# SPACAR Wiki

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parallel_flexure_guide

# SPACAR Light

## Example: Compute motion/deformations of a parallel flexure guide

This example shows the static simulation of a parallel flexure guide which moves from -20 mm deflection up to 20 mm deflection in x-direction (the degree of freedom of the parallel flexure guide). Goal is to evaluate the translational stiffness in the supporting directions (y- and z-direction) of the end-effector over the range of motion.

After a successful simulation, SPACAR Light returns the compliance matrix of each node (per loadstep) - see the Full Syntax List. For a compliance matrix C, it holds that u = C*f. Vector u = [ux;uy;uz;rx;ry;rz] consists of displacement components ux, uy, uz (in the x-,y- and z-direction, resp.) and rotation components rx, ry, rz (about the x-,y- and z-axis, resp.). Vector f = [fx;fy;fz;mx;my;mz] consists of force components fx, fy, fz (in the x-,y- and z-direction, resp.) and moment components mx, my, mz (about the x-,y- and z-axis, resp.). For the 'global' compliance matrix CMglob, the components are resolved in the global coordinate frame. This is the fixed (inertial) frame (which is also the axes frame indicated in SpaVisual with labels x, y and z). For the 'local' compliance matrix CMloc, the components are resolved in the local frame of the node (meaning that the coordinate frame rotates with the node). The stiffness matrix K, for which f = K*u, is the inverse of the compliance matrix C.

The figure bellow provides the stiffness values over the entire range. It can be clearly seen that with increasing deflection, support stiffness drops. Furthermore, the initial stiffness in y-direction is infinite as only the out-of-plane bending and torsion deformations are considered.

An example file for providing the input for SPACAR Light is provided below.

pfg_example.m
clear
clc

% addpath('spacar') %point this to the spacar folder

%% NODE POSITIONS
nodes = [-50e-3 0.0 0       %node 1
-50e-3 100e-3 0    %node 2
0 100e-3 0         %node 3
50e-3 100e-3 0     %node 4
50e-3 0 0];        %node 5

%node 3 is only added so that the system properties can be evaluated right there

%% ELEMENT CONNECTIVITY
elements = [1 2             %element 1: 1st leafspring
2 3             %element 2: 1st half of rigid body
3 4             %element 3: 2nd half of rigid body
4 5];           %element 4: 2nd leafspring

%% NODE PROPERTIES
nprops(1).fix = true;       %fix begin 1st leafspring
nprops(5).fix = true;       %fix end 2nd leafspring

nprops(3).force = [0 -10 0];%load of 10N on node 3 y-direction
nprops(3).mass = 1;         %1kg mass at node 3
nprops(3).displ_initial_x = -20e-3; %initial displacement, node 3 -20mm moved in x
nprops(3).displ_x = 40e-3;  %additional displacement, node 3 40mm moved in x

%% ELEMENT PROPERTIES
eprops(1).elems = [1 4];    %both leafsprings
eprops(1).emod = 210e9;     %steel
eprops(1).smod = 70e9;      %steel
eprops(1).dens = 7800;      %steel
eprops(1).dim = [30e-3 0.5e-3];
eprops(1).cshape = 'rect';
eprops(1).flex = [2 3 4];   %only bending and torsion flexible
eprops(1).orien = [0 0 1];
eprops(1).nbeams = 2;       %2 beams per element
eprops(1).color = [0.8549    0.8588    0.8667];

eprops(2).elems = [2 3];    %rigid body
eprops(2).dens = 2700;      %aluminium
eprops(2).dim = [30e-3 10e-3];
eprops(2).cshape = 'rect';
eprops(2).orien = [0 0 1];
eprops(2).color = [0.1686    0.3922    0.6627];

%% OPTIONAL
opt.gravity = [0 0 -9.81];  %gravity in z-direction
%higher resolution plots

%% DO SIMULATION
out = spacarlight(nodes,elements,nprops,eprops,opt);

%% POST PROCESS RESULTS
%plot stiffness values over the range of motion
Ky(:) = 1./out.node(3).CMglob(2,2,:); %y stiffness evaluated at node 3
Kz(:) = 1./out.node(3).CMglob(3,3,:); %z stiffness evaluated at node 3
x = out.node(3).p(1,:).*1000;         %position of the end-effector in mm (.*1000)

fig1 = figure;
hold on
plot(x,Ky)
plot(x,Kz)
grid minor
xlabel('x-position [mm]')
ylabel('y-stiffness [N/m]')
fig1.Children.YScale = 'log';        %plot with log yscale
fig1.Children.YLim = [1e4 1e10]; 