Brachistochrone for a rolling cylinder

The motion of a heavy homogeneous cylinder is considered as a no-slip rolling along the desired curve. We obtain a functional in the form of the total time of the cylinder rolling and solve the corresponding variational problem of minimizing this functional. We obtain an algebraic equation for the directional line of steepest descent, brachistochrone, in parametric form. We use the equation of motion of the cylinder with constraint reaction to determine the conditions of implementation of its pure rolling without separation and slip with respect to the brachistochrone.     

Limit-cycle vibrations of a rolling cylinder

An explanation for a violent vibration of a high-speed textile yarn winder is proposed based on the hypothesis that variations in radial compliance around the circumference of the yarn cylinder make the smooth winding process unstable due to parametric excitation. The amplitude of the resulting limit cycle depends on the non-linearity in the radial compliance. An analytical procedure for predicting the limit-cycle amplitude is developed and applied to a specially constructed mechanical model of a yarn winder. The actual response of the mechanical model is accurately predicted when its measured dynamic properties are inserted in the analysis.     

Conditions for pure rolling of a heavy cylinder along a brachistochrone

Algebraic equations for the line of steepest descent of a cylinder are derived in parametric form. Conditions for rolling without slipping and separation of the cylinder along a brachistochrone are established based on the equations of motion with constraint reaction. The important conclusion is drawn that the center of mass of a cylinder moving along a brachistochrone describes a cycloid     

Interfacial slip and creep in rolling contact incorporating a cylinder with an elastic layer

A systematic approach for investigating the interfacial behaviour of tyred systems is proposed. A two-dimensional contact model of an elastic strip, shrink-fitted onto a wheel, and subjected to different rolling contact conditions, has been adopted to illustrate the method. The model combines existing techniques to explore individual elastic contact problems and it enables us to characterise the behaviour at the strip/substrate interface caused by loads induced by a quasi-static application of stationary and moving loads on the surface of the layer. The solution is compared to the stationary load case and regimes of local slip, full stick, separation and frictional creep are identified and collated for a variety of loading conditions, materials and geometries. Further, this article presents an investigation of frictional shakedown for layered systems subjected to periodic contact loading. The term shakedown is here referred to as the possibility of developing interfacial residual stresses at the layer/substrate interface such that frictional slip, originally activated by the applied external contact load, ceases after a few loading cycles. The possible applicability of the Melan’s theorem for elastic frictional system is investigated and preliminary results presented.     

Dissipative heating of a viscoelastic cylinder steadily rolling over a rigid foundation

A dissipative-heating problem for an incompressible viscoelastic cylinder rolling over a rigid foundation is solved. The effect of tangential stresses on the dissipative-heating temperature is examined __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 59–69, January 2006.     

Dynamic instability and imperfection sensitivity of a block rolling on a cylinder

Summary An effort is made to present a complete picture of dynamic instability behaviour and imperfection sensitivity of a model consisting of a block rolling without slipping on a cylinder. The formulation is developed in terms of generalized coordinates and uses a stepwise variation of the bifurcation parameter. The model employed is as simple as possible, yet devised to have all dynamic features of a class of mechanical systems having a subcritical pitchfork point of bifurcation.

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Dynamisches Instabilitätsverhalten und Imperfektionsempfindlichkeit eines Schaukelblocks auf einem Halbzylinder

Übersicht In der vorliegenden Arbeit wird ein Ansatz zu einer vollständigen Darstellung des dynamischen Instabilitätsverhaltens eines einfachen und nur der Schwerkraft unterworfenen Systems gemacht. Die Gleichungen sind in verallgemeinerten Koordinaten abgefaßt. Der Verzweigungsparameter wird stufenweise variiert. Das verwendete Modell ist so einfach wie möglich, enthält jedoch alle Merkmale der Klasse mechanischer Systeme, die durch eine subkritische dreiästige Verzweigung gekennzeichnet sind.

