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Closed-form compliance equations of filleted V-shaped flexure hinges for compliant mechanism design
Precision Engineering 34 (2010) 408–418
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Closed-form compliance equations of filleted V-shaped flexure hinges for compliant mechanism design Y. Tian a,b,∗ , B. Shirinzadeh b , D. Zhang a a b
School of Mechanical Engineering, Tianjin University, Tianjin 300072, China Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Australia
a r t i c l e
i n f o
Article history: Received 22 April 2009 Received in revised form 16 September 2009 Accepted 2 October 2009 Available online 31 October 2009 Keywords: Flexure hinge Compliance Accuracy of motion Compliant mechanism
a b s t r a c t This paper presents the closed-form compliance equations for the filleted V-shaped flexure hinges. The in-plane and out-of-plane compliances of the flexure hinges are developed based on the Castigliano’s second theorem. The accuracy of motion, denoted by the midpoint compliance of the flexure hinges, is also derived for optimized geometric design. The influences of the geometric parameters on the characteristics of the flexure hinges are investigated. It is noted that the filleted V-shaped flexure hinges have diverse ranges of compliance corresponding to different filleted radius R and angle . These types of hinges can provide both higher and lower stiffnesses than circular flexure hinges. This makes filleted V-shaped flexure hinges very useful for wide potential applications with different requirements. The finite element analysis is used to verify the established closed-form compliance equations for these filleted V-shaped flexure hinges. © 2009 Elsevier Inc. All rights reserved.
1. Introduction Flexure hinges are commonly utilized in the compliant mechanisms for applications in micro-/nano-instruments, machines and systems such as scanning tunnel microscope, X-ray lithography, mask alignment, and micro-manufacturing [1–8], where high positioning accuracy and resolution are the necessary and crucial requirements to fulfil specified tasks. Flexure-based mechanisms can overcome the shortcomings such as stiction, friction, and backlash which generally exist in the conventional mechanisms with sliding and rolling bearings. Thus, flexure-based mechanisms are capable of smooth motion, free of friction and lubrication [9–14]. The positioning accuracy of flexure-based mechanisms can be further improved by utilizing laser-interferometry-based sensing technique for independent position and displacement measurement and tracking [15,16]. In order to obtain high performance flexure-based mechanisms, CAD (computer aided design) methodology is usually utilized to conduct the mechanical design and optimization, and to develop the fixture structure using 3-2-1 locating and clamping technique for holding such monolithic mechanisms [17,18]. With the aid of CNC (computer numerical control) and WEDM (wire electrical discharge machining) techniques, flexure hinges can be monolithically manufactured with other links of the entire mechanism. This makes it possible to achieve high machining accuracy and eliminate the errors of assembly. Furthermore, the geometric and dimensional tolerances can be exactly controlled, and the positioning accuracy of the entire system is improved. In the precision positioning, the flexure-based mechanisms are also used to enlarge the small displacement of the actuator or the small output force to driving moving platform [19–24]. Unlike conventional revolute joints, flexure hinges have finite stiffness in the output direction. Therefore, the rotational center will offset when the flexure hinges generate output displacement. This is the negative aspect of using flexure-based mechanisms [25,26]. In order to achieve efficient compliant mechanisms for applications in nanomanipulation, it is important to correctly choose the geometric parameters and to predict and optimise the performance of flexure hinges. In flexure-based mechanisms, flexure hinges are generally made of the rectangular blank removing two symmetric cut-outs with the profiles of circular, corner-filleted, and elliptical profiles as shown in Fig. 1. These kinds of flexure hinges have low rotational stiffness about one axis providing displacement and high stiffness in other degrees of freedom. The analytical solutions for compliance in these kinds of flexure hinges have been investigated in the past. Paros and Weisbord [27] presented the closed-form equations and curves of the circular flexure hinges for both symmetric single-axis and two-axis configurations based on the theory of mechanics of materials. The
∗ Corresponding author at: School of Mechanical Engineering, Tianjin University, Tianjin 300072, China. Tel.: +86 22 27405561; fax: +86 22 27406260. E-mail address: [email protected] (Y. Tian). 0141-6359/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2009.10.002
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Fig. 1. Three traditional flexure hinges.
angular and linear compliances were developed for bending and axial loads of the in-plane and out-of-plane, respectively. For each case, the solutions were expressed in both exact and simplified forms. Using the same method, Smith et al. [28] developed the closed-form equations for symmetric single-axis semi-elliptical flexure hinges. It was noted that the performance of the elliptical flexure hinges were within the range constrained by the circular flexure hinges and the leaf hinges in terms of compliance. The finite element analysis and experimental tests were carried out to verify model predictions over a range of the typical geometries for many flexure hinge designs. Using finite element analysis, Xu and King [29] investigated the performance of the circular, corner-filleted and elliptical flexure hinges in terms of motion, stiffness and stress concentrations. Compared with the corner-filleted flexure hinges, the elliptical flexure hinges have less stress when the deflections are the same. The right circular flexure hinges have the highest stiffness, and the corner-filleted flexure hinges have the lowest stiffness. Lobontiu et al. [30] and Lobontiu and Garcia [31] derived the closed-form compliance equations for corner-filleted flexure hinges using Castigliano’s second theorem. Similar to elliptical flexure hinges, corner-filleted flexure hinges also range within the domain confined by the right circular flexure hinges and the leaf flexure hinges in terms of compliance. The finite element analysis and experimental tests were used to confirm the model predictions. It was noted that the corner-filleted flexure hinges are more compliant and induce lower stress, however these are less accurate in rotation compared with the right circular flexure hinges. Utilizing a similar approach, Lobontiu et al. [32,33] developed the closed-form equations for a symmetric conic-section such as circular, elliptic, parabolic, and hyperbolic flexure hinges. The in-plane and out-of-plane compliance equations of the flexure hinges as well as the motion accuracy were derived for specific conic profiles. The finite element analysis and experimental tests were utilized to examine the model predictions. Wu and Zhou [34] introduced a different intermediate variable and presented the concise closed-form equations for circular flexure hinges. The comparison with equations developed by Paros was carried out. Tseytlin [35] presented the closed-form compliance equations for monolithic flexure hinges with circular and elliptical sections. The inverse conformal mapping of circular approximating contour was utilized to derive the analytical solutions. The predictions of the developed models were likely to be much closer to the finite element analysis and experimental data. Schotborgh et al. [36] described the dimensionless design graphs for three flexure hinges
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with the profiles such as circular, corner-filleted beam and cross-section. These plots can be used as a design tool to determine the optimal geometries for flexure-based mechanism designs. Vallance et al. [37] provided a unified geometric model for flexure hinge using quadratic rational Bezier curves. The general representation of the geometry has great advantages in the computer aided design and analysis. The thorough review of flexure hinges has indicated that a number of notch profiles have been utilized to improve the performance, and analytical/empirical equations have also been developed to predict the behaviour of such hinges. This paper presents novel filleted V-shaped flexure hinges and the closed-form compliance equations for the flexure hinges. Based on the Castigliano’s second theorem, the in-plane and out-of-plane compliances of the flexure hinges are developed. The accuracy of motion, denoted by the midpoint compliance of the filleted V-shaped flexure hinges, is also derived for the purpose of optimized mechanical design. The influences of the geometric parameters on the characteristics of these types of flexure hinges are investigated. It is noted that the filleted V-shaped flexure hinges have a large compliance ranges corresponding to different filleted radius R and angle . This makes such flexure hinges capable of being used in wide potential applications with different requirements including high motion accuracy and large displacement. The finite element analysis is used to verify the established closed-form compliance equations for filleted V-shaped flexure hinges. 2. Compliance models The model of the filleted V-shaped flexure hinge is shown in Fig. 2. The symmetric profiles within the flexure hinge are formed by removing two filleted triangular prisms from the rectangular blank. Thus, for the V-shaped flexure hinge: b denotes the width of the flexure hinge, h denotes the height of the flexure hinge, t denotes the minimum thickness of the flexure hinge, l denotes the half length of the flexure hinge, c denotes the height of the profile and is equal to (h − t)/2, R denotes the radius of the circular section, and denotes the separation angle of the profile from the horizontal axis. The Cartesian coordinate frame is utilized, where the origin is located at the minimum thickness of the flexure hinge, and the x and y axes are in the longitudinal and height directions, respectively. The boundary conditions are chosen such that the left end of the flexure hinge is fixed, and the external loads act at the midpoint of right end. The comparisons of the proposed filleted V-shaped flexure hinge with circular and leaf flexure hinges are shown in Fig. 3. It can be seen from Fig. 3(a) that the filleted V-shaped flexure hinge possesses the capability of higher stiffness than circular and leaf flexure hinges when the filleted radius R equals to l/4 and angle equals to /4. As the filleted radius R increases and reaches up to l as shown in Fig. 3(b), the stiffness of the filleted V-shaped flexure hinge will reduce and remain in the range constrained by the circular and leaf flexure hinges. Let the filleted radius R equals to l and change the angle from the range from 0◦ to 90◦ as shown in Fig. 3(c). The filleted V-shaped flexure hinge can change from leaf hinge into right circular one. In order to derive closed-form compliance equations of filleted V-shaped flexure hinges, the Castigliano’s second theorem is adopted and written as follows: =
∂U , ∂P
(1)
where is the deformation due to the applied load, P is the applied load, and U is the deformation energy and given by: U=
1 2
l
−l
M2 dx EI
(for bending moment),
Fig. 2. Geometric parameters and loads of filleted V-shaped flexure hinge.
