Relaxation damping in contacts under superimposed normal and torsional oscillation Текст научной статьи по специальности «Физика»

УДК 539.62

Релаксационное затухание в контактах под воздействием нормальных и торсионных колебаний

М. Попов12, В.Л. Попов123

1 Берлинский технический университет, Берлин, 10623, Германия

2 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия

3 Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия

Ранее было показано, что если контакт двух упругих тел без проскальзывания осциллирует в нормальном и тангенциальном направлениях, то в нем возникает специфическое затухание, связанное с частичной релаксацией тангенциальных напряжений, несмотря на отсутствие проскальзывания и связанных с ним фрикционных потерь энергии. В настоящей статье показано, что такой же механизм действует и в случае суперпозиции нормальных и торсионных колебаний. Найдено аналитическое решение для случая торсионных и комбинированных (торсионных/тангенциальных) колебаний.

Ключевые слова: структурное демпфирование, механика контактного взаимодействия, торсионные колебания, метод редукции размерности

Relaxation damping in contacts under superimposed normal and

torsional oscillation

M. Popov12 and V.L. Popov1-2-3

1 Berlin University of Technology, Berlin, 10623, Germany

2 National Research Tomsk State University, Tomsk, 634050, Russia

3 National Research Tomsk Polytechnic University, Tomsk, 634050, Russia

It was recently shown that if a contact of two purely elastic bodies with no sliding is subjected to oscillations in normal and tangential directions, a kind of damping occurs due to relaxation of tangential stress in areas of intermittent contact, despite the absence of sliding and corresponding frictional work. In the present paper we show that the same mechanism acts in contacts with superimposed normal and torsional oscillations. A closed-form solution for the torsional and combined (torsional/tangential) relaxation dissipation for a contact of arbitrary bodies of revolution is presented.

Keywords: structural damping, contact mechanics, torsional oscillations, method of dimensionality reduction

1. Introduction

Oscillating tangential contacts exhibit partial slip at the border of the contact area and energy dissipation related to this slip. Mindlin et al. studied this frictional damping both analytically and experimentally [1] in 1952. The contact damping plays an important role in numerous applications in structural mechanics [2], tribology [3], materials science [4] and technological processes related to the dynamics of granular media [5-8]. As partial slip is the main cause of this sort of damping, the damping disappears when the coefficient of friction increases and tends toward infinity [1]. However, when a contact oscillates in both normal and tan-

gential directions, the situation changes. Such damping has been recently studied by Davies et al. [9] for smooth two-dimensional profiles and by Putignano et al. [10] for rough surfaces. However, the fact that dissipation exists even in the limiting case of an infinite coefficient of friction, when relative frictional movement of the contacting bodies does not occur, went unnoticed and was first pointed out in [11]. The effect of relaxation damping as proposed in [11] can be easily extended to a contact with torsional (instead of tangential) oscillation superimposed with normal oscillation, which is the purpose of the present paper. As in the original paper [11], we assume perfect stick throughout the contact.

© Popov M., Popov V.L., 2015

2. Analysis

Consider a contact between two axially-symmetric elastic bodies with moduli of elasticity of Ex and E2, Poison's

ratios of Vj and v2, and shear moduli of Gx and G2,

ac-

cordingly. We denote the difference between the profiles of bodies as z = f (r), where z is the coordinate normal to the contact plane, and r is in the in-plane polar radius. The profiles are brought into contact and are subjected to a superposition of normal, tangential, and torsional oscillations with small amplitudes. This contact problem can be reduced to the contact of a rigid profile z = f (r) with an elastic half-space (Fig. 1, a).

Derivations of the present paper are based on the method of dimensionality reduction in contact mechanics (MDR) [12, 13], which recently was extended to torsional contacts [14]. In the framework of the method of dimensionality reduction, two preliminary steps are performed [13]. First, the three-dimensional elastic bodies are replaced by a one-dimensional linearly elastic foundation consisting of an array of independent springs, with a sufficiently small separation Ax and normal and tangential stiffness Akz and Akx defined according to the rules

1 1 -v? 1 — v2

ДЬ = E Ax with

* 12 — Vi

Akx = G Ax with —=-1 +

x G 4G

E? 2 — v?

4G.

(1)

(2)

In the second step, the three-dimensional profile z = =f(r) is transformed into a one-dimensional profile according to

, f '( r )

g ( x) = |x| JfJL d r.

о V x — r

(3)

If now the MDR-transformed profile g(x) is indented into the defined elastic foundation by the indentation depth d and is moved normally and tangentially according to arbitrary law, the force-displacement relations of the equivalent one-dimensional system will reproduce those of the original three-dimensional contact problem (proofs have been done in [12]).

In [14], it was shown that this procedure can be extended to torsional contacts by also allowing movement of the springs in the j-direction and by defining the corresponding stiffness according to the rule

Aky = GAx with -1 = + — G 8G1 8G?

This rule guarantees the correct description of the dependence of the torsional moment on the torsional angle, while the z-axis is considered as axis of rotation.

From the correctness of the force-displacement and torque-angle relations, it follows that the work and the dissipated energy will also be reproduced correctly. This is the reason for using the method of dimensionality reduction for calculation of energy dissipation.

