Two-axis flexure hinges with axially-collocated and symmetric notches

Computers and Structures 81 (2003) 1329–1341 www.elsevier.com/locate/compstruc

Two-axis flexure hinges with axially-collocated and symmetric notches Nicolae Lobontiu *, Ephrahim Garcia Sibley School of Mechanical and Aerospace Engineering, Cornell University, 258 Upson Hall, Ithaca, NY 14850, USA Received 14 October 2002; accepted 6 January 2003

Abstract The paper introduces a new class of two-axis flexure hinges with axially-collocated and symmetric notches as an alternative to the existing flexure designs with serially-disposed notches. A generic formulation is developed in terms of the geometric curves defining the two notches which includes assessing the capacity of rotation, precision of rotation, sensitivity to parasitic effects, stress values, motion efficiency and shearing effects by means of compliance factors. Closed-form compliance equations are derived for a two-axis flexure hinge that is defined by two non-identical parabolic profiles. The analytical model predictions are confirmed by finite element data. A numerical comparison is made of the parabolic flexure with a constant rectangular cross-section flexure hinge in terms of several performance criteria. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Flexure hinge; Compliant mechanisms; Two-axis; Collocation; Parabolic

1. Introduction A compliant mechanism contains at least one link that is designed to be elastically deformable. In most cases the flexible link connects two adjacent rigid members and is usually sensitive to bending, which leads to being called flexure hinge, flexural pivot or simply, flexure. The flexure hinge is actually an elastically-flexible, slender part between two rigid members undergoing relative limited rotation in a mechanism (called ÔcompliantÕ, due to the presence of at least one flexure hinge) that is supposed to achieve a specific task. The compliant mechanisms gain their mobility by transforming an input form of energy (mechanical, electric, thermal, magnetic, etc.) into output motion. The flexure hinge is monolithic with the rest of the mechanism for the vast majority of applications and this is the source of its advantages over classical rotation joints.

*

Corresponding author. E-mail address: [email protected] (N. Lobontiu).

The benefits provided by flexure hinges include lack of friction losses, no need of lubrication, no hysteresis, compactness, ease of fabrication, no maintenance necessary, and no repair for the flexure hinges that are monolithic with their compliant mechanism since the mechanism will operate until a flexure will fail because of fatigue or overloading. There are however certain drawbacks in flexure-based compliant mechanisms and one of them regards the inherent trait of the flexure hinges of providing relatively low rotation levels because of stress limitations. Also, the rotation is not pure because a flexure hinge is sensitive to axial loading, shearing and torsion, in addition to bending. Unlike classical rotation joints that keep their center of rotation in a fixed position, the Ôrotation centerÕ of a flexure hinge (this role is generally attributed to its midpoint) is not fixed during the relative rotation produced by the flexure as it displaces under the action of the combined load. Another disadvantage is that flexure hinges are temperature-sensitive and therefore thermal changes will modify their physical dimensions and, consequently, the compliance/stiffness properties, which will affect the motion precision and repeatability.

0045-7949/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0045-7949(03)00056-7

1330

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

Flexure hinges have numerous applications at both macro- and microscale (microelectromechanical systems––MEMS). They are largely implemented in the auto/aviation industry, in the biomedical industry, in computers and fiber optics applications. The compliant microelectromechanical systems are almost entirely based on generating and transmitting their motions by means of flexure-like members. Examples in this industry comprise optical switches, miniature load cells, flexible mounts for imaging masks, load-sensitive resonators, gyroscopes, disc memory head positioners, wire bonding heads, microaccelerometers, and cantilevers for microscopy. The flexure needs to be compliant about the bending direction (usually called sensitive or compliant axis) and rigid (ideally) about all other axes and deformations. Constructively, a flexure hinge may have one, two or several (multiple) sensitive axes. The sensitive axis defines the operational motion and the main function of a flexure hinge, which is to produce limited relative rotation between two adjacent rigid members. The flexure hinges that belong to the singleaxis category need to be sensitive in rotation about oneaxis and therefore, to bending that generates this type of motion. Fig. 1(a) is a three-dimensional view of a singleaxis flexure hinge that is realized by machining two symmetric cutouts in a constant-thickness blank mate-

rial. The sensitive or compliant axis is uniquely defined as it passes through the center (midpoint) of the flexure and lies in the cross-section of minimum thickness that possesses maximum bending compliance. Generally, such a flexure hinge has a rectangular cross-section with constant width and variable thickness. This flexure configuration is designed for applications where the two adjacent rigid members are expected to experience a planar motion. Although not desired, parasitic motions about the other axes of the flexure hinge accompany the bending about the sensitivity axis. Fig. 1(b) illustrates a multi-axis flexure hinge that has revolute geometry. Because of the rotational symmetry, any axis that passes through the center of the flexure hinge can act as a compliant/sensitive axis and therefore this direction is not uniquely defined. This type of flexure can thus be employed in three-dimensional applications where the direction of the sensitive axis is not a priori specified. Another flexure hinge configuration, which belongs to the two-axis category, is illustrated in Fig. 2. Structurally, this design is a serial chain formed of two possibly different single-axis flexure hinges. Similar to a single-axis configuration, one flexure hinge will preferentially bend about an axis of maximum bending compliance (called the primary sensitive axis) that lies in the cross-section of minimum thickness. The other flexure hinge is designed to bend about its own sensitive axis (called secondary sensitive axis) which is spaced from, and most often perpendicular to the primary sensitive axis. Compared to the primary sensitive axis, the compliance of the secondary axis is slightly smaller in order to enable reaction to higher bending loading that is expected to occur about this direction. Applications include cases where two adjacent rigid members need to perform the relative rotation about the primary sensitive axis on a regular basis, while preserving the capacity of relative rotation about the secondary axis in exceptional situations, such as when they have to react to higher

Fig. 1. Two common flexure hinge designs: (a) single-axis (constant width) and (b) multiple-axis (revolute).

