Calculation and Analysis of Double-Axis
Elliptical-Parabolic Compound Flexure Hinge

Ping' An Liu, Jianqun Cheng, Zhigang Lai

► To cite this version:

Ping' An Liu, Jianqun Cheng, Zhigang Lai. Calculation and Analysis of Double-Axis Elliptical-Parabolic Compound Flexure Hinge. 4th Conference on Computer and Computing Technologies in Agriculture (CCTA), Oct 2010, Nanchang, China. pp.361-367, 10.1007/978-3-642-18369-0_41. hal-01564871

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
¹Calculation and analysis of double-axis
elliptical-parabolic Compound flexure hinge

Ping' an LIU¹, JianqunCHENG², Zhigang LAI³

¹,²School of Mechanical and Electronic Engineering at East China Jiaotong University;
Nanchang 330013; CHN
Chjq84@163.com

³Jiangxi Technical College of Manufacturing, Nanchang, Jiangxi, 330095,CHN

**Abstract.** A double-axis flexure hinges combined with elliptical and parabolic curve is presented in this paper. The design equations for its compliance computation have been deduced by application of Castigliano's displacement theorem after the analysis of the structure, which were analyzed by the use of the software MATLAB, and confirm the validity of the model with finite element analysis software ANSYS.

**Keywords:** flexure hinge ; mixed flexure hinge ; compliance ; Castigliano' theorem

**Introduction**

Flexible hinge eliminates its idling running and mechanical friction by using small
angle elastic deformation and self-recovery characteristics in the transmission process ,
and can get higher resolution of displacement. It is been widely used in various
occasions, such as small angular displacement, high-precision rotation of occasions,
especially in precision measurement, positioning and other fields because of its
small-volume, gapless, no mechanical friction, no lubrication, in-gage and high
sensitivity of the transmission structure[1-3].

People have been making a series of studies on the single-axis oval, chamfered, parabolic and hyperbolic flexure hinge[4-5] since 1965 Paros and Weisbord derived circular flexure hinge flexibility formul. However, they have done little research about

¹ * Supported by Natural Science Foundation of Jiangxi, China, NO. 2008GZC005
the compound flexure hinge in addition to Chengui Min's [6] on the right circular and oval compound flexure hinge. Elliptical flexible hinge has a larger range of rotation but lower accuracy, and Parabolic with small range of rotation but high precise compared to the Circular incision hinge; for both accuracy and range of motion exercise, combining the elliptic and parabolic type flexure hinge will take the advantages of both flexible hinges, so this paper will present elliptic - parabolic compound flexible hinge.

# 1 Elliptic - parabolic compound flexible hinge

Elliptic and parabolic compound flexible hinge is composed of two half elliptical and parabolic form, as shown in figure1, oval-side as a fixed end and the Parabolic side as the rotation-side, and make the following assumptions:

(1) Elliptic and parabolic incision have the same thickness, and both are t.

(2) Elliptic and parabolic are tangented at the intersection, or smooth over.

(3) Two incision length in the X direction are m₀.

(4) Ellipse major axis m is greater than the minor axis n, when m=n, that is circular and parabolic compound flexible hinge.

<figure><img src="image_2.png" /><figcaption>Fig. 1. elliptic - parabolic hinge pro/e model</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2. the structure parameters of compound hinge</figcaption></figure>

## 2 Flexibility Formulas

Assumptions based on cantilever small deformation, bending deformation is generated by the force and moment. The impact of axial load has considered, while the impact of shear and torsion were not. The structure parameters of axial elliptic - parabolic compound flexure hinge were shown in Figure 2, and the force analysis as shown by Figure 3, the flexibility formulas of compound hinge were deduced based on mechanical card's second theorem:

<figure><img src="image_4.png" /><figcaption>Fig. 3. the force analysis of compound hinge</figcaption></figure>

$$ \begin{Bmatrix} \theta_{z1} \\ y_1 \\ x_1 \end{Bmatrix} = \begin{Bmatrix} C_{\theta-M} & C_{12} & 0 \\ C_{12} & C_y & 0 \\ 0 & 0 & C_x \end{Bmatrix} \begin{Bmatrix} M_{z1} \\ F_{y1} \\ F_{x1} \end{Bmatrix} \quad (1) $$

