## 1 Introduction  
Tire/road noise is becoming a major source of vehicle noise as the transition towards electrical vehicles is underway. This has resulted in tire manufacturers refining their tire design to help reduce traffic noise. The basic physical principles of tire/pavement noise are well understood and documented [1]. For the interested reader, a literature review of design considerations on parameters influencing tire/pavement interaction noise can be found in [2] and a complete review of models on tire-pavement interaction noise is also provided in [3]. The tire/pavement noise generation mechanisms can be divided into two categories. The first one is an air-flow mechanism that generates noise at high frequencies (above 1000 Hz) and is commonly referred to as air-pumping. The second one is the tire vibrations mechanism that generates noise at low and medium frequencies (below 1000 Hz). These noise generation mechanisms are well described in [4]. This paper focuses on the latter mechanism. The tire solid vibrations are generated by two excitation sources. The first one is the road roughness for which the resulting vibrations could be well predicted for axi-symmetric tires using an ALE-formulation [5, 6]. This approach consists of using a fixed mesh through which the tire material particles flow. This approach gives a stationary state of the loaded rotating tire. Afterward, the small vibrations due to the road excitation could be computed using a modal superposition technique [7, 8]. Most industrial models found in the literature are based on this approach. The second excitation source is the tire's groove resonance. The small scale tread deformation has an impact on the noise emitted by the tire [9]. The main goal of the grooves is to ensure sufficient grip in wet conditions in order to avoid hydroplaning; however, the tire groove geometry parameters (length, width, angle, depth...) and the material proprieties change the vibratory behavior of the tire [10,11]. The prediction of the vibrations induced by this second mechanism can be more complex to achieve and requires the implementation of a specific methodology. Indeed, obtaining a stationary state is no longer possible and full Lagrangian vision should be adopted to treat the tire's large deformations. Hence, a time integration process seems unavoidable. The vibrations small scale requires using energy conserving schemes [12,13] but remains costly from a CPU time perspective. The main contribution of the proposed study is to promote a numerical methodology based on a two-step approach for solving such problem [14]. Firstly, a non-linear quasi-static simulation is performed to avoid using a high-order stable numerical scheme capable of handling the considered non-linearities. Afterward, the non-stationary equations of motion are linearized using the quasi-static response as a reference point. The method was originally proposed by [PERSON] [15] for linear elastic materials. The effectiveness of the proposed approach is illustrated by predicting the non-linear vibrations of a grooved cylinder rolling on a rigid plane with frictionless contact in the first case and with frictional contact in the second case. Even if special attention is given to the mechanical contact, since the tire vibration noise is mainly caused by the impact between the tire and the pavement [4], the main objective is not to propose a reality of the physical phenomena and the interaction between the vibratory response of the tire and the road, but only to demonstrate the capacity of the numerical methodology put forward to correctly predict the vibratory response of solid undergoing large deformation with frictional contact. This paper is organized as follows. Section 2 presents the mathematical framework used to describe the mechanical behavior of a hyperelastic solid undergoing large deformations in contact with a rigid plane. Afterward, the linearization process is detailed and several hypotheses are made and justified.