Report on Dynamics and Trajectory of Shuttle Cock in Badminton

Four Degree Of Freedom Parallel Manipulator

Four Degree Of Freedom parallel manipulator

Kinematics

Vinu.K.S

Mechanical

ME

6337-410-091-07147


Contents

Sl No

Description Page No

1 Introduction 32 Objective of the project 43 Modeling techniques 44 Assumptions 55 Specifications of model 66 Equations of motion 77 Results 88 Conclusion 99 Matlab Code for simulation of dynamics 910 References 13


1.Introduction

Badminton is a very popular game in Asia. It is played using shuttle cock. A shuttlecock is a high-drag projectile , which has an open conical shape. The cone is formed from sixteen overlapping goose feathers embedded into a rounded cork base. The cork is covered with thin leather. The shuttlecock's shape makes it extremely aerodynamically stable. Regardless of initial orientation, it will turn to fly cork first, and remain in the cork-first orientation.

For playing feathered or synthetic type is used. Synthetic shuttlecock lasts longer, costs less and exhibits less aerodynamic drag compared to feather shuttlecock which is predominantly used by the professional players and have high initial velocity. Unlike most racquet sports, a badminton shuttlecock is an extremely high drag projectile and possesses almost parabolic flight trajectory. The parabolic flight trajectory is generally skewed heavily thus its fall has much steeper angle than the rise.

Many papers have been published which details about aerodynamic properties of shuttle cock, obtained through series of experiments with different initial velocity. Typically drag coefficient for shuttlecock is of the order of 0.6-0.7.

The trajectory of shuttle cock depends on force and angle of stroke. Moreover air resistance force plays an important factor in determining whether linear or quadratic air resistance force is used.

The shuttlecock is very light compared to the area it presents during travel because of which the effect of air resistance on its flight is very pronounced. When struck, the shuttlecock travels cork first, followed by the flexible feathered section. It rapidly loses speed as it travels through the air. An important consequence of its aerodynamic properties is that its fall is significantly steeper than its rise

Shuttlecock


2. Objective:

To study the dynamics and trajectory of shuttlecock. By studying the trajectory one can determine speed, time, direction and path for different type of strokes. The flight of shuttle cock is a complex phenomenon that results from interaction of aerodynamic forces and dynamic effects.

To capture the cork first orientation and study the behavior of trajectory for different aerodynamic drag condition.

3. Modeling Techniques

Shuttle cock is modeled as a 3D object with feather as frustum of a cone with shell thickness of unity and cork base as hemisphere embedded on a cylinder.

From these parameters like centre of pressure, centre of gravity, moment of inertia are computed.

The forces considered for deriving equations of motion are

a. Weight of shuttle cock (mg)b. Buoyant Force (ρ*V*g) –constantc. Aerodynamic Force :1/2* ρ*A*V2

Where ½* ρ*A= bit is computed from free falling condition m*g=b*v2

terminal velocity Vt = (m*g/b)1/2


terminal velocity is computed from experiment and is constant from which air resistance factor ‘b’ is calculated .

Location of CP and CG

The cp(centre of pressure) is located near feather tip and cg (centre of gravity) at cork base.

The aerodynamic drag force creates a torque about cg which is opposed by the pitching moment of inertia of shuttle cock. For stability the cork rotates and aligns with the direction of drag force. This is the principle of cork first direction.

Stable Configuration

Drag Force

Stroke Force

Weight


4.Assumptions for deriving equation of motion

a. Buoyant force on shuttle cock considering it to be frustum of cone is of the order of 6.1x10-4 N. Hence can be neglected.

b. The equation of motion is derived considering i) Drag Force as linear.ii) Drag Force as quadratic.

5.Specifications used for modeling

Mass : 5.19 gms.

Overall Length : 85 mm.

Diameter of feather end : 54 mm.

Diameter of cork end : 25 mm.

Moment of Inertia (Izz) : 1.305x10-5 Kgm2

Centre of Pressure : 40 mm from fore end of cork.

Centre of Gravity : 8mm towards cork end

Equation of Motion

Force Balance

θ

α


(for linear aerodynamic force)

i) Vertical direction

m*y” = -mg - b*y’+ Fb

Fb –buoyant force

m-mass of shuttle cock

ii) Horizontal direction

m*x’’ = -b*x’, b: air resistance factor=m*g/vt2

iii) Moment Balance

Icg*θ’’ = b*[(x’-r *θ’)2 +(y’+r* θ’)2 ]1/2 *r*sin(θ-α) ;

Icg = pitching moment of inertia

r = hcp - hcg

hcp = centre of pressure

hcg = Centre of gravity

θ = angle of rotation of shuttle

α = angle of trajectory

(for quadratic aerodynamic force)

