**3. Linear Stability Analysis**

We now aim at analyzing the system's behavior for small perturbations around the equilibrium. Thus, we linearize it around its trivial solution and we study the stability of this trivial solution. The analysis is performed for both the systems, with and without DVA, in order to assess the beneficial effects of the DVA.

## *3.1. Linear Stability of the Host System without the DVA*

We first linearize Equation (5) around its equilibrium point *x*1e = *µ*(*v*rel = *v*), obtaining

*z* ′′ <sup>1</sup> − 2*ψz* ′ <sup>1</sup> + *z*<sup>1</sup> = 0 , (11)

where *z*<sup>1</sup> = *x*1e + *x*˜<sup>1</sup> and 2*ψ* = *∂µ*/*∂z* ′ 1 *z* ′ <sup>1</sup>=0 − 2*ζ*1. Considering the friction law adopted, we have that

$$\left. \frac{\partial \mu}{\partial z\_1'} \right|\_{z\_1'=0} = \frac{\mu\_s - \mu\_d}{v\_0} \mathbf{e}^{-\frac{v}{v\_0}}\,,\tag{12}$$

(valid for *v* > 0), which is the slope of the friction force coefficient at the belt velocity *v*.

Equation (11) corresponds to a linear oscillator, whose trivial solution is asymptotically stable if and only if *ψ* < 0. According to the friction law utilized, and considering that *ψ* is a monotonically decreasing function of *v*, the trivial solution is stable for

$$v > v\_{\rm h,cr} = v\_0 \ln \left( \frac{\mu\_{\rm s} - \mu\_{\rm d}}{2v\_0 \zeta\_1} \right) \,. \tag{13}$$

This result is well-known and better discussed, for instance, in [1]. The practical consequences are that, if the belt moves at a speed lower than *v*h,cr, the equilibrium of the system is unstable and stick–slip oscillations occur. More details about these stick–slip oscillations are provided below.

#### *3.2. Linear Stability of the Host System with DVA*

To study the stability of the system with the DVA, we linearize Equation (9) around the equilibrium *x*˜<sup>1</sup> = *x*1e and *x*˜<sup>3</sup> = 0. By reformulating Equation (9) in explicit form with respect to *x*˜ ′′ 1 and *x*˜ ′′ 3 and by utilizing the variables and parameters introduced in the previous subsection, we obtain

$$
\begin{bmatrix} z'\_1 \\ z'\_2 \\ z'\_3 \\ z'\_4 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & -\gamma^2 \varepsilon & 2\psi & -2\epsilon \zeta\_2 \gamma \\ -1 & -\gamma^2(\varepsilon+1) & 2\psi & -2(\varepsilon+1)\gamma \zeta\_2 \end{bmatrix} \begin{bmatrix} z\_1 \\ z\_2 \\ z\_3 \\ z\_4 \end{bmatrix} = A \mathbf{z}\_{\prime} \tag{14}$$

where *z*<sup>1</sup> = *x*1e + *x*˜1, *z*<sup>2</sup> = *x*3, *z*<sup>3</sup> = *z* ′ 1 and *z*<sup>4</sup> = *z* ′ 2 .

We analyze the characteristic exponents of the system to determine the stability. The characteristic polynomial is

$$p(\lambda) = \det\left(A - \lambda I\right) = 0.\tag{15}$$

The characteristic polynomial is in the form of

$$p(\lambda) = a\_0 \lambda^4 + a\_1 \lambda^3 + a\_2 \lambda^2 + a\_3 \lambda + a\_4 = 0 \quad (a\_0 > 0) \,. \tag{16}$$

The stability is determined based on the *Liénard–Chipart conditions* (LCC) [24]. For the polynomial *p*(*λ*) to have all roots with negative real parts, it is necessary and sufficient that the coefficients of the polynomial are all positive (*a<sup>i</sup>* > 0, *i* = 1, . . . , 4) and that the determinant inequalities ∆<sup>1</sup> > 0 and ∆<sup>3</sup> > 0 (Hurwitz determinantal inequalities) are verified, which in our case means

$$a\_0 = 1 > 0,\tag{17}$$

$$a\_1 = \mathcal{Z}\left(\gamma\left(1+\varepsilon\right)\zeta\_2 - \psi\right) > 0 \Longleftrightarrow \psi < \left(1+\varepsilon\right)\gamma\zeta\_2\tag{18}$$

$$a\_2 = 1 + \gamma \left(\gamma \left(1 + \varepsilon\right) - 4\zeta\_2\psi\right) > 0 \Longleftrightarrow \psi < \frac{1 + \gamma^2 \left(1 + \varepsilon\right)}{4\gamma\zeta\_2},\tag{19}$$

$$a\_3 = 2\gamma \left(\zeta\_2 - \gamma \psi\right) > 0 \Longleftrightarrow \psi < \frac{\zeta\_2}{\gamma} \,. \tag{20}$$

$$a\_4 = \gamma^2 > 0,\tag{21}$$

$$
\Delta\_1 = a\_1 = 2\gamma \left( 1 + \varepsilon \right) \zeta\_2 - 2\psi > 0 \text{ (Already present in (18)) }, \tag{22}
$$

$$\begin{split} \Delta\_3 &= a\_1 a\_2 a\_3 - a\_0 a\_3^2 - a\_1^2 a\_4 = -4\gamma \left( \gamma^4 (\varepsilon + 1)^2 \zeta\_2 \psi - \gamma^3 \psi^2 \left( 4(\varepsilon + 1)\zeta\_2^2 + \varepsilon \right) \right. \\ &\left. + 2\gamma^2 \zeta\_2 \psi \left( 2(\varepsilon + 1)\zeta\_2^2 + 2\psi^2 - 1 \right) - \gamma \zeta\_2^2 \left( \varepsilon + 4\psi^2 \right) + \zeta\_2 \psi \right) > 0. \end{split} \tag{23}$$

#### 3.2.1. Analytical Optimal Solution

✾✵

✾✶

✾✺ ✾✻ ✾✼ ✾✽

Considering the stability analysis performed in the previous section, we aim at finding the parameter values of the absorber which maximize the stable region. The linear system in Equation (14) has the same mathematical form as the one studied in [25]. Thus, we can follow the same steps in order to optimize the absorber.

