Effect of initial curvature in uniform flexures on position accuracy

Mechanism and Machine Theory 119 (2018) 106–118

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Research paper

Effect of initial curvature in uniform flexures on position accuracy M. Verotti Department of Mechanical and Aerospace Engineering, Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy

a r t i c l e

i n f o

Article history: Received 22 June 2017 Revised 29 August 2017 Accepted 30 August 2017

Keywords: Compliant mechanisms Pseudo-rigid-body model MEMS Flexures Center of rotation Position accuracy

a b s t r a c t Position accuracy is a prerequisite for compliant mechanisms, especially in micro-scale applications. Generally, this feature depends on the flexures ability to imitate the revolute joints of the pseudo-rigid body model. In case of flexible elements, the center of the relative rotation varies its position during the deflections, affecting the position accuracy. In this paper, the role played by the initial curvature is investigated, in case of uniform primitive flexures. Analytical expressions are derived to evaluate the shift of the rotation axis with respect to the flexure centroid. An example shows the performance of four flexures with different curvatures, evaluated by comparing the rotation axis shift and the position error. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Compliant mechanisms are being increasingly used because they offer some advantages with respect to their rigid-body counterparts, such as reduced friction and backlash, no need for lubrication, simplified manufacture, fabricability at macro, meso and micro scale, and compatibility with MEMS-based technologies. Compliant mechanisms have been applied, for example, in the fields of micro-manipulation [1–5], micro-positioning [6–10], precision manufacturing [11,12]. Applications in these areas require high position accuracy, therefore precision of analytical models represents a significant challenge in compliant mechanisms design [13,14]. Generally, the synthesis of a compliant mechanism can be performed by means of freedom and constraint topologies [15–17], topology optimization [18–20], building blocks [21–23], or rigid-body replacement [24–26]. The rigid-body replacement method consists in obtaining a compliant mechanism by replacing the revolute joints of a rigid-body mechanism, or pseudo-rigid body model (PRBM), with flexible elements. In case of straight-axis primitive flexures, the position of the flexible element with respect to the revolute joint can be determined according to different criteria. If the length of the flexure is small if compared to the subsequent rigid links, its midpoint can be placed on the position of the revolute joint. If the length of the flexible beam is comparable to the rigid elements, its position with respect to the revolute joint can be determined by parametric approximation of the beam’s deflection path, by calculating the characteristic radius factor [27,28]. Another possible criteria consists in positioning the center of rotation of the flexure on the revolute joint [29]. The position accuracy of a compliant mechanism depends on the mechanical characteristics of the flexure and on its position with respect to the connecting links. In other words, the motion accuracy of the compliant mechanism depends on the flexure ability to imitate the behaviour of the revolute joint. Whilst the latter guarantees a fixed position for the center

E-mail address: [email protected] http://dx.doi.org/10.1016/j.mechmachtheory.2017.08.021 0094-114X/© 2017 Elsevier Ltd. All rights reserved.

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of the relative rotation of the connecting links, the deformation of the flexure depends on the applied loads. Therefore, the center of rotation varies its position during motion. This condition is referred to as axis shift or parasitic motion [30,31]. To improve the performance of compliant systems in terms of range of motion and precision of rotation, but also in terms of stress concentration or off-axis stiffness to axial stiffness ratio [32], many investigations have been focused on the development of complex flexures. These flexures combine more flexible elements or involve contact systems [33–37]. In fact, the variation of the position of the center of rotation occurs especially in primitive flexures, such as straight- or curved-axis beams, undergoing large deflections. In case of end-moment loads, the center or rotation moves along the axis of symmetry of the flexure, and its distance from the axis centroid increases for increasing loads [29]. Recent investigations focused on the adoption and on the modeling of circular flexures in compliant mechanisms. Wang et al. used the PRB to model corrugated beams [38], and Han et al. designed a quadstable monolithic mechanism based on curved elements [39]. Edwards et al. proposed a PRB model for curved beams spinned at both ends [40], whereas Venkiteswaran and Su developed a PRBM for circular beams subjected to combined loads [41]. With respect to straight-axis beams, curved flexures offer more flexibility in the design phase, introducing a new parameter, that is the radius of curvature. This parameter can be set in order to optimize the geometry of the compliant mechanism, for example to minimize the distance between two subsequent rigid links or to overcome geometrical constraints. However, this parameter has an effect on the position of the center of rotation of the flexure. As a consequence, it affects also the position accuracy of the compliant mechanism. This paper focuses on the role played by the initial curvature of uniform flexures on their ability to mimic the rotational accuracy of revolute joints. The analysis is performed considering constant cross-section beams subjected to end-moment loads. Analytical expressions are derived to determine the position of the center of rotation with respect to the beam axis centroid. Various axis curvatures are considered, including the straight-axis case. 2. Position of the center of rotation of the free-end section Considering a constant-curvature beam with axis length equal to l, the Euler-Bernoulli beam equation can be written as

