L ICE N T IAT E T H E S I S
Luleå University of Technology 2016
Optical Measurements of Rolling Friction Coefficients
ISSN 1402-1757
ISBN 978-91-7583-663-8 (print)
ISBN 978-91-7583-664-5 (pdf)
Yiling Li
Department of Engineering Sciences and Mathematics
Division of Fluid and Experimental Mechanics
Op
ptical measureements oof Rolliing Fricction Cooefficien
nts
Optical Measurements of
Rolling Friction Coefficients
Yiling Li
Y
Yiling Li
L
Luleå Uniiversity of Technology
T
Experimental Mechanics
Departmeent of Enginneering Scieence and Maathematics
Divisio
on of Fluid and Experiimental Mecchanics
A
August 2016
Optical Measurements of
Rolling Friction Coefficients
Yiling Li
Luleå University of Technology
Department of Engineering Sciences and Mathematics
Division of Fluid and Experimental Mechanics
August 2016
Printed by Luleå University of Technology, Graphic Production 2016
ISSN 1402-1757
ISBN 978-91-7583-663-8 (print)
ISBN 978-91-7583-664-5 (pdf)
Luleå 2016
www.ltu.se
CONTENT
CONTENT .............................................................................................................................................. I
PREFACE ............................................................................................................................................ III
ABSTRACT ........................................................................................................................................... V
THESIS ............................................................................................................................................... VII
PART I .....................................................................................................................................................1
1.
INTRODUCTION ............................................................................................................................. 3
2.
EXPERIMENTAL SETUP AND PROCEDURE ....................................................................................... 6
3.
MODELS OF FREE-ROLLING ........................................................................................................... 9
4.
BALL RECOGNITION ALGORITHM AND DATA PROCESSING ............................................................ 12
5.
ESTIMATION OF THE ROLLING FRICTION COEFFICIENTS ............................................................... 17
5.1 Estimation of the rolling friction coefficients of the small ball with small initial angle .......... 17
5.2 Estimation of the rolling friction coefficients of the bigger balls with arbitrary initial angles . 19
6.
CONCLUSIONS AND FUTURE WORK ............................................................................................. 22
7.
REFERENCE................................................................................................................................. 24
PART II ................................................................................................................................................. 27
PAPER 1: .............................................................................................................................................. 29
PAPER 2: .............................................................................................................................................. 51
I
II
PREFACE
This work has been carried out at the Division of Fluid and Experimental Mechanics,
Department of Engineering Science and Mathematics at Luleå University of
Technology (LTU). The research was under the supervision of Prof. Mikael Sjödahl,
LTU and Dr. Erik Olsson, LTU.
Thanks to my supervisor Prof. Mikael Sjödahl and to my co-supervisor Dr. Erik
Olsson.
Thanks also to Dr. Henrik Lycksam in helping me for the preparation of the
experimental equipment.
Thanks to Dr. Yijun Shi for the suggestion of the experiment setup and supply for the
crucial material.
Thanks to my officemate and close friend Dr. Davood Khodadad for our four year’s
together and finally thanks to my great friend Yinhu Xi for the guidance and endless
help.
Yiling Li
Luleå, August 2016
III
IV
ABSTRACT
This thesis presents an optical method to measure the rolling friction coefficients
for balls rolling freely on a cylindrical surface. Two different models of a ball rolling
freely on a cylindrical surface are established, one is an analytical model and the other
is a numerical model derived from Lagrange equation. The rolling friction coefficients
are evaluated from the position data of the steel balls. The positions data are retrieved
from images recorded by a high-speed camera. The locating algorithms including
background subtraction and ball recognition are presented in detail. The rolling
friction coefficients between different diameter steel balls and a cylindrical aluminum
surface are measured. The angular positions of the balls are predicted by the solution
of the equation of motion (EOM), and good agreements are found between the
experimental and theoretical results. The values of rolling friction coefficients
between different diameter steel balls and a cylindrical aluminum surface are
evaluated.
Key words: rolling friction, friction torque, friction coefficient, ball recognition,
high-speed camera
V
VI
THESIS
This thesis consists of a background of the work and the following Papers.
Paper 1
Yiling Li, Yinhu Xi and Yijun Shi, “Estimation of rolling friction
coefficients in a tribosystem using optical measurements”, Industrial
Lubrication and Tribology. The reviewer(s) have recommended
publication.
Paper 2
Yiling Li, Yinhu Xi, “Rolling Friction Coefficient estimation in a
Tribosystem with Ball Recognition Algorithm”, Manuscript.
VII
VIII
Part I
Background of the work
1
2
1. Introduction
Rolling contacts are considered as a low energy loss to motion. The ideal rolling
contact is when two rigid bodies of revolution are pressed together and touch in a
point. The velocity of the contact point between the two bodies is equal in both bodies
[1]. However, in reality, ideal rigid bodies do not exist and the contact region between
two bodies is extended. The definition of rolling is that the relative velocity between
contact surfaces is much less than the bulk velocity [2]. The friction force is
distributed over the contact region which slows down the rolling [2, 3]. When
components in rolling contact have relative motion, rolling friction is inevitable. The
various mechanisms involved to cause the rolling to slow down are often lumped
together into a rolling friction coefficient. The pioneers started to be interested in the
phenomena of the rolling motion at the end of the 19th century and the beginning of
the 20th century [4, 5]. In the middle of the 20th century, Eldredge and Tabor
described the mechanism of rolling friction in the plastic range [6] and the elastic
range [7]. In 1970’s Kalker has presented the mechanism of Three-Dimensional
elastic bodies in rolling contact in detail [2]. With the scientific and technological
progress, measurements of rolling friction coefficients have become critical for
mechanical industry. For example it plays an important role of the development of
microelectromechanical systems (MEMS) [8].
In the recent two decades many models and experiments have been established for
measuring rolling friction coefficients. Ball race models and ball bearing models have
appeared frequently in scientific publications and optical methods are mentioned as
effective methods. A laser vibrometer was used by Fujii (2004) [9] for estimating the
friction force on linear bearings. Lin et al. (2004) developed a vision based system for
characterizing the tribological behavior of linear ball bearings. The setup includes
strain-free steel microballs and a silicon V-groove [10]. Values of dynamic rolling
friction coefficients were estimated at 0.007 in average. Tan et al. (2006) developed a
dynamic viscoelastic friction model to evaluate the rolling friction coefficient between
3
steel microballs and silicon up to 0.007 by using the same setup [11]. The rolling
friction coefficient between 0.285 mm diameter balls and microfabricated silicon was
measured to be 0.02 by Ghalichechian et al. (2008), that was the first demonstration
of a rotary micrometer [12]. An analytical model of freely rolling steel microballs on a
spherical glass surface was developed by Olaru et al. (2009) [13], and a video camera
was used to take record of the position data of the small ball. Similarly, Cross (2016)
[14] established a model based on Coulomb’s law and evaluated the rolling friction
coefficients by the setup with steel balls rolling freely on a concave lens surface. The
limitations of Olaru et al. and Cross are that they can only process the ball
reciprocated in small angular positons, and there was no appropriate algorithm used
for getting accurate center locations of the balls. In addition the sampling frequency
was low.
For increasing the sampling frequency, in a later approach of Olaru et al. (2011)
[15] a high-speed camera was used to determine the rolling friction coefficients of
thrust ball bearings. In 2014 they studied this model with lubricant viscosity
conditions (%ăODQ et al., 2014) [16]. Another usage of the high-speed camera is De
Blasio and Saeter (2009) [17], that recorded a ball rolling on a granular medium and
evaluated the friction coefficient.
The procedure of ball recognition in general contains edge detection and center
location. D’Orazio et al. (2004) developed a ball recognition algorithm based on the
Hough transform and solved the problem in different light conditions [18]. An edge
detection approach combining the Zernike moments operator with the Sobel filter was
proposed by Qu (2005) [19]. A real-time accurate circle fit algorithm based on the
maximum likelihood was presented by Frosio et al. (2008) [20].
In this thesis, an optical method for measuring rolling friction coefficients in an
efficient way is presented. The rolling friction coefficients between steel balls of
different diameters and a cylindrical aluminum surface are measured. In chapter 2,
experiments are designed for measuring the rolling friction coefficients between
4
free-rolling steel balls and a cylindrical surface. In chapter 3 two models of a ball
rolling freely on a cylindrical surface are established. In chapter 4, the positioning
algorithms to retrieve the position data recorded by a high speed camera are presented.
Chapter 5 presents how the rolling friction coefficients are evaluated from the position
data of the steel balls. Chapter 6 is the conclusion.
5
2. Experimental setup and procedure
Figure 2.1 Sketch of the experimental setup.
The ideas to design the experiments for measuring rolling friction coefficients
optically are as follows. Firstly the direction of rolling should be easy to control, and
then the space of rolling has to be limited for taking effective records. The setup of a
ball rolling on a cylindrical surface is a good choice as the ball is rolling back and
forth on the cylindrical surface within a restricted space. Based on these ideas, the
experimental setup was designed. An aluminum (6061-T6 aluminum with anodized
coating) cylindrical surface with radius of curvature 85.3 mm was mounted on an
adjustable screw set with precision 10 Ɋm. The tangent plane of the cylindrical surface
was adjusted parallel to the ground by the adjustable screw. This cylindrical surface
was imaged by a high-speed camera (Dantec Dynamics NanoSense, pixel pitch 12 Ɋm
h 12 Ɋm) at a sampling frequency of 1000 Hz. The exposure time of the camera
was 200 Ɋs and the object pixel size was 0.1343 mm/pixel. The scence was
illuminated by two lamps (COOLH dedocool) located on each side of the camera.