First, we look at the curves where the coefficients and the Hurwitz determinants are zeros; these are the boundaries where certain roots change. At certain boundaries, the stable/unstable transition takes place. Following the steps discussed in [25], we can define specific points on these curves, which helps us find the optimal parameters. Figure 4 shows the stability regions for different values of *ζ*2; the boundary curves of the LCC are also depicted. The gray shaded region is the stable region; we can observe that the stability boundary is not at a constant value of *ψ* as it is the case for the host system without the DVA; instead, it is a function of *γ*, with a pronounced peak for *γ* ≈ 1. This is an expected feature, considering that the DVA usually needs to be tuned at a frequency close to the natural frequency of the primary system [14]. For different values of *ζ*2, the maximum value of *ψ* also changes, thus we need to find the optimal combination of (*ζ*2, *γ*) such that the value of *ψ* generating instabilities is maximized. For the optimization, the mass ratio *ε* is assumed constant; however, the results of the analysis show that larger values of *ε* increase the stable region. ✽✻ ✽✼ ✽✽ ✽✾

**Figure 4.** Stability diagrams for different values of *ζ*<sup>2</sup> and *ε* = 0.05: (**a**) *ζ*<sup>2</sup> = 0.07; (**b**) *ζ*<sup>2</sup> = 1/2p *ε*/ (1 + *ε*) = 0.109109; and (**c**) *ζ*<sup>2</sup> = 0.13.

✾✷ ✾✸ ✾✹ As we can see in Figure 4a, the intersection of the curves *a*<sup>1</sup> = ∆<sup>1</sup> = 0 and *a*<sup>3</sup> = 0 defines a point that we denote with C. Its coordinates in the (*ψ*, *γ*) plane are C = *ζ*2 √ 1 + *ε*, 1/ √ 1 + *ε* . For low values of *ζ*2, Point C marks the maximal value of *ψ* providing stability, which we call *ψ* ∗ . Increasing

the value of *ζ*2, Figure 4c illustrates that Point C does not identify any more *ψ* ∗ and that the stability boundary is defined by Equation ∆<sup>3</sup> = 0. Nevertheless, also for large damping *ζ*2, the value of *γ* which maximizes *ψ* ∗ is very close to the *γ* coordinate of Point C. Therefore, we define Point P as the point of the curve ∆<sup>3</sup> = 0 having the same *γ* coordinate as C. Substituting the *γ* coordinate of C, *γ* = 1/ √ 1 + *ε*, into ∆<sup>3</sup> = 0, we obtain Point P = *ε*/ 4*ζ*<sup>2</sup> √ 1 + *ε* , 1/<sup>√</sup> 1 + *ε* .

Since Points C and (approximately) P alternatively mark *ψ* ∗ , depending on the value of the absorber damping *ζ*2, by choosing *ζ*<sup>2</sup> such that P and C coincide, we can maximize *ψ* ∗ . Imposing equality between the *ψ* coordinates of C and P, we attain

$$
\zeta\_2 \sqrt{1+\varepsilon} = \frac{\varepsilon}{\left(4\sqrt{1+\varepsilon}\zeta\_2\right)}\,\,\,\,\tag{24}
$$

therefore, *ψ* ∗ is maximized for

$$
\gamma = \gamma\_{\rm opt} = \frac{1}{\sqrt{1 + \varepsilon}} \quad \text{and} \quad \zeta\_2 = \zeta\_{2\rm opt} = \frac{1}{2} \sqrt{\frac{\varepsilon}{1 + \varepsilon}} \tag{25}
$$

and the corresponding maximal value of *ψ* ∗ is

*ψ*max = √ *ε* 2 . (26)

Accordingly, if *γ* = *γ*opt and *ζ*<sup>2</sup> = *ζ*2opt, the system is stable if 2*ψ* < √ *ε*, or, in other words, if

$$\frac{\mu\_{\rm s} - \mu\_{\rm d}}{v\_0} \mathbf{e}^{-\frac{\overline{v}}{v\_0}} - 2\zeta\_1 < \sqrt{\varepsilon}. \tag{27}$$

Considering that *ψ* is a monotonically decreasing function of *v*, the equilibrium of the system is stable if

$$v > v\_{\rm cr} = v\_0 \log \left( \frac{\mu\_{\rm s} - \mu\_{\rm d}}{v\_0 \left( \sqrt{\varepsilon} + 2\zeta\_1 \right)} \right) \,. \tag{28}$$

The stability chart corresponding to this case is illustrated in Figure 4b. We remark that the optimal tuning proposed here does not strictly depend on the friction law considered, which could be modeled with alternative functions without varying *γ*opt and *ζ*2opt. This represent a clear advantage in the case of real engineering applications.
