Demonstrations with a magnetically controlled pendulumYaakov Kraftmakhera�

Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel

�Received 7 November 2007; accepted 1 December 2009�

A magnetically controlled pendulum is used to demonstrate self-excited oscillations, parametricexcitation, and multiple-well potential oscillations. Except for the data-acquisition system, theapparatus is simple and inexpensive. © 2010 American Association of Physics Teachers.

�DOI: 10.1119/1.3276412�

I. INTRODUCTION

Recently, a magnetically controlled pendulum was used insome experiments on nonlinear dynamics.1 The same pendu-lum can be used in many other experiments, some of whichare described in this paper.

The magnetically controlled pendulum consists of a thin12 cm long aluminum rod with two permanent magnets at-tached to its ends, as shown in Fig. 1. The pivot of the rod,located 4 cm below the top of the rod, is attached to the shaftof a PASCO Rotary Motion Sensor �CI-6538�. The centers ofthe upper and lower magnets are positioned 2.5 cm aboveand 7.5 cm below the pivot, respectively. The magnets arestrong ceramic magnets 1.2�1.2�1.2 cm3 in size. Themagnetic dipoles can be aligned parallel or perpendicular tothe axis of the pendulum. An aluminum disk, 1.2 cm thickand 4 cm in diameter, attached to the other end of the shaft ofthe Rotary Motion Sensor can be subjected to the magneticfield of another permanent magnet. This magnet modifies thedecay constant of the pendulum.

External magnetic fields can be used to drive the pendu-lum and/or to modify the restoring torque. Properly orientedmagnets and current-carrying coils are used for this purpose.The potential energy U of a magnetic dipole m in a magneticfield B is equal to

U = − m · B . �1�

Since the magnetic fields produced by coils can be calculatedor measured, the forces and torques acting on the pendulumcan be determined. To calculate the effect of an externalmagnetic field, a magnetic dipole can be considered as twoseparated magnetic charges. This approach is recommended,e.g., by Griffiths.2 With this model, one can qualitativelypredict the driving torque and the changes in the restoringtorque of the pendulum. In our case, the magnetic fields con-trolling the pendulum are inhomogeneous, and the propertiesof the pendulum strongly depend on the arrangement of thesetup.

The SCIENCEWORKSHOP data-acquisition system and theDATASTUDIO software from PASCO Scientific3 are used fordata collection and data display. The phenomena demon-strated are described in many textbooks.4–10

II. SELF-EXCITED OSCILLATIONS

With positive feedback, the oscillations of the pendulummay become self-excited like those of a pendulum clock orof some chaotic toys �Ref. 5, p. 323; Ref. 6, p. 284�.

Two methods can be used to compensate the energy lossesin the system. The first method relies on positive feedbackprovided by magnetic induction. An 800 turn driving coil�PASCO, SF-8611� with a magnetic core is positioned 2 cm

above the upper magnet, as shown in Fig. 2�a�. The resis-tance of the coil is about 10 �. The axes of both magneticdipoles are aligned horizontally. A 3200 turn coil �SF-8613�,positioned 2 cm beneath the lower magnet, provides a feed-back voltage with an amplitude of about 0.3 V. The feedbackvoltage is amplified with PASCO’s Digital FunctionGenerator-Amplifier �PI-9587C� and used to drive the driv-ing coil. The sign of the feedback depends on the polarity ofthe two coils. The equation of motion of the pendulum withpositive or negative feedback can be presented in a formsimilar to the Van der Pol equation,

�� + �� � ������� + �02� = 0, �2�

where � depends on the feedback coefficient. A positive ornegative sign in this equation implies negative or positivefeedback, respectively. Equation �2� is a second-order differ-ential equation with a linear restoring force and nonlineardamping, which exhibits a limit cycle behavior �Ref. 6, p.268�. It can be solved numerically.11 The oscillations becomeself-excited when

� − ���� = 0. �3�

To trigger the oscillations, the pendulum is displaced slightlyfrom its equilibrium position. After being released, the am-plitude of the oscillations gradually grows and reaches asteady value. This steady-state motion is called the Poincarélimit cycle �Ref. 6, p. 15; Ref. 10, pp. 153–154�. The limit

Fig. 1. The setup of the magnetically controlled pendulum used to produceself-excited oscillations with a reed switch.