(Figure 2.1)
6
Three steel (52100 steel) balls were used in the free-rolling experiments. One of
diameter 1.58 mm and mass 0.01675 g, one of diameter 10.00 mm and mass 4.07612
g, and one of diameter 12.69 mm and mass 8.35894 g. The masses of the steel balls
were measured by a METTLER TOLEDO AX205 analytical balance. To adjust the
setup, the bigger steel ball was put on the cylindrical surface. The screw set was used
to adjust the position of the cylindrical surface. When the ball stood on the cylindrical
surface steadily, the tangent plane of the cylindrical surface was considered parallel to
the ground. For every measurement series, a background image without the ball was
taken as reference (Figure 2.2). The balls were manually released from a point on the
cylindrical surface with zero initial angular velocity. After release the ball rolls back
and forth on the cylindrical surface until it eventually comes to a stop at the bottom of
the cylindrical surface. During the initial few periods images were recorded by the
high-speed camera at a sampling frequency of 1000 Hz. A typical image recorded by
the camera is shown in Figure 2.3 (12.69 mm diameter ball).
Figure 2.2 Image of the background.
7
Figure 2.3 Image of the 12.69 mm diameter steel ball on the cylindrical surface captured by the
high-speed camera.
8
3. Models of free-rolling
The details of the derivations of the equation of motions are in Paper 2 and Paper
1, respectively. In this section a short summary is presented. Figure 3.1 shows a ball
rolling freely on a cylindrical surface. The plane crossing the center of the cylindrical
surface and is parallel to the ground is defined as the zero potential energy plane.
When the ball is at an angular position ߶ from the vertical direction the potential
energy V is
V=െ݉݃(ܴ െ ‫)ݎ‬cos߶,
(3.1)
Figure 3.1 A ball rolling freely on a cylindrical surface with defined zero potential energy plane.
where ܴ is the diameter of the cylindrical surface, ‫ ݎ‬is the diameter of the ball, ݉
is the mass of the ball, and g is the acceleration due to gravity. The kinetic energy is
given by
1
1
ܶ = ݉‫ ݎ‬ଶ ߮ሶ ଶ + ‫߮ܫ‬ሶ ଶ ,
2
2
(3.2)
ଶ
where ‫ = ܫ‬ହ ݉‫ ݎ‬ଶ is the moment of inertia of the ball and ߮ is the angle of the ball
rotation. Then Equation (3.2) can be rewritten as
଻
ܶ = ଵ଴ ݉‫ ݎ‬ଶ ߮ሶ ଶ .
9
(3.3)
The Lagrangian ‫ ܮ‬is the difference between the kinetic energy and the potential
energy,
଻
௥
‫ = ܮ‬ଵ଴ ݉‫ ݎ‬ଶ ߮ሶ ଶ + ݉݃(ܴ െ ‫)ݎ‬cos ቀோି௥ ߮ቁ.
(3.4)
By calculating the derivative of the corresponding components respectively, The
non-conservative Lagrange equation is given by
ୢ
ቀ
డ௅
ୢ௧ డఝሶ
డ௅
ቁ െ డఝ = ‫ܯ‬௥ ,
(3.5)
where ‫ܯ‬௥ is the rolling friction torque that slows down the rotation speed of the
ball. Under the assumption of the ball rolling without slipping, the relationship
between ߶ and ߮ is
(ܴ െ ‫߮ݎ = ߶)ݎ‬,
(3.6)
which reduces the degree of freedom to one. The final version of the equation of
motion becomes
ሶ
ହ ௚ୱ୧୬థ
ହ ఓ థ
߶ሷ = െ ଻ ோି௥ െ ଻ ோି௥ หథሶห ൫݃cos߶ + (ܴ െ ‫߶)ݎ‬ሶ ଶ ൯,
(3.7)
which can be solved numerically given the initial conditions ߶(0) and ߶ሶ(0). See
Paper 2 for details.
In a special case, when the ball is so small that the diameter is negligible and
the initial angle is small enough that the approximation sin߶ ൎ ߶ is valid,
analytical solutions of the equation of motion do exist. When the ball is rolling
down from ߶ = ߶଴ to ߶ = 0, the initial condition at ‫ = ݐ‬0 is ߶ = ߶଴ , and
݀߶Τ݀‫ = ݐ‬0. The solution is
߶(‫= )ݐ‬
ܾ
ܾ
+ ൬߶଴ െ ଶ ൰ cos(݇‫)ݐ‬,
ଶ
݇
݇
10
(3.8)
where ݇ is a gravity parameter defined by
݇=ඨ
5݃
7ܴ
and ܾ is a friction parameter defined by
ܾ=
5 ‫ܯ‬௥
.
7 ݉‫ܴݎ‬
When the ball is climbing up from ߶ = 0 to ߶ଵ , the initial condition at ‫ = ݐ‬0 is
߶ = 0, and ݀߶Τ݀‫߱ = ݐ‬଴ , where ߱଴ is the maximum angular velocity of the rolling
ball. The solution with these conditions is
߶(‫ = )ݐ‬െ
ܾ
ܾ
߱଴
+ ଶ cos(݇‫ )ݐ‬+
sin(݇‫)ݐ‬.
ଶ
݇
݇
݇
(3.9)
The details of the derivation of Equation (3.8) and Equation (3.9) are found in
Paper 1. For this special case, the rolling friction coefficient is given as [13]
ߤ=
7ܾܴ
.
(17݃ െ 10݃ܿ‫߶ݏ݋‬଴ ) െ 14ܾܴ߶଴
11
(3.10)
4. Ball recognition algorithm and data processing
In order to calculate the angular positions of the ball from the recorded image
sequence, the center of the ball in each frame as well as the center of the cylindrical
surface has to be found. In this section all coordinates are presented in pixels.
The center ൫‫ܥ‬௫ , ‫ܥ‬௬ ൯ of the cylindrical surface is determined from the recorded
images. As the contour of the cylindrical surface was visible in the recorded images,
൫‫ܥ‬௫ , ‫ܥ‬௬ ൯ is calculated as the intersection point of the horizontal line starting from the
left most point inside the contour and the vertical line starting from the lowest point of
the circle.
The conclusion of the center positions of the balls are expressed in detail in Paper
1 and Paper 2. The outline of the ball center determination is as follows.
‹ Background subtraction.
Background subtraction is applied on the recorded images. In practice the
intensity of a relative image is the difference of the background image without the ball
and a recorded image ‫ܫ‬௥௘௟ = ‫ܫ‬௕ െ ‫ܫ‬, is used.
‹ Ball recognition algorithms.
z For the small ball when the diameter is negligible.
The 1 mm diameter steel ball can be considered as a point, a bright spot on the
circle indicated the position of the smallest ball. (Figure 4.1)
12
Figure 4.1 Image of the steel ball on the cylindrical surface captured by the high-speed camera.
Based on the construction of the setup, a sub-image of size 561 pixels h 130 pixels
that included the position of the ball was cut from the recorded image. This sub-image
was traversed by a 15 pixel h 15 pixel window, in which the total intensity is
summed up. When the maximum total intensity was reached, this was taken to
indicate that the bright spot was included in the 15 pixel h 15 pixel window. The
center of the ball (‫ݔ‬௖௖ , ‫ݕ‬௖௖ ) was taken to be the brightest point in this 15 pixel h 15
pixel neighborhood. Figure 4.2 shows the positions of the ball as obtained from the
background subtraction in different time frames.
Figure 4.2 Positions of the ball in different time frames.
13
Figure 4.3 Image of the 10.00 mm diameter steel ball on the cylindrical surface captured by the
high-speed camera.
z For the big ball that the diameter is not negligible.
The processing of a recorded frame of the 10.00 mm diameter ball is taken as an
example.
A sub-image around the two shining spots of size 101 pixels × 101 pixels is cut
from the image. By considering the shadow information a part of the ball including a
fragment of a relatively smooth edge is selected. Then a cursory edge is detected by a
canny operator and a rough center of the ball (‫ݔ‬௖ , ‫ݕ‬௖ ) is determined. By calculating
the intensity gradient in the radial direction, the center of the ball (‫ݔ‬௖௖ , ‫ݕ‬௖௖ ) is
determined and the ball recognition is complete (Figure 4.4).
14
Figure 4.4 The result of ball recognition.
The position data of the balls are validated by substituting the retrieved positions
with randomly selected time ‫( ݐ‬in different period) into corresponding frames
(Figure 4.5).
‹ Calculate angular positions of the ball.
The angular position of the ball ߶(‫ )ݐ‬in each frame was finally evaluated as
sin߶(‫= )ݐ‬
(‫ݔ‬௖௖ െ ‫ܥ‬௫ ) × object pixel size
,
ܴെ‫ݎ‬
(5.6)
which gives ߶(‫)ݐ‬.
‹ Noise elimination.
The high frequency noise has to be reduced. The subsequent processing includes
time derivatives of the angular position information, therefore a low pass filter is
applied on the raw angular position data (Guddei et al. 2013) [21]. A Chebyshev low
pass filter is used in this work. With the filter the data processing method becomes
more robust.
15
Figure 4.5 Validation of the retrieved ball locations. A.10 mm diameter ball acquired from different
time frames. B.12.69 mm diameter ball from different time frames
16
5. Estimation of the rolling friction coefficients
In chapter 3, two versions of the equation of motion for the ball were derived. One
is approximate but gives an analytical solution. The second version is more general
but requires a numerical solution. In this section, the use of both models to estimate
the rolling friction coefficients from the recorded movement of the ball is described.
In both cases, there are three unknowns ߶(0), ߶ሶ(0) and ߤ that are determined
from the experiments.
5.1 Estimation of the rolling friction coefficients of the small ball with small initial
angle
Two initial values for ܾ are evaluated by substituting an experimental value into
the Equation (3.8) and Equation (3.9). Then ܾ is iterated between the two initial
guesses until the relative error between the experimental angles and model angles is
minimized. As shown in Figure 5.1.1, the optimum ܾ was determined to be 1.9 s ିଶ .
Figure 5.1.1 Optimization of the rolling friction parameter ܾ.
With this optimized value, the expressions for ߶(‫ )ݐ‬in the two stages both
converged to ߶(‫ = )ݐ‬0, and the total relative error of these two stages was 1.29%.
The relative error is calculated as
17
Relative error =
σห߶୰ୣୡ୭୰ୢ െ ߶ୟ୬ୟ୪୷୲୧ୡୟ୪ ห
.