(2a)
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Fig. 3. Comparisons of three flexure hinges.
U=
1 2
l
−l
F2 dx EA
(for force),
(2b)
where E is the Young’s modulus, M is the bending moment, I is the moment of area, and A is the cross-sectional area. Based on the established Cartesian coordinate frame and the geometric relationships, the upper profile equation for the filleted V-shaped flexure hinge is given as follows:
t − R2 − x 2 , 2 t x tan sign(x) + R(1 − sec ) + , 2
R+
y(x) =
0 ≤ |x| ≤ R sin ,
(3)
R sin < |x| ≤ l,
where sign(·) is the sign function and defined by:
sign(x) =
1, x ≥ 0, −1, x < 0.
Thus, the moment of area I of the flexure hinge can be written as follows: Iz =
2by3 (x) 3
Iy =
b3 y(x) 6
(about z-axis),
(4a)
(about y-axis).
(4b)
2.1. Angular compliance about z-axis The filleted V-shaped flexure hinge subjected to the bending moment Mz will rotate about z-axis, and the angular displacement is denoted by ˛z . Based on the Castigliano’s second theorem, the compliance equation is developed as follows:
3 ˛z = · Mz 2EbR2 −
1 · 2ˇ + ˇ2
2 cot ˇ2 (1 + )2
+
6(1 + ˇ) (3 + 2ˇ + ˇ2 ) sin + + arctan 2 2 )(1 + ˇ − cos ) 3/2 (2ˇ + ˇ (1 + ˇ − cos ) (2ˇ + ˇ2 ) (1 + ˇ) sin
2+ˇ tan 2 ˇ
cot (1 + ˇ − cos )
2
,
(5)
where ˇ = t/2R, and = t/2c. The application of force Fy at the free end of the filleted V-shaped flexure hinge can generate bending moment acting on the hinge and may be written as follows: M = Fy (l − x),
− l ≤ x ≤ l.
(6)
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Substituting Eq. (6) into Eq. (2) and replacing Iz by Eq. (4a), the compliance equation about z-axis is obtained as follows:
3 + 3(ˇ − ) cos ˛z · = Fy 2EbR sin −
2 cot ˇ2 (1 + )2
1 · 2ˇ + ˇ2
cot
+
(1 + ˇ − cos )
6(1 + ˇ) (3 + 2ˇ + ˇ2 ) sin + + arctan 2 3/2 (2ˇ + ˇ2 )(1 + ˇ − cos ) (1 + ˇ − cos ) (2ˇ + ˇ2 ) (1 + ˇ) sin
2+ˇ tan 2 ˇ
,
2
(7)
where l = (( + (ˇ − ) cos )/ sin )R. 2.2. Linear compliance along y-axis The filleted V-shaped flexure hinge can generate translation along y-axis due to the bending moment Mz . In order to calculate the displacement using the Castigliano’s second theorem, the virtual force Fy should be superimposed at the free end of the flexure hinge. The total bending moment is given by: M = Mz + Fy (l − x),
− l ≤ x ≤ l.
(8)
Substituting Eq. (8) into Eq. (2), the compliance along y-axis due to bending moment Mz is obtained as: 3 + 3(ˇ − ) cos y · = Mz 2EbRsin −
2 cot ˇ2 (1 + )2
+
1 · 2ˇ + ˇ2
cot (1 + ˇ − cos )
6(1 + ˇ) (3 + 2ˇ + ˇ2 ) sin + + arctan 2 3/2 (2ˇ + ˇ2 )(1 + ˇ − cos ) (1 + ˇ − cos ) (2ˇ + ˇ2 ) (1 + ˇ) sin
2+ˇ tan 2 ˇ
.
2
(9)
The application of force Fy can also cause the filleted V-shaped flexure hinges translate along y-axis, and the linear compliance is established as follows: 3 y = · Fy 2Eb
+ (ˇ − ) cos
4(1 + ˇ)
−
·
(2ˇ + ˇ2 ) 2 sin2
[ + (ˇ − ) cos ] − sin2
2
ˇ2 + 2ˇ
−
2
(3 + 2ˇ + ˇ2 ) sin 6(1 + ˇ) + arctan + 2 3/2 (2ˇ + ˇ2 )(1 + ˇ − cos ) (1 + ˇ − cos ) (2ˇ + ˇ2 ) (1 + ˇ) sin
2 cot ˇ2 (1 + )2
2(1 + ˇ) (2ˇ + ˇ2 )
sin2 cot
−
2 (1 + ˇ − cos )
arctan
3/2
cot
2+ˇ tan 2 ˇ
ˇ(1 + ) + + 2 cot ln 2 (1 + ˇ − cos ) (1 + ˇ − cos ) 3
−
2
2+ˇ tan 2 ˇ
3ˇ2 + 6ˇ + 1 − 2 2 (ˇ + 2ˇ)(1 + ˇ − cos ) (1 + ˇ − cos ) 1+ˇ
+ 2 − 2
+ (ˇ − ) cos sin − ˇ(1 + )sin 1 + ˇ − cos
sin
cot2
.
(10)
The shear force induced by Fy can also cause the flexure hinge translate along y-axis, and the linear compliance due to shear force is obtained as follows:
1 y = Fy Gb
2(1 + ˇ)
2ˇ + ˇ2
arctan
2+ˇ tan 2 ˇ
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
,
(11)
where G is the shear modulus. 2.3. Angular compliance about y-axis When the filleted V-shaped flexure hinge is subjected to bending moment My and force Fz , the rotation about y-axis will occur. This is defined as out-of-plane deformation which is undesirable for high precision motion of a flexure hinge. Therefore, it is necessary to reduce the compliance about y-axis for the optimization of mechanical design. The angular compliances about y-axis due to bending moment My and force Fz are respectively developed as follows:
˛y 12 = My Eb3
2(1 + ˇ)
2ˇ + ˇ2
arctan
˛y 12R [ + (ˇ − ) cos ] = Fz Eb3 sin
2+ˇ tan 2 ˇ 2(1 + ˇ)
2ˇ + ˇ2
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
arctan
2+ˇ tan 2 ˇ
,
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
(12)
.