532 532Am. J. Phys. 78 �5�, May 2010 http://aapt.org/ajp © 2010 American Association of Physics Teachers

cycle depends on the energy losses to be balanced. When thependulum is initially positioned at an angle beyond the limitcycle, its oscillations gradually decay until the same limitcycle is reached. The coefficient � decreases with increasing�, and the system reaches the limit cycle due to this nonlin-earity. The existence of limit cycles shows that the nonlin-earity is an inevitable feature of self-excited oscillations.

The second method used to compensate for the energylosses uses a reed switch to provide suitable current pulses tothe driving coil, as shown in Fig. 2�b�. The switch consists ofa pair of contacts on ferrous metal reeds in a hermeticallysealed glass envelope. The state of the switch depends on theexternal magnetic field. We use a dry reed switch �As-semTech GC 3817� that is open when the magnitude of theexternal magnetic field is small. When this method is used,the axis of the lower magnetic dipole is aligned vertically.The switch is located 2 cm away from the equilibrium posi-tion of the pendulum, as shown in Fig. 1, and is triggered bythe lower magnet of the pendulum. The switch is connectedin series with a dc power supply and the driving coil. Everytime the pendulum reaches its lowest point, the switch isclosed and an electric current of about 0.3 A passes throughthe driving coil. The switch breaks the current when the pen-dulum moves away from its lowest point. A 5 �F capacitor,shunting the driving coil, suppresses emf pulses in the driv-ing coil when the switch opens.

To start the oscillations, the dc supply is switched on whenthe pendulum is in its lowest position. If the reed switchstands too far from the pendulum, one needs to push thependulum toward the switch. The amplitude of the oscilla-tions gradually grows until a steady state is reached. Thisamplitude depends on the position of the reed switch and theamplitude of the current pulses passing through the drivingcoil. The proper position of the switch and the suitable volt-age of the dc supply can be found with several trials. Thisapproach is probably the simplest method to balance the en-ergy losses. A drawback of this approach is the limited sta-

Fig. 2. Schematic setup of the demonstrations described in this paper:��a� and �b�� Self-excited oscillators, �c� parametric excitation, and �d� two-and three-well potential oscillations.

Fig. 3. Routes to the limit cycle of oscillators with a feedback coil �top� and a reed switch �bottom�.

533 533Am. J. Phys., Vol. 78, No. 5, May 2010 Yaakov Kraftmakher

bility of the reed switch: The electrical pulses in the drivingcoil are not exactly regular. Nevertheless, the results ob-tained are quite satisfactory.

For both methods, the routes to the limit cycle are easy toobserve, as can be seen in Fig. 3.

III. PARAMETRIC EXCITATION

An oscillator is parametric if its motion obeys the so-called Hill’s equation �Ref. 8, p. 418; Ref. 9, p. 389�,

x� + G�t�x = 0, �4�

where G�t� is a periodic function. An interesting example is amotion that satisfies the Mathieu equation �Ref. 4, p. 81–82;Ref. 8, p. 426�,

x� + �02�1 + h cos �t�x = 0, �5�

where �0 is the natural frequency of the oscillator. It is as-sumed that h1.

Continuous oscillations can be attained by periodicallyvarying the natural frequency of the oscillator. With paramet-ric excitation, the system acquires additional energy, and theoscillations become continuous when the gain from the para-

metric process outweighs the dissipative losses. Parametricoscillations are possible when the frequency of the excitationis close to

� = 2�0/n , �6�

where n is an integer.The frequency range � for which parametric excitation

is possible depends on n. For n=1, the frequency range �is maximized and the necessary value of h can be minimized.Parametric oscillations occur when the system is displacedfrom its equilibrium position. Experiments on parametric ex-citation have been described in a number of papers.12–14

In our case, it is easy to periodically change the restoringtorque by using an ac magnetic field, as shown in Fig. 2�c�. A320 turn coil, about 14 cm in diameter, is positioned 2 cmbeneath the pendulum and connected to PASCO’s PowerAmplifier II �CI-6552A� controlled by the Signal Generatorincorporated into the SCIENCEWORKSHOP 750 Interface. Thedipole of the lower magnet of the pendulum is directed ver-tically. The field produced by the lower coil modifies therestoring torque and thus the natural frequency of the pendu-lum. The upper magnet is not required for this experiment.With the data-acquisition system, the time series and phaseplane plots are measured, as well as the phase relations be-tween the current in the coil and the angular position. Totrigger the parametric oscillations, the pendulum should bedisplaced from its equilibrium point and then released. In ourmeasurements, different frequencies obeying Eq. �6� weretried. Parametric oscillations are readily observable for nequal to 1 and 2, as shown in Fig. 4.

