Study of Anisotropic Friction in Gears of Mechatronic Systems

Abstract

The article discusses the features of anisotropic friction, which can be used to refine the calculation of the efficiency in various friction and gear drives and transmissions in mechatronic systems. Friction processes are considered that determine the level of losses in friction and gear drives, which are complex and heterogeneous in a number of parameters: the contact patch, which depends on the quality of the contacting surfaces; the direction and intensity of sliding; load distribution, etc. A more complete understanding of the features of these processes requires the use of the concept of friction anisotropy, which is well known in tribology of mechatronics systems. The anisotropy effect is caused by the difference in the characteristics of the surface microgeometry and its physical and mechanical properties in relation to the direction of the tool marks remaining on the surface after machining. In the presence of anisotropic friction, in contrast to isotropic, the body moves at a certain angle to the direction of application of the perturbing (external) force. The situation is considered in detail within the framework of the tensor model of anisotropic friction. The model and methodological approaches considered in the paper to the estimation of friction anisotropy can be used to refine the calculations of friction losses. The aim of the work is to create mechanical and analytical models of frictional anisotropy for a more complete understanding of this phenomenon in relation to various friction pairs. This article may be of interest to specialists in the field of friction gears for solving problems related to improving the accuracy of calculations and quantifying friction losses.

1. Introduction

The processes of friction, which eventually define the level of losses during friction and gear transmissions used in manufacturing machines and robots [1], have complicated and non-uniform characters over the contact spot, depending on the quality of contact surfaces, on the direction and the intensity of sliding, the distribution of loading, etc. [2,3]. There are known works aiming to measure the coefficient of, e.g., the sliding friction of pairs of materials by means of a variable inclination tribometer [4].

For a deeper understanding of the peculiarities of these processes, it is useful to introduce the notion of friction anisotropy, widely known in tribology in mechatronics systems. The anisotropy effect [5] is caused by the difference in the microgeometry characteristics of the surface and its physio-mechanical properties, depending on the direction of tracks of a tool, left after mechanical processing of the surface, typical for cutting metals with a defined cutting edge, such as turning, drilling, milling, etc. [6,7,8]. Orthotropic friction (a special case of anisotropic friction) is usually known as the unequal frictional characteristic in two mutually perpendicular directions in the contact plane.

The aim of the work is to create a mechanical and analytical models of frictional anisotropy for a more complete understanding of this phenomenon in relation to various friction pairs.

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 $\overrightarrow{e}$ be the unit vector in the direction of the body motion, disposed at the angle α (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 $\overrightarrow{e}$, it is necessary to apply to the body certain active force P along the line OO₁ at the angle β (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:

$$\{\begin{array}{c}{F}_{trX}=-NK\cos \alpha \\ F_{trY}=-NK\sin \alpha \end{array}$$

(1)

$$\mathrm{Hence} tg=\frac{F_{trY}}{F_{trX}}=\frac{k}{K}tg\alpha ,$$

(2)

and

$$F_{tr}=\sqrt{F_{trX}^2+F_{trY}^2}=N\sqrt{K^2{\cos }^2\alpha +k^2{\sin }^2\alpha } .$$

(3)

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

$$\{\begin{array}{c}{F}_{=}=F_{trX}\cos \alpha +F_{trY}\sin \alpha =-N(K{\cos }^2\alpha +k{\sin }^2\alpha )\\ F_{\perp }=-F_{trX}\sin \alpha +F_{trY}\cos \alpha =N\left(K-k\right)\cos \alpha \sin \alpha \end{array}$$

(4)

For the angle of deviation θ we obtain

$$tg\theta =-\frac{F_{\perp }}{F_{-}}=\frac{\left(K-k\right)tg\alpha }{K+k t{g}^2\alpha } ,$$

(5)

in this case, the angle θ lies within the limits from −θ_max to θ_max, where

$$tg{\theta }_{max}=0.5\left|\sqrt{\frac{K}{k}}-\sqrt{\frac{k}{K}}\right|,0\le {\theta }_{max}<\frac{\pi }{2} .$$

(6)

In the first quadrant (at $0<\alpha <\pi /2$) for the angle of deviation θ, we have $0<\theta \le {\theta }_{max}$ if $K>k$ and −${\theta }_{max}<\theta \le 0$ 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 θ = 0 for all the angles of motion α. 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 φ_i (where i = 1 … n; n = number of supports) with the direction of motion, it allows modeling different classes of indicatrix at unequal φ_i. The orientation of the model is rigidly fixed and its torque about its axis is compensated (the mode of plane-parallel motion).

For the model, moving uniformly and linearly on three supports, the set of equilibrium equations has the following form:

$$\{\begin{array}{c}{F}_{=}=-\Sigma N_j\left[K{\cos }^2\left(\alpha -{\phi }_j\right)+k{\sin }^2(\alpha -{\phi }_j\right]\\ F_{\perp }=\Sigma N_j\left(K-k\right)\sin \left(\alpha -{\phi }_j\right)\cos \left(\alpha -{\phi }_j\right)\\ N=\Sigma N_j\\ -F_{\perp }h+Nr\sin \left(\Psi -\alpha \right)=\Sigma N_j{R}_j\sin \left({\Psi }_j-\alpha \right)\\ -F_{=}h+Nr\cos \left(\Psi -\alpha \right)=\Sigma N_j{R}_j\cos \left({\Psi }_j-\alpha \right)\end{array} ,$$

(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; ${\phi }_j$ 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].

