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УДК 531.43
Моделирование диссипации энергии в контакте волокон иод действием нормальных и касательных колебаний
J. Zhang, A. Butz1, Q. Li1
Бэйханский университет, Пекин, 100191, Китай 1 Берлинский технический университет, Берлин, 10623, Германия
В статье проведено численное исследование фрикционного контакта двух скрещенных волокон под действием нормальных и касательных колебаний. Результаты моделирования на основе метода редукции размерности показывают, что вначале диссипация энергии трения растет с увеличением коэффициента трения, после чего почти симметрично уменьшается до постоянного значения. Отношение длины волокна к его диаметру оказывает значительное влияние на диссипацию энергии, причем данный эффект тем более выражен, чем выше значение коэффициента трения. Результаты моделирования для очень высокого коэффициента трения хорошо согласуются с аналитическим решением для случая бесконечного коэффициента трения.
Ключевые слова: диссипация энергии трения, контакт волокон, метод редукции размерности, коэффициент трения
Simulation of frictional energy dissipation in a fiber contact subjected to normal and tangential oscillation
J. Zhang, A. Butz1, and Q. Li1
Beihang University, Beijing, 100191, China 1 Berlin University of Technology, Berlin, 10623, Germany
This paper presents a numerical study on the frictional contact between two crossed fibers subject to both normal and tangential oscillation. The results from simulation using the method of dimensionality reduction show that the frictional energy dissipation increases firstly with coefficient of friction, and then almost symmetrically decreases to a constant. The fiber aspect ratio has an important effect on the energy dissipation and this effect becomes more significant for larger coefficient of friction. The simulation results for very large coefficient of friction show a good agreement with the analytical solution for the case of infinite coefficient of friction.
Keywords: frictional energy dissipation, fiber contact, method of dimensionality reduction, coefficient of friction
1. Introduction
When a contact of elastic bodies is subjected to tangential oscillation, frictional damping occurs due to slip in the parts of contact. This behavior plays an important role in numerous applications in mechanical engineering [1] and materials science [2], like as the machine elements subject to periodic vibrations [3]. The solutions for frictional damping of elastic spheres under an oscillating tangential force were given by Mindlin et al. [4] in 1952. However, the structure damping under both normal and tangential oscillation has been rarely studied [5-7]. Recently the so-called "relaxation damping" of elastic contact with infinite coefficient of friction due to a superposed normal and tangential oscillation was described by Popov et al. [8]. It was shown for the first time that dissipation exists even in the non-slip
case, when relative movement of contacting bodies does not occur at all. In this case, the energy dissipation in an oscillation cycle is equal to [8]
Q=3 G* TF "f2^2 «h, (i)
3 E duz
where u£0 and u(0) are the amplitudes of tangential and normal oscillation, is the phase shift between the oscillations, Fz is the normal contact force, uz is the indentation depth, v is the Poisson ratio, E and G are effective elastic and shear modulus, which for a contact of two identical materials are given by
E* = , G* =-E-. (2)
2(1 -v2) (1 + v)(2-v)
© Zhang J., Butz A., Li Q., 2015
This analytical solution was later applied to the contact of fibers in a woven material [9]. As in [8], only the limiting case of an infinite coefficient of friction was considered with some simplification that the influence of indentation depth and tangential displacement (compared with the oscillation amplitudes) is neglected.
In the present paper we consider the coefficient of friction in a large range. The frictional damping of fiber contact subject to superposed normal and tangential oscillation is numerically study in the framework of the method of dimensionality reduction for the case of arbitrary coefficient of friction and arbitrary, not necessarily very small, indentation depth. We consider the same contact geometry as described in [9], which we briefly describe in the following.
As shown in Fig. 1, two crossed cylindrical fibers with the same radius R and length 2l are in contact at right angles. Three of the four fiber ends are fixed in the plane at z = 0, while one end can move free in the horizontal and vertical direction (x-, z-plane). From the linear beam theory, it is known that when the free end has a vertical deflection Wz, the beam will be stressed with a contact force Fz in the middle, and the deflection of the central point of the beam with these boundary conditions is given by [10]
F l3 W WIz =-Fl- + WL
I,z 24 EI 2
(3)
where I = %/4 R4 is the area moment of inertia, E is the elastic modulus. For the beam with two fixed ends, it has only the first term:
Wii, z =-
Ff
24EI '
(4)
Then the indentation depth dz in the contact point is the difference between the deflection of these two beams that is equal to
(5)
12 EI 2
Similarly, for a horizontal deflection of the free end Wx, it gives
F/ Wx
W x =—x—+ —L I,x 24EI 2
but Wn,x _ 0. The relation between the tangential displacement dx and tangential force Fx at the contact point is
W
'Wr- (6)
Fl 3 x 24EI
In paper [9], under assumptions, Wz >> dz and Wx >> >> dx , Eqs. (5) and (6) were simplified so that linear relations between the contacting force and the deflection of the free end were obtained. Taking into account the analytical solution of relaxation damping in [8], the energy dissipation in one oscillation cycle with normal motion of the free end according to Wz = Wz,0 + AWz sin(wt) and tangential motion Wx =AWx sin(wt + ^0) is described as [9]
Qo _ 4
^5/3 (1 + V)(2 -v) ^
x E
\ ~ J
R '
l
R
(1
\4/3
2 0/3
V2)
V Wz.0,
AWx2 |AWz|sin2 <|>o,
(7)
where AWz and AWx are normal and tangential oscillation amplitudes of fiber free end. Equation (7) can be expressed as
Qo _ qa-5Wo-4/3E(AWx)2 |AWz|sin2 |o (8)
with a _ l/R being the fiber aspect ratio, W0 _ Wz the normalized initial displacement and the factor
r2 ^^ 213
z ,0
/R
q _ 4
n
,5/3
(1 + v)(2-v)
(i-v¥3 ■
Equation (8) can be only used for infinite coefficient of friction. Now we simulate the frictional damping of this contact using the method of dimensionality reduction while relations (5) and (6) between the contact force and deflection of the free end of the fiber will be used.