According to reciprocal theorem known $\theta_{z1}=C_{11}M_{z1}+C_{12}F_{y1}$, it can be obtained from the left of formula (1) with the use of cartesian displacement :
$$
\left\{
\begin{aligned}
\theta_{z1} &= \frac{\partial u}{\partial M_{z1}} \\
y_1 &= \frac{\partial u}{\partial F_{y1}} \quad (2) \\
x_1 &= \frac{\partial u}{\partial F_{x1}}
\end{aligned}
\right.
$$

By the material mechanics, without considering the shear and torsion, the
deformation energe of the compound flexible hinge is :

$$
U = \frac{1}{2} \left[ \int_{l} \frac{F_{x1}^2}{EA(x)} dx + \int_{l} \frac{M_{z}^2}{EI_{z}(x)} dx \right] \quad (3)
$$

$$
\text{and } F_x = F_{x1}, \quad M_z = M_{z1} + F_{yt}x, \quad A(x) = bt(x), \quad l(x) = (bt(x))^3 / 12
$$

The coordinate system as shown in figure 2, variable thickness equations of
dual-axis flexure hinge as follow :

$$
t(x) = \begin{cases} t + 2 \left[ n - \frac{n}{m} \sqrt{2mx - x^2} \right] & 0 \le x \le m \\ t + 2 \left[ 2p(x - m)^2 \right] & m < x \le l \end{cases} \quad (4)
$$

To the type of substitution (1) (2) (3) can obtain the flexibility formulas :

$$
C_x = \frac{1}{EB} \left\{ m^* \frac{-\pi \sqrt{-t(4n+t)} - (8n+4t) \text{Arc tan} \left( \frac{2n+t}{\sqrt{-t(4n+t)}} \right) + \text{Arc tan} \left( \frac{2m\sqrt{p}}{\sqrt{t}} \right)}{4n\sqrt{-t(4n+t)}} + \frac{2\sqrt{pt}}{2\sqrt{pt}} \right\} \quad (5)
$$

$$
C_y = \frac{12}{EB} \left\{ m^* \frac{16c^3(-80c^4 + 24c^3t + 8c^2(3+2\pi)t^2 + 4c(1+2\pi)t^3 + \pi t^4)}{16c^3t^2(4c+t)^2} + m^* \frac{(2n+t)^3(6n^2 - 4nt - t^2)}{4n^3(-t(4n+t))^2} \text{Arc tan} \left( \frac{\sqrt{-4nt-t^2}}{t} \right) \right. \\
\left. + \frac{m(112p^2m^4 + 56pm^2t - t^2)}{32p^2t^2(4pm^2+t)^2} + \frac{(12pm^2+t)\text{Arc tan}\left(\frac{2m\sqrt{p}}{\sqrt{t}}\right)}{64p^2t^2} \right\} \quad (6)
$$
$$C_{\theta-M} = \frac{12}{EB} \left\{ m \frac{6nt^2 + 4nt + t^2}{t^2(2n+t)(4n+t)^2} - m \frac{6n(2n+t)}{(-4nt - t^2)^{5/2}} \text{Arc tan} \left( \frac{2n+t}{\sqrt{-4nt - t^2}} \right) + \frac{m(12pn^2 + 5t)}{8t^2(4pn^2 + t)^2} + \frac{3\text{Arc tan}(\frac{2m\sqrt{p}}{\sqrt{t}})}{16\sqrt{pt^2}} \right\} \quad (7)$$

From the formulas of elliptical-parabolic compound hinges which have deduced, it can easily see that the flexibility is inversely proportional to the thickness B and modulus of elasticity E.

## 2.1 The verification of Closed-loop compliance formula

Design two dual-axis elliptical – parabolic compound hinges, its parameters as follows: thickness B is 5mm, length of oval long axle 5mm, the minor axes 3mm and 5mm, the thinnest thickness in the middle is 1mm, the total width 10mm. FEM experimental parameters are:

modulus of elasticity : E = 110 × 10⁹ Nm⁻², Poisson ratio : µ = 0.3.

parabol parameters : p=1/16 Fx1 = Fy1 = 1 N Mz1 = 1 N.m.

