Unit 5 of 13

Oscillations, Response and Circuits

  • 4 lessons
  • Videos 27 to 30

Second-order linear equations describe vibrating masses and electric circuits. This unit studies damping, beats and resonance and the steady response to a sinusoidal input.

A mass on a spring with viscous damping is underdamped, critically damped or overdamped according to the sign of a discriminant, and its energy explains why the swings shrink. A drive at a nearby frequency produces beats, and without damping a drive at the natural frequency makes the amplitude grow linearly in time. With damping the steady amplitude stays finite, and it peaks at a positive frequency only under a stronger condition than underdamping. Complex gain gives the amplitude and phase lag of the steady response. The series RLC circuit leads to the same equation for the charge.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Damped and Forced Oscillations

  1. Free and Damped Mechanical Oscillations

    How a mass on a spring with viscous damping is modeled, how one discriminant separates underdamping, critical damping and overdamping, what the damped angular frequency is, and why mechanical energy decays.

    Video 27

  2. Forced Oscillations, Beats, and Resonance

    How a periodic force drives a mass on a spring, why a drive near the natural angular frequency gives beats, why undamped resonance causes secular growth, and when damping gives a finite amplitude peak.

    Video 28

Frequency Response and Circuits

  1. Sinusoidal Steady State and Frequency Response

    How complex gain gives the amplitude ratio and phase lag of the sinusoidal steady state, how the frequency response tracks them across driving frequencies, and how a decaying transient meets the initial data.

    Video 29

  2. RLC Circuits and Electromechanical Analogies

    How Kirchhoff's voltage law gives the series RLC circuit equation for charge and current, what resistance, inductance and capacitance mean in it, and how the electromechanical analogy matches it to a damped spring.

    Video 30