*3.3. Evaluation of the Absorber's Performance*

We now aim at evaluating the performance of the DVA by performing various comparison between the system with and without DVA. As mentioned above, the critical velocities for both cases are

$$v\_{\rm h,cr} = v\_0 \ln \left( \frac{\mu\_{\rm s} - \mu\_{\rm d}}{2v\_0 \zeta\_1} \right) \,, \tag{29}$$

$$v\_{\rm cr} = v\_0 \ln \left( \frac{\mu\_{\rm s} - \mu\_{\rm d}}{v\_0 \left( \sqrt{\varepsilon} + 2\zeta\_1 \right)} \right) \,. \tag{30}$$

Considering the characteristics of a logarithm function, we can state that, if the argument becomes 1, the logarithm function yields 0. Thus, there is a certain parameter set for which the critical velocity becomes 0 (inherent stability). We define *ζ*1,cr as the critical primary damping parameter, for which the critical velocity becomes 0. To obtain *ζ*1,cr, we solve the arguments of the logarithm functions for 1; these yield

$$\frac{\mu\_{\rm s} - \mu\_{\rm d}}{2v\_0 \zeta\_{1\rm h,cr}} = 1 \longrightarrow \zeta\_{1\rm h,cr} = \frac{\mu\_{\rm s} - \mu\_{\rm d}}{2v\_0} \, , \tag{31}$$

$$\frac{\mu\_{\rm s} - \mu\_{\rm d}}{v\_0 \left(\sqrt{\varepsilon} + 2\zeta\_{1, \rm cr}\right)} = 1 \longrightarrow \zeta\_{1, \rm cr} = \frac{\mu\_{\rm s} - \mu\_{\rm d} - v\_0 \sqrt{\varepsilon}}{2v\_0} \,. \tag{32}$$

Utilizing values in Table 3, the numerical values for the critical primary damping are

$$\mathcal{J}\_{1\text{h,cr}} = 0.5\,,\tag{33}$$

$$
\mathcal{L}\_{1,\text{cr}} = 0.388197.\tag{34}
$$

This shows that the application of an optimally tuned DVA with a mass of only 5% of the host system mass enables to reduce the critical primary damping of 22%.

Considering, instead, the critical velocity as a base of comparison, we define the improvement factor *ϕ* such that

$$
\varphi := \frac{\tilde{\upsilon}\_{\rm h,cr} - \tilde{\upsilon}\_{\rm cr}}{\tilde{\upsilon}\_{\rm h,cr}} \times 100\% \,. \tag{35}
$$

The critical velocities for both cases and the improvement factor are plotted against the varying *ζ*<sup>1</sup> in Figure 6, utilizing the parameter values in Table 3. We can observe that the difference in critical velocity is more significant for smaller values of *ζ*1, i.e., for a slightly damped host system. We also notice that, if the host system is completely undamped, then the critical velocity is undefined, meaning that the equilibrium is always unstable. For any value of the host system damping *ζ*1, the improvement factor is almost always above 50%.

Let us also observe what happens if we vary the value of the mass ratio *ε*. Similar to before, the critical velocities for both systems and the improvement factor curve are illustrated in Figure 7. The parameter values are those indicated in Table 3. The critical velocity of the host system is constant because it does not depend on *ε*; however, for the system with the DVA, the critical velocity monotonously decreases with *ε*. Utilizing the parameter values in Table 3, the critical velocities for both the host system and the system with DVA are *v*h,cr = 1.151 and *v*cr = 0.5641, hence the improvement provided by the DVA is of *ϕ* = 51%.

✶✸✾ ✶✹✵

✶✹✶ ✶✹✷ ✶✹✸ ✶✹✹ ✶✹✺

✶✹✻ ✶✹✼ ✶✹✽ ✶✹✾ ✶✺✵ ✶✺✶ ✶✺✷ ✶✺✸

✶✺✹

✶✺✺ ✶✺✻ ✶✺✼ ✶✺✽ ✶✺✾ ✶✻✵

**Figure 6.** Comparison of host system with and without DVA with varying *ζ*<sup>1</sup> , other parameters as in Table 3: (**a**) critical velocities; and (**b**) improvement curve.

**Figure 7.** Comparison of host system with and without DVA with varying *ε*: (**a**) critical velocities; and (**b**) improvement curve.

