Design of symmetric conic-section flexure hinges based on ... - Research

Mechanism and Machine Theory 37 (2002) 477–498 www.elsevier.com/locate/mechmt

Design of symmetric conic-section flexure hinges based on closed-form compliance equations Nicolae Lobontiu a,*, Jeffrey S.N. Paine a, Ephrahim Garcia b,1, Michael Goldfarb b a

Dynamic Structures and Materials, LLC, 205 Williamson Square, Franklin, TN 37064, USA b Center for Intelligent Mechatronics, Vanderbilt University, Nashville, TN 37235, USA Received 2 January 2001; accepted 28 November 2001

Abstract The paper develops closed-form compliance equations for conic-section (circular, elliptic, parabolic and hyperbolic) flexure hinges. Finite element simulation results confirm the theoretical formulation data. The main objectives are to predict the deformation/displacement field of a flexure hinge under loading and to assess the precision of rotation for a specific conic flexure hinge. A non-dimensional analysis is carried out to discuss both problems. Conclusions are formulated regarding the performance of circular, elliptic, parabolic, and hyperbolic flexure hinges.
2002 Elsevier Science Ltd. All rights reserved. Keywords: Flexure hinges; Circular; Elliptic; Parabolic; Hyperbolic; Compliance; Rotation precision

1. Introduction The flexure hinges are increasingly popular with designs requiring one-piece (monolithic) manufacturing, reduced weight, zero backlash, friction and lubrication, motion smoothness, and virtually infinite resolution. Applications include accelerometers, gyroscopes, translation micro-positioning stages, motion guides, piezoelectric actuators and motors, high-accuracy alignment devices for optical fibers, missile-control devices, displacement amplifiers, scanning tunneling microscopes, high-precision cameras, robotic micro-displacement mechanisms, orthotic prostheses, antennas and valves.

*

Corresponding author. Tel.: +1-615-595-6665; fax: +1-615-595-6610. E-mail address: [email protected] (N. Lobontiu). 1 Tel.: +1-615-343-6924; fax: 1+615-343-6687. 0094-114X/02/$ - see front matter
2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 0 0 2 - 2

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Nomenclature

A C E F I M N R U a b c f l t x; y; z b; c h

flexure cross-sectional area, intermediate function compliance Young’s modulus force flexure cross-sectional moment of inertia, integral bending moment normal force radius of a right circular flexure hinge elastic strain energy flexure length parameter flexure cross-sectional width flexure thickness parameter function length flexure thickness reference axes non-dimensional parameters rotation angle

Subscripts a axial b bending c circular e elastic, elliptic h hyperbolic i; j; k counters min, max minimum, maximum p parabolic x; y; z reference axes Superscripts 0 right circular flexure hinge

Functionally, the ideal flexure hinge permits limited relative rotation of the rigid adjoining members while prohibiting any other types of motion. The typical flexure hinge consists of one or two cutouts that are machined in a blank material. Paros and Weisbord [1], in their fundamental work, presented the design equations, both exact and simplified, for calculating the compliances (spring rates) of single-axis and two-axis circular cutout constant cross-section flexure hinges. Ragulskis et al. [2] applied the static finite element analysis to one-quarter of circular flexure hinges in order to calculate their compliances. The analysis results were further used to formulate

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an existence criterion based on the deformation of an initially straight cross-section; this allowed specifying an optimal flexure geometry for minimum bending stiffness. Smith et al. [3] introduced the elliptic cross-section flexure hinge in an approach similar to that of Paros and Weisbord [1]. Closed-form equations were derived for the mechanical compliance of a simple monolithic elliptic cross-section flexure hinge. The elliptical flexure hinge was demonstrated to range in a domain bounded by the circular flexure hinge and the simple beam in terms of its compliance. The model predictions were checked by finite element analysis and experimental measurements. In a recent monograph, Smith [4] presented the basic geometry and analytic models of leaf-type springs such as notch or two-axis flexure hinges. Treated was also the problem of incorporating flexures into mechanisms that are designed for precise motion with fast dynamic control. Lobontiu et al. [5] developed an analytical model of corner-filleted flexure hinges that are incorporated into planar amplification mechanisms. Compliance factors were formulated that allow evaluating the rotation efficiency, precision of motion and stresses. A corner-filleted flexure hinge spans a design space that is limited by the right circular flexure and the simple beam, in terms of its compliance. Xu and King [6] performed static finite element analysis of circular, corner-filleted and elliptic flexure hinges. The results revealed that the corner-filleted flexure is the most accurate in terms of motion, the elliptic flexure has less stress for the same displacement, while the right circular flexure is the stiffest. Ryu and Gweon [7] modeled the motion errors that are induced by machining imperfections into a flexure hinge mechanism. More recently, Lobontiu et al. [8] introduced the parabolic and hyperbolic flexure hinges for planar mechanisms, by using compliance closed-form solutions to characterize their performance in terms of flexibility, precision of motion and stresses. The present work attempts to bring together four types of flexure hinges that share a common trait. By intersecting a cone with a plane in four distinct and non-trivial relative positions, a circle, an ellipse, a parabola or a hyperbola can be produced, respectively (Fig. 1), curves that are usually termed conic-sections. The in-pane closed-form compliances are presented for symmetric right

