Analyzing the Motion of CoupledOscillators Using the WiiMotion Plus

Romulo Ochoa, Cris R. Ochoa, and Michael Erickson Department of Physics, The College of New Jersey, Ewing, NJ 08628

IntroductionThe Wii console utilizes a very powerful controller for game-playing. The Wii Motion Plus was introduced in May of 2010. In addition to the accelerometers and Bluetooth interface found in regular Wiimotes, it contains gyroscopes that measure the rate of rotation along 3-axes, X (pitch), Y (roll), and Z (yaw), up to ±2,000°/s. Open source code, such as GlovePie (http://glovepie.org ) ,allows PCs with Bluetooth capability to detect the information sent out by the controller. Up to three Wiimotes are used simultaneously to measure the motion of a playground swing, a compound pendulum and of coupled pendulums. Results of our experiments compare well with those predicted by Newtonian mechanics.

Experimental MethodsPendulum and playground swing

A wooden support was built to hold the Wiimote safely in place. The setup was fastened to a pulley so it could swing freely on a vertical plane. Angular velocity data were collected for small angles.

A Wiimote was fastened to one of us (CO), at waist level, while on a playground swing and data was collected under various initial conditions. The simplest experiment was releasing the swing at small initial angles and comparing the measurements with those obtained from simple pendulum calculations.

Coupled pendulum motion

Two pendulums supports were coupled using a weak spring (k = 3 N/m) as shown in Figure 1. The two Wiimotes reported the angular velocities of their corresponding pendulum. The studies were extended to three pendulums coupled by two soft springs.

ResultsPendulum and playground swing

Figure 2 shows a sample case of the angular velocities and positions of the Wiimote for a playground swing. We measured the period of motion of the playground swing to be 2.82 ± 0.01 s. The period of a simple pendulum is given by:

where l is the distance from the mass to the pivot point. In our case l = 2.0 m gives a period of 2.8 ± 0.15 s, in agreement with the period found above..

The angular positions are obtained by the Wiimote from the measured angular velocities. The offset observed was reported by the Wiimote itself. Although the graph correctly displays the 90o phase shift between the angular positions and velocities, it incorrectly reports an asymmetry in the swing motion. This is a limitation of the Wiimote’s ability to calculate angular positions. We have found the device unreliable for reporting angular positions, especially when some abrupt motion occurs, while the angular velocities are very stable and correctly reported. Coupled pendulums

Figure 3 shows the out of phase motion and beats of two pendulums. Initially one of the pendulum was displaced from equilibrium while the other was held stationary. The alternating motion of the two pendulum is clearly visible. Figure 4 shows a typical normal mode of two coupled oscillators, that is 180o out of phase oscillation. Figure 5 displays the motion of three pendulums when givien random initial conditions. A complex motion is observed. Normal modes have also been observed but are not shown.

AbstractThe WiiMotion Plus has a dual-axis tuning fork gyroscope and a single-axis gyroscope in addition to three accelerometers. The gyroscopes can measure rotational velocities up to 2000 degrees/s. We present experiments that use the gyroscopes capabilities to follow the motion of a single pendulum, a playground swing, and of coupled pendulums.

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Figure 2. Plots of playground swing motion. The solid line represents the angular velocity and the dashed line the angular position, both as function of time. Notice the 90o phase difference between the two data sets.

Figure 3. Angular velocities of two coupled pendulums demonstrate motion replicates beats.

Figure 4. Graph shows two coupled pendulum motion. The pendulum are moving at 180 degrees out of phase with a slight exchange of energy.

Figure 1. Wii controllers in wooden supports as set up for coupled oscillator motion.

Figure 5. Motion of three coupled pendulum with random initial conditions.

ConclusionsThe experiments demonstrate that the gyroscopes in the Wii MotionPlus are tools that can be very powerful in physics experiments. The Bluetooth communications gives users freedom to design multiple experiments both in the lab and equivalent experiences in a playground environment.

ReferencesS. Van Hook, A. Lark, J. Hodges, E. Celebrezze, and L. Channels, “Playground Physics Determining the Moment of Inertia of a Merry-Go-Round,” Phys. Teach. 45, 85 -87 (2007).R. Taylor, D. Hutson, W. Krawiec, J. Ebert, and R. Rubinstein, “Computer physics on the playground,” Phys. Teach. 33, 332–337 (1995).R. Ochoa, F. G. Rooney, and W. J. Somers, “Using the Wiimote in Introductory Physics Experiments,” Phys. Teach. 49, 16-18 (2011) and references therein.A. Kawam and M. Kouh, “Wiimote Experiments: 3-D Inclined Plane Problem for Reinforcing the Vector Concept,” Phys. Teach. 49, 508

(2011).M. Erickson, C. R. Ochoa, and R. Ochoa, “The Wiimote in the Playground,” Phys. Teach. Accepted for publication.

2013 AAPT Winter Meeting New Orleans, LA, January 5 -9, 2013

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