<figure><img src="image_5.png" /><figcaption>Fig. 4. the ansys geometric model of compound</figcaption></figure>

Models with different parameters in the finite element software to verify compliance formulas, choose the shell63 finite element, making finite element analysis to compound flexure hinge( As shown in figure 4), analytical method(AM) and finite element results were shown in Table 1. Known from the data in the table, finite element method(FEM) and the exact formula is less than 10% deviation.
<table><caption>Table 1, analytical and finite element calculation about flexibility</caption><thead><tr><th>φ</th><th>t/mφ</th><th>m/mφ</th><th>n/mφ</th><th>AMφ</th><th>FEMφ</th></tr></thead><tbody><tr><td rowspan="2">C<sub>11</sub>(N<sup>-1</sup>m<sup>-1</sup>)φ</td><td>0.001φ</td><td>0.005φ</td><td>0.003φ</td><td>1.44φ</td><td>1.37φ</td></tr><tr><td>0.001φ</td><td>0.005φ</td><td>0.005φ</td><td>1.08φ</td><td>1.14φ</td></tr><tr><td rowspan="2">C<sub>22</sub>(N<sup>-1</sup>m<sup>-1</sup>×10<sup>-6</sup>)φ</td><td>0.001φ</td><td>0.005φ</td><td>0.003φ</td><td>6.87φ</td><td>7.24φ</td></tr><tr><td>0.001φ</td><td>0.005φ</td><td>0.005φ</td><td>6.31φ</td><td>6.52φ</td></tr><tr><td rowspan="2">C<sub>33</sub>(N<sup>-1</sup>m<sup>-1</sup>×10<sup>-8</sup>)φ</td><td>0.001φ</td><td>0.005φ</td><td>0.003φ</td><td>1.43φ</td><td>1.37φ</td></tr><tr><td>0.001φ</td><td>0.005φ</td><td>0.005φ</td><td>1.35φ</td><td>1.44φ</td></tr></tbody></table>

## 2.2 Performance Analysis

From the formulases (5) to (7), it is easy to know that the formula of flexibility is inversely proportional to modulus E and thickness B; the relationship between the CX, CY, C0-M and the short axis length n, incision thickness t were shown in figure 5-7.

<figure><img src="image_6.png" /><figcaption>Fig. 5. the relationship between Cx and n, t</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** the relationship between Cy and n, t</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** the relationship between Cθ-M and n, t</figcaption></figure>

(1) We can see from figure 5, the relationships between the flexibility of x-axis direction Cx and t, n are in the overall smooth, keep the t unchanged, with the increase of n, Cx will decrease linearly; if the n unchanged, Cx decrease with t increases and changes in relatively fast as t in the range of 0.2-0.4, Cx changes in relatively fast, in conclusion the stiffness of X axis is large relatively.

(2) The relationship between flexibility Cy and t, n as shown in Figure 6, Cy will reach maximum when t equals to 0.2, and Cy will change sharply as t in the range of 0.2-0.4 and decrease smoothly when t is greater than 0.4, Cy has no significant change with n.

(3) From the figure 7, we can see that the Cθ-M obtains maximum when t equals to 0.2 and decreases as n increase, especially when t in the range of 0.2-0.4, it decreases sharply; when t is greater than 0.6, the Cθ-M decreases slowly and close to zero.

From the analysis above all, this compound hinge has large rigidity in the direction of x-axis and good flexibility in y axis direction and rotate around Z axis, so it is very suitable as rotation flexibility hinge.
## 3 Conclusions

This paper presents a new type of compound hinge oval - parabolic compound flexible hinge. And analyzed its structure, its flexibility formulas were deduced based on card's second law and verified by the finite element software, find the errors are within 8%, which demonstrates the correctness of the flexibility formulas. Then analyzed the relationship between the flexibility and the thickness B and the short axis length with matlab software, and got some conclusions, it could be a theoretical references in future industry and agriculture fields.

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