1. Free Vibrations: mx'' + cx' + kx = 0
Natural frequency: ω0 = √(k/m)
Discriminant: D = c² - 4mk
| Type | Condition | Solution Form | Behavior |
|---|
| Undamped | c = 0 | A cos(ω0t - φ) | Perpetual oscillation |
| Underdamped | c² < 4mk | eαt(c1cos βt + c2sin βt) | Decaying oscillation |
| Critically damped | c² = 4mk | (c1+c2t)ert | Fastest non-oscillatory decay |
| Overdamped | c² > 4mk | c1er1t+c2er2t | Sluggish decay |
2. Forced Vibrations: mx'' + cx' + kx = F0cos(ωt)
General solution = transient (yh, decays) + steady-state (yp, persists)
Resonance (ω = ω0, c = 0)
xp = [F0/(2mω0)] t sin(ω0t) — amplitude grows without bound
Beating (ω ≈ ω0, c = 0)
x(t) = [2F0/(m(ω0²-ω²))] sin[(ω0-ω)t/2] sin[(ω0+ω)t/2]
Beat frequency = |ω0 - ω|
Steady-State Amplitude (c > 0)
C = F0 / √[(k - mω²)² + (cω)²]
3. RLC Circuits: Lq'' + Rq' + (1/C)q = E(t)
Same math, different physics. Current i = q' (differentiate charge).
Impedance: Z = √[R² + (ωL - 1/(ωC))²]
Resonant frequency: ω0 = 1/√(LC)
Classification: R² vs 4L/C (same rule as c² vs 4mk)
4. Mechanical-Electrical Analogy
| Mechanical | Electrical |
|---|
| Mass m | Inductance L |
| Damping c | Resistance R |
| Spring constant k | 1/Capacitance (1/C) |
| Displacement x | Charge q |
| Velocity v = x' | Current i = q' |
| Force F(t) | Voltage E(t) |
| KE: ½mv² | Magnetic: ½Li² |
| PE: ½kx² | Electric: q²/(2C) |