Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 19, No 1, 2021, pp. 105 - 113 

https://doi.org/10.22190/FUME201226017P 

© 2021 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

FRICTION UNDER LARGE-AMPLITUDE  

NORMAL OSCILLATIONS 

Mikhail Popov1,2 

1Technische Universität Berlin, Germany 
2National Research Tomsk State University, Russia 

Abstract. Building on a recently proposed contact-mechanical theory of friction control 

by external vibration, the case of large-amplitude normal oscillation is revisited. It is 

shown that the coefficient of friction can be expressed in particularly simple form if the 

waveform of the displacement oscillation is triangular or rectangular, and the contact 

stiffness is constant. The latter requirement limits the scope of the exact solutions to 

contacts between a plane and a flat-ended cylinder or a curved shape with a wear flat, but 

the adopted methodology also enables efficient numerical solution in more general cases. 

Key words: Contact Mechanics, Vibration, Control of Friction, Large Amplitudes, 

Sliding Friction 

1. INTRODUCTION 

The ability of externally applied vibration to substantially reduce both static and 

sliding friction is well known and enjoys many practical applications. The classical 

examples of wire drawing [1,2] and metal forming [3,4] deserve mention, but a thorough 

review is outside the scope of this paper. While the effect has attracted a fair amount of 

research, most of the works are of an experimental, application-oriented nature [5-7], and 

proposed models are at best semi-empirical [8]. For this reason, no consensus has been 

established concerning the theoretical underpinnings of the phenomenon. A possible 

physical model based on macroscopic contact mechanics was recently proposed by the 

author and colleagues [9]. The mechanism of force reduction in this model is based on 

the observation that stick-slip can arise in an oscillating contact under suitable conditions, 

if the compliance of the contact is taken into account. During the stick phases the lateral 

force is by definition subcritical (i.e. less than what is required to sustain sliding), and 

therefore lowers the average friction force. Multiple extensions of this model have since 

 
Received December 26, 2020 / Accepted February 08, 2021 

Corresponding author: Mikhail Popov  

Technische Universität Berlin, Institut für Mechanik, Fachgebiet Kontinuumsmechanik und Materialtheorie, 
Sekr. MS2, Einsteinufer 5, D-10587 Berlin 

E-mail: mpopov@fastmail.fm 



106 M. POPOV 

been published and were reviewed in a recent paper [10]. Here, the same approach is used 

to analyze the case of large-amplitude normal oscillation, when the amplitude is larger 

than the mean indentation and the body starts to “jump” over the plane. 

2. MODEL 

For a complete description of the model the reader is referred to previous publications 

[9,10], but a short overview is provided here for convenience. First and foremost, it is 

assumed that the contact is quasistatic and that the contact stiffness is independent of 

indentation depth. Both assumptions are nonessential for the model as such, but are 

required for analytical calculations. Together, they allow us to treat the contact as a single 

linearly elastic massless spring (Fig. 1) with normal and lateral stiffness kz and kx, 

respectively. If the modeled contact is a flat-ended cylinder with radius a, the stiffness 

values are given by: 

 

2 2

* 1 2

*

1 2

* 1 2

*

1 2

1 11
2      where     

2 21
2      where     

4 4

z

x

k E a
E EE

k G a
G GG

 

 

− −
= = +

− −
= = +

  (1) 

with Ei, Gi being the elastic and shear moduli of the contacting bodies and νi their Poisson 

numbers. 

 

Fig. 1 A single massless spring, which serves as a minimal model of a sliding frictional 

contact. The sliding velocity is constant, while the vertical coordinate oscillates. 

Amontons friction with the constant coefficient of friction µ0 is assumed in the contact. 

The spring is pulled with a constant velocity v0 while also being subjected to a normal 

oscillation that is parametrized as 

 ( ) ( )
z z z

u t u A w ft= +  (2) 



 Friction Under Large-Amplitude Normal Oscillations 107 

where 
z

u  is the mean indentation, Az the amplitude, f the frequency and w a zero-mean, 

unit-amplitude waveform. The lateral displacement ux is the primary unknown of the 

system. 