dθ μ = , ds l

(1)

where s is the arc-length coordinate system, dθ /ds is the rate of change of angular deflection along the beam, and

μ = μ0 + μ˜

(2)

is a non-dimensional term equal to the sum of

l r

μ0 = ,

(3)

where r is the radius of curvature, and

μ˜ =

M l, EI

(4)

where M is the bending moment and EI is the bending stiffness. With reference to Fig. 1, the application of an external moment to the flexure free-end, represented by μ ˜ , determines a transition from the initial configuration, Cμ0 , to the deformed one, Cμ . The orientation of the free-end section varies from the angle θ E to the angle θ F , and its centroid moves from point E to point F. The position of the pole of the displacement, or center of rotation, Cr , is given by [42]:



θF − θE

xCr 1 = xE + xF − (yF − yE ) cot l 2l





2

yCr 1 = yE + yF + (xF − xE ) cot l 2l

θF − θE 2

,

(5)

,

(6)



where the Cartesian coordinates are normalized with respect to the beam axis length. As demonstrated in Ref. [29], a convenient representation of the Cr coordinates can be obtained by introducing the reference frame RO {O, x, y, z}, whose origin is positioned on the centroid of the clamped section, and whose y−axis is parallel to the arc chord. With respect to RO , the position of the free-end section in the neutral and deformed configurations are given by



p0E = 0, and

p0F =

1 μ

sin α

α

T

,

(cos (α − μ ) − cos α ),

(7)

1

μ

T (sin α − sin (α − μ ) ) ,

(8)

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Fig. 1. Flexure in neutral (Cμ0 ) and deformed (Cμ ) configurations.

respectively. The orientation of the free-end section changes from θE = π /2 + α to θF = π /2 − α + μ, with an angular variation equal to

θ = θ F − θ E = μ − 2 α .

(9)

By substituting (7,8,9) in (5,6), it follows



pC0r

1 cos (μ − α ) − cos α = − 2 μ



sin α + sin (μ − α )

μ

−

sin α

α

 cot

μ − 2α 2

,

sin α

T

α

(10)

In the particular cases of straight (α = 0) and full-angle (α = π ) beams, the position of the center of rotation is given by

pC0r = and

pC0r =

1 2



1 μ

cot

, 0

μ˜

T

2



−

2 , 1 μ˜

T

,

(11)

(12)

respectively. Equations (10,11,12) show that, during the application of the external load, the center of rotation: (a) moves on the axis of symmetry of the flexure in neutral configuration, (b) tends to infinity as the free-end section approaches a full rotation, and (c) tends to the flexure centroid as the external load tends to zero. It is worth noting that these results are valid for the specified load conditions, that are end-moment loads. In case of combined loads, a dedicated study should be conducted, considering the magnitude and the direction of the applied force.

