Abstract
Dynamic sliding friction was studied based on the angular velocity of a golf ball during an oblique impact. This study used the analytical model proposed for the dynamic sliding friction on lubricated and non-lubricated inclines. The contact area A and sliding velocity u of the ball during impact were used to describe the dynamic friction force Fd = λAu, where λ is a parameter related to the wear of the contact area. A comparison with experimental results revealed that the model agreed well with the observed changes in the angular velocity during impact, and λAu is qualitatively equivalent to the empirical relationship, μN + μη′dA/dt, given by the product between the frictional coefficient μ and the contact force N, and the additional term related to factor η′ for the surface condition and the time derivative of A.
Similar content being viewed by others
Introduction
Golf, one of the most popular sports worldwide, has a long history and the physics of golf has been studied for centuries1,2. Aerodynamic studies of golf balls have demonstrated the following fundamental mechanisms. A dimpled golf ball flies much farther than a smooth ball due to turbulence caused by the dimples. The spin of the ball, imparted by the club, changes the direction and flying distance of the ball in the air. The impact dynamics of golf balls3 have also been studied (Fig. 1a). For example, previous studies investigated the oblique impact of the ball with a steel target4 (Fig. 1b) and revealed that the sliding (u) and angular (ω) velocities during impact increased with the inbound velocity Vi. The coefficient of restitution and contact time on the target decreased gradually with Vi. An impact study using a transparent polymethyl methacrylate (PMMA) target5 with dry and oiled surfaces (Fig. 1c) showed that lubrication with oil did not affect the contact area or time. The sliding velocity u increased, while the angular velocity ω decreased.
A study of the angular velocity ω demonstrated the following6: (i) the experimental value of ω increased in the initial phases of contact and then decreased; (ii) there was a significant discrepancy between the experimental results and analytical velocity ω derived from μN; and (iii) the experimental results agreed with the analytical velocity ω given by μN + μη′dA/dt. Many studies have investigated the dynamic contact problem from analytical and experimental perspectives7,8,9,10,11 to elucidate influential factors, including the contact force12,13,14,15,16,17, contact area18,19,20,21, sliding velocity14,15,17,22,23,24,25, surface roughness18, temperature26,27, humidity22,24,28, and interface wear13,29. However, these effects are still not fully understood.
To study influential factors, sliding tests conducted under different surface conditions demonstrated the following: (i) the sliding velocity of polyurethane (PU) rubber on oiled inclines30 was significantly dependent on the contact area; (ii) the contact area of polytetrafluoroethylene (PTFE) spheres on dry inclines31 increased with wear, while the sliding velocity decreased; and (iii) the analytical model indicated that the contact area and sliding velocity are key factors on oiled surfaces30, while the wear of the contact surface can also be an influential factor on dry surfaces31, implying that the dynamic friction force can be expressed as Fd = λAu.
This work studied dynamic sliding friction based on the angular velocity ω of a golf ball during an oblique impact. The applicability of the frictional force Fd = λAu proposed for the dynamic sliding tests30,31 to the impact problem was examined, together with the reported correlation with the empirical relationship6 μN + μη′dA/dt.
Figure 2a shows the contact area A of the ball as a function of time t. For the inbound ball velocity Vi = 32 m s−1, the value of A increased in the early phases of impact, attained a peak value of 150 mm2 at t = 220 μs, and then decreased subsequently. To simplify the analysis, the contact area A was expressed as follows:
where Ao is the peak value of A, ϕ = π/tc, and tc is the contact time. The results in Fig. 2a show that equation (1) agrees with the experimental data. A similar equation was previously used for erosive wear problems to describe the penetration depth of abrasive spherical particles into solid surfaces32.
The rotation angles θ for Vi = 28 and 61 m s−1 are shown in Fig. 2b, where the experimental results are plotted using symbols; the curves derived from a data-fitting procedure are also shown. For two inbound velocities, θ increased slightly in the initial phases of contact, significantly in the subsequent phases, and then gradually in the final phases. The largest values of θ for Vi = 28 and 61 m s−1 were 13 and 26°, respectively. The angular velocity ωe was obtained by differentiating the fitted curve θ(t) with respect to time, t (Methods).