(13)
2.4. Linear compliance along z-axis Similar to the bending moment Mz and force Fy that lead to a linear displacement of the flexure hinge, the bending moment My and force Fz can also result in the linear displacement of the filleted V-shaped flexure hinge along z-axis. However, the displacement of the
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flexure hinge along z-axis belongs to the out-of-plane motion and will reduce the motion accuracy. Therefore, the linear compliances along z-axis due to the bending moment My and force Fz should be limited to a small range in the mechanical design of such a flexure hinge. The linear compliances along z-axis under the bending moment My and force Fz are respectively given as follows:
12R z [ + (ˇ − ) cos ] = My Eb3 sin
z 12R2 [ + (ˇ − ) cos ] = Fz Eb3 2 sin2 6R2 + 3 Eb
2
2(1 + ˇ)
2ˇ + ˇ2
2ˇ + ˇ2
arctan
+ cot
2
[ + (ˇ − ) cos ] − sin2 2 sin2
2+ˇ tan 2 ˇ
2+ˇ tan 2 ˇ
(2 + 2ˇ + cos ) sin − 4(1 + ˇ)
arctan
2(1 + ˇ)
2ˇ + ˇ2 arctan
2
− 2 cot
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
2+ˇ tan 2 ˇ
,
(14)
+ (1 + 4ˇ + 2ˇ2 )
+ (ˇ − ) cos − sin sin
ˇ(1 + ) × (1 + ˇ − sec ) + 2(1 + ˇ − sec ) cot ln . (1 + ˇ − cos ) 2
3
(15)
Further, the linear compliance along z-axis due to the shear force acting on the end of the filleted V-shaped flexure hinge may be obtained as follows:
z 1 = Fz Gb
2(1 + ˇ)
2ˇ + ˇ2
arctan
2+ˇ tan 2 ˇ
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
.
(16)
2.5. Linear compliance along x-axis The filleted V-shaped flexure hinge will exhibit the deformation along x-axis, when the force Fx acts on the end of the flexure hinge. The compliance along x-axis may be developed as follows:
x 1 = Fx Eb
2(1 + ˇ)
2ˇ + ˇ2
arctan
2+ˇ tan 2 ˇ
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
(17)
3. Accuracy of motion The ideal revolute joint would have the stationary rotational center, but the flexure hinges will exhibit deformation along the entire length of the flexible component. This indicates that the rotational center of the flexure hinge drifts during the displacement output. This is undesirable for the purpose of generating precise motion, especially in the amplification mechanism of the displacement and force. This will lead to the actual amplification factors being different from the desired values. The drift of the rotational center also causes the designed mechanism being unable to fulfil the requirements, and may even cause design failure within the mechanism. Therefore, more attentions must be directed towards the motion accuracy of the flexure hinges within the compliant mechanism designs. In order to carry out a quantitative analysis, the compliance of the center of the flexure hinge is utilized to describe the motion accuracy. The larger the compliance of the center point, the lower the motion accuracy of the flexure hinge. In order to calculate the displacements of the center point of the filleted V-shaped flexure hinge, the virtual forces Fx and Fy are added at the rotational center. Based on the Castigliano’s second theorem, the motion accuracy of the flexure hinge can be derived. It must be pointed out that the shear effect is negligible small due to the slender section of the flexure hinges, especially in the calculation of the motion accuracy, where the value is only half of that in the total compliance computation. Thus, the shear effect is neglected in the following calculation.
Fig. 4. Finite element model and boundary conditions of flexure hinge.
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The application of bending moment Mz can induce a displacement of the center of the filleted V-shaped flexure hinge, and thus the compliance of the rotational center may be derived as: ıy 3 = Mz 2EbR −
cot2 2
1−ˇ 1 1+ˇ + − 2 1 + ˇ − cos 2ˇ2 2(1 + ˇ − cos )
1 − ˇ(1 + ) 1 + ˇ − cos
−
cot 2
2 + ( − ˇ) cos ˇ2 (1 + )2 sin
−
sin (1 + ˇ − cos )
,
where ıy denotes the displacement along y-axis of the rotational center.
Fig. 5. Stiffness of filleted V-shaped flexure hinge for different conditions.
2
(18)
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415
Fig. 5. (Continued ).
The application of force Fy at the end of the flexure hinge can also cause the translation of rotational center. The compliance of the rotational center is obtained as follows: ıy 3 = Fy 2Eb
+ (ˇ − ) cos sin
2(1 + ˇ)
−
+
2ˇ + ˇ2
−
1 1+ˇ 1−ˇ − + 2 1 + ˇ − cos 2ˇ2 2(1 + ˇ − cos )
1+ˇ (2ˇ
3/2 + ˇ2 )
arctan
2+ˇ tan 2 ˇ
+ (ˇ − ) cos + 2 sin2 3( + (ˇ − ) cos ) − 2 sin (1 + ˇ − cos ) 2ˇ(1 + ) sin
++
−
3ˇ2 + 6ˇ + 1 − 2 + 4ˇ)(1 + ˇ − cos ) 2 (2ˇ 2(1 + ˇ − cos ) 1+ˇ
+ (ˇ − ) cos − sin2 2
2(1 + ˇ − cos )
ˇ(1 + ) cot + ln cot3 (1 + ˇ − cos ) 2
sin
cot
.
(19)
The closed-form equation for the motion accuracy of the rotational center due to the force Fx is derived as follows:
ıx 1 = Fx 2Eb
2(1 + ˇ)
2ˇ + ˇ2
arctan
2+ˇ tan 2 ˇ
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
,
(20)
where ıx denotes the displacement of the rotational center along x-axis. 4. Finite element analysis and validation In order to verify the established closed-form compliance equations for the filleted V-shaped flexure hinges, the finite element analysis is carried out to calculate the compliances and motion accuracy. The ANSYS software is utilized to perform the finite element analysis. The filleted V-shaped flexure hinge consists of two symmetrical notches with a length of 20 mm on each side. The minimum thickness of the flexure hinge is t that varies among 0.2, 0.5, and 1.0 mm, and each side of the flexure hinge is monolithic with a block 15 mm in length. The width of the flexure hinge is chosen as 10 mm. There are generally two methods to conduct the 3D mechanism/structure analysis: one is using the 3D element and the other is using 2D element with defined thickness. The later method has the advantages of high computation efficiency with acceptable calculation accuracy. Thus, the 2D method is utilized in the compu-
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tational analysis of the developed filleted V-shaped flexure hinges. The finite element model of the flexure hinge is shown in Fig. 4, where the two dimensional 10 node element named Plane 82 is utilized to mesh the flexure hinge and the width 10 mm of the flexure hinge is defined in the attribution of the element. This kind of element has quadratic displacement behavior and is well suited to model irregular shapes such as flexure hinges. Each node has two degrees of freedom which are translations in the x and y directions. In order to improve the computational accuracy, the mapping mesh method is utilized and the element sizes are different for each part of the flexure hinge and about quintile of the minimum dimension of the local region. In each static analysis, the boundary conditions are chosen as: the left block is fixed for all degrees of freedom, and the force is applied at the midpoint of the free end of the
Fig. 6. Motion accuracy of filleted V-shaped flexure hinge for different conditions.
