Study of the Rolling Friction Coefficient Between Dissimilar

materials

Article Study of the Rolling Friction Coeﬃcient between Dissimilar Materials through the Motion of a Conical Pendulum

1, 1 2 Stelian Alaci *, Ilie Muscă and S, tefan-Gheorghe Pentiuc 1 Department of Mechanics and Technologies, Stefan cel Mare University of Suceava, 13 University Str., 720229 Suceava, Romania; [email protected] 2 Computers Department, Stefan cel Mare University of Suceava, 13 University Str., 720229 Suceava, Romania; [email protected] * Correspondence: [email protected]; Tel.: +40-740-261-540

Received: 9 October 2020; Accepted: 5 November 2020; Published: 8 November 2020

Abstract: The rolling friction phenomenon is encountered in a wide range of applications and when two different materials are involved, quantitative characterization is necessary. The parameter to be determined is the coefficient of rolling friction, for whose estimation a methodology is proposed, based on the damped oscillation of a conical pendulum. The pure rolling contact between a sphere and a plane is obtained when a steel ball is the bob of the pendulum, which rolls on an inclined plate made from a second material from the contacting pair. The mathematical model of the motion of a conical pendulum constructed from a revolution body supported on an inclined plane in the presence of the rolling friction is developed. The dynamic equations of the rigid body with fixed point are applied and the differential equation of motion of the pendulum is obtained together with the expressions of the reaction forces in the contact point. For different pairs of materials, tests are performed on a laboratory device. The damped oscillatory motion of the conical pendulum is video-captured for the estimation of the angular amplitude variation. A program for image processing is developed for measuring the values of angular elongations from the analysis of each frame of the video and, finally, the coefficient of rolling friction is obtained. For all the materials tested, a linear decrease in angular amplitude is detected and the slope of angular amplitude can be considered as a characteristic parameter related to the coefficient of rolling friction between the two materials.

Keywords: rolling friction coeﬃcient; conical pendulum; damped oscillations; image processing; OpenCV; motion detection

1. Introduction The elements of any mobile mechanical structure requiring the accomplishment of a definite function must have well expressed motions. To fulfil this condition, the elements of the structure— which are considered rigid bodies as an initial assumption—must come into contact with each other so that, of the six degrees of freedom of the free condition circumstance, some will be cancelled. In the zones of interaction between two elements, according to the action and reaction principle, a system of forces and torques will occur. Their magnitude and orientation are critically influenced by the shape of the contacting surfaces and by the physical properties of the bodies. Adding the condition of smooth surfaces to the hypothesis of rigid bodies, the most general situation is characterized by the theoretical point contact, which is a nonconforming contact. Such contacts are known as higher pairs [1,2] and frequently met in engineering applications. In a higher pair, only the displacement along the common normal in the contact point is not allowed. The possible motions are as follows: sliding along a straight line from the tangent plane and the rotations about the normal (spinning motion), and about a line

Materials 2020, 13, 5032; doi:10.3390/ma13215032 www.mdpi.com/journal/materials Materials 2020, 13, 5032 2 of 25 contained in the tangent plane (rolling motion). To these three motions, the following reactions oppose: the friction force T the spinning moment Ms and the rolling moment Mr. The amplitudes of the three components of the friction torsor depend on numerous factors: the magnitude of the normal force, the material properties of the bodies, the quality of the surfaces, the presence of lubrication, etc., [3–5]. Of the three components of the torsor, the friction force was the first to be studied. For the dry friction case, the magnitude of the friction force is proportional to the normal force and the friction coefficient is the dynamic sliding coefficient. For the fluid friction case, Newton showed that the amplitude of the friction force is proportional to the velocity gradient [6]. Concerning the magnitude of the spinning torque and rolling torque, in order to explain the occurrence of these, the hypothesis of rigid bodies must be disregarded in favor of the assumption of deformable bodies. A contact surface of small dimensions exists in the vicinity of the initial contact point, and the stress state from the points of this surface can be found based on the theory of elasticity [7–10]. The two moments can be regarded as the result of elementary moments generated by the contact area’s stress state components. All of the friction torsor’s components consume energy from the mechanical system where they act, and they generate undesirable phenomena such as wear, vibrations, etc. [11,12]. For this reason, knowing the laws describing these parameters as precisely as possible is an essential condition for the correct design of the system so as to ensure optimum running. Although rolling friction generates less thermal energy and wear compared to sliding friction or spinning friction, its effects are not at all negligible when the contact forces, the angular velocity and the duration of operation of the system have considerable values. The first model for rolling friction is based on the similarity with the dry sliding friction and accepts the proportionality between the rolling friction torque and the normal force from contact [13]. Based on the theory of elasticity, Cherepanov [14] shows that the rolling friction moment is proportional to the normal force raised at power 3/2 for cylinders and at power 4/3 for spheres; the factor of proportionality depends on the elastic characteristics of the materials (Young modulus and Poisson coefficient). Evidently, a complete model should consider all the phenomena influencing the rolling friction: the viscoelastic properties of the two bodies, the presence of adhesion forces, deformations, etc. Quantifying the influence of all phenomena is very difficult; therefore, in engineering applications, a global estimation of rolling friction is expected, accepting that the friction torque depends on the contact loading and on the relative motion from the theoretical point of contact. Two main classes of methods can be applied to evaluate the rolling friction torque: the first implies oscillatory motions, and the second uses non oscillatory motions. The first category uses pendula which make contact with the ground [15–18]. In this case, the estimation of rolling friction moment is based on experimental estimation of the damping of the rotation motion of a pendulum. For the second class, bodies of revolution are used, which roll with angular velocity of constant sign on a support surface, plane or cylindrical. The principle of the method consists in comparing the actual acceleration of the mobile body to the theoretical acceleration, and the difference leads to the estimation of the rolling friction torque [19,20]. Of these two categories of procedures, the oscillatory motion method presents the advantage that the contact zone is reduced; thus, constant parameters of the material and surface are ensured for the entire period of the experiment.

2. Materials and Methods

2.1. Principle of Methodology In a moving mechanical system, it was experimentally observed that the attenuation of the motion depends directly on the magnitude of friction from the system. The quantitative characterization of friction effects is one of the tribology objectives and the classical tribometers are based on sliding friction measurements. Referring to the rolling motion, first, the parameters describing the rolling friction phenomenon should be specified, and, afterwards, the methodology for accurate evaluation of these parameters must be detailed. The present work proposes the coefficient of rolling friction as MaterialsMaterials2020 2020,,13 13,, 5032x FOR PEER REVIEW 33 ofof 2525

coefficient of rolling friction as a quantitative parameter influencing the magnitude of the rolling afriction quantitative torque. parameter For pure rolling inﬂuencing motion, the the magnitude selected moving of the rolling system friction consists torque. in a conical For pure pendulum rolling motion,constructed the selected from a moving ball oscillating system consists on top in of a conical a tilted pendulum plate; see constructed Figure 1. from The arolling ball oscillating friction onphenomenon top of a tilted occurs plate; between see Figure the1 .spherical The rolling bob friction of the phenomenonpendulum and occurs the plate. between the spherical bob of the pendulum and the plate.

7 6

5 4

FigureFigure 1.1. SchemeScheme ofof thethe device.device.

TheThe experimentalexperimental set-upset-up isis describeddescribed inin detaildetail inin SectionSection 2.3 2.3.. TheThe aimaim ofof thethe testtest isis toto estimateestimate thethe coecoefficientfficient of of rolling rolling friction friction between between the rolling the rolling steel ball steel and ball the materialand the to material be tested. to The be experimental tested. The dampingexperimental of the damping angular amplitudes of the angular of the pendulum amplitudes is corroborated of the pendulum with the is theoretical corroborated signal obtainedwith the bytheoretical integrating signal the equationobtained of by motion integrating (deduced the inequa Sectiontion of2.2 motion) and the (deduced value of thein Section coefficient 2.2) of and rolling the frictionvalue of is the obtained. coefficient The of experimental rolling friction analysis is obta isined. based The on experimental the law of damping analysis of is angular based amplitudeon the law ofof adamping pendulum. of angular The theoretical amplitude model of a pendulum. is developed The and theoretical the law ofmodel motion is developed for a conical and pendulum the law of constructedmotion for froma conical a sphere pendulum linked byconstructed a wire to afrom fulcrum a sphere that oscillates linked by on a a tiltedwire to plane a fulcrum is obtained. that Theoscillates equation on ofa tilted theoretical plane model is obtained. permits The consideration equation of of theoretical any type of model dependency permits between consideration the rolling of frictionany type torque of dependency and the normal between reaction. the rolling In Figure friction2, there torque are and presented the normal the signals reaction. obtained In Figure for a2, β powerthere are law presented dependency the signalsMr = sr Nobtained, for β =for1 aand powerβ = 3law/2. Fordependency both examples, Mr=srN theβ, for angular β=1 and amplitudes β=3/2. For decreaseboth examples, linearly the and angular this remark amplitudes gives thedecrease idea forlinearly sketching and this an experimentalremark gives set-upthe idea for for finding sketching the coean ffiexperimentalcient of rolling set-up friction; for finding see Figure the coefficient1. A rectangular of rolling thick friction; flat plate,see Figure 1, made 1. A ofrectangular duralumin thick is supportedflat plate, 1, at made one end of onduralumin a horizontal is supported board 2. Theat one other end end on isa positionedhorizontal atboard a specified 2. The heightother end via is a cylinder,positioned 4, placedat a specified beneath height it. The via plate a cylinder, to be tested, 4, placed 3, is beneath placed onit. topThe ofplate the to tilted be tested, plate, 1.3, Theis placed ball, 5,on fixed top of by the an tilted inextensible plate, 1. wire, The 6,ball, in a5, fulcrumfixed by attached an inextensible to the plate, wire, 36, (condition in a fulcrum required attached for to pure the rollingplate, 3 existence), (condition is required brought intofor contactpure rolling with theexistence), plate, 3. is A brought protractor, into 7, fixedcontact with with a block, the plate, 8, is used3. A forprotractor, measuring 7, thefixed angular with amplitudea block, 8, of is the used wire. for Specifically, measuring for the a pendulum angular thatamplitude oscillates of onthe a planewire. tiltedSpecifically, with respect for a pendulum to the horizontal, that oscillates the slope on of a theplane straight tilted lineswith envelopingrespect to the the horizontal, damped theoretical the slope angularof the straight amplitudes lines of enveloping oscillation the must damped be found theore and,tical from angular here, the amplitudes value of the of coe oscillationfficient of must rolling be frictionfound and, is estimated. from here, the value of the coefficient of rolling friction is estimated. This paper analyzes, collects and originates results obtained for the coefficient of rolling friction between a steel ball and plates made of different materials (steel, glass, copper, polycarbonate, aluminum, carbon fiber composite) using the methodology described next.

