Investigation on Dynamic Response of Rubber in Frictional Contact [original]

ORIGINAL RESEARCH

published: 26 March 2019

doi: 10.3389/fmech.2019.00009

Frontiers in Mechanical Engineering | www.frontiersin.org 1March 2019 | Volume 5 | Article 9

Edited by:

Irina Georgievna Goryacheva,

Institute for Problems in Mechanics

(RAS), Russia

Reviewed by:

Luciano Afferrante,

Politecnico di Bari, Italy

Erik Kuhn,

Hamburg University of Applied

Sciences, Germany

Fedor Stepanov,

Institute for Problems in Mechanics

(RAS), Russia

*Correspondence:

Ken Nakano

[email protected]

Valentin L. Popov

v[email protected]

Specialty section:

This article was submitted to

Tribology,

a section of the journal

Frontiers in Mechanical Engineering

Received: 21 December 2018

Accepted: 04 March 2019

Published: 26 March 2019

Citation:

Nakano K, Kawaguchi K,

Takeshima K, Shiraishi Y, Forsbach F,

Benad J, Popov M and Popov VL

(2019) Investigation on Dynamic

Response of Rubber in Frictional

Contact. Front. Mech. Eng. 5:9.

doi: 10.3389/fmech.2019.00009

Investigation on Dynamic Response

of Rubber in Frictional Contact

Ken Nakano1*, Kai Kawaguchi1, Kazuho Takeshima1, Yu Shiraishi1, Fabian Forsbach2,

Justus Benad2, Mikhail Popov2and Valentin L. Popov 2*

1Faculty of Environment and Information Sciences, Yokohama National University, Yokohama, Japan, 2Department of

System Dynamics and Friction Physics, Technische Universität Berlin, Berlin, Germany

In the present work, we analyze a device for measuring the dynamic response of rubber

in sliding contact. The principle of the device is to measure excited oscillations of a

mechanical system with a sliding contact between a rubber roller and a rigid surface.

Depending on the contact properties, varying oscillation amplitudes are measured. The

goal is to determine dynamic response properties of rubber from oscillating tests. For

this sake, an analytical model is introduced in which the contact problem and system

dynamics are considered in detail. Analytical and numerical results obtained for this model

are compared with some experimental data and discussed both on a qualitative and

quantitative level.

Keywords: rubber, contact, sliding friction, viscoelasticity, contact stiffness, contact damping

INTRODUCTION

Elastomers such as rubber are of great importance in many technical applications, particularly

where large frictional forces occur (Saccomandi and Ogden, 2004; Popov, 2017). Tires,

transportation rollers, shoe soles, seals, contact materials of buttons and keys on small electronic

devices, or viscoelastic vibration dampers in structures and machines (Jones, 2001; Rao, 2003)

are only a few examples. High industrial demands require a profound knowledge of the material

properties of elastomers used in these components. An accurate, yet rapid and simple acquisition of

these material parameters is often desired. Especially difficult is accurate determination of dynamic

response in frictional contacts, which is important for analyzing stability of sliding contacts (Braun

et al., 2009; Nakano and Maegawa, 2009; Amundsen et al., 2012). An efficient acquisition of material

properties and determination of dynamic response of rubber is the goal of the measuring device we

study in this paper.

The principle of the device is to measure excited oscillations of a mechanical system with a

sliding contact between a rubber roller and a rigid surface. Depending on the contact properties,

varying oscillation amplitudes are measured. By introducing an analytical model for the contact

between the rubber roller and the rigid oscillator, material and dynamic response properties that

we seek can be extracted from the experimental data. With such a procedure, the quality of the

results will depend to a large extent on how efficiently and accurately one can model the contact.