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Table 1 Comparison between finite element analysis and analytical solution. R (mm)
(◦ )
t (mm)
2 4 6 8 10 2 4 6 8 10 2 4 6 8 10
15 30 45 60 75 15 30 45 60 75 15 30 45 60 75
1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.2 0.2 0.2 0.2 0.2
Stiffness (×105 N/m)
Motion accuracy (×107 N/m)
FEA
Analytic
Error (%)
FEA
Analytic
Error (%)
6.7824 6.7146 5.6638 4.9317 4.4131 1.4710 1.2534 1.0366 0.9001 0.8058 0.1736 0.1306 0.1071 0.0929 0.0838
7.2653 7.3202 6.0704 5.2144 4.6223 1.5454 1.3136 1.0729 0.9244 0.8229 0.1787 0.1332 0.1085 0.0937 0.0837
6.65 8.27 6.70 5.42 4.53 4.81 4.58 3.38 2.63 2.08 2.85 1.95 1.29 0.85 0.12
0.8122 1.2116 0.9128 0.6835 0.5397 0.3314 0.3589 0.2491 0.1848 0.1454 0.0908 0.0638 0.0425 0.0313 0.0243
0.7218 1.4169 1.0670 0.7752 0.5994 0.3091 0.3934 0.2686 0.1965 0.1537 0.0918 0.0663 0.0438 0.0323 0.0255
12.51 14.49 14.45 11.83 9.95 7.21 8.78 7.25 5.94 5.40 1.12 3.95 2.97 3.10 4.77
flexure hinge. The material for flexure hinge is chosen as the spring steel alloy with Young’s modulus of 210 GPa and Poisson’s ratio of 0.28. The complete set of the results for finite element analysis is shown in Table 1. To examine the compliance predicted from the developed closed-form equations of the filleted V-shaped flexure hinge, the errors between the results of the finite element analysis and the Eqs. (10) and (19) are provided. It must be mentioned that the stiffness, reciprocal of the compliance, is adopted to carry out the comparisons. It can be seen that the results between the finite element analysis and closed-form equations are in good agreement with the maximum deviation being less than 15%. This indicates that the closed-form equations developed for filleted V-shaped flexure hinge are correct and can be used for the computational analysis and design of the compliant mechanisms. 5. Numerical simulations and discussions The established closed-form compliance equations for the filleted V-shaped flexure hinges can be utilized to investigate the influences of the geometric parameters on compliance and motion accuracy. This is important for understanding the characteristics of the filleted V-shaped flexure hinges and the optimized mechanical design of compliant mechanisms to achieve high static and dynamic performance. For the numerical calculations, the length l of the filleted V-shaped flexure hinge remains constant, and the angle , radius R, and minimum thickness t vary within a range for practical flexure hinge. It should be emphasized that the actual height h of the filleted V-shaped flexure hinge varies with the change of the angle and the radius R. The numerical simulations are carried out based on the flexure hinge with the length of 20 mm and width of 10 mm. The material for the filleted V-shaped flexure hinge is also chosen as spring steel with Young’s modulus of 210 GPa. The calculation results for different loads are shown in Fig. 5. It is noted that the output motions of the flexure hinge, rotation about z-axis and translation along y-axis, are more dominant than the motions in other axes, especially in the small range of minimum thickness t. The stiffness of the flexure hinge decreases with the decreasing angle and the ratio ˇ. The filleted V-shaped flexure hinge possesses a wide stiffness range which can be larger than that of the circular flexure hinge, and span between the circular flexure hinges and the leaf flexure hinges. The motion accuracy of the filleted V-shaped flexure hinge is shown in Fig. 6. It is noted that the influences of the ratio ˇ of the flexure hinge on the motion accuracy due to the bending moment and forces are different. For small values of ˇ, there is a little effect on the motion accuracy for the bending moment and force Fy , and large influence on the compliance induced by force Fx . However, larger ˇ has the inverse influence on the motion accuracy, i.e., the motion accuracy for the bending moment Mz and force Fy are more sensitive to the change of ˇ than that for the force Fx . The angle also has significant effect on the motion accuracy of the filleted V-shaped flexure hinge. It is noted that as angle becomes smaller, its influence on the motion accuracy for bending moment and forces is reduced. Whereas, the larger angle has relatively large influence on the motion accuracy, especially for the motion accuracy due to the force Fx . Considering the performance criteria of compliance and motion accuracy, the filleted V-shaped flexure hinge can provide for a trade-off between large deformation and high motion accuracy by changing the geometric parameters such as filleted radius R and angle . 6. Conclusions The closed-form compliance equations for the filleted V-shaped flexure hinge have been developed using the Castigliano’s second theorem. Based on the established models, the computational analysis has been carried out to investigate the influence of the geometric parameters on the characteristics of the filleted V-shaped flexure hinges. It is noted that the stiffnesses of the filleted V-shaped flexure hinge increase with the increasing parameters ˇ and . The stiffness of the filleted V-shaped flexure hinge has diverse sensitivity for change in the parameters ˇ and within the given range. The motion accuracy also shows the similar characteristics for the changes of the parameters ˇ and . The finite element analysis has been used to verify the developed models. The results between the finite element analysis and the analytical solutions are in good agreement, which indicates that it is practical to utilize the closed-form equations for the filleted V-shaped flexure hinge. These form the tools for stiffness prediction in compliant mechanism designs. The proposed filleted V-shaped flexure hinges can converge to right circular hinges (at R = l and = 90◦ ) and leaf hinges (at → 0). The developed closed-form equations for the filleted V-shaped flexure hinge can be utilized as guidelines for the compliant mechanism design, where the relative large deformation and high motion accuracy need to be achieved simultaneously.
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Acknowledgements This research is supported by Australian Research Council (ARC) Discovery under Grant Nos. DP0450944 and DP0666366, ARC Linkage Infrastructure, Equipment and Facilities under Grant Nos. LE0347024 and LE0775692, and National Natural Science Foundation of China under Grant No. 50705064. References [1] Miller JA, Hocken R, Smith ST, Harb S. X-ray calibrated tunneling system utilizing a dimensionally stable nanometer positioner. Precision Engineering 1996;18(2–3):95–102. [2] Smith ST, Chetwynd DG, Bowen DK. Design and assessment of monolithic high precision translation mechanisms. Journal of Physics E: Science Instrument 1987;20(8):977–83. [3] Tian Y, Liu X, Zhang D, Chetwynd DG. Dynamic modeling of the fidelity of random surface measurement by the stylus method. Wear 2009;266(5–6):555–9. [4] Choi KB, Lee JJ. Passive compliant wafer stage for single-step nano-imprint lithography. Review of Scientific Instruments 2005;76(5):075106. [5] Mohd Zubir MN, Shirinzadeh B. 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Laser-interferometry-based tracking for dynamic measurements. Industry Robot 1998;25(1):35–41. [17] Shirinzadeh B. CAD-based design and analysis system for reconfigurable fixture in robotic assembly. Computing and Control Engineering Journal 1994;5(1):41–6. [18] Shirinzadeh B. Flexible and automation workholding systems. Industrial Robot 1995;22(2):29–34. [19] Liaw HC, Shirinzadeh B. Enhanced adaptive motion tracking control of piezo-actuated flexure-based four-bar mechanisms for micro/nano manipulation. Sensors and Actuators A: Physical 2008;147(1):254–62. [20] Jouaneh M, Yang RY. Modeling of flexure-hinge type lever mechanisms. Precision Engineering 2003;27(4):407–18. [21] Liaw HC, Shirinzadeh B, Smith J. Enhanced sliding mode motion tracking control of actuators. Sensors and Actuators A: Physical 2007;138(1):194–202. [22] Choi SB, Han SS, Han YM, Thompson BS. A magnification device for precision mechanisms featuring piezoactuators and flexure hinges: design and experimental validation. Mechanism and Machine Theory 2007;42(9):1184–98. [23] Liaw HC, Shirinzadeh B, Smith J. Robust motion tracking control of piezo-driven flexure-based four-bar mechanism for micro/nano manipulation. Mechatronics 2008;18(2):111–20. [24] Alici G, Shirinzadeh B. Optimum dynamic balancing of planar parallel manipulators based on sensitivity analysis. Mechanism and Machine Theory 2006;41(12):1520–32. [25] Hsiao FZ. Analysis of a novel flexure hinge with three degrees of freedom. Review of Scientific Instruments 2001;72(2):1565–73. [26] Lobontiu N, Paine JSN. Design of circular cross-section corner-filleted flexure hinges for three-dimensional compliant. Journal of Mechanical Design, Transactions of ASME 2002;124(3):479–84. [27] Paros JM, Weisbord L. How to design flexure hinges. Machine Design 1965;37(27):151–6. [28] Smith ST, Badami VG, Dale JS, Xu Y. Elliptical flexure hinges. Review of Scientific Instruments 1997;68(3):1474–83. [29] Xu W, King T. Flexure hinges for piezoactuator displacement amplifiers: flexibility, accuracy and stress consideration. Precision Engineering 1996;19(1):4–10. [30] Lobontiu N, Paine JSN, Garcia E, Goldfarb M. Corner-filleted flexure hinges. Journal of Mechanical Design, Transactions of ASME 2001;123(3):346–52. [31] Lobontiu N, Garcia E. Stiffness characterization of corner-filleted flexure hinges. Review of Scientific Instruments 2004;75(11):4896–905. [32] Lobontiu N, Paine JSN, O’Malley E, Samuelson M. Parabolic and hyperbolic flexure hinges: flexibility, motion precision and stress characterization based on compliance closed-form equations. Precision Engineering 2002;26(2):183–92. [33] Lobontiu N, Paine JSN, Garcia E, Goldfarb M. Design of symmetric conic-section flexure hinges based on closed-form compliance equations. Mechanism and Machine Theory 2002;37(5):477–98. [34] Wu YF, Zhou ZY. Design calculations for flexure hinges. Review of Scientific Instruments 2002;73(8):3010–6. [35] Tseytlin YM. Notch flexure hinges: an effective theory. Review of Scientific Instruments 2002;73(9):3363–8. [36] Schotborgh WO, Kokkeler FGM, Tragter H, Houten FJAM. Dimensionless design graphs for flexure elements and a comparison between three flexure elements. Precision Engineering 2005;29(1):41–7. [37] Vallance RR, Haghighian B, Marsh ER. A unified geometric model for designing elastic pivots. Precision Engineering 2008;32(4):278–88.
