A short workshop on the simulation of a soccer ball in flight
- 1. 1 The trajectory of a soccer ball How to simulate it: A Short Workshop www.physicsandsport.com/en Vassilios M Spathopoulos Lecturer Department of Aircraft Technology Technological Educational Institute of Central Greece
- 2. 2 Motivation for workshop The trajectory of a soccer ball in flight is governed by complex aerodynamic mechanisms similar to those determining the motion of an aircraft
- 3. 3 Personal interest As an aeronautical engineer and a keen football fan, have become interested in soccer ball flight mechanics This has lead to the development of a flight model in the Matlab® environment (see figure above) One can also simulate the trajectory quite easily in the Excel® environment This is what this presentation will describe!
- 4. 4 Airplane vs soccer ball The motion of both airplane and soccer ball is driven by aerodynamic and gravitational forces v mg Fdrag Fmag ω
- 5. 5 Workshop aims Present and familiarise with the basic aerodynamic mechanisms involved Apply basic flight mechanics principles in order to calculate the trajectory of a soccer ball in flight Use a trajectory simulation in the Excel® environment in order to obtain important parameters relating to typical free kicks
- 6. 6 Force diagram Gravitational force Downward direction Drag Direction opposite to velocity Magnus force Direction normal to the plane determined by the velocity and the spin vector Note: This force diagram ignores side forces due to seam orientation! v mg Fdrag FMagnus ω
- 7. 7 Gravitational force Constant direction and magnitude m = 0.43 kg g = 9.81 m/s2 gW m v mg ω
- 8. 8 Drag Varying direction and magnitude Direction opposite to velocity ρ = 1.2 kg/m3 Α =0.038 m2 Cd = Function, primarily, of the Reynolds number, D = 0.22 m μ = 18.2 μPa.s u u uD dCA 2 2 1 μ ρUD Re
- 9. 9 Generation of drag (pressure drag) •The flow separates from the surface producing a low pressure wake behind the ball. •The difference in pressure produces the drag force opposing the motion of the ball.
- 10. 10 Drag crisis •Once a critical Reynolds number, and therefore speed, is exceeded, the flow changes from laminar to turbulent thus delaying the separation and therefore producing a smaller wake and smaller drag coefficient. •The critical Reynolds number (and therefore the speed) at which the drag is drastically reduced, is lower for rough spheres. •It is for this reason that golf balls are designed with dimples, thus increasing their range.
- 11. 11 Magnus force Varying direction and magnitude Direction normal to the plane determined by the velocity and spin vectors Cmag = function, primarily, of the spin parameter U ωR s uω uω uFmag magCA 2 2 1
- 12. 12 Generation of Magnus force •Due to the rotation, at the upper surface we have a later flow separation than at the lower one and as a result the wake is deflected downwards. •As a reaction to this deflection (Newton’s 3rd Law), an upwards force is exerted on the ball.
- 13. 13 Effect of Magnus force By determining the tilt of the rotation axis of the ball, the player can produce the desired ball trajectory!
- 14. 14 Simulation methodology We divide time into small steps, dt = 0.01s If we know the values of x,y,z, vx,vy,vz at time t At time t+dt we have, x(t+dt)=x(t)+ux(t)*dt y(t+dt)=y(t)+uy(t)*dt z(t+dt)=z(t)+uz(t)*dt ux(t+dt)=ux(t)+ax(t)*dt uy(t+dt)=uy(t)+ay(t)*dt uz(t+dt)=uz(t)+az(t)*dt Numerical method: of Euler We need to know ax(t), ay(t), az(t) It is reminded that, ...,, 2 2 xx a dt xd u dt dx
- 15. 15 Equations of motion of a soccer ball in flight 222 dt dz dt dy dt dx U , 2 magmag C m A k , 2 dd C m A k Where, sin2 2 dt dy k dt dx kU dt xd magd dt dy k dt dz dt dx kU dt yd dmag cossin2 2 g dt dy k dt dz kU dt zd magd cos2 2 angleaxisspin
- 16. 16 Drag and Magnus coefficients For speeds greater than 15 m/s we can assume that the drag coefficient has a constant value of 0.15. For spin parameters greater than 0.2 we can assume that the Magnus coefficient is equal to the spin parameter (great simplification!)
- 17. 17 The simulation.. Has several simplifying assumptions: o The drag coefficient is assumed constant throughout a flight o The Magnus coefficient is assumed to be equal to the spin parameter o Side forces due to seam orientation are ignored o The spin rate is assumed constant o A very simple numerical integration technique is employed in order to solve the equations of motion It can portray the basic features of soccer ball flight!
- 18. 18 Simple hand calculation Estimate the lateral deflection of the ball for a free kick 20 yards from goal for 7 rev/s sidespin given an initial velocity of 25 m/s Assume that: o Motion only occurs in the x-y plane (2D) o The drag force is neglected o The Magnus coefficient is constant at 0.2 Use Newton’s 2nd Law to solve! 2.0 25 0.117π2 U Rfπ2 U ωR sCmag
- 19. 19 Solution 98.22.025038.0256.1 2 1 2 1 22 magCAuFmag 2 m/s93.6 43.0 98.2 m F a mag x m U y atax xx 85.1 25 28.18 93.6 2 1 2 1 2 1 22 2 Keep a note of the answer and compare to that obtained from the simulation!
- 20. 20 For more info on the physics of sports.. Please refer to my website: www.physicsandsport.com/en Thank you!