Study of Aerodynamic Force variation with the Velocity and Drag Coefficient

Title: Drag Force variation with the Velocity and Drag Coefficient

Aim

To show the variation of Drag force experienced while cycling with respect to the Velocity and Coefficient also to Plot the variation using MATLAB.

Theory

During cycling, three external forces are applied to the couple bicycle/cyclist. These are the weight, the aerodynamic drag force and the contact force between road and wheels. During a TT stage on a flat track, at an average speed close to 50 km·h–1, nearly 90% of the cyclist power output is used to overcome the aerodynamic drag force (Belluye & Cid, 2001). This aerodynamic drag force is given by the following equation.

R=12(ρACv^2)

where R is the aerodynamic drag force (N), ρ denotes the air density (kg·m–3), AP represents the projected frontal area of the couple bicycle/cyclist (m2), CD is the aerodynamic drag force coefficient and V is the airflow velocity (m·s–1). The drag force coefficient of the couple bicycle/cyclist cannot be directly measured. The above relationship indicates that both terms RD and AP must be known to determine CD. Methods implemented to measure both the drag force and the frontal area are presented in the following section.

Procedure

1. Define all the inputs which is required to calculate the Drag force such
2. The Inputs are density, Drag coeff, Frontal area(as shown in figure),and the Velocity
3. Give the equation R = 12(ACv2)with the available input and the aerodynamic drag force relation the drag force is calculated.
4. The system charecteristics to be studied for this the velocity is changed and drag force is calculated
5. Similarly drag force variation is studied with the variation of drag Coefficient.
6. The results are shown

Program:

Program to calculate the variation of Drag force with the Velocity and Drag Coefficient

clear all

clc



%%Drag foce caculation variation in Velocity



%%input values

c_d= 0.25:1:10 ;  %drag coeff

rho= 1.2;    %Desity 

a=1;         %Frontal area

v= linspace(1,10,10);

for i=1:length(v)

    F1(i)=c_d(1)*rho*a*v(i).^2

     F2(i)=c_d(2)*rho*a*v(i).^2

      F3(i)=c_d(3)*rho*a*v(i).^2

       F4(i)=c_d(4)*rho*a*v(i).^2

        F5(i)=c_d(5)*rho*a*v(i).^2

        

end

hold on

plot(v,F1,'r')

plot(v,F2,'r')

plot(v,F3,'r')

plot(v,F4,'r')

plot(v,F4,'r')



xlabel('Drag coeff')

ylabel('Velocity')



clear all

clc

close all



%%Drag foce caculation variation in Cd



%%input values

c_d= 0.25:1:10 ;  %drag coeff

rho= 1.2;    %Desity 

a=1;         %Frontal area

v= linspace(1,10,5);

for i=1:length(c_d)

    F1(i)=c_d(i)*rho*a*v(1).^2

      F2(i)=c_d(i)*rho*a*v(2).^2

       F3(i)=c_d(i)*rho*a*v(3).^2

        F4(i)=c_d(i)*rho*a*v(4).^2

         F5(i)=c_d(i)*rho*a*v(5).^2

end



plot(c_d,F1,'r')

hold on

plot(c_d,F2,'r')

hold on

plot(c_d,F3,'r')

hold on

plot(c_d,F4,'r')

hold on

plot(c_d,F4,'r')



xlabel('Drag coeff')

ylabel('Velocity')

The C_d , Rho ,A, V are the input given which is assigned as shown in the first half of the program. Here we have used “LINSPACE” to get the range of values for the velocity and drag coefficient and for this variation or number of iteration we have used “for “ loop to vary the drag coefficient and inside that the Drag force is calculated for individual velocities. Then we have used a plot command to plot each and every drag force against the variation of the drag coefficient and Velocity. HOLD ON command is used to hold the plot and incorporate the all variation in single plot.

In other way also the same can be coded using nested for loop for varying both velocity and drag coefficient.

clear all
clc

%%Drag foce caculation

%%input values
c_d= 0.25:1:20 ;  %drag coeff
rho= 1.2;    %Desity 
a=1;         %Frontal area
v= linspace(1,10,10);
for i=1:length(v)
    for j=1:length(c_d)
    F(i,j)=c_d(j)*rho*a*v(i).^2
     
    end      
end
figure(1)
plot(c_d,F,'r')
xlabel('Drag Coeff')
ylabel('Force')
title('Variation of Drag force with Drag Coefficient')
figure(2)
plot(v,F,'r')
xlabel('Velocity')
ylabel('Force')
title('Variation of Drag force with Velocity')

Result:

The below are the plot showing the variation in Drag force with respect to Drag Coeff and Velocity. Here we can see that Variation in of drag force to the drag coeff is linear and the variation of Drag force wrto Velocity is varying parabolically as per our relation.

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