#### **4. Bifurcation Analysis of the Host System without the DVA**

The analysis performed in Section 3 refers to the system linearized around its equilibrium. Therefore, it is able to describe its dynamics only in the vicinity of the equilibrium, while phenomena occurring when the stability is lost are overlooked. Additionally, it provides no information about the stable equilibrium's robustness if the system is subject to non-small perturbations. To investigate the system behavior at the loss of stability and correctly evaluate the DVA performance, we reintroduce the nonlinear terms and analytically perform a bifurcation analysis of the system without and with DVA.

Considering the system in Equation (5), we first center the system around its equilibrium point *x*1e by introducing the variable *z*<sup>1</sup> = *x*1e + *x* ˜1, and then we expand it in Taylor series up to the third order, obtaining

$$
\begin{bmatrix} z'\_1 \\ z'\_3 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & \frac{\mu\_d - \mu\_s}{\nu\_0} \mathbf{e}^{-\frac{\mu}{\nu\_0}} - 2\zeta\_1 \end{bmatrix} \begin{bmatrix} z\_1 \\ z\_3 \end{bmatrix} + \begin{bmatrix} 0 \\ -\left(\frac{\mu\_d - \mu\_s}{2\nu\_0^2} \mathbf{e}^{-\frac{\mu}{\nu\_0}}\right) z\_3^2 - \left(\frac{\mu\_d - \mu\_s}{6\nu\_0^3} \mathbf{e}^{-\frac{\mu}{\nu\_0}}\right) z\_3^3 \end{bmatrix} = A\_\mathbf{h} z\_\mathbf{h} + b\_\mathbf{h} . \tag{36}
$$

For *v* ≈ *v*h,cr , matrix *A*<sup>h</sup> has complex conjugate eigenvalues *λ*1,2h = *α*1h ± i*ω*1h and eigenvectors *s*<sup>1</sup> = *s* ¯2, which are reduced to

$$
\lambda\_{1\text{h}} = \text{i}, \lambda\_{2\text{h}} = -\text{i}, \mathbf{s}\_1 = \begin{bmatrix} -\text{i} \\ 1 \end{bmatrix}, \mathbf{s}\_2 = \begin{bmatrix} \text{i} \\ 1 \end{bmatrix} \tag{37}
$$

for *v* = *v*h,cr.

We then define the transformation matrix

$$\mathbf{T\_h} = \begin{bmatrix} \text{Re}\left(\mathbf{s\_2}\right) & \text{Im}\left(\mathbf{s\_2}\right) \end{bmatrix} \Big|\_{\mathbf{v} = v\_{\mathbf{h}, \mathbf{r}}} \tag{38}$$

we apply the transformation *z* = *T*h*y*<sup>h</sup> and we pre-multiply Equation (36) by *T* −1 h , leading to

$$\mathbf{y}'\_{\mathbf{h}} = T\_{\mathbf{h}}^{-1} A\_{\mathbf{h}} T\_{\mathbf{h}} \mathbf{y}\_{\mathbf{h}} + T\_{\mathbf{h}}^{-1} \mathbf{b}\_{\mathbf{h}} = \mathbf{W}\_{\mathbf{h}} \mathbf{y}\_{\mathbf{h}} + \tilde{\mathbf{b}}\_{\mathbf{h}}.\tag{39}$$

where

$$\mathbf{W\_{h}} = \begin{bmatrix} a\_{1\text{h}}(v) & -1 \\ 1 & a\_{1\text{h}}(v) \end{bmatrix} \text{ and } \tilde{\mathbf{b\_{h}}} = \begin{bmatrix} -\left(\frac{\mu\_{d} - \mu\_{s}}{2v\_{0}^{2}} \mathbf{e}^{-\frac{\mathcal{V}}{v\_{0}}}\right) y\_{2}^{2} - \left(\frac{\mu\_{d} - \mu\_{s}}{6v\_{0}^{3}} \mathbf{e}^{-\frac{\mathcal{V}}{v\_{0}}}\right) y\_{2}^{3} \\ 0 \end{bmatrix}. \tag{40}$$

For *v* = *v*h,cr, *α*1h = 0, *α*1h is kept as a generic function of *v*, since *α*1h is the critical term causing the instability. The system in Equation (39) is in the so-called Jordan normal form.

