# Meshing Stiffness--A Parameter Affecting the Emission of Gearboxes
## 2 Materials and Methods
### Definition of Meshing Stiffness  
Deformations of teeth are generally quantified by tooth stiffness, which is defined as the ratio of load to deformation. We define tooth stiffness as the force per width unit that is required to deform by 1 \(\upmu\)m. During the meshing cycle, the stiffness of the teeth changes, mainly due to a change in the bending arm. For spur gears, the contact ratio CR is in the range \(1<\text{CR}<2\). In the two-pair engagement section, two pairs of teeth, I and II, engage at the same time, which corresponds to a parallel model of two springs (Figure 2b). In general, the resulting stiffness \(c\) is defined:  
\[c=\frac{w}{\delta}=\sum_{p}c_{p} \tag{1}\]  
where \(c\) is meshing stiffness (N/mm.\(\upmu\)m), \(p\) characterizes the order of a pair of teeth in contact, \(w\) is total width load of the gear (N/mm), and \(\delta\) is resulting deformation (\(\upmu\)m). Figure 1: Example of car transmission noise evaluation. However, in the case of parallel connected springs, the resulting stiffness is equal to the sum of the partial stiffnesses according to the second part of the previous equation. In the case of spur gearing, the resulting stiffness along the path of contact, as well as the corresponding deformation, is shown in Figure 2a. Points A, B, D, and E are characteristic points of contact, which are located on the line of action. Since contact cannot occur outside the teeth, it takes place along the line of action only between the points A and E on the line of action and inside both addendum circles. The line segment AE is named the path of contact. Points B and D are called the endpoints of a lone meshing. These are the points at which the meshing of one pair of teeth alternates with the meshing of two pairs of teeth. The resulting meshing stiffness \(c\) and the deformation \(\delta\) of teeth of spur gearing change periodically along the contact path, with a period equal to the pitch of the base circle \(p_{th}\). For approximate calculations, we calculate the mean value of the resulting stiffness \(c_{\gamma}\) for the entire meshing (Figure 2). This stiffness is called the engagement stiffness of the gearing, and it is constant for the given gearing and given by relation (2):  
\[c_{\gamma}=\frac{1}{p_{tb}}\cdot\int\limits_{0}^{p_{th}}c(\xi)d\xi=\int \limits_{0}^{1}c\big{(}\overline{\xi}\big{)}d\overline{\xi} \tag{2}\]  
where \(\xi\) is the coordinate of the path of contact and \(p_{tb}\) is the pitch of the base circle. Typical value of engagement stiffness \(c_{\gamma}=20\) N/mmum[19]. For spur gearing, the stiffness of the gearing was a clear function of the position of the contact point on the line of action. The meshing stiffness of the individual teeth pairs of helical gearing \(c_{p}\) varies not only depending on the position of the contact line in the field of view, but also along this contact line. It is relatively easy to determine the stiffness \(c_{p}\) along the length of the contact line of one pair of teeth of helical gearing. This varies depending on the coordinate of the path contact \(\xi\), as shown in Figure 3a, i.e., for one pair of teeth in contact. The teeth enter the meshing at point E. To a point at a distance equal to \(\varepsilon_{p}\).\(p_{bt}\) from point E, the contact length increases. Then, up to point A, the contact length is constant. Then the contact length is shortened. In the section of constant contact length, the  
Figure 2: The course of the total stiffness and deformation of spur gearing. (**a**) Deformation and meshing stiffness of spur gear; (**b**) parallel model of two-pair teeth contact section. course of meshing stiffness of helical gearing is similar to that of the meshing stiffness of the spur gear. In sections with a change in contact length, the stiffness of helical gearing changes faster.