Elastomers exhibit a time-dependent behavior generally characterized by a large spectrum

of relaxation times, which adds complexity to the already multi-scalar properties of

the contacting surfaces, which makes the numerical modeling a challenge (Kürschner

et al., 2015). Many of these difficulties can be overcome with the application of the

method of dimensionality reduction (Popov and Heß, 2015) with which we can map

the given three-dimensional contact problem to an equivalent contact problem of

a transformed indentation profile with a one-dimensional viscoelastic foundation of

independent rheological elements, for details see Benad (2018); Popov et al. (2018). In order

Nakano et al. Dynamic Response of Rubber

to obtain a first qualitative and quantitative understanding of

the measuring device, we begin in the present investigation with

a simplified model of only one of these rheological elements

consisting of a single spring and a single damper to model the

contact. In this approach, we follow the recent works of Popov

(2016),Mao et al. (2017) and Benad et al. (2018) in which the

influence of oscillations on friction is investigated. It will be

shown that many characteristic features of the system studied

in the present paper show already with such a simplified one-

element model. This paves the way for more detailed studies in

the future.

The parts of this work are organized as follows: First, we

describe the chosen analytical model of the system. We then

discuss two limiting cases for the motion of the system and

present the analytical solutions for these cases. Afterwards, we

turn from the limiting cases to the arbitrary motion of the

system which is investigated numerically. The results are first

discussed on a qualitative level and then compared with the data

obtained from a measuring device. We conclude with a summary

and a brief outlook in which we address open questions, which

are recommended for investigation in future works using more

detailed models.

INVESTIGATED SYSTEM

Analytical Model

Figure 1 is the analytical model showing a mechanism for

measuring dynamic response of rubber in sliding contact. The

rubber roller with the radius Rcontacts with the arc-shaped rigid

surface of a rigid arm at the normal load Wand rotates about

the “rotational axis” with the peripheral velocity V. The rigid arm

is supported by a torsional spring (torsional spring constant: K0

and torsional damping coefficient: C0), which allows the rigid

arm to oscillate about the “torsional axis.” The moment of inertia

of the rigid arm about the torsional axis is J. The distance between

the torsional axis and the contact point is b, which is identical to

the curvature radius of the rigid surface. The rigid arm is pinched

by two translational springs (translational spring constant: k0and

translational damping coefficient: c0) at the distance afrom the

torsional shaft. One of the translational springs is supported by an

actuator (denoted by PZT) mounted to the base, while the other

is supported directly by the base under a certain preload. When

the actuator provides the harmonic displacement excitation

u(t)=u0sin ωt(1)

we measure the steady-state response of the angular displacement

of the rigid arm

θ(t)=αsin (ωt+ϕ)+θ0(2)

where tis the time, u0is the amplitude, ωis the angular

frequency, αis the angular amplitude, ϕis the phase shift, and

θ0is the static angular displacement.

Equation of Motion

The equation of motion of the rigid arm is written as

J¨

θ+C0+2a2c0˙

θ+K0+2a2k0θ=ak0u+ac0˙u+bF (3)

FIGURE 1 | Investigated system: scheme of the measuring system for

dynamic response of rubber in sliding contact.

where (·) is the time derivative and Fis the tangential force

acting on the rigid arm due to the rubber roller. By using the

following notations

K=K0+2a2k0(4)

C=C0+2a2c0(5)

Γ=au0qk02+(c0ω)2(6)

ϕ0=tan−1c0ω

k0(7)

the equation of motion can be rewritten as

J¨

θ+C˙

θ+Kθ=Γsin (ωt+ϕ0)+bF (8)

Note that the phase shift ϕ0is coming from the viscoelastic

properties of the two translational springs (i.e., k0and c0), not

from the viscoelastic properties of the contact.

Limiting Cases

Case I: No Contact

First, let us consider the situation of “no contact” (i.e., W=0 and

V=0). Since the rubber roller does not contact with the rigid

surface, the tangential force vanishes (i.e., F=0). Then we obtain

the steady-state response Equation (2) with αand ϕ

α=Γ

K

q1−Ω22+(2ζΩ)2

(9)

ϕ= −tan−1 2ζ Ω

1−Ω2!+ϕ0(10)

Frontiers in Mechanical Engineering | www.frontiersin.org 2March 2019 | Volume 5 | Article 9

Nakano et al. Dynamic Response of Rubber

respectively, where the dimensionless excitation frequency Ωand

the dimensionless damping ratio ζare given by

Ω=ω

(11)

ζ=C

2√JK(12)

respectively.

Case II: Static Contact

Secondly, let us consider the situation of “stationary contact” (i.e.,

W>0 and V=0). With no slippage at the contact, the tangential

force is given by

F= −bkcθ−bcc˙

θ(13)

where kcand ccare the tangential contact stiffness and the

tangential contact damping, respectively. Then we obtain the

angular amplitude and the phase shift in the same forms as

Equations (9), (10), where Ωand ζare given by

=ω

q(K+b2kc)

(14)

ζ=C+b2cc

2qJK+b2kc(15)

respectively.

Case III: Fully Slipping Contact

Thirdly, let us consider the situation of “fully slipping contact”

(i.e., W>0 and V>0). With considering the frictional force

with the constant friction coefficient µ, the tangential force at the

contact is given by

|F|=µW(16)

This merely shifts the equilibrium position of the oscillator, but

does not influence its frequency and damping (Nakano, 2006).

Thus, both remain the same as the case of “no contact” and are

given by Equations (11, 12).

Arbitrary Case

To examine the arbitrary motion of the system, the equation

of motion Equation (8) can be easily solved numerically using

the Euler time integration procedure. Therein, the calculation

scheme for the contact force is the following: In each time step

of the dynamic simulation, it is first assumed that the immediate

contact between the contact spring and the oscillator is sticking.

Thus, the change of the tangential contact force is equal to

∆F=kcV−b˙

θ−ccb¨

θ∆tif |F|< µW(stick) (17)

This is valid as long as the absolute value of the tangential

force remains smaller than the normal force multiplied with the

coefficient of friction. If this condition is violated in the given

time step, then the tangential force is set to the normal force times

the coefficient of friction with the appropriate sign:

F=sign V−b˙

θµW(slip) (18)

The above equations describe unambiguously the procedure for

determining the tangential force in each time step. We will

now turn to a qualitative discussion of the simulation results.

Thereafter, the simulation results will be compared to some

experimental data.

TRANSITION BETWEEN STICKING AND

SLIPPING REGIMES

In the previous section, we identified essentially two limiting

cases for the motion of the system. For one of them, the velocity of

the roller Vis zero (Case II: static contact), while for the other, it

is very high (Case III: fully slipping contact). It is clear that when

the roller starts rotating, there will be some transition between

the two regimes. Some features of this transition are relatively

simple and robust, which can be discussed on the qualitative

level. We will now illustrate these qualitative features with the

numerical results.

Consider first the dependency of the velocity amplitude of the

rigid arm on the excitation frequency. This is shown in Figure 2

for both the case of static contact with no slip (right resonance

curve, displayed with a blue broken line) and the case of sliding

contact with no stick (upper left resonance curve, displayed with

a red broken line). If the peripheral velocity of the roller is larger

than the velocity amplitude of the rigid arm, then the roller

will continuously slide on the rigid arm surface with a relative

velocity, which keeps changing its absolute value but not the sign.

Therefore, the force of friction will remain constant at any time.

This will introduce some static displacement but will not affect

the resonance curve. Thus, if the rotation velocity of the roller

is larger than the maximum velocity amplitude, the roller does

not influence the oscillations and they occur exactly as in the

non-contact mode. This is similar to the effects which are seen

in the active control of friction by oscillations, where there is also

a critical velocity, after which there is no influence of oscillations

by the force of friction (Popov, 2016).

Let us now consider the case when the roller velocity is smaller

than the maximum oscillation velocity. Consider some particular

roller velocity, as shown for instance with the horizontal green

broken line in Figure 2 and denoted by V. This line has two

intersection points with the resonance curve for the velocity

amplitude at the frequencies f1and f2. For all frequencies smaller

than f1or larger than f2the roller velocity is larger than the

velocity amplitude. This means that in these two regions, the

contact with the roller does not influence the resonance curve.

Between f1and f2, partial stick occurs, and the resonance curve

differs from that without the contact. For the chosen roller

velocity, the corresponding dependency is shown with the green

bold curve.

Accordingly, we can identify three stages during the decrease

of the roller velocity: (1) At very high sliding velocities, the

contact with the roller does not influence the resonance curve. (2)

Frontiers in Mechanical Engineering | www.frontiersin.org 3March 2019 | Volume 5 | Article 9

Nakano et al. Dynamic Response of Rubber

FIGURE 2 | Dependencies of velocity amplitude of rigid arm on excitation

frequency for different rubber roller velocities; blue broken line: case II (static

contact); red broken line: case III (fully slipping contact). Decrease of sliding

velocity (starting from the complete sliding – left high peak) leads first to a

decrease of the resonant peak (with practically unchanged resonant

frequency) followed by a relatively quick shift of the frequency toward the value

corresponding to the stationary contact. For each particular sliding velocity V

the curve under the horizontal line at the level of sliding velocity (between

frequencies f1and f2) remains unchanged compared to the case of complete

sliding.

When the roller velocity touches the maximum of the resonance

curve, first only the top of the curve becomes “cut”—without

significantly changing the resonance frequency. (3) Further

decrease of the roller velocity leads to a shift of the maximum of

the resonance curve to the right; finally, it tends toward the case

of static contact.

Note that in Figure 2, the shape of the resonance curve below

the level of the roller velocity Vremains unchanged compared

with the limiting case III (fully slipping contact). Therefore, the

resonance curve below the roller velocity level does not depend

on the friction coefficient, which only determines the static shift

of the oscillator. Between the frequencies f1and f2, the shape

of the resonance curve is changed, and it could depend on the

friction coefficient. However, most surprisingly, the results of

the numerical simulation of the investigated model show that

there is no such dependence. Figure 3 illustrates this somewhat

counter-intuitive feature. In addition to the resonance curves

for the two limiting cases (broken lines), it shows six additional

resonance curves (black solid lines), all obtained for the same

exemplary sliding velocity of the roller, which is low enough so

as to allow an excitation frequency band in which partial stick

occurs. Different finite values for the friction coefficient were used

to obtain these six additional resonance curves. Not only do all six

curves coincide in the frequency region of continuous sliding, but

also in the middle frequency band, in which the resonance curves

differ from the continuous sliding case due to the periods of stick.

FIGURE 3 | Dependencies of the amplification factor (ratio of rigid arm

amplitude to excitation amplitude) on excitation frequency for six different

friction coefficients in the range from 0.1 to 1.0; blue broken line: case II (static

contact); red broken line: case III (fully slipping contact).

COMPARISON WITH EXPERIMENTAL

DATA

Figure 4 is a photograph of the apparatus developed for

measuring dynamic response of rubber in sliding contact, which

embodies the model shown in Figure 1. The rubber roller was

made of styrene-butadiene rubber with Young’s modulus of 15

MPa. As the counter surface of the rubber roller, an abrasive

paper with an arithmetic roughness of 4.3 µm was affixed to the

curved surface of the rigid arm. The specifications of the system

were as follows: the moment of inertia J=5.0 ×10−3kgm2,

the torsional spring constant K0=9.9 ×102Nm, the torsional

damping coefficient C0=4.3 ×10−1Nms, the translational

spring constant k0=1.6 ×104N/m, the translational damping

coefficient c0=1.2 ×101Ns/m, the length a=30 mm, and

the length b=60 mm. The amplitude and frequency of the

PZT actuator were u0=2.5 µm, and ω/2π=50 to 100 Hz,

respectively. The normal load was W=20 N and the peripheral

velocity of the rubber roller was V=20 mm/s. Note that the

normal load was applied by pulling a coil spring with a low

stiffness to minimize the variation of its value.

Let us now compare the analytical and numerical results with

some experimental data on resonance curves. As in the previous

sections, we first examine the limiting cases (i.e., case I: no

contact, case II: static contact, and case III: fully slipping contact).

It is shown in Figure 5 that the theoretical results for cases I and

II are in very good agreement with the experimental data for no

contact (W=0 N and V=0 mm/s) and the static contact (W=

20 N and V=0 mm/s), respectively, where the contact stiffness

and contact damping are kc=8.9 ×104N/m and cc=26 Ns/m,

Frontiers in Mechanical Engineering | www.frontiersin.org 4March 2019 | Volume 5 | Article 9

Nakano et al. Dynamic Response of Rubber

FIGURE 4 | Apparatus developed for measuring dynamic response of rubber in sliding contact.

FIGURE 5 | Resonance curves showing frequency response of angular

amplitude; black line: theoretical results for case I (no contact) and case III (fully

slipping contact); blue line: theoretical result for case II (static contact) at W=

20 N; the blue line is barely visible due to excellent fitting; black symbols:

experimental data at W=0 N and V=0 mm/s; blue symbols: experimental

data at W=20 N and V=0 mm/s.

respectively, which were determined by fitting experimental data

to the theoretical solution Equation (9) with Equations (14), (15).

The above values of kcand cchave been used for all simulations

in contact: static contact, intermittent slip and full slip.

We now turn again to the transition between the limiting

cases. From our theoretical results, we expect the resonance

FIGURE 6 | Resonance curves showing frequency response of angular

amplitude; black line: theoretical results for case I (no contact) or case III (fully

slipping contact); blue line: theoretical result for case II (static contact) at W=

20 N; red lines: numerical results for arbitrary case at W=20 N and V=0.1 to

0.5 mm/s; red symbols: experimental data at W=20 N and V=20 mm/s.

amplitude to be lowered and the resonance frequency to be

shifted to the right for the case of partial stick, when compared

to limiting case I/III (no contact/fully slipping contact). This

is confirmed with the experimental data, as can be seen in

Figure 6. One can further observe from the graph that the

experimental resonance curve differs substantially from the

Frontiers in Mechanical Engineering | www.frontiersin.org 5March 2019 | Volume 5 | Article 9

Nakano et al. Dynamic Response of Rubber

resonance curve for case I/III in a certain frequency range around

its resonance frequency, while for higher or lower frequencies

the resonance curves still coincide. This is a characteristic

feature in the transition phase which was also observed and

described in detail in the previous section. On a qualitative

level, however, a significant discrepancy of the roller velocities

is evident comparing the experimental data of the transition

with the corresponding numerical results. In order to obtain a

similar resonance frequency and maximum amplitude, the roller

velocity in the numerical simulation would have to be smaller by

an approximate factor of 0.01 than in the experimental setup. A

velocity of V=20 mm/s as it is shown in Figure 6 would fully

comply with the slipping case in the framework of the simplified

model. And yet, in the experimental data, we clearly observe a

tendency which we would expect only at much lower velocities.

At this stage, this leaves an open question which will have to be

investigated further in future works. It is expected that a more

detailed model for the contact will yield more accurate simulation

results and provide insight on why the simplified model chosen

in this work leads to the discrepancies in the transition curve.

CONCLUSION

In this work, we analyzed a measuring device whose basic

principle lies in measuring excited oscillations of a mechanical

system with a sliding contact between a rubber roller and a rigid

surface, which is expected to lead to the extraction of material

properties of rubber. In order to obtain a first qualitative and

quantitative understanding of the device, we introduced a simple

model of the system. Therein, the contact between the rubber

roller and the rigid oscillator was modeled with one rheological

element consisting of a single spring and a single damper. It

was shown that many characteristic features of the system show

already with such a simplified one-element model.

Essentially two limiting cases of the system were identified:

The static contact, where the rotation velocity of the roller is zero,

and the case of continuous sliding, where the velocity of the roller

is so high as to never allow phases of stick between the rubber

roller and the oscillator. It was shown that the latter coincides

with the no contact case. The analytical and numerical results for

the limiting cases obtained with the simplified model are in very

good agreement with the experimental data.

