Friction Influenced by Vibrations: A Refined Contact-Mechanics View on Lateral and Rotational Oscillations

It is known that superposed movements can lower the friction felt at the macroscale. This is well documented for in-plane and normal translatory oscillations. Contact mechanics are a suitable approach to model this effect but so far have not gone beyond single-slider dynamics. In this study, we make use of 3D Boundary Elements Simulations to study the macroscopic friction reduction. This approach allows us to take into account also partial sliding of the contact zone. We first revisit the case of transversal in-plane translatory oscillations. Here, we argue that the behavior at small velocities can best be described when partial slip is indeed taken into account. Next, we investigate the frictional response of a Hertzian indenter when the lateral movement is superposed with a rotational bore movement. An analytical approximation is given for the steady state solution with constant angular velocity. The third case under investigation is an oscillating bore rotation. We present numerical results for the reduction of the macroscopic friction. In two limiting cases, an analytical prediction is given, following the lines used in the translatory case. For extremely large amplitudes, it is based on the idea that a rotational steady state is assumed at every instant. For small velocities we adapt our new approach including partial sliding. We find these predictions to be good but not perfect, slightly underestimating the reduction that rotational oscillations can provide.


INTRODUCTION
When frictional contacts are subjected to external oscillations, the friction felt at the macroscale is generally reduced. Since this constitutes a relatively simple way to control friction, the effect is used in countless manufacturing applications (Siegert and Ulmer, 2001a;Siegert and Ulmer, 2001b;Murakawa, 2001;Egashira and Mizutani, 2002;Ashida and Aoyama, 2007) as well as in noise control (Thomsen, 1999). On the negative side, it can lead to an error in measurements of the coefficient of friction (Kado et al., 2014). The effect has been described experimentally in the middle of the last century (Fridman and Levesque, 1959;Pohlman and Lehfeldt, 1966;Godfrey, 1967;Lenkiewicz, 1969) but modeling was not attempted. In 2002, Storck (2002) investigated translatory in-plane oscillations experimentally and analytically. They argued that the interplay of forward friction and the superposed vibrations can be modeled using the perspective of contact mechanics with the simple Coulomb law of friction of a single slider. Kumar and Hutchings (2004) use a comparable modeling to interpret their experimental data. Tsai and Tseng (2005) employed the Dahl friction model to estimate the reduction for in-plane oscillations and found that rigid slider models overestimate the friction reduction. Teidelt, (2012) reported an impressive series of experimental data on the reduction for oscillations in the two in-place directions as well as the out-of-plane direction and interpreted the results using a model of rigid sliders. Starcevic and Filippov also worked on this dataset. They used the Method of Dimensionality Reduction to study static friction under the influence of in-plane parallel oscillations. Their setup consisted of two coupled parabolic sliders and included micro-slip (Starcevic and Filippov, 2012). The same method was used by Teidelt et al. (2012) to study the performance of Microdrive actuators. The simultaneous acting of normal and in-plane-parallel oscillation was studied by Popov and Li (2018) using an elastic slider model. The case of in-plane oscillation transversal to the macroscopic forward motion was modeled in a recent study by Benad et al., (2019). They used a coulomb-type frictional slider attached to a linear spring. In this work, results are given as a function of normalized input quantities and explicit dependencies are given for two limiting cases, corresponding to very low forward velocities (static friction) on the one hand and very large oscillation amplitudes on the other hand. Their work can be considered the starting point of the current paper. "Transverse Oscillations" section of the current paper will build on their results but will introduce explicit partial slip.
The setup under investigation is the following: We focus solely on superposed in-plane motion, leaving the normal contact unchanged. Unless stated otherwise, the indenter is a sphere with radius R, approximated by a parabola shape. It is compressed elastically against a non-deformable flat counterbody. The resulting contact zone and normal stresses are of the Hertzian type and are assumed to be unaffected by tangential stresses. The elastic sphere and the rigid counterbody interact locally with a constant coefficient of friction µ, such that the effective tangential stress inside the contact zone is always less or equal to the normal stress at that point. While the indentation depth in normal direction (z) is held constant, the sphere is subjected to controlled in-plane displacements. "Transverse Oscillations" section will start with a constant velocity in forward (x) direction and harmonic oscillation in y-direction. In "Continuous Rotation (Bore)" section, the sphere will also move with constant velocity but will also rotate (bore) around its central z-axis at constant angular velocity. "Oscillating Rotation (Bore)" section will cover the case of oscillatory bore rotation. In all cases, the macroscopic forward friction will be decreased. The greatest reduction is generally associated with larger amplitude, frequency or velocity, respectively, of the superposed motion.

