Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  

Series: Mechanical Engineering Vol. 14, N
o
 3, 2016, pp. 335 - 341 

DOI: 10.22190/FUME1603335P 

CRITICAL VELOCITY OF CONTROLLABILITY OF SLIDING 

FRICTION BY NORMAL OSCILLATIONS IN VISCOELASTIC 

CONTACTS 

UDC 539.3 

Mikhail Popov
1,2,3

 

1
Tomsk Polytechnic University, Tomsk, Russia 

 

2
Technische Universität Berlin, Berlin, Germany  

3
Tomsk State University, Tomsk, Russia 

Abstract. Sliding friction can be reduced substantially by applying ultrasonic vibration in 

the sliding plane or in the normal direction. This effect is well known and used in many 

applications ranging from press forming to ultrasonic actuators. One of the characteristics of 

the phenomenon is that, at a given frequency and amplitude of oscillation, the observed 

friction reduction diminishes with increasing sliding velocity. Beyond a certain critical 

sliding velocity, there is no longer any difference between the coefficients of friction with 

or without vibration. This critical velocity depends on material and kinematic parameters 

and is a key characteristic that must be accounted for by any theory of influence of 

vibration on friction. Recently, the critical sliding velocity has been interpreted as the 

transition point from periodic stick-slip to pure sliding and was calculated for purely 

elastic contacts under uniform sliding with periodic normal loading. Here we perform a 

similar analysis of the critical velocity in viscoelastic contacts using a Kelvin material to 

describe viscoelasticity. A closed-form solution is presented, which contains previously 

reported results as special cases. This paves the way for more detailed studies of active 

control of friction in viscoelastic systems, a previously neglected topic with possible 

applications in elastomer technology and in medicine. 

Key Words: Active Control of Friction, Ultrasonic Vibration, Viscoelastic Contact, 

Critical Velocity 

                                                           
Received October 22, 2016 / Accepted November 30, 2016 

Corresponding author: Mikhail Popov  

Technische Universität Berlin, Str. des 17. Juni 135, 10623 Berlin  

E-mail:m@popov.name 



336 M. POPOV 
 

1. INTRODUCTION 

The reduction of static and sliding friction by ultrasonic oscillation in various directions 

is a well-known phenomenon with many applications ranging from wire drawing and press 

forming, stabilization of system dynamics, as in brake squeal suppression, and production of 

directed motion, as in ultrasonic motors and linear actuators. The effect has been studied for 

several decades, both experimentally and theoretically. Among the proposed explanations, 

microscopic theories have historically been prevalent. E.g. Zaloj et al. [1] suggest that the 

effect may be due to the dilatation caused by sliding. V. Popov et al. point to the possible 

importance of the microscopic interaction potential [2]. Although plausible, microscopic 

models could never achieve good, quantitative correspondence between theoretical predictions 

and experimental results, e.g. [3]. Opposite to that stand purely macroscopic models, which 

explain the phenomenon using macroscopic contact mechanics or system dynamics. 

Several system configurations have been considered from that perspective [3, 4, 5] and it 

was found that the macroscopic models can describe the observed behavior of the systems 

without fitting parameters. This result is in fact somewhat surprising, considering that 

these macroscopic theories assume a constant microscopic coefficient of friction and a 

friction law of the form Ff = 0Fn. When the average force of friction is determined by 

integrating the force of friction over time (or integrating stress over time and contact area) 

and dividing by the integral of the normal force, the direct proportionality of the assumed 

law of friction will insure that the integrals of normal force will cancel out, with the end 

result that the average coefficient of friction   must always be equal to 0. This reasoning, 

however, is subtly flawed, in that it assumes sliding in one direction with a nonzero velocity. 

