Critical velocity of controllability of sliding friction by normal oscillations for an arbitrary linear rheology Текст научной статьи по специальности «Физика»

УДК 531.8, 531.44, 528.5-531.7

Critical velocity of controllability of sliding friction by normal oscillations for an arbitrary linear rheology

J.M. Zughaibi, F.H. Schulze, Q. Li

Technische Universität Berlin, Berlin, 10623, Germany

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. In 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.

Keywords: ultrasonic vibration, macroscopic coefficient of friction, active control of friction, arbitrary linear rheology, critical velocity, viscoelastic contact

DOI 10.24411/1683-805X-2018-12009

Критическая скорость при контроле трения скольжения с помощью нормальных колебаний для произвольной линейной реологии

J.M. Zughaibi, F.H. Schulze, Q. Li

Берлинский технический университет, Берлин, 10623, Германия

Ультразвуковые колебания широко применяются в промышленности с целью уменьшения и контроля трения скольжения. Одной из основных особенностей данного метода является невозможность контроля после достижения некоторой критической скорости скольжения несмотря на прикладываемое внешнее ультразвуковое воздействие. В предыдущих работах были получены аналитические решения/выражения для критической скорости в рамках чисто упругой и специфической вязкоупругой моделей. В настоящей работе приводится универсальная формула критической скорости, справедливая для произвольной линейной реологии. Уравнение получено на теоретической основе предыдущих исследований, где уменьшение трения происходит за счет прерывистого скольжения в контакте. В связи с этим, все предыдущие результаты представляют собой предельные и частные случаи полученного универсального уравнения. Во второй части статьи представлен численный анализ предыдущих исследований. Впервые изучено уменьшение трения для вязкоупругого материала Кельвина.

Ключевые слова: ультразвуковые колебания, макроскопический коэффициент трения, активный контроль трения, произвольная линейная реология, критическая скорость, вязкоупругий контакт

1. 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 [1-3]. 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 [5-7].

© Zughaibi J.M., Schulze F.H., Li Q., 2018

Several types of oscillation directions have been investigated, including in-plane 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 models 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 Fr is proportional to the normal force Fn, Fr = Fn, in the sticking case it is smaller, Fr < Fn, where 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. Thus, on the average, a reduction of friction can be observed. At a certain velocity v*, 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 observed—although vibrations are still being imposed on the system.

We denote v* 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 active control of friction. A knowledge of v* 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 model— which is done here for the first time.

as any possible combinations of springs and dampers can be pooled in k'(œ) and k"(œ). Perpendicular to the sliding direction, a displacement-controlled, forced harmonic oscillation is being imposed with the angular frequency œ, the amplitude Auz and the constant mean indentation depth uz ,0:

uz (t ) = uz ,0 + Auz cos(œt ). (2)

In the immediate contact with substrate we assume the classic Coulomb law of friction with a constant coefficient of friction ^ 0. The dynamics of the system is characterized by the horizontal and vertical position of the center of mass ux and uz as well as the position of the contact uxc. The origin of the coordinate system represents the unstressed state of the "rheology element".

3. Generalization of 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 velocity—as already described in the introduction. Above a certain critical velocity v* the macroscopic coefficient of friction of a system with applied vibrations remains identically the same to the system without any applied vibrations—at 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 kr(œ). 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.

3.1. Calculation of critical velocity

We study only the non-jumping case which means that the indenter remains in contact with the substrate for all

2. Formulation of the model

We consider a system having a mass m and system stiffness kx (Fig. 1). The system is pulled with a constant velocity v0 on a substrate. The immediate contact between them is modeled by a linear rheology whose normal and tangential stiffness, kz , kx , as well as the system stiffness kx can be described in the complex form

Jt(ro) = k'(rc) + ik "(rc). (1)

The real part k'(rc) is representative for the elasticity of the material, whereas the imaginary part k" (rc) 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

Fig. 1. Mechanical model of the indenter system. By using the complex stiffnesses any arbitrary linear rheology can be represented

times. This implies on the one hand that the oscillation amplitude Auz must be smaller than the mean indentation depth uz 0 and therefore

Auz < u

z,0

Vt

(3)

is valid. On the other hand, the normal force must remain positive for all times and hence

F > 0 Vt. (4)

Both inequalities (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 ux c remains positive for all times for the first time and thus

Ux,c > 0 Vt (5)

is valid. In order to calculate the critical velocity, it is necessary to set up and solve a differential equation in ux c. For this purpose, one has to set up a differential equation in ux first of all.

