An analytical method for calculating the natural frequencies of spatial compliant mechanisms

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.…”

“…Nevertheless, the traditional transfer matrix approach is mainly suitable for structures with serial configurations. Some improved transfer matrix methods have been developed that can be used for a class of serial–parallel compliant mechanisms 55–59 . All these studies, however, suffer from the limitation of use of a certain class of serial–parallel configurations.…”

“…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.…”