Fig. 2. Two-axis flexure hinge with non-collocated (seriallydisposed) notches.

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

loading. Paros and Weisbord [1] presented two-axis circular flexure hinges designed in this serial configuration. The serial design preserves the convenience of having each flexure hinge designed according to the standard single-axis geometry but also requires the extra-length that is necessary to locate the two flexures in a serial manner. Paros and Weisbord [1] introduced the analytic approach to circular flexure hinges, both single- and serial two-axis, by formulating the compliances (stiffness inverses) in terms of deflections and rotations produced through bending and axial loading. Other configurations of single-axis flexure hinges were also analyzed by approaches similar to that developed by Paros and Weisbord [1]. Smith et al. [2], for instance, extrapolated the procedure introduced by Paros and Weisbord [1] to give approximate compliance equations for symmetric flexure hinges of elliptic geometry. Lobontiu et al. [3] developed exact compliance equations for symmetric corner-filleted flexure hinges and, based on these compliances, discussed the capacity of rotation, precision of motion and limits posed by maximum stress levels. In a similar manner, Lobontiu et al. [4] gave a unitary presentation to conicsection flexure hinges, including circular, elliptic, parabolic and hyperbolic profiles. Smith [5] briefly presented the revolute multiple-axis flexure hinges, while a more extended treatment of single-, two- and multiple-axis flexure hinges can be found in Lobontiu [6]. Flexure hinges have also been approached by means of the finite element method, through either direct simulation by commercially-available software or, more recently, by basic formulation. Ragulskis et al. [7] used a commercial finite element code to analyze a quartermodel of symmetric circular flexure hinges in order to find criteria leading to the optimal profile of this flexure configuration. Xu and King [8] analyzed circular, corner-filleted and elliptic flexure hinges in terms of flexibility, motion accuracy and stress levels. Zhang and Fasse [9] analyzed the constant-width symmetric circular flexure hinges by formulating the generic equations of six non-dimensional stiffness parameters (in terms of three translations and three rotations) to define the stiffness properties of the so-called Ôcenter of stiffnessÕ (actually the flexureÕs centroid). A similar approach was pursued in the monograph of Koster [10], who, amongst other discussed topics, analyzes a circular flexure hinge by means of five non-dimensional stiffness parameters. More recently, Murn and Kutis [11] formulated a threedimensional beam element with its cross-section dimensions varying continuously. The stiffness matrix and nodal forces were formulated and their components were solved numerically. Lobontiu [6] give generic formulations for the elemental stiffness and mass matrices for single-, two- and multiple-axis flexure hinges, as well as explicit formulations of these elemental matrices for single-axis corner-filleted flexure hinges.

1331

Fig. 3. Two-axis flexure hinge with collocated notches.

The subject of this paper is a novel two-axis flexure hinge with symmetric and axially-collocated notches, as shown in Fig. 3, configuration which removes the main disadvantage of a classical two-axis flexure hinge with serially-disposed notches. This design is particularly useful in applications with space restrictions where the serial solution cannot be employed and can be implemented in compliant devices that are utilized in the auto, aviation, biomedical or MEMS industries. The geometry of such a flexure can be tailored to satisfy certain design pre-requisites in terms of the compliances that pertain to the primary and secondary sensitive axes. The paper gives the generic compliance formulation of two-axis flexure hinges with collocated and symmetric notches that are defined by means of analytical curves. Specific closed-form compliance equations are derived for parabolic-profiled flexures by analyzing the static response to axial loading, torsion and direct- and cross-bending about the two principal planes (directions). The formulation also enables inclusion of the shearing effects for short (Timoshenko-type) flexure hinges. The capacity of rotation, precision of rotation, stress limitations and strain energy characterization are unitarily discussed by means of several compliance factors.

2. Compliance-based formulation A generic mathematical model is developed here addressing two-axis flexure hinges that can be defined analytically by various geometric profiles. Several compliances (the compliance being the inverse of the spring rate) will be defined to characterize the performance of a two-axis flexure hinge with collocated and symmetric notches in terms of its capacity to produce the relative rotations (about the primary and secondary axes), the sensitivity to parasitic effects such as axial loading and/ or torsion, the precision of rotation, shearing effects for relatively-short flexures, stress limitations and efficiency of motion––described in terms of the stored strain energy.

1332

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

along the z-direction, whereas the secondary sensitive axis is directed along the y-direction of Fig. 4. The translatory and rotary displacements of point 1 in Fig. 4 can be determined by means of CastiglianoÕs displacement theorem through calculating the corresponding partial derivatives of the strain energy in the generic form: 8 oU > > < u1i ¼ oF1i ; i ¼ x; y; z ð5Þ oU > > : h1i ¼ oM1i

Fig. 4. Schematic representation of the loads and displacements for a two-axis flexure hinge.

