LICENTIATE T H E S I S

Department of Engineering Sciences and MathematicsDivision of Fluid and Experimental Mechanics Optical Measurements of

Rolling Friction CoefficientsISSN 1402-1757ISBN 978-91-7583-663-8 (print)ISBN 978-91-7583-664-5 (pdf)

Luleå University of Technology 2016

Yiling Li O

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Optical Measurements of

Rolling Friction Coefficients

Yiling Li

Luleå University of TechnologyDepartment of Engineering Sciences and Mathematics

Division of Fluid and Experimental MechanicsAugust 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 ( 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 m. 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 m

12 m) at a sampling frequency of 1000 Hz. The exposure time of the camera

was 200 s 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= ( )cos , (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, is 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

=12

+12

, (3.2)

where = is the moment of inertia of the ball and is the angle of the ball

rotation. Then Equation (3.2) can be rewritten as

= . (3.3)

9

The Lagrangian is the difference between the kinetic energy and the potential

energy,

= + ( )cos . (3.4)

By calculating the derivative of the corresponding components respectively, The

non-conservative Lagrange equation is given by

= , (3.5)

where 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

( ) = , (3.6)

which reduces the degree of freedom to one. The final version of the equation of

motion becomes

= cos + ( ) , (3.7)

which can be solved numerically given the initial conditions (0) and (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 is valid,

analytical solutions of the equation of motion do exist. When the ball is rolling

down from = to = 0, the initial condition at = 0 is = , and

= 0. The solution is

( ) = + cos( ), (3.8)

10

where is a gravity parameter defined by

=57

and is a friction parameter defined by

=57

.

When the ball is climbing up from = 0 to , the initial condition at = 0 is

= 0, and = , where is the maximum angular velocity of the rolling

ball. The solution with these conditions is

( ) = + cos( ) + sin( ). (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]

=7

(17 10 ) 14. (3.10)

11

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 , of the cylindrical surface is determined from the recorded

images. As the contour of the cylindrical surface was visible in the recorded images,

, 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 = , is used.

Ball recognition algorithms.

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 130 pixels

that included the position of the ball was cut from the recorded image. This sub-image

was traversed by a 15 pixel 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 15 pixel window. The

center of the ball ( , ) was taken to be the brightest point in this 15 pixel 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.

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 ( , ) is determined. By calculating

the intensity gradient in the radial direction, the center of the ball ( , ) 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 (in different period) into corresponding frames

(Figure 4.5).

Calculate angular positions of the ball.

The angular position of the ball ( ) in each frame was finally evaluated as

sin ( ) =( ) × object pixel size

, (5.6)

which gives ( ).

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), (0) and 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 .

Figure 5.1.1 Optimization of the rolling friction parameter .

With this optimized value, the expressions for ( ) in the two stages both

converged to ( ) = 0, and the total relative error of these two stages was 1.29%.

The relative error is calculated as

17

Relative error = | | . (6.1)

Figure 5.1.2 shows the match between the angular position ( ) obtained from

the recorded images and that evaluated using the optimum rolling friction parameter

= 1.9s . 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°, = 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 = 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 , 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 ( ) and

angular velocities ( ) are experimentally measured. The estimation of 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: ( ) Optimization (1)

subject to: (0), (0), , , ,

in which and are measured before the experiment. (0) and (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 is estimated to be 0.0079 and the rolling friction coefficient between the

bigger ball and the cylindrical surface 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 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 ( ) calculated 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 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 can be identified with adequate precision. ( ) was calculated by22

numerical differentiation of the angular positions ( ) with respect to time . 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

[1] Jacobson, Bo, J J. Kalker, eds. Rolling contact phenomena[M]. Vol. 411. Springer, 2014.

[2] Kalker J J. Three-dimensional elastic bodies in rolling contact[M]. Springer Science & Business

Media, 2013.

[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,

1876, 166: 155-174.

[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,

1955, 229(1177): 181-198.

[7] Tabor D. The mechanism of rolling friction. II. The elastic range[C] Proceedings of the Royal

Society of London A: Mathematical, Physical and Engineering Sciences. The Royal Society, 1955,

229(1177): 198-220.

[8]

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

microball bearings[J]. Instrumentation and Measurement, IEEE Transactions on, 2004, 53(3): 839-846.

[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]

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: aisingiorotsidao@gmail.com, yiling.li@ltu.se

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 ( 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

= + , (1)

where 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, is the variation in

kinetic energy of the ball, is the friction torque, and is the variation in the

ball’s rotation; thus, is the energy lost due to friction torque. is given by

= ( ), (2)

where is the radius of curvature of the cylindrical surface and is 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 ( ) is

=2

+2

, (3)

where is the linear velocity of the ball, is its moment of inertia (for a solid ball,

= ), and is its angular velocity.

Equation (1) can be rewritten as

710

+ ( ) + (cos cos ) = 0. (4)

The linear velocity and angular velocity satisfy the relations = and

= , and 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. can then be considered independent of angular velocity and angular

position (for the experimental conditions in this paper, the variation of is within

5%). On differentiating Equation (4) with respect to time , the equation of motion of

the ball becomes

+ sin = 0, (5)

where is a gravity parameter defined by

=57

and is a friction parameter defined by

=57

.

