# Study of Anisotropic Friction in Gears of Mechatronic Systems
## 2 The Phenomenon of Anisotropic Friction
Figure 2: Calculation scheme of the mechanical model of anisotropic friction: (**a**) geometric parameters for calculation; (**b**) the scheme of application of forces; (**c**) power scheme, reduced to the center of gravity of the CG. For the model, moving uniformly and linearly on three supports, the set of equilibrium equations has the following form:  
\[\left\{ \begin{array}{l}F_{=}=-\Sigma N_{j}\big{[}K\cos^{2}\big{(}\alpha-\varphi_ {j}\big{)}+k\sin^{2}(\alpha-\varphi_{j}\big{]}\\ F_{\perp}=\Sigma N_{j}(K-k)\sin\big{(}\alpha-\varphi_{j}\big{)}\cos\big{(} \alpha-\varphi_{j}\big{)}\\ N=\Sigma N_{j}\\ -F_{\perp}h+Nr\sin(\Psi-\alpha)=\Sigma N_{j}R_{j}\sin\big{(}\Psi_{j}-\alpha \big{)}\\ -F_{=}h+Nr\cos(\Psi-\alpha)=\Sigma N_{j}R_{j}\cos\big{(}\Psi_{j}-\alpha\big{)} \end{array} \right., \tag{7}\]  
where \(N,N_{j}\) is general normal force and constituent normal force, respectively, per the \(j\), which is the support; \(K,k\) are friction coefficients on a support in the orthogonal directions (\(k<<K;R_{j},r\), which is a radius of disposition of the \(j\), i.e., the support and the center of gravity of the model; \(\Psi_{j},\Psi\) is the angle of disposition of the \(j\), which is the support and the center of gravity of the model; \(\varphi_{j}\) is the angle of discrepancy of orientation of the \(j\), which is the support with the direction of motion; and \(h\) is the height of motive force application [20; 21]. The set of Equation (7) was realized through the programming language BASIC, and the integration Gauss method was used. The main graphic dependences of change of force and angular parameters of the model are represented in Figure 3[22; 23]. Figure 3: Change in the power characteristics of the mechanical model depending on the angle of motion in polar (**a**) and decretal (**b**) coordinates. Values \(F_{trX}\) and \(F_{trY}\) are projections of the friction force on the coordinate axes (Figure 1). The changes of extreme values of the angular parameters of conjunction, depending on the factor \(k/K\), are defined from the nomogram (Figure 4). In Figure 4, 1 is the angle of application of the driving force; 2 is angle of movement; and 3 is the slip angle. For real pairs of friction \(\left(k/K\right)_{max}>0.6\), and technologically, it is difficult to achieve its further decrease [24; 25; 26]. The appearance of the materials with controlled friction characteristics allows the better use of the properties of anisotropic friction in various tribo-conjuctions. For friction transmissions (variators), which are kinematically similar to the simulation model (Figure 2), the anisotropy factor reaches the level of \(\left(K_{x}/K_{y}\right)_{min}\approx 0.04\), and its influence becomes rather considerable; therefore, the revealed regularities can be used for designing the mechanisms with such transmissions and for computing their efficiency. The comparison of the model representations according to Equation (7) with the experimental ones was carried out with the help of the mechanical model and special samples [27]. The experiments, conducted with the use of the mechanical model, have confirmed both the qualitative and quantitative correspondence of experimental and calculated results for the deviation angle \(\theta\) and the force of resistance to the motion depending on the angle of motion \(\alpha\). The actual values of friction coefficients have been determined [28]. Thus, when the steel plate was used for the experiment, the following limiting coefficients were determined: \(K_{max}=0.123\) and \(K_{min}=0.05\) and \(K_{min}/K_{max}=0.041\), respectively. The experiments with the use of special samples were carried out for the pairs \"isotropic-anisotropic surface\" without lubricant. For the pair \"steel 45-brass B62\" under the loading \(Fn=1.7\) N, \(\sigma_{K}=2.4\) kPa, and the velocity \(V=45\)-\(50\) mm/min, the experimentally found friction coefficients were: \(K_{max}=0.25\) and \(K_{min}/K_{max}=0.816\).