Fig. 1. Conic-sections.

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circular, right elliptic, parabolic and hyperbolic flexure hinges. A sketch illustrating a generic symmetric conic-section flexure hinge is shown in Fig. 2 while Fig. 3 indicates the parameters that define one flexure’s geometry. The precision of rotation, quantified by the offset of a flexure’s symmetry center, is also discussed in terms of compliance. An analysis is performed in terms of two non-dimensional parameters that allows comparing the performance of elliptic, parabolic and hyperbolic flexure hinges relative to circular flexure hinges. The analytical model predictions are confirmed by finite element simulation results within 10% error margins.

2. Basic assumptions The formulation that follows is based on several assumptions as summarized below: • The flexure hinges consist of two cutouts that are symmetric with respect to both the longitudinal axis and the middle transverse one (Fig. 2). • Each cutout is a conic section, specifically a circle, ellipse, parabola or hyperbola. • The conic-section flexure hinges are designed to be integrated into amplification mechanisms that are geometrically and kinematically two-dimensional; this provision reduces the analysis

Fig. 2. Symmetric conic-section flexure hinge.

Fig. 3. Parameters defining half of a symmetric conic-section flexure hinge.

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Fig. 4. Schematic representation of a flexure hinge with loading.

to three degrees of freedom (two in-plane translations and a rotation normal to the translations’ plane). • The flexure hinges are modeled and analyzed as small-displacement Euler–Bernoulli beams subjected to bending produced by forces and moments; the axial loading is also considered (see Fig. 4) while shearing effects are not taken into account. • The beam is considered to be fixed at one end and free at the other.

3. Compliance equations A generic conic-section flexure hinge is defined by the geometric parameters illustrated in Fig. 3. Based on the assumptions previously stated, this flexure can be assimilated to a fixed-free flextensional element, as the one sketched in Fig. 4. The displacement–loading relationship at the free end 1 is of the form: 8 9 2 9 38 C11 C12 0 < Mz1 = < h1 = ¼ 4 C12 C22 0 5 Fy1 ð1Þ y : 1; : ; x1 Fx1 0 0 C33 and can be formulated by using Castigliano’s second theorem: oUe ; oMz1 oUe y1 ¼ ; oFy1 oUe x1 ¼ ; oFx1 h1 ¼

where the elastic strain energy comprises bending and axial terms: Z

Z 1 M2 N2 dx þ dx : Ue ¼ 2 l EI l EA

ð2Þ

ð3Þ

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The compliance factors of Eq. (1) are expressed as: C11 ¼ 12=ðbEÞI1 ; C12 ¼ 12=ðbEÞI2 ; C22 ¼ 12=ðbEÞI3 ;

ð4Þ

C33 ¼ 1=ðbEÞI4 : For a variable-thickness cross-section, the integrals of Eq. (4) are expressed in the generic form: Z dx=tðxÞ3 ; I1 ¼ Zl I2 ¼ x dx=tðxÞ3 ; Zl ð5Þ 3 2 I3 ¼ x dx=tðxÞ ; l Z I4 ¼ dx=tðxÞ: l

Eqs. (1)–(5) are combined in order to derive the closed-form compliance equations for the conic-section flexure hinges. The formulation that follows is based on two non-dimensional parameters, b and c, that are defined as: t ; 2c t c¼ : 2a b¼

ð6Þ

3.1. Elliptical flexure hinges The variable thickness tðxÞ, as shown in Fig. 3, can be expressed as: n o tðxÞ2yðxÞ ¼ t þ 2c 1  ½1  ð1  x=aÞ2 1=2 :

