# Comparative Analysis of NES and TMD Performance via High-Dimensional Invariant Manifolds

Giuseppe Habib, Francesco Romeo

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Giuseppe Habib, Francesco Romeo. Comparative Analysis of NES and TMD Performance via High-Dimensional Invariant Manifolds. IUTAM Symposium on Exploiting Nonlinear Dynamics for Engineering Systems, Jul 2018, Novi Sad, Serbia. pp.143-153, 10.1007/978-3-030-23692-2_13. hal-04077702

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# Comparative Analysis of NES and TMD Performance via High-Dimensional Invariant Manifolds

Giuseppe Habib and Francesco Romeo

**Abstract** A comparative study between a tuned mass damper (TMD) and a nonlinear energy sink (NES), attached to a linear two-degree-of-freedom (DoF) mechanical system under impulsive excitation, is performed. The analysis involves different scenarios; namely, we consider the cases in which only one or both modes of the primary system are initially excited. First, exploiting a harmonic balance approach, the invariant manifolds describing the slow dynamics of the system are identified. Then, introducing the so-called relative dissipation power, the performance of the two absorbers is carefully compared, based on analytical computations. Results illustrate that the two absorbers have similar performance, albeit resorting to different mechanical properties: the NES achieve a broad frequency band of operation exploiting nonlinearity, while the TMD by increasing damping. An interesting feature, highlighted by the invariant manifold, is that the NES is generally unable to resonate with more than one mode of the primary system at the same time, rather, it experiences a sort of modal cascade.

**Keywords** Nonlinear energy sink · Tuned mass damper · Multi-modal vibration absorber · Impulsive energy dissipation

## 13.1 Introduction

Resonance vibrations have since ever aroused great concern in the engineering community due to their harmful consequences on mechanical and structural systems. Countless mitigation strategies, each with its pros and cons, have been so far proposed

G. Habib (✉)

Department of Applied Mechanics, Budapest University of Technology and Economics, Műegyetem rkp. 5., Budapest 1111, Hungary
e-mail: habib@mm.bme.hu

F. Romeo

Department of Structural Engineering and Geotechnics, Sapienza University of Rome, via Gramsci 53, 00197 Rome, Italy
e-mail: francesco.romeo@uniroma1.it
in the literature; a common classification distinguishes between active, semi-active, passive and hybrid techniques which, in turn, can rely on devices characterized by linear and nonlinear behaviours. Among the passive linear vibration absorbers, tuned mass dampers (TMDs) are undoubtedly an established benchmark for mitigation of resonances [1, 2]. As known, their effectiveness lies in the tuning of their own natural frequency with that of the resonance to be mitigated, which implies that a single TMD can be used to optimally damp only one resonance of the host structure. Such intrinsic limitation of TMDs has prompted the engineering challenge of expanding the absorbers bandwidth of operation, giving the start to an extensive and multifaceted research activity. Among the proposed passive linear strategies, multiple tuned mass dampers (MTMDs) and spatially distributed MTMDs have been considered in literature, as reported in the recent review article [3]. Alternatively, willing to keep a single device in the control system, nonlinear absorbers designed to resonate for broad frequency band have been recently proposed. This brought the development of the nonlinear energy sink (NES), consisting of a small mass connected to the primary system by an essential nonlinear spring [4, 5]. Given the inherent different dynamic regimes involved in the passive linear and nonlinear absorbers during the vibration mitigation processes, a comparison between the performance of the two families of devices is not straightforward. Moreover, the variety of excitations, host structure typology, design constraints and objectives have so far lead to partial and incomplete performance comparisons between the two families of devices. In [6] it is illustrated that, in some specific conditions, the NES can outperform the TMD for resonance mitigation of a single-degree-of-freedom (DoF) linear primary system, exploiting quasiperiodic motions. In [7], a comparative analysis about the relevance of damping for NES and TMD revealed that, for low damping of the absorber, the NES can be more effective than the TMD also in narrow-band energy dissipation; increasing damping, the TMD behaves better than the NES. A comparison between NES and TMD performance in a periodically excited linear beam is performed in [8].

In this work, an attempt is made to qualitatively and quantitatively compare the TMD and the NES capabilities for the mitigation of broadband impulsive energy, considering various engineering scenarios. By exploiting the high-dimensional invariant manifold of a two-DoF host system, an effective performance measure is introduced, the so-called relative dissipation power.