IV. TWO- AND THREE-WELL POTENTIALOSCILLATOR

When the lower current-carrying coil produces a repulsiveaction on the lower magnet, installed with its dipole momentoriented vertically as shown in Fig. 2�d�, it is possible tocreate an oscillator with two or three potential wells. Tomake the two wells observable, it is enough to increase thedc current in the 320 turn coil up to about 1 A. For a specificlocation of the coil under the pendulum, the equilibrium po-sitions of the pendulum depend on the dc current in this coil.

Fig. 4. Parametric oscillations of the pendulum. The Lissajous patterns,showing angular position versus driving current, can be used to determinethe frequency ratio.

Fig. 5. Examples of free oscillations of the two-well potential pendulum.Time series and phase plane plots for two different decay constants areshown.

Fig. 6. Examples of free oscillations of the three-well potential pendulum.Time series and phase plane plots for the three equilibrium positions areshown.

534 534Am. J. Phys., Vol. 78, No. 5, May 2010 Yaakov Kraftmakher

Free oscillations of the pendulum decay to one of two equi-librium positions, as shown in Fig. 5. Magnetic damping isused to increase the decay constant.

The repulsive action of the lower coil on the lower magnetdepends on the gradient of the magnetic field. When thelowest position of the magnet coincides with the center ofthe coil, where the gradient of the magnetic field vanishes,the vertical position of the pendulum also becomes an equi-librium position. In this case, the pendulum has three equi-librium positions: The vertical position and two positions oneither side of the vertical. The system thus becomes a three-well oscillator. Free oscillations of the three-well pendulumdecay to one of the three equilibrium positions, as shown inFig. 6. For the example shown in Fig. 6, the current in the320 turn coil was 0.5 A.

When using the upper magnet and the driving coil aboveit, forced oscillations of the pendulum are also observable.

a�Electronic mail: krafty@mail.biu.ac.il1Y. Kraftmakher, “Experiments with a magnetically controlled pendulum,”Eur. J. Phys. 28, 1007–1020 �2007�.

2D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. �Prentice-Hall,Upper Saddle River, NJ, 1999�, p. 258.

3PASCO Scientific, 10101 Foothills Blvd., Roseville, CA 95747-7100,�htpp://www.pasco.com�.

4L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. �Pergamon, Oxford,1976�.

5A. B. Pippard, The Physics of Vibration �Cambridge U. P., Cambridge,1978�.

6F. C. Moon, Chaotic Vibrations �Wiley, New York, 1987�.7R. C. Hilborn, Chaos and Nonlinear Dynamics �Oxford U. P., New York,1994�.

8J. V. José and E. J. Saletan, Classical Dynamics: A Contemporary Ap-proach �Cambridge U. P., Cambridge, 1998�.

9L. N. Hand and J. D. Finch, Analytical Mechanics �Cambridge U. P.,Cambridge, 1998�.

10S. T. Thornton and J. B. Marion, Classical Dynamics of Particles andSystems, 5th ed. �Brooks/Cole, Belmont, CA, 2004�.

11 W. F. Drish and W. J. Wild, “Numerical solutions of Van der Pol’s equa-tion,” Am. J. Phys. 51, 439–445 �1983�.

12L. Falk, “Student experiments on parametric resonance,” Am. J. Phys.47, 325–328 �1979�.

13W. Case, “Parametric instability: An elementary demonstration and dis-cussion,” Am. J. Phys. 48, 218–221 �1980�.

14F. L. Curzon, A. L. H. Loke, M. E. Lefrançois, and K. E. Novik, “Para-metric instability of a pendulum,” Am. J. Phys. 63, 132–136 �1995�.

Electric Wind. This demonstration device shows that the candle flame is rich in positive ions. The positive terminalof an electrostatic machine is connected to the sharp point, and the resulting electrostatic force pushes the flame awayfrom the point. This demonstration can be easily done today. The apparatus in the picture is at Union College inSchenectady, New York, and has been adapted from a universal discharger by removing the central pillar and the pointon the right. �Photograph and Notes by Thomas B. Greenslade, Jr., Kenyon College�

535 535Am. J. Phys., Vol. 78, No. 5, May 2010 Yaakov Kraftmakher