ABSTRACT

Example Harmonic and Anharmonic Oscillators Demonstrated and Analyzed

I easily modified a commercially available teaching apparatus to make spring and magnetic “hardening” oscillators. That apparatus is demonstrated with position (time) data collected and graphed. Simple approximate fits reveal the character of their dissipations in addition to the anharmonic oscillator's expected period variation and transition to harmonicity. Phase plotting makes the latter especially obvious. I also numerically modeled the spring oscillator with a simple leapfrog algorithm using the force constant found experimentally.

Introduction

While studying the motion of a spring oscillator cart supported pendulum, I noticed the cart’s interesting behavior when nearing the magnetic bumpers I had used as spring fixtures. I realized, with some improvement, I would have a useful hardening oscillator. After developing that apparatus, I made a similar Hookian one for comparison and presented both at the most recent SCAAPT meeting at CSUCI.

The Apparatuses

Both oscillators consist of low friction carts running on a track. The linear (spring) oscillator uses similar springs stretched between the cart and hooks in wood blocks clamped to adjustable brackets fastened to the track to supply the restoring force. The hardening (magnetic repulsion) oscillator uses cylindrical magnets mounted in the cart and in wood bumpers bolted to the track to supply its restoring force. The carts’ masses plus bolted steel cubes supply the necessary inertias. The position of the carts is determined by a linearized rotary motion sensor (optical encoder). The RMS is fastened to a bracket bolted to the track. A heavy thread loop clamped to the cart, looped over the RMS’s pulley, and a small pulley on a bracket at the opposite end of the track transmits the motion of the cart to the RMS. The two adjacent photographs illustrate the preceding.

Plate One, The Hookian (linear spring) Oscillator

Plate Two, The Magnetic (non-linear hardening) Oscillator

Note: All of the commercial apparatus is from Vernier, except the cart clamp, pulley and its bracket, which are from Pasco. I salvaged the eight magnets from linear generator powered flashlights from RAFT. I shimmed the magnets to prevent their movement, because I’d used RTVR for easy removal.

Data Collection and Display

I used Vernier’s LabQuest interface and LoggerPro for data collection, analysis, and display with one exception. Various plotted data sets adjacent illustrate typical behavior of the oscillators.

Notes

My intention was qualitative only, though one may compare, for example, the measured period and that calculated from the mass (Plate 3. 0.66 kg) and a spring constant of 243 N/m. See plate seven.

The fits shown in plate three reveal a likely, and expected, small proportion of Coulomb damping in addition to viscous, as shown by the better fit. In addition, as noted, the oscillator is not strictly linear, but a very slightly hardening one. Plate four shows the expected period change due to hard repulsion, and also suggests a dissipation mixture. To obtain the triple fits in plates five and six I exported the data to Kaleidograph, which is not limited to five fitting coefficients. I used a Vernier force probe to collect the data displayed in plate seven. Plate nine shows the algorithms I used for spring oscillator modeling with a resulting graph.

Conclusion

By stealing from the Freshman Lab. and simple addition and modification, I think, one may create apparatus suitable for an interesting study of oscillators at the intermediate lab. level.

Further

I intend to add turbulent damping to an oscillator and numerically model it, and to find the bugs in my coulomb and magnetic restoring force algorithms.

Plate Three, Spring Oscillator with Two fits

Plate Four, Magnetic Oscillator with Two fits

Plate Five, Magnetic Oscillator with Triple Fit

Plate Six, Magnetic Oscillator Position(time) and Speed

Plate Seven, Spring and Magnetic Restoring Forces

Plate Eight, Phase Plots

Plate Nine, Leapfrog Algorithm and Example Plot