| name | vibration-absorber |
| description | Vibration absorber and tuned mass damper — undamped/damped DVA design, optimal tuning (Den Hartog), frequency ratio, mass ratio, suppression bandwidth, applications. |
| metadata | {"priority":7,"promptSignals":{"phrases":["vibration absorber","tuned mass damper","dynamic absorber","TMD","DVA","Den Hartog"],"minScore":3}} |
Vibration Absorber / Tuned Mass Damper — Complete Skill
Undamped DVA (Dynamic Vibration Absorber)
Principle: Secondary mass-spring system (m₂, k₂) attached to primary system (m₁, k₁)
When tuned correctly, absorber vibrates and imposes zero response on primary mass at target frequency
2-DOF equations of motion:
m₁ẍ₁ + k₁x₁ + k₂(x₁-x₂) = F₀ sinωt
m₂ẍ₂ + k₂(x₂-x₁) = 0
Primary mass response amplitude:
|X₁| = F₀/k₁ × (1 - ω²/ω₂²) / [(1 + μ - ω²/ω₁²)(1 - ω²/ω₂²) - μ(ω/ω₂)²]
where μ = m₂/m₁ (mass ratio); ω₁ = √(k₁/m₁); ω₂ = √(k₂/m₂)
Perfect absorption: tune ω₂ = ω_excitation → X₁ = 0 at ω = ω_excitation
Absorber amplitude: x₂ = -F₀/k₂ (constant regardless of mass ratio)
→ Design absorber spring and mass to keep x₂ within allowable motion
Damped DVA (Lanchester Damper) — Den Hartog Optimal
Adding damping to absorber allows broader suppression band
Parameters:
μ = m₂/m₁ (mass ratio, typically 0.05–0.15 for structures; 0.01–0.05 for machinery)
f = ω₂/ω₁ = tuning ratio
g = ω/ω₁ = excitation frequency ratio
ζ₂ = c₂/(2m₂ω₂) = absorber damping ratio
Den Hartog optimal tuning (minimizes peak response of primary):
f_opt = 1/(1+μ) = ω₂/ω₁ optimal
Den Hartog optimal damping:
ζ_opt = √(3μ/8(1+μ)³)
Typical values:
μ = 0.05: f_opt = 0.952, ζ_opt = 0.134
μ = 0.10: f_opt = 0.909, ζ_opt = 0.187
μ = 0.15: f_opt = 0.870, ζ_opt = 0.226
Performance
Maximum amplification factor (optimally tuned):
X_max = (F₀/k₁) × √(1 + 2/μ) (with optimal damping and tuning)
Increase μ → decrease X_max → better suppression → but more mass added
Half-power bandwidth: ω_2 - ω_1 ≈ 2ζ_opt × ω₁ (approximate suppression bandwidth)
Design Procedure
- Identify excitation frequency ω_exc or frequency band
- Identify primary mass m₁ and stiffness k₁
- Select mass ratio μ (structural: 1–5%; buildings: 0.5–2%; machines: 2–10%)
- Calculate: m₂ = μ m₁; k₂ = m₂ × (f_opt × ω₁)²; c₂ = 2 ζ_opt m₂ ω₂
- Check absorber stroke: x₂ = F₀/k₂ at resonance → must fit in space
- Verify suppression at off-design frequencies if broadband source
Mass Ratio Practical Limits
| Application | μ (typical) | Notes |
|---|
| Building floors (crowd excitation) | 0.5–2% | Mass constraint |
| Bridge TMD (Tacoma type) | 1–5% | Retrofit |
| Skyscraper (wind/seismic) | 0.1–1% | Taipei 101: 660-tonne pendulum |
| Machine tools | 2–10% | Chatter suppression |
| Cutting tools (boring bar) | 5–15% | Tuned boring bar |
Types by Implementation
Pendulum absorber: mass on cable; ω₂ = √(g/L); L = g/(ω₂)² — large stroke, low frequency
Elastomeric TMD: rubber spring + mass; compact; temperature sensitive
Liquid column damper: TLCD; water in U-tube; tuned by column length and orifice damping
Active TMD: actuator replaces passive spring/damper; higher performance; power required
Multiple Absorbers
Two absorbers at ω₂ ± Δω: broader band suppression
Multiple narrow-band absorbers: target multiple harmonics (engine orders)
Parallel DVAs: each tuned to one mode of MDOF primary system
Standards and Codes
ISO 2631: vibration limits for human exposure
AISC Design Guide 11: floor vibration (walking excitation, TMD design)
ASCE 7-22: seismic isolation and supplemental damping
Output
Provide: m₂ [kg], k₂ [N/m], c₂ [N·s/m], ζ_opt, f_opt, absorber frequency [Hz], absorber stroke [mm], primary mass X₁,max [mm] vs. unabsorbed, suppression bandwidth [Hz].