By performing several transformations, namely transformation in complex form, near-identity transformation and transformation in polar coordinates, the bifurcation can be characterized through its normal form

$$r' = a\_{1\text{h}}(v)r + \delta\_{\text{h}}r^3 \,\text{.}\tag{41}$$

where

$$\delta\_{\mathbf{h}} = \frac{\mathbf{e}^{-\frac{\mathbf{v}}{\mathbf{v}\_{0}}}(\mu\_{\mathbf{s}} - \mu\_{d})}{16v\_{0}^{3}}.\tag{42}$$

Details of this standard procedure can be found in [26]. Non-zero real equilibrium solutions of Equation (41) correspond to periodic motion of the system in Equation (5). Linearizing *α*1h(*v*) around *v* = *v*h,cr, we obtain

$$r' = r \left( a\_{1\text{h}}^{\*} \left( v - v\_{\text{h}, \text{cr}} \right) + \delta\_{\text{h}} r^{2} \right), \text{ where } a\_{1\text{h}}^{\*} = \frac{\text{d} \mathfrak{a}\_{1\text{h}}}{\text{d}v} \Big|\_{v = v\_{\text{h}, \text{cr}}} = -\mathbf{e}^{-\frac{v}{v\_{0}}} \frac{\mu\_{\text{s}} - \mu\_{\text{d}}}{2v\_{0}^{2}},\tag{43}$$

which has solutions

$$r = r\_{\rm h0} = 0 \text{ and } r = r\_{\rm h}^\* = \sqrt{-\frac{a\_{\rm 1h}^\* \left(v - v\_{\rm h,cr}\right)}{\delta\_{\rm h}}} = 2\sqrt{2v\_0 \left(v - v\_{\rm h,cr}\right)}.\tag{44}$$

The trivial solution *r*h0 exists for any value *v* and it is stable for *v* > *v*h,cr. Differently, *r* ∗ h is real only if the argument of the square root in Equation (44) is non-negative, which occurs for *v* > *v*h,cr. Since *µ*<sup>s</sup> > *µ*d, in all relevant cases *δ* is positive (see Equation (42)), which, as clearly explained in [26], means that the bifurcation is subcritical. This implies that *r* ∗ h corresponds to unstable solutions of Equation (41). This result is in accordance with [1].

A practical consequence of the subcritical character of the bifurcation is that the system, even within the stable region of the equilibrium (*v* > *v*h,cr), can experience large oscillations. If the system, while in equilibrium, is subject to a sufficiently large perturbation, which makes it cross the unstable periodic solution in the phase space, it will leave its basin of attraction and it will reach another attractor, which in this case consists of stick–slip oscillations.

The bifurcation diagram in Figure 8a clearly illustrates this scenario. The dashed line indicates a branch of unstable periodic solutions generated at the bifurcation (this branch was obtained through time reverse numerical simulations). The solid line, instead, marks the branch of stick–slip oscillation. The thin solid red line represents the branch of unstable periodic solutions obtained from the analytical computation. We remark on the excellent agreement of the analytically computed solution with the numerical one at low amplitudes. For *v* ∈ [1.15, 1.83], the system presents two stable solutions, the

trivial one and a stick–slip periodic solution, and an unstable periodic solution, as illustrated in Figure 8b for *v* = 1.3. Depending on the initial conditions, the system will either converge towards the trivial solution (red curve in Figure 8c) or will undergo stick–slip oscillations (blue curve in Figure 8c). Numerical solutions were computed utilizing the switch model proposed in [4].

**Figure 8.** (a) bifurcation diagram for the host system without DVA; the thin red line marks analytical ✶✽✽ ✶✽✾ **Figure 8.** (**a**) Bifurcation diagram for the host system without DVA; the thin red line marks analytical solutions, black lines numerical ones and dashed lines the numerical unstable solutions. (**b**) Steady state solutions of the system for *v* = 1.3; solid line is the stable solution and dashed line is the unstable solution. (**c**) Time series of the system leading to the steady state solutions represented in (**b**) with initial conditions *z<sup>h</sup>* = [1.367, 0] T (blue line) and *z<sup>h</sup>* = [1.36, 0] T (red line). Other parameter values are as in Table 3.

#### ✶✾✵ ✶✾✶ **5. Bifurcation Analysis of the Host System with the DVA**

✶✾✷ ✶✾✸ ✶✾✹ To evaluate the DVA performance when stability is lost, we analyze the bifurcation behavior of the system with an attached DVA. The analysis is performed assuming that *γ* and *ζ*<sup>2</sup> are tuned approximately according to Equation (25). An analysis of the eigenvalues of matrix *A* illustrates that at the loss of stability, if *γ* and *ζ*<sup>2</sup> are tuned approximately according to Equation (25), a couple of complex conjugate eigenvalues leaves the left-hand side of the complex plane, meaning that their real parts become positive. This scenario corresponds to the occurrence of a Hopf bifurcation. We also notice that, if *ζ*<sup>2</sup> ≤ *ζ*2opt and *γ* = *γ*opt, not one, but two couples of complex conjugate eigenvalues leave the left-hand side of the complex plane. Referring to the stability chart in Figure 4a, the entire unstable region matrix *A* has only one couple of eigenvalues with positive real part, except in the loop delimited by Points C and P, where all four eigenvalues have positive real part. This scenario corresponds to a Hopf–Hopf (or double Hopf) bifurcation. In the following, the case of a single Hopf bifurcation is analyzed.