When the contact point is in a sliding state, its velocity can be shown to be 

 
0

( ) ( )z
x z

x

k
u t A fw ft

k
 =   (3) 

The contact transitions from slip to stick when this velocity vanishes. The point of stick 

onset φ1 = ft1 can therefore be written as: 

 
1

1
( ) ( )w 

−=  (4) 

where β is one of the dimensionless variables that parametrize the behavior of the system: 

 
0

0

,      ,      xz

z z z

k vA
ft

u k A f
  


= = =   (5) 

The Eq. (4) does not necessarily have solutions. For stick-slip to be present, it is 

necessary that 

 max ( ) cw


   =   (6) 

where βc is the maximum positive gradient of the oscillation waveform. If the 

dimensionless velocity β exceeds this threshold value, stick-slip becomes impossible and 

the macroscopic coefficient of friction   is the same as the intrinsic coefficient of friction 

µ0. Otherwise it is reduced by some amount that depends on α, β and the shape of w. 

If condition (6) is satisfied and stick is initiated, the spring continues stretching with 

the constant velocity v0 and the lateral spring force therefore increases linearly with time: 

 
stick 0 1 0 1

( ) ( ) ( )
z x

F t F t k v t t= + −   (7) 

This continues while the stick condition Fstick < µ0Fz(t) holds. Substituting Fz = kzuz and 

rearranging gives the end of the stick phase φ2 in implicit form: 

 
2 1 2 1

( ) ( ) ( )w w    − = −   (8) 

The stick-slip process is visualized in Fig. 2. 

The macroscopic friction force 
x

F  is computed by integrating Fx(t) over both the 

slip and stick periods: 

 
0

1
( )d

T

x x
F F t t

T
=    (9) 



108 M. POPOV 

 

Fig. 2 Stick and slip under the influence of a harmonic oscillation. The dotted line 

represents the tangential force as it would be in pure slip, Fslip = µ0Fz(t). The solid 

line is the actual tangential force in the presence of stick-slip. The stick phases are 

the straight segments, e.g. between t1 and t2, while slip phases are the sinusoidal 

segments, e.g. between 2t  and t1, repeating periodically. Note that Fx ≤ Fslip  

everywhere, which is the origin of friction reduction in our model. 

Since Fx only differs from µ0Fz during the stick phase, it is actually more convenient to 

determine the absolute force reduction 
0x z x

F F F = − : 

 
2

1
0 stick

1
( ( ) ( ))d

t

x z
t

F F t F t t
T

 = −   (10) 

After expanding and rearranging, it is found that ΔFx can be expressed as 

 
0

( )
x z z w

F k A  =    (11) 

where Ψw is a dimensionless “reduction function” that is specific to the waveform w: 

 ( )
2

1
1 1

( ( ) ( ) ( ))d
w

w w



       = − − −   (12) 

The macroscopic coefficient of friction   can then be recovered through 

 
0

0 0
(1 ( ))

z x x

w

z z z

F F F

F k u


    

− 
= = − = −    (13) 

This puts the dependence into a very simple form, with most of the complexity contained 

in a function of one argument, Ψw(β). This function, however, needs to be determined 

numerically in most cases. 

This concludes our whirlwind tour of the model framework that will be used in the 

sequel. A less hurried presentation can be found in [10]. 



 Friction Under Large-Amplitude Normal Oscillations 109 

3. LARGE AMPLITUDES 

In the preceding discussion it was implicitly assumed that the amplitude Az is smaller 

than the mean indentation 
z

u , so that the bodies are permanently in contact and the 

normal force is non-negative. The purpose of this paper is to extend the analysis to 

z z
A u , that is, cases where the bodies lose contact periodically. Equivalently, 

z z z
A u A−   , where we have excluded the trivial no-contact case. This form also makes 

evident the need for a small re-parametrization: 

 
1

z

z

u

A



= =   (14) 

which avoids the singularity at 0
z

u = . 

The first thing to note about the jumping case is that the static coefficient of friction is 

always zero, because the contact obviously cannot sustain a lateral force while it is “in the 

air”, and slow creep will therefore be present at arbitrarily small pulling forces. If 

measurements of the static coefficient of friction under normal oscillation do not go to zero 

at suitably large amplitudes, this probably indicates a misalignment in the measurement 

apparatus. 

