Chaos in a physical pendulum
The driven damped physical pendulum is a classic problem in computational physics. This physical model of that system uses an optical encoder for position measurement and a Helmholtz coil driving a dipole magnet on the pendulum axis for the drive. Coil current, timing, and position sensing are all accomplished via a microcontroller which then sends all information to a computer via serial line at 256 time intervals on the drive waveform.
One can adjust the drive frequency and amplitude as well as the damping coefficient. At some drive parameters, the motion of the particle becomes chaotic. Interesting investigations could include creation of Poincare plots of the strange attractor, mapping bifurcations, and determination of Lyapunov exponent at different drive parameters.
This apparatus was designed and built by Eric Ayars and Brandon Thacker. Brandon (Class of '16) won the inaugural AAPT/ALPhA Award -$4000 plus travel to the national AAPT meeting- for best new lab equipment built by an undergraduate for his work on this apparatus. The prototype worked well, and a production version is now available. A LabVIEW application to run the entire apparatus is now available, see link below.
Reading
- Lab Manual for the apparatus
- Computational Physics 2e by Giordano and Nakanishi, section 3.3 and related sections.
- A unit on oscillations, determinism and choas for introductory physics students
- Read enough of one of the books on chaos in 108 that you know what all the terms in the above description mean.
Questions
- What is phase space, and what should a phase space plot look like for a pendulum in a stable repeating 'orbit'?
- What is a Poincare plot?
- What are bifurcations, and why are they interesting in the context of chaos?
Location
Room 108, Cabinet 108.052.
Equipment needed
- Mechanical Chaotic Oscillator (MCO)
- 0-12VDC, 1A power supply
- +/-12VDC, 2A bipolar power supply
- LabVIEW Controller code
Hazards
No serious hazards. The heat-sink fins on the controller board can become uncomfortably (but not dangerously) hot.