Contents lists available at ScienceDirect
Precision Engineering journal homepage: www.elsevier.com/locate/precision
Closed-form compliance equations of filleted V-shaped flexure hinges for compliant mechanism design Y. Tian a,b,∗ , B. Shirinzadeh b , D. Zhang a a b
School of Mechanical Engineering, Tianjin University, Tianjin 300072, China Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Australia
a r t i c l e
i n f o
Article history: Received 22 April 2009 Received in revised form 16 September 2009 Accepted 2 October 2009 Available online 31 October 2009 Keywords: Flexure hinge Compliance Accuracy of motion Compliant mechanism
a b s t r a c t This paper presents the closed-form compliance equations for the filleted V-shaped flexure hinges. The in-plane and out-of-plane compliances of the flexure hinges are developed based on the Castigliano’s second theorem. The accuracy of motion, denoted by the midpoint compliance of the flexure hinges, is also derived for optimized geometric design. The influences of the geometric parameters on the characteristics of the flexure hinges are investigated. It is noted that the filleted V-shaped flexure hinges have diverse ranges of compliance corresponding to different filleted radius R and angle . These types of hinges can provide both higher and lower stiffnesses than circular flexure hinges. This makes filleted V-shaped flexure hinges very useful for wide potential applications with different requirements. The finite element analysis is used to verify the established closed-form compliance equations for these filleted V-shaped flexure hinges. © 2009 Elsevier Inc. All rights reserved.
1. Introduction Flexure hinges are commonly utilized in the compliant mechanisms for applications in micro-/nano-instruments, machines and systems such as scanning tunnel microscope, X-ray lithography, mask alignment, and micro-manufacturing [1–8], where high positioning accuracy and resolution are the necessary and crucial requirements to fulfil specified tasks. Flexure-based mechanisms can overcome the shortcomings such as stiction, friction, and backlash which generally exist in the conventional mechanisms with sliding and rolling bearings. Thus, flexure-based mechanisms are capable of smooth motion, free of friction and lubrication [9–14]. The positioning accuracy of flexure-based mechanisms can be further improved by utilizing laser-interferometry-based sensing technique for independent position and displacement measurement and tracking [15,16]. In order to obtain high performance flexure-based mechanisms, CAD (computer aided design) methodology is usually utilized to conduct the mechanical design and optimization, and to develop the fixture structure using 3-2-1 locating and clamping technique for holding such monolithic mechanisms [17,18]. With the aid of CNC (computer numerical control) and WEDM (wire electrical discharge machining) techniques, flexure hinges can be monolithically manufactured with other links of the entire mechanism. This makes it possible to achieve high machining accuracy and eliminate the errors of assembly. Furthermore, the geometric and dimensional tolerances can be exactly controlled, and the positioning accuracy of the entire system is improved. In the precision positioning, the flexure-based mechanisms are also used to enlarge the small displacement of the actuator or the small output force to driving moving platform [19–24]. Unlike conventional revolute joints, flexure hinges have finite stiffness in the output direction. Therefore, the rotational center will offset when the flexure hinges generate output displacement. This is the negative aspect of using flexure-based mechanisms [25,26]. In order to achieve efficient compliant mechanisms for applications in nanomanipulation, it is important to correctly choose the geometric parameters and to predict and optimise the performance of flexure hinges. In flexure-based mechanisms, flexure hinges are generally made of the rectangular blank removing two symmetric cut-outs with the profiles of circular, corner-filleted, and elliptical profiles as shown in Fig. 1. These kinds of flexure hinges have low rotational stiffness about one axis providing displacement and high stiffness in other degrees of freedom. The analytical solutions for compliance in these kinds of flexure hinges have been investigated in the past. Paros and Weisbord [27] presented the closed-form equations and curves of the circular flexure hinges for both symmetric single-axis and two-axis configurations based on the theory of mechanics of materials. The
∗ Corresponding author at: School of Mechanical Engineering, Tianjin University, Tianjin 300072, China. Tel.: +86 22 27405561; fax: +86 22 27406260. E-mail address: [email protected] (Y. Tian). 0141-6359/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2009.10.002
Y. Tian et al. / Precision Engineering 34 (2010) 408–418
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Fig. 1. Three traditional flexure hinges.
