Integrate with two dependent variables and solve a system

Hello all!

I will try to explain my problem:

I have two equations and two unknowns "w" and "T". One of the equations is an integral from a to b relative to a variable "r". The integral equation contains two terms that are function of both "w" and "r" and this term calls a value on a interpolated plot. Those terms are defined as Cd and Cl.

The equations are the following, knowing that:

• mult_Cl = CL(-((atan(20/(pi*r)) - pi/18) - atan(w/(omega*r)))*180/pi)* omega*r/(sqrt(omega.^2*r.^2+w.^2));
• mult_Cd = CD(-((atan(20/(pi*r)) - pi/18) - atan(w/(omega*r)))*180/pi)* omega*r/(sqrt(omega.^2*r.^2+w.^2));

and

• CL and CD are interpolated functions

T = integral ( rho/2*(omega.^2*r^2 + w.^2)* ( 4 - 2.35294*(r/50 - 0.1)) *( mult_Cl + mult_Cd ),5,50);

T = 2*pi*rho*R.^2*(Vinf - w)*w

----

Where rho, omega, Vinf are known values

I need to solve both T equations at the same time to find a value of "w" for both.

I have never tried that before, so I am kinda lost. Thank you very much!!

Answer 1

[Torsten]

Use MATLAB's fzero on the function

f=@(w)integral ( rho/2*(omega.^2*r^2 + w.^2)* ( 4 - 2.35294*(r/50 - 0.1)) *( mult_Cl + mult_Cd ),5,50) - 2*pi*rho*R.^2*(Vinf - w)*w

to get w.

Then insert in one of the above equations to get T.

Best wishes

Torsten.

[Ricardo Rodrigues]

Hello Torsten, thank you for your answer!

I tried and got the following error

Error using fzero (line 288)

FZERO cannot continue because user-supplied function_handle ==> @(w)integral(rho/2*(omega.^2*r.^2+w.^2)*(4-2.35294*(r/50-0.1))*(CL(-((atan(20./(pi*r))-pi/18)-atan(w./(omega*r)))*180./pi)*omega*r./(sqrt(omega.^2*r.^2+w.^2))+CD(-((atan(20./(pi*r))-pi/18)-atan(w./(omega*r)))*180./pi)*omega*r./(sqrt(omega.^2*r.^2+w.^2))),5,50)-2*pi*rho*R.^2*(Vinf-w)*w failed with the error below.

Inner matrix dimensions must agree.

Below is the code I have so far. Once again, thanks for the help!

clc; clear all; close all;

%%read Cl and Cd from excel sheet
Y = xlsread('NACA_23012.xlsx');

Separate values

AOA = Y(:,1);
Cl = Y(:,2);
Cd = Y(:,5);
%Cl interpolation
figure(1)
x=AOA;
y=Cl;
xi = -90:0.1:90; 
CL = interp1(x,y,xi); 
plot(AOA,Cl,'o',xi,CL)
%Cd interpolation
figure(2)
CD = interp1(AOA,Cd,xi); 
plot(AOA,Cd,'o',xi,CD)

Parameters

R = 50;
R_cutout = 0.1*R;

r = R_cutout:0.1:R;

c = 4 - 2.35294*(r/50 - 0.1);

%plot blade
figure(3)
plot(r,3/4*c,r,1/4*c)
ylim([-10 10])

plot twited blade

theta = atan(20./(pi.*r)) - pi./18;

figure(4)
plot(r,(3/4).*c.*cos(theta),r,(1/4).*c.*cos(theta))
ylim([-10 10])
% mult_Cl = CL(-((atan(20./(pi*r)) - pi/18) - atan(w./(omega*r)))*180./pi)* omega*r./(sqrt(omega.^2*r.^2+w.^2));
% mult_Cd = CD(-((atan(20./(pi*r)) - pi/18) - atan(w./(omega*r)))*180./pi)* omega*r./(sqrt(omega.^2*r.^2+w.^2));

R = 50;
R_cutout = 0.1*R;
omega = 2*pi/3;
rho = 1.225;
Vinf = 22;

f=@(w)integral ( rho/2*(omega.^2*r.^2 + w.^2)* ( 4 - 2.35294*(r/50 - 0.1)) *( CL(-((atan(20./(pi*r)) - pi/18) - atan(w./(omega*r)))*180./pi)* omega*r./(sqrt(omega.^2*r.^2+w.^2)) + CD(-((atan(20./(pi*r)) - pi/18) - atan(w./(omega*r)))*180./pi)* omega*r./(sqrt(omega.^2*r.^2+w.^2)) ),5,50) - 2*pi*rho*R.^2*(Vinf - w)*w;

Z = fzero(f,1)

[Torsten]

CL and CD are arrays, not functions.

So CL(-atan...) and CD(-atan...) does not make sense to me.

Best wishes

Torsten.

[Ricardo Rodrigues]

You are right!

Now CL and CD are functions of x, but now the error I get is

FZERO cannot continue because user-supplied function_handle ==> @(w)integral(rho./2.*(omega.^2.*r.^2+w.^2).*(4-2.35294.*(r./50-0.1)).*(CL(-((atan(20./(pi.*r))-pi./18)-atan(w./(omega.*r))).*180./pi).*omega.*r./(sqrt(omega.^2.*r.^2+w.^2))+CD(-((atan(20./(pi.*r))-pi./18)-atan(w./(omega.*r))).*180./pi).*omega.*r./(sqrt(omega.^2.*r.^2+w.^2))),5,50)-2.*pi.*rho.*R.^2.*(Vinf-w).*w failed with the error below.

 First input argument must be a function handle.

[Torsten]

The first argument to "integral" must be a function handle, not an expression.

Best wishes

Torsten.