main mass FRF
Interactive Lecture
Tuned Mass Dampers for Structural Vibration Suppression
by Dr Arnaldo Delli Carri
A Tuned Mass Damper (TMD) is a passive vibration control device consisting of a mass, spring, and damper attached to a primary structure. The TMD is "tuned" so that its natural frequency sits close to the primary structure's resonance — Den Hartog's optimum places it slightly below, at \(\omega_2 = \omega_1/(1+\mu)\) — where it absorbs vibrational energy and reduces the motion of the main system.
When properly tuned, a TMD oscillates out of phase with the primary structure, transferring energy to the secondary mass where damping dissipates it. The result is a dramatic reduction in the primary structure's vibration amplitude — try the slider below to see it happen.
The TMD operates on the principle of dynamic vibration absorption. When the primary structure vibrates, the TMD oscillates out of phase with the structure, creating an inertial force that opposes the motion of the main system.
For a 2-DOF system with a primary mass \(m_1\) and TMD mass \(m_2\):
\[m_1\ddot{x}_1 + c_1\dot{x}_1 + k_1x_1 + k_2(x_1-x_2) + c_2(\dot{x}_1-\dot{x}_2) = F_0\cos(\omega t)\]
\[m_2\ddot{x}_2 + k_2(x_2-x_1) + c_2(\dot{x}_2-\dot{x}_1) = 0\]
where:
Mass Ratio: \(\mu = \frac{m_2}{m_1}\) (typically 0.01 to 0.05). The primary/absorber pair \(m_1, m_2\) is written \(m_{structure}, m_{TMD}\) in the design problems and summary — same two masses, so \(\mu = m_{TMD}/m_{structure}\).
Frequency Ratio: \(f = \frac{\omega_2}{\omega_1}\) where \(\omega_1 = \sqrt{\frac{k_1}{m_1}}\) and \(\omega_2 = \sqrt{\frac{k_2}{m_2}}\)
Optimal tuning: \(f_{opt} = \frac{1}{1 + \mu}\)
Optimal damping ratio: \(\zeta_{opt} = \sqrt{\frac{3\mu}{8(1+\mu)^3}}\)
With the primary mass undamped and the absorber carrying damping ratio \(\zeta_2 = c_2/(2m_2\omega_2)\), the magnification factor of the primary mass is (Den Hartog):
\[\frac{X_1}{X_{st}} = \sqrt{\frac{(g^2 - f^2)^2 + (2\zeta_2 f g)^2}{\big[\mu f^2 g^2 - (g^2-1)(g^2-f^2)\big]^2 + (2\zeta_2 f g)^2\,(g^2 - 1 + \mu g^2)^2}}\]
where \(g = \omega/\omega_1\) is the forcing ratio, \(f = \omega_2/\omega_1\) is the tuning ratio (set to \(f_{opt}=1/(1+\mu)\) in the simulation), and \(X_{st} = F_0/k_1\) is the static deflection. \(X_1/X_{st}\) is dimensionless.
A properly tuned TMD creates two fixed points in the frequency response curve whose height is independent of the absorber damping. Slide the damping ratio from zero upwards: the two undamped resonance spikes collapse toward these fixed points, and the optimal \(\zeta_2\) is the value that levels the two peaks so the worst-case response is as small as possible.
Compare the unprotected and TMD-protected systems. Adjust the TMD mass ratio, damping, and forcing frequency to observe their effects on vibration amplitude.
Tuned Mass Dampers are passive vibration control devices with wide-ranging applications across civil, mechanical, and aerospace engineering:
| Concept | Equation | Description |
|---|---|---|
| Mass Ratio | \(\mu = \frac{m_{TMD}}{m_{structure}}\) | Ratio of TMD mass to primary structure mass (typically 1-5%) |
| Optimal Frequency Ratio | \(f_{opt} = \frac{1}{1 + \mu}\) | Ratio of TMD frequency to structure frequency for optimal tuning |
| Optimal Damping Ratio | \(\zeta_{opt} = \sqrt{\frac{3\mu}{8(1+\mu)^3}}\) | Damping ratio that minimises peak response |
| TMD Natural Frequency | \(\omega_{TMD} = \sqrt{\frac{k_{TMD}}{m_{TMD}}}\) | Natural frequency of the TMD subsystem |
| Peak Magnification | \(\left(\frac{X_1}{X_{st}}\right)_{max} = \sqrt{1 + \frac{2}{\mu}}\) | Lowest achievable peak response of the primary at optimal tuning and damping (≈6.4 at μ = 0.05) |
A building with mass m₁ = 500 kg and natural frequency ω₁ = 10 rad/s needs a TMD with mass ratio μ = 0.05. Calculate:
Show Solution
Solution:
a) TMD mass:
\(m_2 = \mu \cdot m_1 = 0.05 \times 500 = 25\) kg
b) Optimal TMD frequency (Den Hartog's tuning places it just below the primary frequency):
\(f_{opt} = \frac{1}{1 + \mu} = \frac{1}{1 + 0.05} = 0.9524\)
\(\omega_2 = f_{opt} \cdot \omega_1 = 0.9524 \times 10 = 9.524\) rad/s
c) Optimal damping ratio:
\(\zeta_{opt} = \sqrt{\frac{3\mu}{8(1+\mu)^3}} = \sqrt{\frac{3(0.05)}{8(1.05)^3}} = \sqrt{\frac{0.15}{9.261}} = 0.127\)
The TMD is built with this optimal damping, so \(\zeta_2 = \zeta_{opt} = 0.127\).
