Resonance Threshold and Bistability in a Parametrically-Driven Pendulum

POSTER

Abstract

We examine the steady-state energy of a weakly-damped pendulum driven into parametric resonance. Motion control and data acquisition of the pendulum are accomplished using a micro-controller, which we use to record the passage time of the pendulum through its equilibrium position and obtain the maximum speed per oscillation as a function of time. As examples of the interesting physics which the experiment reveals, we present contour plots depicting the energy of the system as functions of driven frequency and modulation depth, as well as the bistability of the system for various modulation depths. We observe the transition to steady state oscillation and compare the experimental oscillation threshold with theoretical expectations. The equations of motion can be derived from first principles, and the cost-friendly and easily constructed apparatus makes this robust experiment a method for undergraduate students to thoroughly understand damped harmonic motion and parametric amplification in an upper-division mechanics class environment.

Authors

  • Alison Huff

    Univ of California - Merced

  • C. Jennings

    Stanford University, None, none, Univ of California - Merced, Dr. Remeis-Sternwarte & ECAP, Universität Erlangen-Nürnberg, NASA GSFC, Lawrence Livermore Natl Lab, Lawrence Livermore National Laboratory, Goddard Space Flight Center-NASA, University of Nevada, Reno, Cal Poly - San Luis Obispo, Victoria University of Wellington, Wellington 6021, New Zealand, Penn State, Micron School of Materials Science & Engineering, Department of Physics, Boise State University, Autonomous University of Zacatecas, Micron School of Materials Science & Engineering, Boise State University, Naval Research Laboratory, Independent Researcher, Cornell University, UC Santa Cruz, Middle Georgia State University, University of California, Merced, Stanford University, California, Institute of Medical Biology, Singapore; Stanford University, California, University of California Merced, Kent State University, ORNL, UC Berkeley, LLNL, Physics, Hokkaido University, Japan, Physics, UC San Diego, Physics, California State University, Fresno, Department of Physics and Astronomy, California State University Long Beach, Long Beach, California 90840, USA, Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA, Department of Physics, Southern University of Science and Technology, Shenzen 518055, China, Virginia Tech, Sotera Defense Solutions, Inc., Univ of California - Santa Cruz, College of Staten Island, UC Irvine, Naval Research Laboratory, Washington, DC 20375, Virginia Tech, Blacksburg, VA 24061, University of California, Merced CA 95343

  • C. Jennings

    Stanford University, None, none, Univ of California - Merced, Dr. Remeis-Sternwarte & ECAP, Universität Erlangen-Nürnberg, NASA GSFC, Lawrence Livermore Natl Lab, Lawrence Livermore National Laboratory, Goddard Space Flight Center-NASA, University of Nevada, Reno, Cal Poly - San Luis Obispo, Victoria University of Wellington, Wellington 6021, New Zealand, Penn State, Micron School of Materials Science & Engineering, Department of Physics, Boise State University, Autonomous University of Zacatecas, Micron School of Materials Science & Engineering, Boise State University, Naval Research Laboratory, Independent Researcher, Cornell University, UC Santa Cruz, Middle Georgia State University, University of California, Merced, Stanford University, California, Institute of Medical Biology, Singapore; Stanford University, California, University of California Merced, Kent State University, ORNL, UC Berkeley, LLNL, Physics, Hokkaido University, Japan, Physics, UC San Diego, Physics, California State University, Fresno, Department of Physics and Astronomy, California State University Long Beach, Long Beach, California 90840, USA, Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA, Department of Physics, Southern University of Science and Technology, Shenzen 518055, China, Virginia Tech, Sotera Defense Solutions, Inc., Univ of California - Santa Cruz, College of Staten Island, UC Irvine, Naval Research Laboratory, Washington, DC 20375, Virginia Tech, Blacksburg, VA 24061, University of California, Merced CA 95343

  • C. Jennings

    Stanford University, None, none, Univ of California - Merced, Dr. Remeis-Sternwarte & ECAP, Universität Erlangen-Nürnberg, NASA GSFC, Lawrence Livermore Natl Lab, Lawrence Livermore National Laboratory, Goddard Space Flight Center-NASA, University of Nevada, Reno, Cal Poly - San Luis Obispo, Victoria University of Wellington, Wellington 6021, New Zealand, Penn State, Micron School of Materials Science & Engineering, Department of Physics, Boise State University, Autonomous University of Zacatecas, Micron School of Materials Science & Engineering, Boise State University, Naval Research Laboratory, Independent Researcher, Cornell University, UC Santa Cruz, Middle Georgia State University, University of California, Merced, Stanford University, California, Institute of Medical Biology, Singapore; Stanford University, California, University of California Merced, Kent State University, ORNL, UC Berkeley, LLNL, Physics, Hokkaido University, Japan, Physics, UC San Diego, Physics, California State University, Fresno, Department of Physics and Astronomy, California State University Long Beach, Long Beach, California 90840, USA, Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA, Department of Physics, Southern University of Science and Technology, Shenzen 518055, China, Virginia Tech, Sotera Defense Solutions, Inc., Univ of California - Santa Cruz, College of Staten Island, UC Irvine, Naval Research Laboratory, Washington, DC 20375, Virginia Tech, Blacksburg, VA 24061, University of California, Merced CA 95343

  • C. Jennings

    Stanford University, None, none, Univ of California - Merced, Dr. Remeis-Sternwarte & ECAP, Universität Erlangen-Nürnberg, NASA GSFC, Lawrence Livermore Natl Lab, Lawrence Livermore National Laboratory, Goddard Space Flight Center-NASA, University of Nevada, Reno, Cal Poly - San Luis Obispo, Victoria University of Wellington, Wellington 6021, New Zealand, Penn State, Micron School of Materials Science & Engineering, Department of Physics, Boise State University, Autonomous University of Zacatecas, Micron School of Materials Science & Engineering, Boise State University, Naval Research Laboratory, Independent Researcher, Cornell University, UC Santa Cruz, Middle Georgia State University, University of California, Merced, Stanford University, California, Institute of Medical Biology, Singapore; Stanford University, California, University of California Merced, Kent State University, ORNL, UC Berkeley, LLNL, Physics, Hokkaido University, Japan, Physics, UC San Diego, Physics, California State University, Fresno, Department of Physics and Astronomy, California State University Long Beach, Long Beach, California 90840, USA, Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA, Department of Physics, Southern University of Science and Technology, Shenzen 518055, China, Virginia Tech, Sotera Defense Solutions, Inc., Univ of California - Santa Cruz, College of Staten Island, UC Irvine, Naval Research Laboratory, Washington, DC 20375, Virginia Tech, Blacksburg, VA 24061, University of California, Merced CA 95343

  • Jay Sharping

    Univ of California - Merced, University of California at Merced, University of California Merced, University of California, Merced