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Dedicated to the memory of Professor Horst Leipholz (1919–1988)     

Quickest-descent curve in the problem of rolling of a homogeneous cylinder

The motion of a heavy homogeneous cylinder is considered as rolling without slipping along an unknown curve. A functional in the form of the total time of rolling is found and minimized by solving a variational problem. The algebraic equation of the quickest-descent directrix is derived in parametric form Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 131–138, December 2008.     

Dynamical behavior of a two-component pendulum system with rolling at the boundary of the region of stability

Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 29, No. 4, pp. 78–86, April, 1993.     

Integral manifold in the problem of Andronov-Hopf bifurcation in two-unit pendulum systems with rolling

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of the Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 7, pp. 85–93, July, 1994.     

Motion control of a two-wheeled inverted pendulum with uncertain rolling resistance and angle constraint based on slow-fast dynamics

Nonlinear Dynamics - Rolling resistance and angle constraint are important factors affecting the control performance and dynamics of Two-Wheeled Inverted Pendulums (TWIPs, for short), but they are...     

Dynamics of the rolling and sliding of an elongated cylinder

A snap of a finger on an elongated cylinder produces on it a surprising fast spinning motion during which its mass center rises with very large oscillations. After that, the mass center goes to a long quasi-stationary oscillations state and eventually goes down very slowly. To explain this behavior we present a theoretical and numerical analysis of the dynamics of a spinning elongated cylinder moving with a single point of contact on a horizontal plane under the action of gravity. The study has been made taking into account the rolling and sliding dissipation as well as the Kutta–Joukowski airflow effect. The results of the simulations are in agreement qualitatively with the observed real motion.     

Chaos in a pendulum with feedback control

We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.     

Rotating a pendulum with an electromechanical excitation

The objective of this paper is showing investigation of pendulum rotations via vertical, non-linear electromechanical excitation generated using a RLC-circuit-powered solenoid, which is originally built for an electro-vibro-impact mechanism. Various non-linear phenomena of pendulum dynamics, namely period-1 rotation, period-1 oscillation and period-2 oscillation, have been observed experimentally from the proposed apparatus. A mathematical model has been developed for the experimental rig and the system parameters have also been identified for the mathematical model. Finally, numerical results have been generated using the developed mathematical model and identified parameters, and their correlations with experimental observations have been discussed.     

Torsion problems for a cylinder with a rectangular hole and a rectangular cylinder with a crack

Using the method of singular integral equation and the crack-cutting technique, the rigorous solutions are obtained for a cylinder with a rectangular hole and a rectangular cylinder with a crack, which exactly satisfy the boundary conditions and the conditions at the corner points. After that the torsional rigidities and the stress intensity factors at the crack tip are determined. Next, for the doubly connected circular cylinder with a rectangular hole the expressions for the singular stresses around the concave corner points are derived and the generalized stress intensity factors are then defined. Since the crack-cutting technique is used in this paper, the solution of the matching rectangular cylinder is also obtained and its numerical results coincide with those in references. Thus the method proposed here is verified. The project supported by National Natural Science Foundation of China     

On a toroidal pendulum

We consider the problem of motion of a heavy particle on the surface of a torus with horizontal axis of rotation.     

Vibration energy harvesting for low frequency using auto-tuning parametric rolling pendulum under exogenous multi-frequency excitations

An electromagnetic parametrically excited rolling pendulum energy harvester with self-tuning mechanisms subject to multi-frequency excitation is proposed and investigated in this paper. The system consists of two uncoupled rolling pendulum. The resonance frequency of each the rolling pendulum can be automatically tuned by adjusting its geometric parameters to access parametric resonance. This harvester can be used to harvest the energy at low frequency. A prototype is developed and evaluated. Its mathematical model is derived. A cam with rolling follower mechanism is employed to generate multi-frequency excitation. An experimental study is conducted to validate the proposed concept. The experimental results are confirmed by the numerical results. The harvester is successfully tuned when the angular velocity of the cam is changed from 1.149 to 1.236 Hz.     

Limit cycles of a double pendulum with nonlinear springs

The limit cycles of a double pendulum with hard, soft, or linear springs subject to a follower force are drawn using computer simulation