Materials 2020, 13, 5032 4 of 25 Materials 2020, 13, x FOR PEER REVIEW 4 of 25

FigureFigure 2. 2. TheoreticalTheoretical solutions solutions for for two two different diﬀerent dependencies dependencies of rolling torque. 2.2. Proposed Method: Theoretical Fundamentals This paper analyzes, collects and originates results obtained for the coefficient of rolling friction betweenTwo a surfaces steel ballS1 ,andS2 makeplates non-conformal made of different contact materials in a point, (steelC,, glass, where copper, two points, polycarbonate,C1 and C2, aluminum,belonging tocarbon these, fiber respectively, composite) superimpose; using the methodology see Figure3. Indescribed the contact next. point, the tangent plane Π is deﬁned and the common normal of versor n. The relative motion is characterized by the relative 2.2.velocity ProposedvC1C 2Method:between Theoretical the points FundamentalsC1, C2, contained in the tangent plane and by the angular velocity ω which has two components: a component perpendicular to the tangent plane, ωs (spinning component), Two surfaces S1, S2 make non-conformal contact in a point, C, where two points, C1and C2, and a component contained in the tangent plane, ωr, the angular rolling velocity. Due to external belonging to these, respectively, superimpose; see Figure 3. In the contact point, the tangent plane П applied forces, a reaction torsor occurs in the theoretical contact point. In the absence of friction, is defined and the common normal of versor n. The relative motion is characterized by the relative the interaction between the two surfaces is characterized only by the normal reaction N parallel to the velocity vC1C2 between the points C1, C2, contained in the tangent plane and by the angular velocity ω normal to the tangent plane. The presence of friction between the two surfaces is described by forces which has two components: a component perpendicular to the tangent plane, ωs (spinning and moments that oppose to the three possibilities of motion: component), and a component contained in the tangent plane, ωr, the angular rolling velocity. Due to the tendency of sliding between the points C and C , the sliding friction force T, parallel and to• external applied forces, a reaction torsor occurs in1 the theoretical2 contact point. In the absence of friction,of contrary the interaction direction between of velocity the vtwoC1C2 ,surfaces will oppose; is characterized only by the normal reaction N parallelto theto the spinning normal motion, to the thetangent spinning plane. torque The prMesences, normal of tofriction the tangent between plane the andtwo ofsurfaces contrary is • describeddirection by forces of the and angular moments velocity thatω opposes, will oppose; to the three possibilities of motion: • toto the the tendency angular rolling of sliding velocity betweenωr, the the rolling points friction C1 and torque C2, theM slidingr contained friction inthe force tangent T, parallel plane and and • Materialsofof contrary opposite 2020, 13, directionxdirection FOR PEER of ofREVIEW velocity angular v velocityC1C2, willω oppose;r, will oppose. 5 of 25 • to the spinning motion, the spinning torque Ms, normal to the tangent plane and of contrary direction of the angular velocity ωs, will oppose; • to the angular rolling velocity ωr, the rolling friction torque Mr contained in the tangent plane and of opposite direction of angular velocity ωr, will oppose.

FigureFigure 3. 3.The The friction friction torsor torsor at at a a point point of of contact. contact.

The components of the reaction torsor at a point of contact are:

• The normal force N • The sliding friction force T • The spinning friction moment Ms • The rolling friction moment Mr It was observed that in solving problems from rigid dynamics, the two fundamental theorems (the momentum theorem and the moment of momentum theorem) do not offer sufficient scalar equations for finding all the unknowns of the problem. To surpass this fact, the relations between the components of the friction torsor were demanded. The first one is the unanimously accepted Amonton–Coulomb, which shows proportionality between the modulus of the sliding friction force and the modulus of the normal force: T = μ N (1) where μ is the dimensionless factor named the coefficient of sliding friction. Similar to Equation (1), it was first accepted [13] that the magnitudes of the spinning torque Ms and rolling friction torque Mr can be considered proportional to the modulus of the normal force N: = M s ss N (2) = M r sr N (3)

where the proportionality factors ss and sr have dimensions of length and are known as coefficient of spinning friction and coefficient of rolling friction, respectively. Concerning the rolling friction torques, relatively recent research based on the theory of elasticity showed that sr from Equation (3) depends on its turn on the normal force N. Regarding the rolling friction moment, Mr, Cherepanov [14] shows that for a spherical rolling body, = 4 / 3 M r sr N (4)

and

= 3 / 2 M r sr N (5)

for a cylindrical rolling body. Referring to the importance of finding sr, Cherepanov [14] states, "Experimental determination this coefficient is one of the basic problem of tribology”, and this affirmation justifies the purpose of our work. Recent literature uses a dimensionless coefficient of

Materials 2020, 13, 5032 5 of 25

The components of the reaction torsor at a point of contact are:

The normal force N • The sliding friction force T • The spinning friction moment Ms • The rolling friction moment Mr • It was observed that in solving problems from rigid dynamics, the two fundamental theorems (the momentum theorem and the moment of momentum theorem) do not offer sufficient scalar equations for finding all the unknowns of the problem. To surpass this fact, the relations between the components of the friction torsor were demanded. The first one is the unanimously accepted Amonton–Coulomb, which shows proportionality between the modulus of the sliding friction force and the modulus of the normal force: T = µN. (1) where µ is the dimensionless factor named the coefficient of sliding friction. Similar to Equation (1), it was first accepted [13] that the magnitudes of the spinning torque Ms and rolling friction torque Mr can be considered proportional to the modulus of the normal force N:

Ms = ssN, (2)

Mr = srN (3) where the proportionality factors ss and sr have dimensions of length and are known as coefficient of spinning friction and coefficient of rolling friction, respectively. Concerning the rolling friction torques, relatively recent research based on the theory of elasticity showed that sr from Equation (3) depends on its turn on the normal force N. Regarding the rolling friction moment, Mr, Cherepanov [14] shows that for a spherical rolling body, 4/3 Mr = srN , (4) and 3/2 Mr = srN (5) for a cylindrical rolling body. Referring to the importance of finding sr, Cherepanov [14] states, "Experimental determination this coefficient is one of the basic problem of tribology”, and this affirmation justifies the purpose of our work. Recent literature uses a dimensionless coefficient of rolling friction, expressed as µr = sr/r, dividing the lever arm coefficient by the value of the radius of the rolling body. The above considerations are applied for a fixed surface Σ and a mobile sphere contacting the surface in the C point, as represented in Figure4. A second point O is observed, fixed to the Σ surface which, together with the C point, defines the instantaneous axis of rotation. The instantaneous axis of rotation defines, with respect to an immobile system and to a system attached to the sphere, the axodes of the motion, which are conical surfaces having O as a common point. The relative motion of the two axodes is a pure rolling about the instantaneous axis. For the case of pure rolling, the angular velocity ω is parallel to the axis of rotation. A consequence of the fact that the angular velocity lies in the plane tangent to the surfaces of the two axodes is the fact that the angular velocity does not have a projection on the normal in the contact point. Therefore, the moment of friction in the contact point is characterized only by the rolling torque Mr collinear to the angular velocity, the spinning torque being zero. When the surface Σ is not horizontal and the sphere is tied with an inextensible string in the fulcrum O, a conical pendulum is obtained. Materials 2020, 13, x FOR PEER REVIEW 6 of 25 rolling friction, expressed as μr=sr/r, dividing the lever arm coefficient by the value of the radius of the rolling body. MaterialsThe 2020 above, 13, x considerationsFOR PEER REVIEW are applied for a fixed surface Σ and a mobile sphere contacting6 of the 25 surface in the C point, as represented in Figure 4. A second point O is observed, fixed to the Σ surface rolling friction, expressed as μr=sr/r, dividing the lever arm coefficient by the value of the radius of which, together with the C point, defines the instantaneous axis of rotation. The instantaneous axis the rolling body. of rotation defines, with respect to an immobile system and to a system attached to the sphere, the The above considerations are applied for a fixed surface Σ and a mobile sphere contacting the axodes of the motion, which are conical surfaces having O as a common point. The relative motion of surface in the C point, as represented in Figure 4. A second point O is observed, fixed to the Σ surface the two axodes is a pure rolling about the instantaneous axis. For the case of pure rolling, the angular which, together with the C point, defines the instantaneous axis of rotation. The instantaneous axis velocity ω is parallel to the axis of rotation. A consequence of the fact that the angular velocity lies in of rotation defines, with respect to an immobile system and to a system attached to the sphere, the the plane tangent to the surfaces of the two axodes is the fact that the angular velocity does not have axodes of the motion, which are conical surfaces having O as a common point. The relative motion of a projection on the normal in the contact point. Therefore, the moment of friction in the contact point the two axodes is a pure rolling about the instantaneous axis. For the case of pure rolling, the angular is characterized only by the rolling torque Mr collinear to the angular velocity, the spinning torque velocity ω is parallel to the axis of rotation. A consequence of the fact that the angular velocity lies in being zero. When the surface Σ is not horizontal and the sphere is tied with an inextensible string in theMaterials plane2020 tangent, 13, 5032 to the surfaces of the two axodes is the fact that the angular velocity does not have6 of 25 the fulcrum O, a conical pendulum is obtained. a projection on the normal in the contact point. Therefore, the moment of friction in the contact point is characterized only by the rolling torque Mr collinear to the angular velocity, the spinning torque being zero. When the surface Σ is not horizontal and the sphere is tied with an inextensible string in the fulcrum O, a conical pendulum is obtained.

FigureFigure 4. 4. ConicalConical pendulum. pendulum.

Assuming a rigid body, as in Figure5, subjected to p external torques M , i = 1..p and to q external Assuming a rigid body, as in Figure 5, subjected to p external torques Mi i, i=1..p and to q external concentrated forces F , j = 1..q applied in the points with position vectors r’ , j = 1..q, the resultant concentrated forces Fj, j=1..q applied in the points with position vectors r’jj, j=1..q, the resultant appliedapplied moment moment calculated calculated with with respect respectFigure to to O4.O Conicalisis as as follows: follows: pendulum. p q Assuming a rigid body, as in Figure 5, subjectedX toX p external torques Mi, i=1..p and to q external M0 = Mp + r0 Fj. (6) concentrated forces Fj, j=1..q applied in the points withj ×position vectors r’j, j=1..q, the resultant i= j= applied moment calculated with respect to O is1 as follows:1

Figure 5. Rigid body subjected to generalized forces.

FigureFigure 5. 5. RigidRigid body body subjected subjected to to generalized generalized forces. forces.