We assume the motions of the mass and the contact point in the sliding case is

ux = u

+ v0t + Aux cos (rot + 9),

(6)

ux,c = uXx + V + Aux>x COS (rot + (7)

9 and * are phase shift relative to the forced harmonic oscillation (2) which can be also written in the following form

(8)

uz,0

+ Au,

irot . -¡rot

e + e

The normal force is calculated using the complex stiffness as [14]

Fn = kz,c (0)uz,0 + [kz,c (ro)e'rot + kz,c (-ro)e"'rot ] =

= kz,c (0)uz,0 + Auz [k' c (ro) cos (rot) -

- kZ,c (ro)sin (rot )]. (9)

The Newton's equation of motion for the mass reads as follows:

= Fs -^0Fn, (10)

where Fs is the pulling force. Substitution of (6) into Eq. (10) provides

-mAuxro2 cos (rot + 9) = kx (0)uX0) - Aux X x [kX (ro)cos(rot + 9) - k"x (ro) sin (rot + 9)]- X X kz,c (0)uz,0 - ^0Auz k,C (ro)COs(rot) -- kZ,c (ro) sin (rot)]. (11)

Considering only the oscillation part of Eq. (11), we have [Aux (-mro2 + k'x (ro) + ik"x (ro))e'9 +

+ ^0 Auz (kZ,c(ro) + ik;,c(ro))]eirot + + [Aux (-mro2 + kx (ro) - ik"x (ro)) e-9 + + ^ Auz (kz,c(ro) -ik;,c(ro))]e-irot = 0, (12)

Aux (-mro2 + k'x (ro) + ik"x (ro))e'9 + + ^0 Auz (k'zc (ro) + ik"z,c (ro)) =0, (13)

or

Auxe9 = -

Auxe~'9 = -

(14)

(15)

Aux (-mro2 + kx (ro) - ikx (ro))e"i9 + + M-o Auz (kZc (ro) - ikZc (ro)) = 0. From above equation system we obtain M-0 Auz (k'z c(ro) + ik"zc(ro)) -mro2 + kx (ro) + ikx (ro) M-0 Auz (kZ,c(ro) - ikZ'c( ro)) -mro2 + kx (ro) - ikx (ro) Now let us consider the motion of contact point ux c. Since in the sliding case all horizontal forces remain in equilibrium, one has:

M0Fn = kx,c (0)(ux0) + V - - V) +

+ kx,c (ro)[Aux cos(rot + 9) - Aux,c cos(rot + ^)] = = kx,c (0)(uxo) - ufc) + Aux [kx,c (ro) cos (rot + 9) -- k' (-ro) sin (rot + 9)] - Aux,c [kx,c (ro)x

x cos (rot + - kXfi (ro) sin(rot + Similar to (12), we consider the oscillation parts M0Auz [kz ,c (ro) cos (rot) - k"z c (ro) sin (rot)] = = Aux [k' (ro) cos(rot + 9) - kx,c (-ro) sin (rot + 9)] --Aux c [kx,c (ro) cos(rot + - kx,c (ro) sin (rot + or

[(kx,c (ro) + ikx,c (ro))(Auxei9 - Aux^) -

-M0Auz (k'z ,c (ro) + ik"2fi (ro))]etot + + [(kx,c(ro) - ikxjro))(Auxe-'9-Aux,ce^) -

(16)

(17)

- ^Auz(kX,c(ro) - ik'' (ro))]e_iœt = 0.

