This walkthrough given an example of the simulation of a parallel flexure guide.

Let's consider a parallel flexure guide consisting of two parallel leafsprings and an intermediate rigid body. For this problem, we consider L=100mm and W=50mm. Furthermore, we consider steel leafsprings with a width of 50mm and a thickness of 0.5mm. The rigid intermediate body is made from aluminium and has a cross-section of 50x50mm. A schematic overview is given in the figure below.

Figure: Parallel flexure guide (from janssenprecisionengineering.com)

To set up the simulation, we first need to define the Node postions, Element connectivity, Node properties and the Element properties.

We start by defining the positions of the nodes where we place the origin at node 1. nodes is an n_n×3 matrix for defining the positions of the nodes. Each row represents a node (there are n_n nodes). The row number determines the number a node gets. The first column contains the x-coordinate of the node position, the second column the y-coordinate, and the third column the z-coordinate.

clc    %clear command window
clear  %clear workspace
 
%% NODE POSITIONS
L = 0.1;    %[m]
W = 0.05;   %[m]
 
%% NODE POSITIONS
nodes = [0 0 0;     %node 1
         0 L 0;     %node 2
         W L 0;     %node 3
         W 0 0];    %node 4

Next the connectivity between elements is defined. Element connectivity is defined by an n_e×2 matrix for defining elements. Each row represents an element (there are n_e elements). The row number determines the number an element gets. Each element connects two nodes. The first column contains the number of the node that the element is connected to on one side, the second column the node number that the element is connected to on the other side.

%% ELEMENT CONNECTIVITY
elements= [1 2;     %leafspring between node 1 and 2 
           2 3;     %intermediate body
           3 4];    %leafspring between node 3 and 4 

Node properties are defined by a structure array to assign properties to nodes, such as boundary conditions, applied loads, and inertia. The usage is nprops(i).field = value; to assign property field with value value to node i. For a full syntax list with all possible inputs, see SPACAR Light syntax.

First of all, node 1 and 4 are fixed:

%% NODE PROPERTIES
nprops(1).fix = true;       %fix node 1
nprops(4).fix = true;       %fix node 4

Furthermore, we would like for the parallel flexure guide to start 10mm deflected to the left and let it move 20mm to the right. For this purpose, we pre-describe the position of node 2 in x-direction:

nprops(2).displ_initial_x =-0.01; %start with node 2 displaced 10mm to the left
nprops(2).displ_x =         0.02; %displace node 2 20mm to the right

At last we apply a load of 5N in Z-direction on node 2 and 3

nprops(2).force_initial = [0 0 5]; %initial load of 5N i z-direction 
nprops(3).force_initial = [0 0 5]; %initial load of 5N i z-direction 

Element properties are defined by a structure array for assigning properties to elements, such as dimensions, material constants, flexibility, and visual appearance. Properties are created in sets, such that the same set of properties can be assigned to multiple elements.

The usage is eprops(i).field = value; to assign property field with value value to set i. For a full syntax list with all possible inputs, see SPACAR Light syntax.

The first element property set is assigned to both leafsprings (element 1 and 3), selected material is steel, width = 50mm, thickness 0.5mm and torsional and out-of-plane bending deformations are considered flexible. Furthermore, two spacar-beams are used to model flexibility of the leafsprings.

%% ELEMENT PROPERTIES
 
%first element property set
eprops(1).elems = [1 3];        %assing property set 1 to element 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).dim = [0.05 0.0005];  %crossectional dimension [w t] in [mm]
eprops(1).cshape = 'rect';      %crossectional shape rectangular
eprops(1).flex = [2 3 4];       %torsional (2) and out-of-plane bending (3,4) deformations are flexible
eprops(1).orien = [0 0 1];      %width-direction of leafspring oriented in z-direction
eprops(1).nbeams = 2;           %Leafspring simulated with 2 spacar-beams for increased accuracy. 

The second element property set is assigned to the intermediate body (element 1 and 3), selected material is aluminium, width = 50mm and thickness = 50mm.