σ|߶୰ୣୡ୭୰ୢ |
(6.1)
Figure 5.1.2 shows the match between the angular position ߶(‫ )ݐ‬obtained from
the recorded images and that evaluated using the optimum rolling friction parameter
ܾ = 1.9sିଶ. The negative and positive signs of the angle indicate the relative angular
position of the ball with respect to the zero angle. The analytical method presented
here is valid as long as the initial angle is sufficiently small.
Figure 5.1.2 Comparison of the recorded angular positions and those calculated using the optimized
rolling friction parameter ܾ = 1.9 for ߶଴ = െ12.07°, ߶ଵ = 8.49°. Error bars (each 10 samples)
indicate errors in the evaluation of the center of the ball.
Using the evaluated rolling friction parameter ܾ, the value of the rolling friction
coefficient between the 1.58 mm diameter steel ball and the cylindrical aluminum
surface was estimated using Equation (3.10), with the result ߤ = 0.0164. This lies in
the same range as obtained by Ghalichechian et al. (2008) and Olaru et al. (2009).
The differences between the values of the rolling friction coefficients evaluated in
those papers and in the present study are associated with the different materials that
18
were used.
5.2 Estimation of the rolling friction coefficients of the bigger balls with arbitrary
initial angles
Either when the diameter of the ball cannot be neglected or the initial angle of the
ball rolling is not within the valid scope of sin߶ ൎ ߶, the analytical model does not
work. The 10 mm diameter ball and the 12.69 mm diameter ball rolling on the
cylindrical surface are processed by the method presented in chapter 4. For both the
10 mm diameter ball and the 12.69 mm diameter ball, the angular positions ߶(‫ )ݐ‬and
angular velocities ߶ሶ(‫ )ݐ‬are experimentally measured. The estimation of ߤ is a
differential equation parameter estimation problem of Equation (3.7). In order to
estimate the rolling friction coefficients, the following optimization problem is stated.
The objective of the optimization is:
minimize: σ(߶୰ୣୡ୭୰ୢ െ ߶୬୳୫ୣ୰୧ୡୟ୪ )ଶ
Optimization (1)
subject to: ߶(0), ߶ሶ(0), ߤ, ܴ, ‫ݎ‬,
in which ܴ and ‫ ݎ‬are measured before the experiment. ߶(0) and ߶ሶ(0) are the
starting angle and starting angular velocity of the rolling, and they are initially
measured from the experiment.
The rolling friction coefficient between the smaller ball and the cylindrical
surface ߤ௦ is estimated to be 0.0079 and the rolling friction coefficient between the
bigger ball and the cylindrical surface ߤ௕ is estimated to be 0.0065. The comparison
between the numerical results and the experimental results are shown in Figure 5.2.1.
The red dot lines are the experimental results and the blue lines are the numerical
results. The relative error for the 10 mm diameter ball is 5.8% and for the 12.69
diameter ball is 4.6%.
19
Figure 5.2.1 Comparison of the angular positions evaluated from the EOM and from the experiments.
A.The 10 mm diameter ball. B. The 12.69 mm diameter ball.
The rolling friction coefficient of the bigger diameter ball and the cylindrical
surface is smaller than the rolling friction coefficient of the smaller ball as it should be.
The rolling friction coefficient between a 9.525 mm diameter steel ball and a glass
spherical surface was measured by Olaru [12] to be 0.004 on average. The rolling
20
friction coefficient between a 10 diameter ball and a glass concave lens was reported
around 0.002 by Cross [14]. The rolling friction coefficients between 0.285 mm
diameter balls and a silicon groove were measured by Lin et al. (2004) [10] and Tan et
al. (2006) [11] to be around 0.007. Our results are in the same order of magnitude
compared to theirs results. The surface roughness of the aluminum surface is bigger
than the surface roughness of the glass surface. It is therefore expected that the rolling
coefficient measured between the steel balls and the aluminum surface is bigger than
that was measured between a similar diameter steel ball and a glass surface.
21
6. Conclusions and future work
The purpose of this thesis is to present an optical method to measure the rolling
friction coefficient for a ball rolling freely on a cylindrical surface. As proof of
principle the rolling friction of three steel balls of different diameters rolling freely on
an ionized aluminium surface are evaluated. Two models for the equation of motion
of a spherical ball rolling freely on a cylindrical surface are developed. One is an
analytical model, one is a numerical model derived from the Lagrange Equation. The
analytical model accepts a ball rolling freely on the cylindrical surface with small
initial angle, the approximation sin߶ ൎ ߶ was employed, and so a small initial angle
was necessary. In case the ball is small enough to be considered as a point, it was
assumed that the rolling friction torque is independent of angular velocity and angular
position. The model was validated by comparing ߶(‫ )ݐ‬calculated from the evaluated
rolling friction parameter ܾ with the angular positions obtained from experimental
data. The experimental data agreed well both with the analytical solution for a small
starting angle and with the numerical solution for a larger starting angle. The relative
error was less than 6%. The rolling friction coefficient ߤ between a 1.58 mm
diameter steel ball and a cylindrical aluminum surface was estimated to be 0.016. For
bigger balls that could not be considered as a point, a ball recognition algorithm and a
data processing method were presented in detail. The position data retrieved from the
recorded images is validated by the match of the reconstructed edges of the ball to the
original recorded images. The rolling friction coefficient between an aluminum
cylindrical surface and a 10 mm diameter steel ball is evaluated to be 0.0079 and the
rolling friction coefficient between the bigger ball and the cylindrical surface is
evaluated to be 0.0065.
Profit from the high sampling frequency of the camera, the accuracy of the
starting angles is high and synchronization of the release of the ball and the sampling
is not necessary. At the sampling frequency of 1000 Hz, the initial angle ߶଴ and final
angle ߶ଵ can be identified with adequate precision. ߶ሶ(‫ )ݐ‬was calculated by
22
numerical differentiation of the angular positions ߶(‫ )ݐ‬with respect to time ‫ݐ‬. The
high sampling frequency was necessary to obtain precise position and velocity signals.
In order to deal with high-frequency noise when obtaining the velocity signals from
the position signals, a low-pass filter is necessarily used to restrain the high-frequency
noise.
In the future, the rolling friction coefficients of the balls and the cylindrical
surface in different lubrication conditions can be measured.
23
7. Reference
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[2] Kalker J J. Three-dimensional elastic bodies in rolling contact[M]. Springer Science & Business
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[3] Kalker J J. T Simplified theory of rolling contact[J]. Mechanical and Aeronautical Engineering and
Shipbuilding, 1973,1:1-10.
[4] Reynolds O. On rolling-friction[J]. Philosophical Transactions of the Royal Society of London,
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[5] Stribeck R. Article on the evaluation of ball-bearings[J]. Zeitschrift Des Vereines Deutscher
Ingenieure, 1901, 45: 1421-1422.
[6] Eldredge K R, Tabor D. The mechanism of rolling friction. I. The plastic range[C] Proceedings of
the Royal Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society,
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[7] Tabor D. The mechanism of rolling friction. II. The elastic range[C] Proceedings of the Royal
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229(1177): 198-220.
[8] %ăODQ05'6WDPDWH9&+RXSHUW/HWDO7KHLQIOXHQFHRIWKHOXEULFDQWYLVFRVLW\RQWKHUROOLQJ
friction torque[J]. Tribology International, 2014, 72: 1-12.
[9] Fujii Y. An optical method for evaluating frictional characteristics of linear bearings[J]. Optics and
lasers in engineering, 2004, 42(5): 493-501.
[10] Lin T W, Modafe A, Shapiro B, et al. Characterization of dynamic friction in MEMS-based
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[11] Tan X, Modafe A, Ghodssi R. Measurement and modeling of dynamic rolling friction in linear
24
microball bearings[J]. Journal of Dynamic Systems, Measurement, and Control, 2006, 128(4): 891-898.
[12] Ghalichechian N, Modafe A, Beyaz M I, et al. Design, fabrication, and characterization of a rotary
micromotor supported on microball bearings[J]. Journal of Microelectromechanical Systems, 2008,
17(3): 632-642.
[13] Olaru D N, Stamate C, Prisacaru G. Rolling friction in a micro tribosystem[J]. Tribology letters,
2009, 35(3): 205-210.
[14] Cross R. Coulomb's law for rolling friction[J]. American Journal of Physics, 2016, 84(3): 221-230.
[15]Olaru D N, Stamate C, Dumitrascu A, et al. New micro tribometer for rolling friction[J]. Wear,
2011, 271(5): 842-852.
[16] %ăODQ05'6WDPDWH9&+RXSHUW/HWDO7KHLQIOXHQFHRIWKHOXEULFDQWYLVFRVLW\RQWKH
rolling friction torque[J]. Tribology International, 2014, 72: 1-12.
[17] De Blasio F V, Saeter M B. Rolling friction on a granular medium[J]. Physical Review E, 2009,
79(2): 022301.
[18] D'Orazio T, Guaragnella C, Leo M, et al. A new algorithm for ball recognition using circle Hough
transform and neural classifier[J]. Pattern Recognition, 2004, 37(3): 393-408.
[19] Ying-Dong Q, Cheng-Song C, San-Ben C, et al. A fast subpixel edge detection method using
Sobel–Zernike moments operator[J]. Image and Vision Computing, 2005, 23(1): 11-17
[20] Frosio I, Borghese N A. Real-time accurate circle fitting with occlusions[J]. Pattern Recognition,
2008, 41(3): 1041-1055.
[21] Guddei B, Ahmed S I U. Rolling friction of single balls in a flat-ball-flat-contact as a function of
surface roughness[J]. Tribology Letters, 2013, 51(2): 219-226.
25
26
Part II
Papers
27
28
Paper 1:
Estimation of rolling friction coefficients in a tribosystem
using optical measurements
29
30
Estimation of rolling friction coefficients in a tribosystem
using optical measurements
Yiling Li*1, Yinhu Xi2, Yijun Shi2
* Corresponding author. Tel +46 725625615, E-mail: [email protected], [email protected]
1. Department of Engineering Sciences and Mathematics Division of Fluid and Experimental Mechanics, Luleå University
of Technology.