## 13.2 Mathematical Model and Invariant Manifold Derivation

We consider the simple model shown in Fig. 13.1. The dynamics of this system is governed by the equations
<figure><img src="image_2.png" /><figcaption>Fig. 13.1 A three-DoF system consisting of two coupled symmetric linear oscillators and an NES connected to one of them</figcaption></figure>

$$ 
\begin{aligned}
M x_1'' + k x_1 + k (x_1 - x_2) + k_{nl} (x_1 - x_a)^3 + c_a (x_1' - x_a') &= 0 \\
M x_2'' + k x_2 + k (x_2 - x_1) &= 0 \\
m x_a'' + k_{nl} (x_a - x_1)^3 + c_a (x_a' - x_1') &= 0
\end{aligned} 
\quad (13.1) $$

where $x_1$ and $x_2$ refer to the displacements of the primary 2 DoF system, while $x_a$ refers to the displacement of the absorber; $m$ is assumed much smaller than $M$ and the prime denote differentiation with respect to time $t$. The choice of considering an undamped hosting structure reflects results obtained in previous works [9], where it is illustrated that small damping in the primary system does not affect the overall qualitative dynamics.

We divide the system of Eqs. (13.1) by $M$, introduce the dimensionless time $T = \omega_n t$ and dimensionless variables $y_1 = \sqrt{\lambda_3} (x_1 + x_2)/2$, $y_2 = \sqrt{\lambda_3} (x_1 - x_2)/2$ and $y_3 = \sqrt{\lambda_3} (x_1 - x_a)$, where $\omega_n = \sqrt{k/M}$ and $\lambda_3 = k_{nl} / (m\omega_n^2)$, obtaining the governing equations in primary system dimensionless modal coordinates, i.e.

$$ 
\begin{aligned}
\ddot{y}_1 + y_1 &= \varepsilon \left( -\frac{y_3^3}{2} - \mu_a \dot{y}_3 \right) \\
\ddot{y}_2 + 3y_2 &= \varepsilon \left( -\frac{y_3^3}{2} - \mu_a \dot{y}_3 \right) \\
\ddot{y}_3 + y_1 + 3y_2 + (1 + \varepsilon) (y_3^3 + 2\mu_a \dot{y}_3) &= 0
\end{aligned} 
\quad (13.2) $$

where $\varepsilon = m/M$, $\mu_a = c_a / (2m\omega_n)$ and the overdots denote differentiation with respect to $T$. Starting from Eqs. (13.2), by considering $\varepsilon \ll 1$ as a perturbation parameter, an analytical framework enabling to design the NES and to optimize its performance is derived.

In order to study the slow dynamics of the system, we collect terms of order $\varepsilon^0$, reducing the system to
$$ \ddot{y}_1 + y_1 = 0 \qquad (13.3) $$

$$ \ddot{y}_2 + 3y_2 = 0 \qquad (13.4) $$

$$ \ddot{y}_3 + 2\mu_a \dot{y}_3 + y_3^3 = -y_1 - 3y_2. \qquad (13.5) $$

We define an approximate solution by adopting the harmonic balance method [10, 11], assuming 1:1 resonance between the primary system and the absorber. The solutions of Eqs. (13.3) and (13.4) are $y_1 = A_1 e^{iT}$ + c.c. and $y_2 = A_2 e^{\sqrt{3}iT}$ + c.c., where $A_1$ and $A_2$ are complex and c.c. stands for complex conjugate. The approximate solution of Eq. (13.5) is expressed by $y_3 = B_1(t_1) e^{iT} + B_2(t_1) e^{\sqrt{3}iT}$ + c.c.

We substitute the approximate solutions of $y_1$, $y_2$ and $y_3$ into Eq. (13.5) and collect harmonics of $e^{iT}$ and $e^{\sqrt{3}iT}$, obtaining

$$ \begin{aligned} (e^{iT}) : & -B_1 + A_1 + 3B_1^2 \bar{B}_1 + 6B_1 B_2 \bar{B}_2 + 2\mu_a i B_1 = 0 \\ (e^{\sqrt{3}iT}) : & -3B_2 + 3A_2 + 3B_2^2 \bar{B}_2 + 6B_1 \bar{B}_1 B_2 + 2\sqrt{3}\mu_a i B_2 = 0. \end{aligned} \qquad (13.6) $$