2.1. Capacity of rotation Irrespective of its defining geometry, a flexure hinge can be characterized by the elastic properties of one of its ends (generally assumed free) in terms of the other end (usually assumed fixed), as sketched in Fig. 4. From this standpoint, the free end 1 of the flexure hinge represented in generic form in Fig. 4 will possess six degrees of freedom (DOF) namely: three translations and three rotations produced through bending, axial loading and torsion, respectively. The deformation vector at 1 is connected to the load vector at the same point, under the small-displacement assumptions, according to the equation: fu1 g ¼ ½C1 fL1 g

with 8 Z l F1x2 > > dx > Ua ¼ > > 0 2EAðxÞ > > Z > l > > ðF1y x þ M1z Þ2 > > dx < Ubz ¼ 2EIz ðxÞ 0 Z l 2 > ðF1z x þ M1y Þ > > dx Uby ¼ > > > 2EIy ðxÞ 0 > > Z > l 2 > M1x > > : Ut ¼ dx 0 2GIt ðxÞ

ð3Þ

For a two-axis flexure hinge, as the one pictured in Fig. 3, the symmetric compliance matrix ½C1  of Eq. (1) is defined as: 2 3 0 0 0 0 0 C1;xFx 6 0 C1;yFy 0 0 0 C1;yMz 7 6 7 6 0 0 C 0 C 0 7 1;zFz 1;zMy 6 7 ½C1  ¼ 6 0 0 7 0 0 C1;hx Mx 6 0 7 4 0 0 C1;hy My 0 5 0 C1;zMy 0 C1;yMz 0 0 0 C1;hz Mz ð4Þ where the second subscript indicates the displacement– load correlation. It will be assumed in all further development that the primary sensitive axis is directed

ð6Þ

ð7Þ

By combining Eqs. (5)–(7) the compliance factors of Eq. (4) can be written in generic form as: C1;xFx ¼

1 I1 E

ð8Þ

C1;yFy ¼

12 I2 E

ð9Þ

C1;yMz ¼

12 I3 E

ð10Þ

C1;hz Mz ¼

12 I4 E

ð11Þ

ð2Þ

and the load vector is: fL1 g ¼ fF1x ; F1y ; F1z ; M1x ; M1y ; M1z gT

U ¼ Ua þ Ubz þ Uby þ Ut

ð1Þ

where the displacement vector is: fu1 g ¼ fu1x ; u1y ; u1z ; h1x ; h1y ; h1z gT

The total strain energy U comprises fractions produced through axial loading, bending about the primary (z) and secondary (y) sensitive axes and torsion, namely:

C1;zFz ¼

12 I5 E

ð12Þ

C1;zMy ¼

12 I6 E

ð13Þ

C1;hy My ¼

12 I7 E

ð14Þ

C1;hx Mx ¼

I8 G

ð15Þ

The newly-introduced I1 through I8 factors of Eq. (8) through (15) are integrals depending on the variable

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

cross-section parameters that define a particular collocated two-axis flexure hinge. The generic formulation which is developed here considers that the cross-section is rectangular and is defined by two parameters tðxÞ– thickness and wðxÞ–width that vary continuously over the length of the flexure. Fig. 5 gives a representation of the geometric parameters for a collocated two-axis flexure hinge, and the I1 through I8 integrals that define the compliances previously formulated are:

I1 ¼

Z

l

0

I2 ¼

Z

l

Z

l

Z

l

Z

l

Z

l

I7 ¼

l

Z 0

dx tðxÞwðxÞ3

0

I8 ¼

x dx tðxÞwðxÞ3

0

Z

x2 dx tðxÞwðxÞ3

0

I6 ¼

dx tðxÞ3 wðxÞ

0

I5 ¼

x dx tðxÞ3 wðxÞ

0

I4 ¼

x2 dx tðxÞ3 wðxÞ

0

I3 ¼

dx tðxÞwðxÞ

l

dx It ðxÞ

1333

ð16Þ

ð17Þ

ð18Þ

ð19Þ

ð20Þ

ð21Þ

ð22Þ

ð23Þ

where It ðxÞ, the conventional torsion moment of inertia for a two-axis flexure hinge, can be defined based on the formulation presented by Young [12] as:

tðxÞ ð24Þ It ðxÞ ¼ tðxÞ3 wðxÞ a  b wðxÞ with the constants a and b defined as:  a ¼ 0:333 b ¼ 0:21

ð25Þ

Eq. (24) is valid for the cases where the minimum width w of the flexure hinge is larger than the minimum thickness t, as is the case analyzed here, but for the opposite situation, one just has to switch w and t in Eq. (24). The original formulation given by Young [12] also contains a fourth-degree term of the form tðxÞ4 =wðxÞ4 in the bracket of Eq. (24). The errors induced by neglecting the fourth-degree term in the original equation given by Young [12] were shown to be less than 0.8% when the thickness-to- width ratio spans the range from 0.001 to 1 for a constant rectangular cross-section flexure hinge–– Lobontiu [6], and therefore the approximate Eq. (24) will be utilized here as well. 2.2. Short flexures––shearing effects Fig. 5. Geometric parameters defining a two-axis flexure hinge with symmetric notches.

The formulation so far has implicitly assumed that, in terms of bending, the collocated two-axis flexure