Force Balance

i) Vertical direction

m*y” = -mg-b*y’2+Fb

ii) Horizontal direction

m*x’’ = -b*x’2

iii) Moment Balance

Icg*θ’’ = b*[(x’-r *θ’*sinα)2 +(y’+r* θ’* cosα)2 ] *r*sin(θ-α) ;

8.Results


Comparing the trajectory of two cases it can be seen that modeling aerodynamic forces as quadratic is much more accurate than other. For angle of projection of 60 deg with initial velocity of 20 m/s the range of the shuttle cock is nearly 6 m as compared to 25 m in which case the shuttle will go out of the court for most period of time

9.Conclusion


Formulation of equation of motion using quadratic aerodynamic drag model is much more accurate than linear. Since the range obtained is 6 m (which is with in the badminton court length 13.4m)

A higher air resistance factor has been used than that predicted through experiment to capture the rotation of shuttle cock in 2D.

10.Matlab Code

% main program for animating the shuttle cock trajectoryglobal a b L xgclose all;clcL=60e-3;% Length of shuttlecock featherb=54e-3;%Feather top diametera=23e-3;%Cork base diameterr=12.5e-3;%cork base radiusxg=-8.00e-3;% x coordinate of cg of shuttle cockyg=0;% y coordinate of cg of shuttle cockalpha=pi/4;% initial angle of rotation for shuttle cockalphadot=0;% angular velocity of shuttle cockv0=20;% Initial Velocity of shuttle cocktheta=pi/6; % Initial angle of projectionx0=0.1;% Initial position of shuttle cocky0=0.1;% Initial position of shuttle cockvx=v0*cos(theta);%initial horizontal velocity of shuttle cockvy=v0*sin(theta);%initial vertical velocity of shuttle cockp0=[x0 vx y0 vy alpha alphadot]';% initial condition for ode solveroptions = odeset('Events',@event1);[t,z]=ode45(@shuttle3,0:.001:2,p0,options);%ode solver% plot(t,z(:,7)*180/pi) xsol=z(:,1);%x coordinate displacement solutionvxs=z(:,2);% velocity solutionysol=z(:,3);%y coordinate displacement solutionvys=z(:,4);% velocity solutionthsol=z(:,5);% angular velocity of shuttle cock xc=[0 0 L L 0 0];% x vector to draw shuttle cockyc=[0 a/2 b/2 -b/2 -a/2 0];% y vector to draw shuttle cock% commands for plotting shuttle cock motion with respect to cgxs=[0,-r,r*cos(pi/2:pi/20:3*pi/2)-r,-r,0,0];ys=[r,r,r*sin(pi/2:pi/20:3*pi/2),-r,-r,r];xn=[xc xs]-xg;yn=[yc ys]-yg;shuttle = plot(xn,yn,'-');hold on;plot(0,0,'b+')axis equal;axis([min(xsol)-.05,max(xsol)+0.05,min(ysol)-0.05,max(ysol)+0.05]);pos=[1 35 1366 660];set(gcf,'pos',pos)% axis([min(xsol1)-.05,max(xsol1)+0.05,min(ysol1)-0.05,max(ysol1)+0.05]);% axis([xsol(j)-0.1,xsol(j)+0.1,ysol(j)-0.05,ysol(j)+0.1]);


m(1)=getframe;% commands for animating shuttle cock motionfor i=2:length(t)% theta=pi/2; xcg=xsol(i); ycg=ysol(i); theta=thsol(i); Rot=[cos(theta) -sin(theta); sin(theta) cos(theta)]; news=Rot*[xn;yn]; newxs=xcg+news(1,:); newys=ycg+news(2,:); set(shuttle,'xdata',newxs,'ydata',newys); plot(xsol(i-1:i),ysol(i-1:i),'b-'); xlabel('Horizontal Distance(m)') ylabel('Vertical Distance(m)')% drawnow; m(i)=getframe;% pause(.01);end% movie(m,5)movie2avi(m,'bnm3.avi','quality',100)% commands for capturing movie

function file

function dz=shuttle3(t,z)global a b L xg% function for solving dynamics of equations of motion for shuttle cockm=0.00519;% mass of shuttlecockg= 9.81; % acceleration due to gravity in metre/sec2vx=z(2);%horizontal component of velocityvy=z(4);%vertical component of velocitythetadot=z(6);%angular rotation rate of shuttle cockvt=5;%terminal velocity of shuttle cock measured from experimentIg=1.3051e-5;% moment of inertia of shuttle cock about pitching axishcg=xg;%centre of gravity of shuttle cockhcp=50e-3;% centre of pressure of shuttle cock% vt=sqrt(vx^2+vy)^2;bv= (m*g)/(vt)^2;% constant depends on property of air% i) Modelling resistance force as quadraticFv=bv*(vx^2+vy^2);theta=atan2(vy,vx);dz(6,1)= (1/Ig)*0.25*(((vx-(hcp-hcg)*thetadot*sin(z(5)))^2+(vy+(hcp-hcg)*thetadot*cos(z(5)))^2)*((hcp-hcg)*sin(z(5)-theta)));dz(1,1)=z(2);dz(2,1)=-Fv*cos(theta)/m;dz(3,1)=z(4);dz(4,1)=-g-(Fv*sin(theta)/m);dz(5,1)=z(6);