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However, in first approximation, these results can be considered acceptable when, in non-dimensional terms, the applied force is small if compared to the applied moment. For further details, see Ref. [29]. 3. Effect of the initial curvature Property (c) can be exploited in order to evaluate the effect of the initial curvature on the position of the center of rotation. With reference to Fig. 1, let p0 be the vector of coordinates of the generic point P with respect to R0 {O0 , x, y, z}. Consider then a new reference frame, R1 {O1 , x1 , y1 , z1 }. The position of P with respect to R1 can be expressed as

p¯ 1 = D10 p¯ 0 ,

(13)

where the notation p¯ denotes the homogeneous representation of vector p, and



D10

=

R01

−R01 o01

0T

1



(14)

is the homogeneous transformation matrix. In (14), R01 is the rotation matrix of R0 with respect to R1 , and o01 is the position vector of O1 in R0 . Let the origin of R1 be the centroid of the flexure, and let the axes x1 , y1 , z1 , be parallel to x, y, z, respectively. It follows

 o10

=

1 2α



sin α

α

 − cos α

sin α , 2α

T .

(15)

The homogeneous transformation described by (13) can be applied to vector pC0r as

p¯ C1r = D10 p¯ C0r .

(16)

Substituting (10,15) into (16) yields



pC1r



sin α 1 cos (α − μ ) − cos α 1 = + cos α − 2 μ α α



 +

sin α − sin (α − μ )

μ

−

sin α



α

2α − μ cot , 0 2

T .

(17)

Equation (2) can be rewritten as

μ = 2α + μ˜

(18)

and substituted into (17), resulting in



pC1r



sin α 1 cos (α + μ ˜ ) − cos α 1 = + cos α − 2 2α + μ ˜ α α



 −

sin α sin (α + μ ˜ ) + sin α − 2α + μ ˜ α

 cot

μ˜ 2

T , 0

.

(19)

In the particular cases of straight and full-angle beams, the position of the centroid is given by o10 = [0, 1/2]T and o10 =

T [1/2π , 0] , respectively. By applying the homogeneous transformation (16) to the vectors pC0r defined in (11) and (12), it follows

pC1r

1 = 2

and

pC1r =

1 2



μ˜



2 cot − , 0 2 μ˜





T

1 1 − , 0 2π + μ ˜ 2π

(20)

T

,

(21)

respectively. Fig. 2 shows the non-dimensional x coordinate of the center of rotation with respect to μ ˜ for different initial curvatures, from α = 0 to α = π with steps of π /6. Three different cases can be considered, depending on the intervals of the domain of the function. 3.1. Case 1 Figure 2 shows that, for a given value of μ ˜ in the interval (0, 2π ), the absolute values of xCr /l decreases as the curvature increases. Therefore, in case of applied loads that tend to increase the flexure curvature, higher values of the initial curvature reduce the shift of the rotation axis along the x−axis. However, a maximum value of the initial curvature should be also considered. In fact, as this parameter increases, the range of motion decreases because of planar nature of the flexure, that constrain the free-end section not to overstep the fixed one.

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Fig. 2. Non-dimensional x coordinate of Cr with respect to μ ˜ for different initial curvatures, from α = 0 to α = π .

Fig. 3. Case 2: non-dimensional x coordinate of Cr for μ ˜ ∈ (−2π , 0 ).

3.2. Case 2 For a given value of μ ˜ ∈ (−2π , 0 ), xCr /l decreases as the curvature increases. In this interval then, the trend of the nondimensional abscissa is generally similar to the trend showed in the interval (0, 2π ). Even in this case, high values of initial curvature limit the rotation axis shift along the x−axis. As the absolute value of μ ˜ increases, the initial curvature decreases gradually, until the flexure assumes the straight configuration. As |μ| ˜ increases further, the curvature changes its sign and start to increase. However, as μ ˜ approaches the value −2π , the free-end section approaches a full rotation. For μ ˜ = −2π , the displacement becomes a pure translation and the pole of the planar displacement lies at infinity. As represented in Fig. 3, for μ ˜ approaching the asymptote or for initial curvatures corresponding to α ∈ [0, π /2], the xCr /l curves become very close and a dedicated comparison should be performed. For example, Fig. 4 compares the non-dimensional x coordinate of Cr corresponding to α = 0 (straight axis) and α = π /2 (semi-circular arc). The two curves intersect at Pint = (−4.070, 0.496 ). For μ˜ > −4.070, the rotation axis drift is less for semi-circular flexures than for straight flexures. Otherwise, for μ˜ < −4.070, straight flexures guarantees less rotation axis drift. The scatter plot in Fig. 5 shows the values of xCr /l on the μ ˜ − α plane, from the straight axis flexure to the full-angle beam, and for μ ˜ ∈ (−π , −π /2 ).