2. Department of Engineering Sciences and Mathematics Division of Machine Elements, Luleå University of Technology.
Abstract
Purpose
This paper presents a method to measure the rolling friction coefficient in an easy and fast way. The
aim is to measure the rolling friction coefficient between a small steel ball and a cylindrical aluminum
surface.
Design/methodology/approach
An analytical model of the tribosystem of a freely rolling ball and a cylindrical surface is established.
The rolling friction coefficient is evaluated from images recorded by a high-speed camera. The
coefficient between a 1.58 mm diameter steel ball and a cylindrical aluminum surface is measured. A
background subtraction algorithm is used to determine the position of the small steel ball.
Findings
The angular positions of the ball are predicted using the analytical model, and good agreement is found
between the experimental and theoretical results.
31
Originality/value
An optical method for evaluating the rolling friction coefficient is presented, and the value of this
coefficient between a small steel ball and a cylindrical aluminum surface is evaluated.
Key words: rolling friction coefficient, rolling friction torque, optical measurement, high-speed camera
1. Introduction
Rolling motions are widespread in industrial production and indeed in daily life.
In the last two decades, micro ball bearings have been the subject of much attention
from researchers because of their role in the development of microelectromechanical
systems (MEMS) (Li et al., 2015). There have been many studies of rolling friction
coefficients (see, e.g., Padgurskas et al., 2008; Liu et al., 2015), and in particular with
regard to microball rolling friction, a number of models have been developed and
experimental investigations performed.
In practice, for measuring friction characteristics of MEMS tribosystems, optical
measurement is a suitable method. Tan et al. (2006) developed a dynamic viscoelastic
friction model for the contact between strain-free steel microballs and a silicon
V-groove. Values of the rolling friction coefficient up to 0.007 were obtained. A rotary
micrometer was first demonstrated by Ghalichechian et al. (2008) using microball
bearing technology, and the rolling friction coefficient between 0.285 mm diameter
balls and microfabricated silicon was measured to be 0.02. Olaru et al. (2009)
developed an analytical model of freely rolling steel microballs on a spherical glass
surface. Their experimental procedure was recorded by a video camera and the rolling
32
friction coefficient between a 1 mm diameter ball and the glass surface was measured,
with an average value of 0.025 being obtained. Later, they developed another
analytical approach, a spin-down method, to measure the rolling friction coefficient of
thrust ball bearings (Olaru et al., 2011). Two models were considered in that paper: in
one, the rolling friction coefficient is independent of rotational velocity, while in the
other, it is linearly dependent on rotational velocity. The experiment was recorded by
a high-speed camera, and rolling friction coefficients between 0.0002 and 0.0004
were obtained. More recently, they have also studied a thrust ball bearing model with
lubricant viscosity (%ăODQ et al., 2014). Fujii (2004) estimated the friction force on
linear bearings by measuring velocity using a laser vibrometer and described in detail
the data processing method used. De Blasio and Saeter (2009) recorded a sphere
rolling on a granular medium using a high-speed camera and evaluated the friction
coefficient.
The advantages of optical measurement are as follows. First, it is a noncontact
measurement. Second, it has zero response time, it does not need to synchronize the
measurements, and the recorded images can be processed by image processing
algorithms. Third, it does not require specialized apparatus, but only readily available
equipment such as cameras.
This paper describes a method to evaluate the rolling friction coefficient of a
freely rolling microball on a cylindrical aluminum surface using optical measurements.
An analytical model based on that of Olaru et al. (2009) is developed, with a
33
corrected solution of the differential equation in the original model. The model is
validated by assuming the rolling friction torque to be independent of angular velocity
and of angular position within a small range of the latter. The angular positions of the
microball are acquired from images recorded by a high-speed camera with a sampling
frequency of 1000 Hz. A background subtraction algorithm is used to retrieve the
position information of the microball. Compare to the previous study of Olaru et al., a
more precisely analytical model is introduced. Base on the improved sampling
frequency, the angular position can be measured more accurately.
2. An analytical model for a ball rolling on a cylindrical surface
Figure 1 Freely rolling ball on a cylindrical surface.
Based on energy conservation law and the analysis by Olaru et al. (2009), when a
ball is freely rolling on a cylindrical surface from an angular position ߶଴ with zero
initial angular velocity to an angular position ߶, as shown in Figure 1, the following
energy equation is satisfied:
34
݉݃ȟ݄ = ȟ‫ ܧ‬+ ‫ܯ‬௥ ȟ߮,
(1)
where ȟ݄ is the change in height of the ball relative to that at its original position, ݉
is the mass of the ball, g is the acceleration due to gravity, ȟ‫ ܧ‬is the variation in
kinetic energy of the ball, ‫ܯ‬௥ is the friction torque, and ȟ߮ is the variation in the
ball’s rotation; thus, ‫ܯ‬௥ ȟ߮ is the energy lost due to friction torque. ȟ߮ is given by
ο߮ =
ܴ
(߶ െ ߶),
‫ ݎ‬଴
(2)
where ܴ is the radius of curvature of the cylindrical surface and ‫ ݎ‬is the radius of
the ball.
Taking as t = 0 the time at which the ball starts rolling at angular position ߶଴
with initial angular and linear velocities both zero, the kinetic energy variation of the
ball from ߶଴ to ߶(‫ )ݐ‬is
ȟ‫= ܧ‬
݉‫ ݒ‬ଶ ‫߱ܫ‬௦ଶ
+
,
2
2
(3)
where ‫ ݒ‬is the linear velocity of the ball, ‫ ܫ‬is its moment of inertia (for a solid ball,
ଶ
‫ = ܫ‬ହ ݉‫ ݎ‬ଶ), and ߱௦ is its angular velocity.
Equation (1) can be rewritten as
7
ܴ
݉‫ ݒ‬ଶ + ‫ܯ‬௥ (߶଴ െ ߶) + ܴ݉݃(cos߶଴ െ cos߶) = 0.
10
‫ݎ‬
(4)
The linear velocity ‫ ݒ‬and angular velocity ߱ satisfy the relations ‫ ܴ߱ = ݒ‬and
߱ = ݀߶Τ݀‫ ݐ‬, and ‫ܯ‬௥ is a function of the normal force and the rolling friction
coefficient. When the diameter of the ball is small, the mass of the ball is small and
the initial angle ߶଴ is small, the approximation proposed by Olaru et al. (2009) is
35
valid.
‫ܯ‬௥ can then be considered independent of angular velocity and angular
position (for the experimental conditions in this paper, the variation of ‫ܯ‬௥ is within
5%). On differentiating Equation (4) with respect to time ‫ݐ‬, the equation of motion of
the ball becomes
݀ଶ߶
+ ݇ ଶ sin߶ െ ܾ = 0,
݀‫ ݐ‬ଶ
(5)
where ݇ is a gravity parameter defined by
݇=ඨ
5݃
7ܴ
and ܾ is a friction parameter defined by
ܾ=
5 ‫ܯ‬௥
.
7 ݉‫ܴݎ‬
If the variation of the angle is small, the approximation sin߶ ൎ ߶ can be applied.
Then Equation (5) is a nonhomogeneous differential equation with the general
analytic solution
߶(‫= )ݐ‬
ܾ
+ ‫ܥ‬ଵ cos(݇‫ )ݐ‬+ ‫ܥ‬ଶ sin(݇‫)ݐ‬,
݇ଶ
(6)
where ‫ܥ‬ଵ and ‫ܥ‬ଶ are constant determined by the initial conditions ߶ = ߶଴ , and
݀߶Τ݀‫ = ݐ‬0, the time ‫ ݐ‬at ߶ = ߶଴ is set to be 0. The solution is
߶(‫= )ݐ‬
ܾ
ܾ
+ ൬߶଴ െ ଶ ൰ cos(݇‫)ݐ‬.
ଶ
݇
݇
(7)
When the ball reaches the bottom of the cylindrical surface, it will climb up the other
36
side of the surface.
Taking as t =0 the time at which the ball is at angular position ߶ = 0 with linear
velocity ‫ݒ = ݒ‬଴ , the following energy equation can be established:
7
ܴ
݉(‫ ݒ‬ଶ െ ‫ݒ‬଴ଶ ) + ܴ݉݃(1 െ cos߶) + ‫ܯ‬௥ ߶ = 0.
10
‫ݎ‬
(8)
On differentiating Equation (8) with respect to time ‫ݐ‬, the following differential
equation is obtained.
݀ଶ߶
+ ݇ ଶ sin߶ + ܾ = 0.
݀‫ ݐ‬ଶ
(9)
With the approximation sin߶ ൎ ߶, the general solution of Equation (9) is given by
߶(‫ = )ݐ‬െ
ܾ
+ ‫ܥ‬ଷ cos(݇‫ )ݐ‬+ ‫ܥ‬ସ sin(݇‫)ݐ‬.
݇ଶ
(10)
The constants ‫ܥ‬ଷ and ‫ܥ‬ସ are determined by the initial conditions ߶(‫ = )ݐ‬0, and
݀߶Τ݀‫߱ = ݐ‬଴ , where ߱଴ is the maximum angular velocity of the rolling ball, the time
‫ ݐ‬at ߶ = 0 is set to be 0. The solution with these conditions is
37
߶(‫ = )ݐ‬െ
ܾ
ܾ
߱଴
+ cos(݇‫ )ݐ‬+
sin(݇‫)ݐ‬.
݇ଶ ݇ଶ
݇
(11)
When the ball passes through the lowest point of the cylindrical surface,
߶(‫ = )ݐ‬0, the only tangential force acting on it is friction, and therefore the friction
coefficient can be taken to be
ߤ=
‫)ݐ(ܨ‬
ฬ
,
ܰ(‫ )ݐ‬థ(௧)ୀ଴
(12)
where ܰ(‫ )ݐ‬and ‫ )ݐ(ܨ‬are respectively the normal and tangential forces acting on
the ball. When ߶(‫ = )ݐ‬0, according to the analysis by Olaru et al. (2009),
‫|)ݐ(ܨ‬థ(௧)ୀ଴ =
5 ‫ܯ‬௥
7 ‫ݎ‬
and
ܰ(‫|)ݐ‬థ(௧)ୀ଴ =
1
10 ‫ܯ‬௥
݉݃(17 െ 10cos߶଴ ) െ
߶ .
