Empirical compliance equations for conventional single-axis flexure hinges
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In consideration of the stress concentration, unified compliance equations for conventional single-axis hinges are presented. The relationship between the stress concentration and the compliance of corner-filleted flexure hinges is first analyzed. Considering the stress concentration, coupled with a wide range of geometrical parameters, empirical compliance equations for conventional flexure hinges, are then obtained by using the exponential model. Subsequently, the proposed equations are unified. To verify the validity and accuracy of these equations, the characteristics of a bridge-type flexure-based mechanism are then analyzed by the proposed equations and finite element analysis, respectively. The results of compliances and displacement amplification ratios obtained by these two methods are in good agreement. It demonstrates that the empirical compliance equations could be obtained by exponential model, and these equations can be unified.
KeywordsFlexure hinge Empirical compliance equations Exponential model Compliance Compliant mechanism
A flexure hinge, coupled with elastically regions and rigid beams, is a thin member that provides the relative rotation between two adjacent rigid beams . These hinges possess notable benefits such as no hysteresis, no friction losses, no need for lubrication, and ease of fabrication [1, 2, 3, 4, 5, 6, 7]. Therefore, flexure hinges have been widely used in various areas, such as automobile and aviation industries, inertial navigation industries, biomedical industries, computers and fiberoptics industries, and so on [8, 9, 10, 11, 12, 13]. These hinges are the basic elements of the flexure-based mechanisms, which can be applied to a wide range of applications. For example, accelerometers are key components of inertial navigation systems. In order to guarantee the accuracy of accelerometers, these flexure hinges are used to test accelerometer transverse sensitivity. In biomedical industries, they have been used to be key components of scanners. They are designed based on three piezoelectric actuators and several flexure hinges. In order to obtain high frequency in all three axes, a compact and rigid structure should be adopted.
The compliance of flexure hinges can influence the mechanical design, topology optimization, and dynamic accuracy of flexure-based mechanisms . Thus, many compliance equations for flexure hinges have been obtained to reduce the compliance modeling errors of flexure-based mechanisms, including polynomial approximation method, Castigliano’s theorem, and empirical equations obtained from FEA [15, 16]. Paros and Weisbord  introduced compliance equations for circular hinges, and these equations were simply and accurate. Smith et al.  presented empirical equations for elliptical hinges, and the characteristics were then verified by FEA. The prototype hinges were fabricated by a CNC milling machine. Subsequently, the bending moment was applied on the developed hinges, and the compliance of the hinge obtained by experiments was in good agreement with the theoretical arithmetic derived from those equations. Tian  introduced dimensionless empirical equations and graph expressions of three kinds of flexure hinges. The relationship between performances and geometrical parameters were then discussed, and empirical equations were obtained by the least square polynomial approximation method. At last, the characteristics of the three hinges were compared which could provide designers with a thorough understanding of these hinges and flexure-based mechanisms. Lobontiu et al.  investigated the characteristics of corner-filleted hinges. And compliance equations were obtained by employing the castigliano’s first theorem. The relationship between the performance derived from these equations and geometrical parameters was then discussed. It indicated that the proposed equations were accurate and cost-effective. In addition, the theoretical results were compared to experimental values, and the errors were less than 8%. Meng  focused on the corner-filleted flexure hinge, and its stiffness/compliance equations were presented. According to the FEA results, three stiffness/compliance equations with a wide range of geometrical parameters were obtained to overcome the influence induced by shearing. The comparisons with FEA indicated that the proposed empirical stiffness/compliance equations could enlarge the range of hinge thickness to hinge length. In addition, it could also improve the accuracy under large deformation. Yong  focused on the characteristics of circular hinges, and discussed the difference of the various compliance equations by using FEA. These equations were derived by different methods, and could be used in certain conditions. A proper scheme was then proposed to choose accurate equations according to the comparison. Subsequently, the empirical compliance equations, coupled with a large range of hinge parameters (hinge thickness/hinge radius), were obtained.
However, these proposed methods ignore the influence of the stress concentration caused by changes in cross-section of flexure hinges, and few have got unified equations for conventional single-axis flexure hinges. In this work, the influence of the stress concentration caused by changes in cross-section is taken into account, and the empirical compliance equations can be determined by using FEA. Subsequently, the equations for different hinges determined by this method can be unified. In addition, the characteristics of a bridge-type flexure-based mechanism are discussed, which can verify the method and the corresponding equations.
The remaining sections are organized as follows. In Sect. 2, the compliance matrices of single-axis hinges are introduced. In Sect. 3, the relationship between stress concentration and the compliance of corner-filleted ones is discussed, and then the derivation of the empirical compliance equations is described. In Sect. 4, the compliances of circular and rectangular ones are determined. In Sect. 5, unified empirical compliance equations are derived. In Sect. 6, amplification ratios of a flexure-based mechanism determined by FEA and empirical compliance equations are compared.
2 Single-axis flexure hinges
3 Corner-filleted flexure hinges
3.1 The influence of the stress concentration
3.2 Empirical compliance equations
According to the analysis above, we can see that the stress concentration should be considered when deriving the axial compliance equation, while it can be ignored when calculating the other compliance components. Subsequently, the compliance components of corner-filleted flexure hinges are then obtained by using FEA, the derivation of these equations can be shown as follows.
3.2.1 Axial compliance
3.2.2 Bending compliance
3.2.3 Shear compliance
4 Circular and rectangular flexure hinges
5 Conventional single-axis flexure hinges
Characteristics of the developed mechanism
Young’s modulus, E (GPa)
Poisson’s ratio, ν
For single-axis flexure hinges, stress concentration is in relation to the axial compliance but not in relation to the bending compliance and shear compliance.
For single-axis flexure hinges, empirical compliance equations determined by exponential model can be unified.
The amplification ratios calculated by FEA are in accordance with the theoretical arithmetic derived from empirical equations. It indicates that the compliance calculation method and the corresponding equations are valid.
This work was supported by Beijing Natural Science Foundation (3194044), and Beijing Postdoctoral Research Foundation of China (2017-ZZ-034).
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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