# Study of Anisotropic Friction in Gears of Mechatronic Systems
## 3 Estimation of Anisotropy of Friction Properties on Flat and Cylindrical Surfaces  
For the estimation of anisotropy of the friction coefficient on the contact surfaces of the elements of friction and gear transmissions, the following technique can be used [28]: on the control surface, the indenter is moved along the circular trajectory and the components of the total vector of resistance forces are measured, these measurements are used for building up indicatrix. In tribometry, for the pairs \"anisotropic-anisotropic body\", which include the majority of elements of friction and gear transmissions, the motion of the indenter over the surface should be arranged in a way that the direction of anisotropy would never change its position when the moving along the given trajectory on the control body. For the pair of samples from steel 45 with different roughness and with use of the indenter 2H, the friction coefficients \(K_{max}=0.162\) and \(K_{min}=0.134\) were obtained. The components of the friction force, acting upon the indenter, were determined in two perpendicular directions with the help of tensometric yoke. In the experiments the indenter was moved, while the mutual position of the selected axes of anisotropy was kept constant in a special device [29; 30]. The accuracy of measurements in this technique would be better if the radius of the indenter itself were as small as possible and the radius of rotation of the indenter over the control surface were as large as possible. This technique can be used for flat surfaces or the surfaces with a small curvature; in the latter case, the indenter should be spring-loaded. In this way, it is possible to estimate the anisotropy of friction properties over the working surface, for instance, over the surface of teeth of gears or the non-cylindrical elements of friction variators [19; 20; 21]. The estimation of the anisotropy of friction on cylindrical surfaces should be considered more thoroughly. Let the cylindrical body 1 with the anisotropy of frictional surface properties move along the axis **A**-**A** with the velocity \(V_{c}\) (Figure 6). The isotropic strip 2 is pressed to the surface of this body with force \(P\) in the area of the element **a**-**a** at an angle \(\xi\); the isotropic  
Figure 5: Change in the value of the slip angle from the angle of motion for full-scale samples in polar (**a**) and Cartesian (**b**) coordinates. strip only has the possibility of moving in the direction of the axis **B**-**B**. In this case, as the result of the contact interaction, the strip moves along the axis **B**-**B** with the velocity \(V_{s}\). Following is the equilibrium equation for this case on the plane of tangency for the point **O**:  
\[F_{=}\cos\xi_{=}-F_{\perp}\sin\xi_{=}=0, \tag{8}\]  
The values of \(F_{=}\) and \(F_{\perp}\) can be presented in the form  
\[\left\{ \begin{array}{l}F_{=}kNe_{=}\\ F_{\perp=}kNe_{\perp}\end{array} ,\right. \tag{9}\]  
where \(e_{=}\) and \(e_{\perp}\) are the projections of the unit vector \(e\) on the direction along the axis of the cylinder and perpendicular to it. They are determined as  
\[\left\{ \begin{array}{l}e_{=}=V_{=}/\sqrt{V_{\perp}^{2}+V_{\perp}^{2}}\\ e_{\perp}=V_{\perp}/\sqrt{V_{\perp}^{2}+V_{\perp}^{2}}\end{array} ,\right. \tag{10}\]  
Figure 6: Calculation scheme of device for estimation of friction anisotropy on cylindrical surface (**a**) The design scheme of the device (Figure 7) and (**b**) the scheme of the application of forces. Figure 7: Kinematic scheme of device for the estimation of friction anisotropy on the cylindrical surface. Here, the velocities \(V_{=}\) and \(V_{\perp}\) are the constituents of the relative velocity \(V_{rel}\) of the motion of the cylinder and the strip [31], which can be found from the relationship:  
\[\left\{ \begin{array}{l}V_{=}=V_{c}-V_{s}\cos\xi,\\ V_{\perp}=-V_{s}\sin\xi,\end{array}