%second element property set
eprops(2).elems = [2];          %assing property set 2 to element 2
eprops(2).dens = 2700;          %density   [kg/m^3]
eprops(2).dim = [0.05 0.05];    %crossectional dimension [w t] in [mm]
eprops(2).cshape = 'rect';      %crossectional shape rectangular
eprops(2).orien = [0 0 1];      %width-direction of leafspring oriented in z-direction

At this moment we have defined the basic configuration for our simulation and we can analyze the overconstraints in our model. For an initial suggestion for the releases to resolve the overconstraints, we can call spacarlight with

 out = spacarlight(nodes,elements,nprops,eprops);

When the system is properly defined, the number of overconstraints and the suggestion for released deformation modes is provided in the command window like below:

Warning: System is overconstrained; releases are required in order to run static simulation.
A suggestion for possible releases is given in the table below.

 
Number of overconstraints: 7

    Element    def_1    def_2    def_3    def_4    def_5    def_6
    _______    _____    _____    _____    _____    _____    _____

    1          1        1        1        1        1        1    
    2          1        1        1        1        1        1    
    3          1        1        1        1        1        1    

It appears this problem contains 7 overconstraints and a suggestion is provided. Now we can add a single released deformation mode with the help of table and redo the overconstrained analyses. Therefore, we define the rls structure array which assigns 'released' deformation modes. The usage is rls(i).def = value; to assign the released deformations modes value to element i. More information on using the rls structure is provided at SPACAR Light syntax.

To release the second deformation mode of element two, we use:

%% RELEASES
rls(1).def = [2];

If we now redo the overconstraint analysis (call out = spacarlight(nodes,elements,nprops,eprops,rls);), it will show 6 overconstraints and an updated release table. By repeating this process six more times and by adding a single released deformation at each step, a full set of releases can be obtained (multiple solutions/combinations are possible):

%% RELEASES
rls(1).def = [1 2 3 4 5 6];
rls(3).def = [3];

When an exact constrained model is obtained, spacarlight automatically runs the static simulation when it is called an a visualization of the model will appear. For more information on the visualization GUI, see ????.

Furthermore, simulation results are stored in the output argument out.

example.m

clear
clc
 
L = 0.1;    %[m]
W = 0.05;   %[m]
 
%% NODE POSITIONS
nodes = [0 0 0;     %node 1
         0 L 0;     %node 2
         W L 0;     %node 3
         W 0 0];    %node 4
 
 
%% ELEMENT CONNECTIVITY
elements= [1 2;     %leafspring between node 1 and 2 
           2 3;     %intermediate body
           3 4];    %leafspring between node 3 and 4 
 
 
 
%% NODE PROPERTIES
nprops(1).fix = true;       %fix node 1
nprops(4).fix = true;   %fix node 4
 
nprops(2).displ_initial_x =-0.01; %start with node 2 displaced 10mm to the left
nprops(2).displ_x =         0.02; %displace node 2 20mm to the right
 
nprops(2).force_initial = [0 0 5]; %initial load of 5N in z-direction 
nprops(3).force_initial = [0 0 5]; %initial load of 5N in z-direction 
 
 
%% ELEMENT PROPERTIES
 
%first element property set
eprops(1).elems = [1 3];        %assing property set 1 to element 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).dim = [0.05 0.0005];  %crossectional dimension [w t] in [mm]
eprops(1).cshape = 'rect';      %crossectional shape rectangular
eprops(1).flex = [2 3 4];       %torsional (2) and out-of-plane bending (3,4) deformations are flexible
eprops(1).orien = [0 0 1];      %width-direction of leafspring oriented in z-direction
eprops(1).nbeams = 2;           %Leafspring simulated with 2 spacar-beams for increased accuracy. 
 
%second element property set
eprops(2).elems = [2];          %assing property set 2 to element 2
eprops(2).dens = 2700;          %density   [kg/m^3]
eprops(2).dim = [0.05 0.05];    %crossectional dimension [w t] in [mm]
eprops(2).cshape = 'rect';      %crossectional shape rectangular
eprops(2).orien = [0 0 1];      %width-direction of leafspring oriented in z-direction
 
 
%% RELEASES
rls(1).def = [1 2 3 4 5 6];
rls(3).def = [3];
 
 
%% DO SIMULATION
out = spacarlight(nodes,elements,nprops,eprops,rls);