By defining $B_1 = 1/2b_1 e^{i\beta_1}$, $B_2 = 1/2b_2 e^{i\beta_2}$, $A_1 = 1/2a_1 e^{i\alpha_1}$ and $A_2 = 1/2a_2 e^{i\alpha_2}$, and separating real and imaginary parts of the first equation of (13.6), we have

$$ \begin{aligned} \frac{1}{2} a_1 \cos \alpha_1 &= \frac{1}{2} b_1 \left( 1 - \frac{3}{4} b_1^2 - \frac{3}{2} b_2^2 \right) \cos \beta_1 + \mu_a b_1 \sin \beta_1 \\ \frac{1}{2} a_1 \sin \alpha_1 &= \frac{1}{2} b_1 \left( 1 - \frac{3}{4} b_1^2 - \frac{3}{2} b_2^2 \right) \sin \beta_1 - \mu_a b_1 \cos \beta_1. \end{aligned} \qquad (13.7) $$

We calculate the squares of the two equations of (13.7) and we sum them up attaining

$$ a_1^2 = b_1^2 \left( 1 - \frac{3}{4}b_1^2 - \frac{3}{2}b_2^2 \right)^2 + 4\mu_a^2 b_1^2. \qquad (13.8) $$

Repeating the same operation with the second equation of (13.6) we obtain

$$ a_2^2 = b_2^2 \left( 1 - \frac{1}{4}b_2^2 - \frac{1}{2}b_1^2 \right)^2 + \frac{4}{3}\mu_a^2 b_2^2. \qquad (13.9) $$

Equations (13.8) and (13.9) describe the invariant manifold that relates the slow dynamics of $y_3$ with respect to $y_1$ and $y_2$. A detailed analysis of the obtained manifold is performed in the following sections by considering separately the cases in which the excitation involves either a single or both modes.

We notice that, although the absorber is characterized by the three parameters $\varepsilon$, $\mu_a$ and $\lambda_3$, thanks to the performed non-dimensionalization the only parameter left in the manifold equation is $\mu_a$. $\lambda_3$ simply scales the amplitude of the variables, while $\varepsilon$ is proportional to the energy dissipation rate, but, if it is kept small, it has no
<figure><img src="image_3.png" /><figcaption>Fig. 13.2 a Invariant manifold for the first mode ($\mu_a = 0.1, \gamma = 1$ for TMD); b Estimated relative dissipation power based on the invariant manifold ($\mu_a = 0.1, \varepsilon = 0.01, \gamma = 1$ for TMD). Blue and orange lines refer to analytical results of NES and TMD, respectively, black lines refer to numerical results</figcaption></figure>

qualitative effect on the slow dynamics of the system, which can be captured by the adopted approach.

## 13.3 Single Mode Dynamics

We consider at first the case when only the first mode of the primary system is initially excited. In this case, $y_2$ is assumed of order $\varepsilon$, therefore the invariant manifold is defined only by Eq. (13.8), with $b_2 = 0$, i.e.

$$a_1^2 = b_1^2 \left(1 - \frac{3}{4}b_1^2\right)^2 + 4\mu_a^2 b_1^2. \qquad (13.10)$$

The relative manifold is illustrated in Fig. 13.2a for $\mu_a = 0.1$. The black line in the figure is the result of a numerical simulation, which qualitatively confirms analytical results. Modal amplitudes were obtained through a wavelet transformation of the system time series. Although stability of the manifold was not studied, it can be guessed that, if for a single $a_1$ value there are three different $b_1$ values, the middle one is unstable.

The invariant manifold enables one to predict the amplitude of oscillation of $y_3$ depending on the oscillation amplitude $y_1$. Large values of $y_3$ (i.e. $b_1$) correspond to high dissipation power. Adopting the hypothesis of single harmonic response, dissipation power in one period is given by $\varepsilon\mu_a b_1^2$. However, from an engineering point of view, it is more significant to indicate the energy dissipated in one period with respect to the energy present in the primary system, that is (if only the first mode
is activated) $P_r = \varepsilon\mu_a b_1^2/a_1^2$. This curve is illustrated in Fig. 13.2b for $\mu_a = 0.1$ and $\varepsilon = 0.01$.