2. Method of dimensionality reduction
The method of dimensionality reduction is used for fast calculation and simulation of contact problems for elastic and viscoelastic bodies [11]. With this method, three-dimensional contacts are mapped onto one-dimensional ones with properly defined elastic or viscoelastic foundations. The method of dimensionality reduction provides exact solutions for the normal contact problem of axially symmetric bodies [12], as well as for tangential contact problems with Coulomb friction [13, 14]. Compared with popular finite element method or boundary element method, the computing time using the method of dimensionality reduction is sharply reduced by several orders of magnitude.
In the framework of the method of dimensionality reduction, the contacting bodies are replaced by one-dimensional models. Firstly, a rotationally symmetric three-dimensional profile z = f (r), is transformed into a equivalent one-dimensional profile according to the rule [15]
|x| f\r)
g ( x) _ x
Fig. 1. Contact between two crossed fibers
r
Vx -/
rd r.
(9)
Secondly, the elastic half-space is replaced by an elastic foundation consisting of independent springs with a sufficiently small spacing Ax. The normal and tangential stiffness of the springs kz and kx is defined as
kz = E*Ax, kx = GAx. (10)
If the profile is pressed into the elastic foundation with indentation depth dz and moved tangentially by dx, the springs will be stressed both in the normal and tangential direction. The resulting normal displacements of springs are given by uz (x) = dz - g (x). The radius a of contact area is given by the condition uz (a) = 0 or
g (a) = dz. (11)
The normal force of springs is equal to
fz (x) = kzuz (x) = E *uz (x)Ax. (12)
If Ax ^ 0, the total normal load Fn can be written as an integral
Fz = J fz (x)dx = 2JE*[dz - g(x)]dx
(13)
With the Coulomb friction law, the radius c of the stick area will be determined by the condition that the tangential force fx = kxdx is equal to the coefficient of friction ^ multiplied with the normal force fz (c) = kzuz (c) which results in the relation
G*dx =^E*(dz-g (c)). (14)
If the tangential displacement of springs is ux (x), the tangential force is calculated as
fx (x) = kxux (x). (15)
When this value of some spring is smaller than the local maximum friction force f (x), this spring sticks to the indenter and its tangential movement is same as the inden-ter's. After achieving the maximum value, fx (x) > yfz (x), the spring has a relative sliding against the indenter. These conditions can be written as following:
^ux (x) = Adx , if 1 fx 1 = 1 kx Aux (x) 1 <f ( x)
/ X , f (x) (16)
ux ( x ) = ,if sliding.
Akx
With Eq. (16), the tangential displacement ux (x) of any spring can be determined for arbitrary load history. The tangential force Fx can be then calculated as
Fx = J fx ( x)dx = 2 J G* dx dx + 2 J^E*[ dz - g( x)]d x. (17)
3. Frictional energy dissipation
From contact mechanics [16], it is known that contact between two crossed cylinders with radius R and R2 as shown in Fig. 1, is equivalent to the contact between a sphere with radius R* = ^R, R2 and an elastic half space with effective elastic modulus E as defined in Eq. (2). The three-dimensional profile of the equivalent sphere for this contact problem can be expressed with a parabolic function
Fig. 2. Sketch of elastic contact under the normal and tangential force
f ( r) = r2/2 R* when the indentation depth is much smaller than the radius of the sphere. Its corresponding one-dimensional profile according to Eq. (9) is calculated as g(r) = = r2/R*. Now this oscillating contact can be solved in the framework of the method of dimensionality reduction.