For the pendulum bob, the forces are as follows: the weight of the ball G, the reaction from the wire R (its direction passes through O and the moment is zero), the normal reaction N and the friction force T acting in the point of contact; the moments are the spinning moment Ms parallel to the normal in the contact point (identical zero here) and the rolling friction moment Mr in the tangent plane. The torsion moment from the wire acts along the wire and can be neglected, Mtw 0 as shown in the AppendixA. The non-zero forces and moments are represented in Figure6. We examine an inclined plane at the angle α with the vertical direction, as shown in Figure6. It supports a sphere of m mass and r radius. An inextensible wire ensures the constant distance d between the origin O of the system and the center of the ball. The motion of the sphere is modeled by the motion of a rigid with ﬁxed point and described by the moment of momentum theorem [21]:

J0ε + ωeJ0ω = M0. (7) Materials 2020, 13, x FOR PEER REVIEW 7 of 25

p q +× M0 ='M pjjrF (6) ij==11 For the pendulum bob, the forces are as follows: the weight of the ball G, the reaction from the wire R (its direction passes through O and the moment is zero), the normal reaction N and the friction force T acting in the point of contact; the moments are the spinning moment Ms parallel to the normal in the contact point (identical zero here) and the rolling friction moment Mrin the tangent plane. The torsion moment from the wire acts along the wire and can be neglected, Mtw≅0 as shown in the Appendix. The non-zero forces and moments are represented in Figure 6. We examine an inclined plane at the angle α with the vertical direction, as shown in Figure 6. It supports a sphere of m mass and r radius. An inextensible wire ensures the constant distance d between the origin O of the system and the center of the ball. The motion of the sphere is modeled by the motion of a rigid with fixed point and described by the moment of momentum theorem [21]: ε +ω~ ω = J0 J0 M0 . (7)

In the above equation, J0 is the inertia matrix of the ball with respect to the system Ox’y’z’ attached to the ball having the axis Oz’ oriented along the wire; ω and ε are the column matrices attached to the angular velocity and acceleration, respectively. ῶ is the anti-symmetric matrix attached to the angular velocity vector. Due to the rotational symmetry of the body, the J0 matrix is a diagonal one of the following form:

Jx 0 0  =   J0  0 Jy 0  , (8)   Materials 2020, 13, 5032  0 0 Jz  7 of 25

Materials 2020, 13, x FOR PEER REVIEW 7 of 25

p q M0 ='Mp+ r j F j (6) ij==11 For the pendulum bob, the forces are as follows: the weight of the ball G, the reaction from the wire R (its direction passes through O and the moment is zero), the normal reaction N and the friction force T acting in the point of contact; the moments are the spinning moment Ms parallel to the normal in the contact point (identical zero here) and the rolling friction moment Mrin the tangent plane. The torsion moment from the wire acts along the wire and can be neglected, Mtw≅0 as shown in the Appendix. The non-zero forces and moments are represented in Figure 6. We examine an inclined plane at the angle α with the vertical direction, as shown in Figure 6. It supports a sphere of m mass and r radius. An inextensible wire ensures the constant distance d between the origin O of the system and the center of the ball. The motion of the sphere is modeled by the motion of a rigid with fixed point and described by the moment of momentum theorem [21]: Figure~ 6. The reference systems used for the model. J0 +FigureJ0 6.= TheM0 . reference systems used for the model. (7)

In the above equation,whereIn the J0 aboveis the equation, inertia matrixJ0 is the of inertia the ball matrix with of respect the ball to with the respect system to O thex’y’z’ system Ox0y0z0 attached attached to the ball havingto the ball the having axis Oz’ the oriented axis Oz0 alongoriented the along wire; the ω wire;and2 εω areand theε are column the column matrices matrices attached to the J = mr 2 , (9) attached to the angularangular velocity velocity and and acceleration, acceleration, respectively. respectively. zῶ isis5 the the anti-symmetric anti-symmetric matrix matrix attached to the angular attached to the angularvelocity velocity vector. vector. Due Due to to the the rotational rotational symmetry symmetry of of the the body, body, the theJ J00 matrixmatrix is a diagonal one of the diagonal one of the follfollowingowing form: form:    Jx 0 0  Jx 0 0     J =  J  J = 0 J 0 , O  0 y 0 , (8) 0  y    (8)   0 0 Jz  0 0 Jz  where 2 J = mr2, (9) z 5 and applying the Steiner theorem, the next relation is found:

2 Jx = Jy = J = Jz + md . (10)

The term M0 from Equation (7) is a column matrix that has as elements the projections of the resultant moment, on the axes of the mobile frame Ox0y0z0 attached to the rigid body. Since the scalar equations of motion are needed, the following coordinate systems are considered (Figure7):

The Ox y z system that has the Oz axis in the vertical direction; • 0 0 0 0 The Ox y z frame obtained by revolving the frame Ox y z with an angle α about the axis • 1 1 1 0 0 0 Oy Oy . The ball is sustained by the plane Ox y ; 0 ≡ 1 1 1 The frame Ox y z obtained by rotating the system Ox y z around the axis Oz Oz with an • 2 2 2 1 1 1 1 ≡ 2 angle ψ; The coordinate system Ox y z obtained by rotating the frame Ox y z with an angle (π/2 δ), • 3 3 3 2 2 2 − as shown in Figure3, about the axis Oy Oy ; 2 ≡ 3

Figure 6. The reference systems used for the model. where 2 J = mr 2 , (9) z 5

Materials 2020, 13, x FOR PEER REVIEW 8 of 25 and applying the Steiner theorem, the next relation is found: = = = + 2 J x J y J J z md . (10)

The term M0 from Equation (7) is a column matrix that has as elements the projections of the resultant moment, on the axes of the mobile frame Ox’y’z’ attached to the rigid body. Since the scalar equations of motion are needed, the following coordinate systems are considered (Figure 7):

• The Ox0y0z0 system that has the Oz0 axis in the vertical direction; • The Ox1y1z1 frame obtained by revolving the frame Ox0y0z0 with an angle α about the axis Oy0≡Oy1. The ball is sustained by the plane Ox1y1; • The frame Ox2y2z2 obtained by rotating the system Ox1y1z1 around the axis Oz1≡Oz2 with an angle ψ; • The coordinate system Ox3y3z3 obtained by rotating the frame Ox2y2z2 with an angle (π/2-δ), as shown in Figure 3, about the axis Oy2≡Oy3; • The coordinate system Ox’y’z’ obtained by rotating the frame Ox3y3z3 with an angle φabout the axis Oz3≡Oz’. The orientation of the axes of the reference system Ox’y’z’ attached to the rigid with respect to the fix frame is stipulated using three scalar parameters. The system attached to the rigid can be brought into the chosen position if a set of three successive rotations is applied to the rigid body, about three stipulated axes obeying the condition that two successive axes of rotation must not be parallel. The most known sequence is rot(Z)—>rot(X)—>rot(Z) according to Euler’s angles. Another sequence corresponds to Bryant’s angles, rot(X)—>rot(Y)—>rot(Z). Obviously, the use of another rotation combination leads to the same results. In the present work, the rotation sequence was chosen in a manner that, at least in one frame, any of the positional parameters ψ and φ, kinematical ω, ε parameters, constructive parameters α, δ, r, d, , and the exterior forces and moments G, N, T, Mr mustMaterials appear2020, 13 in, 5032 the actual size. The absolute motion of the pendulum could be studied with respect8 of 25 to the fixed reference system Ox1y1z1. However, the use of the fixed coordinate system Ox0y0z0 was necessary in order to express the gravity force G in a simple way. In Figure 7, there are presented the The coordinate system Ox0y0z0 obtained by rotating the frame Ox3y3z3 with an angle ϕ about the four• rotations applied to the system Ox0y0z0 in order to bring it to the final position Ox’y’z’, passing axis Oz3 Oz0. through the auxiliary≡ (intermediate) systems Ox1y1z1, , Ox2y2z2 and Ox3y3z3.

FigureFigure 7. 7. TheThe coordinate coordinate systems systems and and the the rotations rotations applied applied to to the the ball ball..

The orientation of the axes of the reference system Ox0y0z0 attached to the rigid with respect to the fix frame is stipulated using three scalar parameters. The system attached to the rigid can be brought into the chosen position if a set of three successive rotations is applied to the rigid body, about three stipulated axes obeying the condition that two successive axes of rotation must not be parallel. The most known sequence is rot(Z)—>rot(X)—>rot(Z) according to Euler’s angles. Another sequence corresponds to Bryant’s angles, rot(X)—>rot(Y)—>rot(Z). Obviously, the use of another rotation combination leads to the same results. In the present work, the rotation sequence was chosen in a manner that, at least in one frame, any of the positional parameters ψ and ϕ, kinematical ω, ε parameters, constructive parameters α, δ, r, d, and the exterior forces and moments G, N, T, Mr must appear in the actual size. The absolute motion of the pendulum could be studied with respect to the fixed reference system Ox1y1z1. However, the use of the fixed coordinate system Ox0y0z0 was necessary in order to express the gravity force G in a simple way. In Figure7, there are presented the four rotations applied to the system Ox0y0z0 in order to bring it to the final position Ox0y0z0, passing through the auxiliary (intermediate) systems Ox1y1z1, Ox2y2z2 and Ox3y3z3. Based on Figure7, the relations between the versors of the pairs of frames are deduced: The rotation with an angle α about the axis Oy Oy : 1 ≡ 0    i1 = i0 cos α k0 sin α  i0 = i1 cos α + k1 sin α  −   j1 = j0 or  j0 = j1 (11)    k = i sin α + k cos α  k = i sin α + k cos α 1 0 0 0 − 1 1 The rotation with an angle ψ about the axis Oz Oz : 1 ≡ 2    i2 = cos ψ i1 + sin ψ j  i1 = cos ψ i2 sin ψ j  1  − 2  j = sin ψ i + cos ψ j or  j = sin ψ i + cos ψ j (12)  2 1 1  1 1 1  −  k2 = k1 k1 = k2 Materials 2020, 13, x FOR PEER REVIEW 9 of 25

Based on Figure 7, the relations between the versors of the pairs of frames are deduced: The rotation with an angle α about the axis Oy1≡Oy0: = α − α = α + α i1 i0 cos k0 sin i0 i1 cos k1 sin  =  =  j1 j0 or  j0 j0 (11)  = α + α  = α − α k1 i0 sin k0 cos k0 i1 sin k1 cos

The rotation with an angle ψψ about the axis Oz1≡Oz2: = ψ + ψ = ψ − ψ Materials 2020, 13, 5032 i2 cos i1 sin j1 i1 cos i2 sin j2 9 of 25  = − ψ + ψ  = − ψ + ψ  j2 sin i1 cos j1 or  j1 sin i1 cos j1 (12)  =  = The rotation with ank2 anglek1 (π/2 δ) about the axisk1 Oyk2 Oy : − 2 ≡ 3 The rotation with an angle (π/2-δ) about the axis Oy2≡Oy3:  i3 = cos(π/2 δ) i2 sin(π/2 δ) k2  i2 = cos(π/2 δ) i3 + sin(π/2 δ) k3  i = cos(π / 2 −δ )i −−sin(π / 2 −−δ )k  i = cos(π / 2 −−δ )i + sin(π / 2 − δ−)k  3 j =2 j 3 or  2 j 3= j 3 (13)   = 3 2   = 2 3   j3 j2 or   j2 j3 (13)  k3 = sin(π/2 δ) i2 + cos(π/2 δ)k2  k2 = sin(π/2 δ) i3 cos(π/2 δ)k3  = π −− δ + π −− δ  = − π − δ− + − π − δ− k3 sin( / 2 )k3 cos( / 2 )k3 k2 sin( / 2 )i3 cos( / 2 )k3 The rotation with an angle ϕ about the axis Oz3 Oz’: The rotation with an angle φ about the axis Oz3≡≡Oz’:  i' ==cosϕ i + sin+ ϕ j i = cos=ϕ i'+ sinϕ j'  i0 cos ϕ3 i3 sin3ϕ j3 3 i3 cos ϕ i0 sin ϕ j0   = − ϕ + ϕ   = − ϕ + −ϕ  j0j'= sinsin ϕi3i3 +coscosjϕ' jor0 or j3 j3 =sinsini'ϕ cosi0 + cosj' ϕ j0 (14)(14)  −    = k = k   = k = k k' k3 0 3 k3 k' 3 0 TheThe aboveabove relationsrelations allow allow for for expressing expressing any any vector vector in anyin any of theof the frames frames considered considered and canand alsocan bealso obtained be obtained via tensorial via tensorial calculus calculus [22,23 [22,23].]. InIn FigureFigure8 8,, itit cancan bebe seenseen thatthat sinδδ == r/dr/ doror δ =δ asin(r/d)= asin(r /andd) and from from here, here, the thecondition condition r/d ≤r /1d. The1. ≤ Thephysical physical significance signiﬁcance is obvious: is obvious: in order in order to toco constructnstruct the the pendulum, pendulum, the the length length of of the the pendulum, measuredmeasured fromfrom thethe centercenter ofof thethe ballball toto thethe fulcrum,fulcrum, mustmust bebe longerlonger thanthanthe theradius radiusof of the the ball. ball.