TRANSVERSE OSCILLATIONS
The case of transverse oscillations consists of a frictional indenter which is moved with constant velocity in x-direction while its position in y-direction is given by a sine-function The simplest approach is to assume that Eq. 1 describe not only the macroscopic motion of the indenter but also the exact motion of the contact spot. This approach neglects the lateral elasticity of the indenter. In contrast, the model of Benad et al. (2019) consists of a single slider but connected to the coordinates given in Eq. 1 by a linear spring. They showed that the stiffness has a considerable influence on the system behavior, in particular for small oscillation amplitudes. We agree with this assessment. Because the system is quasistatic, it makes sense to formulate in problem in a way that eliminates the time. Similar to Benad et al. (2019), we employ a set of two dimensionless variables for such system including elasticity. The amplitude of the oscillation is described as where u lim μd E p G p is the maximum unidirectional tangential displacement of the indenter before full sliding occurs, see (Mindlin, 1949;. Here d is the indentation depth in normal direction, E*, G* are the reduced moduli of shear and elasticity Eq. 1, ] is Poisson ratio.
The forward speed is normalized with respect to the speed of the oscillation and is expressed as the dimensionless variable.] The definitions] 0 andỹ 0 coincide with those from Benad et al. (2019) when l 0 of their paper equals u lim , which is the most reasonable interpretation since "l 0 is the elongation at which sliding starts." The dependent variable is the apparent coefficient of frictioñ μ macro which is determined from the tangential force F x in x-direction and the normal force F z Here 〈/〉 means averaging over one period of oscillation, after the system has tuned in. Figure 1 shows a typical evolution of the resistive tangential force in x-direction. Such curves were obtained for a variety of combinations for parameters] 0 andỹ 0 . The resulting macroscopic coefficient of friction is depicted in Figure 2 and mostly reaffirms known results (Tsai and Tseng, 2005). In Benad et al. (2019), an almost identical graph is not given in the paper but can be found on the cover page of the journal's issue.

Limiting Solution for Large Oscillation Amplitudes
An estimate for the apparent coefficient of friction at large oscillation amplitudes is given as In the range covered in this current work, we find the same as (Benad et al., 2019): We basically confirm (6) and find it slightly overestimates the reduction inμ macro for finiteỹ 0 , see the red curve in Figure 2. This is to be expected, since the turning points in the oscillation delay the reducing effect.

Limiting Solution for Small Driving Velocities
The limit of small driving velocities can be considered the case of static friction. From a macroscopic point of view, this describes simply the value of the tangential force below which the indenter does not advance. Looking at the situation within an oscillation cycle, the limit translates into the requirement that at least during one infinitesimal instant of the cycle, a forward motion is possible.
In the elastic-single-slider model of (Benad et al., 2019), these instants coincide with the extremal values of the oscillations and a value for the tangential force can be associated. For μ macro a closed form solution at ] 0 10 is deduced This solution is applicable forỹ 0 ≤ 1 and is shown in Figure 2. For any larger values, static macroscopic friction vanishes.
Our numerical simulations show that Eq. 7 overestimates the macroscopic friction. In other words, the reduction resulting from the oscillation is underestimated. Here we'd like to argue that this due to the exclusion of partial slip in the model. Partial slip does not occur in a single slider element but requires a finite contact zone and distribution of normal stress.
For a Hertzian indenter in the absence of a forward motion, the oscillation with amplitudeỹ 0 ≤ 1 gives rise to the formation of a ring-shaped slip zone and an inner circle of stick. According to Mindlin theory, the radius of the stick region c depends onỹ 0 according to where a is the contact radius. From this state, what condition must be met in order to achieve a forward motion? Since] 0 10, the oscillations are at high frequency in the outer ring of slip and are very effective in reducing, even suppressing the friction there. Therefore it is reasonable to assume that the driving forward motion only has to overcome the friction inside the remaining stick zone. With the total normal force F z and the normal force only acting inside the stick zone F z,c , we can conjecturẽ Here p(r) is the distribution of normal contact stress. In the particular case of a Hertzian stress distribution, Eq. 9 reduces tõ Furthermore in the particular case of translatory oscillations using Eq. 8 it reduces tõ which is also plotted in Figure 2 and coincides much better with direct simulation results. Thus we see reason that unless the result is zero or one,μ macro,static cannot be predicted by a single slider approach.

