Critical Velocity of Controllability of Sliding Friction by Normal Oscillations for an Arbitrary Linear Rheology

The application of ultrasonic vibrations is an established procedure in industry in order to significantly reduce and control sliding friction. One of the main characteristics of this phenomenon is that, beyond a certain critical sliding velocity, the friction is no longer controllable—although oscillations are still being externally applied. an a previous series of related studies, closed-form solutions of the critical velocity have been derived with respect to pure elastic and specific viscoelastic models. In the present paper we present a universal formula of the critical velocity which is valid for arbitrary linear rheology. The derivation relies on the same theoretical basis of the previous studies, where the reduction of friction is ascribed to a stick-slip motion of the contact. Therefore, all previous results represent limiting and special cases of this universal equation. In the second part of this paper we pursue the numerical analysis of the previous studies by investigating the reduction of friction for a viscoelastic Kelvin material for the first time.


INTRODUCTION
Applying ultrasonic vibrations is a well-known phenomenon to significantly reduce static and sliding friction. It is of great practical importance as it is used in many industrial applications such as wire drawing and press forming [13]. Another important field of application is located in ultrasonic motors and actuators as described in detail in [4]. In recent decades, various types of experiments have been performed in order to study this phenomenon [57]. Several types of oscillation directions have been investigated, including inplane oscillations (oscillations in the contact plane along and perpendicular to the sliding direction) and out-of-plane oscillations (oscillations perpendicular to the contact surface). However, in the following we merely consider out-of-plane oscillations. Since microscopic models were not able to achieve good results concerning theoretical predictions and experimental results [8], a macroscopic approach was introduced in [8,9]. It was shown in [10] that macroscopic mod-els entailed small deviations between theoretical predictions and experimental data.
This approach is based on the explicit consideration of the stick-slip motion of the contact. While in the sliding case the friction r . is proportional to the normal force n , . r 0 n , . . = µ in the sticking case it is smaller, r 0 n , .
. ≤ µ where 0 µ is the local coefficient of friction. This inequality allows the macroscopic coefficient of friction to be less or equal to the constant coefficient of friction 0 .
µ Thus, on the average, a reduction of friction can be observed. At a certain velocity * 0 , L the stick-slip motion of the contact is transferred into a pure sliding motion. As a consequence, the phenomenon of abated friction can no longer be observedalthough vibrations are still being imposed on the system.
We denote * 0 L as the critical velocity of the system, since the controllability of friction is not given for sliding velocities beyond this value. This effect is one of the main characteristics regarding the phenomenon of PHYSICAL MESOMECHANICS Vol. 21 No. 4 2018 active control of friction. A knowledge of * 0 L for specific materials is therefore absolutely essential. In [11,12] closed-form solutions of the critical velocity for pure elastic contacts has been presented. While the authors of [11] investigate the influence of contact stiffness, Mao and collagues [12] additionally take system dynamics into account. M. Popov extends these results by investigating the behavior for a simple viscoelastic Kelvin rheology in [13]. However, these equations are featured by their restricted scope. In the present paper, we deduce a universal equation for the critical velocity, which is valid for arbitrary linear rheology and thus contains all previous solutions as limiting cases. This enables prospective research on more complex rheology models allowing an improvement for theoretical predictions. In the second part of the paper we conduct a numerical analysis investigating the active control of friction for a simple viscoelastic modelwhich is done here for the first time.

.ORMULATION O. THE MODEL
We consider a system having a mass m and system stiffness x k (.ig. 1). The system is pulled with a constant velocity 0 L on a substrate. The immediate contact between them is modeled by a linear rheology whose normal and tangential stiffness, ,c ,c , , z x k k as well as the system stiffness x k can be described in the complex form k′ ω is representative for the elasticity of the material, whereas the imaginary part ( ) k′′ ω represents pure viscose properties. In this way any arbitrary linear rheology containing the models of the previous studies as special cases can be described by the complex stiffness ( ), k ω as any possible combinations of springs and dampers can be pooled in ( ) k′ ω and ( ).
k′′ ω Perpendicular to the sliding direction, a displacement-controlled, forced harmonic oscillation is being imposed with the angular frequency ω, the amplitude z u ∆ and the constant mean indentation depth In the immediate contact with substrate we assume the classic Coulomb law of friction with a constant coefficient of friction 0 . µ The dynamics of the system is characterized by the horizontal and vertical position of the center of mass x u and z u as well as the position of the contact ,c .
The origin of the coordinate sys-tem represents the unstressed state of the rheology element.