FigureFigure 8.8. Schematics for the kinematic and dynamic study.study.

EquationsEquations (11)–(14) permit permit expres expressingsing any any vector vector in inthe the frame frame Ox’y’z’Ox0y attached0z0 attached to the to rigid. the rigid. The Theangular angular velocity velocity of the of ball the ballcan canbe written be written as follows: as follows: ω =. ψ +.ϕ  k1  k3 , ω = ψ k1 + ϕ k3, (15)(15) having in the system Ox’y’z’ the following components: having in the system Ox0y0z0 the following components: −ψ cosϕ cosδ  . ω =  ψ ϕ δ   ψ cossinϕ coscos δ  (16)  −.  ω =  ψ sinψ sinϕ cosδ +δϕ . (16)  .   .  The contact point C has the following positionψ sin vector:δ + ϕ The contact point C has the following position vector:    cos ϕ sin δ    rC = i2d cos δ =  sin ϕ sin δ d cos δ. (17)  −   cos δ 

The pure rolling condition [24,25] in the point C is as follows:

v = v + ω rc = 0. (18) C 0 × Materials 2020, 13, 5032 10 of 25

Since, in this case, the velocity of the point O is zero (the point O is immobile), this means that for the pure rolling situation, the position vector of the contact point is parallel to the angular velocity of the ball ω with respect to the plane. After the calculus is made, the pure rolling condition in the point C is obtained: . . ψ = ϕ sin δ. (19) − Equations (19) and (16) permit ﬁnding the expression of the angular velocity for the pure rolling case:  .   ϕ cos ϕ sin δ   .  ω =  ϕ sin ϕ sin δ  cos δ. (20)  − .   ϕ cos δ  In order to obtain the vector of angular acceleration, the derivative with respect to time of Equation (20) is made:

 .. . 2   ϕ cos ϕ sin δ ϕ sin ϕ sin δ    .  .. − . 2  ε = ω =  ϕ sin ϕ sin δ ϕ cos ϕ sin δ  cos δ. (21)  − .. −   ϕ cos δ 

The expression of the left member of Equation (7) can be obtained using Equations (20) and (21):

 h .. 2 2 . 2 i   Jϕ cos ϕ (J sin δ + Jz cos δ)ϕ sin ϕ cos δ sin δ   h .. − . 2 i  J ε + ωJ ω =  ( 2 + 2 ) . (22) 0 e 0  Jϕ sin ϕ J sin δ Jz cos δ ϕ cos ϕ cos δ sin δ   − − .. 2  Jzϕ cos δ

The right member of Equation (7) must be in explicit form. To this end, the forces and torques acting upon the ball must be identiﬁed. The sole external force is G, the weight of the ball,

G = mgk , (23) − 0 which acts in the ball’s center of mass, having the vector of position:

rG = dk3. (24)

In the contact point C act the following forces:

The normal reaction N, parallel to k : • 2

N = Nk2; (25)

The friction force T in the support plane and normal to the angular rolling velocity ω: •

T = Tj2; (26)

The rolling torque Mr, collinear to the angular rolling velocity and of opposite orientation: •

Mr = Mrsgn(ω)i . (27) − 2 The spin torque from the contact point is zero since the angular velocity lacks a component along • the normal to the support plane.

The reaction of the wire, R, passes through the origin of the system of axes and therefore the moment of the reaction with respect to the origin is zero. The unknowns occurring in the system of Materials 2020, 13, 5032 11 of 25 scalar equations of projection are as follows: one of the angles ϕ(t) or ψ(t) (the two angles are related by the pure rolling condition; see Equation (19)), the values of the normal force, the friction force and the rolling friction torque. In order to attain a compatible system of equations, it is assumed that the value of the friction torque is proportional to the value of normal force raised to a certain power.

β Mr = srN . (28)

This results in the equation of motion, a nonlinear ordinary diﬀerential equation ODE,

β + 2 2 . 2 Jz md sin δ ψ + mg tan δ sin α cos ψ .. sin ψ d sin δ . ϕ = mg sin α + sr sgnψ. (29) + 2 2 2 2 Jz md sin δ (Jz + md sin δ) cos δ d sin δ cos δ and the reactions from the contact point:

2 2 Jz + md sin δ . 2 sin α sin δ cos ψ + cos α cos δ N = sin δ ϕ + mg, (30) d cos δ

J . z " 2 2 #β 2 s Jz + md sin δ . 2 sin δsgnψ T = md sin α sin ψ mg r ψ + tan δ sin α cos ψ mg (31) Jz + 2 − d d sin δ Jz + 2 md2 sin δ md2 sin δ In order to integrate the equation of motion, Equation (29), the derivative of Equation (19) with respect to time, must be considered: .. .. ψ = ϕ sin δ. (32) − The ﬁnal form of the equation of motion for pure rolling is as follows:

β + 2 2 . 2 Jz md sin δ ψ + mg tan δ sin α cos ψ .. sin ψ d sin δ . ψ = mg sin α tan δ sr tan δ sgnψ. (33) + 2 2 2 2 − Jz md sin δ − (Jz + md sin δ) d sin δ

In the solutions (30), (31) and (33) of Equation (7), the second derivative of the position parameter of the system ψ occurs only in the last equation. In addition, the expressions of the two reactions N and T also lack precessional angular acceleration (the second derivative with respect to time of ψ angle). This observation permits a substantial simplification of the numerical integration of the equation of motion. When the planar motion pendula are used, the angular acceleration can be found only for the assumption of proportionality between the rolling friction torque and the normal force. Thus, one can conclude that one of the main advantages of the proposed model is that it permits the integration of the equation of motion regardless of the accepted dependency between the rolling friction moment and normal force. Equation (33) is a nonlinear ordinary differential equation (ODE). The Runge–Kutta method [26] is applied in order to integrate it. After the numerical integration, the dependency on time of the precession angle ψ and of the precession velocity is found. With these dependencies, the time variations of the friction force and normal reaction are found. These values are required for the validation of pure rolling condition: T | | < µ, (34) N where µ is the coefficient of sliding friction. Materials 2020, 13, 5032 12 of 25

2.3. ExperimentalMaterials 2020, 13 Method, x FOR PEER REVIEW 12 of 25

2.3.1.2.3. Experimental Experimental DeviceMethod

The2.3.1. principle Experimental scheme Device of the experimental device presented in Figure1 is detailed in Figure9. Since the pure rolling condition is fulfilled, between the precession angle ψ and the intrinsic rotation The principle scheme of the experimental device presented in Figure 1 is detailed in Figure 9. angleSinceϕ, a relationthe pure existsrolling (Equation condition is (19)), fulfilled, and thebetween position the precession of the pendulum angle ψ willand bethe completely intrinsic rotation described by anyangle of theφ, atwo relation parameters. exists (Equation From an(19)), experimental and the position point of of the view, pendulum the precession will be completely angle is more convenientdescribed to beby any measured of the two (between parameters. the fixedFrom axisan exxperimental1 and mobile point axis of view,x2) while the precession the intrinsic angle rotation is anglemore is measured convenient between to be measured two mobile (between axes the (x3 fixedand xaxis0). Fromx1 and Figure mobile9 ,axis it is x noticed2) while the that intrinsic the axes x1 and xrotationare normal angle is to measured the axis betweenz z .two The mobile same axesψ angle (x3 and is x’ formed). From byFigure the 9, two it is planes noticed made that the by the 2 1 ≡ 2 axis zaxes zx1 and x the2 are axis normalx and to thex axis, respectively; z1≡z2. The same see ψ Figure angle 9is. formed The ball, by 5,the oscillates two planes on made the plateby the to be 1 ≡ 2 1 2 tested,axis 3; z a1≡ rubberz2 and the plate, axis 8, x1 is and positioned x2, respectively; with the see edge Figure to 9. the The center ball, of5, oscillates the fulcrum, on theO .plate A rectangular to be tested, 3; a rubber plate, 8, is positioned with the edge to the center of the fulcrum, O. A rectangular prismatic steel body, 9, is positioned on top of the rubber plate. A protractor, 7, is attached to the body, prismatic steel body, 9, is positioned on top of the rubber plate. A protractor, 7, is attached to the 9, by means of two magnets, 10. The magnets permit positioning the protractor, 7, both in the plane body, 9, by means of two magnets, 10. The magnets permit positioning the protractor, 7, both in the parallelplane to theparallel plate, to 3,the and plate, on 3, the and normal on the directionnormal direction to the plate,to the forplate, a minimum for a minimum distance distance from from the wire, 6, andthe for wire, obtaining 6, and thefor obtaining actual size the of actual the ψ sizeangle. of the The ψ oscillations angle. The oscillations of the wire of are the filmed wire are with filmed a camera, 11. Thewith actual a camera, device 11. isThe shown actual in device Figure is shown10. in Figure 10.

Materials 2020, 13, x FOR PEERFigure REVIEWFigure 9. 9. Scheme Scheme for for measurementmeasurement of of precession precession angle. angle. 13 of 25

Figure 10. Experimental device. Figure 10. Experimental device. The steps made for an experimental test are as follows: • The plate to be tested, 3, is positioned on top of the duralumin plate, 1; • The tilting angle of the device to the imposed angle α is obtained by moving the adjusting cylinder, 5 (the cylinder is moved until the short mobile edge of the plate, 1, reaches the height h=l∙sinα); • The ball, 5, is removed from the equilibrium position by a small ψ0 angle (less than 15deg) and let to oscillate freely while rolling over the plate, 3; it is observed that the angular amplitude of the oscillation decreases in time and finding the manner of the angular amplitude damp in time is the goal of the test; • The displacement of the wire, 6, with respect to the protractor, 7, is filmed using a camera, 11, focused on the region where the distance of wire–protractor is minimum; • The film is transferred to a computer and the images are analyzed frame by frame to obtain the values of the extreme elongations and the instants when they occur; • In the first stage, these experimental values for time and amplitude were measured manually by a human operator, using QuickTime software; this step was time-consuming and tedious; • To overcome this inconvenience, an image analysis code was developed (presented in Section 2.3.2) and thus the position of the wire, the times and the extreme angular elongations could be accurately found. The last step concerns obtaining the value of the coefficient of rolling friction sr from the experimental data. The proposed method must be validated by comparison to other results from the literature. Since most of the papers present results for the hypothesis of proportionality between the rolling friction torque and normal force, we accepted the same hypothesis for comparing the results. On the same graph, there were plotted both the experimental data for the variation in time of angular amplitude and the theoretical plot of the equation of motion, corresponding to different values of the theoretical coefficient of rolling friction. The solution that best interpolates the experimental data was kept as the value of the experimental coefficient of rolling friction.

