1 / 8

Dynamic Equilibrium

Dynamic Equilibrium. Orbital Potentials. Kepler orbits involve a moving system. Effective potential reduces to a single variable Second variable is cyclic. V eff. r 0. r. r 0. r. q. A perturbed orbit varies slightly from equilibrium. Perturbed velocity

fraley
Download Presentation

Dynamic Equilibrium

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. Dynamic Equilibrium

  2. Orbital Potentials • Kepler orbits involve a moving system. • Effective potential reduces to a single variable • Second variable is cyclic Veff r0 r r0 r q

  3. A perturbed orbit varies slightly from equilibrium. Perturbed velocity Track the difference from the equation of motion Apply a Taylor expansion. Keep first order Small perturbations are stable with same frequency. Radial Perturbation

  4. Modified Kepler • Kepler orbits can have a perturbed potential. • Not small at small r • Two equilibrium points • Test with second derivative • Test with dr Veff r0 r rA stable unstable

  5. An inverted pendulum may have an oscillating support. Driving frequency W Moment of inertia I The apparent acceleration of gravity is adjusted by the oscillation. Inverted Pendulum m l

  6. Substitute variables to get a standard form. a compares natural frequency to driving frequency q is relates to the amplitude of oscillation a0/W2 x is a dimensionless time variable Mathieu’s Equation

  7. The Mathieu equation is soluble as an infinite series. Infinite Fourier series Solutions are unstable for real or complex m. Builds up exponentially Purely imaginary m has stable motion. Dominant term n = 0 Fundamental frequency w’ Infinite Series

  8. There are an infinite set of stable regions. Mirror behavior in negative q There is well defined for q = 0. Stable for a > 0 (normal pendulum) Unstable for a < 0 (inverted pendulum) Stability Regions next March and Hughes, Quadrupole Storage Mass Spectrometry

More Related