*2.1. Primary System*

The primary system considered in this study is the classical mass-on-moving-belt model, which is an archetypal system for studying FIVs [1,4,21]. As shown in Figure 1, this single-DoF system consists of a mass *m*1, a linear spring *k*<sup>1</sup> and a linear damper *c*1. The mass of the system is in contact with the belt, which moves at a constant driving speed *v*, while the friction coefficient *µ*(*v*rel ) of the contact is a function of the relative velocity *v*rel = *v* − *x* ˙1. ✻✶ ✻✷

**Figure 1.** (**a**) The host system without absorber; and (**b**) free body diagram of the host system.

Figure 1b illustrates the free body diagram (FBD) of the host system. The forces acting upon the lumped mass *m*<sup>1</sup> are the normal forces *F*<sup>N</sup> that cancel each other; the damping and spring forces *Fc*<sup>1</sup> and *Fk*<sup>1</sup> , respectively; and the friction force *F*R. (For the represented FBD, it is assumed that *v*rel ≥ 0.) Based on Newton's second law of motion and considering that *F*<sup>R</sup> = *µ* (*v*rel) *F*N, *Fc*<sup>1</sup> = *c*1*x* ˙<sup>1</sup> and *Fk*<sup>1</sup> = *k*1*x*1, we obtain that the second order differential equation describing the motion of the one DoF system is

$$m\_1\ddot{\mathbf{x}}\_1 + c\_1\dot{\mathbf{x}}\_1 + k\_1\mathbf{x}\_1 = F\_\mathbf{R} \,\prime \tag{1}$$

with

$$\begin{cases} \quad F\_{\rm R} = \mu \left( v\_{\rm rel} \right) F\_{\rm N} & v\_{\rm rel} \neq 0 \\ \quad |F\_{\rm R}| \le \mu\_{\rm s} F\_{\rm N} & v\_{\rm rel} = 0 \end{cases} \tag{2}$$

where the overdot indicates derivation with respect to the time *t*. The system is in stick condition when the relative velocity is zero (*v*rel = 0). In this case, the friction force is smaller or equal to *µ*s*F*N, where *µ*<sup>s</sup> is the static friction. In sliding condition (*v*rel 6= 0), the direction of the friction force *F*<sup>R</sup> depends on the sign of the relative velocity *v*rel , which is included in the mathematical formulation of *µ* (*v*rel). Additional details about the friction coefficient utilized are provided in Section 2.3. Let us introduce the following expressions

$$\mathcal{L}\_1 = \frac{c\_1}{2\sqrt{m\_1 k\_1}} ; \; \omega\_{\mathbf{n}1} = \sqrt{\frac{k\_1}{m\_1}} ; \; \mathbf{x}\_0 = \frac{F\_\mathbf{N}}{k\_1} ; \; \mathbf{\tau} = \omega\_{\mathbf{n}1} \mathbf{t} . \tag{3}$$

By dividing Equation (1) by the mass *m*1, applying the expressions from Equation (3) and dividing it by *ω*<sup>2</sup> n1 , we obtain

$$\mathbf{x}\_1^{\prime\prime} + 2\boldsymbol{\zeta}\_1 \mathbf{x}\_1^{\prime} + \mathbf{x}\_1 = \frac{F\_\mathbf{R}}{k\_1} \, , \tag{4}$$

where prime ′ indicates derivation with respect to the dimensionless time *τ*. Then, introducing the dimensionless displacement *x* ˜ <sup>1</sup> = *x*1/*x*0, the system is eventually reduced to

$$\mathfrak{X}\_1^{\prime\prime} + 2\mathfrak{z}\_1 \mathfrak{x}\_1^{\prime} + \mathfrak{x}\_1 = \mathfrak{F}\_{\mathbb{R}} \,\,\, \, \tag{5}$$

where *F* ˜ <sup>R</sup> = *F*R/*F*N.

✻✹

✻✺

✻✾
