1 / 11

Understanding Oscillatory Systems: Linear vs Nonlinear Restoring Forces

Explore the integration of equations of motion for parabolic trajectories, from linear restoring forces to nonlinear springs. Learn about the Ideal Gas Law in springs of air and Taylor Series Expansions for stability analysis.

hanzila
Download Presentation

Understanding Oscillatory Systems: Linear vs Nonlinear Restoring Forces

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. Real Oscillators … constant forces  integrate EOM  parabolic trajectories. … linear restoring force  guess EOM solution  SHM … nonlinear restoring forces  ? nonlinear spring? linear spring F F x x

  2. 0 0 The spring of air : use Ideal Gas Law: PV=NRT Patm chamber volume: V=Ax WTF! (whoa there, fella) m EOM A +x Stable Equilibrium at xeq = NRT / (mg + APatm) P, V 0

  3. Taylor Series Expansions: Turns a function into a polynomial near x = a Example:

  4. Expand around x = -3: 2nd order 0th order 1st order

  5. Expand around x = 2: 0th order 1st order 2nd order

  6. Expand NRT/x around xeq: Is it safe to linearize it? Better check a unitless ratio. How about: (Yes, excellent choice Dr. Hafner!)

  7. .. Displacement 5% of xeq: 0 .05 .0025 …. Perhaps you would prefer…. SHM with

  8. -x Simple Pendulum: Stable Equilibrium: Length: L Mass: m Q Displace by Q: mg cosQ T mg cosQ sinQ mg cosQ EOM: mg Expand it! mg

  9. Derivatives:

  10. Now express as a unitless ratio of the dependent variable and some parameter of the system: Displacement 5% of length: 0 .05 0 .0000625 … SHM with

  11. The world is not linear. However, one can use a Taylor expansion to linearize an EOM by assuming only small perturbations around a point of stable equilibrium (which may not be the origin).

More Related