ð7Þ

Eq. (7) is used to solve the integrals of Eqs. (5) that are subsequently substituted into Eqs. (4). The compliance equations for an elliptical flexure hinge are: " # 6 b 3 þ 4b þ 2b2 6ð1 þ bÞ 1=2 þ 1=2 ; ð8Þ ArcTanð1 þ 2=bÞ C11;e ¼ Ebt2 ð2 þ bÞ2 c 1þb b ð2 þ bÞ1=2 t C11;e ; 2c 3 1 ¼ ½fe1 ðbÞ þ fe2 ðbÞ 2Eb ð1 þ bÞð2 þ bÞ2 c3

C12;e ¼  C22;e

ð9Þ ð10Þ

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with; fe1 ¼ bf3 þ b½6 þ bð11 þ 8b þ 2b2 þ pð1 þ bÞð2 þ b2 ÞÞg; fe2 ¼ 2b1=2 ð2 þ bÞ1=2 ð1 þ bÞ4 ð3  4b  2b2 Þ ArcTanð1 þ 2=bÞ1=2 : The compliance C33;e is; i 1 bh 2ð1 þ bÞb1=2 ð2 þ bÞ1=2 ArcTanð1 þ 2=bÞ1=2  p=2 : C33;e ¼ Eb c

ð11Þ

ð12Þ

3.2. Circular flexure hinges The variable thickness tðxÞ, as shown in Fig. 3, can be expressed as; tðxÞ2yðxÞ ¼ 2R þ t  ½xð2R  xÞ1=2 :

ð13Þ

For a circular flexure hinge the two non-dimensional parameters b and c are equal since c ¼ a ¼ R. Eq. (13) is combined with Eqs. (4) and (5) to formulate the compliance equations for a circular flexure hinge, namely; " # 6 1 3 þ 4b þ 2b2 6ð1 þ bÞ þ 1=2 ð14Þ ArcTanð1 þ 2=bÞ1=2 ; C11;c ¼ Ebt2 ð2 þ bÞ2 1þb b ð2 þ bÞ1=2 t C11;c ; ð15Þ 2b 3 1 ½fe1 ðbÞ þ fe2 ðbÞ; ð16Þ C22;c ¼ 2Eb ð1 þ bÞð2 þ bÞ2 b3 i 1 h ð17Þ 2ð1 þ bÞb1=2 ð2 þ bÞ1=2 ArcTanð1 þ 2=bÞ1=2  p=2 : C33;c ¼ Eb The fact should be mentioned here that the circle is an ellipse with equal semi-axes ðc ¼ aÞ. In this case, since the non-dimensional parameters b and c are equal, the compliance equations for a circular flexure hinge should be retrieved from the corresponding compliance equations for an ellipse by taking b ¼ c. This is indeed so, as simply noticed by comparing Eqs. (8) and (14), Eqs. (9) and (15), Eqs. (10) and (16), and Eqs. (12) and (17), respectively. Another check was performed by comparing the compliance equations derived here for a right circular flexure hinge with those provided by Paros and Weisbord [1]. They were identical and the interested reader can easily verify this by simply taking c ¼ 1 þ b in Eqs. (1), (3), (9) and (25) of [1]. C12;c ¼ 

3.3. Parabolic flexure hinges For a parabolic flexure hinge, the variable thickness tðxÞ, as shown in Fig. 3, can be expressed as; tðxÞ2yðxÞ ¼ t þ 2cð1  x=aÞ2 :

ð18Þ

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Eq. (18) is now used in conjunction with Eqs. (4) and (5) to formulate the compliance closedform equations. They are: i 3 h 1=2 2 1=2 1 bð3 þ 5bÞ þ 3b ð1 þ b Þ ArcCotb ð19Þ C11;p ¼ c ð1 þ bÞ2 ; 2 2Ebt C12;p ¼ t=ð2cÞC11 ; i 3 bh 1=2 1=2 C22;p ¼ ð3 þ bÞb ArcCotb þ ½3  ð6 þ bÞbð1 þ bÞ2 ; 8Eb c3 C33;p ¼

1 1=2 b c ArcCotb1=2 : Eb

ð20Þ ð21Þ ð22Þ

3.4. Hyperbolic flexure hinges The variable thickness tðxÞ, as shown in Fig. 3, of a hyperbolic flexure hinge is; 2 1=2

tðxÞ2yðxÞ ¼ ½t2 þ 4cðc þ tÞð1  x=aÞ 

:

ð23Þ

Eqs. (23), (4) and (5) yield the closed-form compliance expressions for a hyperbola: C11;h ¼

12 bc1 ð1 þ bÞ1 ; 2 Ebt

C12;h ¼ t=ð2cÞC11;h ; C22;h

C33;h

 3 1 3=2 3 1=2 bð1 þ bÞ ð1 þ 2bÞ ¼ c 2ð1 þ 2bÞ ½1 þ bð2  bÞ 2Eb pffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 þ b  1 þ 2b pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;  b2 ð1 þ bÞð1 þ 2bÞ1=2 Log 1 þ b þ 1 þ 2b n o 1 bc1 ð1 þ 2bÞ1=2 Log 1 þ b1 ½1 þ ð1 þ 2bÞ1=2  : ¼ 2Eb

ð24Þ ð25Þ

ð26Þ ð27Þ

4. Precision of rotation The relative rotation of two mechanical members that are connected by a conventional rotation joint is produced along an axis that passes through the geometric center of the joint, which is fixed provided one member is also fixed. In the case of a symmetric flexure hinge, the center of rotation (the geometric symmetry center of the flexure) is no longer fixed since the forces and moments acting on the flexure produce elastic deformations that alter its position. The displacement of the rotation center of a flexure hinge (point 2 in Fig. 4), can be assessed by applying two fictitious loads, a horizontal one, Fx2 and a vertical one, Fy2 in addition to the actual load vector made up of Mz1 , Fy1 and Fx1 . The Castigliano’s second theorem is again utilized to find the displacements of the rotation center in the form:

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oUe ; oFy2 oUe x2 ¼ : oFx2

y2 ¼

ð28Þ

The elastic strain energy Ue is given in Eq. (3) and includes now the formal effects of Fx2 and Fy2 . The matrix-form equation that relates deformations to the load vector is similar to Eq. (1), namely; 8 9 2 9 38 0 0 0 < Mz1 = ¼ 1;kc ; f ðb; cÞ ¼ C11;k > 11;c > 11;k > > < f12;k ðb; cÞ ¼ C12;k ¼ f2;k2ðbÞ ; C12;c c k ¼ e; p; h: ð44Þ C22;k f3;k ðbÞ > f ðb; cÞ ¼ ¼ ; > 22;k C22;c c3 > > > : f ðb; cÞ ¼ C33;k ¼ f4;k ðbÞ ; 33;k C33;c c In doing so, the following conclusions can be derived that are valid for all the conic-section flexure hinges (see Figs. 6–10):

Fig. 6. Compliance ratio C11;e =C11;c .

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Fig. 7. Compliance ratios C11;k =C11;c : (a) elliptic flexures; (b) parabolic flexures; (c) hyperbolic flexures; (d) comparison of elliptic, parabolic and hyperbolic flexures for b ¼ 0:01; (e) f1;k functions.

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Fig. 8. Compliance ratios C12;k =C12;c : (a) elliptic flexures; (b) parabolic flexures; (c) hyperbolic flexures; (d) comparison of elliptic, parabolic and hyperbolic flexures for b ¼ 0:01; (e) f2;k functions.

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Fig. 9. Compliance ratios C22;k =C22;c : (a) elliptic flexures; (b) parabolic flexures; (c) hyperbolic flexures; (d) comparison of elliptic, parabolic and hyperbolic flexures for b ¼ 0:01; (e) f3;k functions.

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Fig. 10. Compliance ratios C33;k =C33;c : (a) elliptic flexures; (b) parabolic flexures; (c) hyperbolic flexures; (d) comparison of elliptic, parabolic and hyperbolic flexures for b ¼ 0:01; (e) f4;k functions.

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0 0 Fig. 11. Compliance ratios C12;k =C12;c : (a) elliptic flexures; (b) parabolic flexures; (c) hyperbolic flexures; (d) comparison 0 of elliptic, parabolic and hyperbolic flexures for b ¼ 0:01; (e) f1;k functions.

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0 0 Fig. 12. Compliance ratios C22;k =C22;c : (a) elliptic flexures; (b) parabolic flexures; (c) hyperbolic flexures; (d) comparison 0 of elliptic, parabolic and hyperbolic flexures for b ¼ 0:01; (e) f2;k functions.

