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spatial flexure mechanism analysis - energy-based geometric stiffness modeling

Marijn Nijenhuis is a PhD student in the research group Applied Mechanics (AM). His supervisor is D.M. Brouwer from the Faculty of Engineering Technology.

Flexure mechanisms are used in precision applications for their ability to guide motion with high repeatability. While traditional rigid-link mechanisms move due to rolling or sliding components that induce hysteresis, flexure mechanisms move solely due to elastically deforming components. This means that flexure mechanisms operate without friction, backlash, stick-slip and wear, resulting in low hysteresis. The elastic deformation also presents challenges. The accompanying stress limits the range of motion by yielding and fatigue, and the changing configuration decreases the support stiffness. This thesis presents research on modeling techniques and design principles for flexure mechanisms in order to improve the stiffness characteristics of flexure mechanisms with a large range of motion.

The first thesis topic, on modeling techniques, concerns the sheet flexure, which is a common elastic component of flexure mechanisms. While it is often analyzed in a simplified form, e.g. by assuming planar deformation or linearized stiffness, the deformation in practice is spatial and sufficiently large that nonlinear effects due to the geometric stiffness are significant. This thesis describes the formulation of a parametric spatial stiffness model for sheet flexures at moderate deformations. This closed-form model is expressed in terms of design parameters, such as error motions, degrees of freedom and applied loads. A mixed variational principle with a specific set of interpolation functions is shown to be well suited for the derivation of the third-order terms that govern the support stiffness decrease. The geometrically nonlinear effects of bending, shear, elongation, torsion and warping are included, so that sheet flexures with a large width-to-length ratio can be accurately modeled. The energy-based discretization approach is also extended to model parallel assemblies of sheet flexures.

The sheet flexure model is implemented numerically as a beam element in multibody software. It enables faster simulations of flexure mechanisms containing a large number of sheet flexures, because a single such beam element per sheet flexure generally suffices. This is demonstrated with various numerical models of practical large range of motion flexure mechanisms.

The second thesis topic concerns the exact-constraint design principle, which is commonly applied to flexure mechanisms to ensure deterministic behavior, but at the cost of reduced robustness, support stiffness, load capacity and usually an increased complexity of design. To explore the potential benefit of overconstrained design in flexure mechanisms, this thesis investigates the misalignment sensitivity associated with overconstrained design. An elementary two-flexure cross-hinge with a single overconstraint serves as a case study in an experimental, numerical and analytical analysis of the stiffness effects of inadvertent stress due to misalignment. It is shown that small, typically unintended, misalignments in the overconstrained direction can affect the system behavior significantly. Numerical models show that the effect can be accurately predicted. The models indicate that detailed aspects, such as flexure thickness variations, parasitic compliance and constrained warping deformation, favorably decrease the misalignment sensitivity. An analytical buckling analysis provides an explicit upper bound for the critical misalignment of a variety of cross-hinge designs.

A second case study demonstrates the principle of dynamically stiffened exact-constraint design. It enables the use of overconstrained design for its increased dynamic performance by reducing the misalignment sensitivity with a custom synthesized viscoelastic material. Experiments show that the elastomer can compensate for unintended misalignments without significant stress buildup, while improving the dynamic performance in terms of a higher first parasitic natural frequency. Analytical and numerical model predictions match with the measurements.