Finding the zeros of two functions

Hello,

I am trying to balance a ship in water and for that I need to find the value where the ship is balanced in the water.

I am first using fzero to balance the ship in the Z direction (up and down) and then I am using fzero again to balance it in rotation around the Y axis (pitch). Since the boyancy of the ship changes when the pitch is changed I would like to be able to do fzero in the Z direction and around the Y axis.

Is there a way to find the zeros of both these variables?

Thank you in advance!

    
% This is where the fzeros are used
% eta3 is the Z coordinate and eta5 is the rotation around the Y axis.
eta3start = -1;
[eta3] = fzero(@(eta3) ForceVertical(eta3,ship,state),eta3start);
state.eta(3) = eta3;
 
eta5start = 0;
[eta5] = fzero(@(eta5) MomentVertical(eta5,ship,state), eta5start);
state.eta(5) = eta5;

% This is how ForceVertical and MomentVertical look
function [F3] = ForceVertical(eta3,ship,state)
    state.eta(3) = eta3;
    [HS] = CalculateHydrostatics(ship,state);
    FB = HS.FB0(3);
    FG = ship.m * state.g * -1; % equal to F G in equation (11) in Rosén (2017) – remember the minus sign!;
    F3 = FB + FG; % according to (12)&(13) in Rosén (2017)
end

function [M3] = MomentVertical(eta5,ship,state)
    state.eta(5) = eta5;
    [HS] = CalculateHydrostatics(ship,state);
    M3 = cosd(eta5)*(-HS.CoB(1)+ship.CoG(1));
    
end
    
    

[Davide Masiello]

It seems you might be dealing with a system of non-linear equations.

To solve this kind of problem, you could use fsolve.

[Sam Chak]

Hi @Jonas Andersson,

Based on the code you provided, it appears that you are attempting to find the equilibrium point for a marine vessel in water. To begin, would it be possible for you to utilize a trial-and-error approach in order to identify solution sets that are relatively close to zero? This could help narrow down the search for the equilibrium point.

Answer 1

[Drishti]

Hi Jonas,

To utilize the MATLAB ‘fzero’ function to find the balanced point, you need to consider the convergence of both ‘eta3’ representing vertical displacement and ‘eta5’ representing pitch angle of the ship.

While utilizing the ‘fzero’ function, it is necessary to use iterative approach to further enhance the solution.

You can refer to the following code snippet to check the convergence of both the variables to find the balanced point:

%upto number of iterations you want to run this section
while ~converged && iteration < maxIterations     
    iteration = iteration + 1;
    % Solve for eta3
    [eta3] = fzero(@(eta3) ForceVertical(eta3, ship, state), eta3start);
    state.eta(3) = eta3;
    % Solve for eta5
    [eta5] = fzero(@(eta5) MomentVertical(eta5, ship, state), eta5start);
    state.eta(5) = eta5;
    % Check for convergence
    if abs(eta3 - eta3start) < tolerance && abs(eta5 - eta5start) < tolerance
        converged = true;
    else
        % Update starting points for the next iteration
        eta3start = eta3;
        eta5start = eta5;
    end
end

Furthermore, considering the interaction between vertical displacement and pitch resulting from changes in buoyancy, another workaround would be to use ‘fsolve’ function to find equilibrium in both the Z direction and around the Y-axis.

To utilize ‘fsolve’ function, you can refer to the below given code:

% Solve the system of equations
[solution, fval, exitflag, output] = fsolve(@(vars) BalanceEquations(vars, ship, state), initialGuess, options);
% Extract the solutions
eta3 = solution(1);
eta5 = solution(2);
% Update the state with the found solutions
state.eta(3) = eta3;
state.eta(5) = eta5;

For more information, you can refer to the MATLAB documentation of the ‘fsolve’ and ‘fzero’ functions:

https://www.mathworks.com/help/releases/R2023b/optim/ug/fsolve.html

https://www.mathworks.com/help/releases/R2023b/matlab/ref/fzero.html

I hope this resolves the issue.