GENERALIZATION O. THE CRITICAL VELOCITY
One of the main characteristics of the phenomenon of reduction of friction by normal oscillations is that the extent of the reduced friction decreases with increasing sliding velocityas already described in the introduction. Above a certain critical velocity * 0 L the macroscopic coefficient of friction of a system with applied vibrations remains identically the same to the system without any applied vibrationsat a given frequency and amplitude of the oscillation. Closed-form solutions of the critical velocity for simple elastic models have been deduced in previous studies. In the present paper, we deduce an equation of the critical velocity for arbitrary linear rheology, which is mainly characterized by the time-dependent stiffness k(t) or the complex frequency-dependent stiffness ( ).
k ω This enables the investigation of systems with high complexity and thus a higher accuracy can be achieved, which in turn is of great interest for many possible applications regarding the controllability of friction.

Calculation of Critical Velocity
We study only the non-jumping case which means that the indenter remains in contact with the substrate for all times. This implies on the one hand that the oscillation amplitude z u ∆ must be smaller than the mean indentation depth ,0 z u and therefore On the other hand, the normal force must remain positive for all times and hence .ig. 1. Mechanical model of the indenter system. By using the complex stiffnesses any arbitrary linear rheology can be represented. (3) and (4) must be considered as constraints in all calculations, particularly in numerical evaluations.
The critical velocity is characterized by the condition that the contact velocity ,c x u remains positive for all times for the first time and thus is valid. In order to calculate the critical velocity, it is necessary to set up and solve a differential equation in ,c .
x u .or this purpose, one has to set up a differential equation in x u first of all. We assume the motions of the mass and the contact point in the sliding case is +∆ ω +φ L (7) ϕ and φ are phase shift relative to the forced harmonic oscillation (2) which can be also written in the following form The normal force is calculated using the complex stiffness as [14] n The Newtons equation of motion for the mass reads as follows: .
.rom above equation system we obtain 0 , c , c 2 ( ( ) ( )) , Now let us consider the motion of contact point ,c .
x u Since in the sliding case all horizontal forces remain in equilibrium, one has: Similar to (12), we consider the oscillation parts [ ( )cos( ) .
Substitution of solution (15) leads to .
Differentiating it with respect to time provides the velocity of the lower point of the spring: This velocity remains positive if the sliding velocity exceeds the critical velocity * 0 . L By taking condition (5) into account the critical velocity is given by Thus, the knowledge of the complex stiffness ( ), k ω and ,c ( ) z k ω allows to calculate the critical velocity of controllability of sliding friction for an arbitrary linear rheology.

Validation for a Kelvin Material
The Kelvin rheology represents one of the simplest models for a viscoelastic material. A closed-form solution of the critical velocity for such a model has been derived in [13]. We will illustrate that by application of the universal equation for the critical velocity (24) exactly the same solution as in [13] can be obtained.
.or a Kelvin material the complex stiffness of the contact is given by [14]: In [13], the investigated model consists of a body with mass m, which is pulled with a constant velocity 0 L throughout a pure elastic system spring .
x k Therefore ( ) x k ω is equal to . ( ) By assuming an isotropic material and hence is valid, this implies which is exactly the same result developed in [13]. By considering the limiting case ,c 0 z γ → and hence a pure elastic contact, the results in [12] is again obtained In the case of a very stiff system x k → ∞ and without mass, the critical velocity in [11] is further constituted Moreover, Eq. (24) can be applied for complicated model such as the Prony series with different relaxation times.

Theoretical Analysis
In the following analysis we numerically investigate the reduction of sliding friction by ultrasonic vibrations considering out-of-plane oscillations for a simple viscoelastic model as illustrated in .ig. 2. Previous numerical analyses in [11,12] considered solely pure elastic contacts. However, in industrial applications viscoelastic materials take on an important role.
The investigated model is a Kelvin element composed by a spring and a damper as described in Eqs. (25) and (26). In this very simple modelling the contact is not associated with any mass (m = 0) and we assume a stiff system ( ) x k → ∞ meaning that the con- At the moment 1 t the sticking case begins and thus the velocity of the contact diminishes ,c 1 ( ) 0;    .ig. 3. Qualitative illustration of stick-slip cases. Please note that the linear slope during sticking is only valid, when system dynamics are being neglected.  To conduct a numerical analysis, we formulate all equations in a dimensionless way with following variables: