# Empirical compliance equations for conventional single-axis flexure hinges

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## Abstract

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.

## Keywords

Flexure hinge Empirical compliance equations Exponential model Compliance Compliant mechanism## 1 Introduction

A flexure hinge, coupled with elastically regions and rigid beams, is a thin member that provides the relative rotation between two adjacent rigid beams [1]. 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 [14]. 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 [17] introduced compliance equations for circular hinges, and these equations were simply and accurate. Smith et al. [18] 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 [19] 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. [20] 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 [21] 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 [22] 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

*xyz*is located at the free end of the left rigid beam. The

*x*-axis is in the longitudinal direction of the hinge, while the

*y*-axis is in the height direction. Generally, the flexure hinge between the two rigid beams can be applied by a load with six components: two shearing forces,

*F*

_{y},

*F*

_{z}; two bending moments,

*M*

_{y},

*M*

_{z}; a force along the

*x*-axis,

*F*

_{x}; and a moment around the

*x*-axis,

*M*

_{x}. For two-dimensional applications, where all active and resistive loads are planar, only the in-plane components

*M*

_{z},

*F*

_{y}and

*F*

_{x}have substantive effects on the flexure operation. The other components specified are out-of-plane agents that usually have a lesser magnitude, and therefore impact, on the flexure [1].

*a*of the proposed hinge could be written as

*δ*denotes the total deformation generated at the point

*a*,

*F*denotes an external load at the end point

*a*, and

*C*denotes the compliance.

**δ**^{in}= [

**δ**_{x},

**δ**_{y},

*α*

_{z}]

^{T}and

**δ**^{out}= [

**δ**_{z},

*α*

_{y}]

^{T}are the in-plane and out-of-plane deformation at the end point

*o*,

*F*

^{in}= [

*F*

_{x},

**F**_{y},

*M*

_{z}]

^{T}and

*F*

^{out}= [

**F**_{z},

*M*

_{y}]

^{T}are the in-plane and out-of-plane external load, respectively.

**C**_{m−n}denotes the compliance generated along the direction

*m*caused by the load

*n*.

## 3 Corner-filleted flexure hinges

*R*denotes the radius of flexure hinge,

*t*denotes the thickness,

*L*denotes the length,

*W*is the width,

*D*is the depth, and

*H*is the height.

### 3.1 The influence of the stress concentration

*L*= 5 mm,

*W*= 30 mm,

*R*= 5 mm,

*t*= 2 mm, and

*D*= 10 mm. FEA model is carried on by the software ANSYS 14.0. As shown in Fig. 3, the left rigid beam is fixed, while the right rigid beam is free, and the load (1 N) is applied on the free end. The stress and strain of the surfaces with same colors are equal, and the surfaces with equal stress and strain are not vertical to the

*x*-axis, and they are also not parallel to each other, which is inconsistent with the basic theoretical stress assumptions. The reason is that the cross-section changes along the flexure hinge, which can affect the stress distribution, so that the basic theoretical stress analysis equations are no longer applied. Such changes in cross-section cause a local increase of stress, referred to as stress concentration. Therefore, the proposed stress concentration can lead to the compliance calculation errors.

*x*-axis of a selected flexure hinge, can be compared in Fig. 4. The load (1 N) is applied on the free end, and the FEA stress is used as a benchmark for comparing with the theoretical stress. Compared with the theoretical values, the FEA values are larger at the locations where cross-section changes, which is in accordance with the distribution of the stress concentration.

*C*

_{α-Mz}, shear compliance

*C*

_{y-Fy}and axial compliance

*C*

_{x-Fx}, as illustrated in Fig. 6a. And the axial compliance proportions are much larger than other’s. As depicted in Fig. 6b, axial compliance calculation errors are relative large. While shear and bending compliance calculation errors increase first and then decrease sharply. In addition, when the value of

*t*/

*L*is more than 0.9, these two compliance calculation errors increase again.

*R*/

*L*changed due to the proposed reason, but not illustrated herein.

### 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

*x*-axis. We can see that it is comprised of two regions. One is a variable region (above the straight line) related to the corner-filleted hinge. The other one is a constant region (below the straight line) which is almost in accordance with the basic theoretical stress curve. The uniform distributed load is applied on the surface D of the hinge for keeping the deformation of the rigid beams constant, and thus the deformation of the constant region can be derived easily. Meanwhile, according to the theories of mechanisms of materials, the equation involved all the geometrical parameters affecting the deformation of the variable region can also be obtained. Without loss of generality,

*α*represents the total deformation of the selected hinge,

*α*

_{s}denotes the deformation of the constant region (the rigid beams), and

*α*

_{x}denotes the deformation of the variable region (the corner-filleted hinge). Then, the proposed deformations can be expressed as, respectively

*L*

_{t}denotes the total length,

*F*is the axial load,

*S*is the area,

*E*is the Young’s modulus, and

*k*denotes the constant coefficient which is independent of geometrical parameters, material properties and the loads;

*m*,

*i*and

*j*are the indexes corresponding to the geometrical parameters.

*W*= 30 mm,

*D*= 10 mm. Firstly, the hinge length is 5 mm, the hinge radius is 5 mm, and the value of

*t*is varied from 0.5 mm to 5 mm to analyze the relationship between

*t*and

*α*

_{x}, as shown in Fig. 9. It indicates that they have the exponential relationship. The logarithm of the values of the horizontal and vertical coordinate are obtained. And then least square method is applied. To obtain the unified dimension, the index is revised, and

*j*= 0.6890. Then, the hinge length is 5 mm,the hinge thickness is 1 mm, and the value of

*R*is varied from 1 mm to 10 mm to analyze the relationship between

*R*and

*α*

_{x}. Finally, the hinge radius is 5 mm, the hinge thickness is 1 mm, and the value of

*L*is varied from 0.5 to 5 mm to analyze the relationship between

*L*and

*α*

_{x}. Similarly, the index

*i*and

*m*can be obtained as 1.3129 and 0.3761, respectively.

*k*,

*f*(

*k*) could be expressed as

*f*(

*k*),

*k*could be given as

*k*

_{t}= 6.6585,

*k*

_{R}= 7.0307 and

*k*

_{L}= 6.9544 in terms of the relationship

*α*

_{x}−

*t*,

*α*

_{x}

*R*and

*α*

_{x}

*L*, and then the average value can be calculated as

*k*= 6.8812. Hence, the total axial deformation and the axial compliance can be given as follows

#### 3.2.2 Bending compliance

*θ*can be written as

*k*is the constant coefficient which is independent of the geometrical parameters, material properties and the loads;

*m*,

*i*and

*j*are the indexes corresponding to the geometrical parameters.

*x*-axis can be read by FEA directly. Geometrical parameters are selected as

*W*= 30 mm,

*D*= 10 mm. Firstly, the hinge length is 5 mm,the hinge radius is 5 mm, and the value of

*t*is varied from 1 mm to 5 mm to analyze the relationship between

*t*and

*θ*, as shown in Fig. 10. To obtain the unified dimension, the index is revised, and

*j*= 2.6676. The index

*m*and

*i*can be obtained as 0.4833 and 0.1843, respectively. According to the principle of minimizing variance, coupled with the relationships

*θ*−

*t, θ*−

*R*and

*θ*−

*L*, the constant coefficients

*k*

_{t},

*k*

_{R}and

*k*

_{L}are derived. And then the average value can be calculated as

*k*= 29.7502.

