Elastic Averaging in Flexure Mechanisms: a Multi-beam Parallelogram Flexure Case-study

Over-constraint is an important concern in mechanism design because it can lead to a loss in desired mobility. In distributed-compliance flexure mechanisms, this problem is alleviated due to the phenomenon of elastic averaging, thus enabling performance-enhancing geometric arrangements that are otherwise unrealizable. The principle of elastic averaging is illustrated in this paper by means of a multi-beam parallelogram flexure mechanism. In a lumped-compliance configuration, this mechanism is prone to over-constraint in the presence of nominal manufacturing and assembly errors. However, with an increasing degree of distributed-compliance, the mechanism is shown to become more tolerant to such geometric imperfections. The nonlinear load-stiffening and elastokinematic effects in the constituent beams have an important role to play in the over-constraint and elastic averaging characteristics of this mechanism. Therefore, a parametric model that incorporates these nonlinearities is utilized in predicting the influence of a representative geometric imperfection on the primary motion stiffness of the mechanism. The proposed model utilizes a beam generalization so that varying degrees of distributed compliance are captured using a single geometric parameter. INTRODUCTION In the design of precision mechanisms, the choice of flexure-based mechanisms over traditional linkage mechanisms is obvious. Lack of friction and backlash in the former provides an exceptional level of precision [1]. However, the biggest drawback in employing a flexure-based design is its reduced range of motion. This may be improved, to some extent, by choosing distributed-compliance over lumped-compliance in the flexure mechanism topology. However, each choice has its own set of distinct advantages and disadvantages. Distributed compliance results in a better distribution of strains, lower stresses, and therefore relatively larger motion ranges. On the other hand, lumped-compliance mechanisms provide excellent stiffness along constraint directions, which leads to improved performance in terms of error motions and stiffness. This tradeoff between range and performance is representative of flexure mechanism design in general [2]. In this paper, we consider the example of a multiparallelogram mechanism. Fig.1 illustrates three potential embodiments of this mechanism – traditional linkage, lumpedcompliance flexure, and distributed-compliance flexure designs. For the flexure-based designs of Fig.1 to be effective motion guides, it is generally desirable to maximize the stiffness along the axial or constraint direction, while keeping the transverse or primary direction stiffness as low as possible. However, any attempt to increase the axial stiffness proportionally by increasing the flexure beam thickness results in a cubic order increase in the transverse stiffness, which is highly undesirable. Therefore, it is common design knowledge to introduce additional parallel beams [3-4], which result in a proportional increase in the axial as well as the transverse stiffness values. Even though a multi-parallelogram mechanism is desirable for these reasons, mobility remains to be a concern. While it appears reasonable that the linkage mechanism of Fig.1a should have a single degree of mobility, an application of Grubler’s criterion [5] reveals zero Degree of Freedom (5 links, 6 Rjoints). Although counter-intuitive, it should be recognized that this generic criterion does not take into account the fact that the