1 / 6

Inverted Pendulum

Inverted Pendulum. 2008 Summer Dan Brunski. Theory. Mathieu equation. A pendulum with an oscillating base can be described by the Mathieu equation Regions of stability exist in the Mathieu equation for δ < 0 These regions correspond to an inverted pendulum position.

malcolm-larsen
Download Presentation

Inverted Pendulum

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Inverted Pendulum 2008 Summer Dan Brunski Inverted Pendulum

  2. Theory Mathieu equation • A pendulum with an oscillating base can be described by the Mathieu equation • Regions of stability exist in the Mathieu equation for δ < 0 • These regions correspond to an inverted pendulum position Regions of stability of the Mathieu equation Regular pendulum Inverted pendulum Inverted Pendulum

  3. Equation Development b(t) • The coordinates of a pendulum with mass m on an oscillating base b(t) are given by • The Lagrangian is then • Using the Euler-Lagrange equation and substituting in b(t), the equation of motion is • Making the following substitutions, the Mathieu equation is obtained Inverted Pendulum

  4. What Didn’t Work • Pendulums powered by either Black and Decker jigsaw or Dewalt reciprocating saw • Pendulum dimensions: • 16-1/4” x 1-1/4” x 1/8” • Length measured from pivot center to end • Lengths down to 6-3/4” • Tried adding weights to various places • Either the frequency wasn’t high enough, amplitude not great enough, modes of vibration in unintended directions, or a function of all three Inverted Pendulum

  5. What Worked • Short piece of straw (44.5mm from pivot to end) on a Pasco SF-9324 wave driver • Foam pad under wave driver to keep noise down and stop it from bouncing around • Function generator signal amplified to wave driver limit of 16Vpp • Pivot made from 0.021” steel wire welded to a washer • Confirmed stable settings: • 41Hz – 77Hz • 16Vpp ± 0.5V • 5.39Vrms ± 0.1V • 377mA ± 1mA • 4mm ± 0.5mm amplitude Inverted Pendulum

  6. Wiring Schematic Function Generator Amplifier Frequency Counter Ammeter Voltmeter Wave Driver Oscilloscope • Ammeter, Voltmeter, and oscilloscope are not necessary, but recommended to prevent damaging the wave driver Inverted Pendulum

More Related