VD-05Vibration & Structural Dynamics

Tuned Mass Damper

Den Hartog optimal tuning and damping of a dynamic vibration absorber.

Primary structure

Modal mass M

Natural frequency fₙ

Absorber

Mass ratio μ = m₂/M

Tuning ratio f = fₐ/fₙ

Absorber damping ζ

Peak amplification

9.88×

vs 25× without (ζ₁=2%)

Reduction

8.1dB

Optimum tuning

0.952

fₐ = 3.81 Hz

Optimum damping

0.127

2-DOF model

Absorber mass m₂

50kg

Absorber stiffness k₂

26,730N/m

Absorber damping c₂

201.1N·s/m

Absorber frequency

3.68Hz

Primary mass response

With TMDWithout TMD (ζ = 2 %)

§ Theory of operation

Fighting resonance with resonance

A tuned mass damper (TMD) is a small mass m₂ attached to a structure through a spring and damper. Tuned near the structure's natural frequency, it oscillates out of phase and applies a counter-force that suppresses the main resonance. Used in skyscrapers (Taipei 101's 660-tonne pendulum), footbridges (London Millennium Bridge), chimneys, power lines and machine tools.

Response of the primary mass

With $g = \omega/\omega_n$ (forcing ratio), $f = \omega_a/\omega_n$ (tuning ratio), $\mu = m_2/M$ and $\zeta = c_2/(2m_2\omega_n)$ , the amplification of the undamped primary is:

\dfrac{X_1}{x_\mathrm{st}} = \sqrt{ \dfrac{(2\zeta g)^2 + (g^2 - f^2)^2}{(2\zeta g)^2\,(g^2 - 1 + \mu g^2)^2 + \big(\mu f^2 g^2 - (g^2 - 1)(g^2 - f^2)\big)^2} }

Den Hartog's optimum (1928)

All response curves, whatever the damping, pass through two fixed points P and Q. The optimum design makes these points equal in height (tuning) and places the curve's maxima at them (damping):

\begin{aligned} f_\mathrm{opt} &= \dfrac{1}{1 + \mu} \\ \zeta_\mathrm{opt} &= \sqrt{\dfrac{3\mu}{8(1 + \mu)^3}} \\ \text{Peak} &\approx \sqrt{1 + \dfrac{2}{\mu}} \end{aligned}

Too little damping: two sharp peaks either side of the original resonance.
Too much damping: the absorber locks to the structure and a single peak returns at a slightly lower frequency.
Larger mass ratios give wider bandwidth and lower peaks, but cost weight. Typical $\mu$ = 1–10 %.

This model assumes force excitation of an undamped primary. For base excitation or a damped primary, optimal values differ slightly.