| name | natural-frequency |
| description | Vibration analysis — natural frequency (SDOF/MDOF), damping ratio, forced response, resonance, Dunkerley, Rayleigh-Ritz, modal analysis, transmissibility, vibration isolation. |
| metadata | {"priority":8,"promptSignals":{"phrases":["natural frequency","vibration","resonance","damping","mode shape","SDOF","harmonic","transmissibility","isolation"],"minScore":4}} |
Vibration Analysis — Complete Skill
SDOF Free Vibration
Equation of motion: mẍ + cẋ + kx = 0
Natural frequency: ω_n = √(k/m) [rad/s], f_n = ω_n/(2π) [Hz]
Period: T = 1/f_n
Critical damping: c_cr = 2√(km) = 2mω_n
Damping ratio: ζ = c/c_cr
Underdamped (ζ < 1):
x(t) = e^(-ζω_n·t)·[A·cos(ω_d·t) + B·sin(ω_d·t)]
Damped natural frequency: ω_d = ω_n·√(1-ζ²)
Logarithmic decrement: δ = ln(x_n/x_{n+1}) = 2πζ/√(1-ζ²) → ζ = δ/√(4π²+δ²)
Overdamped (ζ > 1): No oscillation, exponential decay
Critically damped (ζ = 1): Fastest return to equilibrium without oscillation
Natural Frequency from Static Deflection
ω_n = √(g/δ_st) [rad/s] where δ_st = static deflection under weight
f_n = (1/2π)·√(g/δ_st) ≈ 0.498/√(δ_st[m]) Hz
Quick reference:
δ_st = 1mm → f_n = 15.8 Hz
δ_st = 10mm → f_n = 5.0 Hz
δ_st = 25mm → f_n = 3.1 Hz
Common System Stiffnesses
| System | k |
|---|
| Axial rod | AE/L |
| Simply supported beam (center) | 48EI/L³ |
| Cantilever (tip) | 3EI/L³ |
| Fixed-fixed beam (center) | 192EI/L³ |
| Coil spring | Gd⁴/8D³N |
| Parallel springs | k₁+k₂ |
| Series springs | k₁k₂/(k₁+k₂) |
Forced Harmonic Response
Force: F(t) = F₀·sin(ωt)
x(t) = X·sin(ωt - φ)
Amplitude ratio (DMF — dynamic magnification factor):
X/(F₀/k) = 1/√((1-r²)² + (2ζr)²)
Where r = ω/ω_n (frequency ratio)
Phase angle: φ = arctan(2ζr/(1-r²))
Resonance (r=1): X = F₀/(2ζk) → amplitude limited only by damping
Regions:
r << 1: quasi-static, X ≈ F₀/k
r = 1: resonance, X = F₀/(2ζk)
r >> 1: inertia-controlled, X → 0
Base Excitation (Transmissibility)
Displacement transmissibility: T_R = √((1+(2ζr)²)/((1-r²)²+(2ζr)²))
Force transmissibility = same formula
At r = √2: T_R = 1 for all ζ (isolation begins above this)
For isolation: r > √2 (operate above √2·f_n, low damping helps in isolation region)
Vibration isolation design:
Choose mount stiffness: k = (2πf_exc)²·m / (r²) where r = 3-5 for good isolation
T_R = 0.1 → r ≈ 3.2 (90% isolation)
T_R = 0.01 → r ≈ 10 (99% isolation)
MDOF Systems
2-DOF: [M]{ẍ} + [K]{x} = {0}
det([K] - ω²[M]) = 0 → ω₁, ω₂
Mode shapes from ([K] - ω_i²[M]){X} = {0}
Dunkerley's Equation (lower bound estimate):
1/ω_n² ≈ 1/ω₁² + 1/ω₂² + ... + 1/ω_n²
Where ω_i = natural frequency if only mass i were present
Rayleigh's Quotient (upper bound):
ω² ≤ {X}ᵀ[K]{X}/{X}ᵀ[M]{X}
Use assumed mode shape → best estimate of ω₁
Critical Speed (Rotating Machinery)
n_cr = 30/π · √(k/m) rpm = 30·f_n [rpm] / π
Or from static deflection: n_cr = 946/√(δ_st[mm]) rpm
Rule of thumb:
Operating speed: n < 0.75·n_cr OR n > 1.4·n_cr
If passing through critical speed: do it quickly (low damping → high amplitude)
Modal Damping
Typical ζ values:
- Welded steel: 0.01-0.02 (1-2%)
- Bolted steel: 0.02-0.03
- Reinforced concrete: 0.02-0.05
- Timber: 0.04-0.06
- Soil/foundation: 0.05-0.20
- Rubber mounts: 0.05-0.10
- Viscoelastic: 0.10-0.30
Random Vibration (PSD)
RMS response: σ_x = √(∫S_x(f)df) = √(π·f_n·S_0·(1+4ζ²)/(4ζ))
For white noise input S_0: σ_x = √(π·f_n·S_0/(4ζ))
Peak factor: x_peak ≈ 3σ_x (3-sigma rule)
Output
Provide: ω_n [rad/s], f_n [Hz], ζ, critical speed if applicable, resonance check (r vs. 1.0), isolation T_R if relevant.