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 ${(k/K)}_{max}>0.6$, 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 ${(K_x/K_y)}_{min}\approx 0.04$, 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, σ_K = 2.4 kPa, and the velocity V = 45–50 mm/min, the experimentally found friction coefficients were: $K_{max}=0.25$ and $K_{min}/K_{max}=0.816$. The deviation angle at different angles of the application of force changed within the range ±3°. In the preliminary experiments with the pairs “anisotropic–anisotropic surface”, in some cases the deviation angle was up to 20°.

The comparison of experimental and calculated data shows qualitative coincidence of the model representations with the real ones in both the deviation angle (Figure 5) and the character of change of the motive force depending on the angle of motion. The exact approximating functions are the following:

• on the deviation angle $\theta =-0.25+2.60\sin \left(2{\alpha }_{rad}\right)$;
• on the motive force $F=60.99+13.18{\sin }^2\left({\alpha }_{rad}\right)$.

The quantitative coincidence of the model representations together with the real representations regarding the change of the components F_x and F_y of the friction force, depending on motion angle, has also been found. Experimental dependence is approximated by the curve of $F_x=0.28+0.05{\sin }^2\left({\alpha }_{rad}\right)$.

It has been shown that the increase of anisotropy factor leads to the increase of the deviation angle, which deviated in the range ±12° in the last case.

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 $\mathsf{\xi }$; the isotropic 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:

$$F_{=}\cos \mathsf{\xi }-F_{\perp }\sin \mathsf{\xi }=0,$$

(8)

The values of $F_{=}$ and $F_{\perp }$ can be presented in the form

$$\{\begin{array}{c}{F}_{=}kN{e}_{=}\\ F_{\perp =}kN{e}_{\perp }\end{array},$$

(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

$$\{\begin{array}{c}{e}_{=}=V_{=}/\sqrt{V_{=}^2+V_{\perp }^2}\\ e_{\perp }=V_{\perp }/\sqrt{V_{=}^2+V_{\perp }^2}\end{array} ,$$

(10)

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:

$$\{\begin{array}{c}{V}_{=}=V_c-V_s\cos \mathsf{\xi }\\ V_{\perp }=-V_s\sin \mathsf{\xi }\end{array} ,$$

(11)

Substituting (8) into (9), taking into account (10) and (11), after transformation we obtain:

$$\frac{K}{k}=\left[V_C/V_S\sin \mathsf{\xi } \mathrm{ctg}\mathsf{\xi }\right]\mathrm{ctg}\mathsf{\xi } ,$$

(12)

Thus, there is the possibility of determining the anisotropy factor experimentally with the help of the measured velocities $V_C$ and $V_S$.

With the rotation of the cylindrical body with the velocity $\omega R$ ($R$-cylinder radius), the component $V_{\perp }$ in the expression (13) should be calculated according to the formula

$$V_{\perp }=-V_S\sin \mathsf{\xi }\pm \omega R,$$

(13)

In expression (13), the plus (+) sign is used when there is a coincidence in directions of the vector of rotation velocity and the vector of the velocity of the motion of the strip [31], otherwise the minus (−) sign is taken. In this case, Formula (12) assumes the following form:

$$\frac{K}{k}=\left[(V_C/-V_S\sin \mathsf{\xi }\pm \omega R)-\mathrm{ctg}\mathsf{\xi }\right]\mathrm{ctg}\mathsf{\xi } ,$$

(14)

In Figure 7, the device for the realization of the offered technique is shown.

The device in Figure 1 consists of a frame (1) with the axis (2) and guider (3). Regarding the axis (2) on the bearing block (4) through suspension (5), the standard strip (6) with isotropic roughness of the surface is attached; this strip is pressed with force P by the pressure roller (7) through the fork (8) to the control cylindrical sample (10) from the load center (9). To provide stability of motion, the iron rule (6) is fitted with a balance beam (11).

4. Conclusions

A simulation of a mechanical model has been developed in the form of a platform with oriented-controlled cylindrical rolling bearings, which makes it possible to simulate the effect of anisotropic friction when moving in different directions.

Based on the model and its mathematical representation, the relationships of the main characteristic parameters of tribosystems with anisotropic friction are obtained. Modeling the anisotropy of varying complexity (symmetries) is provided through the mutual misorientation of the supports.

The considered features of anisotropic friction can be used to improve the efficiency of the calculations of various friction and gear drives in mechatronic systems.

The proposed model and methodological approaches to estimating friction anisotropy can be used to improve the calculation of friction losses and develop control paths [31].

The authors suggest the continuation of experimental research in relation to gears in mechatronic systems.

The considered features of anisotropic friction, the proposed model, and methodological approaches to estimating the anisotropy of friction can be used for more accurate and reasonable calculations of the efficiency in various friction and gear drives and transmissions. This is also true for extremely loaded gears with a high coefficient of friction as presented by the authors in paper [32]. The application of the proposed method of calculation makes it possible to take into account the influence of a number of parameters that are omitted from the reasoning when conducting a significant number of calculations. The use of the tensor model of anisotropic friction has made it possible to refine a number of calculation formulas and provide a more accurate estimate.