2.3.2. Software for Elongation Measurement Using Automatic Image Processing Since the methodology for angular amplitude estimation by a human operator requires cumbersome work, it was decided that a computer program that uses image processing techniques should be written to estimate the angular elongations of the conical pendulum. The software processes the video and outputs a file with the values of time t and angular elongations at the t instant. The file format is CSV to be straightforwardly used by spreadsheet applications. The software accomplishes the following processing steps:

Materials 2020, 13, 5032 13 of 25

The steps made for an experimental test are as follows:

The plate to be tested, 3, is positioned on top of the duralumin plate, 1; • The tilting angle of the device to the imposed angle α is obtained by moving the adjusting cylinder, • 5 (the cylinder is moved until the short mobile edge of the plate, 1, reaches the height h = l sinα); · The ball, 5, is removed from the equilibrium position by a small ψ angle (less than 15deg) and let • 0 to oscillate freely while rolling over the plate, 3; it is observed that the angular amplitude of the oscillation decreases in time and finding the manner of the angular amplitude damp in time is the goal of the test; The displacement of the wire, 6, with respect to the protractor, 7, is filmed using a camera, 11, • focused on the region where the distance of wire–protractor is minimum; The film is transferred to a computer and the images are analyzed frame by frame to obtain the • values of the extreme elongations and the instants when they occur; In the first stage, these experimental values for time and amplitude were measured manually by a • human operator, using QuickTime software; this step was time-consuming and tedious; To overcome this inconvenience, an image analysis code was developed (presented in Section 2.3.2) • and thus the position of the wire, the times and the extreme angular elongations could be accurately found.

The last step concerns obtaining the value of the coefficient of rolling friction sr from the experimental data. The proposed method must be validated by comparison to other results from the literature. Since most of the papers present results for the hypothesis of proportionality between the rolling friction torque and normal force, we accepted the same hypothesis for comparing the results. On the same graph, there were plotted both the experimental data for the variation in time of angular amplitude and the theoretical plot of the equation of motion, corresponding to different values of the theoretical coefficient of rolling friction. The solution that best interpolates the experimental data was kept as the value of the experimental coefficient of rolling friction.

2.3.2. Software for Elongation Measurement Using Automatic Image Processing Since the methodology for angular amplitude estimation by a human operator requires cumbersome work, it was decided that a computer program that uses image processing techniques should be written to estimate the angular elongations of the conical pendulum. The software processes the video and outputs a ﬁle with the values of time t and angular elongations at the t instant. The ﬁle format is CSV to be straightforwardly used by spreadsheet applications. The software accomplishes the following processing steps: Splits the video into constituent frames. For each frame, it performs the following operations:

detects the moving object, i.e., the wire, partitions it into zones of interest(rectangles) and surrounds • the pixels of the wire with a red quadrangle (as shown in Figure 11); due to binarization of image, it is possible that the wire is represented by several disconnected portions, so a number of morphological operations of dilation and erosion are executed; in order to establish the orientation of the wire, the coeﬃcients of the regression line passing • through the pixels cloud of the wire image in the frame (located in the red polygon on Figure5) will be determined; these coeﬃcients will be computed based on the coordinates of the wire pixels within the screen (xi, yi); thus, having the equation of the regression line,

yi = A xi + B (35) the angular elongation relative to the screen is determined. Materials 2020, 13, x FOR PEER REVIEW 14 of 25

Materials 2020, 13, x FOR PEER REVIEW 14 of 25

Splits the video into constituent frames. For each frame, it performs the following operations: • detects the moving object, i.e., the wire, partitions it into zones of interest(rectangles) and surrounds the pixels of the wire with a red quadrangle (as shown in Figure 11); due to binarization of image, it is possible that the wire is represented by several disconnected portions, so a number of morphological operations of dilation and erosion are executed; • in order to establish the orientation of the wire, the coefficients of the regression line passing through the pixels cloud of the wire image in the frame (located in the red polygon on Figure 5) will be determined; these coefficients will be computed based on the coordinates of the wire pixels within the screen (xi, yi); thus, having the equation of the regression line, = ⋅ + yi A xi B , (35) Figure 11. Experimental measurement of the angular amplitude of the pendulum. the angularFigure elongation 11. Experimental relative to the measurement screen is determined. of the angular amplitude of the pendulum. Subsequently,The following all information values in theis recorded file are scaledin the CSV considering file: the time the relativein seconds orientation (with three of decimals) the wire in Thetaken following from the timestamp information of the is recorded analyzed inframe the and CSV the file: angle the in time radians in seconds relative (withto the threeOX axis decimals) of its equilibrium position. The image processing program detects the position of the wire for each takenthe from frame. the timestamp of the analyzed frame and the angle in radians relative to the OX axis of theframe frame. of the video. To this purpose, the software uses the OpenCV [27] library and the JavaCV [28] interface required to achieve the link between the program written in Java code and the OpenCV Subsequently, all values in the file are scaled considering the relative orientation of the wire in library. In Figure 12, there are some values marked with red circles that are erroneous from the its equilibrium position. The image processing program detects the position of the wire for each estimation point of view. These values prove the limits of the method, which is influenced by the frame of the video. To this purpose, the software uses the OpenCV [27] library and the JavaCV [28] quality of the video. interface required to achieve the link between the program written in Java code and the OpenCV library. In Figure 12, there are some values marked with red circles that are erroneous from the estimation point of view. These values prove the limits of the method, which is influenced by the quality of the video. The following approach was adopted to correct such values:

the program establishes for each frame the value of the angular elongation after image processing; • an estimated value is also determined depending on the value from the previous frame; if the • diﬀerence in absolute value between the established value and the estimated one is greater than a

certain threshold, then the automatic processing stops at this frame, which is displayed together with the graphicalFigure 11. representation Experimental measurement of the two of lines the an (thegular regression amplitude of one the and pendulum. the one corresponding to the estimated value); Subsequently, all values in the file are scaled considering the relative orientation of the wire in the operator examines the frame at which the processing stopped; if there is a discrepancy • its equilibrium position. The image processing program detects the position of the wire for each betweenframeFigure of the12. the Plotvideo. orientation of theTo thisangular of purpose, the elongation wire the and software variation the calculated uses versus the time OpenCV regression as resulted [27] line, libra in the thery firstand operator stagethe JavaCV of can image manually [28] selectinterfaceprocessing two required marks program to by achieveelaboration. clicking the on link the between image ofthe the program wire, written and the in program Java code calculates and the OpenCV the angular elongationlibrary. In Figure for this 12, frame there basedare some on values the line mark passinged with through red circles the twothat chosenare erroneous marks from and resumesthe theestimation automatic point processing. of view. These values prove the limits of the method, which is influenced by the quality of the video.

FigureFigure 12. Plot 12. Plot of the of the angular angular elongation elongation variationvariation versus time time as as resulted resulted in the in thefirst ﬁrst stage stage of image of image processingprocessing program program elaboration. elaboration.

Materials 2020, 13, 5032 15 of 25

This methodology was chosen as it was considered preferable to work with unprocessed data. The filtering algorithm for the calculus of estimated value was used only to decide to stop the video at a frame where there was a noticeable deviation from the accepted threshold. For the completed experiments, the operator’s involvement is reduced to 10–20 frames, mainly where the image of the wire is less clear. These are small in number compared to the thousands of frames which should otherwise be estimated manually. The protractor was initially used for the manual reading, frame by frame, of angular elongation, by a human operator. In the automatic image processing for the detection of the wire, the image of the protractor is not taken into consideration, since the moving object is analyzed and the protractor is immobile. However, the protractor was used to validate the results produced by the computer program. Subsequently, when an acceptable degree of precision of the delivered results was proven, the protractor was ignored. Using the data from the CSV file which contains the values of angular elongations and the corresponding instants, the time moments of maximum elongation and the values of angular amplitudes for each oscillation can be easily identified. This program that has at the input a video clip with the oscillating motion of the pendulum and whose output is a CSV file containing the angular elongations and various time moments was executed on a laptop with microprocessor Intel (R) Core (TM) i7-8550U CPU @ 1.80GHz, and 8M RAM, running the Windows operating system.

2.4. Experimental Results The materials used for the method are a bearing steel ball (radius r = 5/4” = 0.03175 m) and a plate to be tested, TP. For the ﬁrst tests, a 6 mm thick glass plate was used. In order to obtain a well-controlled inclination of the plate, a much longer aluminum plate, BP, of length l = 1.500 m was used to support it. A cylindrical part, P, was placed beneath the aluminum base and the level diﬀerence between the two ends of the plate was h = 0.142 m. Consequently, the angle of inclination (Figure 13) is well-controlled and has the following value: 0.142 α = a sin = 5.432◦, (36) Materials 2020, 13, x FOR PEER REVIEW 1.500 16 of 25

Figure 13. The angle of the inclined plane—truthful valuation. Figure 13. The angle of the inclined plane—truthful valuation. The length of the pendulum is denoted by d. Tests were performed for three didifferfferentent valuesvalues ofof thethe pendulumpendulum lengthlength d.d. The d parameter is difficultdifficult to size, so so the the distance distance L L between between the the fulcrum fulcrum and and the the ball-glass ball-glass contact contact point point is is measured, measured, thesethese two two points being reachable. Then, Then, the the simple simple relation relation is used for finding finding the d length: d =p L2 + r 2 (37) d = L2 + r2. (37) The values found for d for the three experimental tests are as follows: d=0.140m; d=0.195m; d=0.240mThe. values The experimental found for d forvalues the of three maximum experimental oscillation tests amplitudes are as follows: are representedd = 0.140 m; graphically.d = 0.195 m; Ind = Figure0.240 m14a,. The there experimental are plotted values the results of maximum for the first oscillation test (d=0.140m amplitudes). Establishing are represented the equilibrium graphically. position of the pendulum is a major issue of the method. To solve it, the following function is created: n ψ = ψ −ψ 2 F( ) ( k ) . (38) k =1

The searched equilibrium position corresponds to the value ψ0 that gives the minimum of the function. The value ψ0 results from the following equation: dF(ψ ) = 0 , ψ (39) d ψ =ψ o where n represents the number of experimental points. Now, with known value ψ0 for the position of equilibrium, the decrease in the angular amplitude of the pendulum can be represented, as in Figure 14b.

(a) (b)

Materials 2020, 13, x FOR PEER REVIEW 16 of 25

Figure 13. The angle of the inclined plane—truthful valuation.

Tests were performed for three different values of the pendulum length d. The d parameter is difficult to size, so the distance L between the fulcrum and the ball-glass contact point is measured, these two points being reachable. Then, the simple relation is used for finding the d length:

d = L2 + r 2 (37) The values found for d for the three experimental tests are as follows: d=0.140m; d=0.195m; d=0.240m. The experimental values of maximum oscillation amplitudes are represented graphically. In Figure 14a, there are plotted the results for the first test (d=0.140m). Establishing the equilibrium position of the pendulum is a major issue of the method. To solve it, the following function is created: n ψ = ψ −ψ 2 F( ) ( k ) . (38) k =1 Materials 2020, 13, 5032 16 of 25 The searched equilibrium position corresponds to the value ψ0 that gives the minimum of the function. The value ψ0 results from the following equation: In Figure 14a, there are plotted the results fordF the(ψ ) ﬁrst test (d = 0.140 m). Establishing the equilibrium = 0 , position of the pendulum is a major issue of theψ method. To solve it, the following function is created:(39) d ψ =ψ o n where n represents the number of experimental Xpoints. Now, with known value ψ0 for the position F(ψ) = (ψ ψ)2. (38) of equilibrium, the decrease in the angular amplitudek −of the pendulum can be represented, as in k=1 Figure 14b.

(a) (b)

Figure 14. Experimental evidence of the decrease in angular amplitude of the pendulum: (a) initial values; (b) with respect to the equilibrium position.