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Fig. 4. Non-dimensional x coordinate of Cr corresponding to α = 0 and α = π /2 for μ ˜ ∈ (−3π /2, 0 ).

Fig. 5. Non-dimensional x coordinate of Cr values for α ∈ [0, π ] and μ ˜ ∈ (−π , −π /2 ).

3.3. Case 3 For a given value of μ ˜ ∈ (−4π , −2π ), the absolute value of xCr /l generally increases with the increasing of the curvature, as depicted in Fig. 6. Values of μ ˜ in this interval correspond to rotations of the free-end section greater than the full-angle, therefore to flexures undergoing extremely large deflections. According to this theoretical model, minimum curvature should guarantee minimum axis drift. As in the previous case, in proximity of the asymptote μ ˜ = −2π , the xCr /l curves become very close and a dedicated comparison should be performed.

4. Position error To evaluate the effect of the initial curvature on position accuracy, it is possible to consider the curved flexure and its corresponding PRBM depicted in Fig. 7. The PRBM consists of a rigid link connected to the frame by a revolute joint, whose

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M. Verotti / Mechanism and Machine Theory 119 (2018) 106–118

Fig. 6. Case 3: non-dimensional x coordinate of Cr for μ ˜ ∈ (−4π , −2π ).

Fig. 7. Flexure and its corresponding PRBM.

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113

Fig. 8. Position of the center of rotation and position error.

axis passes through the centroid of the flexure, O1 . The free end of the rigid body represents the free-end section of the flexure. The reference frame is R1 , therefore superscripts can be omitted in the notation. Due to the rotation μ ˜ , the free end of the link moves from E to F , following a circular path with center in O1 and radius equal to pE . The position of F can be obtained as

pF  = Rμ˜ pE ,

(22)

where Rμ˜ is the rotation matrix associated with the rotation angle μ ˜. Considering the flexible element, its free-end section moves from E to F. Therefore, the position error can be written as

e = pF − pF 

(23)

or, by making use of (23), as

e = pF − Rμ˜ pE .

(24)

More specifically, the transition of the free-end section from the neutral configuration to the deformed one corresponds to a rotation μ ˜ around Cr . With reference to Fig. 8, the position of the free-end section in the deformed configuration can be written as

pF = pCr + rF ,

(25)

where rF , that is the position vector of F with respect to Cr , is equal to

rF = Rμ˜ rE ,

(26)

and rE is the position vector of E with respect to Cr . Substituting (26) into (25), and substituting the result into (24), yields

e = pCr + Rμ˜ (rE − pE ) ,

(27)

and, considering that rE − pE = −pCr , it follows





e = I − Rμ˜ pCr .

(28)

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Fig. 9. Rigid-body replacement method: PRBM and flexures with different curvatures.

Finally, substituting (19) into (28), the position error can be written as

1 e= 2



cos (α + μ ˜ ) − cos α 1 + 2α + μ ˜ α



cos α −

sin α

α





−

sin α sin (α + μ ˜ ) + sin α − 2α + μ ˜ α

 cot

μ˜ 2





1 − cos μ ˜ . − sin μ ˜

(29)

In the particular case of straight flexure, the position error can be determined substituting (20) into (28), obtaining



2 1 μ˜ e= cot − 2 2 μ˜



1 − cos μ ˜ − sin μ ˜



.

(30)

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Analogously, in case of full-angle flexure, substituting (21) into (28), it follows



1 1 1 e= − 2 2π + μ ˜ 2π



1 − cos μ ˜ − sin μ ˜

 .

