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Lecture 4: Forced Oscillations: Resonance & Transient Phenomena

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Lecture Topics

A tuning fork sits on a block of wood. The professor holds down one end with his hand.
  • Forced Oscillations
  • Power at Resonance (Resonance Absorption, Resonance Width, Quality Q)
  • Transient Phenomena (General Solutions including Initial Conditions)

Learning Objectives

By the end of this lecture, you should:

  • understand the general solution of a differential equation as the sum of homogeneous and inhomogeneous equation solutions.
  • understand that a damped system needs energy input to attain a steady state solution.
  • calculate the average power input into a steady state damped oscillation, and its phase.
  • understand the limiting cases of power input into a damped oscillation, and maximum power case.
  • understand the resonance and power in an RLC circuit.
  • know what electromagnetic radiation is, and the relation to atomic energy level transitions.
  • know that red light is of low relative frequency, while blue/violet are of higher frequency.
  • understand the production of an absorption spectrum by resonant absorption of light.

Lecture Activities

Check Yourself

  • Explain mathematically why the undriven (transient) solution is an unimportant part of the motion of a driven oscillator for large values of time.

› View/hide answer

Since the undriven or transient solution is \(x = X{e^{ - \gamma t/2}}\cos (\omega 't + \alpha )\) where the constants X and α arise from initial conditions, and \(\omega '\)is based on the natural frequency and damping rate, which are assumed constant, this part decays exponentially and becomes negligibly small for large values of time.

 

  • Explain roughly why power input to a driven, damped system would be maximal near resonance.

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The opposing force is proportional to the speed, which, along with the amplitude of motion, increases at resonance. The work done in one cycle is roughly that force times the amplitude: since both are peaked near resonance, the power (work per unit time) will also peak near resonance.

 

  • It is stated in the lecture that for an RLC circuit with R = 50 Ω, L = 50 mH, and C = 0.5 μF, Q has the rather small value 6.3, and the decay time is 2 ms. Write the formulas involved in calculating Q for an RLC circuit, and verify it and the decay time.

› View/hide answer

From Lecture 2, we had that the number of oscillations that go by in one exponential decay time is \(\frac{Q}{\pi }\). In turn, \(Q = \frac{{{\omega _0}}}{\gamma }\), and for a circuit, \[{\omega _0} = \sqrt {\frac{1}{{LC}}} \]and \(\gamma = \frac{R}{L}\) Combining these two, \[Q = \sqrt {\frac{1}{{LC}}} \frac{L}{R} = \frac{1}{R}\sqrt {\frac{L}{C}} = 6.3\] In Lecture 2, it was found that \(s = \frac{2}{\gamma }\), and this was the decay exponent, also given as the exponential decay time \({\tau _{\tfrac{1}{e}}}\). Here, \[\gamma = \frac{R}{L} = 1000,\] so \({\tau _{\tfrac{1}{e}}} = s = 2/1000 = 2ms.\)

 

  • The dark lines in the spectra of normal stars do not vary with time and are referred to as "absorption lines". Why is this name a misnomer, in that it is incorrect to infer that cooler gas above the surface of a star "absorbs" light?

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If the gas was actually only "absorbing" the radiation, it would heat up. It is in fact re-radiating it in all directions, removing most of it from the line of sight. This allows the process to go on indefinitely without the temperature of the gas changing.

 

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