If the variation of the angle is small, the approximation sin can be applied.

Then Equation (5) is a nonhomogeneous differential equation with the general

analytic solution

( ) = + cos( ) + sin( ), (6)

where and are constant determined by the initial conditions = , and

= 0, the time at = is set to be 0. The solution is

( ) = + cos( ). (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 = , the following energy equation can be established:

710

( ) + (1 cos ) + = 0. (8)

On differentiating Equation (8) with respect to time , the following differential

equation is obtained.

+ sin + = 0. (9)

With the approximation sin , the general solution of Equation (9) is given by

( ) = + cos( ) + sin( ). (10)

The constants and are determined by the initial conditions ( ) = 0, and

= , where is the maximum angular velocity of the rolling ball, the time

at = 0 is set to be 0. The solution with these conditions is

37

( ) = + cos( ) + sin( ). (11)

When the ball passes through the lowest point of the cylindrical surface,

( ) = 0, the only tangential force acting on it is friction, and therefore the friction

coefficient can be taken to be

=( )( ) ( )

, (12)

where ( ) and ( ) are respectively the normal and tangential forces acting on

the ball. When ( ) = 0, according to the analysis by Olaru et al. (2009),

( )| ( ) =57

and

( )| ( ) =17

(17 10cos )107

.

Equation (12) can then be rewritten as

=7

(17 10 cos ) 14 . (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 m. 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 m 12 m) 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 at which its angular velocity was zero.

This roll-down procedure was then repeated from an initial position = (| | <

| |). 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 s and the image

pixel size was 0.1343 mm/pixel. A typical image recorded by the camera is shown in

Figure 2. The center , (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 , 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 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 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 15 pixel window. The center of

the ball ( , ) (in pixels) was taken to be the brightest point in this 15 pixel 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,

( ), in each frame was evaluated from

sin ( ) =( ) × 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 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 ( ) reached 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

( ) can be established. This is given by Equation (7) for the roll-down stage and by

Equation (11) for the climbing stage. At time = 0, the initial angle at which the

angular velocity was zero was 12.07°. With the approximation of sin ( ) by ( ),

41

the maximum error induced by the approximation was |sin ( ) | |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

1.55×10 6. 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 ( ) = 7.48° was randomly

selected, ( ) and were substituted into Equation (7), from which a value of

was calculated as 1.321 s . Using this value of in Equation (7) gave the behavior

of the angular position as a function of time, ( ). This is compared in Figure 4 with

the experimental values of ( ) obtained from the recorded images. For =

1.321 s , 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 = | | . (14)

The ball’s climbing stage started when the ball passed through ( ) = 0 with

42

angular velocity = . It then climbed up to the other side of the cylindrical

surface to an angle at which the angular velocity was zero. In the experiment

described here, = 8.49°. The initial condition = | was evaluated

from the angular positions obtained from the recorded images by numerical

differentiation with respect to time . 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 ( ) = and time into Equation (11) gave a value of the rolling friction

parameter = 2.355 s . Using this in Equation (11) gave the behavior of the

angular position as a function of time ( ). Figure 5 compares this with the behavior

determined from the recorded images. The relative error was 0.68%, so this value of

= 2.355 s can be used as another initial value.

43

Figure 4 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.

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 ( ) in the two stages must both

converge to ( ) = 0. 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 in steps of 0.0001 s . As shown in Figure 6, the

optimum was determined to be 1.9012 s .

44

Figure 6 Optimization of the rolling friction parameter .

With this optimized value, the expressions for ( ) in the two stages both converged

to ( ) = 0, and the total relative error of these two stages was 1.29%. Figure 7

shows the match between the angular position ( ) obtained from the recorded

images and that evaluated using the optimum rolling friction parameter =

1.9012 s . 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°, = 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°, = 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 s was substituted into Equations (7) and (11) with

= 34.14° and = 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 ( ) obtained 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 ( ) = 0. 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 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 = 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 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 was

employed, and so a small initial angle was necessary. The model was validated by

comparing ( ) calculated 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 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 ( )

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 can be identified with adequate

precision. was calculated by numerical differentiation of the angular positions

( ) with respect to time . 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

References

, M.R.D., Stamate, V.C., Houpert, L. et al. (2014), “The influence of the lubricant viscosity on

the rolling friction torque”, Tribol. Int., Vol. 72, 1-12.

De Blasio F.V. and Saeter, M.B. (2009), “Rolling friction on a granular medium”, Phys. Rev. E, Vol.

79 No. 2, 022301.

Fujii, Y. (2004), “An optical method for evaluating frictional characteristics of linear bearings”, Opt.

Lasers Eng., Vol. 42 No. 5, pp. 493-501.

Ghalichechian, N., Modafe, A., Beyaz, M.I. et al. (2008), “Design, fabrication, and characterization of

a rotary micromotor supported on microball bearings”, J. Microelectromech. Syst., Vol. 17 No. 3, pp.

632-642.