angular and linear compliances were developed for bending and axial loads of the in-plane and out-of-plane, respectively. For each case, the solutions were expressed in both exact and simplified forms. Using the same method, Smith et al. [28] developed the closed-form equations for symmetric single-axis semi-elliptical flexure hinges. It was noted that the performance of the elliptical flexure hinges were within the range constrained by the circular flexure hinges and the leaf hinges in terms of compliance. The finite element analysis and experimental tests were carried out to verify model predictions over a range of the typical geometries for many flexure hinge designs. Using finite element analysis, Xu and King [29] investigated the performance of the circular, corner-filleted and elliptical flexure hinges in terms of motion, stiffness and stress concentrations. Compared with the corner-filleted flexure hinges, the elliptical flexure hinges have less stress when the deflections are the same. The right circular flexure hinges have the highest stiffness, and the corner-filleted flexure hinges have the lowest stiffness. Lobontiu et al. [30] and Lobontiu and Garcia [31] derived the closed-form compliance equations for corner-filleted flexure hinges using Castigliano’s second theorem. Similar to elliptical flexure hinges, corner-filleted flexure hinges also range within the domain confined by the right circular flexure hinges and the leaf flexure hinges in terms of compliance. The finite element analysis and experimental tests were used to confirm the model predictions. It was noted that the corner-filleted flexure hinges are more compliant and induce lower stress, however these are less accurate in rotation compared with the right circular flexure hinges. Utilizing a similar approach, Lobontiu et al. [32,33] developed the closed-form equations for a symmetric conic-section such as circular, elliptic, parabolic, and hyperbolic flexure hinges. The in-plane and out-of-plane compliance equations of the flexure hinges as well as the motion accuracy were derived for specific conic profiles. The finite element analysis and experimental tests were utilized to examine the model predictions. Wu and Zhou [34] introduced a different intermediate variable and presented the concise closed-form equations for circular flexure hinges. The comparison with equations developed by Paros was carried out. Tseytlin [35] presented the closed-form compliance equations for monolithic flexure hinges with circular and elliptical sections. The inverse conformal mapping of circular approximating contour was utilized to derive the analytical solutions. The predictions of the developed models were likely to be much closer to the finite element analysis and experimental data. Schotborgh et al. [36] described the dimensionless design graphs for three flexure hinges
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with the profiles such as circular, corner-filleted beam and cross-section. These plots can be used as a design tool to determine the optimal geometries for flexure-based mechanism designs. Vallance et al. [37] provided a unified geometric model for flexure hinge using quadratic rational Bezier curves. The general representation of the geometry has great advantages in the computer aided design and analysis. The thorough review of flexure hinges has indicated that a number of notch profiles have been utilized to improve the performance, and analytical/empirical equations have also been developed to predict the behaviour of such hinges. This paper presents novel filleted V-shaped flexure hinges and the closed-form compliance equations for the flexure hinges. Based on the Castigliano’s second theorem, the in-plane and out-of-plane compliances of the flexure hinges are developed. The accuracy of motion, denoted by the midpoint compliance of the filleted V-shaped flexure hinges, is also derived for the purpose of optimized mechanical design. The influences of the geometric parameters on the characteristics of these types of flexure hinges are investigated. It is noted that the filleted V-shaped flexure hinges have a large compliance ranges corresponding to different filleted radius R and angle . This makes such flexure hinges capable of being used in wide potential applications with different requirements including high motion accuracy and large displacement. The finite element analysis is used to verify the established closed-form compliance equations for filleted V-shaped flexure hinges. 2. Compliance models The model of the filleted V-shaped flexure hinge is shown in Fig. 2. The symmetric profiles within the flexure hinge are formed by removing two filleted triangular prisms from the rectangular blank. Thus, for the V-shaped flexure hinge: b denotes the width of the flexure hinge, h denotes the height of the flexure hinge, t denotes the minimum thickness of the flexure hinge, l denotes the half length of the flexure hinge, c denotes the height of the profile and is equal to (h − t)/2, R denotes the radius of the circular section, and denotes the separation angle of the profile from the horizontal axis. The Cartesian coordinate frame is utilized, where the origin is located at the minimum thickness of the flexure hinge, and the x and y axes are in the longitudinal and height directions, respectively. The boundary conditions are chosen such that the left end of the flexure hinge is fixed, and the external loads act at the midpoint of right end. The comparisons of the proposed filleted V-shaped flexure hinge with circular and leaf flexure hinges are shown in Fig. 3. It can be seen from Fig. 3(a) that the filleted V-shaped flexure hinge possesses the capability of higher stiffness than circular and leaf flexure hinges when the filleted radius R equals to l/4 and angle equals to /4. As the filleted radius R increases and reaches up to l as shown in Fig. 3(b), the stiffness of the filleted V-shaped flexure hinge will reduce and remain in the range constrained by the circular and leaf flexure hinges. Let the filleted radius R equals to l and change the angle from the range from 0◦ to 90◦ as shown in Fig. 3(c). The filleted V-shaped flexure hinge can change from leaf hinge into right circular one. In order to derive closed-form compliance equations of filleted V-shaped flexure hinges, the Castigliano’s second theorem is adopted and written as follows: =
∂U , ∂P
(1)
where is the deformation due to the applied load, P is the applied load, and U is the deformation energy and given by: U=
1 2
l
−l
M2 dx EI
(for bending moment),
Fig. 2. Geometric parameters and loads of filleted V-shaped flexure hinge.
(2a)
Y. Tian et al. / Precision Engineering 34 (2010) 408–418
411
Fig. 3. Comparisons of three flexure hinges.
U=
1 2
l
−l
F2 dx EA
(for force),
(2b)
where E is the Young’s modulus, M is the bending moment, I is the moment of area, and A is the cross-sectional area. Based on the established Cartesian coordinate frame and the geometric relationships, the upper profile equation for the filleted V-shaped flexure hinge is given as follows:
t − R2 − x 2 , 2 t x tan sign(x) + R(1 − sec ) + , 2
R+
y(x) =
0 ≤ |x| ≤ R sin ,
(3)
R sin < |x| ≤ l,
where sign(·) is the sign function and defined by:
sign(x) =
1, x ≥ 0, −1, x < 0.
Thus, the moment of area I of the flexure hinge can be written as follows: Iz =
2by3 (x) 3
Iy =
b3 y(x) 6
(about z-axis),
(4a)
(about y-axis).
(4b)
2.1. Angular compliance about z-axis The filleted V-shaped flexure hinge subjected to the bending moment Mz will rotate about z-axis, and the angular displacement is denoted by ˛z . Based on the Castigliano’s second theorem, the compliance equation is developed as follows:
3 ˛z = · Mz 2EbR2 −
1 · 2ˇ + ˇ2
2 cot ˇ2 (1 + )2
+
6(1 + ˇ) (3 + 2ˇ + ˇ2 ) sin + + arctan 2 2 )(1 + ˇ − cos ) 3/2 (2ˇ + ˇ (1 + ˇ − cos ) (2ˇ + ˇ2 ) (1 + ˇ) sin
2+ˇ tan 2 ˇ
cot (1 + ˇ − cos )
2
,
(5)
where ˇ = t/2R, and = t/2c. The application of force Fy at the free end of the filleted V-shaped flexure hinge can generate bending moment acting on the hinge and may be written as follows: M = Fy (l − x),
− l ≤ x ≤ l.
(6)
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Substituting Eq. (6) into Eq. (2) and replacing Iz by Eq. (4a), the compliance equation about z-axis is obtained as follows:
3 + 3(ˇ − ) cos ˛z · = Fy 2EbR sin −
2 cot ˇ2 (1 + )2
1 · 2ˇ + ˇ2
cot
+
(1 + ˇ − cos )
6(1 + ˇ) (3 + 2ˇ + ˇ2 ) sin + + arctan 2 3/2 (2ˇ + ˇ2 )(1 + ˇ − cos ) (1 + ˇ − cos ) (2ˇ + ˇ2 ) (1 + ˇ) sin
2+ˇ tan 2 ˇ
,
2
(7)
where l = (( + (ˇ − ) cos )/ sin )R. 2.2. Linear compliance along y-axis The filleted V-shaped flexure hinge can generate translation along y-axis due to the bending moment Mz . In order to calculate the displacement using the Castigliano’s second theorem, the virtual force Fy should be superimposed at the free end of the flexure hinge. The total bending moment is given by: M = Mz + Fy (l − x),
− l ≤ x ≤ l.
(8)
Substituting Eq. (8) into Eq. (2), the compliance along y-axis due to bending moment Mz is obtained as: 3 + 3(ˇ − ) cos y · = Mz 2EbRsin −
2 cot ˇ2 (1 + )2
+
1 · 2ˇ + ˇ2
cot (1 + ˇ − cos )
6(1 + ˇ) (3 + 2ˇ + ˇ2 ) sin + + arctan 2 3/2 (2ˇ + ˇ2 )(1 + ˇ − cos ) (1 + ˇ − cos ) (2ˇ + ˇ2 ) (1 + ˇ) sin
2+ˇ tan 2 ˇ
.
2
(9)
The application of force Fy can also cause the filleted V-shaped flexure hinges translate along y-axis, and the linear compliance is established as follows: 3 y = · Fy 2Eb
+ (ˇ − ) cos
4(1 + ˇ)
−
·
(2ˇ + ˇ2 ) 2 sin2
[ + (ˇ − ) cos ] − sin2
2
ˇ2 + 2ˇ
−
2
(3 + 2ˇ + ˇ2 ) sin 6(1 + ˇ) + arctan + 2 3/2 (2ˇ + ˇ2 )(1 + ˇ − cos ) (1 + ˇ − cos ) (2ˇ + ˇ2 ) (1 + ˇ) sin
2 cot ˇ2 (1 + )2
2(1 + ˇ) (2ˇ + ˇ2 )
sin2 cot
−
2 (1 + ˇ − cos )
arctan
3/2
cot
2+ˇ tan 2 ˇ
ˇ(1 + ) + + 2 cot ln 2 (1 + ˇ − cos ) (1 + ˇ − cos ) 3
−
2
2+ˇ tan 2 ˇ
3ˇ2 + 6ˇ + 1 − 2 2 (ˇ + 2ˇ)(1 + ˇ − cos ) (1 + ˇ − cos ) 1+ˇ
+ 2 − 2
+ (ˇ − ) cos sin − ˇ(1 + )sin 1 + ˇ − cos
sin
cot2
.
