A generic approximation model for analyzing large nonlinear deflection of beam-based flexure joints

Precision Engineering 34 (2010) 607–618

Contents lists available at ScienceDirect

Precision Engineering journal homepage: www.elsevier.com/locate/precision

A generic approximation model for analyzing large nonlinear deflection of beam-based flexure joints Tat Joo Teo a,b,∗ , I-Ming Chen a , Guilin Yang b , Wei Lin b a b

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore Mechatronics Group, Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore

a r t i c l e

i n f o

Article history: Received 3 July 2009 Received in revised form 8 December 2009 Accepted 5 March 2010 Available online 27 March 2010 Keywords: Semi-analytic model Large deflection analysis Nonlinear modeling Beam-based flexure Compliant mechanism

a b s t r a c t This paper introduces a semi-analytic model that provides a simple and generic solution for approximating the large nonlinear deflection of any beam-based flexure configuration, i.e., a beam-based flexure coupled with a rigid-link. The accuracy and robustness of the proposed model is ensured by considering two crucial factors while adopting the principle of using a torsional spring to represent the stiffness of a beam-based flexure joint. The first factor is the parasitic shifting of the ‘pivot’ joint of a beam-based flexure joint during large deflection, while the second factor is the changing angular stiffness of the torsional spring. Comparisons with past theoretical models show that the semi-analytic model can accurately approximate these two factors. Consequently, this proposed model offers an accurate approximation on the large nonlinear deflections of any beam-based flexure configuration with deviations ranging from 0.1% to 5.6% when compared to the experimental results. © 2010 Elsevier Inc. All rights reserved.

1. Introduction Implementation of compliant joints on positioning mechanisms has been one of the most popular approach to achieve ultrahigh precision motions [1–4]. Through elastic deformation, these compliant joints overcome the limitations of the conventional bearing-based joints such as coulomb friction, mechanical play, backlash, and wear-and-tear [5]. Consequently, the mechanisms articulated by such compliant joints offer smooth, highly repeatable, and nanometric resolution motion, which make them suitable for fine-positioning applications. In general, a compliant joint can be classified into two elementary types, i.e., a notch-hinge [6] and a leaf-spring (or beam-based flexure joint) [7]. A notch-hinge flexure joint has a ‘necked-down’ design that offers high stiffness ratio, i.e., low stiffness along the desired driving direction while high out-of-axis stiffness in the non-driving directions. Yet, such a design possess high stress concentrations and relatively small elastic deflections, which result in a limited motion range. On the other hand, a beam-based flexure joint, which has a thin slender design, offers a larger deflection within the elastic region of a material. Hence, it becomes a promising solution for the next generation of ultra-high precision mechanisms to achieve a large

∗ Corresponding author at: Mechatronics Group, Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore 638075, Singapore. Tel.: +65 67938285. E-mail address: [email protected] (T.J. Teo). 0141-6359/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2010.03.003

workspace of a few millimeters and degrees [8,9]. Unfortunately, a beam-based flexure joint exhibits nonlinear deflection behavior due to the parasitic shifting of the ‘pivot’ point, P, that causes the actual deflection to exceed the desired deflection [10] as shown in Fig. 1. For the past five decades, tremendous amount of efforts were made to model such beam-based flexure joints and to analyze their nonlinear deflections. These modeling approaches can be classified into four main categories; the closed-form solution [10–16], the finite-element analysis [17–20], the solid-mechanics theorem [21,22] and the Pseudo-Rigid-Body (PRB) approximation technique [23]. Among these approaches, PRB approximation technique offers the simplest modeling approach by using a rigid-link attached to a revolute joint to model the deflection of a flexure joint, while adding a torsional spring to model the flexure joint stiffness (Fig. 2a). Compliant mechanisms (Fig. 2c) are often developed through a basic kinematic-chain, which is an assemblage of rigid-links connected by kinematic joints. A PRB approximation technique allows the analysis of such mechanisms to be conducted at rigidbody mechanism level (Fig. 2b). First introduced by Howell and Midha [24], this technique was used to analyze the deflection and stiffness of a small-length flexure pivot, i.e., a short beam-based flexure joint coupled with a long rigid-link. The authors [25] later modeled the large deflection of a beam-based flexure joint subjected to a single vertical end load using the same technique and achieved a high-degree of accuracy with a deviation of less than 0.5% throughout a deflection angle of up to 70◦ when compared to