It is also possible for the body to temporarily cease motion (e.g. due to increasing normal 

force or more complicated reasons relating to system dynamics). During such stick 

phases, the law of friction needs to be written in its static form: Ff  0Fn. Note the less 

than or equal in this formula, which breaks the proportionality and allows 
 
to be less 

than 0. To the author’s knowledge, the possibility that the influence of normal oscillations 

on sliding friction may be explained entirely by the presence of intermittent stick phases 

has not been made explicit before the publication of the two part-study [6, 7]. In these 

papers, the stick-induced reduction of friction force was studied in a displacement-

controlled setting with and without in-plane system dynamics. Although a closed-form 

solution for the actual force of friction under the action of normal vibrations does not 

exist in either case, it has turned out to be possible to calculate the critical velocity vc for a 

broad class of problems. This critical velocity refers to the maximum sliding velocity, above 

which vibration no longer has any influence on friction (at a given frequency and amplitude). 

This is illustrated in Fig. 1, which qualitatively describes the behavior of the average 

coefficient of friction, as it increases from its static value to 0 with increasing sliding 

velocity. In the theory presented in [6] it was argued that this critical velocity is related to 

the disappearance of stick in the contact. Also in [6], the following expression was obtained 

for the critical velocity in an entirely displacement-controlled system: 

 
0

*

c z *

E
v u

G
    , (1) 



 Critical Velocity of Controllability of Sliding Friction by Normal Oscillations in Viscoelastic Contacts 337 

where uz is the amplitude of velocity oscillation and E
*
/G

*
 is the ratio of the normal 

and tangential stiffness of the contact (the so-called Mindlin-ratio). Since this ratio is 

generally of the order of unity, one can roughly say that the critical sliding velocity is 

equal to the maximum velocity in the normal direction (due to the oscillation) times the 

microscopic coefficient of friction. This critical velocity also enters into the primary 

dimensionless parameter characterizing the behavior of the system, which makes accurate 

analysis of this quantity doubly important. 

 

Fig. 1 Qualitative dependence of the average coefficient of friction (COF) on sliding 

velocity under action of normal oscillations. Of particular interest are the “static 

COF” at zero velocity, the monotonous increase of the COF with increasing sliding 

velocity and the critical velocity of controllability, above which the average COF 

is equal to the microscopic COF, 0, with or without oscillations. 

If the model is augmented with a system spring and a contact mass, thus enabling in-

plane system dynamics, the expression for the critical velocity becomes [7]: 

 

2

0 2

z ,c x ,c x

c z

x ,c x

k | k k m |
v u

k | k m |

  
  

 
 , (2) 

where kx,c and kz,c are the tangential and normal stiffness of the contact (in this model, the 

contact stiffness is assumed to be constant), kx is the tangential stiffness of the surrounding 

system and m is the mass of the sliding body. The only difference compared to Eq. (1) is 

the additional dependence on the two natural frequencies of the system. Indeed, if kx tends 

to infinity, Eq. (2) reduces to the previous result. Another notable feature is the presence 

of two resonant frequencies, in particular xk m   where vc becomes infinite. Numerical 

experiments show that in this case, the coefficient of friction reaches a plateau (which is 

less than 0) at fairly low sliding velocities and does not change thereafter. For the full 

analysis, the reader is referred to [6, 7]. 

In the present paper these previous results are extended to also include viscoelastic 

contacts. Active control of friction and system stability seems to be an underexplored 

topic when viscoelastic contacts are concerned, despite many possible applications in 

conjunction with the ubiquitous use of elastomers and the rising demands placed on 

devices in contact with biological tissues in medical technology. With this paper we 



338 M. POPOV 
 

would like to begin establishing a quantitative framework for the analysis of viscoelastic 

friction under oscillation, by proposing that the same methods used in [6, 7] can be 

applied in viscoelastic contacts in order to calculate the critical velocity in closed form. 

2. MODEL AND ANALYSIS 

2.1. Formulation of the model 

The model that will be analyzed in this paper is very similar to the one presented in 

[7]. It consists of a mass m that is pulled with a constant velocity v0 through a system 

spring with a constant stiffness kx (see Fig. 2). In addition, a displacement-controlled 

harmonic oscillation is imposed in the direction normal to the plane. The oscillation is 

defined by: 

 
0z z , z

u u u cos t   , (3) 

where uz is the coordinate of the body in the normal direction, uz,0 the mean indentation 

depth, uz the oscillation amplitude and  the frequency. The body is connected to the 

substrate through a contact point, in which Amontons’ law of friction with a constant 

coefficient of friction 0 is assumed. The main difference is that the contact is not elastic 

but viscoelastic and characterized not only by the constant tangential and normal spring 

stiffness kx,c and kz,c, but also by the dynamic viscosities γx,c and γz,c. This corresponds to 

the Kelvin material, the simplest model of viscoelasticity. The relevant dynamics of the 

resulting system is confined to the sliding plane and is characterized by ux, the position of 

the body and ux,c, the position of the contact point. 