CONTINUOUS ROTATION (BORE)
We now investigate a rotationally symmetric indenter which is rotated around the z-axis while sliding in the x-y-plane.
Since the problem is quasistatic, it can be reduced to a single system parameter and dimensionless timẽ In this configuration, the instantaneous center of rotation in the undeformed remote parts of the body is situated at x center −]a, y center ] 0 t.
See Figure 3A. The steady state can be analyzed using the following simplifications.
• The velocities of all spots on the surface are assumed to be the same as their corresponding spots in the remote body • For the calculation of frictional torque, surface stresses are assumed to act at the undeformed spot.
In particular, the above simplifications are valid for small µ. The amplitudes of the frictional stresses τ are given by τ(x, y) μp(x, y) and they are directed opposite to the velocity in each surface point. This can be expressed as See Figure 3B for an example. The resulting tangential forces and torque can then be calculated as wherein 0 dA means integrating over the contact area and we introduced Q Y (r) 2π 0r cos φ + 1 The additional rotational movement eases the forward movement and effectively reduces the apparent coefficient of friction. Vice versa, the forward motion eases the rotation. Therefore, the required torque M z also depends on]. A macroscopic coefficient of friction can be formulated for both the lateral movement and the rotation.
where M z,0 is the steady state torque at zero forward velocity.
FIGURE 3 | Hertzian indenter slipped in positive y-direction and rotated. Normalized forward speedṽ 0.5. Quiver plots of (A) local velocity field, (B) tangential surface stresses. The stress vector field also rotates around x/a −0.5 but its amplitude is related to the contact pressure.
Frontiers in Mechanical Engineering | www.frontiersin.org October 2020 | Volume 6 | Article 566440 For a Hertzian Indenter the dependencies of Eqs 21 and 22 oñ ] are shown in Figure 4. In our transient numerical simulations, these values are approached after the influence of the initial conditions has vanished. Typical evolutions of the tangential forces and torque are shown in Figure 5. It can be seen that the macroscopic forces and torque approach a steady state which is indeed well predicted by Eqs 21 and 22.
Interestingly, our simulations never showed zones of stick when the steady state was reached, even when instantaneous center of rotation is located inside the contact region.

OSCILLATING ROTATION (BORE)
In this section, we will set a continuous forward motion and the additional bore rotation will be oscillating. Let us impose a displacement of the indenter according to Some normalization is again required for the oscillation amplitude. In "Transverse Oscillations" section, it was divided by the maximum tangential deflection before the start of macroscopic slip, which is when the inner stick region fully vanishes. In the case of pure rotation of a Hertzian indenter however, there is no such maximum angle at which the stick region vanishes. According to Lubkin (1951) (indexed "lk") , the stick radius c and the torsion angle φ are related as where we introduced With the complete elliptical integrals of the first and second kind Indeed Eq. 24 tends to infinity for small c, meaning that the very center of the contact will always be in stick state. Normalization of the rotational amplitude will be instead be done with respect to the angle necessary to achieve a stick radius of c a/2 in the absence of lateral displacement.
The normalization of the forward velocity is done with respect to the maximum speed of the contact zone edge at r ã With these definitions, simulations and evaluation can be done in analogy to "Transverse Oscillations" section. Figure 6 shows the results of μ macro for a range of parametersṽ andφ. At first sight, the general appearance resembles the dependencies shown in Figure 2 of "Transverse Oscillations" section. Forφ 0, no reduction is found and thusμ macro 1. For large amplitudes of the rotational amplitude, friction is greatly reduced, especially for small forward velocitiesṽ. Please note however, that even in the limit ofṽ10, the macroscopic friction is not reduced to zero.

Limiting Solution for Large Oscillation Amplitudes
For the limiting case of large amplitudes, a similar expression to Eq. 6 can be found, when it is assumed that the contact is always in a rotational-transversal steady state such as "Continuous Rotation (Bore)" section has introduced. Then, the results found in Eq. 21 can be averaged over one period with momentary values of ω(t) φ 0 Ω cos(Ωt) yielding This finding is shown in Figure 6 with the red line. Please note the resemblance to Eq. 6 in Figure 2.