7
7 ‫ ݎ‬଴
Equation (12) can then be rewritten as
ߤ=
7ܾܴ
.
(17݃ െ 10݃cos߶଴ ) െ 14ܾܴ߶଴
(13)
3. Experimental setup and procedure
An aluminum (6061-T6 aluminum with anodized coating) cylindrical surface with
38
radius of curvature 85.3 mm (Figure 2) was mounted on an adjustable screw set with
precision 10 Ɋm. The tangent plane of the cylindrical surface was adjusted parallel to
the ground by the adjustable screw. This cylindrical surface was imaged by a
high-speed camera (Dantec Dynamics NanoSense, pixel pitch 12 Ɋm h 12 Ɋm) at a
sampling frequency of 1000 Hz. The normal of the lowest point of the cylindrical
surface was taken as angular position zero (߶ = 0). The free-rolling procedure for the
ball was as follows. A small steel (52100 steel) ball of diameter 1.58 mm and mass
0.01675 g (as determined by a METTLER TOLEDO AX205 analytical balance) was
manually released from a point on the cylindrical surface at angle ߶଴ with zero
initial angular velocity. After release, the ball started freely rolling on the cylindrical
surface. At its first passage through the lowest point of the cylindrical surface (zero
angle), the angular velocity was ߱଴ . It then climbed to the other side of the
cylindrical surface, reaching another angle ߶ଵ at which its angular velocity was zero.
This roll-down procedure was then repeated from an initial position ߶ = ߶ଵ (|߶ଵ | <
|߶଴ |). The whole of this process was recorded by the high-speed camera at a sampling
frequency of 1000 Hz. The exposure time of the camera was 200 Ɋs and the image
pixel size was 0.1343 mm/pixel. A typical image recorded by the camera is shown in
Figure 2. The center ൫‫ܥ‬௫ , ‫ܥ‬௬ ൯ (in pixels) of the cylindrical surface was determined
from the recorded images. The contour of the cylindrical surface was visible in the
recorded images, so ൫‫ܥ‬௫ , ‫ܥ‬௬ ൯ can be considered as the intersection point of the
horizontal line in the circle of maximum width with the vertical line starting from the
lowest point of the circle. A bright spot on the circle indicated the position of the ball.
39
The position of the bright spot in each frame was determined by a background
subtraction algorithm. From the known way in which the setup was constructed, the
rough location of the ball was known. A sub-image of size 561 pixels h 130 pixels
that included the position of the ball was cut from the recorded image. A further
sub-image was then obtained by subtracting the corresponding background from this
sub-image. This subtracted sub-image was traversed by a 15 pixel h 15 pixel
window. When the maximum total intensity was reached, this was taken to indicate
that the bright spot was included in the 15 pixel h 15 pixel window. The center of
the ball (‫ݔ‬, ‫( )ݕ‬in pixels) was taken to be the brightest point in this 15 pixel h 15
pixel neighborhood. Figure 3 shows the positions of the ball as obtained from the
background subtraction in different time frames. The angular position of the ball,
߶(‫)ݐ‬, in each frame was evaluated from
sin߶(‫= )ݐ‬
(‫ ݔ‬െ ‫ܥ‬௫ ) × image pixel size
.
ܴ
Figure 2 Image of the steel ball on the cylindrical surface captured by the high-speed camera.
40
Figure 3 Positions of the ball in different time frames.
In the experiment described here, the diameter of the ball is small, and the errors
arising from taking the brightest point in the 15 pixel h 15 pixel neighborhood of
the shining spot as the geometric center of the small sphere (maximum 1 pixel
difference) are indicated by the error bars in Figures 4, 5, 7, and 8. The error bars are
calculated from the absolute difference between the angle ߶(‫ )ݐ‬reached by the
brightest point and that reached by its 1 pixel neighborhood. Profit from the high
sampling frequency, the neighboring records can be considered as the results of
different experiments. The difference between two neighboring records is within 1
pixel. The maximum errors for the angles evaluation are induced by maximum 1
pixel’s differences, within the range of the error bars.
4. Results and discussion
According to the analysis above, the behavior of the angular position of the ball
߶(‫ )ݐ‬can be established. This is given by Equation (7) for the roll-down stage and by
Equation (11) for the climbing stage. At time ‫ = ݐ‬0, the initial angle ߶଴ at which the
angular velocity was zero was 12.07°. With the approximation of sin ߶ (‫ )ݐ‬by ߶(‫)ݐ‬,
41
the maximum error induced by the approximation was |sin(߶଴ ) െ ߶଴ |Τ|sin(߶଴ )| =
0.74%. In the experiment described here, at the sampling frequency of 1000 Hz, the
maximum relative error in the starting angle ߶଴ arising from the lack of
synchronization between the release of the ball and the recording was less than
1.55×10í6. The initial value of the rolling friction parameter ܾ can be evaluated by
substituting an angular position measured from one of the recorded images. Two
initial values of ܾ were obtained from Equations (7) and (11). The optimum value of
ܾ was then determined from iteration in the neighborhood of the initial values.
For the ball’s roll-down stage, the initial value of the rolling friction parameter ܾ
was evaluated from Equation (7). A value of ܾ can be calculated from an arbitrary
recording in the roll-down stage. The recording at ߶(‫ = )ݐ‬7.48° was randomly
selected, ߶(‫ )ݐ‬and ‫ ݐ‬were substituted into Equation (7), from which a value of ܾ
was calculated as 1.321 sିଶ . Using this value of ܾ in Equation (7) gave the behavior
of the angular position as a function of time, ߶(‫)ݐ‬. This is compared in Figure 4 with
the experimental values of ߶(‫ )ݐ‬obtained from the recorded images. For ܾ =
1.321 sିଶ , the relative error was 1.56%, so this value of ܾ can be used as an initial
value. The relative error is defined as
Relative error =
σห߶୰ୣୡ୭୰ୢ െ ߶ୟ୬ୟ୪୷୲୧ୡୟ୪ ห
.
σ|߶୰ୣୡ୭୰ୢ |
(14)
The ball’s climbing stage started when the ball passed through ߶(‫ = )ݐ‬0 with
42
angular velocity ߱ = ߱଴ . It then climbed up to the other side of the cylindrical
surface to an angle ߶ଵ at which the angular velocity was zero. In the experiment
described here, ߶ଵ = 8.49°. The initial condition ߱଴ = ݀߶Τ݀‫|ݐ‬థୀ଴ was evaluated
from the angular positions obtained from the recorded images by numerical
differentiation with respect to time ‫ݐ‬. Numerical differentiation of position signals is
sensitive to high-frequency noise, and in order to obtain an accurate value of ߱଴ , a
low-pass filter was applied to the angular position data evaluated from the recorded
images. Guddei et al. applied the same treatment to the position data in order to obtain
velocity signals. ߱଴ was measured as 1.58 rad/s. Substitution of this ߱଴ and the
recorded ߶(‫߶ = )ݐ‬ଵ and time ‫ ݐ‬into Equation (11) gave a value of the rolling friction
parameter ܾ = 2.355 s ିଶ . Using this ܾ in Equation (11) gave the behavior of the
angular position as a function of time ߶(‫)ݐ‬. Figure 5 compares this with the behavior
determined from the recorded images. The relative error was 0.68%, so this value of
ܾ = 2.355 sିଶ can be used as another initial value.
43
Figure 4 Angular positions of the freely rolling ball with initial conditions ߶(0)
= ߶଴ , ‫ = ݒ‬0. Error
bars (each 10 samples) indicate errors in the evaluation of the center of the ball.
Figure 5 Angular positions of the freely rolling ball with initial conditions ߶(0) = 0, ߱ = ߱଴ . Error
bars (each 10 samples) indicate errors in the evaluation of the center of the ball.
As ܾ was assumed to be constant for both stages of the rolling, it has to satisfy
the following conditions. First, the expressions for ߶(‫ )ݐ‬in the two stages must both
converge to ߶(‫ = )ݐ‬0. Moreover, the total relative error of both stages should be
within an acceptable range. The optimum ܾ was assumed to lie between the initial
value evaluated from Equation (7) and that evaluated from Equation (11). ܾ was
iterated from 1.321 to 2.355 s ିଶ in steps of 0.0001 s ିଶ . As shown in Figure 6, the
optimum ܾ was determined to be 1.9012 s ିଶ .
44
Figure 6 Optimization of the rolling friction parameter ܾ.
With this optimized value, the expressions for ߶(‫ )ݐ‬in the two stages both converged
to ߶(‫ = )ݐ‬0, and the total relative error of these two stages was 1.29%. Figure 7
shows the match between the angular position ߶(‫ )ݐ‬obtained from the recorded
images and that evaluated using the optimum rolling friction parameter ܾ =
1.9012 s ିଶ. The negative and positive signs of the angle indicate the relative angular
position of the ball with respect to zero angle.
45
Figure 7 Comparison of the recorded angular positions and those calculated using the optimized rolling
friction parameter ܾ = 1.9012 for ߶଴ = െ12.07°, ߶ଵ = 8.49°. Error bars (each 10 samples) indicate
errors in the evaluation of the center of the ball.
Figure 8 Comparison of the recorded angular position and that calculated using the optimized rolling
friction parameter ܾ = 1.9012 for ߶଴ = െ34.14°, ߶ଵ = 31.56°. Error bars (each 10 samples) indicate
errors in the evaluation of the center of the ball.
