About Constraint for Difference between two Displacements

This relative displacement constraint is used in the stiffness optimization task of the shift fork model, see the following figure.

The two arms of the shift fork have unequal displacements. This is caused by the different lengths of the arms. In order to get a working component, the displacements of the two arms have to be nearly identical. If the displacements are different, then the gear wheel is not able to be moved with the shift-fork.

Formulation of the Optimization Problem

The optimization problem will be solved with the sensitivity-based topology optimization. The optimization task is formulated as

$\min (U)$

$u_{1} - u_{2} \leq u_{\max D i f f e r e n c e}$

$V o l \leq v o l_r e s t r i c t$

with U being the sum of the strain energy over all elements, $u_{1}$ and $u_{2}$ the displacements of the nodes in a given direction (x, y or z), Vol the relative material volume, vol_restrict the value of the volume constraint and $u_{\max D i f f e r e n c e}$ the constraint value.