iterative equation two unknown for parachute

I need to calculate the radius of a recovery main parachute and I have two equation with two unknowns.

v=sqrt(2*Weight/(C_d*Rho*A)); % terminal velocity C_d-drag coefficient

r=sqrt((2*m*g)/(pi*C_d*Rho*velocity^2)); radius of the chute

As you can see, I need the terminal velocity to solve for radius and vice versa. I am a complete beginner at MATLAB and I have no idea how to compute it iteratively using MATLAB. Anyu help would be greatly appreciated. I tried sth myself and ofc it does not work. You can see it below.

%calculates the radius and terminal velocity of parachute during recovery

m=40; %mass of engine [kg]

g=9.81; % gravitational acceleration [m/s^2]

v=0; %terminal velocity [m/s]

C_d=1.2; % drag coefficient

Rho=1.229; % air density kg/m^3

Weight=m*g;

r=0; %radius of chute [m]

Drag= C_d*(Rho*v^2/2)*pi*r;

Drag=Weight; %at terminal velocity

A=3.2; %projected area of the chute [m^2]

tol= 0.05 %tolerance

while error <= tol

v_old=v;

r_old=r;

v=v+sqrt(2*Weight/(C_d*Rho*A));

r=r+sqrt((2*m*g)/(pi*C_d*Rho*velocity^2));

error=abs(v-1-v)

if error <= tol

break

end

end

[Suleyman Sahin]

I made some changes. the code works now but does not converge at all even though I change the tolerance

Answer 1

[William Rose]

@Suleyman Sahin,

[edit: correct typographical errors]

If you have 2 independent equations and 2 unknowns, you can solve. Matlab's fsolve() could do it.

Your equations are

and , where .

(I assume A in your first equation is . Your code does not do this, but I think it should. )

As you can see, your two equations are not independent. They are two versions of the same equation. Therefore you really have one equation. It defines a reciprocal relationship between radius ( r) and velocity (v). For a given v, there is a corresponding value of r. And vice versa. Therefore, if you have a maximum acceptable terminal velocity, you can use that to compute the chute size needed.

[Suleyman Sahin]

Thanks a lot. Your answer was very helpful.

[William Rose]

@Suleyman Sahin, you are welcome.