When the free end of fiber oscillates with a normal motion Wz (t) = Wz,0 + AWz sin(wt) and tangential motion Wx (t) = A Wx sin(wt + ^0), the contacting force and displacement (or indentation depth) at the contact point cannot be directly obtained with these given deflections Wz and Wx according to Eqs. (5) and (6). For the normal movement, the dependence of indentation depth on normal force at the contact point is known as
d^ =
3F„
2/3
4E
fî I * VR
(18)
Substituting into Eq. (5), we get the analytical relation between the normal contacting force and deflection of fiber end
3F
2/3
4E
*
VR
Fzl3 +W 12 EI 2
(19)
For the tangential movement, such a relation between tangential deflection Wx and the tangential contact force
Fig. 3. Dependence of frictional energy dissipation on the coefficient of friction for different phases
- a
- a
Fig. 4. Dependence of frictional energy dissipation on the ratio for three small coefficients of friction, | = 0.2, 0.5, 0.8 (a) and a large one | = 20 (b)
Fx can only be solved by the numerical simulation using the method of dimensionality reduction. Firstly from Eq. (19), the normal force on the contact point Fz is obtained for a given normal deflection of fiber end Wz, then the indentation depth dz and contact radius a are determined in one-dimensional contact as described in Eqs. (11) and (13). If the tangential displacement dx at the contact point is known, the stick and slip region with condition (16) will be checked, and the tangential displacement of each spring ux as well as the tangential force Fx can be calculated. Although dx is initially unknown, with one more relation (6) for a given tangential deflection of free end Wx, the oscillating tangential force Fx and displacement of fiber at the contact point dx can be determined by numerical iteration. The tangential deflection known, the energy dissipation in a time step can be calculated as
AQ(t) = Fx Adx, (20)
where Fx is the tangential force and Adx incremental displacement of the indenter at time moment t. After the same procedure for all time steps in a period, the frictional energy dissipation Qf in an oscillation cycle can be obtained as the sum of Eq. (20).
Figure 3 shows simulation results with parameters a = ¡¡R = 10 and W0 = Wz0/ R = 2. For comparison, the
01-
1
50.0 0.5 1.0 1.5 2.0 2.5 3.0 Coefficient of friction |u
Fig. 5. Power P as a function of the different coefficient of friction
dimensionless frictional energy dissipation Qf /Q0 is used as y-axis. It is clear that the energy dissipation increases firstly for very small coefficient of friction, and then almost symmetrically drops to a constant. For the large values of coefficient of friction, it is seen that Qf/Q0 = 1 ~ 2, the energy dissipation is very close to the analytical solution (7) in [9], where the coefficient of friction was considered to be infinite large.
An important parameter of interest in practice is fiber aspect ratio a_ l/R which has a significant effect on the contact behavior of crossed fibers. The analytical solution (5) in [9] shows that the non-frictional energy dissipation is dependent of parameter l/R with a power -5. Here we consider this relation for frictional case. To investigate the influence of ratio a_ l/R, the radius of fibers R is kept constant and the length l varies in the simulation. In Fig. 4 examples for three small coefficients of friction | = 0.2, 0.5, 0.8 and a large one | = 20 show that the frictional energy dissipation decreases with the ratio a_ l/R. The fitting (solid line) in Fig. 4, b indicates that this relation can be described well with a power function
and this power for the large coefficient of friction | = 20 is equal to P = - 4.8, close to the analytical solution for the infinite large friction.
The dependence of power P on coefficient of friction | is shown in Fig. 5. It can be seen that the absolute value of power P increases gradually from 2.5 to 5.0. The influence of parameter l/R of contacting fibers on the frictional dissipation with smaller coefficient of friction is weaker.
4. Conclusion
In the present paper, we simulated the two crossed fiber contact subject to an oscillation of one free end in the framework of the method of dimensionality reduction. The use of the method of dimensionality reduction allows one to obtain the missing additional relationships without simpli-
fications as in Ref. [9] but using the analogy between the system of two crossed cylinders and the contact of spheres. This problem can be studied for all values of coefficient of friction. The numerical results show that frictional energy dissipation increases firstly with the coefficient of friction, and then decreases to a constant value which is very close to the analytical solution in [9] for the non-frictional case. Furthermore, the influence of the important parameter— fiber aspect ratio a = l/R on the energy dissipation was studied. It is found that this influence is weaker for smaller coefficient of friction. For large coefficient of friction the simulation results show a good agreement with the analytical solution in [9]. With the advantage of fast computing time using the method of dimensionality reduction the study in this paper can be further developed to the multiple contacts of fibers in complicated woven structures.
Acknowledgement
The authors acknowledge many valuable discussions with V.L. Popov, M. Popov and N. Popov. This work was partially supported by Deutsche Forschungsgemeinschaft (DFG) and Deutscher Akademischer Austausch Dienst (DAAD).
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Поступила в редакцию 04.06.2015 г.
Сведения об авторах
Jie Zhang, MSc., Beihang University, China, jiezhanghn@gmail.com
Adam Butz, BSc., Berlin University of Technology, Germany, adam.butz@gmail.com
Qiang Li, MSc., Dr.-Ing., Researcher, Berlin University of Technology, Germany, qiang.li@tu-berlin.de