1334

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

hinge behaves as a long member and therefore the shearing effects were not taken into account. However, when the flexureÕs length is short relative to the other two dimensions of the cross-section, the shearing force introduces stresses and related strains (deformations) that can no longer be ignored. The threshold length-tothickness ratio which separates long from short beams is assigned values ranging from 3 to 5 in most of the dedicated literature (a value of 3 is given in Young [12] while Den Hartog [13] assumes a limit value of 5). It is also recognized that the deflection produced by shearing in a short beam becomes comparable to the regular deflection produced by bending. Lobontiu [6] discusses several aspects regarding the distinction between ÔlongÕ and ÔshortÕ flexure hinges and proposes a criterion that discriminates between short and long beams by evaluating the shearing-to-bending deflection ratio and comparing it to a limit value (called ÔerrorÕ). The situation where this ratio exceeds the error limit for a particular geometry will place that particular beam into the ÔshortÕ category, otherwise into the ÔlongÕ class. The decision over the short/long character of a flexure hinge directly from the error limit that is set by a specific length-tothickness ratio is a bit more involved, and aspects regarding the cross-section type and material properties need be taken into account. When shearing is taken into consideration, the total strain energy of Eq. (6) will include an extra term, which, in the case of a two-axis flexure hinge, is of the form: Z l Z l aF1y2 aF1z2 Us ¼ dx þ dx ð26Þ 0 2GAðxÞ 0 2GAðxÞ where a is a factor that depends on the shape of the cross-section. The compliances that will change through taking into account the shearing effects are those where the partial derivatives in terms of either F1y or F1z have to be calculated, as indicated by Eq. (26). Because the total strain energy, as expressed in Eq. (6), will now also include the shearing terms of Eq. (26), the compliances that will change due to this inclusion will be larger than the corresponding ones that do not have the shearing effects considered. It can easily be shown by inspecting Eqs. (8), (9) and (26) that the new compliances are: E C1;xFx G

ð27Þ

E C1;xFx G

ð28Þ

s C1;yF ¼ C1;yFy þ a y

s ¼ C1;zFz þ a C1;zF z

The displacements u1y and u1z that define a deformation state where the shearing effects are disregarded can be written explicitly from Eq. (1) through (4) as: u1y ¼ C1;yFy F1y þ C1;yMz M1z and

ð29Þ

u1z ¼ C1;zFz F1z þ C1;zMz M1y

ð30Þ

Similar equations can be formulated for the same displacements when shearing effects are taken into account, namely: s us1y ¼ C1;yF F þ C1;yMz M1z y 1y

ð31Þ

and s us1z ¼ C1;zF F þ C1;zMy M1y z 1z

ð32Þ

As mentioned previously, a criterion that enables to characterize whether or not the shearing effects need to be accounted for in compliance calculations is to compare the deflections about the y- and z-axes, by means of the ratios: rdy ¼

us1y u1y

ð33Þ

us1z u1z

ð34Þ

and rdz ¼

By combining Eqs. (27) through (32), the deflection ratios of Eqs. (33) and (34) become: rdy ¼ 1 þ

F1y aE C1;xFx C1;yFy F1y þ C1;yMz M1z G

ð35Þ

F1z aE C1;xFx C1;zFy F1z þ C1;zMy M1y G

ð36Þ

and rdz ¼ 1 þ

It is therefore obvious that deciding whether a flexure hinge responds like a long or short beam with respect to bending depends on the specific geometry of the analyzed configuration as well as on the material of the flexure. 2.3. Precision of rotation One drawback of utilizing flexure hinges to produce limited relative rotation between two rigid links, instead of using classical rotation joints, is that the so-called center of rotation of the flexure (customarily identified with the flexureÕs midpoint, as previously mentioned) displaces during the deformation of the flexure hinge. The precision of rotation for a three-dimensional twoaxis flexure hinge can be quantified by the translations of the geometric center 2 along the x-, y- and z-directions, as indicated in Fig. 4. By following a development that is similar to the one carried out when defining the displacements/rotations registered at the flexureÕs free end 1, the displacements at the flexureÕs midpoint can be assessed by means of several compliances that define the motion of point 2 with respect to the fixed point 3. Application of the CastiglianoÕs displacement theorem

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

results in the following equation that connects the displacement vector at point 2 to the load vector at point 1: fu2 g ¼ ½C2 fL1 g

ð37Þ

I30 ¼

I40 ¼

ð38Þ

ð39Þ

C2;xFx ¼

1 0 I E 1

ð40Þ

C2;yFy ¼

  12 0 l 0 I2  I3 E 2

ð41Þ

By following a similar reasoning to the one detailed when discussing the capacity of rotation for relativelyshort beams where shearing is taken into account, it can be shown that the corresponding shearing-affected compliance in terms of motion about the primary axis is:   12 0 l 0 a s I I C2;yF  þ I10 ¼ ð42Þ y E 2 2 3 G

s C2;zF ¼ z

ð43Þ

  12 0 l 0 I5  I6 E 2

ð44Þ

  12 0 l 0 a I5  I6 þ I10 E 2 G

ð45Þ

(for relatively-short beams where shearing is taken into account)   12 0 l 0 C2;zMy ¼ ð46Þ I6  I7 E 2 The I10 through I70 integrals entering the compliance equations that characterize the precision of rotation are: Z l dx ð47Þ I10 ¼ l=2 tðxÞwðxÞ I20 ¼

Z

l

x2 dx

l=2

tðxÞ3 wðxÞ

l

I50 ¼

I60 ¼

Z

ð48Þ

dx tðxÞ3 wðxÞ x2 dx

l=2

tðxÞwðxÞ3

Z

l

l=2

I70 ¼

x dx tðxÞ3 wðxÞ

l

Z

l

l=2

The compliances of Eq. (39) can be expressed as:

C2;zFz ¼

Z

l=2

while the compliance matrix of the same equation is: 2 3 0 0 0 0 0 C2;xFx C2;yFy 0 0 0 C2;yMz 5 ½C2  ¼ 4 0 0 0 C2;zFz 0 C2;zMy 0

The other compliances are:   12 0 l 0 I3  I4 C2;yMz ¼ E 2

l

l=2

The displacement vector of Eq. (37) is: fu2 g ¼ fu2x ; u2y ; u2z gT

Z

x dx tðxÞwðxÞ3 dx tðxÞwðxÞ3

1335

ð49Þ

ð50Þ

ð51Þ

ð52Þ

ð53Þ

2.4. Stress considerations When neglecting the effects produced through direct shearing forces and torsion, the stresses will only be normal to the cross-section since they are produced through bending and axial effects. The stresses produced by axial loading are constant over the cross-section while the stresses generated through bending on each of the two perpendicular planes vary linearly over the cross-section. As a consequence, the maximum stress will occur at one vertex of the rectangular cross-section where both stresses coming from the double bending are maximum and can be expressed, according to Lobontiu [6], as:  1 Ktb;z rmax ¼ ðF1y l þ M1z Þ Kta F1x þ 6 t wt

 Ktb;y ðF1z l þ M1y Þ ð54Þ þ w where Kta , Ktb;z and Ktb;y denote stress concentration factors in axial loading and bending about the z- and yaxes, respectively, and are presented in more detail in the works of Peterson [14] and Pilkey [15], for instance. Eq. (54) is useful when the loading on a flexure hinge is known. For cases where the displacement is rather available, the maximum stress can be expressed in terms of stiffnesses instead of compliances, as detailed by Lobontiu et. al. [3] for instance, and the following equation can be written:  1 6Ktb;z  rmax ¼ ðlK1;yFy þ K1;hz Mz Þu1y Kta K1;xFx u1x þ t wt  þ ðlK1;yMz þ K1;hz Mz Þh1z 6Ktb;z  ðlK1;zFz þ K1;hyz My Þu1z þ w   ð55Þ þ ðlK1;zMy þ K1;hyz My Þh1y

1336

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

where the individual stiffnesses of this equation are the components of the 6  6 stiffness matrix ½K1  which is the inverse of the compliance matrix ½C1  defined in Eq. (4). For a given flexure design and a specified displacement/deformation field, Eq. (55) can be utilized to compare the maximum stress with the yield stress corresponding to the design material. Alternatively, the same Eq. (55) can be employed in dimensioning the variable cross-section of the two-axis flexure based on the yield stress or another (reduced) allowable stress threshold. In other words, the capacity of rotation, as defined by one of the compliances that express bending/ rotation in Eq. (55) is clearly limited by the maximum stress value that is set for a given configuration and material parameters.

Utot ¼ 12½C1;xFx F1x2 þ C1;yFy F1y2 þ C1;zFz F1z2 2 2 þ C1;hx Mx M1x þ C1;hy My M1y þ C1;hz Mz M1z2

þ 2ðC1;zMy F1z M1y þ C1;yMz F1y M1z Þ Uhy ¼ 12ðC1;zMy F1z þ C1;hy My M1y ÞM1y

ð63Þ

Uhz ¼ 12ðC1;yMz F1y þ C1;hz Mz M1z ÞM1z

ð64Þ

In the case of an overall unit loading, Eqs. (57) and (58), via Eqs. (62) through (64), transform into: ghy ¼ ðC1;zMy þ C1;hy My Þ=ðC1;xFx þ C1;yFy þ C1;zFz þ C1;hx Mx þ C1;hy My þ C1;hz Mz þ 2ðC1;yMz þ C1;zMy ÞÞ

2.5. Efficiency in terms of strain energy In the case of a collocated two-axis flexure hinge, it is possible to evaluate its motion efficiency by calculating the strain energy stored during deformation produced by the overall load vector. As mentioned previously, the loading at the flexureÕs free end consists of bending on two perpendicular planes (planes of axial-symmetry), axial tension/compression and torsion. The explicit deformation–load equations are: 8 u1x ¼ C1;xFx F1x > > > > u1y ¼ C1;yFy F1y þ C1;yMz M1z > > < u1z ¼ C1;zFz F1z þ C1;zMy M1y ð56Þ h ¼ C1;hx Mx M1x > > > 1x > h ¼ C1;zMy F1z þ C1;hy My M1y > > : 1y h1z ¼ C1;yMz F1y þ C1;hz Mz M1z Two types of efficiency qualifiers are introduced here in terms of either the y- or the z-rotation, namely: ghy ¼

Uhy Utot

ð57Þ

ghz ¼

Uhz Utot

ð58Þ

ð65Þ

ghz ¼ ðC1;yMz þ C1;hz Mz Þ=ðC1;xFx þ C1;yFy þ C1;zFz þ C1;hx Mx þ C1;hy My þ C1;hz Mz þ 2ðC1;yMz þ C1;zMy ÞÞ

ð66Þ

For short flexure hinges that take into consideration the shearing effects, the compliance factors C1;yFy and C1;zFz s are substituted by the corresponding factors C1;yFy and s C1;zFz , respectively.

3. Application: Two-axis flexure hinge with parabolic notches An application is now analyzed by considering that both notches have the same length and are defined through parabolic profiles, as shown in Fig. 6 where three different parabolic profiles are drawn for three different values of the parameter c (root parameter) which dictates the maximum thickness/width of a given

7

The total strain energy under static (quasi-static) loading is:

6

Utot ¼ 12ðF1x u1x þ F1y u1y þ F1z u1z þ M1x h1x þ M1y h1y þ M1z h1z Þ

4

c=2

3 2

c=1

1 1 M h 2 1y 1y

ð60Þ

Uhz ¼ 12M1z h1z

ð61Þ

0 0

By substituting the displacements of Eq. (56) into Eqs. (59)–(61) the respective energy factors become:

c=3

5

t(x)/2

ð59Þ

whereas the strain energy factors that produce the y- or z-rotations are: Uhy ¼

ð62Þ

2

4

6

8

10

x

Fig. 6. Parabolic profiles for different values of the root parameter c.

