| 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.
