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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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A swing will move higher if the legs are drawn back under the child when at the farthest back position, and pushed out ahead when at the front of the swinging motion. This means that the forced frequency of the legs is exactly that of the swing. In this situation, the driving is resonant and causes the amplitude of the swing to become maximized.

 

  • 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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A set of rules for computing results based on determinants, which are in turn readily calculated using simple rules.

 

  • 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?

› View/hide answer

Moving the suspension point by hand, it is difficult to produce driving at the exact resonant frequency. In addition, for high Q, the resonance is narrow, so that attaining a driving frequency extremely close to the resonant frequency is required.

 

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