# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling  
[PERSON]  
Dipartimento di Ingegneria Meccanica e Aerospaziale (DIMEAS), Politecnico di Torino, Corso Duca degli Abbruzzi, 24, 10129 Torino, Italy; [EMAIL_ADDRESS] (E.P. ); [EMAIL_ADDRESS] (S.Z.) [PERSON]  
The dynamic coupling is a direct consequence of the variations of tooth flexibility (mesh stiffness) because of the different contact conditions during rotation, but also because of the variations of the contact ratio. Thus, time-varying mesh stiffness causes the system to be non-linear. It is worth remembering that here the term \"non-linear\" is adopted to highlight the fact that the system is not the usual Linear Time-Invariant System (LTIS), but a Linear Time-Variant System (LTVS). Being the system of the type LTVS, it is not possible to compute the forced response by inverting the dynamic stiffness matrix as for a LTIS, since the latter is a time-variant parameter inside the equation of motion. Nevertheless, the dependent variable of the equation of motion does not show elevation to powers or other non-linearities (e.g., Duffing equation). Then, the superimposition effect principle is valid for such a system and it will be used in the following discussion. As a consequence, the adoption of a \"non-linear\" method is needed to compute the forced response of the system. The dynamics of gear systems have been extensively studied by researchers for decades and still represent an important matter of interest for the understanding of phenomena affecting the dynamics of such systems. The evaluation of the mesh stiffness variations and the related non-linear aspects have a primary importance for researchers who provide several modelling solutions of the phenomenon [1] for mathematical definition. According to the different types of modelling of the mesh stiffness, many works provided different methodologies for the computation of the response of the system, according to different levels of complexity. Most of the works focused on the combined effect of mesh stiffness variation and backlash between the meshing teeth, which affects largely the response of the meshing gears, developing non-linear methodologies for the iterative and numerical computation of the response [2; 3; 4; 5; 6; 7; 8; 9]. Other studies focused on the analysis of the instability conditions, which can be caused by the fluctuations of the mesh stiffness involving sometimes wide operational speed ranges of the gears and can lead to failures. Works on instability provide an analytical solution, using perturbation methods (e.g., method of multiple time-scales, MMTS) to establish relations between the analyzed instability conditions and the entity of the mesh stiffness fluctuations [10]. Recently, instability analyses and forced response studies were extended to more complex systems like planetary gear systems [11; 12; 13; 14]. Most of the cited works analyze the dynamics of a gear system by considering the gears as rigid bodies connected by the mesh stiffness and introducing the transmission error between two meshing gears, which consider the fluctuations of an equivalent tooth compliance that excites the system. In this paper, the aim is to consider the gears as compliant bodies and compute analytically the forced response of the system excited both parametrically and externally. The backlash phenomenon is not considered at this stage in order to focus the attention on the phenomenon of the dynamic coupling and on the method to be developed to study the phenomenon without the nonlinearity introduced by intermitting contacts during meshing. Here, transmission error cannot be used anymore since the gear bodies are considered as compliant. The gears, which constitute the overall system, are linked together by means of a time-variant mesh stiffness, which acts on the nodes of the teeth, where the contact takes place. In other words, the system sees both a parametric excitation and an external force exciting the system.