Matlab code for event detectionfunction [value,isterminal,direction] = event1(t,y)value=y(3);isterminal=1;direction=-1;

matlab code for comparision of trajectory of linear and quadratic drag


main program

global a b L xgclose all;clcL=60e-3;b=54e-3;a=23e-3;r=12.5e-3;xg=-8.00e-3;yg=0;alpha=pi/10;alphadot=0;v0=20;% Initial Velocity of shuttle cocktheta=pi/6; % Initial angle of projectionx0=0.5;y0=0.5;% theta=pi/20;vx=v0*cos(theta);vy=v0*sin(theta);%(6.51/0.015)*sin(theta);p0=[x0 vx y0 vy alpha alphadot]';[t,z]=ode45(@shuttle3,0:.001:2,p0);[t1,y]=ode45(@shuttlelinear,0:.001:1,p0);k=z(:,3);m=z(:,1);j=y(:,3);n=y(:,1);p=z(:,5);q=y(:,5);for i=1:1:length(t) if(k(i)>0.01) k1(i)=k(i); t2(i)=t(i); m1(i)=m(i); endendfor l=1:1:length(t1) if(j(l)>0.01) j1(l)=j(l); t3(l)=t1(l); n1(l)=n(l); endendlength(k1);length(k);length(t);length(t1); plot(t2,k1,'-.r',t3,j1,'-b','LineWidth',2)% hold on xlabel('time (seconds)') ylabel('Vertical Height(m)') title('Height attained by shuttle cock') h = legend('quadratic drag force','Linear Drag Force',2); set(h,'Interpreter','none') figure


plot(m1,k1,'-.r',n1,j1,'-b','LineWidth',2)% hold on% plot(n1,j1,'-b','LineWidth',2) xlabel('Range (m)') ylabel('Height(m)') title('Horizontal Range of shuttle cock') h = legend('quadratic drag force','Linear Drag Force',2); set(h,'Interpreter','none') figure plot(m,(180/pi)*p,'-.r','LineWidth',2) hold on plot(n,(180/pi)*q,'-b','LineWidth',2) xlabel('Range (m)') ylabel('angle of rotation (deg)') title('angle of rotation of shuttle cock') h = legend('quadratic drag force','Linear Drag Force',2); set(h,'Interpreter','none')

function file

function dz=shuttlelinear(t1,y)global a b L xg% function is matrix formulation for computing horizontal velocity and% displacement for linear resistance/drag forcem=0.00519;% mass of shuttlecockg= 9.81; % acceleration due to gravity in metre/sec2vx=y(2);%horizontal component of velocityvy=y(4);%vertical component of velocitythetadot=y(6);%angular rotation rate of shuttle cockvt=5.61;%terminal velocity of shuttle cock measured from experiment% Ig=1.3051e-4;`Ig=1.3051e-4;% moment of inertia of shuttle cockhcg=xg;%centre of gravity of shuttle cockhcp=50e-3;% vt=sqrt(vx^2+vy)^2;bv= (m*g)/(vt)^2;% constant depends on property of air% i) Modelling resistance force as linearFv=bv*(vx^2+vy^2)^(1/2);% angle of projection of shuttle cocktheta=atan2(vy,vx);dz(6,1)= (1/Ig)*1*((vx-(hcp-hcg)*thetadot*sin(y(5)))^2+(vy+(hcp-hcg)*thetadot*cos(y(5)))^2)^(1/2)*((hcp-hcg)*sin(y(5)-theta));dz(1,1)=y(2);dz(2,1)=-Fv*cos(theta)/m;dz(3,1)=y(4);dz(4,1)=-g-(Fv*sin(theta)/m);dz(5,1)=y(6);

10. References:

1.www.cookeassociates.com/researchretail.html


2.www.wikipedia.org3.www.doi.wiley.com4.The engineering of sport 7-Maxine Kwan, Michael Skipper Andersen

5.Aerodynamic properties of badminton shuttle cock

Firoz Alam¹, Harun Chowdhury¹, Chavaporn Theppadungporn¹, Aleksandar Subic¹ and M Masud Kamal Khan²

6. Effect of Local Conditions on the Flight Trajectory of an Indoor Badminton ShuttlecockRaghavan Subramaniyan, Bangalore, India

Documents

Report on Dynamics and Trajectory of Shuttle Cock in Badminton