(10)
The shear force induced by Fy can also cause the flexure hinge translate along y-axis, and the linear compliance due to shear force is obtained as follows:
1 y = Fy Gb
2(1 + ˇ)
2ˇ + ˇ2
arctan
2+ˇ tan 2 ˇ
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
,
(11)
where G is the shear modulus. 2.3. Angular compliance about y-axis When the filleted V-shaped flexure hinge is subjected to bending moment My and force Fz , the rotation about y-axis will occur. This is defined as out-of-plane deformation which is undesirable for high precision motion of a flexure hinge. Therefore, it is necessary to reduce the compliance about y-axis for the optimization of mechanical design. The angular compliances about y-axis due to bending moment My and force Fz are respectively developed as follows:
˛y 12 = My Eb3
2(1 + ˇ)
2ˇ + ˇ2
arctan
˛y 12R [ + (ˇ − ) cos ] = Fz Eb3 sin
2+ˇ tan 2 ˇ 2(1 + ˇ)
2ˇ + ˇ2
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
arctan
2+ˇ tan 2 ˇ
,
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
(12)
.
(13)
2.4. Linear compliance along z-axis Similar to the bending moment Mz and force Fy that lead to a linear displacement of the flexure hinge, the bending moment My and force Fz can also result in the linear displacement of the filleted V-shaped flexure hinge along z-axis. However, the displacement of the
Y. Tian et al. / Precision Engineering 34 (2010) 408–418
413
flexure hinge along z-axis belongs to the out-of-plane motion and will reduce the motion accuracy. Therefore, the linear compliances along z-axis due to the bending moment My and force Fz should be limited to a small range in the mechanical design of such a flexure hinge. The linear compliances along z-axis under the bending moment My and force Fz are respectively given as follows:
12R z [ + (ˇ − ) cos ] = My Eb3 sin
z 12R2 [ + (ˇ − ) cos ] = Fz Eb3 2 sin2 6R2 + 3 Eb
2
2(1 + ˇ)
2ˇ + ˇ2
2ˇ + ˇ2
arctan
+ cot
2
[ + (ˇ − ) cos ] − sin2 2 sin2
2+ˇ tan 2 ˇ
2+ˇ tan 2 ˇ
(2 + 2ˇ + cos ) sin − 4(1 + ˇ)
arctan
2(1 + ˇ)
2ˇ + ˇ2 arctan
2
− 2 cot
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
2+ˇ tan 2 ˇ
,
(14)
+ (1 + 4ˇ + 2ˇ2 )
+ (ˇ − ) cos − sin sin
ˇ(1 + ) × (1 + ˇ − sec ) + 2(1 + ˇ − sec ) cot ln . (1 + ˇ − cos ) 2
3
(15)
Further, the linear compliance along z-axis due to the shear force acting on the end of the filleted V-shaped flexure hinge may be obtained as follows:
z 1 = Fz Gb
2(1 + ˇ)
2ˇ + ˇ2
arctan
2+ˇ tan 2 ˇ
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
.
(16)
2.5. Linear compliance along x-axis The filleted V-shaped flexure hinge will exhibit the deformation along x-axis, when the force Fx acts on the end of the flexure hinge. The compliance along x-axis may be developed as follows:
x 1 = Fx Eb
2(1 + ˇ)
2ˇ + ˇ2
arctan
2+ˇ tan 2 ˇ
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
(17)
3. Accuracy of motion The ideal revolute joint would have the stationary rotational center, but the flexure hinges will exhibit deformation along the entire length of the flexible component. This indicates that the rotational center of the flexure hinge drifts during the displacement output. This is undesirable for the purpose of generating precise motion, especially in the amplification mechanism of the displacement and force. This will lead to the actual amplification factors being different from the desired values. The drift of the rotational center also causes the designed mechanism being unable to fulfil the requirements, and may even cause design failure within the mechanism. Therefore, more attentions must be directed towards the motion accuracy of the flexure hinges within the compliant mechanism designs. In order to carry out a quantitative analysis, the compliance of the center of the flexure hinge is utilized to describe the motion accuracy. The larger the compliance of the center point, the lower the motion accuracy of the flexure hinge. In order to calculate the displacements of the center point of the filleted V-shaped flexure hinge, the virtual forces Fx and Fy are added at the rotational center. Based on the Castigliano’s second theorem, the motion accuracy of the flexure hinge can be derived. It must be pointed out that the shear effect is negligible small due to the slender section of the flexure hinges, especially in the calculation of the motion accuracy, where the value is only half of that in the total compliance computation. Thus, the shear effect is neglected in the following calculation.
Fig. 4. Finite element model and boundary conditions of flexure hinge.
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Y. Tian et al. / Precision Engineering 34 (2010) 408–418
The application of bending moment Mz can induce a displacement of the center of the filleted V-shaped flexure hinge, and thus the compliance of the rotational center may be derived as: ıy 3 = Mz 2EbR −
cot2 2
1−ˇ 1 1+ˇ + − 2 1 + ˇ − cos 2ˇ2 2(1 + ˇ − cos )
1 − ˇ(1 + ) 1 + ˇ − cos
−
cot 2
2 + ( − ˇ) cos ˇ2 (1 + )2 sin
−
sin (1 + ˇ − cos )
,
where ıy denotes the displacement along y-axis of the rotational center.
Fig. 5. Stiffness of filleted V-shaped flexure hinge for different conditions.
2
(18)
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415
Fig. 5. (Continued ).
The application of force Fy at the end of the flexure hinge can also cause the translation of rotational center. The compliance of the rotational center is obtained as follows: ıy 3 = Fy 2Eb
+ (ˇ − ) cos sin
2(1 + ˇ)
−
+
2ˇ + ˇ2
−
1 1+ˇ 1−ˇ − + 2 1 + ˇ − cos 2ˇ2 2(1 + ˇ − cos )
1+ˇ (2ˇ
3/2 + ˇ2 )
arctan
2+ˇ tan 2 ˇ
+ (ˇ − ) cos + 2 sin2 3( + (ˇ − ) cos ) − 2 sin (1 + ˇ − cos ) 2ˇ(1 + ) sin
++
−
3ˇ2 + 6ˇ + 1 − 2 + 4ˇ)(1 + ˇ − cos ) 2 (2ˇ 2(1 + ˇ − cos ) 1+ˇ
+ (ˇ − ) cos − sin2 2
2(1 + ˇ − cos )
ˇ(1 + ) cot + ln cot3 (1 + ˇ − cos ) 2
sin
cot
.