The shape of the $P_r$ function gives important information about the performance of the NES. For very high values of $a_1$, $P_r$ is very low ($P_r$ tends to zeros for $a_1 \to \infty$); however, at large amplitudes additional harmonics, overlooked by the adopted analytical approach, might become more relevant. Reducing $a_1$, $P_r$ increases until it reaches a high peak (called P in the figure), whose exact position can be calculated and it is $P = (4\mu_a/\sqrt{3}, \varepsilon/(4\mu_a))$. For $a_1 < 4\mu_a/\sqrt{3}$, $P_r$ has a sudden decrease and it reaches an almost constant plateau until $a_1 = 0$ (for $a_1 \to 0$, $P_d = \varepsilon\mu_a/(1 + 4\mu_a^2)$). This lower limit corresponds to the minimum energy threshold below which the NES is not activated, a feature which has been extensively studied [5]. Numerical results, represented by black dots in the figure, agree very well with the analytical prediction. The main difference consists in the slightly mismatching position of the peak.

During the design of an NES, the position of point P is clearly a key parameter for defining the performance of the device and the range of operation. Since the amplitude was normalized with respect to $\sqrt{\lambda_3}$, this parameter can be tuned to adjust the energy level of optimal operation. Furthermore, the $a_1$-coordinate of point P grows linearly with $\mu_a$. Increases of $\mu_a$ also enlarge the peak and lower P, reducing the maximal dissipation power, but widening the amplitude range of operation.

The same procedure performed for the NES, can be analogously computed for the TMD, in order to compare the two absorbers. The analysis leads to the invariant manifold described by the equation

$$a_1^2 = b_1^2 (1 - \gamma^2)^2 + 4\mu_a^2 b_1^2 \quad \text{or} \quad a_1 = b_1 \sqrt{(1 - \gamma^2)^2 + 4\mu_a^2}, \quad (13.11)$$

where $\gamma$ is the ratio between the natural frequency of the TMD and of the primary system. Orange lines in Fig. 13.2a and b illustrate the manifold and the relative dissipation power for the TMD, which can be directly compared with the NES. It is particularly interesting that, if $\gamma = 1$, the relative dissipation power of the TMD is always equal to the maximum $P_r$ of the NES. This clearly highlights the superiority of the TMD over the NES when it is properly tuned and only one mode is involved. Also in this case, numerical results qualitatively confirm analytical predictions.

## 13.4 Two Modes Dynamics

In the hypothesis that both modes of the primary system are activated, Eqs. (13.8) and (13.9) form a unique system of equations. This defines the invariant manifold, that is a 2-dimensional surface in the 4-dimensional space ($a_1, a_2, b_1, b_2$). Figure 13.3 illustrates the manifold exploiting two projections: x- and y-axes mark the modal amplitude in the primary system of the first ($a_1$) and second ($a_2$) mode, respectively; z-axis indicates the modal amplitude related to the first ($b_1$, Fig. 13.3a) and to the second mode ($b_2$, Fig. 13.3b) in the NES.
<figure><img src="image_4.png" /><figcaption>**Fig. 13.3** Invariant manifold projected on the ($a_1, a_2, b_1$) (**a**) and on the ($a_1, a_2, b_2$) (**b**) spaces for $\mu_a = 0.1$. The black dots mark the instantaneous modal amplitude of the time series in Fig. 13.4 calculated through a wavelet transformation. Subplots **c** and **d** offer top views of subplots **a** and **b**, respectively</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 13.4** **a** Time series for system (13.2) with $\mu_a = 0.1$ and $\varepsilon = 0.01$; **b** wavelet transformation of $y_3$</figcaption></figure>

We notice the interesting feature that, for given $a_1$ and $a_2$ values, either $b_1$ is large and $b_2$ is small, or the opposite is verified. An ideal line, splitting in two the $a_1, a_2$ space, divides the two regions where either $b_1$ or $b_2$ is larger. This consideration has important practical consequences, indeed it means that the NES, although is able to interact with modes at different frequencies, it works well with only one at a time.