608

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

Fig. 1. Parasitic shift of ‘pivot’ point, P, of a deflected cantilever beam.

the classical closed-form solutions. Due to its accuracy and simplicity, PRB approximation technique became widely recognized as a quick analytical tool for compliant mechanisms [26–28]. Currently, the PRB model for small-length flexure pivots is a popular choice for modeling a beam-based flexure configuration, i.e., a beam-based flexure joint coupled with a rigid-link. Such a beam-based flexure configuration is often found in a compliant mechanism for displacement amplification purposes. From Fig. 2a, the resultant moment, M1 , and the applied moment, M2 , of a PRB model for small-length flexure pivots must be almost equivalent. By equating these two moments yield [23] l M1 =1+ M2 L

(1)

Eq. (1) suggests that the length of the rigid-link, L, must be much longer than the length of the beam-based flexure joint, l, to fulfil the requirement, i.e., M1  M2 . Such a condition limits this PRB model to only one form of beam-based flexure configuration, i.e., L  l, as shown in Fig. 2c. Unfortunately, most compliant mechanisms also adopt other forms of beam-based flexure configurations, e.g., L = 0, L ≤ l, or L > l, etc., to achieve the desired motion stiffness as shown in Fig. 3. In addition, various forms of beam-based flexure configurations will also cause the vertical end load to pose different loading conditions on the beam-based flexure joints. Thus, the PRB model for small-length flexure pivots is unsuitable for modeling all forms of beam-based flexure configurations. Although other forms of PRB models such as the fixed-guided beam (for L  l with translation motion), the cantilever beam with

Fig. 3. (a) Prismatic and (b) universal compliant joint-modules formed by various beam-based flexure configurations.

an end moment load (for L = 0 with pure moment loading) [23], etc., can be used to model other beam-based flexure configurations, selecting and applying these PRB models require good knowledge of their modeling approach. This is because each PRB models has specific locations to place the revolute joints, the specific torsional spring constant value, and other specific requirements to the rigidlink attached to each revolute joint. Hence, any misjudgment and inappropriate selection of PRB models often lead to inaccurate analyses, which become more significant during large deflection analyses. In addition, pairing a suitable PRB model with a beam-based flexure configuration during the design stage is extremely restrictive and troublesome since each PRB model is targeted to a specific form of beam-based flexure configuration or loading condition. In this stage, designing a compliant mechanism often goes through an iterative process of changing the dimensions of the rigid-links or the beam-based flexure joints to achieve desired motion stiffness and out-of-axis stiffness within a given size constrain. Thus, selecting a suitable PRB model and re-modeling the beam-based flexure configuration based on the selected PRB model requirements also become an iterative process to conduct proper and accurate deflection and stiffness analyses. Although a PRB approximation technique has successfully bridged the rigid-body modeling and compliant joint modeling together, it still cannot provide a simple and generic solution for all forms of beam-based flexure configurations. In this paper, a semi-analytic model that offers a generic, simple and quick solution for analyzing large nonlinear deflection, and deflection stiffness of any form of beam-based flexure configuration is presented. The detailed derivations of this semi-analytic model are presented in Section 2. Theoretical comparisons between this proposed model and traditional nonlinear models found in past literature are discussed in Section 3. Lastly, accuracy of this proposed model is verified through both numerical simulations and experimental investigations in Sections 3 and 4 respectively. 2. A semi-analytic model

Fig. 2. (a) Schematic diagram of a PRB model for a small-length flexure pivot. The conversion of (b) a rigid-body mechanism to (c) a compliant mechanism and its stiffness modeling approach using PRB modeling.

A semi-analytic model is formulated using two crucial factors while adopting the principle of using a torsional spring to represent the stiffness of a compliant joint to model a beam-based flexure configuration. The first factor enables the proposed model to approximate the parasitic shift of the ‘pivot’ point and the deflection along the x- and y-axes of a beam-based flexure configuration. The second factor identifies the moment arm of the torsional spring with respect to any form of beam-based flexure configuration so as to approximate the stiffness of the compliant joint. In this section, the derivations of both factors are presented.