The searched equilibrium position corresponds to the value ψ0 that gives the minimum of the function. The value ψ0 results from the following equation:

dF(ψ) = 0, (39) dψ ψ=ψ0 where n represents the number of experimental points. Now, with known value ψ0 for the position of equilibrium, the decrease in the angular amplitude of the pendulum can be represented, as in Figure 14b. Next, taking β = 1, the value of sr is found imposing the condition that the theoretical curve passes as closely as possible to the experimental points. Figure 15 presents the experimental and theoretical data and their excellent superimposition. From Figure 15, it can be observed that the amplitude of the theoretical signal decreases linearly with time, and this tendency is supported by the experimental data. Dzhiladvari [16] found the same dependency for the final stage of oscillations. On the same plot, there were represented the two regression lines that interpolate the experimental data. These lines are also the envelopes of the theoretical signal. In Figure 15, the theoretical signal is shown for the coefficient of rolling friction 6 sr = 3.2 10 m. It should be mentioned that the method is strongly dependent on the coefficient sr, · − as seen in Figure 16, where the values were modified in addition and subtraction with small quantities and the plots (blue) detach from the regression lines (black dash). Materials 2020, 13, x FOR PEER REVIEW 17 of 25

Figure 14. Experimental evidence of the decrease in angular amplitude of the pendulum: (a) initial values; (b) with respect to the equilibrium position.

Next, taking β=1, the value of sr is found imposing the condition that the theoretical curve passes as closely as possible to the experimental points. Figure 15 presents the experimental and theoretical data and their excellent superimposition.

Materials 2020, 13, x FOR PEER REVIEW 17 of 25

Figure 14. Experimental evidence of the decrease in angular amplitude of the pendulum: (a) initial values; (b) with respect to the equilibrium position.

Figure 15. Theoretical angular elongation and values of experimental angular amplitude.

From Figure 15, it can be observed that the amplitude of the theoretical signal decreases linearly with time, and this tendency is supported by the experimental data. Dzhiladvari [16] found the same dependency for the final stage of oscillations. On the same plot, there were represented the two regression lines that interpolate the experimental data. These lines are also the envelopes of the theoretical signal. In Figure 15, the theoretical signal is shown for the coefficient of rolling friction sr=3.2∙10-6m. It should be mentioned that the method is strongly dependent on the coefficient sr, as seen in Figure 16, where the values were modified in addition and subtraction with small quantities

and the plotsFigure (blue) 15. Theoreticaldetach from angular the regression elongation lines and values(black ofdash). experimental angular amplitude. Figure 15. Theoretical angular elongation and values of experimental angular amplitude.

(b) (a)

Figure 16. Theoretical signals for different values of the dimensional coefficient of rolling friction: (a) 6 6 Theoretical signal for sr = 3.7 10 m; (b) Theoretical signal for sr = 2.7 10 m. × − × − The variation of the normal force is presented in Figure 17 and the extreme and mean values are also given. The normal force can be considered approximately constant because it was demonstrated that the maximum variation of the normal force during the oscillation process is 0.054% from the medium value. From Equation (28), it results that the rolling friction torque will have a similar behavior, regardless of the value of the β exponent. The ratio between the friction force and the normal force T/N is represented in Figure 18 in order to estimate the validity of the pure rolling(b) condition, expressed (a) by Equation (34). From the plot, it is remarked that the values of the ratio are one hundred times smaller than the values of the coefficient of sliding friction. This is an important feature of the conical pendulum compared to the planar pendula, [17,18] for which, for amplitude values greater than 15◦..20◦, the sliding friction replaces the rolling friction. MaterialsMaterials 2020 2020, ,13 13, ,x x FOR FOR PEER PEER REVIEW REVIEW 1818 of of 25 25

FigureFigure 16. 16. Theoretical Theoretical signals signals for for different different values values of of the the dimensional dimensional coefficient coefficient of of rolling rolling friction: friction: ( a(a) ) -6 -6 TheoreticalTheoretical signal signal for for s rs=3.7r=3.7∙10∙10-6mm; ;( b(b) )Theoretical Theoretical signal signal for for s rs=2.7r=2.7∙10∙10-6mm. .

TheThe variation variation of of the the normal normal force force is is presented presented in in Figure Figure 17 17 and and the the extreme extreme and and mean mean values values are are alsoalso given. given. The The normal normal force force can can be be considered considered a approximatelypproximately constant constant because because it it was was demonstrated demonstrated thatthat the the maximum maximum variation variation of of the the normal normal force force during during the the oscillation oscillation process process is is 0.054% 0.054% from from the the mediummedium value.value. FromFrom EquationEquation (28),(28), itit resultsresults thatthat thethe rollingrolling frictionfriction torquetorque willwill havehave aa similarsimilar behavior,behavior, regardless regardless of of the the value value of of the the β β exponent. exponent. The The ratio ratio between between the the friction friction force force and and the the normalnormal force force T/N T/N is is represented represented in in Figure Figure 18 18 in in order order to to estimate estimate the the validity validity of of the the pure pure rolling rolling condition,condition, expressed expressed by by Equation Equation (34). (34). From From the the plot plot, ,it it is is remarked remarked that that the the values values of of the the ratio ratio are are oneone hundred hundred times times smaller smaller than than the the values values of of the the coefficient coefficient of of sliding sliding friction. friction. This This is is an an important important Materialsfeaturefeature2020, 13of of, 5032the the conical conical pendulum pendulum compared compared to to the the planar planar pendula, pendula, [17,18] [17,18] for for which, which, for for amplitude amplitude18 of 25 valuesvalues greater greater than than 15°..20° 15°..20°, ,the the sliding sliding friction friction replaces replaces the the rolling rolling friction. friction.

(a(a) ) (b(b) )

FigureFigure 17.FigureThe 17. 17. variation The The variation variation of the of of normalthe the normal normal force: force: force: (a )( a( plota) )plot plot of of of normal normal normal forceforce force variation; variation; ( b (b) )Mathcad Mathcad Mathcad results results results for for extreme and mean values of normal force extreme and mean values of normalfor extreme force. and mean values of normal force

FigureFigureFigure 18. 18. The18. The The ratio ratio ratio between between between frictionfriction friction force forceforce and and normal normal normal force force force ( T/N(T/N (T)./).N ).

In TableInIn TableTable1, there 1,1, therethere are are presentedare presentedpresented the thethe values valuesvalues obtained obtaobtainedined forforfor the thethe coefficients coecoefficientsfficients ofof rolling ofrolling rolling friction,friction, friction, dimensionaldimensionaldimensionalsr [13 s rs, r17[13,17,29–33] [13,17,29–33],29–33] and and and dimensionless dimensionless dimensionless µμ μrr r[34,35]:[ [34,35]:34,35]: μμ ==s / r rr srr / r (40)(40) µr = sr/r (40) between a steel bearing ball (r = 0.03175 m) and different materials: plates made of quenched and tempered steel, glass, aluminum, copper, polycarbonate, carbon fiber composite and glass fiber fabric covered glass plate. The parameter a is the modulus of the slope of the straight lines which envelope the theoretical signal.

Table 1. Values of rolling friction characteristics for a steel ball rolling over diﬀerent materials.

s 106 µ 103 a 103 Material of the Plate r r (m)· (-)· (rad· /s) Steel 4.508 0.142 1.656 Glass 3.714 0.117 1.556 Aluminum 4.445 0.140 1.811 Copper 18.986 0.598 7.587 Polycarbonate 18.986 0.598 7.823 Carbon ﬁber composite 5.810 0.183 2.189 Glass ﬁber fabric over glass plate 53.023 1.670 22.65 Materials 2020, 13, x FOR PEER REVIEW 19 of 25

between a steel bearing ball (r = 0.03175m) and different materials: plates made of quenched and tempered steel, glass, aluminum, copper, polycarbonate, carbon fiber composite and glass fiber fabric covered glass plate. The parameter a is the modulus of the slope of the straight lines which envelope the theoretical signal.

Table 1. Values of rolling friction characteristics for a steel ball rolling over different materials.

6 3 3 s ⋅10 μ ⋅10 a ⋅10 Material of the Plate r r (m) (–) (rad/s) Steel 4.508 0.142 1.656 Glass 3.714 0.117 1.556 Aluminum 4.445 0.140 1.811 Copper 18.986 0.598 7.587 Polycarbonate 18.986 0.598 7.823 Carbon fiber composite 5.810 0.183 2.189 Materials 2020, 13, 5032Glass fiber fabric over glass plate 53.023 1.670 22.65 19 of 25

Figure 19 presents the dependency of the coefficient of rolling friction μr on the modulus of the Figure 19 presents the dependency of the coefficient of rolling friction µr on the modulus of theslope slope a ofa ofthe the straight straight line line enveloping enveloping the the theoreti theoreticalcal signal. signal. The The plot plot shows shows the proportionalityproportionality betweenbetween parameters.parameters. ThisThis resultresult substantiallysubstantially simplifies simplifies the the methodology methodology of of finding finding the the coe coefficientfficient of rollingof rolling friction: friction: practically, practically, for thefor samethe same test devicetest device geometrical geometrical parameters parameters (inclination (inclination of the of plate, the ballplate, radius, ball radius, pendulum pendulum length), twoleng pointsth), two are points sufficient arefor sufficient representing for representing the plot. For athe material plot. For to be a tested,material the to experimental be tested, the data experimental obtained fromdata obtained oscillations from of theoscillations pendulum of th (thee pendulum slope of the (the regression slope of the regression line of angular amplitudes) together with the μr(a) plot permit us to obtain the value line of angular amplitudes) together with the µr(a) plot permit us to obtain the value of the rolling frictionof the rolling coefficient. friction This coefficient. direct methodology This direct avoidsmethodology the guess avoids steps the used guess in integration steps used of in the integration dynamic Equationof the dynamic (33). Equation (33).

1.8

1.6

1.4

1.2 3 μ ⋅10 1 rk 0.8

0.6

0.4

0.2

0 024681012141618202224 a k Figure 19. The coefficient coefficient of rolling friction versus thethe slopeslope ofof thethe envelopeenvelope straightstraight line.line. 3. Discussions 3. Discussions The commercial tribometers are dedicated to the estimation of friction and wear, the measurements The commercial tribometers are dedicated to the estimation of friction and wear, the being based on the pin-on disc scheme, where sliding friction and wear are evaluated [36]. Even for measurements being based on the pin-on disc scheme, where sliding friction and wear are evaluated cases where standard methodology and testers exists, there are scientists who prefer to design and [36]. Even for cases where standard methodology and testers exists, there are scientists who prefer to build new equipment for tribological tests [37]. Concerning the rolling friction, only for the tire–road design and build new equipment for tribological tests [37]. Concerning the rolling friction, only for pair there are standard tests [38,39]—that is, for compliant materials which exhibit large deformations. the tire–road pair there are standard tests [38,39]—that is, for compliant materials which exhibit For the pairs of materials frequently met in mechanical applications which present important normal large deformations. For the pairs of materials frequently met in mechanical applications which contact stiffness (steel, glass, carbon composites, plastics, etc.), there are no standard methods and instruments for determining the parameters characteristic to rolling resistance. Therefore, over time, a series of methods and devices have been proposed by researchers [32–35,40,41]. Our work joins this trend for finding a simple, rapid and inexpensive method and device for estimating a parameter that characterizes the rolling resistance between two different materials. The equation of motion is deduced for the spatial problem. The paper presents a method for the estimation of the rolling friction torque, based on the experimental study of the oscillatory motion of a conical pendulum. The conical pendulum is made of a metallic ball attached to an inextensible wire. The ball rests on a flat surface inclined with respect to the horizontal direction. The conical pendulum is obtained when the end of the wire is fixed in a point from the surface. Attaching the end of the wire to the plane causes the spinning moment between the ball and the plane to be zero. When the ball is removed from the equilibrium position and then set free, it oscillates. For small initial amplitudes, the motion of the ball with respect to the plane is pure rolling. The motion of the ball can be entirely determined by the angle between the current position of the wire and the equilibrium position. Materials 2020, 13, 5032 20 of 25

