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Notes on the underdamped harmonic oscillator {{courses:lecture:oslec:damped_oscillator.ppt|}}




===== The damped harmonic oscillator =====

  * The damped harmonic oscillator is an extension of the previous discussion by the addition of another force term to model damping.

  * Clearly identify "the forces", the restoring force, \[F\left( x \right)=-m\omega _{0}^{2}x\] and and the damping force, \[F_{d}\left( x \right)=-b \dot{x}\].  Set the sum equal to the acceleration, \(a=\frac{d^{2}x}{dt^{2}}\). (Check that the \(\dot{x}\) notation is familiar).

  * Discuss other types of damping forces briefly. Also metion overdamped and critically damped cases, but focus on underdamped case.

  *  Solve the differential equation postulating a solution \(Ce^{pt}\), and obtain a solution in the "C-form" with \(e^{\pm i\left( \omega _{0}+i\beta \right)t}\) terms. Find exponential damping, discuss limits etc. 

  * Class discussion about relationship to the pendulum laboratory.  Is this a good model of that damping?

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