tuned-mass-damper

name	tuned-mass-damper
description	Tuned mass damper (TMD) — Den Hartog optimal tuning (f_opt = 1/(1+µ), ζ_opt = √(3µ/(8(1+µ)³))), mass ratio µ, frequency response with TMD, stroke limits, multi-degree TMD (MTMD), pendulum TMD, active and semi-active TMD (MR damper), TMD for wind-induced vibration (buildings, bridges), vortex-induced vibration (chimneys, cables), and examples (Taipei 101, Citigroup Center, Millennium Bridge).
metadata	{"priority":7,"promptSignals":{"phrases":["tuned mass damper","TMD","dynamic vibration absorber","Den Hartog","vibration suppression TMD","mass damper building"],"minScore":3}}

Tuned Mass Damper (TMD) — Complete Skill

TMD Concept and Equations of Motion

System Model

Primary structure + TMD: Main mass M (SDOF approximation of structure); stiffness K; inherent damping C TMD: mass m; spring k_d; dashpot c_d; relative displacement z = x_d − x_s

Equations of motion (harmonic excitation F₀ sin(ωt)): M × ẍ_s + C × ẋ_s + K × x_s + c_d × (ẋ_s − ẋ_d) + k_d × (x_s − x_d) = F₀ × sin(ωt) m × ẍ_d + c_d × (ẋ_d − ẋ_s) + k_d × (x_d − x_s) = 0

Dimensionless parameters: Mass ratio: µ = m / M [typically 0.01–0.05; higher µ → better reduction but heavier] Frequency ratio: f = ω_d / ω_n [tuning ratio; ω_d = √(k_d/m); ω_n = √(K/M)] Damping ratio of TMD: ζ_d = c_d / (2 × m × ω_d)

Frequency response (amplitude of main mass): H(Ω) = |X_s / X_static| [Ω = ω/ω_n = excitation frequency ratio; X_static = F₀/K]

Den Hartog Optimal TMD

Undamped Primary Structure

For undamped primary (C = 0), Den Hartog found optimal TMD tuning:

Optimal frequency ratio: f_opt = 1 / (1 + µ) [tune TMD slightly below primary natural frequency]

Optimal damping ratio of TMD: ζ_opt = √(3µ / (8(1 + µ)³))

Resulting peak response (at optimal tuning): H_max = √(1 + 2/µ) [maximum amplitude with optimal TMD; undamped structure] Or: H_max = √((2+µ)/(2µ)) [equivalent form; valid for undamped primary]

Example calculation: µ = 0.02 (2% mass ratio); ω_n = 2π × 0.5 = 3.14 rad/s (f_n = 0.5 Hz building) M = 10,000 tonnes → m_TMD = 200 tonnes

f_opt = 1/(1 + 0.02) = 0.9804 → ω_d = f_opt × ω_n = 0.9804 × 3.14 = 3.078 rad/s k_d = m × ω_d² = 200,000 × 3.078² = 1.895×10⁶ N/m = 1,895 kN/m

ζ_opt = √(3×0.02 / (8×1.02³)) = √(0.06 / 8.489) = √(0.00707) = 0.0841 c_d = 2 × m × ω_d × ζ_opt = 2 × 200,000 × 3.078 × 0.0841 = 103,500 N·s/m

Peak amplitude with TMD: H_max = √(1 + 2/0.02) = √101 = 10.05 [vs. resonance without TMD: H = 1/(2ζ_s) → ∞ for undamped] Reduction = 10.05 / (1/(2×0.02)) = 10.05 / 25 = 0.40 → 60% amplitude reduction at resonance

Damped Primary Structure

For damped primary (ζ_s > 0): Den Hartog equations are approximate; numerical optimization preferred Warburton optimal (1982) — tabulated corrections for damped primary: f_opt ≈ (1 − µ/2) / (1 + µ) [Warburton approximation for small ζ_s] ζ_opt ≈ √(µ(3−√(µ/2)) / (8(1+µ)(1−µ/2)))

Detuning sensitivity: 1% frequency detuning (f off by 1%): 10–20% increase in peak response Requires accurate identification of primary ω_n; re-tuning provisions preferred

Stroke Analysis

TMD relative displacement (stroke z = x_d − x_s): Z_max / X_static = (1+µ)/(µ) × √(µ/(2+µ)) [at optimal tuning] For µ = 0.02: Z_max / X_s ≈ 50 × √(0.02/2.02) = 50 × 0.099 = 5.0 If X_s = 10 mm (building drift): Z_TMD = 50 mm

Stroke limits design: TMD requires space for maximum stroke + safety factor (1.5–2.0) Taipei 101 ball TMD: stroke ±0.9 m (90 cm peak-to-peak = 1.8 m) Mechanical stops: spring buffers or hydraulic snubbers when TMD exceeds design stroke (rare events, extreme wind)

TMD Types

Translational (Linear) TMD

Mass-spring-dashpot: simplest; most common for structures Sliding surface: PTFE on polished steel; µ_friction < 0.01 to prevent stiction Spring: coil springs or rubber isolators (combined stiffness + damping) Damper: viscous fluid dashpot; hydraulic; oil-filled cylinder

Taipei 101 ball TMD: m = 660 tonnes (steel sphere 5.5 m diameter); suspended on 8 cables from 91st to 87th floor Passive viscous dampers (16 dashpots around perimeter) Tuned to 0.168 Hz (building fundamental); µ ≈ 0.0056 (0.56%); reduces acceleration by 30–40% Horizontal stroke: ±0.9 m; visible to visitors on observation deck

Pendulum TMD

Pendulum as mass + spring: Natural frequency: f_n = (1/2π) × √(g/L_effective) [L_effective = effective pendulum length] Advantage: no spring required; self-restoring; adjust L to tune ω_n

