Bend it like Zhang Jike Spin in Table Tennis

Bend It Like Zhang Jike:Spin In Table Tennis?

Levi Dudte

11-27-2012

Abstract

We discuss the role of aerodynamic forces in table tennis shots with topspin and backspin,solving for the trajectories of these shots numerically.We use our numerical

model to characterize the e?ect of spin on shot accuracy and speed.

1Introduction

The game of table tennis would be much di?erent were it played in a vacuum or with frictionless paddles.A player’s ability to impart spin onto the ball fundamentally alters the characteristic trajectory of his or her shot and represents a distinct competititive advantage over spinless opponents.We discuss here this physical origins of this e?ect,and attempt to quantify its role in shot selection in the game of table tennis.

2Model

We can write the drag force F D(Fig.1)on a translating sphere as

F D=?1

2

πρR2U2C D?v.(1)

A typical Reynolds number of a competitive table tennis shot is2ρRv0/μ=0.5×105,where

ρ=1.2kg

m3,R=0.02m,v0=20m

s

andμ≈1.98×10?5kg

ms

.In this high Reynolds number

regime,we can write the transverse force F L(Fig.1)on a translating,spinning sphere as

F L=2πρR3ω0UC n?n(2) where?n=ω0×v/|ω0×v|.We assume that the angular velocity of the ball remains constant at its initial valueω0through the shot trajectory.See[3]for a discussion of the time evolution of angular velocity due to viscous torque.The derivation and experimental con?rmation of this expression is discussed in[2]and[7].There are two widespread explanations for the onset of

F L,termed the Magnus e?ect after the German physicist Heinrich Magnus,who described the e?ect in1852.1The?rst explanation attributes the transverse force to Bernoulli’s principle. The reasoning for this is quite simple:the pressure is greater on the side of the ball spinning opposite the direction of the surrounding?ow,because the?uid on this side is made to slow ?Zhang Jike,a member of the Chinese team that dominated the men’s table tennis competition in the2012 Summer Olympics,won singles and doubles gold medals.

1See[2]for an engaging account of Magnus’s investigation of the e?ect,which he conducted using curved musket barrels.

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F G F D F D

F L

v

v ω0Figure 1:The forces on backspin (left)and topspin (right)shots in table tennis:drag force F D ,lift force F L and gravitational force F G .

down by the ball’s rotation.Bernoulli’s principle states that along streamlines the following expression is constant:

v 22+gz +p ρ

(3)where z is the height.It follows that a decrease in ?ow speed on one side of the ball would yield an increased pressure relative to the other side.This simple explanation su?ers from one

serious ?aw:even in ?ow regimes dominated by inertial forces an extremely small viscosity can cause a breakdown of the no-slip condition at the boundary of an object,which predicates Bernoulli’s principle.This breakdown occurs by ?ow separation,wherein the viscous boundary layer ?ow around an object becomes detached and vortical.The separated ?ow exerts a profound in?uence over what could otherwise be considered simple inviscid ?ow outside the boundary layer.The Magnus e?ect can be understood in terms of asymmetric ?ow separation on opposite sides of the ball.On the side spinning opposite the direction of the surrounding ?ow,the boundary layer ?ow separates earlier than on the side spinning with the surrounding ?ow,which may not even undergo separation.The separated ?ow carries momentum away from the surface of the ball,inducing our transverse force F L .Predicting the onset of ?ow separation remains a challenge in boundary layer theory.The basic argument for early ?ow separation on side spinning opposite the surrounding ?ow is that the adverse pressure gradient along this side is greater.We can write the Navier-Stokes equations for two-dimensional rectilinear ?ow along a plate as

u ?u ?x +v ?u ?y =?1ρ?p ?x +ν?2u ?2y (4)

where x and y are coordinates along and normal to the plate,respectively,and u and v are the corresponding velocities.At the plate boundary this reduces to

v ?u ?y +1ρ?p ?x =ν?2u ?2y (5)

assuming the no-slip condition.Assuming v <0at y =0,then we can see from this expression that as the pressure rises along x the value of ?u/?y must at some point become negative.

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Flow separation will occur at this transition point,whose location is clearly dependent on the value of?p/?x.The higher adverse pressure gradient on the side of the ball spinning opposite the wind will then produce an earlier?ow separation than on the other side of the ball.