(31)

5. Example of application An example of the rigid-body replacement method is depicted in Fig. 9, where a single revolute joint, as basic element of the PRBM, is considered. The rotation of the rigid link is defined by the angle μ ˜ . The replacement can be performed positioning the centroid of the flexure on the center of rotation of the revolute joint, according to property (c). Four flexures are considered, with equal axis length and semi-subtended angles α equal to 0, π /4, π /2, and 3π /4, schematically depicted in Fig. 9. The positions of the center of rotation and the position error of the flexures with respect to the PRBM are evaluated by means of the theoretical approach presented in the previous Sections and by finite element analyses. The static structural analyses are performed with the commercial software Ansys. Each flexure is simulated as a cantilever beam by introducing a fixed support on the anchored section, and by applying an external moment to the free-end section, as illustrated in Fig. 9. The axis length is equal to 83.3 mm, whereas width and height of the cross section are equal to 2.5 mm and 5 mm, respectively. Two-node beam elements and geometric nonlinearities (large deflections) are considered in the analysis setup. For each flexure, the generated mesh is composed of 701 nodes and 350 elements. An isotropic, linear elastic material is considered, with Young’s modulus equal to 3.3 GPa. The external moment M ranges from -404.97 Nmm to 404.97 Nmm, with steps of 22.5 Nmm. This set of loads correspond to a set of values of μ ˜ ranging from −π /2 to π /2 with steps of 0.087 rad. Theoretical and simulations results are reported in Figs. 10, 11, and 12, showing the Cartesian coordinates of Cr and the position error magnitude for the different loads and curvatures considered. More specifically, Fig. 10 shows the x-coordinates of the center of rotation for the different applied loads. For μ ˜ varying from 0 to −π /2, xCr varies from 0 mm to 11.38 mm in case of α = 0, and from 0 mm to 7.20 mm in case of α = 3π /4. The position error magnitude at μ ˜ = −π /2 is equal to 16.10 mm and 10.18 mm for α = 0 and α = 3π /4, respectively. Therefore, the rigid-body replacement performed with a curved flexure with α = 3π /4 determines a reduction of the axis drift of 4.18 mm with respect to the straight axis flexure, and a reduction of the position error magnitude of 5.92 mm. For μ ˜ ∈ [0, π /2], xCr varies from 0 mm to -11.38 mm and from 0 mm to -5.31 mm, in case of α = 0 and α = 3π /4, respectively. The position error magnitude at μ ˜ = π /2 is equal to 16.10 mm for α = 0 and to 7.51 mm for α = 3π /4. Therefore, replacement performed with a curved flexure with α = 3π /4 determines a reduction of the axis drift of -6.07 mm and a reduction of the position error magnitude of 8.59 mm with respect to the straight axis flexure.

Fig. 10. X-coordinate of Cr for μ ˜ ∈ [−π /2, π /2] in case of α = 0, α = π /4, α = π /2, and α = 3π /4, evaluated by means of theoretical approach and FEA.

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Fig. 11. Y-coordinate of Cr for μ ˜ ∈ [−π /2, π /2] in case of α = 0, α = π /4, α = π /2, and α = 3π /4, evaluated by means of FEA, and theoretical values (dashed line).

Fig. 12. Position error magnitude for μ ˜ ∈ [−π /2, π /2] in case of α = 0, α = π /4, α = π /2, and α = 3π /4, evaluated by means of theoretical approach and FEA.

Finite element simulations confirm the results of the theoretical approach, as it can be seen in Figs. 10, 11, and 12. Regarding the y-coordinates of Cr , their values with respect to the theoretical value 0 mm are on the order of magnitude of 10−3 mm, as showed in Fig. 11.

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6. Conclusions In this paper, the effect of the initial curvature of flexible elements on the position accuracy is investigated. Uniform primitive flexures are considered, ranging from the straight axis case to the full-angle case. The obtained analytical expressions permit to evaluate the rotation axis shift with respect to the flexure centroid. Three specific intervals for the applied load are discussed, depending on the effects of these loads on the initial curvature. An example shows how the choice of different initial curvature values in the rigid-body replacement phase affects the performance of the flexure, in terms of rotation axis shift and position accuracy.

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