Limiting Solution for Small Driving Velocities
Following the approach laid out in "Limiting Solution for Small Driving Velocities," it is possible to formulate an approximation for the static friction in the limit ofṽ10. Let us take the exact same steps for the rotational problem. In a situation without forward motion, the rotational oscillation will give rise to a zone of constant stick. The oscillation amplitude φ 0 and the static stick radius c are related by Eq. 24 which we can now write in normalized form, introducingc c/ã This is a monotonous dependency with D(1/2) ≈ 0.9455. Let c lk (φ) be the inverse functioñ We can then use this stick radius with Eq. 10 to find Which is shown in Figure 6 with the green line. It can be seen that Eq. 33 constitutes a reasonably good prediction of static friction, but not an exact one. It appears to accurately predict the static friction very well forφ 0 greater than approximately 2. At this point one should note that our judgment on the accuracy is limited, because in order to reach a steady state, the velocities used in the simulation must be strictly positive and smaller values require more computation time. Therefore, the minimum value represented in Figure 6 isṽ 0 0.005. For φ 0 < 2 however, Eq. 33 clearly underestimates the reduction of friction. This means that the forward force does not need to overcome the entire normal stress inside the oscillatory stick zone multiplied with μ, but less. This in turn means that during the forward motion of the indenter, not the entire inner circle slides forward. Indeed, the time series of these simulations indicate that at no point in time there is full sliding in the contact zone. Instead, patches of stick alternate between the left and the right hand side of the contact zone.

CONCLUSIONS
In this paper, we investigate friction under the influence of superposed motion using explicit contact mechanics. In all cases, the superposed motion reduces friction. The stronger the motion (faster, higher amplitude) the greater the reduction. Under the influence of any given superposed motion, the forward friction increases with the forward velocity, stabilizing the frictional system.
Based on the findings presented here for transversal oscillations, the spring model given in Benad et al. (2019) provides satisfactory results, except for very small forward velocities, so-to-speak for static friction when the oscillations provoke some partial slip, which should be taken into account. We postulate that in order to start a forward motion, only the friction inside this stick region must be overcome.
For friction under the influence of additional continuous bore movement, an approximation for the steady state is identified and given in "Continuous Rotation (Bore)" section. Both the friction opposing forward motion and opposing the rotation are reduced.
When the bore movement oscillates, a reduction of the macroscopic forward friction is again found and the dependency on forward velocity and oscillation amplitude resembles the transversal case (Compare Figures 2, 6). Here, two limiting cases are again identified and analyzed. At large amplitudes, assuming steady state in every instant leads to a good approximation. For small velocities near static friction, a new estimate with good agreement is suggested. It is again based on the stick zone caused by the oscillations alone. However, simulations show that the reduction of forward friction is even greater than predicted, hinting to a more elaborate mechanism.
Rotational bore oscillations are a promising way to control static friction or micro-propulsion due to two favorable properties over transversal oscillations. First,μ macro,static does not sharply saturate at zero for increasing amplitudes but is instead very well controllable in this region. Second, the reduction inμ macro,static is very effective even at small amplitudes.

METHOD
We employ the Boundary Elements Method for contact problems (Pohrt and Li, 2014). The normal contact is solved following standard procedures. Interaction of normal and in-plane deformations are not considered. Interactions between the two in-plane directions are considered. For the tangential contact, an iterative scheme is employed that ensures each point satisfies the following conditions in each time step.
• Tangential stress is below the frictional threshold |τ| < μp ("stick state") or, • Tangential stress |τ| μp and the difference in point coordinates including deformation, the local velocity, is directed opposite of tangential stress ("slip state") where τ is the tangential stress, having x and y components, p is the local pressure and μ is the coefficient of friction. For the stress inequality we typically allowed an error of 0.1% and for the alignment of local stresses and velocities we typically allowed an error of 4°. Achieving convergence in the iterative scheme proved be time-consuming. Furthermore, the investigations of low forward speeds required the use of up to 8,000 time steps. As a consequence we limited the spatial resolution of the contact region to 32 times 32 points in bulk simulations such as those shown in Figures 2, 6. We used v 0 for the Poisson ratio and μ 0.2.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
RP developed the BEM code, performed the simulations and created the text and figures.

FUNDING
This research was funded by regular financing of researchers at Technische Universität Berlin.