The first step of the analysis consists of transforming the system in Equation (9) into first-order form, similar to Equation (14), but including nonlinear terms up to the third order, which leads to

$$
\begin{bmatrix} z'\_1 \\ z'\_2 \\ z'\_3 \\ z'\_4 \\ z'\_4 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & -\gamma^2 \varepsilon & 2\up{\wp} & -2\varepsilon\zeta\_2\gamma \\ -1 & -\gamma^2(\varepsilon+1) & 2\up{\wp} & -2(\varepsilon+1)\gamma\zeta\_2 \end{bmatrix} \begin{bmatrix} z\_1 \\ z\_2 \\ z\_3 \\ z\_4 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ -\frac{z\_3^2(\mu\_4-\mu\_4)e^{-\frac{\mu\_3}{60}(3\eta\_0+z\_3)} - \varepsilon k\_{\text{n}\Omega}2\_2^3}{6\upsilon\_0^3} \\ -\frac{z\_3^2(\mu\_4-\mu\_4)e^{-\frac{\mu\_3}{60}(3\eta\_0+z\_3)} - (\varepsilon+1)\varepsilon k\_{\text{n}\Omega}2\_2^3}{6\upsilon\_0^3} \end{bmatrix} \tag{45}
$$
 
$$= A\underline{z} + b$$
.

In the vicinity of the loss of stability, *A* has two couples of complex conjugate eigenvalues *λ*1,2 = *α*<sup>1</sup> ± i*ω*<sup>1</sup> and *λ*3,4 = *α*<sup>2</sup> ± i*ω*2. To decouple the linear part of the system, we define the transformation matrix

$$T = \begin{bmatrix} \operatorname{Re}\left(\mathfrak{s}\_1\right) & \operatorname{Im}\left(\mathfrak{s}\_1\right) & \operatorname{Re}\left(\mathfrak{s}\_3\right) & \operatorname{Im}\left(\mathfrak{s}\_3\right) \end{bmatrix} \\ \text{ }\tag{46}$$

where *s*<sup>1</sup> and *s*<sup>3</sup> are two of the eigenvectors of *A*, and we apply the transformation *z* = *Ty*, obtaining

$$\dot{y} = T^{-1}Ay + T^{-1}b = \mathcal{W}y + \tilde{b} \,. \tag{47}$$

where

$$\mathbf{W} = \begin{bmatrix} \omega\_1 & \omega\_1 & 0 & 0 \\ -\omega\_1 & \omega\_1 & 0 & 0 \\ 0 & 0 & \omega\_2 & \omega\_2 \\ 0 & 0 & -\omega\_2 & \omega\_2 \end{bmatrix} \tag{48}$$

For the sake of brevity, the explicit formulation of *b*˜ is omitted here.

In the case of a single Hopf bifurcation, only *α*<sup>1</sup> becomes positive at the loss of stability, while *α*<sup>2</sup> remains negative. Therefore, only the first two equations of Equations (47) are linearly related to the bifurcation, while *y*<sup>3</sup> and *y*<sup>4</sup> have minor local effect at the bifurcation. Next, we aim at reducing the dynamics of the system to the so-called *center manifold*, which is a two-dimensional surface tangent at the bifurcation point to the subspace spanned by the two eigenvectors *s*<sup>1</sup> and *s*<sup>2</sup> related to the bifurcation. To do so, we approximate *y*<sup>3</sup> and *y*<sup>4</sup> by *y*<sup>3</sup> = *η*320*y* 2 <sup>1</sup> + *η*311*y*1*y*<sup>2</sup> + *η*302*y* 2 2 and *y*<sup>4</sup> = *η*420*y* 2 <sup>1</sup> + *η*411*y*1*y*<sup>2</sup> + *η*402*y* 2 2 , reducing the system to

$$\begin{aligned} y\_1 &= \alpha\_1 y\_1 + \omega\_1 y\_2 + \sum\_{j+k=2,3} a\_{jk} y\_1^j y\_2^k + \text{h.o.t.}\\ y\_2 &= -\omega\_1 y\_1 + \alpha\_1 y\_2 + \sum\_{j+k=2,3} b\_{jk} y\_1^j y\_2^k + \text{h.o.t.} \end{aligned} \tag{49}$$

where *j* and *k* are non-negative integers (more details on this procedure can be found in [26]) and h.o.t. stands for higher order terms.