 

Fig. 2 Schematic representation of the considered system, consisting of a mass,  

a system spring and a viscoelastic contact with the sliding plane. 

2.2. Analysis of the model 

2.2.1. Normal force 

The normal force in the spring-damper combination is given by: 

 
0

( )
N z ,c z z ,c z z ,c z , z z ,c z

F k u u k u u cos t u sin t         . (4) 



 Critical Velocity of Controllability of Sliding Friction by Normal Oscillations in Viscoelastic Contacts 339 

To ensure that the body is always in contact with the plane, the normal force must always 

remain positive. This is the case if: 

 
2 2 2

0z ,c z , z z ,c z ,c
k u u k     . (5) 

Only this “non-jumping” case is considered in the following.  

The static force of friction (the force at zero sliding velocity) can be calculated easily 

by noting that, according to Eq. (4), the amplitude of the oscillation of the normal force is 

equal to: 

 
2 2 2

N z z ,c z ,c
F u k      . (6) 

The static force of friction is equal to the minimal normal force during an oscillation 

cycle, multiplied with the coefficient of friction: 

 
2 2 2

0 ,0 0 , ,0 , ,
( ) ( )

s N N z c z z z c z c
F F F k u u k         . (7) 

2.2.2 Tangential movement 

Under the assumption that the immediate contact point is always in the sliding state, 

the equation of motion of mass m reads: 

 
0 0

( )
x x x N

mu k v t u F   . (8) 

The equilibrium condition for the “foot point” of the spring-damper combination reads: 

 
0

( ) ( )
x ,c x x ,c x ,c x x ,c N

k u u u u F      , (9) 

where FN is given by Eq. (4). 

Equation (8), after inserting Eq. (4) on the right hand side, can be easily solved with 

respect to ux: 

 
0

0 0 0 2
( )

z ,c z
x z , z ,c z ,c

x x

k u
u v t u k cos t sin t

k m k

 
       

 
. (10) 

In our analysis we assume that the material of the contacting elastomer body is isotropic, 

with a constant (frequency-independent) Poisson number. Under these conditions, we have: 

 
x ,c x ,c

z ,c z ,c

k

k





. (11) 

Equation (9) can also be solved with respect to (ux – ux,c): 

 
0 0

( )
z ,c

x x ,c z . z

x ,c

k
u u u u cos t

k
      . (12) 

From Eqs. (10) and (12) we can first determine ux,c: 



340 M. POPOV 
 

0
0 0 0 02

1 1
( )

z ,cz
x ,c z ,c z , z ,c z ,c z

x x ,c x ,cx

ku
u v t k u k cos t sin t u cos t

k k km k

   
                

  (13) 

and finally 
x ,c

u :  

 

20

0 02

2

20

0 02 2

( )

( )

z ,cz
x ,c z ,c z ,c z

x ,cx

z ,c x ,c xz
z ,c z

x ,cx x

ku
u v k sin t cos t u sin t

km k

k k k mu
v cos t u sin t

km k m k

 
            

 

    
        

   

  (14) 

The critical velocity of controllability is given by the condition that the amplitude of 

the oscillating part of this solution becomes equal to constant sliding velocity vc: 

 

2

2 20

2
( ) ( )

z ,cz

c z ,c x ,c x

x ,cx

ku
v m k k

k| m k |

   
            

. (15) 

Note that the critical velocity depends on the oscillation amplitude but not on the average 

indentation. 

