A tuned-mass damper, or TMD, is a device often used to add damping to a structural resonance that is causing problems. There are many types of application. In civil engineering TMDs are used on lightweight bridges and on tall buildings, especially ones in earthquake zones. The discussion in section 9.4 described the application in archery bows. And then there is the application we are particularly interested in here, as a wolf note “eliminator” for cellos and other bowed-string instruments.
For all these applications, we can get a good idea of how a TMD works and how it might best be designed from a very simple model. We can represent the underlying structural resonance by a mass-spring-dashpot oscillator with effective mass $m_1$, stiffness $k_1$ and dashpot strength $c_1$. Its loss factor (see section 2.2.7) is then $\eta_1=c_1/\sqrt{k_1 m_1}$.
The effective mass $m_1$ is not a fixed number for a given mode of the cello body (or whatever the underlying structure may be). Instead, it depends on the mode shape at the position we choose to fit our TMD. From section 2.2.5 we know that if the normalised modal amplitude at this point is $u_n$, then $m_1=1/u_n^2$. So for a location close to a node line of the mode $u_n$, the effective mass would become very large. But we would not want to fit a TMD close to a node line: it will work best when fitted at a position where the relevant mode has significant amplitude. The actual value of the effective mass then depends on the mass of the structure, via the normalisation condition (equation (12) of section 2.2.5). For the application to a cello wolf note, as a guide we would expect this to be somewhere between a quarter and a half of the mass of the top plate.
The TMD then consists of a second mass-spring-dashpot oscillator, sitting on top of the first one, as shown schematically in Fig. 1. The TMD oscillator has effective mass $m_2$, stiffness $k_2$ and dashpot strength $c_2$, and its loss factor if the mass $m_1$ was held fixed would be $\eta_2=c_2/\sqrt{k_2 m_2}$.
To see how the coupled system behaves, we can simply write down Newton’s law for the two masses, to deduce
$$m_1 \ddot{x_1} = – k_1 x_1 – k_2 (x_1 -x_2) \tag{1}$$
and
$$m_2 \ddot{x_2} = – k_2 (x_2 -x_1) \tag{2}$$
where $x_1$ and $x_2$ are the displacements of the two masses away from their equilibrium positions. We can deduce the mass, stiffness and damping matrices:
$$M=\begin{bmatrix}m_1 & 0\\ 0 & m_2\end{bmatrix},~~K=\begin{bmatrix}k_1+k_2 & -k_2\\ -k_2 & k_2\end{bmatrix},~~C=\begin{bmatrix}c_1 & 0\\ 0 & c_2\end{bmatrix} . \tag{3}$$
The case of interest arises when the TMD is tuned to the same natural frequency as the original resonance, so we can choose to set
$$m_2=\epsilon m_1,~~~k_2=\epsilon k_1 \tag{4}$$
where the mass ratio $\epsilon$ is expected to be a small number. The absolute values of $m_1$ and $k_1$ will not influence the behaviour except to set the specific value of the resonance frequency, and act as an overall scale factor in any frequency response functions. For maximum simplicity, we can treat the non-dimensionalised case with $m_1=1,~~k_1=1$ so that when we calculate the coupled natural frequencies they will be normalised by the original resonance frequency.
The mass and stiffness matrices then take the form
$$M=\begin{bmatrix}1 & 0\\ 0 & \epsilon\end{bmatrix},~~K=\begin{bmatrix}1+\epsilon & -\epsilon\\ -\epsilon & \epsilon\end{bmatrix} . \tag{5}$$
We can now calculate the undamped natural frequencies and mode shapes in the usual way (see equations (8) and (9) of section 2.2.5). The (normalised) natural frequencies satisfy
$$\begin{vmatrix}1-\epsilon-\omega^2 & -\epsilon \\ -\epsilon & \epsilon-\epsilon \omega^2\end{vmatrix} =0 \tag{6}$$
which simplifies to the quadratic equation
$$\omega^4 – (2-\epsilon) \omega^2 + (1-2\epsilon) = 0 \tag{7}$$
so that
$$\omega^2 = 1 -\dfrac{\epsilon}{2} \pm \sqrt{\epsilon – \epsilon^2 /4} \approx 1 \pm \sqrt{\epsilon} \tag{8}$$
where the final approximation uses the fact that $\epsilon$ is small.