According to the assumption made in this paper, ܾ is independent of the initial
angle. If ܾ is correctly evaluated, it should be appropriate for cases with different
initial angles. In order to validate the evaluated rolling friction parameter ܾ, it should
be tested with a rolling process started with a different initial angle. In the validation
process, ܾ = 1.9012 sିଶ was substituted into Equations (7) and (11) with
߶଴ = െ34.14° and ߶ଵ = 31.56°. Because the angle ߶଴ is larger, the approximation
of sin ߶଴ as ߶଴ when solving Equation (5) would result in a larger error. In this
case, the relative error of the analytical solution was 6.02% and that of the numerical
solution (Runge–Kutta) was 2.59%. Instead of using the analytical solution, the
Equation (5) and Equation (9) were solved by the Runge–Kutta method. Figure 8
46
compares the ߶(‫ )ݐ‬obtained by numerical solution of Equations (5) and (9) with the
recorded angular positions. The relative error was 2.59% and the numerical solution
and the experimental values both converged to ߶(‫ = )ݐ‬0. The evaluated rolling
friction parameter ܾ was therefore considered reliable. To evaluate ܾ, if the initial
angle is small, both the analytical and numerical solutions are available. However, if
the initial angle is large, the analytical solution introduces a larger error resulting from
the approximation sin߶ ൎ ߶ and the numerical solution is more accurate. Therefore,
if the analytical method presented here is to be used to evaluate the rolling friction
parameter ܾ, it is necessary that the initial angle be sufficiently small. The ball used
in the experiment is a solid steel ball, the density of the ball is big, and the speed of
the ball is low. Compared to the gravity of the ball, the air resistance is quite small, so
the air resistance is negligible.
Using the evaluated rolling friction parameter ܾ, the value of the rolling friction
coefficient between the 1.58 mm diameter steel ball and the cylindrical aluminum
surface was estimated using Equation (13), with the result ߤ = 0.0164. This lies in
the same range as obtained by Ghalichechian et al. (2008) and Olaru et al. (2009).
The differences between the values of the rolling friction coefficients evaluated in
those papers and in the present study may be due to the different materials that were
used.
5. Conclusions
This paper introduced an easy and fast optical method for evaluating the rolling
47
friction parameter ܾ and rolling friction coefficient ߤ by monitoring the angular
positions of a steel ball having 1.58 mm diameter, which is freely rolling on a
cylindrical aluminum surface. An analytical model for a ball rolling freely on a
cylindrical surface with a small starting angle was developed in which it was assumed
that the rolling friction torque is independent of angular velocity and angular position.
When solving the equation of motion analytically, the approximation sin߶ ൎ ߶ was
employed, and so a small initial angle was necessary. The model was validated by
comparing ߶(‫ )ݐ‬calculated from the evaluated rolling friction parameter ܾ with the
angular positions obtained from experimental data. The experimental data agreed well
both with the analytical solution for a small starting angle and with the numerical
solution for a larger starting angle. The relative error was less than 3%. The rolling
friction coefficient ߤ between a 1.58 mm diameter steel ball and a cylindrical
aluminum surface was estimated to be 0.0164.
The accuracy of the evaluation depends on the accuracy of the evaluated ߶(‫)ݐ‬
and ߱଴ . Because of the high sampling frequency of the camera, synchronization of
the release of the ball and the sampling is not necessary. At the sampling frequency of
1000 Hz, the initial angle ߶଴ and final angle ߶ଵ can be identified with adequate
precision. ߱଴ was calculated by numerical differentiation of the angular positions
߶(‫ )ݐ‬with respect to time ‫ݐ‬. The high sampling frequency was necessary to obtain
precise position and velocity signals. In order to deal with high-frequency noise when
obtaining the velocity signals from the position signals, a low-pass filter was needed.
48
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49
Padgurskas, J., Rukuiza, R., Amulevicius, A. et al. (2008), “Influence of fluor-oligomers on the
structural and tribological properties of steel surface at the rolling friction”, Ind. Lubr. Tribol., Vol. 60
No. 5, pp. 222-227.
Tan, X., Modafe, A. and Ghodssi, R. (2006), “Measurement and modeling of dynamic rolling friction
in linear microball bearings”, J. Dyn. Syst. Meas. Contr, Vol. 128 No. 4, pp. 891-898.
Guddei, B., Ahmed, S. I. U. (2013) “Rolling friction of single balls in a flat-ball-flat-contact as a
function of surface roughness”, Tribology Letters, Vol. 51 No.2, pp. 219-226.
50
Paper 2:
Rolling Friction Coefficients Estimation in A Tribosystem
with Ball Recognition Algorithm
51
52
Rolling Friction Coefficient estimation in a Tribosystem
with Ball Recognition Algorithm
Yiling Li1, Yinhu Xi*2
* Corresponding author. E-mail: [email protected]
1. Department of Engineering Sciences and Mathematics Division of Fluid and Experimental Mechanics, Luleå University
of Technology.
2. Department of Engineering Sciences and Mathematics Division of Machine Elements, Luleå University of Technology.
Abstract
Purpose
This paper presents an optical method with ball recognition algorithm to measure rolling friction
coefficients in an efficient way. The aim is to measure the rolling friction coefficients between steel
balls of different diameters and a cylindrical aluminum surface.
Design/methodology/approach
A model of the tribosystem for a freely rolling ball and a cylindrical surface is presented by Lagrange
equation. The rolling friction coefficients are evaluated from the position data of the steel balls. The
position data are retrieved by the ball recognition algorithm from images recording by a high-speed
camera. The rolling friction coefficients between the different diameter steel balls and the cylindrical
aluminum surface are measured.
Findings
53
The angular positions of the balls are predicted by the solution of the equation of motion (EOM)
derived from Lagrange equation, and good agreements are found between the experimental and
theoretical results.
Originality/value
A model of rolling friction based on Lagrange equation was established and an optical method with a
ball recognition algorithm for the rolling friction coefficients evaluation is presented. The values of
rolling friction coefficients between the different diameter steel balls and the cylindrical aluminum
surface are evaluated.
Key words: rolling friction, friction coefficient, ball recognition, high-speed camera
54
1. Introduction
Rolling contacts are considered as a low energy loss to motion. The pioneers
started to be interested in the phenomena of the rolling motion at the end of the 19th
century and in the beginning of the 20th century [1, 2]. In the middle of the 20th
century, Eldredge and Tabor described the mechanism of rolling friction in the plastic
range [3] and the elastic range [4].
In the recent two decades many models and experiments have been established for
measuring rolling friction coefficients. Ball race models and ball bearing models were
frequently appeared in scientific publications and optical methods were mentioned as
effective methods. A laser vibrometer was used by Fujii (2004) [5] for estimating the
friction force on linear bearings and the data processing method was presented. Lin et
al. (2004) developed a vision based system for characterizing the tribological
behavior of linear ball bearings by the setup includes strain-free steel microballs and a
silicon V-groove [6]. Values of dynamic rolling friction coefficients were estimated to
be 0.007 on average. Tan et al. (2006) developed a dynamic viscoelastic friction
model to evaluate the rolling friction coefficient between steel microballs and silicon
up to 0.007 by using the same setup [7]. The rolling friction coefficient between 0.285
mm diameter balls and microfabricated silicon was measured to be 0.02 by
Ghalichechian et al. (2008), that was the first demonstration of a rotary micrometer
[8]. An analytical model of freely rolling steel microballs on a spherical glass surface
was developed by Olaru et al. (2009) [9], and a video camera was used to take record
55
of the position data of the small ball. Similarly, Cross (2016) [10] established a model
by Coulomb’s law and evaluated the rolling friction coefficients by the setup with
steel balls rolling freely on a concave lens surface. The limitations of Olaru et al. and
Cross are that they can only process the ball reciprocated in small angular positons,
and there was no appropriate algorithm used for getting accurate center locations of
the balls, and the sampling frequency was low.
In a later approach of Olaru et al. (2011) [11], a high-speed camera was used for
taking records of the rolling friction coefficients measurement of thrust ball bearings.
In 2014 they studied this model with lubricant viscosity conditions (%ăODQ et al., 2014)
[12]. Another usage of the high-speed camera is De Blasio and Saeter (2009) [13],
and they recorded a ball rolling on a granular medium and evaluated the friction
coefficient.
The great advantage of using a high-speed camera in these measurements is rather
more samples are obtained throughout the period of the experiment, which recudes
the influence of random noise and increase the positional accuracy. In addition, if an
automatic ball recognition algorithm is applied, the processing of the evaluation will
be more efficient. The procedures of ball recognition contain the edge detection and
the center location. D’Orazio et al. (2004) developed a ball recognition algorithm
based on Hough transform and solved the problem in different light conditions [14].
An edge detection approach in subpixel level combing Zernike moments operator with
Sobel was proposed by Qu (2005) [15]. A real-time accurate circle fit algorithm based
56
on the maximum likelihood was presented by Frosio et al. (2008) [16].
In this paper a model of a ball rolling on a cylindrical surface based on Lagrange
equation is presented. The model is appropriate for arbitrary angular positions of the
ball. The images including the position information of the ball are recorded by a
high-speed camera with a sampling frequency of 1000 Hz. A ball recognition
algorithm is used to retrieve the accurate position of the ball.
2. The equation of motion of a ball rolling on a cylindrical surface
Figure 1 A ball rolling freely on a cylindrical surface.
Figure 1 shows the geometry of a ball rolling freely on a cylindrical surface. The
plane through the center of the cylindrical surface and parallel to the ground is defined
as the zero potential energy plane. With relative to the plane,when the ball at an
angular positon ߶ the potential energy V is
V=െ݉݃(ܴ െ ‫)ݎ‬cos߶,
57
(1)
where ܴ is the diameter of the cylindrical surface , ‫ ݎ‬is the diameter of the ball, ݉
is the mass of the ball, and g is the acceleration due to gravity. The kinetic energy is
given by
ܶ=
1
1
݉‫ ݎ‬ଶ ߮ሶ ଶ + ‫߮ܫ‬ሶ ଶ ,
2
2
(2)
ଶ
where ‫ = ܫ‬ହ ݉‫ ݎ‬ଶ is the moment of inertia of the ball and ߮ is the angle of the ball
rotation. Equation (2) can be rewritten as
଻
ܶ = ଵ଴ ݉‫ ݎ‬ଶ ߮ሶ ଶ .