The second thing to note is that, in general, the simplicity of Eq. (13) can no longer be 

maintained. The clean separation between α and β is only possible because the stick-slip 

process is completely independent of mean indentation, so long as the normal force Fz  is 

positive throughout. However, when uz(t) becomes negative in the jumping case, this causes 

Fz to become “clipped” at zero (assuming no adhesion). This destroys the invariance w.r.t. zu , 

because the waveform w effectively becomes “cut off”, and has to be renormalized to 

maintain the properties of zero mean and unit amplitude. Thus, w(φ) should properly be w(γ,φ) 

in the jumping case. Overall, this leads us to expect the coefficient of friction to be a nonlinear 

function of two parameters (in addition to the waveform dependence): 

 
jmp 0

( , )
w

g   =   (15) 

In general, the function g needs to be computed numerically. There are, however, a few 

cases of some practical importance that can be treated analytically. These include square 

and triangle waves, for which solutions can be obtained in closed form due to their 

simplicity; and certain self-similar oscillations, for which asymptotic behavior can be 

deduced. These cases are considered next. 

3.1. Special case 1: Sawtooth and triangle wave 

Of the possible waveforms with triangular shape, here we consider the left-leaning 

sawtooth function (stl), the right-leaning sawtooth function (str) and the symmetric triangle 

wave (tri). The normalized functions w for these waveforms can be defined on the unit 

interval (with periodic extension understood) as: 

 

stl

str

tri

1 2

2 1

4 1, 1 / 2

3 4 , 1 / 2

w

w

w





 

 

= −

= −

− 
= 

− 

  (16) 



110 M. POPOV 

From geometrical considerations (which come down to determining the area between the 

waveform and a straight line with the slope β as in Fig. 2), it is easy to show that the 

corresponding reduction functions Ψw(β) in the simple non-jumping case are given by: 

 
str stl

2 4
( ) 1 ,     ( ) ,      ( )

2 2 4
tri

 
  

 

−
 = −  =  =

+ +
  (17) 

The triangular waves have the unique property that clipping the waveform does not 

affect the coefficient of friction. To appreciate this, refer once again to Fig. 2. The 

coefficient of friction is given by the ratio of the area under Fx to the area under µ0Fz. 

This ratio changes continuously as the waveform is clipped from below by increasingly 

large amplitudes. If the waveform is triangular, however, then the only effect from the 

cutoff is that the ramp of the stick phase starts later and later (in the point of first contact). 

The area ratio is not affected, which means that the coefficient of friction remains constant, 

despite the fact that Ψw formally depends on γ. This means that, for triangular waves, 

 
0

( , 1) ( , 1) (1 ( ))
w

        = = = −   (18) 

Using the reduction functions given in Eq. (17), this provides the following simple results 

for the coefficient of friction under large-amplitude oscillation: 

 
str 0 stl 0 0

2
( ) ,      ( ) ,      ( )

2 2 4
tri

  
        

 
= = =

+ +
  (19) 

3.2. Special case 2: Self-similar waveforms, Square wave 

The triangular waves are a special case of what could be termed self-similar 

waveforms. By this we mean that a cut-off waveform can be rescaled in such a fashion as 

to be identical to the original waveform. Assuming that the waveform is also convex 

ensures that stick is precipitated in the point of first contact, as in the case of the 

triangular wave. This means that the stick-slip graph of a cut-off waveform can be 

rescaled (together with the stick ramp) to have the same area ratio – and therefore the 

same coefficient of friction – as the same waveform at another cutoff. Of course, this 

rescaling also changes the slope β, which must be adjusted accordingly. Usually, it is 

convenient to choose the coefficient of friction at γ = 1 as a reference point, so that the 

large-amplitude coefficient of friction of a self-similar waveform can be expressed as: 

 
0

( , ) (1 ( ( , )))
w

      = −   (20) 

The function ξ which provides the remapping of β is specific to the waveform. 