The angular velocities ωe for Vi = 28 and 61 m s−1 are shown in Fig. 3a and b, respectively. For both inbound velocities, ωe increased in the initial phases of contact and then decreased in the subsequent phases. The peak value of ωe for Vi = 28 m s−1 was 6,700 rpm at t = 260 μs, and it decreased significantly to 4,300 rpm just before rebounding at tc = 500 μs. As Vi increased to 61 m s−1, the peak value of ωe increased to 15,400 rpm at t = 240 μs, and then decreased markedly to 9,800 rpm just before rebounding at tc = 460 μs.
To study the rotation behaviour, the analysis used the following assumptions: (i) the target is rigid and the ball with mass m (46 g) and radius r (21.3 mm) is treated as a hard sphere; (ii) the contact area A is represented with equation (1); and (iii) the gravity force and rolling friction acting on the ball can be disregarded. The angular velocity ω* is given as follows:
where I (=2 mr2/5) is the inertia moment of the ball about the centre of rotation.
For lubricated friction30, we expressed the dynamic friction force, Fd = τA, using the Couette flow shear stress τ = (η/h)u, where η and h are the viscosity and thickness of the oil layer on the contact area, respectively. Therefore, the dynamic friction force Fd can be given as follows:
where λ = η/h and the units of λ are Pa∙s∙m−1.
The oblique impact of a golf ball caused wear of the ball surface in the contact area; hence, this analysis used equation (3), assuming that λ is related to the wear property of the contact area30 during impact. The sliding motion of the ball is expressed as follows:
Using the separation of variables and integrating equation (4), assuming that λ is constant, we obtain:
Integrating equation (5) with respect to t for u = uo at t = 0, the sliding velocity u can be expressed as follows:
where b (=λAo/m ϕ) is a dimensionless parameter. Under the assumption of a relatively large mass m and a very short duration tc, we approximated exp(b cos ϕ t − b) in equation (6) as (1 + b cos ϕt − b) using a series expansion, and represented the sliding velocity u as follows:
Substituting equations (1) and (7) into equation (3), equation (2) can be rewritten as follows:
Integrating equation (8) with respect to t for ω* = 0 at t = 0, the angular velocity ω*can be expressed as follows:
where
Equation (10) corresponds to the solution given by the hard-sphere model and equation (11) relates to the effect of the change in contact area6.
The analytical results of equation (9) for Vi = 28 and 61 m s−1 are shown in Fig. 3a and b, respectively. The assumed values of the coefficients for Vi = 28 m s−1 were uo = 12 m s−1, tc = 500 μs, Ao = 130 mm2, and λ = 1.81 × 106 Pa∙s∙m−1 (therefore, b = 0.81). For Vi = 61 m s−1 they were uo = 27 m s−1, tc = 460 μs, Ao = 270 mm2, and λ = 0.96 × 106 Pa∙s∙m−1 (therefore b = 0.83). Equation (9) agreed well with the experimental result for Vi = 28 m s−1. However, the model yielded slight discrepancies from the experimental result for Vi = 61 m s−1, probably due to the influence of ball deformation during impact. Two things should be noted: (i) the values of uo were slightly smaller than those determined theoretically with uo = Vi sinθi, which can be attributed to the energy loss due to the internal friction of the ball at impact4, and (ii) similar values of b were obtained for both inbound velocities, suggesting that there is an inverse relationship between Ao and λ.
To study the effect of λ on A and u, we made a cursory examination of the values of ω* at t = tc/2 based on the following approximations: (i) from equation (9) ω* ~ 5uob(2 − b)/4r; (ii) for different state ωs* ~ 5usbs(2 − bs)/4r, where bs = λsAs/mϕs; and (iii) assuming that ϕs ~ ϕ and bs ~ b, ωs*/ω* ~ usbs/uob was obtained. The effect of λ was examined based on ωs*/ω* = 1. For As > Ao and us = uo, ωs*/ω* ~ λsAs/λAo, resulting in λs < λ; while for As = Ao and us < uo, ωs*/ω* ~ λsus/λuo, resulting in λs > λ. Similar relations can also be determined using equation (3), i.e., λAouo = λsAsus. This suggests that λ, A, and u are correlated with each other during dynamic sliding.