Equivalent spring stiffness: k_eff = m × g / L [for small angles; identical to linearized pendulum]

Tuning example: Target f_TMD = 0.5 Hz → L = g/(4π² × f²) = 9.81/(4π² × 0.25) = 9.81/9.87 = 0.994 m ≈ 1 m

Bi-directional pendulum: pivoted to swing in any horizontal direction; effective for wind from any direction Taipei 101: ball pendulum works omnidirectionally

Conical pendulum / TLCD (Tuned Liquid Column Damper): water in U-tube; effective for low f_n; lower cost

Active TMD (ATMD)

Active control force applied to TMD mass: u(t) = −G_pos × z − G_vel × ż [PD controller; z = TMD relative displacement] Or LQR: minimize J = ∫(x^T Q x + u^T R u) dt → u = −Kx

ATMD advantages: Effective even when detuned (robust to frequency variation); wider bandwidth; handles non-stationary loads Disadvantages: power required; sensor/actuator failure mode; more expensive; maintenance

ATMD actuator: hydraulic cylinder (high force, 10–500 kN) or electromagnetic linear motor

Semi-Active TMD (Magnetorheological Damper)

MR (Magnetorheological) fluid: iron particles in carrier fluid; viscosity changes with magnetic field Range: c_off (passive-off); c_on (saturated); 100–1,000× ratio Control: clipped optimal LQR; sky-hook; ground-hook strategies

MR damper fail-safe: MR damper reverts to passive damper at c_passive if power loss → always provides passive protection Applications: cable-stayed bridge stay cables; building retrofits with variable loading

Multi-TMD (MTMD) and Distributed TMDs

MTMD: multiple TMDs tuned to slightly different frequencies (bandwidth distribution) Robustness: if primary ω_n uncertain ±10%, MTMD provides better average performance Frequency spread: distribute TMD frequencies ±10–15% around primary ω_n

Distributed TMDs: multiple smaller masses at different floor levels Targets higher modes (second, third modes); floor-mounted small TMDs; combined effect

Wind-Induced Vibration Control

Along-Wind (Gust) Response

Equivalent static wind force method (ASCE 7): G_f = gust effect factor; larger G_f → more dynamic amplification → TMD reduces G_f TMD effect: reduces effective G_f from 0.85 to 0.65 (typical 30% reduction in dynamic response) Performance criterion: peak acceleration < 10–25 mg (0.01–0.025 g) for occupant comfort (ISO 6897)

Natural frequency of tall buildings: f₁ ≈ 46/H [Hz; H = building height [m]; approximate for rectangular towers] 500 m building: f₁ ≈ 0.09 Hz; 300 m: f₁ ≈ 0.15 Hz

Vortex-Induced Vibration (Chimneys, Cables)

Vortex shedding frequency: f_s = S × V / D [Strouhal; S ≈ 0.2 for circular cylinder; V = wind speed; D = diameter] Lock-in: structural resonance if f_s ≈ f_n → large oscillations → TMD critical

Lock-in wind speed: V_lock-in = f_n × D / S [structural natural frequency locks vortex shedding] At V_lock-in: without damping → severe oscillation; chimney amplitude limit: D_chimney / 80

Chimney TMD: pendulum or mass-spring; at tip of chimney where amplitude largest µ = 0.01–0.03 for chimney; vibration amplitude reduction 60–80%

Notable Applications

Taipei 101, Taiwan: 660-tonne pendulum ball; wind and earthquake; 30–40% acceleration reduction Citigroup Center, New York: 400-tonne sliding TMD (retrofit); 40% wind response reduction; sliding on pressure-balanced bearings Millennium Bridge, London: lateral TMDs + vertical TMDs (retrofit 2002); eliminated crowd-induced lateral resonance; 3-tonne TMDs at 37 locations CNTower, Toronto: 18 TMDs on support ring; tuned to 0.13–0.30 Hz; 15% wind response reduction

Standards and References

Standard	Scope
ISO 6897	Evaluation of human exposure to vibration in buildings (comfort)
ISO 10137	Bases for design of structures — serviceability
ASCE 7-22	Structural loads; wind provisions; dynamic analysis
AISC 360	Steel structure design (for TMD support structure)
Den Hartog "Mechanical Vibrations" (1956)	Classic TMD theory derivation
Warburton (1982) Earthquake Engineering	Extended optimal tuning for damped structures

Output

Provide: structure description (type: building/bridge/chimney/equipment; mass M [tonnes]; natural frequency f_n [Hz] or T_n [s]; inherent damping ζ_s [%]; mode shape; excitation: wind/earthquake/machinery), mass ratio (µ = m_TMD/M; recommended 0.01–0.05; basis: stroke space, cost, effectiveness), Den Hartog optimal parameters (f_opt = 1/(1+µ); ζ_opt = √(3µ/(8(1+µ)³))); ω_d = f_opt × ω_n [rad/s]; k_d = m × ω_d² [kN/m]; c_d = 2×m×ω_d×ζ_opt [kN·s/m]), stroke analysis (Z_max at design wind/earthquake; clearance available in structure; stop mechanism if Z > Z_clearance), TMD type (translational/pendulum/liquid column; spring system; damper type; material), response reduction (peak acceleration without TMD [mg]; with TMD [mg]; vs. comfort limit [mg]; structural drift reduction [%]), ATMD or semi-active (if ATMD: actuator force [kN]; power [kW]; control law; fail-safe; MR damper alternative), detuning sensitivity (performance if f_n shifts ±5% from design; MTMD recommendation if uncertainty large), installation (floor level; mounting provisions; maintenance access; inspection interval), and applicable standard (ISO 6897 for comfort; ASCE 7 for wind loads; ISO 10137 serviceability).