Required stiffness: \(k_2 = m_2\omega_2^2 = 25 \times 9.524^2 = 2268\) N/m
Required damping: \(c_2 = 2\zeta_2 m_2\omega_2 = 2 \times 0.1273 \times 25 \times 9.524 = 60.6\) N·s/m
Answer: \(m_2 = 25\) kg; \(\omega_2 = 9.524\) rad/s (\(k_2 = 2268\) N/m); \(\zeta_2 = 0.127\) (\(c_2 = 60.6\) N·s/m).
A primary structure is fitted with an optimally tuned and damped TMD of mass ratio μ = 0.05. Using Den Hartog's result, find the peak magnification factor \((X_1/X_{st})_{max}\) of the primary mass, and describe what the TMD does to the original single resonance.
Show Solution
Solution:
At optimal tuning \(f_{opt}=1/(1+\mu)\) with optimal damping, the two fixed points sit at equal height, and that height is the lowest achievable peak response:
\[\left(\frac{X_1}{X_{st}}\right)_{max} = \sqrt{1 + \frac{2}{\mu}}\]
For μ = 0.05:
\[\left(\frac{X_1}{X_{st}}\right)_{max} = \sqrt{1 + \frac{2}{0.05}} = \sqrt{41} = 6.40\]
Without the absorber, the lightly damped structure has a single tall resonance whose peak is limited only by its small inherent damping. The TMD splits that spike into two much lower equal peaks, each about 6.4 times the static deflection \(X_{st}=F_0/k_1\). A heavier absorber (larger μ) lowers this peak further.
Answer: \((X_1/X_{st})_{max} = \sqrt{1 + 2/\mu} \approx 6.40\).
A building has a mass of 5,000,000 kg and a natural frequency of 0.5 Hz. A TMD with mass 50,000 kg is to be installed. Calculate:
Show Solution
(a) Mass ratio:
\[\mu = \frac{m_{TMD}}{m_{structure}} = \frac{50{,}000}{5{,}000{,}000} = 0.01 = 1\%\]
(b) Optimal absorber frequency (\(f_2\) is an absolute frequency in Hz — the tuning ratio \(f_{opt}=f_2/f_n=1/(1+\mu)\) stays dimensionless):
\[f_2 = \frac{f_n}{1 + \mu} = \frac{0.5}{1.01} = 0.495 \text{ Hz}\]
\[\omega_2 = 2\pi f_2 = 2\pi \times 0.495 = 3.110 \text{ rad/s}\]
(c) Required stiffness:
\[k_{TMD} = m_{TMD}\,\omega_2^2 = 50{,}000 \times (3.110)^2 = 4.838 \times 10^{5} \text{ N/m} = 483.8 \text{ kN/m}\]
(d) Damping coefficient estimate:
For optimal performance, the damping ratio \(\zeta_{opt}\) is approximately:
\[\zeta_{opt} = \sqrt{\frac{3\mu}{8(1+\mu)^3}} = \sqrt{\frac{3 \times 0.01}{8(1.01)^3}} = \sqrt{\frac{0.03}{8.242}} = 0.0603\]
\[c_{cr} = 2\sqrt{k_{TMD}\,m_{TMD}} = 2\sqrt{4.838 \times 10^{5} \times 50{,}000} = 3.110 \times 10^{5} \text{ N·s/m}\]
\[c_{TMD} = \zeta_{opt}\,c_{cr} = 0.0603 \times 3.110 \times 10^{5} = 1.877 \times 10^{4} \text{ N·s/m}\]
Verification: With a 1 m stroke limit and building frequency of 0.5 Hz, the maximum velocity is approximately \(v_{max} = 2\pi f_n A = 2\pi \times 0.5 \times 1 = 3.14\) m/s. The damping force would be \(F_d = c_{TMD}\,v_{max} = 1.877 \times 10^{4} \times 3.14 = 5.90 \times 10^{4}\) N, appropriate for a civil structure of this scale.
Answer: (a) μ = 0.01; (b) \(f_2 = 0.495\) Hz (\(\omega_2 = 3.110\) rad/s); (c) \(k_{TMD} = 483.8\) kN/m; (d) \(c_{TMD} \approx 1.88 \times 10^{4}\) N·s/m.