Then

(kx,c (ro) + iK,c (ro))(Auxe'9 - Auxe ) --^0 Auz ( kz ,c(ro) + iK,c (ro)) = 0,

(kx,c(ro)- ikx)c(ro))(Auxe-9 -Aux,ce-*) -

- ^A^ (kz c (ro) - ikz c (ro)) = 0. Substitution of solution (15) leads to

|kz,c(ro)l

(18)

(19)

(20)

Aux,c = M0Auz

I ^x,c(ro)|

1

(k' (ro) + k'x (ro) - mro2 )2 + (kx,c (ro) + kx(ro))

-. (21)

(k'x(ro) - mro2)2 + (kx(ro))2

The time dependence of the coordinate of the contact point is thus equal to

|kz,c(ro)|

ux,c =ux°C + v0t + M0Auz

|kx,c(ro)|

cos(rot + *) X

1

(k'xc (ro) + k'x (ro) - mro2 )2 + (kx,c (ro) + kx (ro ))

(22)

(k'x(ro) - mro2)2 + (kx(ro))2 Differentiating it with respect to time provides the velocity of the lower point of the spring:

ux c (t) = v0 - ^0œAuz | Kz'c( )| sin (œt + x

I k^x , c(œ)|

1

(k'x,c (œ) + k'x (œ) - mœ2 )2 + (k' (œ) + k'x (œ))

. (23)

(k'x(œ) - mœ2)2 + k»)2 This velocity remains positive if the sliding velocity exceeds the critical velocity v0. By taking condition (5) into account the critical velocity is given by

I kz,c (œ) I

v* = ^0œAuz

kx,c(œ)|

1

(k' (œ) + k'x (œ) - mœ2 )2 + (k' (œ) + k'x (œ))

. (24)

(kX (ro) - mro2)2 + (kX (ro))2 Thus, the knowledge of the complex stiffness kx (ro), kx c (ro) and kz c (ro) allows to calculate the critical velocity of controllability of sliding friction for an arbitrary linear rheology.

3.2. 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.

For a Kelvin material the complex stiffness of the contact is given by [14]:

kx,c(ro) = kx,c + i'Y x,cro (25)

and

K ,c (œ) = kz + iY z „œ.

(26)

In [13], the investigated model consists of a body with mass m, which is pulled with a constant velocity v0 throughout a pure elastic system spring kx. Therefore kx (ro) is equal to

kx • By plugging in K (ro), kx,c(ro) and K,c(ro) in Eq.(24) one has:

v* = M-0œAuz

I kz,c + iYz,c œ I 1 kx,c + i'Yx,cœ I

x

(kx,c + kx -mœ2)2 + (y_œ)2

(kx - mœ2)2

By assuming an isotropic material and hence

Yx,c _ kx,c Y z ,c kz ,c

is valid, this implies

(27)

(28)

v0 =-

I kx - mœ

(Y z,cœ)2 +

~T-(kx,c + kz,c - mœ )

(29)

which is exactly the same result developed in [13]. By considering the limiting case Yz,c ^ 0 and hence a pure elastic contact, the results in [12] is again obtained

* kz c | kx c + kx - mro2 |

v0 ^oroAUz-^ x'; x . • (30)

kx,c I kx - mro2|

In the case of a very stiff system kx ^^ and without mass, the critical velocity in [11] is further constituted

v* .

(31)

Moreover, Eq. (24) can be applied for complicated model such as the Prony series with different relaxation times.

4. Qualitative analysis

4.1. 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 Fig. 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 (kx ^ ro) meaning that the contact stiffness is small in comparison with the stiffness of the indenter. Thus, system dynamics are negligible. The movement of the body and lower point, is then equal to

Ux(t) _ Vot, (32)

kz c kz c Ux,c(t) _ Vot-ho ^Uz,o -ho AUzy-COS(ro0, (33)

kV f kV f

ux c (t ) = v0 + ^-0œAuz —^sin (œt ).

(34)

Fig. 2. Mechanical model of a Kelvin material. A stiff system is assumed

Considering the non-jumping condition (3) and (4), the following inequality must be fulfilled for the latter one for all times:

kzu,0 >Auz*\klc + Y2X Vt,

(35)

where (3) implies that the oscillation amplitude is smaller than the mean indentation depth, while (35) is an inference of the requirement that the normal force remains positive for all times. Here the normal force is given by

Fn = kz,cuz + Yz,cuz =

= kz,c(uz,0 +Au2 cos(rot))-Yz,croAuz sin(rot). (36) In the horizontal movement, the lower point can be either in slip or stick states. Thereby t, represents the moment when the sliding phase ends and a sticking phase begins, while t2 is the moment when the sticking phase ends and accordingly the sliding phase begins. This cycle is repeated continuously and is qualitatively illustrated in Fig. 3. To calculate the macroscopic coefficient of friction defined by

M n

1

ro

<Fn > 2n

t2 t,+2rc/œ

I Fxsfdt + J F*d t

t, t,

(37)

one has to determine the tangential forces occurring during the stick-slip motion of the contact.

