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

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Presentation transcript:

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

WTF! 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

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

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

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

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

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

Beware of assumptions and approximations!

Clerk = person who derived an approximation Clouseau = You Clerk = person who derived an approximation Dog = domain of interest Clerk’s dog = domain of his approximation Bite = result of mistaken assumption that you are working within the Clerk’s domain

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

Derivatives:

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

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).