The dynamic sliding friction was related to μN + μη′dA/dt in a previous study6. To clarify the correlation with the present model, a cursory examination was made of equation (3) using the following approximations: (i) u ~ uo (1 − b + b cos ϕt) in equation (7); (ii) differentiating equation (1) with respect to t, we obtain cos ϕt = (ϕ/Ao)dA/dt; and (iii) the relation Fd ~ λuo (1 − b)A + λuo bϕ (A/Ao)dA/dt was obtained. This approximate expression for equation (3) is qualitatively equivalent to the previous relationship, since A depends on N. This implies that equation (3) more closely describes the dynamic friction force during an oblique impact.
Methods
A golf ball was launched horizontally with an air gun so that it obliquely struck a target rigidly clamped and vertically inclined at an angle of 30°. This study used three-piece golf balls, with a mass of 46 g and diameter of 42.6 mm. A high-speed video camera (HPV-1; Shimadzu) was used to record 100 frames as bitmap graphics (size 312 × 260 pixels) at a framing interval of 10 μs. The impact tests were performed at room temperature for a ball inbound at a velocity between 28 and 61 m s−1. The target surface was degreased with alcohol, and new balls were used to minimise the change in roughness of the ball surface. On oblique impact with the transparent PMMA target (size 130 × 170 mm2, thickness 20 mm), the contact area became dark due to diffuse reflection, and the concave surfaces of the dimples on the ball surface barely contacted the target. The contact area was determined by subtracting the noncontact area due to the dimples using image analysis. The rotation angle of the ball on the steel target (diameter 40 mm, thickness 10 mm) was determined as follows: (i) an image of the ball before impact was selected; (ii) two vertical lines were drawn from the ball centre parallel to the target; (iii) the four markers closest to these two lines and farthest from the ball centre were selected, and the rotation angles of the four markers were measured as a function of time; and (iv) we assumed that the average value of the four angles was the rotation angle of the ball. To minimise data scattering in the evaluation of the angular velocity, we used a data-fitting procedure based on the least-squares method; the measured values of ball rotation were expressed as a ninth-order polynomial of time to fit the observed values. The angular velocity was determined from the first time derivative of the fitted curve.
Additional Information
How to cite this article: Arakawa, K. An analytical model of dynamic sliding friction during impact. Sci. Rep. 7, 40102; doi: 10.1038/srep40102 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
Tait, P. G. Some points in the physics of golf. Nature 42, 420–423 (1890).
Thomson, J. J. The dynamics of a golf ball. Nature 85, 251–257 (1910).
Daish, C. B. The physics of Ball Games. (Hazell Watson & Viney Ltd, Aylesbury Bucks, UK, 1972).
Arakawa, K. et al. Dynamic contact behavior of a golf ball during an oblique impact. Exp. Mech. 46, 691–697 (2006).
Arakawa, K. et al. Dynamic contact behavior of a golf ball during oblique impact: Effect of friction between the ball and target. Exp. Mech. 47, 277–282 (2007).
Arakawa, K. Effect of time derivative of contact area on dynamic friction. Appl. Phys. Lett. 104, 241603 (2014).
Persson, B. N. J. Sliding Friction: Physical Principles and Applications (Springer-Verlag, Berlin, 1998).
Kietzig, A. M., Hatzikiriakos, S. G. & Englezos, P. Physics of ice friction. J. Appl. Phys. 107, 081101 (2010).
Goldsby, D. L. & Tullis, T. E. Flash heating leads to low frictional strength of crustal rocks at earthquake slip rates. Science 334, 216–218 (2011).
Lockner, D. A., Morrow, C., Moore, D. & Hickman, S. Low strength of deep San Andreas fault gouge from SAFOD core. Nature 472, 82–85 (2011).
Lyashenko, I. A. & Popov, V. L. Impact of an elastic sphere with an elastic half space revisited: Numerical analysis based on the method of dimensionality reduction. Scientific Reports 5, 8479 (2015).
