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Lecture 6: Driven Coupled Oscillators

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Lecture Topics

A red and a purple square sit on a metal strip.
  • Solving Linear Equations
  • Driven Coupled Oscillators
  • Amplitude Ratios in Normal Modes

Learning Objectives

By the end of this lecture, you should:

  • derive the (small angle) motion of the double pendulum.
  • use Cramer's Rule to solve an algebraic system.
  • modify the equation of motion of the double pendulum to incorporate driving.
  • find the solutions for the steady state driven double pendulum.
  • understand how to find a general solution for a coupled system.
  • understand the relationship of driving, transients, steady state, and normal modes.
  • describe the normal modes of the triple pendulum.

Lecture Activities

Check Yourself

  • A child sitting on a swing set has the ability to drive her dangling legs, and to some extent her body, to "pump" the swing. How does the child get the swing to go higher?

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  • It turns out that Cramer's rule is not a preferred method in linear algebra, especially with large sets of equations being solved on computers. For moderate sized systems as treated in this course, it is useful because it is basically:

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  • Pendulums usually have a large Q factor since there is little friction and a low damping rate γ, allowing large values for Q = \frac{{{\omega _0}}}{\gamma }. For high Q, the amplitude in the system is much larger than the driving amplitude. Why, then, is it difficult to demonstrate the resonant phenomena discussed in the lecture?

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