*C*

_{α−Mz}can be expressed as

*t*,

*L*, and

*R*can be obtained as 2.6720, 0.7693 and 0.9027, respectively. While the constant coefficient can be obtained by the principle of minimizing variance, and it is 43.9226. The compliance equation of

*C*

_{y-Mz}can be expressed as

#### 3.2.3 Shear compliance

*C*

_{y-Fy}, the indexes corresponding to the geometrical parameters

*t*,

*L*, and

*R*can be obtained as 2.6547, 1.0770 and 1.5777, respectively. The constant coefficient is 75.3279. In addition, for the compliance

*C*

_{α-Fy}, the indexes corresponding to the geometrical parameters

*t*,

*L*, and

*R*can be obtained as 2.6757, 0.7712 and 0.9044, respectively. The constant coefficient is 43.6875. Thus,

*C*

_{y-Fy}and

*C*

_{α-Fy}can be expressed as, respectively

### 3.3 Validation

*R*= 10 m,

*L*= 5 mm,

*W*= 30 m,

*D*= 10 mm, and the value of

*t*is varied from 0.5 to 10 mm. Errors between the results derived from the empirical equations of main compliance components and FEA are illustrated in Fig. 11. The calculation errors of the three main compliance components are all below 10%, and it proves that the values calculated by empirical equations are almost coincident with the FEA values.

## 4 Circular and rectangular flexure hinges

*L*, the hinge thickness

*t*, the side height

*h*, the width

*W*, the total height

*H*and the total depth

*D*[23].

*t*, the hinge radius

*R*, the width

*W*, the total height

*H*and the total depth

*D*[6].

## 5 Conventional single-axis flexure hinges

*c*

_{x}is the constant coefficient,

*H*is the total height of rigid beams,

*L*

_{z}is the total length of flexure hinge, and

*f*

_{x}(

*r*,

*R*,

*t*) is the exponential model related to corresponding geometrical parameters.

*c*

_{y}is the constant coefficient, and

*f*

_{y}(

*r*,

*R*,

*t*) is the exponential model related to corresponding geometrical parameters.

*c*

_{z}is the constant coefficient, and

*f*

_{z}(

*r*,

*R*,

*t*) is the exponential model related to corresponding geometrical parameters.

*C*

_{y-Mz}and

*C*

_{α-Fy}can be expressed as

*c*

_{yz}is the constant coefficient, and

*f*

_{yz}(

*r*,

*R*,

*t*) is the exponential model related to corresponding geometrical parameters.

## 6 Applications

Characteristics of the developed mechanism

Characteristics | |
---|---|

Young’s modulus, | 210 |

Poisson’s ratio, | 0.30 |

| 30 |

| 37.5 |

| 30 |

| 145 |

| 5 |

| 2 |

| 5 |

| 5 |

| 10 |

*θ*when the length of the mechanism is fixed. The comparison between the theoretical value derived from the empirical compliance equations and the FEA value is studied, and the results are shown in Fig. 16. As shown in Fig. 16a, an exponential relationship exists between the input compliance and the angle

*θ*. As illustrated in Fig. 16b, the output compliance of the mechanism will increase when the value of

*θ*increases, and they maintain a linear relationship. Generally, the theoretical results are in accord with FEA results, and it proves that the empirical compliance equations are accurate.

*θ*is less than 2.5, and then they decrease when the value of

*θ*is more than 2.5. For the blue and red lines, they almost coincide, and they increase first and then decrease. In addition, when the value of

*θ*is less than 3, the theoretical ratios determined in terms of the rigid beams is smaller than the ratios determined by FEA. By contrast, when the value of

*θ*is over 3, the theoretical ratios are larger. What’s more, the theoretical ratios determined in terms of the rigid beams are in accordance with the ratios determined by FEA, while the theoretical ratios determined without the rigid beams are much larger than their ratios. It demonstrates that the rigid beams should be considered when calculating the amplification ratios of this developed flexure-based mechanism, and the proposed compliance method and the corresponding equations are accurate.

## 7 Conclusions

- 1.
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.

- 2.
For single-axis flexure hinges, empirical compliance equations determined by exponential model can be unified.

- 3.
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.

## Notes

### Acknowledgements

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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