Nonlinear dynamics of a sinusoidally driven pendulum in a repulsivemagnetic fieldAzad SiahmakounDepartment of Physics and Applied Optics, Rose-Hulman Institute of Technology, Terre Haute,Indiana 47803Valentina A. FrenchDepartment of Physics, Indiana State University, Terre Haute, Indiana 47809Jeffrey PattersonDepartment of Physics and Applied Optics, Rose-Hulman Institute of Technology, Terre Haute,Indiana 47803~Received 24 July 1996; accepted 24 October 1996!The dynamics of a sinusoidally driven pendulum in a repulsive magnetic field is investigatedtheoretically and experimentally. The experimental data are acquired using a shaft encoderinterfaced to a PC which measures the angular displacement of the pendulum as a function of time.Both the theoretical simulations and the experimental measurements exhibit regions of periodic andchaotic behavior, depending on the system parameters. Amplitude jumps, hysteresis, and bistablestates are also observed. The simplicity of the apparatus makes this experiment suitable for anadvanced undergraduate laboratory. ©1997 American Association of Physics Teachers.I. INTRODUCTIONIn recent years several laboratory experiments demonstrat-ing chaotic motion in a pendulum system have beenpublished.1–6Experiments using a passive doublependulum1,2demonstrated that a slight change in the initialrelease position of the pendulum led to the exponential di-vergence of the pendulum’s trajectories in a chaotic regime.A driven inverted pendulum3experiment, in which the driv-er’s frequency was the control parameter, showed how thepower spectrum changed during a transition from periodic tochaotic motion. A simple pendulum whose pivot executedhigh frequency vertical oscillations4,5was used to demon-strate stable inverted states by adjusting the amplitude of thepivot’s motion. A magnetic pendulum6whose deflection wascontrolled by the currents in an electromagnet at its tip andthree others equally spaced around the pendulum was usedfor analog demonstrations of first- and second-order phasetransitions.In this paper we present the results of an investigation of adriven physical pendulum in a repulsive magnetic field. Thisfield is in opposition to the restoring gravitational force ~see393 393Am. J. Phys. 65 ~5!, May 1997 © 1997 American Association of Physics TeachersFig. 1! resulting in a damping of the pendulum’s motion. Thesystem consists of a physical pendulum coupled to a sinusoi-dally varying driving force, and a pair of magnets, one posi-tioned at the end of the pendulum and the other directlybelow the pendulum’s vertical position. The poles of themagnets are oriented such that a repulsive force exists be-tween them. This apparatus possesses five control param-eters: the frequency and the amplitude of the driver, the ver-tical separation between the magnets ~z direction!, and therelative position of the magnets in a horizontal plane ~x andy directions!. All of these parameters are experimentally ac-cessible without any modifications to the setup, and theycontribute to the rich dynamics of the system. The simplicityof this apparatus coupled with its rich dynamics make itsuitable for an advanced undergraduate laboratory experi-ment.The dynamics of this system is investigated both theoreti-cally and experimentally. In the theoretical aspect of our in-vestigation, we use simple Newtonian physics to derive theequations of motion that describe the dynamics of the sys-tem.MATHEMATICA™ is then used to solve these equationsfor specified initial conditions and system parameters ~fre-quency of the driver, amplitude of the driver, and magneticstrength of the magnets!. From these solutions, time seriesand phase space plots are constructed and discussed. Theseplots exhibit regions of both periodic and chaotic behavior,depending on the parameters of the system.The experimental data are acquired using a shaft encoderinterfaced to a PC which measures the angular displacementof the pendulum. These measurements are recorded in theform of time series. By numerically differentiating thosetime series, phase space plots are constructed. Both periodicand chaotic behaviors are observed, depending on the fre-quency of the driver and the distance between the two mag-nets. Amplitude jumps, hysteresis, and bistable states occurfor a range of frequencies near the natural frequency of thephysical pendulum.In Sec. II, we derive the equations of motion and presentthe results of theoretical simulations. In Sec. III, we describethe experimental apparatus and the measurements obtainedfrom it. Our conclusions are presented in the final section.II. THEORYA theoretical model is developed considering the fourforces acting on the pendulum: the restoring gravitationalforce, the repulsive magnetic force, the sinusoidal drivingFig. 1. Schematic diagram of the physical pendulum and the forces actingon it.Fig. 2. ~a! Time series and ~b! phase space plots showing periodic motionfor d570 mm. ~c! Time series and ~d! phase space plots showing morecomplicated periodic orbits when the distance, d, between the magnets isdecreased to 66.70 mm.394 394Am. J. Phys., Vol. 65, No. 5, May 1997 Siahmakoun, French, and Pattersonforce applied by horizontally displacing the pivot, and thedamping force. Figure 1 shows a diagram of the pendulumand the forces acting on it. The damping force is assumed tobe proportional to the angular velocity of the pendulum,v.The magnets are considered to be point magnets. For sim-plicity of our theoretical model, we assume that the magneticforce is a repulsive force between two point magnets with aninverse squared dependence on distance, and it is thus afunction of the angular displacement of the pendulumu,Fmagnetic5m04pm1m2ru2rˆ. ~1!Here,m0is the permeability of vacuum, m1,m2are the polestrengths of each magnet, and ruis the distance between themagnets.As can be seen in Fig. 1,ru5A~L sinu!21 hu2, ~2!hu5 d1 L~12 cosu!, ~3!where L is the length of the pendulum and d is the minimumseparation between the two magnets. The horizontal dis-placement of the pivot is negligibly small compared to L andhu. Newton’s second law is applied to the rotating rigid bodyand thus the equation of motion takes the form given belowFig. 3. As the distance d is further decreased, transition to chaos occurs neard566.55 mm. ~a! Time series and ~b! phase space plots showing chaoticmotion for d566.55 mm.Fig. 4. ~a! Time series and ~b! phase space plots for d566.54 mm and initialconditionsu~0!50.1