# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 4 Equations of Motion and Construction of the Matrices  
Having adopted a lumped parameters model, it is convenient to write, for clarity's sake, an equation of motion (considering at this stage the un-damped unforced system) of one dof connected  
Figure 5: (**a**) Mesh Stiffness traveling on \(\mathrm{G_{1}}\). (**b**) Mesh Stiffness traveling on \(\mathrm{G_{2}}\). to the \(i_{th}\) tooth of Gear-1. Then, let us consider the previous example of a system constituted by one gear (Gear-1) having 10 teeth (\(Z_{1}\)) and a second gear (Gear-2) having 20 teeth (\(Z_{2}\)). Let us write the equation of motion of tooth-\(I\) of Gear-1. By looking at Figure 7 below, neglecting at this stage the presence of damping and excitation force, the resulting equation of motion is:  
\[m_{b}\ddot{x}_{t_{1},1}+k_{b}\big{(}x_{t_{1},1}-x_{c_{1},1}\big{)}+K_{M_{1,1}}( t)\ \big{(}x_{t_{1},1}-x_{t_{2},1}\big{)}+K_{M_{1,1}}(t)\ \big{(}x_{t_{1},1}-x_{t_{2},11}\big{)}=0, \tag{8}\]  
where \(m_{b}\) refers to mass of the teeth; \(k_{b}\), nominal stiffness of the teeth; \(K_{M_{1,1}}(t)\) and \(K_{M_{1,11}}(t)\) mesh stiffness coupling the 1-1 teeth pair and 1-11 teeth pair respectively. It is possible to write the latter equation in a clearer form as follows:  
\[m_{b}\ddot{x}_{t_{1},1}+k_{b}x_{t_{1},1}+\Big{(}K_{M_{1,1}}(t)+\ K_{M_{1,11}}( t)\Big{)}x_{t_{1},1}-K_{M_{1,1}}(t)\ x_{t_{2},1}-K_{M_{1,11}}(t)\ x_{t_{2},11}-k_{b}x_{c_{1},1}=0. \tag{9}\]  
In Equation (9), two types of time-variant stiffness can be highlighted. The first type includes \(K_{M_{1,1}}(t)\) and \(K_{M_{1,11}}(t)\). The two terms refer to a specific teeth pair that meshes during operation. They are characterized by a rectangular waveform having a period \(T_{pair}\) that is strictly dependent on the number of teeth of the two gears. It is easy to derive \(T_{pair}\) and to relate it to the revolution periods of the two gears, \(T_{1}\) and \(T_{2}\) respectively. In general, \(T_{pair}\) is a multiple of both the periods \(T_{1}\) and \(T_{2}\). It is convenient to write the latter relation in a mathematical form:  
\[T_{pair}=n_{T_{1}}T_{1}=n_{T_{2}}T_{2} \tag{10}\]  
where the coefficients \(n_{T_{1}}\) and \(n_{T_{2}}\) define the multiple of the respective revolution periods. In the example analyzed here it is easy to note that \(n_{T_{1}}=2\) and \(n_{T_{2}}=1\). In other words, a specific pair meshes after two revolution of the Gear-1 as well as after one revolution of Gear-2. The other type of variable stiffness term which appears into Equation (9) is the term \(K_{M_{1,1}}(t)+K_{M_{1,11}}(t)\). This sum creates a rectangular waveform different from the previous one. As a matter of fact, \(K_{M_{1,1}}(t)+K_{M_{1,11}}(t)\) is a waveform with the same constant value \(k_{t}\) but with a period exactly equal to the revolution period \(T_{1}\), as shown in Figure 8.  
Then, for all the equations of motion of the dof belonging to Gear-1, two different sets of harmonics appear into the equation. The fundamental frequencies, which describe the two sets, are respectively:  
\[\Omega_{1}=\frac{2\pi}{T_{1}}, \tag{11}\]  
Figure 7: Linearized system showing meshing stiffnesses involving tooth-1 of G\({}_{1}\). The gears are constrained to the ground by means of the stiffness \(k_{a}\). \[\Omega_{3}=\frac{2\pi}{T_{pair}} \tag{12}\]  
Following the same approach, it is possible to write the equations of motion for the dof belonging to Gear-2. For those equations, still two types of variable stiffness can be distinguished into the resulting equation (here the equation of motion is not reported because is similar to Equation (9). The first type is again the rectangular waveform of the single pair of teeth described by the period \(T_{pair}\), so by the fundamental frequency \(\Omega_{3}\). The second type of time-variant stiffness term is the sum of all the other _pair_ waveforms involved into the equation, but now the resulting rectangular waveform has a base period equal to the revolution period of Gear-2, i.e., \(T_{2}\).