(3)
‫ ܶ = ܮ‬െ ܸ,
(4)
The Lagrangian is
The friction torque is considered to slow down the rotation speed of the ball, so
the generalized coordinate ߮ is picked for the following processing. Under the
assumption of the ball rolls without slipping, the relationship between ߶ and ߮
is
(ܴ െ ‫߮ݎ = ߶)ݎ‬
(5)
Substitute Equation (2) and Equation (3) into Equation (4), and consider the
relationship presented in Equation (5), the Lagrangian can be rewritten as
଻
௥
‫ = ܮ‬ଵ଴ ݉‫ ݎ‬ଶ ߮ሶ ଶ + ݉݃(ܴ െ ‫)ݎ‬cos ቀோି௥ ߮ቁ
The non-conservative Lagrange equation is given by
58
(6)
ୢ
ቀ
డ௅
ୢ௧ డఝሶ
డ௅
ቁ െ డఝ = ‫ܯ‬௥ ,
(7)
where ‫ܯ‬௥ is the generalized force. Due to ߮ is an angular coordinate, the
dimension of ‫ܯ‬௥ is the dimension of torque, that is to say ‫ܯ‬௥ is the rolling
friction torque slowing down the rotation speed of the ball. The relationship
between the rolling friction force ‫ܨ‬௙ and ‫ܯ‬௥ is
‫ܯ‬௥ = ‫ܨ‬௙ ή ‫ݎ‬,
(8)
and
థሶ
‫ܨ‬௙ = െߤ หథሶห ‫ܨ‬ே ,
(9)
where ‫ܨ‬ே = ݉݃cos߶ + ݉(ܴ െ ‫߶)ݎ‬ሶ ଶ is the normal force on the contact point
between the cylindrical surface and the small ball, and ߤ is the rolling friction
coefficient.
Substitute Equation (6) into Equation (7), gives the equation of motion
଻
ହ
௥
݉‫ ݎ‬ଶ ߮ሷ + ݉݃‫ݎ‬sin ቀோି௥ ߮ቁ = ‫ܯ‬௥ .
(10)
The data retrieved from the measurements are the angular position ߶(‫)ݐ‬. To
enable compassion with experiments, the variable of the equation of motion is
cast into ߶ by Equation (6). By substituting Equation (8) and Equation (9) into
Equation (10), the final version of the equation of motion becomes
ሶ
ହ ௚ୱ୧୬థ
ହ ఓ థ
߶ሷ = െ ଻ ோି௥ െ ଻ ோି௥ หథሶห ൫݃cos߶ + (ܴ െ ‫߶)ݎ‬ሶ ଶ ൯,
59
(11)
which can be solved numerically by given the initial conditions ߶(0) and ߶ሶ(0).
3. Experimental setup
An adjustable screw set with precision 10 Ɋm was mounted on a steel support on
the table. An aluminum (6061-T6 aluminum with anodized coating) cylindrical
surface with radius of curvature 85.3 mm (Figure 3) was mounted on the screw set.
This setup was imaged by a high-speed camera (Dantec Dynamics NanoSense, pixel
pitch 12 Ɋm h 12 Ɋm). The sampling frequency of the high-speed camera was 1000
Hz. The exposure time of the camera was 200 Ɋs. The image pixel size was
calculated from the quotient of object size and the corresponding pixel number in the
images and it was 0.1343 mm/pixel. The illumination came from two lamps (COOLH
dedocool) one on each side of the camera. A piece of white paper was pasted into the
back of the cylindrical surface in order to make diffuse reflection and make it easier to
distinguish the ball, the shadow and the background. Two steel (52100 steel) balls
were used in this free-rolling experiment. One steel ball has diameter 10.00 mm and
mass 4.07612 g, the other steel ball has diameter 12.69 mm and mass 8.35894 g. The
mass of the steel balls were measured by a METTLER TOLEDO AX205 analytical
balance. Figure 2 shows the sketch of the experimental setup.
60
Figure 2 Sketch of the experimental setup.
To adjust the setup, the bigger steel ball was put on the cylindrical surface. The
screw set was used to adjust the position of the cylindrical surface. While the ball
stood on the cylindrical surface steadily, the tangent plane of the cylindrical surface
was then considered parallel to the ground. For every measurement series, a
background image without the ball was taken as reference. The balls were manually
released from a point on the cylindrical surface with zero initial angular velocity.
After release the ball rolls back and forth on the cylindrical surface while it eventually
courses to a stop at the bottom of the cylindrical surface. During the initial few
periods images were recorded by the high-speed camera at a sampling frequency of
1000 Hz. A typical sequence takes about 4 seconds, which includes around 4000
images of the ball rolling back and forth on the cylindrical surface. In each time frame,
the position of the ball ߶(‫ )ݐ‬can be retrieved from the corresponding recorded image.
Due to the high sampling frequency, the angular position with zero velocity can be
61
retrieved from the recorded sequence, so the synchronization of the release of the
balls and the starting of the record is not necessary. Time ‫ = ݐ‬0 can be chosen as the
time the ball at a peak of the angular position with zero velocity. In practice, the
selection of time ‫ = ݐ‬0 is at the first effective peak of the angular position in order to
include as much as effective record as possible. A typical image of the 10.00 mm
diameter ball rolling on the cylindrical surface recorded by the camera is shown in
Figure 3.
Figure 3 Image of the 10.00 mm diameter steel ball on the cylindrical surface captured by the
high-speed camera.
4. Ball recognition algorithm and data processing
In order to calculate the angular positions of the ball from the recorded image
sequence, the center of the ball in each frame as well as the center of the cylindrical
62
surface has to be found out. In this section all coordinates are presented in pixels. The
determination of the center ൫‫ܥ‬௫ , ‫ܥ‬௬ ൯ of the cylindrical surface is relatively easy. In
each recorded image the contour of the cylindrical surface was clearly visible.
Looking for the horizontal line in the circle with maximum width and the vertical line
started from the lowest point of the circle. The intersection point of these two lines is
considered as the center ൫‫ܥ‬௫ , ‫ܥ‬௬ ൯ of the cylindrical surface.
In Figure 3, a ball attached to the inner side of the cylindrical surface can be seen
clearly. The center of the ball has to be retrieved from each frame of the recorded
images. There are thousands of images, an automatic method is necessary. The
algorithm of the ball recognition and the center location is described as follows. The
processing of a recorded frame of the 10.00 mm diameter ball is taken as an example.
Step 1: There are two shining spots on the ball originally from specular reflection
from the two lamps respectively. The two shining spots are close to the center of the
ball. A sub-image around the two shining spots of size 101 pixels × 101 pixels
(Figure 4) is cut from the recorded image.
63
Figure 4 A 101 pixels × 101 pixels sub-image around the two shining spots includes the ball.
Step 2: The corresponding background is subtracted from this image and a new
sub-image ‫ܫ‬௥௘௟ is obtained (Figure 5).
Figure 5 The background subtracted image ‫ܫ‬௥௘௟ on which the reviewing image processing is
performed
Step 3: According to the position information of the ball, a part of the ball
64
includes a fragment of relative smooth edge is selected. The size of the selected part is
about 1/4 of the ball (Figure 6). The shadow position of the ball is considered.
If ߶ > 0
The upper left part is selected
If ߶ < 0
The upper quarter part is selected
The ball appeared in Figure 3 located at the right part of the cylindrical surface,
so the upper left part of the ball was selected.
Figure 6 A part of the subtracted image include relatively smooth edge.
Step 4: Perform a binaryzation on the image in Figure 6 by an appropriate
threshold and locate a cursory edge (Figure 7) using canny operator. Before the
binaryzation, the diameter information of the ball is considered and the dark part near
the center is masked. Because the light condition is stable in the measurement, the
65
threshold is considered the same in each frame.
Figure 7 Results of binaryzation and the edge calculated from canny operator. The white curve
indicates the edge.
Step5: Calculate a rough center of the ball (‫ݔ‬௖ , ‫ݕ‬௖ ). By the cursory edge
calculated in Step 4, substituting the radius and the coordinates of the points on the
cursory edge calculated in Step 4 into Equation (12) , a rough center can be
determined (Figure 8)
‫ = ݎ‬ඥ(‫ݔ‬௜ െ ‫ݔ‬௖ )ଶ + (‫ݕ‬௜ െ ‫ݕ‬௖ )ଶ ,
(12)
where (‫ݔ‬௜ , ‫ݕ‬௜ ) is the point on the cursory edge was calculated in Step 4, ‫ ݎ‬is the
diameter of the ball presented in pixel.
66
Figure 8 The rough center determined by the cursory edge and the diameter information. The
white curve indicates the edge and the white point indicates the rough center.
Step 6: Looking for an accurate center of the ball. For each point in a 15h15
pixel neighborhood of the center determined by Step 5 ‫ݔ‬௧ ‫ݔ[ א‬௖ െ 7, ‫ݔ‬௖ + 7 , ‫ݕ‬௖ െ
7, ‫ݕ‬௖ + 7 ], a circle with radius ‫ ݎ‬is
‫ݔ = ݔ‬௧ + ‫ݎ‬cosߠ
, (13)
൜
‫ݕ = ݕ‬௧ + ‫ݎ‬sinߠ
where ߠ is the radius angle of the circle. A pixel on the circle is taken as the integer
value of the coordinate (Int (‫)ݔ‬, Int (‫))ݕ‬, the intensity of the pixel is denoted by ‫ܫ‬௖ .
The neighbor pixel in radius direction of this pixel out of the circle is
(Int (‫ ݔ‬ᇱ ), Int (‫ ݕ‬ᇱ )). ‫ ݔ‬ᇱ and ‫ ݕ‬ᇱ is calculated by
‫ ݔ‬ᇱ = ‫ݔ‬௧ + (‫ ݎ‬+ 1)cosߠ
൜ ᇱ
, (14)
‫ݕ = ݕ‬௧ + (‫ ݎ‬+ 1)sinߠ
if (Int (‫ ݔ‬ᇱ ), Int (‫ ݕ‬ᇱ )) and (Int (‫)ݔ‬, Int (‫ ))ݕ‬are overlapped, ‫ ݔ‬ᇱ and ‫ ݕ‬ᇱ is
calculated by
67
‫ ݔ‬ᇱ = ‫ݔ‬௧ + (‫ ݎ‬+ 2)cosߠ
൜ ᇱ
. (15)
‫ݕ = ݕ‬௧ + (‫ ݎ‬+ 2)sinߠ
The intensity of (Int (‫ ݔ‬ᇱ ), Int (‫ ݕ‬ᇱ )) is denoted by ‫ܫ‬௢ . If |‫ܫ‬௢ െ ‫ܫ‬௖ | > ݄ܶ‫݈݀݋݄ݏ݁ݎ‬, it is
considered as an intensity jump. The pixels on the circle centered (‫ݔ‬௧ , ‫ݕ‬௧ ) traversing
(the step of ߠ is 0.025) and the number of intensity jumps ‫ܫ‬௝௨௠௣ in radius direction
is counted. The point in ‫ݔ‬௧ ‫ݔ[ א‬௖ െ 7, ‫ݔ‬௖ + 7 , ‫ݕ‬௖ െ 7, ‫ݕ‬௖ + 7 ] has maximum ‫ܫ‬௝௨௠௣
value is considered as the center of the ball (Figure 9).