For the motion of a rigid body with a fixed point, Euler’s equations must be used. Applying Euler’s equations for the conical pendulum, there are found the positional parameters (precession angle and intrinsic rotation angle) and the expressions of normal and tangential reaction forces acting in the point of contact. The pure rolling condition is expressed as proportionality between the two angles and allows for complete solving of the dynamics of the pendulum. Especially important is the dependency of the reaction forces on the precessional angle, namely the lack of the second derivative of this parameter. (It should be mentioned that for planar pendula, this second derivative occurs in expressions of the reaction forces, as shown in the planar cycloidal pendulum). This is an important advantage because one can accept any dependency between the moment of rolling friction and normal reaction without obtaining intricate expressions from which the second derivative cannot be analytically solved, which is a requirement of the numerical methodology of Runge–Kutta for the integration of the equation of motion. Additionally, from the analysis of the equation of motion, it is noticed that the moment of rolling friction appears explicitly in the expression of the angular acceleration of the pendulum. Thus, the theoretical model becomes a useful theoretical instrument for future research. One can accept any expression for the rolling friction moment as function of velocity, position and time and, finally, the constants from the expression are found from the condition of optimum interpolation of experimental data with the theoretical signal resulting from the theoretical model. The rolling friction torque was assumed to depend on the normal force obeying a power law function. The hypothesis of proportionality between the rolling friction torque and normal force was accepted as encountered in other works [42–45] and confirmed by experimental results. In the literature, the dimensionless coefficient of rolling friction is widely met [34,35,40–43], but it is related to the dimensional one by the radius of the ball. The motion of the center of the ball is a damped one. A video camera records the motion of the wire with respect to a protractor, and afterward, using software, the moments of the wire’s maximum angular elongation are found. In the first stage, these instances were established by human operators, which was a tedious task. To overcome this drawback, a computer code was written in Java to analyze each frame of the film with the oscillating motion of the conical pendulum. By applying digital image processing techniques, the angular elongations were determined at various times. This approach for measuring the angular elongation considerably reduced the effort of the human operator and made possible the running of more experiments. The accuracy of the measurements was satisfactory for the study. The program detects the few possible errors caused by the poor quality of some frames and prompts the operator’s decision. The experimental data are represented graphically, where it is observed that the decrease in the wire’s angular amplitude is linear, as predicted by the theoretical model. Several experiments are made to find the values of the coefficient of rolling friction between a steel ball and a plate made of a different material (glass, copper, aluminum, polycarbonate, etc.) and the experimental data are confirmed by a plot of the theoretical elongation of the wire. The theoretical model is especially sensitive to the variation of the coefficient of rolling friction. The validation of the method was performed by testing the steel–steel pair of materials, widely used in the rolling bearing industry. The studies on the subject consider numerous parameters (radii of rolling bodies, elastic characteristics, surface roughness, adhesions, thermal treatment, etc.) and the variety of the values of these parameters makes the comparison possible as qualitative aspect. For instance, the steel–steel pair may present values of 0.002 for untreated steel [35] or 0.0002–0.0004 for thrust bearing parts [32], in good agreement with the results of the present work, 0.000147. Tests for steel–polycarbonate pairs of materials achieved the value of 0.00075 in [33], which is comparable to the value of 0.0006 achieved in the present tests. An interesting observation was revealed after plotting the dependence between the coefficient of rolling friction and the slope of the enveloping lines of the theoretical signal which interpolates the experimental data. The modulus of the slope of the enveloping straight line is proportional to the coefficient of rolling friction and this can simplify the methodology of finding the coefficient of rolling friction between two materials. In fact, two points from the plot obtained via the initial Materials 2020, 13, 5032 21 of 25 methodology are sufficient for obtaining the constant of proportionality of the device. Subsequently, for any other material used as plate from the ball–plate pair, from the experimental data, the slope of the regression line concerning the decrease with time in the angular amplitude of the pendulum is enough for calculating the coefficient of rolling friction.

4. Conclusions The coefficient of rolling friction is an important parameter influencing the tribological behavior of mechanical systems. The coefficient of rolling friction is a consequence of the deformations of the bodies in the contact region. The majority of the studies concerning the rolling friction discuss the materials used for vehicle tires. For these materials, the deformations for nominal forces are large and comparable to the dimensions of the contacting bodies, and finding the deformations is not a difficult task. For other materials used in engineering (steel, iron, metallic materials, glass, a range of composites, etc.) which are less compliant, finding the coefficient of rolling friction is a difficult objective. Actually, due to the complexity of the subject, there is not yet a parameter or methodology widely recognized for the estimation of this parameter. The present work proposes a simple, inexpensive, quick method for experimental measurement of the coefficient of rolling friction between a steel ball and a plate, made of similar or dissimilar materials. Two main stages must be carried out for obtaining the coefficient of rolling friction. In the first step, a conical pendulum is constructed using the ball and the inclined plate and it is observed that the oscillations are damped. The experiments show that the damping with time of the oscillations amplitudes is a linear one. This operation was initially carried out by a human operator by direct observation of the motion of the pendulum’s wire with respect to a protractor. A computer code was written in Java to analyze each frame of the film with the oscillating motion of the conical pendulum. The program detects the few possible errors caused by the poor quality of some frames and prompts the operator’s decision. In the second stage, the theoretical law of oscillation of the pendulum is found using a theoretical model, depending on the coefficient of rolling friction. From the condition that the experimental data and the theoretical signal must superpose, the actual value of the coefficient of rolling friction is found. In order to validate the method, the hypothesis of the model is that the moment of rolling friction is proportional to the normal force from the contact. Thus, the results from our work are in agreement with the values presented in other papers in the literature. It is confirmed that for less compliant bodies, the coefficient of rolling friction can be even 10,000 times smaller than the characteristic dimensions of the bodies and the difficulty of estimating this parameter is evidenced.

Author Contributions: Conceptualization, S.A.; methodology, S.A.; software, S, .-G.P.; investigation, S.A. and

S, .-G.P.; writing—review and editing, S.A and I.M.; visualization, S.A. and S, .-G.P.; supervision, I.M. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A In order to estimate the rigidity of the wire, the torsion pendulum method is applied. The torsion pendulum is made with the same ball and the same wire used for the conical pendulum; see Figure A1. By denoting the torsional rigidity of the wire kt, when the pendulum rotates with a ϕ angle with respect to the equilibrium position, the wire is subjected to the moment of torsion Mt = ktϕ that opposes the torsion of the wire. Materials 2020, 13, x FOR PEER REVIEW 22 of 25

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix In order to estimate the rigidity of the wire, the torsion pendulum method is applied. The torsion pendulum is made with the same ball and the same wire used for the conical pendulum; see Figure A1. By denoting the torsional rigidity of the wire kt, when the pendulum rotates with a φ angle with respect to the equilibrium position, the wire is subjected to the moment of torsion Mt=ktφ that opposes the torsion of the wire. Materials 2020, 13, 5032 22 of 25

= ϕ M t kt

J z

FigureFigure A1. A1. TorsionalTorsional pendulum. pendulum.

TheThe equation equation of of motion motion of of the the pendulum pendulum is J.. ϕ = −M , Jzϕz = Mt,t (A1) (A1) − or or J.. ϕ + k ϕ = 0 Jzϕz+ ktϕt = 0.. (A2) (A2) TheThe equation equation is is written written under under the the following following form: form: ϕ + kt ϕ = ..  kt 0 . (A3) ϕ + Jϕz = 0. (A3) Jz The torsional oscillation pulsation is ω and is given by the following ratio: The torsional oscillation pulsation is ω and is given by the following ratio: π 2 kt = ω 2 = 4 2 , (A4) ktJ 4Tπ =z ω2 = , (A4) J T2 where T is the torsional oscillation period. Itz results in the following: where T is the torsional oscillation period. It resultsπ in2 the following: = 4 kt J z . (A5) T 2 4π2 kt = Jz (A5) The moment of inertia of the ball Jz is known (EquationT2 (A4)) and for finding the kt constant, the period T must be obtained by experimental means. The device is presented in Figure A2: the end of the wireThe is moment attached of to inertia the mobile of the carrier, ball Jz 1,is whic knownh moves (Equation vertically (A4)) and and is for fixed finding by the the rosette,kt constant, 2. A thrustthe period bearing, T must 3, is be placed obtained on the by board experimental of the devi means.ce, coaxial The device to the iswire, presented 4. After in the Figure ball, A25, takes: the endthe restof the position, wire is it attached is slowly to lowered the mobile till it carrier, contacts 1, the which upper moves ring verticallyof the bearing. and is Next, fixed the by upper the rosette, ring of 2. theA thrust bearing bearing, is rotated 3, is placeda certain on number the board of of rotations. the device, The coaxial rotation to the motion wire, 4.is filmed After the with ball, a 5,camera, takes the 7. Therest carrier position, is gradually it is slowly raised lowered until till the it contactsball detaches the upper from ringthe ring. of the The bearing. ball starts Next, to the rotate upper about ring the of the bearing is rotated a certain number of rotations. The rotation motion is filmed with a camera, 7. The carrier is gradually raised until the ball detaches from the ring. The ball starts to rotate about the wire and the mark, 6, glued to the superior part of the ball helps in identifying the rotations. The film is used to find the value of the torsional oscillation period. 4 2 For Jz = 4.217 10− kg m and a wire length of 0.2 m, the experimental torsional oscillation period × · 7 is T = 205.3 s and the resulting value of rigidity constant is kt = 3.95 10 N m/rad. For an initial × − · precession amplitude ψ0 = 15 deg, an amplitude of self-rotation ϕ0 = 94.5 deg = 1.65 rad corresponds. 7 This maximum angle of torsion of the wire gives the moment of torsion Mtmax = ktϕ = 6.51 10 N m. 0 × − · For the conical pendulum, the smallest value of rolling friction was obtained for the steel–glass pair, 6 sr = 4.5 10 m, and the corresponding rolling friction torque calculated according to the proposed × − Materials 2020, 13, 5032 23 of 25 Materials 2020, 13, x FOR PEER REVIEW 23 of 25

5 wiremodel and is theMr =mark,srmg 6,= 4.61glued to10 theN superiorm. Calculating part of the ratioball helps (Mtmxa in/M identifyingr) 100 = 1.41%, the rotations. it can be concluded The film × − · · isthat used the to e ﬀfindect ofthe the value moment of the of torsional torsion ofoscillation the wire period. is negligible.

FigureFigure A2 A2.. Set-upSet-up for for torsional torsional oscillations. oscillations.