After the triangle, the next-simplest example of a self-similar waveform is the square 

wave, which alternates between 1 and -1 in equal intervals. It is easy to show that the 

remapping function for such an oscillation is given by: 

 
sqr

2
( , )

1


  


=

+
  (21) 

Using this mapping and the reduction function Ψsqr of the square wave (see Eq. (36) in [10]), 

the coefficient of friction under large-amplitude square wave oscillations can be written as: 



 Friction Under Large-Amplitude Normal Oscillations 111 

 
sqr

0

, 2(1 )
4(1 )

1
1 , 2(1 )


 

 


 




 + +

= 
+ −  +



  (22) 

This result is shown in Fig. 3 for the entire range of γ from 1 (starting to separate) to -1 

(barely touching). 

 

Fig. 3 Coefficient of friction under large-amplitude square wave oscillations with 11 

different normalized indentations γ covering the entire jumping range from -1 to 1. 

The concept of self-similar waveforms also applies to the harmonic oscillation, to a 

limited extent. While the entire sine wave is not self-affine according to our definition, it 

can be approximated piecewise by a parabola over some of its domain. Since the parabola 

is indeed a self-affine function, we can expect the coefficient of friction under harmonic 

oscillation to have the described behavior asymptotically, although it will not be valid for 

values of γ close to 1. The remapping function in this case can be shown to be 

 0
2

1
( , )

1


   



+
=

+
  (23) 

where γ0 is the value of γ at the point where the self-affine scaling behavior started. More 

generally, for a waveform that can be asymptotically approximated by a power law φn, 

the corresponding remapping can be shown to be 

 

1/

0
1

( , )
1

n

n


   



+ 
=  

+ 
  (24) 



112 M. POPOV 

3.3. Numerical example: Harmonic oscillation 

As an example of asymptotic scaling, Fig. 4 shows numerically determined coefficients 

of friction under large-amplitude harmonic oscillation. One thing to note is that for γ in the 

range of approximately -0.2 to 1, the coefficient of friction depends only weakly on γ, with 

all curves bunching fairly closely together. The dependence on γ is also non-monotonous in 

this range, leading to lower coefficients of friction at first (from γ = 1 to approx. 0.6), and 

then increasing again (from γ = 0.6 to -1). The value around γ = -0.3 is the point from which 

the remaining part of the cropped waveform can be regarded as roughly parabolical, and the 

subsequent behavior of the coefficient of friction can be described by the scaling given in 

Eq. (23). This is also shown in Fig. 4 with black dots. 

 

Fig. 4 Numerically computed coefficient of friction under large-amplitude harmonic 

oscillations with 11 different normalized indentations γ covering the entire 

jumping range from -1 to 1. Note the non-monotonous dependence on γ: The dark 

red line corresponds to the critical value γ = 1, which separates the jumping and 

non-jumping regions. From there, the coefficient of friction is first reduced with 

diminishing γ (red lines and arrow) and then increases again (blue lines and arrow) 

starting somewhere around γ = 0.6. Black dots indicate the expected scaling 

behavior according to Eq. (23) relative to γ0 = -0.3. 

4. CONCLUSIONS 

The influence of large-amplitude normal oscillation on sliding friction, which has not 

previously received much attention in the literature, was analyzed in this work, based on a 

model proposed by the authors in a previous publication. It was shown that the coefficient of 

friction in the jumping case depends on the same dimensionless variables as in the low-

amplitude case, but in a more complicated fashion. At low amplitudes, the influence of the 

two main variables, α and β is cleanly separated, while at large amplitudes they become 



 Friction Under Large-Amplitude Normal Oscillations 113 

entangled and influence the coefficient of friction in a nontrivial manner. This was 

demonstrated on the example of the harmonic oscillation, where the amplitude-dependence is 

non-monotonic and can only be determined numerically. However, some simple cases such as 

triangular, rectangular and more general self-similar waveforms yield relatively simple results, 

which allow the coefficient of friction to be expressed either in closed form or as an 

asymptotic scaling relation. 

Acknowledgement: This work was supported in part by the Tomsk State University competitiveness 

improvement programme, which the author gratefully acknowledges. 

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