The system in Equation (49) has the same form as Equation (39); therefore, exactly the same steps can be performed to reduce the system to its normal form, that is

$$r' = \mathfrak{a}\_1(\upsilon)r + \delta r^3,\tag{50}$$

where [26]

$$\delta = \frac{1}{8} \left( \frac{1}{\omega\_1} \left( (a\_{20} + a\_{02}) \left( -a\_{11} + b\_{20} - b\_{02} \right) + (b\_{20} + b\_{02}) \left( a\_{20} - a\_{02} + b\_{11} \right) \right) + \left( 3a\_{30} + a\_{12} + b\_{21} + 3b\_{03} \right) \right) . \tag{51}$$

Imposing *ε* = 0.05, *γ* = *γ*opt and *ζ*<sup>2</sup> = 1.05 *ζ*2opt, we obtain

$$
\delta = 0.00474 + 0.68 \, k\_{\text{nl2}} \, . \tag{52}
$$

Proceeding as done for the host system without DVA, we have that the non-trivial solutions of Equation (50) is given by

$$r = r^\* = \sqrt{-\frac{\mathfrak{a}\_1^\* \left(v - v\_{\rm cr}\right)}{\delta}},\tag{53}$$

where *α* ∗ <sup>1</sup> = d*α*1/d*v*|*<sup>v</sup>* = *v*cr. We notice that *δ* is positive if the DVA is linear (*k*nl2 = 0), which means that also in this case the bifurcation is subcritical, and it generates unstable periodic solutions. Analyzing other values of *ζ*<sup>2</sup> and *ε*, we verified that the subcritical characteristic persists for a relatively large parameter value range. The corresponding bifurcation diagram is illustrated in Figure 9a. Comparing Figures 9a and 8a, we notice that, although the linear DVA does not change the characteristic of the bifurcation, the advantages in terms of vibrations suppression persist also in the nonlinear range. In fact, for the considered parameter values, in the host system without DVA, stick–slip oscillations exist for *v* ∈ (0, 1.83], while, with the addition of the absorber, they are limited to the range *v* ∈ (0, 0.768].

✷✵✸

**Figure 9.** Bifurcation diagrams for the host system with the DVA for parameter values as in Table 3, *γ* = *γ*opt and *ζ*<sup>2</sup> = *ζ*2opt: (**a**) *k*nl2 = 0, (**b**) *k*nl2 = −0.01; and (**c**) *k*nl2 = 0.01. Solid lines are stable solutions, dashed lines are unstable solutions and thin red lines are analytical solutions.

**Figure 9.** Bifurcation diagrams for the host system with the DVA for parameter values are as in Table 3,

✷✵✹ ✷✵✺ ✷✵✻ ✷✵✼ ✷✵✽ ✷✵✾ ✷✶✵ ✷✶✶ ✷✶✷ ✷✶✸ ✷✶✹ Equation (52) suggests that, if *<sup>k</sup>*nl2 <sup>&</sup>lt; <sup>−</sup>0.00697, *<sup>δ</sup>* becomes negative, making therefore the bifurcation supercritical. This scenario is confirmed by the bifurcation diagram depicted in Figure 9b, for *k*nl2 = −0.01. Although at first sight it seems that the bifurcation is subcritical, the inset illustrates that the bifurcation is indeed supercritical; however, the branch of periodic solutions bends rapidly to the right in correspondence of a fold bifurcation, making the overall scenario similar to the case of *k*nl2 = 0. The figure confirms the correctness of the analytical computation; nevertheless, it also points out that the performed local analysis is unable to capture the global behavior of the system, which is not qualitatively affected by the variation of the nonlinear characteristic of the DVA's spring. Furthermore, we notice that the addition of the softening nonlinear spring enlarges the bistable range, making stick–slip oscillations exist up to *v* < 0.86, instead of 0.768 as in the case of *k*nl2 = 0.