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

609

2.1. Bernoulli–Euler’s law Based on the classical Bernoulli–Euler’s law, its nonlinear ordinary differential form is expressed as M = EI

d2 y/dx2

(2)

2 3/2

[1 + (dy/dx) ]

Integrate Eq. (2) yields Mx = EI



dy/dx

1 + (dy/dx)

2

+C

(3)

The constant, C, can be determined from the boundary condition, whereby at the fixed end, i.e., x = 0, the slope will be zero, i.e., dy/dx = 0. Hence, this leads to C = 0. Given that dy/dx = tan , Eq. (3) is written as Mx = EI



tan 

1 + (tan )

2

(4)

For a deflection angle, , between 0◦ and 90◦ , tan  = sin / cos . Hence, Eq. (4) can be further simplified as Mx = sin  EI

(5)

Due to the limitation of the material yield strength, a compliant joint usually operates within a deflection angle from 0◦ to 90◦ . Hence, Eq. (5) is sufficient to describe a compliant joint that deflects within this range. 2.1.1. For small deflections When the small deflection theorem is applied to Eq. (5), i.e.,  → 0, sin() = , an expression to determine the deflection angle due to a moment at the free end is obtained as Mx = EI

(6)

2.1.2. For large deflections Using Taylor’s series expansion, Eq. (5) can be re-expressed as Mx 3 5 =− + − ··· EI 3! 5! Thus, the large deflection caused by a moment at free end is expressed as Mx =  EI

(7)

4 2 + 3! 5!

The derived factor, , will be used up to the second term of the Taylor’s series expansion.  is also considered as a Sinc function, which can be expressed as =

sin  

In this work, it is assumed that the deflection along the y-axis of the semi-analytic model to be expressed as ıy = l sin 

(8)

2.2. Parasitic shift of ‘pivot’ point For the semi-analytic model to be effective in analyzing large nonlinear deflection of a beam-based flexure joint, the parasitic shifting of the ‘pivot’ point must be identified. In this work, a beambased flexure configuration can be treated as a cantilever beam subjected to a moment end load as shown in Fig. 4. except for L = 0. The force, F, applied to the free end of the rigid-link is considered as the moment, i.e., M ≡ F × L, applied at the free end of the beam as shown in Fig. 4b. Hence, the proposed model will be derived based on this loading condition.

(9)

where l represents the distance from the tip of the beam-based flexure joint to the ‘pivot’ point, P (Fig. 4a). To determine , it is first assumed that ˛ is extremely small, i.e., ˛ → 0, so that the magnitude of the parasitic shift of ‘pivot’ point, PP can be projected on l. In addition, this assumption also allow the deflection angle of the semi-analytic model, , to be equivalent to the deflection angle, f , derived from the classical nonlinear solution (see Appendix A). Consequently, by equating Eq. (9) with (41) (see Appendix A) yields EI (1 − cos f ) = l sin  ML

(10)

Substituting Eq. (7) into (10) and with the maximum deflection angle occurring at x = l gives 1 

where  =1−

Fig. 4. (a) A general case of a beam-based flexure configuration subjected to a vertical end load treated as (b) a cantilever beam with a moment end loading.



1 − cos  



=  sin 

(11)

In Eq. (11), (1 − cos )/ can be approximated using sin /2 (see Appendix C) and expressed as =

1 2

(12)

From Eq. (12),  represents a dynamic ratio that changes according to the deflection angle due to the presence of , which is used to approximate the parasitic shifting of the ‘pivot’ point (refer to Section 3.1 for more details). 2.3. Large deflection analysis of a beam-based flexure with moment end load Based on the small deflection theory, the ‘pivot’ point of a cantilever subjected to a pure moment end load is located at the middle of the undeflected beam. Hence, the principle of semi-analytic model analysis is based on the hypothesis that the ‘pivot’ point will always be at the middle of the undeflected beam-based flexure joint while l accounts for the distance between the tip of the

610

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

flexure joint to the middle due to the parasitic shift of ‘pivot’ point as shown in Fig. 4b. Thus, by substituting Eq. (12) into (9) yields ıy =

l sin  2

(13)

Consequently, applying the Sinc function from Eq. (8) on (13) will give the semi-analytic model for deflection along the y-axis ıy =

l  2

(14)