4.1.1. Case of sliding

In this case, the tangential force is equal to the normal force multiplying m0:

Fsc =M0 Fn = M0kz,c (uz,0 + Auz cos (rot)) --M0 Yz,croAuz sin(rot ). (38)

4.1.2. Case of sticking During sticking states Coulomb's law of static friction

stick

< M0Fn is valid, where

FxS,cck = kx,c(ux -ux,c(t1)) + Yx,c(ux -u&x,c(t1)). (39)

At the moment t1 the sticking case begins and thus the velocity of the contact diminishes iixc(tj) = 0; t1 is also the moment when the sliding ends, so ux,c (t1) can be solved by the Eq. (33):

Mok Au

Mo^i M0kz,cAuz,0

Sticking stiding

Fig. 3. Qualitative illustration of stick-slip cases. Please note that the linear slope during sticking is only valid, when system dynamics are being neglected

ux,c(ti) = v0ti -M0 uz,0 -M0AuzTz,^cos(rot1), (40)

kx ,c kx ,c

thus Eq. (39) becomes

FxS,cck = kx,c V0 (t - t1) + M0kz,c (uz,0 +

+ Auz cos (rot)) + Yx,cV0. (41)

Now it is still necessary to be determine the bounds of integration t1 and t2. As t1 represents the moment when the sticking case begins and thus the velocity of the contact diminishes, one has

ux,c(t1) = 0. (42)

Applying (42) to (34) leads to

v0 + M0roAuz^ccos(rot1) = 0.

kxc

(43)

The bound t2 represents the moment, when the sliding phase begins. Thus, one has uix c(t2) = 0 and ux c(t2) =

= ux,c(t1):

v0 + M0roAuz^csin(rot2) = 0,

kxc

v0t, -M0Auz-^cos(rot,) =

kxc

= v0t2 -M0cos(rot2)

(44)

(45)

(46)

or write both together

v0(t, - t2) = M0 Auz (cos(rot,) - cos(rot2)) +

kx,c

+ v0 + M0roAuz sin(rot2).

kx,c

Substitute Fxslc, Fxs,t(!ck (Eqs. (38), (41)) and t1; t2 obtained from (43), (46) into Eq. (37), the coefficient of friction can be then calculated.

To conduct a numerical analysis, we formulate all equations in a dimensionless way with following variables: - v0

v =-*, T = rot, v0

which have been already introduced in [12]. The critical velocity in this case is given by Eq. (31). Additionally, we define two new dimensionless variables:

8 Yx,cro 8 Yz,cro Sx = 7-' z =•

kx,c kz,c

Under the assumption of an isotropic material with a constant Poisson's ratio (Eq. (28)), we can rewrite 8 x =

= 8z =8.

The normalized Eqs. (37), (38), (41), (43), (46) then

correspondingly become

M 1 t2 1 2n+T1

^^ = — JF1(T)dT + — J F2(T)dT, (47)

M0 2nt1 2n t2

Fig. 4. The dependence of the reduced coefficient of friction ^macro/on the dimensionless velocity v for 6 linearly increasing oscillation amplitudes varying from 0 to the maximal permissible value (Auzjuz 0)max: 5 = 1 and (Auzjuz 0)max = 0.71 (a); 5 = 2 and (Auz/uz,0)max = 0.45 (b); 5 = 10 and (A^/u^ax = 0.1 (c); 5 = 100 and (A^/u^Ux = 0.01 (d) '

. 1 Auz . Auz ?Auz_ //lo.