When the roller starts rotation, there is a transition between

the limiting cases. This was also confirmed experimentally.

Some features of this transition are relatively simple and robust

and were discussed on a qualitative level: At very high sliding

velocities, the contact with the roller does not influence the

resonance curve. When the roller velocity touches the maximum

of the resonance curve of the velocity oscillations, first only the

upper portion of the curve is slightly altered, without significantly

changing the resonance frequency. A further decrease of the

roller velocity leads to a shift of the maximum of the resonance

curve to the right before finally it tends toward the static

contact case.

The qualitative features of the transition between the limiting

cases upon decreasing the roller velocity which were obtained

with the simplified model could also be observed in the available

experimental data of the transition. On a quantitative level,

however, a significant discrepancy of the roller velocities became

evident when comparing the experimental data of the transition

with the corresponding numerical results. In order to obtain

a system response similar to the experimental data, the roller

velocity in the numerical simulation had to be smaller than in

the experiment by an approximate factor of 0.01. This leaves

an open question which will have to be investigated further in

future works.

It is expected that a more detailed model for the contact

will yield more accurate simulation results and provide

insight on why the simplified model chosen in this work

leads to the discrepancies in the transition of the resonance

curves. A first consideration in future works could be the

friction law used in the model. As suggested and discussed

in detail in Woodhouse et al. (2015), the Coulomb model

in its simplest form (Coulomb, 1821), which we adopted

in this work, may be too crude for predicting details of

friction-driven vibration. More complex friction laws, for

example as they are discussed in Barber (2018), may lead

to better results. Therein, a critical component may be

the dependence of the friction coefficient on the sliding

velocity, see Grosch (1963). Another consideration should

be the adoption of a more sophisticated contact model.

Roller and oscillator of the apparatus are practically two

wheels in contact. Such a rolling contact can be modeled

more accurately than in the present paper using more than

one rheological element with the method of dimensionality

reduction, see Popov et al. (2015); Li and Popov (2017).

AUTHOR CONTRIBUTIONS

KN and VP conceived the study. KN designed the apparatus. YS,

KT, and KK conducted experiments. MP, JB, and FF conducted

numerical simulations. All authors contributed to writing the

manuscript and approved it.

FUNDING

The authors are grateful for financial support of the Japan

Society for the Promotion of Science (VP) and the Deutsche

Forschungsgemeinschaft (MP).

ACKNOWLEDGMENTS

The authors thank Dr. Y. Osawa, Dr. K. Hagiwara, Mr. S.

Hatanaka, and Ms. L. Zimmermann for valuable discussions.

REFERENCES

Amundsen, D. S., Scheibert, J., Thøgersen, K., Trømborg, J., and Malthe-

Sørenssen, A. (2012). 1D model of precursors to frictional stick-slip motion

allowing for robust comparison with experiments. Tribol. Lett. 45, 377–369.

doi: 10.1007/s11249-011-9894-3

Barber, J. (2018). Contact Mechanics. New York, NY: Springer.

doi: 10.1007/978-3-319-70939-0

Frontiers in Mechanical Engineering | www.frontiersin.org 6March 2019 | Volume 5 | Article 9

Nakano et al. Dynamic Response of Rubber

Benad, J. (2018). Fast numerical implementation of the MDR transformations.

Fact. Univ. Mech. Eng. 16, 127–138. doi: 10.22190/FUME180526023B

Benad, J., Popov, M., Nakano, K., and Popov, V. L. (2018). Stiff and soft

active control of friction by vibrations and their energy efficiency. Forschung

Ingenieur. 82, 331–339. doi: 10.1007/s10010-018-0281-1

Braun, O. M., Barel, I., and Urbakh, M. (2009). Dynamics of

transition from static to kinetic friction. Phys. Rev. Lett. 103:194301.

doi: 10.1103/PhysRevLett.103.194301

Coulomb, C. (1821). Theorie des Machines Simple (Theory of Simple Machines).