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

flexure hinge. The variable thickness tðxÞ and width and wðxÞ, according to Fig. 5, are: tðxÞ ¼

2at  2 b2t  x  2l

ð67Þ

8 l < 8cw C1;hx Mx ¼ 32G : awð2cw þ wÞ2 ðcw t  ct wÞ þ

4cw ½cw ð3at þ 4bwÞ  7act w

a2 w2 ð2cw þ wÞðcw t  ct wÞ2 " pffiffiffiffiffiffiffi þ 2 2cw ð8b2 c2w w2 þ 4abcw wðcw t  5ct wÞ

with 8   tl2 t > > > a 1 þ ¼ < t 8 2ct rffiffiffiffiffiffiffiffiffiffiffiffiffiffi > l t > > 1þ : bt ¼ 2 2ct

ð68Þ

and wðxÞ ¼

1337

2aw  2 b2w  x  2l

ð69Þ

with 8   wl2 t > > > a ¼ 1 þ < w 8 2cw rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > l t > > 1þ : bw ¼ 2 2cw

rffiffiffiffiffiffiffi# 2cw þa w ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q , 5 i h p ffiffiffiffiffi ffi 16 2ðac t  bcw Þ 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a3 ðcw t:  ct wÞ w5 þ a3 ðct w  cw tÞ3 at  bw ) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðact  bcw Þ ð75Þ  arctan at  bw 2

ð3c2w t2  10ct cw tw þ 15c2t w2 ÞÞarctan

C1;zFz ¼ fð4l3 ½2112c2w wð5ct þ 3tÞ þ 231t3 ð6ct þ 5tÞ þ 198cw w2 ð32ct þ 21tÞÞ ð70Þ

=ð1155Ew3 tð2cw þ wÞ3 ð2ct þ tÞÞg þ fð256l3 c3w ð96ct þ 55tÞÞ

The closed-form compliance equations that describe the capacity of rotation, together with sensitivity to parasitic (non-rotational) effects, as well as the precision of rotation are given next.

=ð1155Ew3 tð2cw þ wÞ3 ð2ct þ tÞÞg

ð76Þ

C1;zMy ¼ fð2l2 ½105w3 ð4ct þ 3tÞ þ 252cw w2 ð8ct þ 5tÞ þ 288c2w wð12ct þ 7tÞÞ

3.1. Capacity of rotation

=ð105Ew3 tð2cw þ wÞ3 ð2ct þ tÞÞg

The compliances that characterize the capacity of rotation are: C1;xFx ¼

l½32ct cw þ 20ðcw t þ ct wÞ þ 15wt 15Ewtð2ct þ tÞð2cw þ wÞ 3

C1;yFy ¼ fð4l

½2112c2t tð5cw

ð71Þ

3

þ fð256l2 c3w ð16ct þ 9tÞÞ =ð105Ew3 tð2cw þ wÞ3 ð2ct þ tÞÞg

2 C1;hy My ¼ C1;zMy l

ð77Þ

ð78Þ

þ 3wÞ þ 231t ð6cw þ 5wÞ

ð72Þ

The closed-form compliance equations that were formulated within this section were also used to run a numerical simulation and the results were compared to finite element simulation data (the ANSYS finite element software has been used) based on the following geometric and material parameters: l ¼ 0:0025 m, t ¼ 0:0004 m, w ¼ 0:0006 m, ct ¼ 0:001 m, cw ¼ 0:0015 m, E ¼ 200 MPa, l ¼ 0.3. The numerical results are given in Table 1 whereby it can be seen that the relative errors between the two sets of numerical simulations are less than 6%.

ð73Þ

3.2. Precision of rotation

ð74Þ

The compliances that define the precision of rotation are:

2

þ 198ct t ð32cw þ 21wÞÞ =ð1155Ewt3 ð2ct þ tÞ3 ð2cw þ wÞÞg þ fð256l3 c3t ð96cw þ 55wÞÞ =ð1155Ewt3 ð2ct þ tÞ3 ð2cw þ wÞÞg C1;yMz ¼ fð2l2 ½105t3 ð4cw þ 3wÞ þ 252ct t2 ð8cw þ 5wÞ þ 288c2t tð12cw þ 7wÞÞ =ð105Ewt3 ð2ct þ tÞ3 ð2cw þ wÞÞg þ fð256l2 c3t ð16cw þ 9wÞÞ =ð105Ewt3 ð2ct þ tÞ3 ð2cw þ wÞÞg 2 C1;hz Mz ¼ C1;yMz l

1338

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

Table 1 Analytic versus finite element simulation for a two-axis parabolic-profile flexure hinge

Analytic FEA % error

C1;yFy [N1 m]

C1;yMz [N1 ]

C1;hzMz [N1 m1 ]

C1;zFz [N1 m]

C1;zMy [N1 ]

C1;hyMy [N1 m1 ]

C1;xFx [N1 m]

C1;hxMx [N1 m1 ]

3.01e)6 2.85e)6 5.31

2.20e)3 2.10e)3 4.50

1.76 1.68 4.54

1.06e)6 1.00e)6 5.66

9.70e)4 9.20e)6 5.15

7.76e)1 7.40e)1 4.64

3.04e)8 2.90e)8 4.61

1.01 0.95 5.94

C2;yFy ¼ fðl3 ½66c2t tð443cw þ 306wÞ þ 231t3 ð23cw þ 25wÞÞ 3

3

=ð4620Ewt ð2ct þ tÞ ð2cw þ wÞÞg þ fðl3 ½c3t ð15184cw þ 9746wÞ þ 99ct t2 ð204cw þ 161wÞÞ =ð4620Ewt3 ð2ct þ tÞ3 ð2cw þ wÞÞg

tation of a parabolic-profile two-axis collocated flexure hinge, Eqs. (71)–(82), were utilized to run several numerical simulations. The parabolic flexure hinge is compared to a constant cross-section flexure hinge that is defined by the minimum thickness t and width w by means of the generic compliance ratio:

ð79Þ rC1;dL ¼

C2;yMz ¼ ð3l2 ½cw ð16c3t þ 30c2t t þ 20ct t2 þ 5t3 Þ þ 5wðct þ tÞð2c2t þ 2ct t þ t2 ÞÞ =ð10Ewt3 ð2ct þ tÞ3 ð2cw þ wÞÞ

ð80Þ

C2;zFz ¼ fðl3 ½66c2w wð443ct þ 306tÞ þ 231w3 ð23ct þ 25tÞÞ =ð4620Ew3 tð2cw þ wÞ3 ð2ct þ tÞÞg þ fðl3 ½c3w ð15184ct þ 9746tÞ þ 99cw w2 ð204ct þ 161tÞÞ =ð4620Ew3 tð2cw þ wÞ3 ð2ct þ tÞÞg

C2;zMy ¼

ð81Þ

3l2 ½ct ð16c3w þ 30c2w w þ 20cw w2 þ 5w3 Þ 10Ew3 tð2cw þ wÞ ð2ct þ tÞ þ

15l2 tðcw þ wÞð2c2w þ 2cw w þ w2 Þ 10Ew3 tð2cw þ wÞ3 ð2ct þ tÞ

ð82Þ

A limit check was applied to all the compliances that have been derived for a two-axis parabolic flexure hinge in order to verify their correctness. As Fig. 5 indicates it, a two-axis parabolic flexure hinge becomes a constant rectangular cross-section flexure when the symmetric profiles in the two mutually-perpendicular sensitive planes are transformed to lines. Mathematically, this condition requires that the two parameters ct and cw go to zero. These limits have been applied to all the compliances derived within this section and the results were the corresponding compliance equations for a constant rectangular cross-section flexure hinge. 3.3. Numerical simulation The closed-form compliance equations that characterize the capacity of rotation and the precision of ro-

ð83Þ

where the Ô Õ superscript indicates the constant crosssection flexure hinge while the Ôd  LÕ subscript denotes a specific deformation–load relationship describing the capacity of rotation. As illustrated in Figs. 7 and 8, the ratio of Eq. (83) decreases (meaning that the parabolicprofile two-axis flexure hinge approaches the elastic behavior of its constant cross-section counterpart) when the minimum thickness t and width w increase, as well as when the root parameters ct and cw decrease. Similar conclusions can be drawn with respect to the following compliance ratios: rC2;dL ¼

3

C1;dL C1;dL

C2;dL C2;dL

ð84Þ

which characterize the precision of rotation of a constant cross-section flexure hinge versus its corresponding parabolic-profile two-axis counterpart, and Fig. 9 illustrates these trends. The influence of shearing can be monitored by constructing compliance ratios that are similar to those of Eqs. (83) and (84), and that compare constant crosssection flexure hinges to two-axis parabolic-profile flexures in terms of their compliant response. As Fig. 10 indicates it, by decreasing the minimum thickness t and width w and/or by increasing the root parameters ct and cw , the constant-to-parabolic compliance ratios decrease when shearing effects are taken into consideration. The strain energy efficiency of a two-axis parabolic flexure hinge, as defined in Eqs. (65) and (66) in terms of unit loads acting at the free end of the flexure is represented graphically in Fig. 11 (a) and (b) in terms of the minimum thickness t and the root parameter cw , respectively.

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341 2.35

2.36 w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

2.35 2.34

2.25

rC 1,y-Mz

rC 1,y-Mz

2.3

2.2

2.33 2.32 2.31 w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

2.3 2.15

2.29

2.1 0.0005 0.00075

(a)

0.001

2.28 0.001

0.00125 0.0015

5.1 4.9 4.7

9 8.5 8

rC 1, θ x-Mx

10 9.5

4.5 4.3 4.1 3.9

0.00125 0.0015

t [m]

(c)

0.003

0.004

0.005

cw [m]

5.3

w = 0.0003 [m] w = 0.0004 [m] 3.7 w = 0.0005 [m] 3.5 0.0005 0.00075 0.001

0.002

(b)

t [m] 5.5

rC 1, θ x-Mx

1339

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

7.5 7 6.5 6 5.5 5 0.001

0.002

0.003

0.004

0.005

cw [m]

(d)

Fig. 7. Constant-to-parabolic compliance ratios characterizing the capacity of rotation: (a) and (b) compliance of the moment-produced deflection about the y-axis; (c) and (d) torsion compliance.

1.82

1.76

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

1.74

1.8

rC1,x-Fx

rC1,x-Fx

1.72 1.7

1.78 1.76

1.68

1.64 0.0005

0.00075

(a)

0.001

0.00125

1.72 0.001

0.0015

0.004

0.005

2.84 2.83

2.7

2.82

rC1,y-Fy

rC1,y-Fy

0.003

cw [m] 2.85

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

2.75

2.65

2.81 2.8 2.79

2.6

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

2.78

2.55 2.5 0.0005 0.00075

0.002

(b)

t [m] 2.8

(c)

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

1.74

1.66

2.77 0.001

t [m]

2.76 0.001

0.00125 0.0015

(d)

0.002

0.003

0.004

0.005

cw [m]

Fig. 8. Constant-to-parabolic compliance ratios characterizing the capacity of rotation: (a) and (b) axial compliance; (c) and (d) compliance of the force-produced deflection about the y-axis.