^j(t) = 1 + —- v (t-t1) + —z cos t1 +5—z v, (48)

uz ,0 uz ,0 uz ,0

7- / x 1 Auz z Auz . F2( t) = 1 + —- cos T-5—- sin t,

uz ,0 uz

(49)

^z ,0

v = -sin t1, (50)

v ( t 2 -t1) = cos t2 - cos t1 -5(v + sin t2), (51)

and non-jumping condition (35) is

AuzV^< 1

Vt.

(52)

^z ,0

[.2. Numerical results

Numerical results with 5 = 0 for a pure elastic contact validates the dependence of coefficient of friction on the velocity in [11], where this system configuration was studied. Close inspection reveals that for all investigated configurations no reduction of friction can be observed for sliding velocities beyond v*, since ^macro remains identically to ^ 0.

Eventually, we mainly investigate the controllability of friction for a viscoelastic material ( 5 ^ 0). Please note that non-jumping condition must be considered as a constraint in all evaluations. The macroscopic coefficient of friction is the function of following three variables:

^0

Au.

uz ,0

(53)

The numerical results are shown in Fig. 4. It is obvious that with an increasing influence of the damper the oscillation amplitude inevitably has to decrease due to the non-jumping condition. Thus, Auzjuz ,0 = 1 as a maximum relative oscillation amplitude is no longer a sufficient condition for the non-jumping case with respect to viscoelastic materials. We exemplify this by investigating a few cases. In the first two cases we keep the influence of the damper relatively small, whereas in the last cases we investigate a predominantly viscos material.

The results are qualitatively similar to the pure elastic case. For an increasing dimensionless velocity, the normalized macroscopic coefficient of friction tends towards the plateau of 1. The main difference is that the initial reduction of friction is less distinct compared to the purely elastic configuration. This behavior is ascribed to condition (52), as the relative oscillation amplitudes are additionally constrained. With a small 5 = 1, the maximal permissible oscillation amplitude due to non-jumping condition is ( Auz/uz 0 )max = 1/V2 = 0.71. The behaviors for six linearly increasing oscillation amplitudes from zero to the maximal permissible value Auz/uz0 = 0.71 are shown in Fig. 4, a,

where the corresponding curves for pure elastic cases are plotted with dashed lines for a comparison. At a higher value of 8 = 100 (Fig. 4, d), the maximal permissible amplitude is very small, only Auz/uz ,0 = 0.01, where the curves for elastic contact 8 = 0 approach to 1. From Fig. 4, c and d we can observe that the reduction of friction is still strong for higher viscosity of material although the oscillation is quite tiny.

5. Conclusion

The critical velocity of controllability is one of the main characteristics of the phenomenon of reduction of friction. Thus, the knowledge of that is indispensable for industrial applications. In the present paper we obtained an equation to calculate the critical velocity for arbitrary linear rheol-ogy. Hence, this equation can be applied to more complex material modelings in future investigations.

In the second part of this paper we conducted a numerical analysis for the case of a Kelvin material. The influence of viscoelasticity on the phenomenon of reduction of friction has been studied regarding the non-jumping case. It was shown that with increasing influence of viscosity, the effect of reduction of friction declines. This effect is ascribed to the fact that oscillation amplitudes are restricted additionally in order to keep the indenter in contact with the substrate for all times. Therefore, future investigations considering the jumping case might be very interesting. In comparison with a pure elastic contact at the same oscillation amplitude, the effect of friction reduction is stronger.

For simplicity, we have considered a frictional contact with a constant (independent on the normal load) stiffness, which can physically be realized as a cylindrical indenter. However, within the method of dimensionality reduction, it is shown that any contact of elastic or viscoelastic bodies can be equivalently represented by a contact of an equivalent shape with a properly defined Winkler foundation [15]. This means that our analysis can be easily generalized to an indenter having arbitrary shape.

Acknowledgments

The authors would like to thank V.L. Popov for his advice and contributions to develop the theoretical modeling. Furthermore, we also thank M. Popov for his support.

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Поступила в редакцию 31.10.2017 г., после переработки 09.01.2018 г.

Сведения об авторах

Jasan M. Zughaibi, Technische Universität Berlin, Germany, [email protected] Felix H. Schulze, Technische Universität Berlin, Germany, [email protected] Qiang Li, Dr.-Ing., Technische Universität Berlin, Germany, [email protected]