A numerical validation of this phenomenon is illustrated in Fig. 13.4. Figure 13.4a shows time series for the three system coordinates, while Fig. 13.4b depicts the instantaneous frequency of the NES motion ($y_3$), obtained through a wavelet transformation. For $T < 1500$, Fig. 13.4b clearly illustrates that the NES has a strong coupling with the second mode (at 1.732 rad/s), while it does not interact with the first one. This is consistent with the manifold in Fig. 13.3; in fact, for the adopted
initial conditions ($y_1(0) = 6$ and $y_2(0) = 4$), $b_1$ is relatively small, while $b_2$ is much larger. This causes a relatively rapid dissipation of energy on the second mode of the primary system, while the first one is almost unaffected by the absorber, as it can be verified from the time series in Fig. 13.4a. At $T \approx 1500$, the NES abruptly disengages from the second mode and it couples with the first one, forming a sort of modal cascade. In terms of invariant manifold, this coincides to reach the boundary dividing the regions where either $b_1$ or $b_2$ is large. The correspondence between analytical and numerical results is given by the black dots in Fig. 13.3a and b: although the black dots, indicating the modal amplitude in the NES $b_1$ and $b_2$, do not exactly lie on the manifold, the transition between second and first modes is well predicted in terms of modal amplitudes $a_1$ and $a_2$. For $1500 < T < 8000$ the energy decreases on both modes; however, the dissipation rate for the second mode is significantly diminished, in virtue of the discussed modal transition. At $T \approx 8000$ the NES disengages also from the first mode, causing a sudden drop of its oscillation amplitude and of the energy dissipation on the first mode. Considering that at this point most of the energy was only on the first mode, this phenomenon practically coincides with the drop illustrated in Fig. 13.2 for the case of single mode dynamics. Super- and sub-harmonic resonances, visible in Fig. 13.4b, are overlooked by the analytical framework adopted.

We notice that, if the system is initially in the region where the first mode is prevalent but there is some energy also on the second mode, the NES will have a stronger coupling with the first mode, but it will still dissipate some energy on the second mode. This cancels the modal cascade, which seems to occurs only from higher modes to lower ones. We also remark that the ideal line dividing the two regions of modal coupling follows a somehow different trend for low amplitude. The minimum energy threshold of each mode and the S shape observed in Fig. 13.2a is dominant over the interaction between the two modes, causing a drop of modal amplitude for the first mode at $a_1 = 0.23$ and for the second mode at $a_2 = 0.23$. Indeed, the best dissipation performance are obtained for $a_1$ and $a_2$ only slightly larger than these limits.

### 13.4.1 NES-TMD Comparison

In the following, we compare the NES and TMD performance while operating on the primary system with both modes activated. We consider the initial conditions $y_1(0) = y_2(0) = 0$, $\dot{y}_1(0) = \dot{y}_2(0) = v_0$ and identify parameters providing the minimum dissipation time, given $\epsilon = 0.01$. For the NES, the only parameter to be optimized is $\mu_a$ as a function of the initial velocity $v_0$; we remind that the amplitude is scaled with $\sqrt{\lambda_3}$, therefore, in principle, the system can be set to any energy level. For the TMD, optimization is performed tuning $\mu_a$ and $\gamma$; because of the linearity of the system, dynamics is invariant with respect to $v_0$.

As optimal conditions, we obtain for the NES that 70% of the initial energy is dissipated in 147 time units for $\mu_a = 0.1248$, if $v_0 = 0.504$. Regarding the TMD,
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 13.5** a, b Energy decrement on the first and on the second mode for system (13.2) with $\mu_a = 0.1248$, $\varepsilon = 0.01$, $\dot{y}_1(0) = \dot{y}_2(0) = 0.504$ and $y_1(0) = y_2(0) = 0$; c wavelet transformation of $y_3$; d, e, f energy decrement comparison with TMD ($\mu_a = 0.204$, $\gamma = 1.04$) for the initial conditions $y_1(0) = y_2(0) = 0$ and $\dot{y}_1(0) = \dot{y}_2(0) = 0.504$ (d), $\dot{y}_1(0) = 1$ and $\dot{y}_2(0) = 0.5$ (e), $\dot{y}_1(0) = 0.5$ and $\dot{y}_2(0) = 1$ (f)</figcaption></figure>

minimum dissipation time of 172 time units is obtained for $\mu_a = 0.204$ and $\gamma = 1.04$. In these conditions the NES outperform the TMD by 17% in terms of dissipation time.

Extensive numerical simulations show that, for optimal parameter values, the NES has a strong interaction with the first mode, keeping, at the same time, a weak interaction with the second one, therefore the modal cascade shown in Fig. 13.4 is no longer present. This can be clearly seen from the wavelet transformation of $y_3$ in Fig. 13.5c, which refers to the optimal NES. Figure 13.5a and b depict the energy decrement on the first and on the second mode of the primary system, respectively. It can be noted that energy decreases on both modes at a similar rate, until $T \approx 150$, when the NES disengages from both of them almost simultaneously.

