop rotation for the periodic excitation along two axes was studied by the method of direct separation of motion in [3]. In this study, we considered the hula! hoop excita! tion along two axes corresponding to an elliptic trajec! tory of the motion of the athlete’s waist. For identical excitation amplitudes, exact solutions corresponding to the hula! hoop rotation with a constant angular velocity equal to the excitation frequency are obtained. The stability of these solutions is investi! gated. The conditions of the inseparable hula! hoop rotation, both stable and unstable, are derived. The case of close excitation amplitudes corre! sponding to the motion of the athlete’s waist along an ellipse close to a circle is considered. The solutions of the problem on stable hula! hoop rotation in the first, second, and third approximation are obtained by the averaging method. The comparison with the numeri! cal solution obtained with high accuracy shows that the third approximation practically coincides with it. The conditions of coexistence of stable rotation modes with opposite directions are obtained. An interesting case when the athlete’s waist rotates oppositely to the rotation of the hula! hoop is investigated.

On rotational solutions for elliptically excited pendulum

The author considers the planar rotational motion of the mathematical pendulum with its pivot oscillating both vertically and horizontally, so the trajectory of the pivot is an ellipse close to a circle. The analysis is based on the exact rotational solutions in the case of circular pivot trajectory and zero gravity. The conditions for existence and stability of such solutions are derived. Assuming that the amplitudes of excitations are not small while the pivot trajectory has small ellipticity the approximate solutions are found both for high and small linear damping. Comparison between approximate and numerical solutions is made for different values of the damping parameter.

On Optimal Harvesting in Age-Structured Populations

The problem of optimal harvesting (in a fish population as a benchmark) is stated within a model that takes into account the age-structure of the population. In contrast to models disregarding the age structure, it is shown that in case of selective harvesting mode (where only fish of certain sizes are harvested) the optimal harvesting effort may be periodic. It is also proved that the periodicity is caused by the selectivity of the harvesting. Mathematically, the model comprises an optimal control problem on infinite horizon for a McKendrick-type PDE with endogenous and non-local dynamics and boundary conditions.

Dynamics of a Pendulum of Variable Length and Similar Problems

In this chapter we study three mechanical problems: dynamics of a pendulum of variable length, rotations of a pendulum with elliptically moving pivot and twirling of a hula-hoop presented in three subsequent sections. The dynamics of these mechanical systems is described by similar equations and is studied with the use of common methods. The material of the chapter is based on publications of the authors [1-7] with the renewed analytical and numerical results. The methodological peculiarity of this work is in the assumption of quasi-linearity of the systems which allows us to derive higher order approximations by the averaging method. All the approximate solutions are compared with the results of numerical simulation demonstrating good agreement. Supplementary, in Appendix (section 5) we briefly presented the method of averaging with higher order approximations which is used in sections 2, 3, and 4.

Homoclinic, subharmonic, and superharmonic bifurcations for a pendulum with periodically varying length

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the simplest model of child’s swing. Melnikov’s analysis is carried out to find bifurcations of homoclinic, subharmonic oscillatory, and subharmonic rotational orbits. For the analysis of superharmonic rotational orbits, the averaging method is used and stability of obtained approximate solution is checked. The analytical results are compared with numerical simulation results.

How to twirl a hula hoop

We consider the twirling of a hula hoop when the waist of a gymnast moves along an elliptical trajectory close to a circle. For a circular trajectory, two families of exact solutions are obtained, corresponding to twirling of the hula hoop with a constant angular speed equal to the speed of the excitation. We show that one family of solutions is stable, and the other one is unstable. These exact solutions allow us to obtain approximate solutions for a slightly elliptical trajectory of the waist. We demonstrate that to twirl a hula hoop the waist needs to rotate with a phase difference between π/2 and π. An interesting effect of inverse twirling is described when the waist moves in a direction opposite to the hula hoop rotation. The approximate analytical solutions are compared with the results of a numerical calculation.

Optimal cyclic harvesting of renewable resource

The paper obtains existence of a solution and necessary optimality conditions for a problem of optimal (long run averaged) periodic extraction of a renewable resource distributed along a circle. The resource grows according to the logistic law, and is harvested by a single harvester periodically moving around the circle.

Excitation of oscillations by a small limited control force

Linear oscillator with limited excitation force (control) is under consideration. The optimal control, which led oscillatory system to a certain energy level from any initial conditions at minimum time, is found. The control synthesis is made. I. e. time in control function is excluded by current system phase variables (coordinate and velocity). Quasi-optimal synthesized control function is obtained for one-dimensional oscillatory system with unknown parameters. Multidimensional case is considered on the supposition that excitation forces are small.