CODEWITHSUNDEEP
matlab
Kieran J
Kieran J

Reputation: 63

Iteratively solving a non-linear equation

I'm trying to iteratively solve the following non-linear equation, as shown in the below extract from "Propeller Slipstream Model for Small Unmanned Aerial Vehicles" W. Khan, M. Nahon, R. Caverly pp3

paper extract

So far I have the following code which I hope is relatively self-explanatory, although I have a feeling I am using the wrong method by trying to use the solve function:

% III A Induced Velocity at Propeller Plane
%Data from Table I
Sno = [1,2,3,4,5,6,7] 
radialLoc = [5,20,40,60,80,100,127]                             % r
chord = [12.374,16.230,22.850,28.215,28.902,26.310,13.066]      % c
chordlinePitchAngle = [25,26.5,21.89,14.39,9.80,4.75,6.59]      % theta
zeroLiftAngleOfAttack = [0,0,-9.66,-8.70,-7.40,-10.70,-8.40]    % a0
liftCurveSlope = 6.28                                           % Cla
dragCoefficient = 0.02                                          % Cd

%Other data
angularVel = 1710                                                  % Omega (1710RPM /6710RPM)
numBlades = 2                                                   % N
forwardVel = 0                                                  % Vx

%data to be determined
phi = 0

eqn = angularVel*numBlades*chord(1)*(liftCurveSlope*(chordlinePitchAngle(1)-zeroLiftAngleOfAttack(1)-phi)-dragCoefficient*tan(phi))-8*pi*sin(phi)*(angularVel*radialLoc(1)*tan(phi)-forwardVel)
a = solve(eqn,phi)

any help with solving the equation will be appreciated.

note I realise I will end up with 7 phi values, currently I am just trying to find the first one and will go from there.

Full paper can be accessed at https://www.researchgate.net/profile/Waqas_Khan15/publication/264773240_Propeller_Slipstream_Model_for_Small_Unmanned_Aerial_Vehicles/links/54c289670cf2911c7a4922cd/Propeller-Slipstream-Model-for-Small-Unmanned-Aerial-Vehicles.pdf

Companion paper (reference 21) can be accessed at the following link: (although I have read the relevant sections and it does not appear to have any additional relevant information) https://ieeexplore.ieee.org/document/6523983

Upvotes: 0

Views: 110

Answers (1)

joni
joni

Reputation: 7157

The solve method is for solving an equation symbolically. If you want to solve it numerically, you have to use fzero instead.

By the way, there are a few inconsistencies (assuming I understood your problem correctly)

  • With angularVel=0 your equation would become 0=0.

  • There's phi(1) in your definiton of eqn but in your case phi is just a variable, not an array.

  • Why are there just 6 values for radialLoc? I guess one is missing.

Anyway, here's a way to do it with fzero:

%Other data
angularVel = 1710/6710;   % Just a guess                % Omega (1710RPM /6710RPM)
numBlades = 2;                                          % N
forwardVel = 0;                                         % Vx

%data to be determined
phi_sol = zeros(7,1);

for i=1:6
    % Define the function handle
    eqn = @(phi) angularVel*numBlades*chord(i)*(liftCurveSlope * ...
    (chordlinePitchAngle(i)-zeroLiftAngleOfAttack(i)-phi) - dragCoefficient ...
    * tan(phi))-8*pi*sin(phi)*(angularVel*radialLoc(i)*tan(phi)-forwardVel);

    % Find phi such that eqn(phi) = 0 (uses phi=0 as initial guess for fzero)
    phi_sol(i) = fzero(eqn, 0);
end

Upvotes: 1

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