**1. Introduction**

Friction-induced vibrations (FIVs) are a peculiar type of oscillations generated by the friction acting between two bodies in relative motion. They consist of either the successions of stick and slip phases between the two bodies [1], or of quasi-harmonic oscillations having an approximately sinusoidal displacement–time relation [2,3]. Although for some specific applications these kinds of vibrations are intentionally generated, such in the case of violin strings [4] and singing wine glasses [5], typically they are seen as a detrimental phenomenon, as in the cases of brake squeal [6] or earthquakes [7,8].

Several methods exist for suppressing FIVs. One possibility is to reduce the friction force at the interface utilizing a lubricant. This method is efficient if friction is not required for the device to operate, such as in the case of hinge squeaking; however, it cannot be adopted for brake squeal mitigation, where high friction is strictly required. Most brake squeal suppression methods consist of increasing the system damping, which is obtained with various techniques [6,9]. Experimental observations also illustrated that isolating the natural frequencies of the brake system's components at low frequencies tends to reduce the occurrence of audible brake squeal [10]; however, in many cases, this strategy is not effective [11]. For active methods for suppressing FIV, Cunefare and Graf [12] proposed adopting a dither exciting the system at non-audible frequencies, which can suppress brake squeal. Papangelo and Ciavarella [13] proposed to mitigate FIVs by normal load variation, for which they provided a closed-form solution.

The dynamic vibration absorber (DVA) is a practical tool for suppressing undesired vibrations in several engineering applications. Its classical design [14] consists of a mass attached to the host structure through a spring and a damper. By tuning its natural frequency in correspondence of the frequency to be damped, it is able to dynamically interact with the host structure dissipating vibration energy. It is successfully employed in several engineering fields for the suppression of various kinds of vibrations, such as flutter instabilities [15,16], rolling motions in ships [17], helicopter rotor oscillations [18] and machine tool vibrations [19,20]. Although DVAs are a mature technology, which was first proposed more than one hundred years ago [17], to the authors' knowledge, there are only a few and relatively recent studies addressing its implementation to suppress FIVs. Popp and Rudolph [21] numerically and experimentally analyzed the performance of a DVA for FIV suppression; by utilizing a single-degree-of-freedom (DoF) primary system, they illustrated its beneficial effect. Chatterjee [22] studied the stability properties of an undamped DVA attached to a two-DoF primary system. Very recently, Niknam and Farhang [23] proposed a study similar to that of Chatterjee [22], where they also provided some numerical simulations of the full system, missing in [22]. Despite the promising results obtained in [21–23], a clear tuning strategy of the DVA's parameters for maximizing its performance is still missing. This paper aims to fill this gap by providing a precise tuning of the absorber parameters for optimizing stability properties and studying the behavior of the host system with the attached DVA while stability is lost.

The rest of the paper is organized as follows. In Section 2, the mechanical model, consisting of the host mass-on-moving-belt system and the attached DVA, is introduced. In Section 3, the stability analysis of the host system, without and with the DVA, is performed, providing explicit equations for the optimal tuning of the absorber parameters. In Sections 4 and 5, the bifurcations occurring at the loss of stability of the host system, without and with absorber, are analytically studied. Furthermore, the effect of the addition of a cubic term in the absorber's restoring force is analytically investigated; analytical results are integrated by numerical simulation, illustrating the system's behavior at high amplitudes. In Section 6, conclusions about the benefits and limitations of the DVA are presented.