3Signi?cance of Spin

While the competitive advantages of spin are evident to anyone who has played table tennis, there are several quantitative measures by which we can convince ourselves of its signi?cance. As discussed in[4],we compute the spin number S p=F L/F D as

S p=4Rω0

v0

C n

C D

.(6)

Additionally,we compute the ratio L i=F L/F G for the case of coplanar F L and F G(i.e.for topspin and backspin shots in table tennis)which we call the lift number:

L i=3

2

ρ

ρb

ω0U0

g

C n.(7)

A typical spin number in table tennis is.36,higher than in any other sport analyzed in[4](the authors disregard the dimensionless term4C n/C D).A typical lift number in table tennis is.46,

calculated using linear(20m

s )and angular(100m

s

)velocities from[6]and with C n=.1,

C D=.4andρb/ρ=67[4].These ratios indicate that lift forces due to spin play a fundamental role in the game of table tennis,as they are comparable in magnitude to both drag and gravitational forces.

4Numerical Model

Our?rst-order system for shot velocity takes the form

m˙v=F D(v)+F L(v)+F G.(8) Because of the dependence of F G onθwe cannot integrate this expression directly,so we solve numerically using the explicit Euler numerical integration scheme:

y n+1=y n+hf(y n,t)(9) where h is the time step size.This yields a discrete solution for velocity of a shot over time, which we integrate using the trapezoidal rule to determine a trajectory(Fig.2).Our simulation results illuminate several aspects of the role of spin in table tennis.Looking at Fig. 3a,we can see the importance of modulating the linear velocity of a shot especially when hitting with backspin,as these shots tend to travel further than their spinless and topspin counterparts.We can also see why topspin shots are the mainstay of aggressive or attacking table tennis play,as their?nal shot velocities tend to be higher than shots with no spin or backspin(Fig.3b).And although we did not quantify this,the variance of the shot length distributions(Fig.3a)seems to decrease with an increase inω0.Verifying this quantitatively would o?er an explanation for the use of backspin in conventionally conservative or defensive play:adding backspin automatically increases your accuracy.

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X Y Figure 2:Sample shot trajectories from simulation.Each shot has θ0=10?and v 0=20m

s .The

only varied

parameter is ω0.

P (L e n g t h )Length (m)a

P (F i n a l V e l o c i t y )Final Velocity (m/s)b

Figure 3:a)distribution of shot length,b)distribution of ?nal shot velocity.Each probability distribution function aggregates data from 200,000simulation trajectories with v 0=20m

s and

an initial height of 0.We model human error in θ0by a Gaussian distribution with μ=10?and σ=2?.

5Conclusion

We have looked brie?y at the ?uid dynamics which govern the onset of the Magnus e?ect in

table tennis and we have initiated a simulation-based study of the e?ects of these dynamics on shot selection.Much work remains in understanding the role of spin in table tennis.From a theoretical standpoint,we could develop a model of the initiation of ?ow separation on a

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spinning,translating sphere and establish a quantitative relationship between?ow separation and the lift force.We have also only considered two-dimensional trajectories.It would be interesting to allow for a tilt in the initial angular velocity of a shot,and to study these

three-dimensional trajectories in simulation.

References

[1]D.J.Acheson.Elementary Fluid Dynamics.Oxford Applied Mathematics and Computing

Sciences Series,(1990).

[2]L.J.Briggs.“E?ect of Spin and Speed on the Lateral De?ection(Curve)of a Baseball;and

the Magnus E?ect for Smooth Spheres.”American Journal of Physics,27(1959).

[3]G.Dupeux,A.Le Go?,D.Quere and C.Clanet.“The spinning ball spiral.”New Journal of

Physics,12(2010).

[4]G.Dupeux,C.Cohen,A.Le Go?,D.Quere and C.Clanet.“Football curves.”Journal of

Fluids and Structures,27(2011).

[5]A.M.Nathan.“The e?ect of spin on the?ight of a baseball.”American Journal of Physics,

76(2008).

[6]R.Seydal.“Determinant Factors of the Table Tennis Game:Measurement and Simulation

of Ball-Flying Curves.”International Journal of Table Tennis Sciences,1(1992).

[7]R.G.Watts and R.Ferrer.“The lateral force on a spinning sphere:Aerodynamics of a

curveball.”American Journal of Physics,55(1987).

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