ReferencesFor Jz=4.217∙10-4kg∙m2and a wire length of 0.2m, the experimental torsional oscillation period is T=205.3sec and the resulting value of rigidity constant is kt=3.95∙10-7N∙m/rad. For an initial precession 1. Uicker, J.; Pennock, G.; Shigley, J. The World of Mechanisms. In Theory of Machines and Mechanisms, 5th ed.; amplitude ψ0=15deg, an amplitude of self-rotation φ0=94.5deg=1.65rad corresponds. This maximum Oxford University Press: Oxford, UK, 2003; pp. 3–32. angle of torsion of the wire gives the moment of torsion Mtmax=ktφ0=6.51∙10-7N∙m. For the conical 2. Alaci, S.; Ciornei, C.; Filote, C. Considerations upon a new tripod joint solution. Mechanics 2013, 19, 567–574. pendulum, the smallest value of rolling friction was obtained for the steel–glass pair, sr=4.5∙10-6m, [CrossRef] and3. theJohnson, corresponding K.; Kendall, K.;rolling Roberts, friction A. Surface torque energy calculated and the contact according of elastic to solids. the Proc.proposed R. Soc. model Lond. 1971 is , -5 Mr=srmg=4.61324, 301–313.∙10 N∙m. Calculating the ratio (Mtmxa/Mr)∙100=1.41%, it can be concluded that the effect of4. thePersson, moment B. of Rolling torsion friction of the for wire hard is cylinder negligible. and sphere on viscoelastic solid. Eur. Phys. J. E 2010, 33, 327–333. [CrossRef][PubMed] References5. Zhang, Y.; Wang, X.; Yang, W.; Li, H. A numerical study of the rolling friction between a microsphere and a 1. Uicker,substrate J.; Pennock, considering G.; theShigley, adhesive J. The eﬀ Worldect. J. Phys.of Mechanisms. D Appl. Phys. In Theory2016, 49 of ,Machines 025501. [andCrossRef Mechanisms] , 5th ed.; 6. OxfordHibbeler, University R. Fundamental Press: Oxford, Concepts. UK, 2003; In Fluid pp. Mechanics3–32. ; Pearson Prentice Hall: Boston, MA, USA, 2015; 2. Alaci,pp. 3–45. S.; Ciornei, C.; Filote, C. Considerations upon a new tripod joint solution. Mechanics 2013, 19, 7. 567–574.Hertz, H. Über die berührung fester elastischer Körper (On the contact of rigid elastic solids). In Miscellaneous 3. Johnson,Papers; Jones, K.; Kendall, D.E., Schott, K.; Roberts, G.A., Eds.;A. Surfac J. reinee energy und angewandteand the contact Mathematik of elastic solids. 92; Macmillan: Proc. Roy. London, Soc. London UK, 19711896;, 324 pp., 301–313. 146–162. 4.8. Persson,Johnson, B. K. Rolling Normal friction contact offor elastic hard solids:cylinder Hertz and theory. sphere In onContact viscoelastic Mechanics solid.; Cambridge Eur. Phys. University J. E 2010 Press:, 33, 327–333.Cambridge, doi.10.1140/epje/i2010-10678-y UK, 1985; pp. 84–106. 5.9. Zhang,Hills, D.; Y.; Nowell,Wang, X.; D.; Yang, Sackﬁeld, W.; Li, A. EllipticalH. A numerical contacts. stud InyMechanics of the rolling of Elastic friction Contact between; Butterworth a microsphere Heinemann and aLtd.: substrate Oxford, UK,considering 1993; pp. 291–319.the adhesive effect. J. Phys. D Appl. Phys. 2016, 49, 025501. 10. doi:101088/0022-3727/49/2/025501.Popov, V. Qualitative Treatment of Contact Problems—Normal Contact without Adhesion. In Contact 6. Hibbeler,Mechanics R. and Fundamental Friction, Physical Concepts. Principles In: Fluid and Mechanics Applications; Pearson; Springer: Prentice Berlin Hall:/Heidelberg, Boston, MA, Germany, USA, 2015; 2010; pp.pp. 3–45. 9–23. 7.11. Hertz,Halme, H. J.; Über Andersson, die berührung P. Rolling fester contact elastischer fatigue and Kö wearrper fundamentals(On the contact for rollingof rigid bearing elasticdiagnostics— solids). In: Miscellaneousstate of the art. PapersProc.. Jones Inst. and Mech. Schott, Eng. PartEds.; J J. reine Eng. Tribol.und angewandte2010, 224, 377–393. Mathematik [CrossRef 92: Macmillan,] London, 12. UK,Popp, 1896; K.; pp. Stelter, 146–162. P. Nonlinear Oscillations of Structures Induced by Dry Friction. In Nonlinear Dynamics in 8. Johnson,Engineering K. Normal Systems. co Internationalntact of elastic Union solids: of Theoretical Hertz theory. and Applied In Contact Mechanics Mechanics; Schiehlen,; Cambridge W., Ed.; University Springer: Press:Berlin Cambridge,/Heidelberg, UK, Germany, 1985; pp. 1990. 84–106. [CrossRef ] 9.13. Hills,Mangeron, D.; Nowell, D.; Irimiciuc, D.; Sackfield, N. Dinamica A. Elliptical miscarii absolutecontacts. aIn rigidului. Mechanics In ofMecanica Elastic RigidelorContact; cuButterworth Aplicatii in HeinemannInginerie. Mecanica Ltd.: Oxford, rigidului UK, (in 1993; Romanian) pp. 291–319.; Editura Tehnică: Bucuresti, Romania, 1978; Volume 1, pp. 233–436. J. Appl. Mech. Tech. Phys. 55 10.14. Popov,Cherepanov, V. Qualitative G. Theory Treatment of rolling: of Solution Contact ofProblems—Normal the Coulomb problem. Contact without Adhesion. In 2014Contact, , Mechanics182–189. [andCrossRef Friction,] Physical Principles and Applications; Springer: Berlin, Heidelberg, Germany, 2010; pp. 9–23.

Materials 2020, 13, 5032 24 of 25

15. Dzhilavdari, I.; Riznookaya, N. An Experimental Assessment of the Components of Rolling Friction of Balls at Small Cyclic Displacements. J. Frict. Wear 2008, 29, 330–334. [CrossRef] 16. Dzhilavdari, I.; Riznookaya, N. Measurement of the friction characteristics on materials surfaces using the pendulum microoscillations method. J. Frict. Wear 2007, 28, 446–451. [CrossRef] 17. Alaci, S.; Ciornei, F.; Ciogole, A.; Ciornei, M. Estimation of coefficient of rolling friction by the evolvent pendulum method. IOP Conf. Ser.: Mater. Sci. Eng. 2017, 200, 0122005. [CrossRef] 18. Ciornei, M.; Alaci, S.; Ciornei, F.; Romanu, I. A method for the determination of the coefficient of rolling friction using cycloidal pendulum. IOP Conf. Ser. Mater. Sci. Eng. 2017, 227, 012027. [CrossRef] 19. Ciornei, F.; Pentiuc, R.; Alaci, S.; Romanu, I.; Siretean, S.; Ciornei, M. An improved technique of finding the coefficient of rolling friction by inclined plane method. IOP Conf. Ser. Mater. Sci. Eng. 2019, 514, 012004. [CrossRef] 20. Zaspa, Y. Internal Synthesis of Motion and the Dynamic Characteristics of External Friction. J. Frict. Wear 2011, 32, 167–178. [CrossRef] 21. Spiegel, M. Space motion of rigid bodies. In Theoretical Mechanics: Schaum’s Outline Series; McGraw-Hill Education: New York, NY, USA, 1967; pp. 253–281. 22. Fischer, I. Coordinate transformation. In Dual-Number Methods in Kinematics, Statics and Dynamics; CRC Press: Boca Raton, FL, USA, 1998; pp. 9–40. 23. McCarthy,J.; Soh, G. Spherical kinematics. In Geometric Design of Linkages (Interdisciplinary Applied Mathematics), 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 179–201. 24. O’Reilly, O. The dynamics of rolling disks and sliding disks. Nonlinear Dyn. 1996, 10, 287–305. [CrossRef] 25. Scaraggi, M.; Persson, B. Rolling Friction: Comparison of Analytical Theory with Exact Numerical Results. Tribol. Lett. 2014, 55, 15–21. [CrossRef] 26. Kopchenova, N.; Maron, I. Approximate Solutions of Ordinary Differential Equations. In Computational Mathematics, 3rd ed.; Mir Publishers: Moscow, Russia, 1984; pp. 198–243. 27. OpenCV. Open Source Computer Vision Library. Available online: https://github.com/opencv/opencv (accessed on 19 August 2020). 28. JavaCV. Java interface to OpenCV, FFmpeg, and more. Available online: https://github.com/bytedeco/javacv (accessed on 19 August 2020). 29. Huang, P.; Yang, Q. Theory and contents of frictional mechanics. Friction 2014, 2, 27–39. [CrossRef] 30. Beer, F.; Johnston, R.; Mazurek, D.; Eisenberg, E. Vector Mechanics For. Engineers: Statics, 9th ed.; McGraw-Hill: New York, NY, USA, 2010. 31. Wen, S.; Huang, P. Rolling friction and its applications. In Principles of Tribology, 2nd ed.; Tsinghua University Press: Beijing, China; John Wiley & Sons: Hoboken, NJ, USA, 2018; pp. 252–281. 32. Koczan, G.; Ziomek, J. Research on rolling friction’s dependence on ball bearings’ radius. Engl. Transl. Przegl ˛adMech. 2018, 1, 21–26. [CrossRef] 33. Domenech, A.; Domenech, T.; Cebrian, J. Introduction to the study of rolling friction. Am. J. Phys. 1987, 55, 231–235. [CrossRef] 34. Olaru, D.; Stamate, C.; Dumitrascu, A.; Prisacaru, G. New micro tribometer for rolling friction. Wear 2011, 271, 842–852. [CrossRef] 35. Sosnovskii, L.; Komissarov, V.; Shcherbakov, S. A Method of Experimental Study of Friction in a Active System. J. Frict. Wear 2012, 33, 136–145. [CrossRef] 36. Available online: https://www.sciencedirect.com/topics/physics-and-astronomy/tribometers (accessed on 22 October 2020). 37. Zhang, X.; Zhang, Y.; Du, S.; Yang, Z.; He, T.; Li, Z. Study on the Tribological Performance of Copper-Based Powder Metallurgical Friction Materials with Cu-Coated or Uncoated Graphite Particles as Lubricants. Materials 2018, 11, 2016. [CrossRef] 38. ISO 28580-2018. Available online: https://www.iso.org/standard/67531.html (accessed on 27 October 2020). 39. SAE J2452-2017. Available online: https://standards.globalspec.com/std/10170311/SAE%20J2452 (accessed on 27 October 2020). 40. Minkin, L.; Sikes, D. Coefficient of rolling friction—Lab experiment. Am. J. Phys. 2018, 86, 77. [CrossRef] 41. Cross, C. Effects of surface roughness on rolling friction. Eur. J. Phys. 2015, 36, 065029. [CrossRef] 42. Ketterhagen, W.; Bharadwaj, R.; Hancock, B. The coefficient of rolling resistance (CoRR) of some pharmaceutical tablets. Int. J. Pharm. 2010, 392, 107–110. [CrossRef][PubMed] Materials 2020, 13, 5032 25 of 25

43. Bonhomme, J.; Moll, V. A Method to Determine the Rolling Resistance Coeﬃcient by Means of Uniaxial Testing Machines. Exp. Tech. 2015, 39, 37–41. [CrossRef] 44. Abdalsalam, M.; Gutierrez, M. Comprehensive study of the eﬀects of rolling resistance on the stress–strain and strain localization behaviour of granular materials. Granul. Matter 2010, 12, 527–541. [CrossRef] 45. Andersen, L. Rolling Resistance Modelling: From Functional Data Analysis to Asset Management System. Ph.D. Thesis, Roskilde University, Roskilde, Denmark, April 2015.

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