From Fig. 4b, a = l cos . Hence, the deflection along the x-axis can be expressed as l ıx = 2



cos  1− 



(15)

2.4. Large deflection analysis of a beam-based flexure joint coupled with a rigid-link Large deflection analysis of a beam-based flexure configuration is essential because swift deflection and stiffness analyses of the compliant mechanisms can be conducted by breaking down each limb into individual beam-based flexure configurations. As mentioned in previous sections, l represents the distance from the tip of the beam-based flexure joint to the ‘pivot’ point (Fig. 4b). As the flexure joint is coupled with a rigid-link, the deflection of a beam-based flexure configuration along the y-axis is expressed as y =



L+

 l

2

sin 

(16)

x =

L+

  l

2

− L+

 l

2

cos 

(17)

Based on Eq. (16), the deflection along the y-axis can be determined directly based on a desired deflection angle. On the other hand, a Newton–Raphson method is required to approximate the deflection angle based on a desired deflection along the y-axis. 2.5. Angular stiffness of a torsional spring Representing a torsional spring as a compliant joint significantly reduces the complexity in analyzing the deflection stiffness of the beam-based flexure configuration. Hence, the angular stiffness of the torsional spring becomes essential for approximating the amount of torque required to achieve the desired deflection angle. In semi-analytic modeling, this angular stiffness is derived based on the hypothesis that the relationship between the applied torque and the deflection angle of the torsional spring is governed by a moment arm. This moment arm is formed by the rigid-link and a portion of the compliant joint as shown in Fig. 5a. This portion of the compliant joint represents the distance from the center of the torsional spring to the coupling point between the rigid-link and the compliant joint. In this work, this portion of the compliant joint is termed as the changing-arm, S, and can be expressed as S = l where  is introduced as √ l 1.8 + L = l+L

flexure configuration. Subsequently, the changing torque, T , is recognized as a tangential force, Ft , applied to the moment arm and is expressed as T = FT

(18)

(19)

From Fig. 5b, the changing-arm varies according to different beambased flexure configurations even with similar compliant joint lengths. To address this issue,  is an empirical factor that is used to determine the changing-arm with respect to any beam-based



L+

l 2

 (20)

The changing angular stiffness of a torsional spring is given as K =

Similarly, the deflection along the x-axis is given as



Fig. 5. (a) Torsional spring with a moment arm that changes with respect to (b) different beam-based flexure configurations.

T 

(21)

By substituting Eq. (20) and (6) with x → l (when maximum deflection occurs) into Eq. (21), it is re-expressed as F=

EI l(L + (l/2)) sin((/2) − )

(22)

In semi-analytical modeling, the changing angular stiffness of a torsional spring is expressed in Eq. (22). With FT = F sin((/2) − ), the vertical force, F, applied on the moment arm can be determined directly based on a known deflection angle as shown in Fig. 5a. 3. Theoretical comparisons 3.1. Approximating the parasitic shift of ‘pivot’ point Based on Eq. (9), the shifting factor, , is used to determine the distance from the tip of the beam-based flexure joint to the ‘pivot’ point (Fig. 4b). l =

l 2

(23)

Hence, from Eq. (23), the initial distance when  = 0 is given l =

l 2

(24)

Consequently, based on Eqs. (12) and (24), the dimensionless magnitude of the parasitic shift of the ‘pivot’ point is expressed as PP semi-analytic l

=

1 1 − 2 2

(25)

To investigate whether  accounts for the parasitic shift, Eq. (25) was compared against Haringx’s model, which was used to predict the parasitic shift of the ‘pivot’ point for a cantilever beam subjected to a pure moment end loading [29]. The Haringx’s model

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

611

Fig. 6. Approximated parasitic shift of the ‘pivot’ point obtained from the semianalytic and Haringx’s models.

is given as PP Haringx l

=

2 sin(/2) − cos(/2) 

(26)

