# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 3 Definition of Mesh Stiffness  
During meshing, many factors can induce fluctuations of the stiffness characteristics of the teeth. As explained before, the fluctuations of the mesh stiffness can be due to different contact conditions given by different contact ratios and contact positions along the tooth face. Hertzian contact phenomena can also influence the stiffness of the teeth. The combined effect of all these fluctuation sources produces a time history of the mesh stiffness acting on a single tooth. In this paper the time history of the mesh stiffness is not investigated in detail and it is approximated to a rectangular waveform traveling from one tooth to another one. In more detail, the mesh stiffness, which couples the \(n^{th}\) tooth pair, is assumed to have a constant value \(k_{t}\) when the \(n^{th}\) tooth pair is in contact and a null value when the contact is missing. Within the meshing time interval, the constant value, \(k_{t}\), assumed by the mesh stiffness can represent an equivalent mean value of a real trend during meshing. Since rotating reference systems are used, in each gear the mesh stiffness rotates with the same speed as the gear but in the opposite direction. In Figure 4 the time history of a generic mesh stiffness \(K_{\text{M}}\)(t) is shown with equivalent value \(k_{t}\) and unitary contact ratio, acting on a single tooth of a gear, with \(Z\) sectors (teeth) rotating at certain speed with revolution period \(T\). In this qualitative example of mesh stiffness, one can distinguish between the meshing time interval, when the tooth is in contact, from the rest of the time history when the tooth is not in contact and the mesh stiffness assumes a null value. Once a full revolution is performed, so after a period \(T\), the tooth experiences again the mesh stiffness. The rectangular waveform can be translated into a sum of harmonics by developing the Fourier Series of time trend:  
\[K_{M}(t)=K_{c}+\sum_{s=1}^{\infty}[K_{Va}^{s}\cos(s\Omega t)+K_{Vb}^{s}\sin(s \Omega t)],\]where \(K_{c}\) is the mean value of the function, \(s\) is the harmonic index, \(\Omega\) is the speed of the gear (\(T=2\pi/\Omega\)), while \(K_{Va}^{s}\) and \(K_{Vb}^{s}\) are the coefficients of the \"cosine\" harmonics and the \"sine\" harmonics, respectively. The expression of the Fourier series can be further manipulated by means of the Euler formula, to redefine Equation (5) as the real-valued form of the complex notation of the Fourier Series which will be used in the following discussion (Equation (6)). \[K_{M}(t)=K_{c}+K_{V}(t)=K_{c}+\sum_{s=1}^{\infty}\left[K_{V}^{s}e^{i\,\Omega\,t }+\overline{K_{V}^{s}}\,e^{-i\,\Omega\,t}\right], \tag{6}\]  
where:  
\[K_{V}^{s}=\frac{1}{2}(K_{Va}^{s}-iK_{Vb}^{s});\,\overline{K_{V}^{s}},\,\,\text {complex conjugate of }K_{V}^{s}. \tag{7}\]  
In Figure 5a,b a numerical example is shown (with \(Z_{1}=10\), \(Z_{2}=20\) and \(k_{t}=10^{6}\)\(N/m\) ) where T\({}_{1}\) and T\({}_{2}\) are the revolution periods of Gear-1 and Gear-2 respectively, and they are different from one another since \(Z_{1}\
eq Z_{2}\). By looking at the previous example, it is easy to note that the periodic time history of the mesh stiffness of the two gears is different. As a matter of fact, the rectangular waveform is the same, but its period differs in the two gears, being T\({}_{1}\) for Gear-1 and T\({}_{2}\) for Gear-2. Here, this concept is clarified with an example. Let us consider the previous system with \(Z_{1}=10\) teeth and \(Z_{2}=20\) teeth. In Figure 6a the couple 1-1 (tooth-1 of G\({}_{1}\)--tooth-1 of G\({}_{2}\)) starts meshing for an angle \(\theta_{1}=0^{\circ}\) of G\({}_{1}\). After a full revolution of G\({}_{1}\) (Figure 6e) tooth-1 of G\({}_{1}\) meshes again but now with tooth-11 of G\({}_{2}\). The same is for the other couples 2-2 and 2-12 (Figure 6b-f) and so on. Thus, a certain couple (i-j) meshes with a base period that is twice the base period T\({}_{1}\). Of course, the latter relation changes for systems with different number of teeth.