Optimising bracket with holes

I am optimizing the thickness of a bracket in MATLAB while minimizing mass . The bracket has: A central hole (20 mm diameter) for actuator mounting. Four corner holes (10 mm diameter each) for attachment. when calculating the constraints and objective function do take into consideration the holes? and if i was to, how would i modify the deflection equation? as deflection has to not exceed 15mm with a fctor of safety of 1.5mm

Answer 1

[Hitesh]

Hi Rea,

Optimizing the thickness of a bracket while minimizing its mass requires both the geometry and the constraints, including the presence of holes. The primary objective is to minimize the mass of the bracket.

Constraints

• Deflection Constraint: The deflection must not exceed 15 mm with a factor of safety of 1.5.
• Deflection Calculation: For a plate with holes, the deflection calculation becomes more complex. However, a simplified approach is to use the deflection formula for a rectangular plate and adjust for the holes using correction factors or finite element analysis (FEA) for more precise results.
• The correction factor (K) is determined through empirical testing or simulations, as it depends on the size and location of the holes.

Kindly refer to the following MATLAB code and mentioned comments for better understanding of optimizing bracket with holes.

function optimize_bracket_thickness
    % Material properties
    rho = 7800; % Density in kg/m^3
    E = 210e9; % Young's modulus in Pa

    % Bracket dimensions
    L = 0.1; % Length in meters
    W = 0.1; % Width in meters

    % Initial guess for thickness
    t0 = 0.01; % Initial thickness in meters

    % Optimization
    options = optimoptions('fmincon', 'Display', 'iter');
    t_opt = fmincon(@(t)objFun(t, L, W, rho), t0, [], [], [], [], 0.001, [], @(t)nonlcon(t, L, W, E));

    fprintf('Optimal thickness: %.4f m\n', t_opt);
end

function mass = objFun(t, L, W, rho)
    % Calculate volume of the bracket
    % L and W are the length and width of the bracket.
    % t is the thickness of the bracket (the variable to be optimized).
    % V is the volume of the material removed due to the holes.
    
    V = (L * W * t) - (pi * (0.01)^2 * t + 4 * pi * (0.005)^2 * t);
    % Calculate mass
    mass = rho * V;
end

function [c, ceq] = nonlcon(t, L, W, E)
    % Deflection constraint
    q = 1000; % Load per unit area in N/m^2
    C = 0.013; % Example constant for boundary conditions
    % The deflection (delta) for a uniformly loaded rectangular plate can be calculated by 
    delta = (q * L^4) / (C * E * t^3);
    
    % Adjust for holes (example correction factor)
    K = 1.1;
    delta_adjusted = K * delta;

    % Nonlinear inequality constraints
    c = delta_adjusted - 0.01; % Deflection must be less than 10 mm
    ceq = [];
end

optimize_bracket_thickness

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

Optimal thickness: 0.0016 m

For more information regarding "fmincon" function, kindly refer to the following MATLAB documentation:

https://www.mathworks.com/help/optim/ug/fmincon.html