Referring to an optimal TMD, because the value of $\gamma$ is only slightly larger than 1, $y_1$ oscillation amplitude undergoes a much rapid decrement than $y_2$; this was verified through direct numerical simulations, not shown here for the sake of brevity. Nevertheless, energy of the second mode is still dissipated thanks to the quite large damping ($\mu_a = 0.204$).

A comparison of the energy decrement obtained by the NES and by the TMD is illustrated in Fig. 13.5d, which shows that the NES only slightly outperforms the TMD. However, after dissipating 70% of the energy, it becomes almost ineffective, while the TMD works also for small amplitude. Figure 13.5e and f, depict the energy decrement adopting the same absorbers, but changing initial conditions. For $\dot{y}_1(0) = 1$ and $\dot{y}_2(0) = 0.5$ (more energy on the first mode, Fig. 13.5e) the NES has a significant deterioration of its performance, while the TMD an improvement. Conversely, increasing the initial energy on the second mode ($\dot{y}_1(0) = 0.5$ and $\dot{y}_2(0) = 1$,
<figure><img src="image_12.png" /><figcaption>**Fig. 13.6** Relative dissipation power for the NES (colored surface) and the TMD (orange surface) with $\epsilon = 0.01$ and $\mu_a = 0.12$ (NES) or $\gamma = 1.04$ and $\mu_a = 0.204$ (TMD)</figcaption></figure>

Fig. 13.5f), the TMD has slightly worst performance, while the NES has an important improvement.

This behavior can be better understood by plotting the relative dissipation power $P_r$ as a function of $a_1$ and $a_2$. This is defined by the equation $P_r = \epsilon\mu_a(b_1^2 + 3b_2^2) / (a_1^2 + 3a_2^2)$ and it is illustrated in Fig. 13.6 (colored surface) for $\epsilon = 0.01$ and $\mu_a = 0.1248$. In the figure, we notice that there are two peaks; the smaller is related to the first mode and the higher one to the second mode. These correspond to the points with maximum dissipation power (their height is $\epsilon/(4\mu_a)$) and $3\epsilon/(4\mu_a)$, respectively, as it can be computed by the hypothesis of single mode dynamics). For the initial conditions $\dot{y}_1(0) = \dot{y}_2(0) = 0.504$ the NES fully exploits the smaller peak, obtaining a good energy dissipation for the first mode. Increasing the initial energy on the first mode ($\dot{y}_1(0) = 1$), the peak of the second mode is still not exploited, while some additional time is required before reaching the peak of the first mode. This results in a deterioration of the performance. On the contrary, increasing the energy on the second mode ($\dot{y}_2(0) = 1$), the NES first exploits the peak relative to the second mode and then the one relative to the first one. This can be recognized observing that the energy decrement approximately consists of three linear segments. These correspond to dissipation of the energy on the second mode, then on the first mode and finally slowly dissipated residual energy.

Overlapping the relative dissipation power surface of the TMD (orange surface in Fig. 13.6) over the one of the NES, we can directly estimate the relative performance of the two absorbers. In general, for the same $a_1, a_2$ values, the surface with the higher $P_r$ value is the one providing better performance.
13.5 Conclusions

The dynamics of an NES attached to a two-DoF linear oscillator was investigated, evaluating the performance of the absorber against impulsive excitations. Invariant manifolds, describing the slow dynamics of the system, proved to be an effective tool to predict quite accurately the behavior and the performance of the absorber. In particular, they allowed us to explain how the NES couples with the primary system when more than one mode is activated. If the energy content on both modes is suffi- cient, the NES first dissipates energy on the higher mode and then, once a threshold is reached, it couples with the lower one abruptly decoupling from the higher one. A direct comparison between the NES and the TMD illustrated which conditions are required for the NES to outperform the TMD. Prediction of the comparative behavior based on the invariant manifold was confirmed by direct numerical simulations.

The verification of the existence of the described modal cascade for primary
systems with large number of DoF will be the subject of a future study.

**Acknowledgements** GH acknowledges the financial support of the European Union, H2020 Marie Skłodowska-Curie Individual Fellowship, Grant Agreement 704133. FR acknowledges the financial support of PRIN 2015 N. 2015TTJN95.

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