Static and Dynamic Compliance Analyses of Curved-Axis Flexure Hinges: A Discrete Beam Transfer Matrix

Abstract

A discrete beam transfer matrix method is introduced to enhance the existing approaches for the static and dynamic compliance solutions of curved-axis flexure hinges with variable curvatures and nonuniform profiles. An idea of discretizing curved-axis flexure hinges as a series of constant beam segments parallel to the centroidal axis is developed. As a result, only a concise beam transfer matrix with decoupled longitudinal and transverse components is needed to establish the compliance model. A step-by-step modeling procedure with simple formulas is provided as well qualifying for curved-axis and folded hinges. With this modeling idea, the small-deflection compliance matrix in the common sense of statics and particularly in a viewpoint of frequency-dependent dynamics can be simultaneously obtained. A typical curved-axis flexure hinge available in literature is analyzed and compared as a study case. In addition, the static and dynamic design for a compliant guiding mechanism comprised of folded flexure hinges is efficiently implemented as a case study.

Enabling the transfer matrix method to model serial–parallel compliant mechanisms including curved flexure beams

“…Some improved transfer matrix methods have also been proposed for a limited class of serial-parallel structures. [55][56][57][58][59] However, it would be necessary to derive complex formulas in these methods, or such methods require reliance on other approaches, such as the dynamic stiffness matrix method.…”

“…The curved-axis beams (2), ( 4), ( 6), ( 8), ( 10), ( 12), ( 14), ( 16), (18), and ( 20) are further discretized into a series of constant beam members along the centroidal axis with the discrete number of 100. 56 Consequently, the transfer matrix of every corrugated flexure hinge is calculated with the length l i and orientation angle θ i of each beam segment listed in Table 3 as follows:…”

“…Some improved transfer matrix methods have been developed that can be used for a class of serial-parallel compliant mechanisms. [55][56][57][58][59] All these studies, however, suffer from the limitation of use of a certain class of serial-parallel configurations. Castigliano's second theorem has been adopted to analytically analyze curved flexure hinges as well, 60 but it relies on cumbersome inner force analysis and is still not suitable for complex serial-parallel configurations.…”

“…Noticeably, successful attempts on the use of the transfer matrix method have been reported in some previous studies for curved flexure hinges and a limited class of serial-parallel compliant mechanisms by deriving additional formulas. [55][56][57][58][59] Differing from these past efforts, a generalized modeling procedure is developed with a single transfer matrix of Timoshenko straight beams without deriving laborious formulas or combining with other methods, for example, the dynamic stiffness matrix method. Detailed derivation could be avoided with the proposed method, which is applicable for fast parametric modeling of small-deformation compliant mechanisms and beam structures with curved flexure hinges in serial-parallel configurations.…”

“…In addition, a closed-loop chain is considered in the model topology, which is relatively complicated and even challenging to formulate using some of the previous transfer matrix methods. [55][56][57][58][59] For example, rigid-body beam structures with closed-loop chains were modeled by making cuts with the requirement of an extra modeling process. 67,68 Such a modeling idea provides a novel approach to deal with the closed-loop chains, which are not straightforward to handle in the analysis of compliant mechanisms with curved flexure beams in serial-parallel configurations.…”