# Study of Anisotropic Friction in Gears of Mechatronic Systems
## 2 The Phenomenon of Anisotropic Friction  
In the common case in the presence of anisotropic friction, in contrast to isotropic friction, the body moves at a certain angle to the direction of the application of the disturbing (imposed) force. We shall consider this situation within the framework of the tensor model of the anisotropic friction [9; 10]. At the orthotropic friction there are two directions on the surface of the body, when the isotropic body is moving in these directions it is possible to know the maximal and minimal values of friction coefficient, respectively. We place the axes \(X\) and \(Y\) in these directions (Figure 1) and designate the corresponding friction coefficients as \(K\) and \(k\). Let \(\overset{\rightarrow}{e}\) be the unit vector in the direction of the body motion, disposed at the angle \(\alpha\) (the angle of motion) with the axis \(X\). In order for the body, pressed to the surface by normal force \(N\), to move in the direction \(\overset{\rightarrow}{e}\), it is necessary to apply to the body certain active force \(P\) along the line \(OO_{1}\) at the angle \(\beta\) (the angle of the application of the force) with the axis \(X\)[11; 12; 13]. The active force \(P\) compensates the originating friction force \(F_{tr}=-P\). We shall designate \(\theta=\alpha-\beta\) (the angle of deviation). A normal force \(N\) is applied to a point O in a direction orthogonal to the drawing plane. Strength not shown. In the accepted model of anisotropic friction for the constituents of the friction force along the axes \(X\) and \(Y\) we have:  
\[\left\{ \begin{aligned} F_{trX}&=-NK\cos\alpha\\ F_{trY}&=-NK\sin\alpha\end{aligned} \right. \tag{1}\]  
\[\text{Hence }tg=\frac{F_{trY}}{F_{trX}}=\frac{k}{K}tg\alpha, \tag{2}\]  
and  
\[F_{tr}=\sqrt{F_{trX}^{2}+F_{trY}^{2}}=N\sqrt{K^{2}\cos^{2}\alpha+k^{2}\sin^{2 }\alpha}. \tag{3}\]  
Figure 1: Scheme for the analysis of the process of anisotropic friction. The constituents of the friction force \(F_{tr}\), projected in the direction of the \(\overrightarrow{e}\) motion and on the direction normal to \(\overrightarrow{e}\)[13; 14; 15], have the form  
\[\left\{ \begin{array}{l}F_{=}=F_{trX}\cos\alpha+F_{trY}\sin\alpha=-N(K\cos^{2} \alpha+k\sin^{2}\alpha)\\ F_{\perp}=-F_{trX}\sin\alpha+F_{trY}\cos\alpha=N(K-k)\cos\alpha\sin\alpha\end{array} \right. \tag{4}\]  
For the angle of deviation \(\theta\) we obtain  
\[tg\theta=-\frac{F_{\perp}}{F_{-}}=\frac{(K-k)tg\alpha}{K+k\;tg^{2}\alpha}\, \tag{5}\]  
in this case, the angle \(\theta\) lies within the limits from \(-\theta_{max}\) to \(\theta_{max}\), where  
\[tg\theta_{max}=0.5\Bigg{|}\sqrt{\frac{K}{k}}-\sqrt{\frac{k}{K}}\Bigg{|},0 \leq\theta_{max}<\frac{\pi}{2}. \tag{6}\]  
In the first quadrant (at \(0<\alpha<\pi/2\)) for the angle of deviation \(\theta\), we have \(0<\theta\leq\theta_{max}\) if \(K>k\) and \(-\theta_{max}<\theta\leq 0\) if \(K<k\). In both cases the extreme value of the angle of deviation (\(\theta_{max}\) or \(-\theta_{max}\)) is obtained at \(tg\alpha=\sqrt{K/k}\). If \(K=k\), then \(\theta=0\) for all the angles of motion \(\alpha\). Here at \(k/K\Rightarrow 1\), friction degenerates into the isotropic one and at \(k/K\Rightarrow 0\) it tends towards hypothetically absolute anisotropy. In Figure 2, a mechanical simulation model is schematically presented for visual study of the peculiarities of friction and motion of the couple of bodies \"anisotropic-isotropic surface\". On the carrying plate of the model the supports of anisotropic friction are mounted as rolling bearings, having different resistance to the motion along to orthogonal directions [16; 17; 18; 19]. Each support is given the angle of discrepancy of orientations among the supports \(\varphi_{i}\) (where \(i=1\dots\ n;\ n=\) number of supports) with the direction of motion, it allows modeling different classes of indicatrix at unequal \(\varphi_{i}\). The orientation of the model is rigidly fixed and its torque about its axis is compensated (the mode of plane-parallel motion).