In the limit of a very stiff system spring, kx, the critical velocity, Eq. (15), is reduced 

to Eq. (1), which thus appears to be valid independently of the viscoelastic properties of the 

medium. According to the Method of Dimensionality Reduction (MDR) [8], any rotationally 

symmetric contact can be equivalently represented by a model consisting of a series of 

independent springs (note that an equivalent one-dimensional model can in fact be constructed 

for almost arbitrary, e.g. rough, contacts, although there may be no closed-form mapping 

rule in the general case). As has been argued in [6], the existence an equivalent model with 

uncoupled spring elements, together with the indentation-independence of Eq. (15), implies 

that the obtained result in Eq. (15) is valid not only for the simple considered model with a 

single spring-damper combination, but also for quite general contacts (so long as the 

amplitude of oscillation remains small). 

3. CONCLUSION 

While the details of the influence of oscillation on friction may be very complicated at 

intermediate sliding velocities [7], there are still two simple and nearly universal (except 

in resonant cases) characteristic points: First, the velocity-dependences all start from the 

static value at vanishing velocity. Second, the coefficient of friction increases monotonically 

(again, barring exceptional system-dynamical circumstances) until it reaches the microscopic 

value at some critical velocity. These two points, the static coefficient of friction, and the 

critical velocity of controllability of friction, are the most important characteristics of any 

oscillating frictional system. It so happens that both of these points can be determined 

analytically for very general classes of contacts with and without system dynamics.  

In the present paper, the critical velocity of controllability was determined for the 

simplest possible viscoelastic rheology (Kelvin body) and the simplest possible contact 

geometry (contact with constant contact stiffness, e.g. cylindrical punch). Eq. (15) provides 



 Critical Velocity of Controllability of Sliding Friction by Normal Oscillations in Viscoelastic Contacts 341 

an explicit analytical solution. Even under these simple assumptions, the critical velocity 

depends on almost all system and loading parameters: the local coefficient of friction 0, 

mass m of the system, the stiffness of the contact and of the system, the frequency of 

oscillations, the damping coefficient of the contact, and on the amplitude of oscillations. 

However, it does not depend on absolute indentation, which permits easy generalization 

to more realistic contact geometry. 

Further, in the case of displacement-controlled horizontal movement (corresponding 

to an infinitely stiff surrounding system, which eliminates system dynamics in the contact) 

it was found that the critical velocity is given by Eq. (1), without dependence on the 

rheological properties of the contact: only the ratio of the contact stiffness (Mindlin ratio) 

appears in the expression for this critical velocity. 

In the future, the critical velocity could also be considered for materials with more general 

rheology. The Method of Dimensionality Reduction [8] provides a natural theoretical 

framework for this and for further generalizations to arbitrary contact geometries and loading 

histories. 

Acknowledgements: The author would like to thank V. L. Popov and A. E. Filippov for inspiring 

discussions concerning the present work. 

REFERENCES  

1. Zaloj, V, Urbakh, M, Klafter, J., 1999, Modifying friction by manipulating normal response to lateral motion. 

Physical review letters 82(24), pp. 4823, 1999. 

2. Popov, VL, Starcevic, J, Filippov, AE., 2010, Influence of ultrasonic in-plane oscillations on static and sliding 

friction and intrinsic length scale of dry friction processes. Tribology Letters, 39 (1), 25-30, 2010. 

3. Teidelt, E, 2015, Oscillating contacts: friction induced motion and control of friction, Dissertation, TU Berlin, 

Germany, 151 p. 

4. Milahin, N., Starcevic, J., 2014, Influence of the normal force and contact geometry on the static force of 

friction of an oscillating sample. Physical Mesomechanics, 17(3), pp. 84-8. 

5. Milahin, N., Li, Q., 2016, Friction and wear of a spherical indenter under influence of out-of-plane ultrasonic 

oscillations. Physical Mesomechanics, 19(2), pp. 149-153. 

6. Popov, M., Popov, VL, Popov, NV, 2016, Reduction of friction by normal oscillations. I. Influence of contact 

stiffness, arXiv:1611.07017. 

7. Mao, X, Popov, VL, Starcevic, J., Popov, M, 2016, Reduction of friction by nor-mal oscillations. II. In-plane 

system dynamics, arXiv:1611.07018. 

8. Popov, VL., Heß, M. 2015, Method of dimensionality reduction in contact mechanics and friction, Springer, 

Berlin, Heidelberg, 265 p.