Mechanical Vibrations, Forced Oscillations, and Resonance
Stub. The mass-spring-damper equation as the physical model behind second-order linear ODEs; undetermined coefficients for nonhomogeneous equations, including the resonant case; forced oscillations, resonance, and the damped forced oscillator.
Applies the second-order linear machinery to the mechanical vibration model (mass, spring, damper): free vibrations from the homogeneous equation, and forced vibrations from the nonhomogeneous one, solved by undetermined coefficients — including the special (resonant) case where the forcing frequency matches a natural frequency and the ansatz needs an extra factor of . Culminates in forced oscillations and resonance for the undamped case and the more realistic damped, forced oscillator, whose steady-state response motivates representing a general periodic forcing term as a Fourier series.
Outline (TODO — flesh out each)
- Mechanical vibrations — the mass-spring-damper equation
- Nonhomogeneous equations and undetermined coefficients
- Nonhomogeneous equations: the resonant case
- Forced oscillations and resonance
- The damped, forced oscillator