This example provides a shape optimization of a single cross flexure by using matlab's
fmincon function. For this optimization a crosshinge is considered with a mass and inertia connected to node 3. Furthermore, the optimization objective is to maximize the first parasitic frequency of the system (note: rotation of the cross-hinge is prescribed to de desired rotation angle. Therefore, the first parasitic frequency of the system is given by the first eigenfrequency). A rotation of 0.5rad, a maximum stress of 300MPa and and a maximum actuation force of 2.5Nm is considered.
The cost function of the optimization is defined by the inverse of the first eigenfrequency. Additionally, a limit on the maximum stress and actuation force is implemented by applying a penalty on the value of the cost function. The matlab function which calls spacarlight and computes the cost function is defined by:
function [cost, out] = run_sim(P, Silent)% EXAMPLE SCRIPT FOR RUNNING SPACAR LIGHT if nargin == 1 Silent = true; %run in silent mode when called by optimizer end % DESIGN PARAMETERS H = P(1); B = P(2); w = P(3); t = P(4); % Optimization parameters stressmax = 300e6; %Max stress [MPa] Mmax = 2.5; %Max actuation moment [Nm] Stroke = 0.5; %Max stroke [Rad] %% NODE POSITIONS % x y z nodes = [ 0 0 0; %node 1 0 H 0; %node 2 B H 0; %node 3 B 0 0]; %node 4 %% ELEMENT CONNECTIVITY % p q elements = [ 1 3; %element 1 2 3; %element 2 2 4]; %element 3 %% NODE PROPERTIES %node 1 nprops(1).fix = true; %Fix node 1 %node 3 nprops(3).rot_z = Stroke; nprops(3).mass = 1; nprops(3).mominertia = [1e-3 0 0 1e-3 0 1e-3]; %node 4 nprops(4).fix = true; %Fix node 4 %% ELEMENT PROPERTIES %Property set 1 eprops(1).elems = [1 3]; %Add this set of properties to elements 1 and 3 eprops(1).emod = 210e9; %E-modulus [Pa] eprops(1).smod = 70e9; %G-modulus [Pa] eprops(1).dens = 7800; %Density [kg/m^3] eprops(1).cshape = 'rect'; %Rectangular cross-section eprops(1).dim = [w t]; %Width: 50 mm, thickness: 0.2 mm eprops(1).orien = [0 0 1]; %Orientation of the cross-section as a vector pointing along "width-direction" eprops(1).nbeams = 2; %Number of beams used to model this element eprops(1).flex = 1:6; %Consider all deformation modes (1 to 6) as flexible eprops(1).color = 'grey'; %Color eprops(1).opacity = 0.7; %Opacity eprops(1).cw = true; %Constrain warping at nodes (typical condition for flexure clamped at both ends). For a "free" end, warping is typically not constrained %Property set 2 eprops(2).elems = 2; %Add this set of properties to element 2 eprops(2).dens = 3000; %Density [kg/m^3] eprops(2).cshape = 'rect'; %Rectangular cross-section eprops(2).dim = [w 10e-3]; %Width: 50 mm, thickness: 10 mm eprops(2).orien = [0 0 1]; %Orientation of the cross-section as a vector pointing along "width-direction" eprops(2).nbeams = 1; %1 beam for simulating this element (as it is rigid an no more elements are required) eprops(2).color = 'darkblue'; %% OPTIONAL ARGUMENTS opt.filename = 'crosshinge'; %Filename opt.calccompl = false; %Disable calculation of compliance matrices (can reduce computation time for large simulations) opt.silent = Silent; %Run in silent mode %% CALL SPACAR_LIGHT AND COMPUTE COST FUNCTION try out = spacarlight(nodes, elements, nprops, eprops, opt); cost = 1/min(out.freq(1,:)); %cost function (inverse of first frequency ) stress = max(out.stressmax); %max stress Mact = max(out.node(3).Mreac(3,:));%max actuation moment %add penalty if limits are exceeded if stress > stressmax cost = cost * (1 + (stress-stressmax)/stressmax)^5; end if Mact > Mmax cost = cost * (1 + (Mact-Mmax)/Mmax)^5; end catch %if sumulation fails, cost is inf cost = inf; end
An example to conduct the optimization is provided below, taking into account additional boundary conditions on the design parameters of the cross-hinge. See matlab's documentation on
fmincon for more information.
Note: prescribing input rotation is only supported for SPACAR light version 1.27 or higher.