In past literature, an extended model derived from Eq. (26) was used to predict the parasitic shift of ‘pivot’ point for a cross strip pivot [22,29] and compared against the experimental results [30,31]. These experimental results were obtained from the actual cross strip pivot, which was loaded with a pure moment and measured up to a deflection angle of /6 rad (or 30◦ ) [31]. The analytical results obtained from the extended model are found to be consistent with those experimental values. Hence, it is assumed that Eq. (26) will be effective in predicting the parasitic shift of the ‘pivot’ point for a cantilever beam subjected to a pure moment end loading. As Eq. (23) is derived based on the condition of a moment end loading, Eq. (26) will be sufficient to be used as a benchmark for investigating the effectiveness of . Subsequently, the theoretical comparison between Eq. (25) and (26) is also conducted up to 30◦ (similar to the experimental range [31]) and plotted in Fig. 6. It shows that  does provide a good approximation on the parasitic shift of the ‘pivot’ point when compared against the Haringx’s model. 3.2. Semi-analytic model vs. nonlinear model The semi-analytic models used in predicting the nonlinear deflection of a cantilever beam subjected to a moment end loading are evaluated in this section. Based on Eq. (14) and (15), the non-dimensional deflection ratio along the y- and x-axes are given as ıy,semi-analytic l ıx,semi-analytic l

=

 2

1 − (cos /) = 2

Fig. 7. Dimensionless deflection ratio along the y- and x-axes obtained from the nonlinear model, semi-analytic model, and the FEA simulation.

Assuming that the FEA results are accurate, Fig. 7 shows that the classical nonlinear model is extremely accurate in predicting the deflection along the y- and x-axes. Most importantly, the deflections obtained from both the semi-analytic models and the nonlinear models are consistent within a deflection ratio ranging from 0 to 0.5 or a deflection angle ranging from 0◦ to 80◦ . Results also suggest that the hypothesis of the parasitic shifting of the ‘pivot’ point being in the same direction as the deflection slope is valid up to 80◦ (Fig. 4b). Hence, this evaluation shows that Eqs. (27) and (28) can provide a simple and accurate approximation on the nonlinear deflection of a cantilever beam subjected to a moment end load within 0–80◦ . 3.3. Moment arm of the torsional spring This section evaluates the effectiveness of  in predicting the changing-arm, which plays a crucial role in obtaining the moment arm of the torsional spring with respect to different beam-based flexure configurations. Based on Eq. (20), the moment arm is expressed as S=L+

l 2

(29)

By excluding L and , Eq. (29) is re-expressed as l  2

(27)

S=

(28)

With the exclusion of , Eq. (30) will express a changing-arm where the parasitic shifting of the ‘pivot’ point is not considered. Hence, it becomes similar to the effective deflection-length, which is the distance from the free end of the compliant joint to the effective ‘pivot’ point. In the small deflection theory, this effective deflection-length is used to determine the deflection of a cantilever beam [22]. However, the changing-arm of the semi-analytic model is only used to identify the changing angular stiffness of the torsional spring with respect to any beam-based flexure configuration. Unlike past literature, the changing-arm does not play any role in approximating the deflections. Nevertheless, the effective deflection-length can be used as benchmark to evaluate

On the other hand, the non-dimensional deflection ratio along the y- and x-axes based on the classical nonlinear modeling are expressed in Eq. (45) and (46) respectively. For comparison, the analytical results obtained from Eqs. (27), (28), (45) and (46) are plotted in Fig. 7. In addition, the results obtained from the finiteelement analysis (FEA) are also plotted against these analytical results. These FEA results are obtained through a simulated cantilever beam subjected with a moment end load and the FEA parameters including an element type of BEAM3, a Young’s modulus of 71GPa, and a nonlinear large displacement solution.

(30)

612

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

Fig. 8. Comparison between the changing-arm length and the effective deflectionlength obtained from the semi-analytic and Thorpe’s models.

the effectiveness of  in identifying the changing-arm based on Eq. (30). 3.3.1. Case 1 For L = 0, the vertical force applied to the rigid-link free end is transferred to the free end of the compliant joint. Based on Eq. (30), the changing-arm when L = 0 is given as S = 0.67l

(31)

Eq. (31) is consistent with past literature [22], which suggested that a cantilever beam subjected to a vertical force at free end has an effective deflection-length equivalent to 2/3 of the beam. 3.3.2. Case 2 For L  l, L becomes dominating over l, thus  → 1. Based on Eq. (30) the changing-arm when L  l is given as l S= 2

(32)