Now we can deduce the ratio of modal amplitudes from equation (2):
$$\dfrac{x_2}{x_1} = \dfrac{1}{1-\omega^2} \approx \pm \dfrac{1}{\sqrt{\epsilon}} . \tag{9}$$
A physical interpretation of equation (9) is that the two masses have approximately equal kinetic energy in both modes.
Equation (8) tells us that there are two natural frequencies, one on either side of the original resonance. The bigger the mass ratio $\epsilon$, the further apart these two frequencies are split. Equation (9) tells us the corresponding mode shapes. Both of them have large motion of the TMD mass, because of the factor $1/\sqrt{\epsilon}$. The mode at lower frequency has the two masses vibrating in phase, the one at higher frequency has them in antiphase (just as we would have expected from the requirement of orthogonality of these two modes). Figure 2 shows an animation of a typical case of these two modes, for the value $\epsilon = 0.03$ so that the TMD mass is 3% of the original resonator mass.
Finally, we want to include the effects of damping, and see what influence the TMD has on the frequency response for driving on the mass $m_1$. We can write equations (1) and (2) in the frequency domain, for the case of harmonic response driven by forces $F_1 e^{i \omega t}$ on mass $m_1$ and $F_2 e^{i \omega t}$ on mass $m_2$. If $x_1 =X_1 e^{i \omega t}$ and $x_2 =X_2 e^{i \omega t}$, then
$$[-\omega^2 M + i \omega C +K] \textbf{X} = \textbf{F} \tag{10}$$
where $\textbf{X}$ is the vector $[X_1~~X_2]^t$ and $\textbf{F}$ is the vector $[F_1~~F_2]^t$. The term in square brackets $[…]$ in equation (10) is called the dynamic stiffness matrix $D$. The simplest way to compute the frequency response we want is to invert it, because
$$\textbf{X} = D^{-1} \textbf{F} \tag{11}$$
so that the (1,1) element of $D^{-1}$ is the frequency response we want: the displacement response of mass $m_1$ to a force $F_1$.
Figure 3 shows some examples. The blue curve shows the original response of the resonance without the added TMD, assuming a value $\eta_1 = 0.01$. The red curves show the result of adding two alternative TMDs, both with the value $\eta_2 = 0.15$. The dashed curve has $\epsilon = 0.01$, while the solid red curve has $\epsilon = 0.03$ (the case shown in Fig. 2). In the solid curve, the two separate peaks are very clear. In the dashed curve, with weaker coupling, the frequencies are close enough together that the peaks overlap to give a single, slightly distorted, peak.
This simple analysis gives a good impression of what you would need to consider when designing a practical TMD. There is an obvious advantage to making $\epsilon$ as small as possible. On paper, you can achieve a good outcome with any value of $\epsilon$, however small. But there is a snag: the smaller $\epsilon$ becomes, the larger the amplitude of vibration required of the TMD mass. There are always practical limitations on this amplitude, for two different reasons. One relates to constraints on available space: there needs to be room for this large amplitude of vibration without hitting something. But there is another reason. We have done this calculation assuming linear behaviour of the springs and dashpots. But if the TMD moves at very large amplitude, nonlinear effects are likely to become apparent. Most springs, of whatever kind, have some limit on how far they can be stretched or compressed. The combined effect of these two things is to impose a minimum practical limit on $\epsilon$.
The examples shown in Fig. 3 use deliberately extreme values, to illustrate the behaviour clearly. The height of the peak is reduced by some 20 dB in both cases. This is a far bigger effect than you would need to suppress a wolf note, which is fortunate because it would be hard to achieve such a large effect with something you could realistically fit to a cello. In addition to the two constraints just described, there is another issue concerning the assumed damping. We have assumed a rather low value of the cello loss factor $\eta_1$, and a very high value of the TMD loss factor $\eta_2$. In practice these two loss factors would not be so different, and the effect of the TMD on the peak height would be reduced.
Finally, we should note that the smaller the value of $\epsilon$, the more careful we need to be to tune the TMD frequency to match the wolf frequency. When the mass ratio is very tiny, the strength of coupling between the two oscillators is very weak. As the tuning is varied, the two coupled frequencies will go through a “veering” pattern (like Fig. 3 of section 7.3). In order for the TMD to work, the tuning has to be accurate enough to achieve a state well within the veering region, and this region will be narrow when the coupling is weak.