Math 3321 Vibrating Mech. Sys. Solutions

Free vibration

no external force

Forced vibration

external force

Undamped vibration

no resistance

Damped vibration

resistance

General mechanical vibration model

my'' + cy' + ky = F(t), m - mass, c - damping, k - spring constant, F(t) - external force

I. Free Undamped Vibrations

my'' + ky = 0

Natural frequency and period

ω = √(k/m), T = 2π/ω

Free undamped solution

y = C₁sin(ωt) + C₂cos(ωt) = A sin(ωt + φ)

II. Forced Undamped Vibrations

y'' + ω²y = αcos(γt), ω - natural frequency, γ - applied frequency

No resonance

γ ≠ ω, z = α/(ω² − γ²) cos(γt)

Resonance

γ = ω, z = αt/(2ω) sin(ωt), unbounded oscillation

III. Damped Free Vibrations

y'' + αy' + βy = 0, α = c/m, β = k/m

Underdamped

α² − 4β < 0, y = e^(−αt/2)[C₁cos(ωt) + C₂sin(ωt)], oscillatory

Overdamped

α² − 4β > 0, y = C₁e^(r₁t) + C₂e^(r₂t), nonoscillatory

Critically damped

α² − 4β = 0, y = C₁e^(−αt/2) + C₂te^(−αt/2), nonoscillatory

IV. Forced Damped Vibrations

y'' + αy' + βy = acos(γt)

Transient and steady-state solutions

y = Yc + Z, Yc - transient/homogeneous and approaches 0, Z - steady-state/particular and remains