(19)
The closed-form equation for the motion accuracy of the rotational center due to the force Fx is derived as follows:
ıx 1 = Fx 2Eb
2(1 + ˇ)
2ˇ + ˇ2
arctan
2+ˇ tan 2 ˇ
ˇ(1 + ) − + cot ln (1 + ˇ − cos )
,
(20)
where ıx denotes the displacement of the rotational center along x-axis. 4. Finite element analysis and validation In order to verify the established closed-form compliance equations for the filleted V-shaped flexure hinges, the finite element analysis is carried out to calculate the compliances and motion accuracy. The ANSYS software is utilized to perform the finite element analysis. The filleted V-shaped flexure hinge consists of two symmetrical notches with a length of 20 mm on each side. The minimum thickness of the flexure hinge is t that varies among 0.2, 0.5, and 1.0 mm, and each side of the flexure hinge is monolithic with a block 15 mm in length. The width of the flexure hinge is chosen as 10 mm. There are generally two methods to conduct the 3D mechanism/structure analysis: one is using the 3D element and the other is using 2D element with defined thickness. The later method has the advantages of high computation efficiency with acceptable calculation accuracy. Thus, the 2D method is utilized in the compu-
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tational analysis of the developed filleted V-shaped flexure hinges. The finite element model of the flexure hinge is shown in Fig. 4, where the two dimensional 10 node element named Plane 82 is utilized to mesh the flexure hinge and the width 10 mm of the flexure hinge is defined in the attribution of the element. This kind of element has quadratic displacement behavior and is well suited to model irregular shapes such as flexure hinges. Each node has two degrees of freedom which are translations in the x and y directions. In order to improve the computational accuracy, the mapping mesh method is utilized and the element sizes are different for each part of the flexure hinge and about quintile of the minimum dimension of the local region. In each static analysis, the boundary conditions are chosen as: the left block is fixed for all degrees of freedom, and the force is applied at the midpoint of the free end of the
Fig. 6. Motion accuracy of filleted V-shaped flexure hinge for different conditions.
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Table 1 Comparison between finite element analysis and analytical solution. R (mm)
(◦ )
t (mm)
2 4 6 8 10 2 4 6 8 10 2 4 6 8 10
15 30 45 60 75 15 30 45 60 75 15 30 45 60 75
1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.2 0.2 0.2 0.2 0.2
Stiffness (×105 N/m)
Motion accuracy (×107 N/m)
FEA
Analytic
Error (%)
FEA
Analytic
Error (%)
6.7824 6.7146 5.6638 4.9317 4.4131 1.4710 1.2534 1.0366 0.9001 0.8058 0.1736 0.1306 0.1071 0.0929 0.0838
7.2653 7.3202 6.0704 5.2144 4.6223 1.5454 1.3136 1.0729 0.9244 0.8229 0.1787 0.1332 0.1085 0.0937 0.0837
6.65 8.27 6.70 5.42 4.53 4.81 4.58 3.38 2.63 2.08 2.85 1.95 1.29 0.85 0.12
0.8122 1.2116 0.9128 0.6835 0.5397 0.3314 0.3589 0.2491 0.1848 0.1454 0.0908 0.0638 0.0425 0.0313 0.0243
0.7218 1.4169 1.0670 0.7752 0.5994 0.3091 0.3934 0.2686 0.1965 0.1537 0.0918 0.0663 0.0438 0.0323 0.0255
12.51 14.49 14.45 11.83 9.95 7.21 8.78 7.25 5.94 5.40 1.12 3.95 2.97 3.10 4.77
flexure hinge. The material for flexure hinge is chosen as the spring steel alloy with Young’s modulus of 210 GPa and Poisson’s ratio of 0.28. The complete set of the results for finite element analysis is shown in Table 1. To examine the compliance predicted from the developed closed-form equations of the filleted V-shaped flexure hinge, the errors between the results of the finite element analysis and the Eqs. (10) and (19) are provided. It must be mentioned that the stiffness, reciprocal of the compliance, is adopted to carry out the comparisons. It can be seen that the results between the finite element analysis and closed-form equations are in good agreement with the maximum deviation being less than 15%. This indicates that the closed-form equations developed for filleted V-shaped flexure hinge are correct and can be used for the computational analysis and design of the compliant mechanisms. 5. Numerical simulations and discussions The established closed-form compliance equations for the filleted V-shaped flexure hinges can be utilized to investigate the influences of the geometric parameters on compliance and motion accuracy. This is important for understanding the characteristics of the filleted V-shaped flexure hinges and the optimized mechanical design of compliant mechanisms to achieve high static and dynamic performance. For the numerical calculations, the length l of the filleted V-shaped flexure hinge remains constant, and the angle , radius R, and minimum thickness t vary within a range for practical flexure hinge. It should be emphasized that the actual height h of the filleted V-shaped flexure hinge varies with the change of the angle and the radius R. The numerical simulations are carried out based on the flexure hinge with the length of 20 mm and width of 10 mm. The material for the filleted V-shaped flexure hinge is also chosen as spring steel with Young’s modulus of 210 GPa. The calculation results for different loads are shown in Fig. 5. It is noted that the output motions of the flexure hinge, rotation about z-axis and translation along y-axis, are more dominant than the motions in other axes, especially in the small range of minimum thickness t. The stiffness of the flexure hinge decreases with the decreasing angle and the ratio ˇ. The filleted V-shaped flexure hinge possesses a wide stiffness range which can be larger than that of the circular flexure hinge, and span between the circular flexure hinges and the leaf flexure hinges. The motion accuracy of the filleted V-shaped flexure hinge is shown in Fig. 6. It is noted that the influences of the ratio ˇ of the flexure hinge on the motion accuracy due to the bending moment and forces are different. For small values of ˇ, there is a little effect on the motion accuracy for the bending moment and force Fy , and large influence on the compliance induced by force Fx . However, larger ˇ has the inverse influence on the motion accuracy, i.e., the motion accuracy for the bending moment Mz and force Fy are more sensitive to the change of ˇ than that for the force Fx . The angle also has significant effect on the motion accuracy of the filleted V-shaped flexure hinge. It is noted that as angle becomes smaller, its influence on the motion accuracy for bending moment and forces is reduced. Whereas, the larger angle has relatively large influence on the motion accuracy, especially for the motion accuracy due to the force Fx . Considering the performance criteria of compliance and motion accuracy, the filleted V-shaped flexure hinge can provide for a trade-off between large deformation and high motion accuracy by changing the geometric parameters such as filleted radius R and angle . 6. Conclusions The closed-form compliance equations for the filleted V-shaped flexure hinge have been developed using the Castigliano’s second theorem. Based on the established models, the computational analysis has been carried out to investigate the influence of the geometric parameters on the characteristics of the filleted V-shaped flexure hinges. It is noted that the stiffnesses of the filleted V-shaped flexure hinge increase with the increasing parameters ˇ and . The stiffness of the filleted V-shaped flexure hinge has diverse sensitivity for change in the parameters ˇ and within the given range. The motion accuracy also shows the similar characteristics for the changes of the parameters ˇ and . The finite element analysis has been used to verify the developed models. The results between the finite element analysis and the analytical solutions are in good agreement, which indicates that it is practical to utilize the closed-form equations for the filleted V-shaped flexure hinge. These form the tools for stiffness prediction in compliant mechanism designs. The proposed filleted V-shaped flexure hinges can converge to right circular hinges (at R = l and = 90◦ ) and leaf hinges (at → 0). The developed closed-form equations for the filleted V-shaped flexure hinge can be utilized as guidelines for the compliant mechanism design, where the relative large deformation and high motion accuracy need to be achieved simultaneously.
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Acknowledgements This research is supported by Australian Research Council (ARC) Discovery under Grant Nos. DP0450944 and DP0666366, ARC Linkage Infrastructure, Equipment and Facilities under Grant Nos. LE0347024 and LE0775692, and National Natural Science Foundation of China under Grant No. 50705064. References [1] Miller JA, Hocken R, Smith ST, Harb S. X-ray calibrated tunneling system utilizing a dimensionally stable nanometer positioner. Precision Engineering 1996;18(2–3):95–102. [2] Smith ST, Chetwynd DG, Bowen DK. Design and assessment of monolithic high precision translation mechanisms. Journal of Physics E: Science Instrument 1987;20(8):977–83. [3] Tian Y, Liu X, Zhang D, Chetwynd DG. Dynamic modeling of the fidelity of random surface measurement by the stylus method. Wear 2009;266(5–6):555–9. [4] Choi KB, Lee JJ. Passive compliant wafer stage for single-step nano-imprint lithography. Review of Scientific Instruments 2005;76(5):075106. [5] Mohd Zubir MN, Shirinzadeh B. 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