Corwin, A. D. & deBoer, M. P. Effect of adhesion on dynamic and static friction in surface micromachining. Appl. Phys. Lett. 84, 2451 (2004).
Gotsmann, B. & Lantz, M. A. Atomistic wear in a single asperity sliding contact. Phys. Rev. Lett. 101, 125501 (2008).
Riedo, E. et al. Interaction potential and hopping dynamics governing sliding friction. Phys. Rev. Lett. 91, 084502 (2003).
Tambe, N. S. & Bhushan, B. Nanoscale friction mapping. Appl. Phys. Lett. 86, 193102 (2005).
Lessel, M. et al. Impact of van der Waals interactions on single asperity friction. Phys. Rev. Lett. 111, 035502 (2013).
Dias, R. A., Coura, P. Z. & Costa, B. V. Velocity, temperature, normal force dependence on friction: an analytical and molecular dynamics study. Phys. Status Solidi B247, 98–103 (2010).
Dieterich, J. H. & Kilgore, B. D. Direct observation of frictional contacts: new insights for state-dependent properties. Pure Appl. Geophys. 143, 283–302 (1994).
Borovsky, B., Booth, A. & Manlove, E. Observation of microslip dynamics at high-speed microcontacts. Appl. Phys. Lett. 91, 114101 (2007).
Bayart, E., Svetlizky, I. & Fineberg, J. Fracture mechanics determine the lengths of interface ruptures that mediate frictional motion. Nature Phys. 12, 166–170 (2016).
Maegawa, S., Suzuki, A. & Nakano, K. Precursors of global slip in a longitudinal line contact under non-uniform normal loading. Tribol. Lett. 38, 313–323 (2010).
Riedo, E., Lévy, F. & Brune, H. Kinetics of capillary condensation in nanoscopic sliding friction. Phys. Rev. Lett. 88, 185505 (2002).
Reimann, P. & Evstigneev, M. Nonmonotonic velocity dependence of atomic friction. Phys. Rev. Lett. 93, 230802 (2004).
Chen, J., Ratera, I., Park, J. Y. & Salmeron, M. Velocity dependence of friction and hydrogen bonding effects. Phys. Rev. Lett. 96, 236102 (2006).
Li, Q. W. et al. Speed dependence of atomic stick-slip friction in optimally matched experiments and molecular dynamics simulations. Phys. Rev. Lett. 106, 126101 (2011).
Schirmeisen, A., Jansen, L., Hölscher, H. & Fuchs, H. Temperature dependence of point contact friction on silicon. Appl. Phys. Lett. 88, 123108 (2006).
Dias, R. A., Rapini, M., Costa, B. V. & Coura, P. Z. Temperature dependent molecular dynamic simulation of friction. Braz. J. Phys. 36, 741–745 (2006).
Ando, Y. The effect of relative humidity on friction and pull-off forces measured on submicron-size asperity arrays. Wear 238, 12–19 (2000).
Merkle, A. P. & Marks, L. D. Friction in full view. Appl. Phys. Lett. 90, 064101 (2007).
Arakawa, K. Dynamic sliding friction and similarity with Stokes’ law. Tribol. Int. 94, 77–81 (2016).
Arakawa, K. An analytical model for dynamic sliding friction of polytetrafluorethylene (PTFE) on dry glass inclines. Wear 356–357, 40–44 (2016).
Argatov, I. I., Dmitriev, N. N., Petrov, Yu. V. & Smirnov, V. I. Threshold erosion fracture in the case of oblique incidence. Journal of Friction and Wear 30, 176–181 (2009).
Acknowledgements
The author would like to express his gratitude to Prof. G. Etoh and Prof. K. Takehara from Kinki University for their advice in operating the high-speed video camera, and Mr. T. Mada from Kyushu University for his help in performing the experiments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The author declares no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Arakawa, K. An analytical model of dynamic sliding friction during impact. Sci Rep 7, 40102 (2017). https://doi.org/10.1038/srep40102
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep40102
This article is cited by
-
A review of dynamic models and measurements in golf
Sports Engineering (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.