On Systematic Errors of Two-Dimensional Finite Element Modeling of Right Circular Planar Flexure Hinges

Abstract

This paper discusses the finite element method (FEM) based modeling of the behavior of typical right circular flexure hinges used in planar compliant mechanisms. Such hinges have traditionally been approximated either by simple beams in the analytical approach or very often by two-dimensional (2D) plane stress elements when using the FEM. The three-dimensional (3D) analysis presented examines these approximations, focusing on systematic errors due to 2D modeling. It is shown that the 2D models provide only the lower (assuming the plane stress state) or the upper (assuming the plane strain state) limits of the hinge’s stiffness. The error of modeling a particular hinge by 2D elements (with either the plane stress or the plane strain assumptions) depends mainly on its depth-to-height ratio and may reach up to about 12%. However, this error becomes negligible for hinges with sufficiently high or sufficiently low depth-to-height ratios, in which either the plane strain or stress states dominate respectively. It is also shown that the computationally intensive 3D elements can be replaced, without sacrificing accuracy, by numerically efficient 2D elements if the material properties are appropriately manipulated.

Understanding coupled factors that affect the modelling accuracy of typical planar compliant mechanisms

“…This finding is different from the result in Refs. [10,11] and betrays our expectation. The influence from beam length of the CBPM is stronger than that of the CDPM, due to the fact that the CBPM exhibits loadstiffening behavior.…”

“…Since the required actuation force is proportional to the used modulus under the same primary motion [10,11], the following two equations can be obtained:…”

“…If an analytical model for this modulus ratio is hard to derive, an empirical equation based on finite element analysis (FEA) is desired. References [10,11] reported interesting work on equivalent modulus of right circular planar flexure hinges (short beams) using an FEA method, and proposed an empirical equation of the equivalent modulus (or modulus ratio as defined above). It is shown that the equivalent modulus of the right circular planar flexure hinges is larger than the Young's modulus but smaller than the plate modulus, and that it is reasonable to use planar stress (or strain) assumption if beam out-of-plane thickness compared to the in-plane thickness is small enough (or large enough).…”

Fatigue testing of flexure hinges for the purpose of the development of a high-precision micro manipulator

“…The other stress components are negligibly small. σ x and σ z are reduced on the outsides (Zettl et al, 2005). Consequently, equivalent von Mises stress (σ v ) is at maximum on the corners of the thinnest cross section (Fig.…”

New empirical stiffness equations for corner-filleted flexure hinges

“…So far, the flexure hinges can be classified into two categories: the primitive flexure hinges, such as circular flexure hinges, corner filleted flexure hinges, elliptical flexure hinges, parabolic flexure hinges, hyperbolic flexure hinges, "V" shape flexure hinges, right circular elliptical flexure hinges, right circular corner-filleted flexure hinges, two axes flexure hinges, multiple axes flexure hinges, and the complex flexure hinges such as cross axis flexure hinges, cartwheel flexure hinges. The shapes of these flexure hinges can be found in the literatures (Meng et al, 2012;Zettl et al, 2005;Chen et al, 2005;Lobontiu and Paine, 2002;Lobontiu et al, 2002aLobontiu et al, , 2001Lobontiu et al, , 2002b, this paper will not present them in detail again. Among these flexure hinges, circular flexure hinges, elliptical flexure hinges and corner-filleted flexure hinges as shown in Fig.…”