1340

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341 5.5

5.4 w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

5.2

5.45 5.4

rC2,y-Mz Fy

rC2,y-Fy

5 4.8 4.6

5.3 w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

5.25

4.4

5.2

4.2 0.0005

0.00075

0.001

0.00125

5.15 0.001

0.0015

t [m]

(a)

0.002

0.003

0.004

0.005

cw [m]

(b) 4.6

4.5 4.4

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

4.3

4.55 4.5

rC2,y-Mz

4.2

rC2,y-Mz

5.35

4.1 4

4.45 4.4

3.9 3.8

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

4.35

3.7 3.6 0.0005

0.00075

0.001

0.00125

t [m]

(c)

4.3 0.001

0.0015

0.002

0.003

0.004

0.005

cw [m]

(d)

Fig. 9. Constant-to-parabolic compliance ratios characterizing the precision of rotation: (a) and (b) compliance of the force-produced deflection about the y-axis; (c) and (d) compliance of the moment-produced deflection about the y-axis.

1.0356

1.3

1.0354

1.2

1.0353

rCs1,y-Fy

rCs1,y-Fy

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

1.0355

w = 0.0003 [m] 1.25

1.15

1.0352 1.0351

1.1

1.035

1.05 1.0349

1 0.0005 0.00075

0.001

1.0348 0.001

0.00125 0.0015

t [m]

(a)

0.002

1.00075

0.003

0.004

0.005

cw [m]

(b) 1.000059

w = 0.0003 [m] w = 0.0004 [m] w = 0.0005 [m]

1.0000588

w = 0.0003 [m]

1.0000586

rCs1,y-Mz

rCs1,y-Mz

1.0005

1.00025

1.0000584 1.0000582 1.000058 1.0000578 1.0000576

1 0.0005 0.00075

(c)

0.001 t [m]

1.0000574 0.001

0.00125 0.0015

(d)

0.002

0.003

0.004

0.005

cw [m]

Fig. 10. Constant-to-parabolic compliance ratios characterizing the influence of the shearing effects: (a) and (b) compliance of the force-produced deflection about the y-axis; (c) and (d) compliance of the moment-produced deflection about the y-axis.

N. Lobontiu, E. Garcia / Computers and Structures 81 (2003) 1329–1341

1341

its constant rectangular cross-section counterpart in terms of several performance criteria, based on the analytical model.

0.8 0.79 0.78

η

0.77 0.76

Acknowledgement

0.75

0.73 0.0005

0.00075

0.001

0.00125

0.0015

t [m]

(a)

η

The authors acknowledge support for this work by the NSF grant CMS-0241456.

y-rotation z-rotation

0.74

0.76 0.75 0.74 0.73 0.72 0.71 0.7 0.69 0.68 0.67 0.66 0.001

(b)

References y-rotation z-rotation

0.002

0.003

0.004

0.005

cw [m]

Fig. 11. Strain energy efficiency: (a) in terms of the minimum thickness t; (b) in terms of the width parameter cw .

4. Conclusions The paper introduces the axially-collocated and symmetric two-axis flexure hinge as an alternative to the functionally-equivalent serial design. The formulation is generic and therefore enables the selection of various geometric profiles that can be expressed in analytic form. By defining several compliances, this type of flexure hinge can be characterized in terms of its capacity of rotation, precision of rotation, stress levels, strain energy efficiency and influence of shearing effects. An application is considered whereby the two-axis flexure hinge is designed by means of two non-identical parabolic profiles and for which closed-form compliance equations are derived to quantify the capacity and precision of rotation. The results of the simulation that is based on the analytical model are in good agreement with the finite element simulation data (errors less than 6%). The parabolic-profiled two-axis flexure is also compared to

[1] Paros JM, Weisbord L. How to design flexure hinges. Mach Des 1965:151–6. [2] Smith ST, Badami VG, Dale JS, Xu Y. Elliptical flexure hinges. Rev Scientific Instrum 1997;68(3):1474–83. [3] Lobontiu N, Paine JSN, Garcia E, Goldfarb M. Cornerfilleted flexure hinges. ASME J Mech Des 2001;123:346– 52. [4] Lobontiu N, Paine JSN, Garcia E, Goldfarb M. Design of symmetric conic-section flexure hinges based on closedform compliance equations. Mech Mach Theory 2002;37(5):477–98. [5] Smith ST. Flexures-elements of elastic mechanisms. Amsterdam: Gordon and Breach Science; 2000. [6] Lobontiu N. Compliant mechanisms: Design of flexure hinges. Boca Raton: CRC Press; 2002. [7] Ragulskis KM, Arutunian MG, Kochikian AV, Pogosian MZ. A study of fillet type flexure hinges and their optimal design. Vib Eng 1989;3:447–52. [8] Xu W, King TG. Flexure hinges for piezo-actuator displacement amplifiers: flexibility, accuracy and stress considerations. Prec Eng 1996;19(1):4–10. [9] Zhang S, Fasse ED. A finite-element-based method to determine the spatial stiffness properties of a notch hinge. ASME J Mech Des 2001;123(1):141–7. [10] Koster M. Constructieprincipies voor het nauwkeurig bewegen en positioneren (in Dutch). Twente: Twente University Press; 1998. [11] Murn J, Kutis V. 3D-beam element with continuous variation of the cross-sectional area. Comput Struct 2002;80(3–4):329–38. [12] Young WC. RoarkÕs formulas for stress and strain. New York: McGraw-Hill; 1989. [13] Den Hartog JP. Advanced strength of materials. New York: Dover; 1987. [14] Peterson RE. Stress concentration factors. New York: John Wiley and Sons; 1974. [15] Pilkey WD. PetersonÕs stress concentration factors. New York: John Wiley and Sons; 1997.