Figure 9 The result of ball recognition. The white circle line indicated the ball edge and the white
point in the circle is the location of the center.
Step 7: The coordinate of the center and the edge of the ball is reflected to the
image in Figure 3. The center of the ball (‫ݔ‬௖௖ , ‫ݕ‬௖௖ ) is determined and the ball
recognition is done (Figure 10).
68
Figure 10 The result of ball recognition.
Each frame was processed from Step 1 to Step 7 automatically and the angular
position of the ball ߶(‫ )ݐ‬in each frame was evaluated from
sin߶(‫= )ݐ‬
(‫ݔ‬௖௖ െ ‫ܥ‬௫ ) × image pixel size
.
ܴെ‫ݎ‬
(14)
The processing of the bigger ball is basically the same. The only difference between
the processing of the bigger ball and that of the smaller ball is the diameter difference.
After the ball recognition work was down, the high frequency noise has to be
eliminated, and even prepare for the subsequent processing includes time derivatives
of the angular position information, a low pass filter has to be applied on the angular
position data (Guddei et al. 2013) [17]. A Chebyshev low pass filter is used in this
paper. With the filter the data processing method becomes more robust.
5. Determination of the rolling friction coefficient
Benefit from the cylindrical surface, the balls could do reciprocating motion on
69
the surface. In a limited field of view, more experimental values can be recorded. For
both the 10 mm diameter ball and the 12.69 mm diameter ball, the angular positions
߶(‫ )ݐ‬and angular velocities ߶ሶ(‫ )ݐ‬are experimentally measured. The estimation of ߤ
is a differential equation parameter estimation problem of Equation (11). In order to
estimate the rolling friction coefficients, the following optimization problem is stated.
The objective of the optimization is:
minimize: σ(߶୰ୣୡ୭୰ୢ െ ߶୬୳୫ୣ୰୧ୡୟ୪ )ଶ
Optimization (1)
subject to: ߶(0), ߶ሶ(0), ߤ, ܴ, ‫ݎ‬.
In which ܴ and ‫ ݎ‬are measured before the experiment. ߶(0) and ߶ሶ(0) are the
starting angle and staring angular velocity of the rolling, and they are measured from
the experiment. The only unknown is ߤ. The optimization problem is solved by Auto
Fit 5.0 (Trail version). Limited by the function of the trail version, the parameter
estimation of Equation (11) is need the input ߶୰ୣୡ୭୰ୢ , ߶ሶ୰ୣୡ୭୰ୢ and ߶ሷ୰ୣୡ୭୰ୢ , so the
following optimization problem is solved at first.
Optimization (2)
minimize:
ଶ
ଶ
෍ ቂ(߶୰ୣୡ୭୰ୢ െ ߶୬୳୫ୣ୰୧ୡୟ୪ )ଶ + ൫߶ሶ୰ୣୡ୭୰ୢ െ ߶ሶ୬୳୫ୣ୰୧ୡୟ୪ ൯ + ൫߶ሷ୰ୣୡ୭୰ୢ െ ߶ሷ୬୳୫ୣ୰୧ୡୟ୪ ൯ ቃ
subject to: ߶(0), ߶ሶ(0), ߶ሷ(0), ߤ, ܴ, ‫ݎ‬.
An initial value of ߤ denoted as ߤ୧୬୧ is estimated from the Optimization (2).
However, some random errors can be induced to the estimation of ߶୰ୣୡ୭୰ୢ and
70
߶ሶ୰ୣୡ୭୰ୢ from the recorded images. In order to get rid of the random errors and solve
Optimization (1), the value of ߤ in the interval [ߤ୧୬୧ െ 0.001, ߤ୧୬୧ + 0.001] a step
length
0.0001,
the
value
of
߶(0)
in
the
interval
[ ߶(0)୰ୣୡ୭୰ୢ െ 0.0016, ߶(0)୰ୣୡ୭୰ୢ + 0.0016] with a step 0.0001 and the value of
߶ሶ(0) in the interval [െ0.2 ‫݀ܽݎ‬/s, 0.2 ‫݀ܽݎ‬/s] with a step 0.01 rad/s were substituted
in to Equation (11) iteratively. The Equation (11) in each iterative step with the
substituted ߤ, ߶(0) and ߶ሶ(0) values is solved numerically by Runge-Kutta method.
Until the minimum value of σ(߶୰ୣୡ୭୰ୢ െ ߶୬୳୫ୣ୰୧ୡୟ୪ )ଶ is found, the ߤ value is
considered as the optimum solution of the rolling friction coefficient.
6. Results and discussion
The red dots lines in Figure 12 are the angular positions retrieved from the
recorded images by the algorithm described in last section. In order to validate the
retrieved positions of the balls, the retrieved positions with randomly selected time ‫ݐ‬
were substituted into corresponding frames. First, the center of the ball in the
corresponding time frame was located by the retrieved center, and then the edge of the
ball was calculated by the diameter information. If the edge reconstructed by the
retrieved center position and the diameter information was matched to the original
image, the position retrieved is considered reliable. The results of the validation are
shown in Figure 11, the frames are selected from different periods. The red dots
(Instead a 1 pixel dot, each dot is taken 4 pixels for better visibility, in real case each
center dot is taken 1 pixel.) indicate the retrieved centers of the balls in each time
71
frame and the white circles are the edges reconstructed by the diameter information.
Figure 11 A shows the results of the 10 mm diameter ball and Figure 11 B shows the
results of the 12.69 mm diameter ball. The error of center in each time frame of the
balls were retrieved by the algorithm in last section are within 1 pixel. These errors
are indicated by the error bars in Figure 12.
72
Figure 11 Validation of the retrieved ball locations. A.10 mm diameter ball in different time frames.
B.12.69 mm diameter ball in different time frames
By the method in section 5, the rolling friction coefficient between the smaller
ball and the cylindrical surface ߤ௦ is estimated 0.0079 and the rolling friction
coefficient between the bigger ball and the cylindrical surface ߤ௕ is estimated 0.0065.
The comparison of numerical results and the experimental results are shown in Figure
12. The red dot lines are the experimental results and the blue lines are the numerical
results. The relative error for the 10 mm diameter ball is 5.8372% and for the 12.69
diameter ball is 4.5882%. The relative error is defined as
Relative error =
σ|߶୰ୣୡ୭୰ୢ െ ߶୬୳୫ୣ୰୧ୡୟ୪ |
.
σ|߶୰ୣୡ୭୰ୢ |
73
(14)
Figure 12 Comparison of the angular positions evaluated from the EOM and from the experiments.
A.The 10 mm diameter ball. B. The 12.69 mm diameter ball.
The range of the errors of both balls is acceptable. The rolling friction coefficient of
the bigger diameter ball and the cylindrical surface is smaller than the rolling friction
coefficient of the smaller ball as it should be. The rolling friction coefficient between
a 9.525 mm diameter steel ball and a glass spherical surface was measured by Olaru
[9] 0.004 in average. The rolling friction coefficient of between a 10 diameter ball and
a glass concave lens was reported around 0.002 by Cross [10]. The rolling friction
coefficients between 0.285 mm diameter steel balls and a silicon groove were
measured by Lin et al. (2004) [6] and Tan et al. (2004) [7] around 0.007. Our results
are in the same order of magnitude compare to theirs results. The surface roughness of
the aluminum surface is bigger than the surface roughness glass surface, the rolling
coefficient was measured between the steel balls and the aluminum surface is bigger
74
than that was measured between a similar diameter steel ball and a glass surface is
reasonable.
7. Conclusions
This paper approached an efficient and accurate optical method for evaluating the
rolling friction coefficient ߤ for steel balls rolling freely on a cylindrical aluminum
surface. A model based on Lagrange equation (of the 2nd kind) for a ball rolling
freely on a cylindrical surface was developed. A ball recognition algorithm and the
data processing method were presented in detail for retrieve the accurate location of
the ball in each frame. The position data retrieved from the recorded images is
validated by the match of the reconstructed edges of the ball to the original recorded
images. The rolling friction coefficient between the balls and the cylindrical surface
are evaluated by solving the optimization problem the differential equation parameter
estimation. The rolling friction coefficient between an aluminum cylindrical surface
and a 10 mm diameter steel ball is evaluated 0.0079 and the rolling friction coefficient
between the bigger ball and the cylindrical surface is evaluated 0.0065. These two
estimated rolling friction coefficient values and initial angles of the balls were
substituted into the Lagrange model and the Lagrange model agreed well with the
experimental data. This rolling friction coefficient evaluation method is available for
arbitrary initial angles and arbitrary ball diameters (small enough that can rolling on
the cylindrical surface).
Profit from the high sampling frequency of the camera, the starting angles can be
75
read from the record sequence, synchronization of the release of the ball and the
sampling is not necessary. The high sampling frequency is necessary to obtain precise
position and velocity signals. The experimental ߶ሶ(‫ )ݐ‬were retrieved from the
numerical differentiation of the angular positions ߶(‫ )ݐ‬with respect to time ‫ݐ‬, a
low-pass filter was used to restrain the high-frequency noise.
76
8. Reference
[1] Reynolds O. On rolling-friction[J]. Philosophical Transactions of the Royal Society of London,
1876, 166: 155-174.
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