forced-vibration

name	forced-vibration
description	Forced vibration — harmonic excitation, dynamic magnification factor, resonance, periodic forcing (Fourier), arbitrary excitation (Duhamel integral), MDOF modal superposition.
metadata	{"priority":6,"promptSignals":{"phrases":["forced vibration","harmonic excitation","dynamic magnification","resonance","Duhamel integral","frequency response","vibration response"],"minScore":4}}

Forced Vibration — Complete Skill

SDOF Equation of Motion

mẍ + cẋ + kx = F(t) Normalized: ẍ + 2ζω_nẋ + ω_n²x = F(t)/m

Parameters:

• ω_n = √(k/m) [natural frequency, rad/s]
• ζ = c/(2mω_n) = c/(2√(km)) [damping ratio]
• ω_d = ω_n√(1-ζ²) [damped natural frequency]
• c_cr = 2√(km) [critical damping]

Harmonic Excitation: F(t) = F₀ sin(Ωt)

Steady-state response: x(t) = X sin(Ωt - φ)

Amplitude: X = (F₀/k) × D where D = DMF = 1/√[(1-r²)² + (2ζr)²] r = Ω/ω_n (frequency ratio) φ = arctan(2ζr/(1-r²)) [phase angle]

DMF at resonance (r=1): D_max = 1/(2ζ) [at r=1 exactly] Actual peak: D_peak = 1/(2ζ√(1-ζ²)) at r = √(1-2ζ²)

Limiting cases: r << 1 (quasi-static): X ≈ F₀/k = static deflection r = 1 (resonance): X = F₀/(c×ω_n) (limited by damping only) r >> 1 (mass-controlled): X ≈ F₀/(mΩ²) → 0 as Ω → ∞

Rotating Unbalance: F₀ = meΩ²

(m = total mass, e = eccentricity, m_e = unbalanced mass) MX/(m_e×e) = r²/√[(1-r²)² + (2ζr)²]

Peak at r ≈ 1/√(1-2ζ²), approaches 1 as r → ∞

Transmissibility (Force Transmitted to Support)

T_R = F_T/F₀ = √[1 + (2ζr)²] / √[(1-r²)² + (2ζr)²]

For isolation: need T_R < 1 → requires r > √2 At r = √2: T_R = 1 regardless of ζ At r > √2: lower ζ gives better isolation (less damping is better!)

Periodic Non-Harmonic: Fourier Series

F(t) = a₀/2 + Σ(a_n cos(nΩ₀t) + b_n sin(nΩ₀t)) a_n = (2/T)∫₀ᵀ F(t)cos(nΩ₀t)dt b_n = (2/T)∫₀ᵀ F(t)sin(nΩ₀t)dt

Response: superpose responses to each harmonic separately (linear system) Watch for: harmonics near ω_n can excite resonance even if fundamental isn't near ω_n

Arbitrary Excitation: Duhamel (Convolution) Integral

For F(t) = arbitrary loading, x(0) = ẋ(0) = 0:

x(t) = (1/(mω_d)) ∫₀ᵗ F(τ) × e^(-ζω_n(t-τ)) × sin(ω_d(t-τ)) dτ

Undamped (ζ=0): x(t) = (1/(mω_n)) ∫₀ᵗ F(τ) × sin(ω_n(t-τ)) dτ

Numerical evaluation: trapezoidal rule, recursion algorithm (Nigam-Jennings)

Step Load F₀ (t ≥ 0)

Undamped: x(t) = (F₀/k)(1 - cos(ω_n t)) Max response: x_max = 2F₀/k (Dynamic Amplification = 2)

Damped: x(t) = (F₀/k)[1 - e^(-ζω_nt)(cos(ω_d t) + ζ/√(1-ζ²) sin(ω_d t))] Max: x_max = (F₀/k)(1 + e^(-πζ/√(1-ζ²)))

Impulse Response (Dirac δ)

h(t) = (1/(mω_d)) × e^(-ζω_n t) × sin(ω_d t) [impulse response function] x(t) = ∫F(τ)h(t-τ)dτ [convolution = Duhamel] H(ω) = FT{h(t)} = 1/(-mω² + icω + k) [frequency response function]

Shock Spectrum (Response Spectrum)

S_d = max|x(t)| [spectral displacement] S_v = ω_n × S_d (pseudo-velocity) S_a = ω_n² × S_d (pseudo-acceleration ≈ max a for low ζ)

Used in earthquake engineering: base shear V = S_a × W/g

MDOF — Modal Superposition

Equations: [M]{ẍ} + [C]{ẋ} + [K]{x} = {F(t)}

Modal transformation: {x} = [Φ]{q} (Φ = mode shapes, q = modal coords) Uncoupled equations (for proportional damping): q̈_r + 2ζ_rω_r×q̇_r + ω_r²q_r = Γ_r × f_r(t)/m_r*

Where: m_r* = {φ_r}ᵀ[M]{φ_r} [modal mass] Γ_r = {φ_r}ᵀ{F} [modal force participation]

Solve each modal equation (SDOF), recover physical response: x(t) = Σ {φ_r}q_r(t)

CQC combination (for random/seismic): σ_x = √(ΣΣ ρᵢⱼ × σᵢ × σⱼ) SRSS (ρᵢⱼ = δᵢⱼ): σ_x ≈ √(Σσᵢ²) for widely spaced modes

Proportional Damping (Rayleigh)

[C] = α[M] + β[K] ζ_r = α/(2ω_r) + β×ω_r/2 Select α, β to get target ζ at two key frequencies ω₁, ω₂

Output

Provide: DMF at operating speed, x_max [mm], transmissibility T_R, phase angle φ [°], modal frequencies and participation factors for MDOF.