Analyzing overconstrained design of compliant mechanisms

Whereas the use of compliant mechanisms is favorable for high precision applications, the constraints must be dealt with carefully. In an overconstrained design the actual natural frequencies and stiffnesses can differ considerably from their intended values. For this reason the awareness and possibly the avoidance of an overconstrained condition is important. We have developed a kinematic analysis with which underconstraints and overconstraints can be detected. A finite element based multibody approach is applied which offers a flexible beam element that is particularly suited to model the wire and sheet flexures frequently encountered in compliant mechanisms. For each element a fixed number of independent discrete deformations are defined that are invariant under arbitrary rigid body motions of the element. In the kinematic analysis only deformations associated with low stiffnesses are allowed, whereas the remaining deformations are prescribed zero. A singular value decomposition is used to determine the rank of the Jacobian matrix associated with the dependent nodal coordinates. Column and row rank deficiency indicate an underconstrained and overconstrained system, respectively. For an overconstrained system a statically indeterminate stress distribution ca n be derived from the left singular matrix. In this way the overconstraints can be visualized clearly as is illustrated with examples of compliant straight guidance mechanisms. The possible solutions to eliminate the overconstraints are found easily from the visualization.