Paris: Bachelier.

Grosch, K. (1963). The relation between the friction and visco-elastic

properties of rubber. Proc. R. Soc. Lond. A Math. Phys. Sci. 274, 21–39.

doi: 10.1098/rspa.1963.0112

Jones, D. (2001). Handbook of Viscoelastic Vibration Damping. New York, NY: John

Wiley & Sons

Kürschner, S., Popov, V. L., and He,ß, M. (2015). “Contacts with elastomers,”

in Method of Dimensionality Reduction in Contact Mechanics and

Friction. (Berlin, Heidelberg: Springer Berlin Heidelberg), 99–113.

doi: 10.1007/978-3-642-53876-6_7

Li, Q., and Popov, V. L. (2017). Normal line contact of finite length cylinders. Fact.

Univ. Series Mech. Eng. 15, 63–71. doi: 10.22190/FUME170222003L

Mao, X., Popov, V. L., Starcevic, J., and Popov, M. (2017). Reduction of friction

by normal oscillations. II. In-plane system dynamics. Friction 5, 194–206.

doi: 10.1007/s40544-017-0146-x

Nakano, K. (2006). Two dimensionless parameters controlling the occurrence of

stick-slip motion in a 1-DOF system with Coulomb friction. Tribol. Lett. 24,

91–98. doi: 10.1007/s11249-006-9107-7

Nakano, K., and Maegawa, S. (2009). Stick-slip in sliding systems

with tangential contact compliance. Tribol. Int. 42, 1771–1780.

doi: 10.1016/j.triboint.2009.04.039

Popov, M. (2016). Critical velocity of controllability of sliding friction by normal

oscillations in viscoelastic contacts. Fact. Univ. Mech. Eng. 14, 335–341.

doi: 10.22190/FUME1603335P

Popov, M., Benad, J., Popov, V. L., and Heß, M. (2015). “Acoustic emission in

rolling contacts,” in Method of Dimensionality Reduction in Contact Mechanics

and Friction (Berlin: Springer), 207–214. doi: 10.1007/978-3-642-

53876-6_14

Popov, V. L. (2017). Contact Mechanics and Friction. Berlin: Springer.

doi: 10.1007/978-3-662-53081-8

Popov, V. L., and Heß, M. (2015). Method of Dimensionality Reduction in

Contact Mechanics and Friction. Berlin: Springer. doi: 10.1007/978-3-642-

53876-6

Popov, V. L., Willert, E., and Heß, M. (2018). Method of dimensionality

reduction in contact mechanics and friction: a user’s handbook III. Viscoelastic

contacts. Fact. Univ. Mech. Eng. 16, 99–113. doi: 10.22190/FUME180

511013P

Rao, M. (2003). Recent applications of viscoelastic damping for noise control

in automobiles and commercial airplanes. J. Sound Vibration 262, 457–474.

doi: 10.1016/S0022-460X(03)00106-8

Saccomandi, G., and Ogden, R. (2004). Mechanics and Thermomechanics of

Rubberlike Solids. Wien: Springer. doi: 10.1007/978-3-7091-2540-3

Woodhouse, J., Putelat, T., and McKay, A. (2015). Are there reliable

constitutive laws for dynamic friction? Philos. Transact. A 373, 1–21.

doi: 10.1098/rsta.2014.0401

Conflict of Interest Statement: The authors declare that the research was

conducted in the absence of any commercial or financial relationships that could

be construed as a potential conflict of interest.

Popov and Popov. This is an open-access article distributed under the terms of

the Creative Commons Attribution License (CC BY). The use, distribution or

reproduction in other forums is permitted, provided the original author(s) and the

is cited, in accordance with accepted academic practice. No use, distribution or

reproduction is permitted which does not comply with these terms.

Frontiers in Mechanical Engineering | www.frontiersin.org 7March 2019 | Volume 5 | Article 9