✷✶✺ ✷✶✻ ✷✶✼ ✷✶✽ ✷✶✾ ✷✷✵ ✷✷✶ ✷✷✷ Figure 9c illustrates the bifurcation diagram obtained for a hardening absorber's spring (*k*nl2 = 0.01). In this case, the range of existence of stick–slip oscillations is further enlarged, persisting up to *v* < 1.035. We also remark that increasing the value of *k*nl2 above 0.01 or decreasing it below −0.01 provided only worse performance than those illustrated in Figure 9. This result suggests that any low order nonlinearity of the absorber's stiffness is detrimental concerning the DVA effectiveness. This finding is somehow surprising, considering that in similar applications the addition of a properly tuned nonlinear term in the DVA's stiffness provided some advantages [20,25,27].

✷✷✸ ✷✷✹ ✷✷✺ ✷✷✻ ✷✷✼ ✷✷✽ ✷✷✾ ▼❛t❈♦♥t ▼❆❚▲❆❇ ✷✸✵ Regarding Figure 9b, we notice that the branch of stick–slip oscillations presents two folds for *v* ≈ 0.48. However, an analysis of the system's steady state solutions before and after the folds did not reveal any particular detail relevant from an engineering point of view; therefore, the phenomenon was not analyzed in further detail. We also remark that, in Figure 9b,c, the branches of stable and unstable solutions do not encounter each other at a well defined point, as happens in Figure 8a, for instance. This is probably related to the fact that the branches of unstable solutions in Figure 9 were obtained adopting the shooting method (employing MatCont [28], a MATLAB-based toolbox for numerical continuation) of the system smoothed assuming that *v*rel is always positive. This assumption makes the considered system unable to exhibit stick–slip oscillations, but keeps it equivalent to the original system for *v* > *z*3. In contrast, the stable branches were obtained from direct numerical simulations of the full system. Therefore, inaccuracies of the smoothed system in the proximity of the onset of stick–slip motions are possible.

As mentioned at the beginning of this section, for *ζ*<sup>2</sup> < *ζ*2opt and *γ* = *γ*opt, the system undergoes a Hopf–Hopf bifurcation. However, acknowledging the fact that the bifurcation analysis seems to be an inefficient tool for investigating the post-bifurcation behavior of the system, which is dominated by large amplitude oscillations, and considering that the analysis of such a bifurcation requires a significant analytical effort, the detailed investigation of this case is omitted in this study.

#### **6. Conclusions**

In this study, the problem of suppressing FIVs through a DVA was addressed. Possibly the simplest system exhibiting FIVs was considered, i.e., the mass-on-moving-belt system, to which a classical DVA was attached. The optimal tuning of the absorber parameters was defined through an analytical procedure, which enabled us to reduce the critical velocity by approximately 50%, with an additional mass of only 5% of the primary system's mass.

The post-bifurcation behavior analysis illustrated that, although a linear DVA is unable to change the bifurcation character at the loss of stability, it can still significantly reduce the extent of the bistable region. Globally, the area of existence of stick–slip oscillations is reduced by 58%, with a DVA mass of only 5%. The bifurcation analysis proved that it is possible to change the bifurcation character if a small softening term is included in the absorber. However, this has only a local beneficial effect, while, globally, it enlarges the region of existence of stick–slip motions. The performance also worsens if an additional hardening term is introduced, suggesting that the spring characteristic should be maintained as linear as possible. Large order nonlinearities, such as non-smoothness, might have beneficial effects; nevertheless, their analysis was not addressed in this study, and it is left for future developments. Other possible future developments of the present study include the analysis of the Hopf–Hopf bifurcation occurring at the loss of stability for *ζ*<sup>2</sup> < *ζ*2opt and the analysis of the performance of the DVA if the primary system has two DoF, encompassing, therefore, coupling instabilities as well [22].

**Author Contributions:** Conceptualization, G.H.; Data curation, J.L.H. and G.H.; Formal analysis, J.L.H. and G.H.; Funding acquisition, G.H.; Investigation, J.L.H. and G.H.; Methodology, J.L.H. and G.H.; Resources, G.H.; Software, J.L.H. and G.H.; Supervision, G.H.; Validation, J.L.H. and G.H.; Visualization, J.L.H. and G.H.; Writing—original draft, J.L.H. and G.H.; and Writing—review and editing, J.L.H. and G.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Hungarian National Science Foundation under grant number OTKA 134496 and by the NRDI Fund (TKP2020 IES, Grant No. BME-IE-NAT, and TKP2020 NC, Grant No. BME-NC) based on the charter of bolster issued by the NRDI Office under the auspices of the Ministry for Innovation and Technology.

**Conflicts of Interest:** The authors declare no conflict of interest.