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0 0 Fig. 13. Compliance ratios C33;k =C33;c : (a) elliptic flexures; (b) parabolic flexures; (c) hyperbolic flexures; (d) comparison 0 of elliptic, parabolic and hyperbolic flexures for b ¼ 0:01; (e) f3;k functions.

• The elliptic, parabolic and hyperbolic flexure hinges are generally more compliant than the baseline right circular flexure hinge, particularly for larger b; this is demonstrated by the non-dimensional compliance ratios that were always greater than 1 in Figs. 6–10.

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• The compliance ratios increase when b increases and c decreases (see Fig. 6 for a particular case, but the note is valid for all other situations). • The elliptic flexure hinges are more compliant than the parabolic, hyperbolic and circular flexures for smaller values of c and large values of b. • The compliance ratios vary non-linearly with c (in a steep fashion for small c) which indicates that the elliptic parabolic and hyperbolic flexure hinges are more compliant than the circular ones for large length-to-thickness ratios. • The compliance ratios vary quasi-linearly with b. • When b and c are relatively small, the circular flexure hinges are more compliant than the other conic-section flexure hinges. 6.2. Precision of rotation The effect of E; b and t on the compliance factors that quantify the precision of rotation is identical to the one described in the previous sub-section. In order to analyze the influence of the non-dimensional variables b and c, the following compliance ratio functions are introduced; 0 ¼ fij;k

0 Cij;k ; 0 Cij;c

i; j ¼ 1; 2; 3; k ¼ e; p; h:

Eq. (45) is reformulated as; 8 C0 f 0 ðbÞ > 0 > f12;k ðb; cÞ ¼ C12;k ¼ 1;kc2 ; > 0 > > 12;c < 0 C22;k f 0 ðbÞ 0 f22;k ðb; cÞ ¼ C0 ¼ 2;kc3 22;c > > 0 0 ðbÞ > C33;k f3;k > 0 > : f33;k ðb; cÞ ¼ C0 ¼ c ;

k ¼ e; p; h:

ð45Þ

ð46Þ

33;c

The elliptic, parabolic and hyperbolic flexure hinges are again compared to the right circular flexures baseline. It can be seen (Figs. 11–13) that the hyperbolic flexures perform best in terms of preserving the center of rotation position.

7. Conclusions The conic-section (circular, elliptic, parabolic, and hyperbolic) flexure hinges are presented in a unitary manner by means of their closed-form compliances. The flexibility and precision of rotation are the main subjects that are addressed. The analysis is performed in terms of two nondimensional parameters and this allows comparing the performance of elliptic, parabolic and hyperbolic flexure hinges relative to circular flexure hinges. The elliptic, parabolic and hyperbolic flexure hinges (in this order) are more compliant than the circular ones for large lengthto-thickness ratios. The hyperbolic flexures perform best in terms of preserving the center of rotation position.

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References [1] J.M. Paros, L. Weisbord, How to design flexure hinges, Machine Design 25 (1965) 151–156. [2] K.M. Ragulskis, M.G. Arutunian, A.V. Kochikian, M.Z. Pogosian, A study of fillet type flexure hinges and their optimal design, Vibration Engineering 3 (1989) 447–452. [3] T.S. Smith, V.G. Badami, J.S. Dale, Y. Xu, Elliptical flexure hinges, Review of Scientific Instruments 68 (3) (1997) 1474–1483. [4] S. Smith, Flexures: Elements of Elastic Mechanisms, Gordon and Breach Science Publishers, New York, 2000. [5] N. Lobontiu, J.S.N. Paine, E. Garcia, M. Goldfarb, Corner filleted flexure hinges, ASME Journal of Mechanical Design 123 (2001) 346–352. [6] W. Xu, T.G. King, Flexure hinges for piezo-actuator displacement amplifiers: flexibility, accuracy and stress considerations, Precision Engineering 19 (1) (1996) 4–10. [7] J.W. Ryu, D.-G. Gweon, Error analysis of a flexure hinge mechanism induced by machining imperfection, Precision Engineering 21 (1997) 83–89. [8] N. Lobontiu, J.S.N. Paine, E. O’Malley, M. Samuelson, Parabolic and hyperbolic flexure hinges: flexibility, motion precision and stress characterization based on compliance closed-form equations, Precision Engineering (in press).