In the following, we consider a rigid conical indenter

z = f (r) = r tg e (5)

in a contact with a half-space (Fig. 1, a). This choice means no restriction as the results can be generalized very easily to an arbitrary axis-symmetrical shape (see discussion below). The one-dimensional MDR-image of the conical profile (5), according to (3), is

g ( x) = |x| ?tg 0 = c|x|,

(6)

where c = n/2tg e is the slope of the one-dimensional equivalent profile (Fig. 1, b). The generalization for an arbitrary axis-symmetrical shape is possible due to the fact that the central area of permanent contact does not contribute to relaxation damping. Only the edge of the contact needs to be considered, and if the amplitude of normal oscillation is sufficiently small compared to the curvature of the in-denter, the shape of the edge of the contact will always be approximately linear. In this case, all axially-symmetric in-denters will behave like conical indenters and the slope c at the edge of the contact of the one-dimensional MDR-transformed profile becomes the only shape-related parameter. For example, for a parabolic indenter z = r2/ (2R), the MDR-transformed profile is z = g (x) = x2/R and the edge slope is c = 2a/R where a is the contact radius. As shown already in [11], the parameter c can be represented in a universal form which does not depend on the profile shape. The incremental contact stiffness is known to be equal to dFn/dd = 2aE [15]. Deriving this equation once more gives d2 Fn/ dd2 = 2 E*9a/dd = 2E*/c. Thus, the slope of the MDR-transformed profile can be calculated as

1=-1- (7)

c 2E* dd2 ■

Fig. 1. Contact of a rigid cone with a half-space (a) and the corresponding MDR-transformed one-dimensional profile (b)

Consider a point of the rigid indenter with initial distance z(0) from the plane and let us assume that the oscillations in the normal and tangential direction are given by

z(t) = -z(0) +uZ0) cos(wt),

(8)

y (t) = x^(0) cos(wt + 90), where ^(0) is the amplitude of the angle of torsion. All points of the medium which are in contact with the indenter follow this motion. If |uZ0) | > z^0), the point of the rigid surface will come into contact with one of the springs of the elastic foundation when the coordinate of the indenter will be y1 and the indenter will drag it along to point y2, where contact is lost and the spring relaxes over the distance s = y2 - y1. The coordinates y1 and y2 are determined by setting z = 0. After simple calculations we get

s = 72 - У1 = W

1

'z(0)

2

,(0)

Sin

(9)

The energy lost in one oscillation cycle (in the entire contact) by a conical indenter is

W = G J s 2dx,

amin

where

z0 = g (x) - d = cx - d

and the minimal and maximal contact radii are

d-u(0) d + u(0)

The integral (10) evaluates to

2( d uz

W =16 G Ф(0)2 3

sin2 90

5c3

(10)

(11)

(12)

(13)

Introducing the "average contact radius", a = d/c, we can rewrite this as

w = I6 G ( аф(0) )2 luflsin2 Ф0 3 c 11

1 + ^

.(0)2 _z_

5d2

(14)

Using equation (7), the dissipated energy during one cycle in a contact under action of normal and torsional oscillations will therefore be given by:

^ ( u(0)2 A

1 + . (15)

W = a 2ф(0)2 |uZ0)|sin2 Фо

3 E* dd2 ' ' ^ ' 5d2

V

For the example case of a contact of a rigid indenter with an elastic half-space having the Young modulus E and Poisson ratio v, we have

GL = 8G =

~E* ~ 2G(1 + v) = T+v and the dissipation equation takes the form

W = -

32 d2Fn

2na2ф(0)2 uZ0) sin2 ф0

1+

.(0)2 _z_

5d 2

(16)

3(1+v) dd2

\ y

Note that this result can be further generalized to superimposed oscillations in the normal and tangential direction as well as torsion. As the tangential force does not influ-

Fig. 2. A point of the rigid surface with the initial coordinate z = -z(0) oscillates around this position. It comes into contact with a spring in point y1 and loses contact in point y2

ence the torsional moment, the force-displacement and the moment-angle relations will be independent, which means that the corresponding relaxation contributions can be just added. For oscillations described by

z (t ) = -z(0) +40) cos (œt ),

x(t ) = u X0) cos( œt + 91),

^(t) = 9(0) cos (wt + 92) the energy dissipation per cycle will be 1 d 2 F

(17)

(18) (19)

W = -

3 E dd2

G~uf)2 sinz Ф1 + Ga Фwz sinz Ф2

u(0) x

^«(0)2 ,

1+

.(0)2 _z_

5d2

\\

. (20)

J J

If the oscillation amplitude is small Up << d, then the second term in the last brackets can be dropped, therefore 1 d 2 Fn

W = --

u(0) x

3 E dd2

x (G \ f2 sin2 ф1 + G a 2 ф(0)2 sin2 ф2).

(21)

3. Conclusion

In the present paper, the effect of relaxation damping described in [11] was extended for the superimposed normal, tangential and torsional contact of arbitrary axis-symmetric elastic bodies with infinite friction in the contact area. The assumption of the infinite coefficient of friction was made only to study the effect in the pure. However, all results are also applicable to systems with a finite coefficient of friction provided that the changes in the radius of the stick region are much smaller than those due to changing indentation. The dissipated energy is proportional to the amplitude of the normal oscillation and to square of the amplitude of torsional oscillations and to second derivative of the normal force with respect to the indentation depth.

Acknoledgements

This work was supported in part by Tomsk State University Academic D.I. Mendeleev Fund Program, project No. 8.2.19.2015.

References

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Поступила в редакцию 19.06.2015 г.

Сведения об авторах

Mikhail Popov, Researcher of Berlin University of Technology, [email protected]

Valentin L. Popov, Prof. Dr. of Berlin University of Technology, Prof. of Tomsk State University, Prof. of Tomsk Polytechnic University, [email protected]