Li, L., Yang, J. and Liu, W. (2015), “Effect of random surface roughness on squeeze film damping

characteristics in damper of linear rolling guide with a fractal-based method”, Ind. Lubr. Tribol., Vol.

67 No. 6, pp. 549-556.

Liu, H., Zhang, Y., Zhang, S. et al. (2015), “Preparation and evaluation of tribological properties of

oil-soluble rice-like CuO nanoparticles”, Ind. Lubr. Tribol., Vol. 67 No. 3, pp. 276-283.

Olaru, D.N., Stamate, C. and Prisacaru, G. (2009), “Rolling friction in a micro tribosystem”, Tribol.

Lett., Vol. 35 No. 3, pp. 205-210.

Olaru, D.N., Stamate, C., Dumitrascu, A. et al. (2011), “New micro tribometer for rolling friction”,

Wear, Vol. 271 No. 5, pp. 842-852.49

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: easonxiyinhu@gmail.com

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.

Findings53

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 ( 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= ( )cos , (1)

57

where is the diameter of the cylindrical surface , is 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

=12

+12

, (2)

where = is the moment of inertia of the ball and is the angle of the ball

rotation. Equation (2) can be rewritten as

= . (3)

The Lagrangian is

= , (4)

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

( ) = (5)

Substitute Equation (2) and Equation (3) into Equation (4), and consider the

relationship presented in Equation (5), the Lagrangian can be rewritten as

= + ( )cos (6)

The non-conservative Lagrange equation is given by 58

= , (7)

where is the generalized force. Due to is an angular coordinate, the

dimension of is the dimension of torque, that is to say is the rolling

friction torque slowing down the rotation speed of the ball. The relationship

between the rolling friction force and is

= , (8)

and

= , (9)

where = cos + ( ) is the normal force on the contact point

between the cylindrical surface and the small ball, and is the rolling friction

coefficient.

Substitute Equation (6) into Equation (7), gives the equation of motion

+ sin = . (10)

The data retrieved from the measurements are the angular position ( ). 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

= cos + ( ) , (11)

59

which can be solved numerically by given the initial conditions (0) and (0).

3. Experimental setup

An adjustable screw set with precision 10 m 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 m 12 m). The sampling frequency of the high-speed camera was 1000

Hz. The exposure time of the camera was 200 s. 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 ( ) can 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 = 0 can be chosen as the

time the ball at a peak of the angular position with zero velocity. In practice, the

selection of time = 0 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 , 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 , 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 is obtained (Figure 5).

Figure 5 The background subtracted image 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 ( , ). 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)

= ( ) + ( ) , (12)

where ( , ) is the point on the cursory edge was calculated in Step 4, is the

diameter of the ball presented in pixel.

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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 15 15

pixel neighborhood of the center determined by Step 5 [ 7, + 7 ,

7, + 7 ], a circle with radius is

= + cos= + sin , (13)

where is the radius angle of the circle. A pixel on the circle is taken as the integer

value of the coordinate (Int ( ), Int ( )), the intensity of the pixel is denoted by .

The neighbor pixel in radius direction of this pixel out of the circle is

(Int ( ), Int ( )). and is calculated by

= + ( + 1)cos= + ( + 1)sin , (14)

if (Int ( ), Int ( )) and (Int ( ), Int ( )) are overlapped, and is

calculated by

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= + ( + 2)cos= + ( + 2)sin . (15)

The intensity of (Int ( ), Int ( )) is denoted by . If | | > , it is

considered as an intensity jump. The pixels on the circle centered ( , ) traversing

(the step of is 0.025) and the number of intensity jumps in radius direction

is counted. The point in [ 7, + 7 , 7, + 7 ] has maximum

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 ( , ) is determined and the ball

recognition is done (Figure 10).

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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 ( ) in each frame was evaluated from

sin ( ) =( ) × image pixel size

. (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

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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

( ) and angular velocities ( ) are experimentally measured. The estimation of

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: ( ) Optimization (1)

subject to: (0), (0), , , .

In which and are measured before the experiment. (0) and (0) are the

starting angle and staring angular velocity of the rolling, and they are measured from

the experiment. The only unknown is . 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 , and , so the

following optimization problem is solved at first.

minimize: Optimization (2)

( ) + +

subject to: (0), (0), (0), , , .

An initial value of denoted as is estimated from the Optimization (2).

However, some random errors can be induced to the estimation of and

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from the recorded images. In order to get rid of the random errors and solve

Optimization (1), the value of in the interval [ 0.001, + 0.001] a step

length 0.0001, the value of (0) in the interval

[ (0) 0.0016, (0) + 0.0016] with a step 0.0001 and the value of

(0) in the interval [ 0.2 /s, 0.2 /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 , (0) and (0) values is solved numerically by Runge-Kutta method.

Until the minimum value of ( ) is found, the 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

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

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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.

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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 is estimated 0.0079 and the rolling friction

coefficient between the bigger ball and the cylindrical surface 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 =| |

| | . (14)

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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

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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 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

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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 ( ) were retrieved from the

numerical differentiation of the angular positions ( ) with respect to time , a

low-pass filter was used to restrain the high-frequency noise.

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