Eq. (32) is consistent with the small deflection theory and the PRB model for small-length flexure pivots [23] that assume the effective deflection-length is half of the compliant joint. 3.3.3. Case 3 For other mixtures of compliant joint and rigid-link lengths, Eq. (30) is compared with Thorpe’s model [32], which proposed that the effective deflection-length of any beam-based flexure configuration (loaded with a single vertical end load at the rigid-link free end) is given by SThorpe =

lL + (2/3)l2 l + 2L

(33)

By fixing the length of the compliant joint at 3 mm, a set of changing-arm lengths is obtained from Eq. (30) and a set of effective deflection-lengths is obtained from Eq. (33) based on a rigid-link length of 1 mm, 3 mm, 5 mm, 10 mm, and 20 mm respectively. Both sets of data are plotted in Fig. 8 and show that the changingarm lengths are relatively close to the effective deflection-lengths with an average deviation of 2.4% between both data sets. Both data sets also display similar trends, shifting towards 2/3 of the undeflected compliant joint when the rigid-link becomes shorter than the compliant joint. Also, the obtained changing-arm length

Fig. 9. Different beam-based flexure configurations used for comparison between the analytical models and FEA simulations.

and the effective deflection-length shift towards half of the undeflected compliant joint when the rigid-link becomes longer than the compliant joint. Consequently, this theoretical comparison can conclude that  is effective in identifying the changing-arm with respect to different beam-based flexure configurations. 3.4. Semi-analytic and PRB models vs. FEA results With  and  proven to be effective in identifying the moment arm of the torsional spring and the parasitic shift of ‘pivot’ point respectively, the semi-analytic model could provide a quick analysis on any form of beam-based flexure configuration. In this section, the results obtained from the FEA simulation are used to validate this claim and the simulated beam-based flexure configurations are illustrated in Fig. 9. The parameters used in the FEA simulation are listed in Table 1. The analyses were carried out in the following manner; based on a set of desired deflections along the y-axis, the force required for achieving these desired deflections are obtained from the semi-analytic model, i.e., Eqs. (16) and (22), and the PRB model, i.e., Eqs. (48) and (49) (see Appendix B). Subsequently, each set of forces was input to the FEA simulator, i.e., ANSYS10. The deflection along the y-axis of each beam-based flexure configuration was thus computed. Fig. 10 shows a diagram of the steps used to obtain the deflection along the y-axis from both the semi-analytic and PRB models where the results are listed in Table 1 Parameters used in the FEA analysis. Young’s modulus Poisson ratio Beam thickness Rigid-link thickness Width Element type Solution

71 GPa (aluminum) 0.33 0.2 mm 20 mm 10 mm BEAM3 (2D elastic) Nonlinear large displacement

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

613

Table 2 FEA-based deflection along the y-axis of different beam-based flexure configurations obtained from the stiffness models of the semi-analytic and PRB approximations. Desired, ıy (mm)

Ratio, ıy /l

Rigid-link length, L = 1 mm and flexure joint length, l = 3 mm 0.25 0.0625 0.5 0.1250 1.0 0.2500 2.0 0.5000

PRB model

Semi-analytic model

Output, ıy (mm)

Error (%)

Output, ıy (mm)

Error (%)

0.28066 0.56385 1.1509 2.5013

12.2640 12.7700 15.0900 25.0650

0.24301 0.48704 0.98384 2.0389

2.7960 2.5920 1.6160 1.9450

Rigid-link length, L = 3 mm and flexure joint length, l = 3 mm 1.0 0.1667 1.0425 2.0 0.3333 2.1172 3.0 0.5000 3.2560

4.2500 5.8600 8.5333

0.9835 1.9802 2.9995

1.6500 0.9900 0.0167

Rigid-link length, L = 5 mm and flexure joint length, l = 3 mm 1.0 0.1250 2.0 0.2500 3.0 0.3750 4.0 0.5000

1.0196 2.0499 3.1012 4.1829

1.9600 2.4950 3.3733 4.5725

0.9892 1.9823 2.9823 3.9911

1.0800 0.8850 0.5900 0.2225

Rigid-link length, L = 10 mm and flexure joint length, l = 3 mm 1.0 0.0769 1.0067 2.0 0.1538 2.0140 3.0 0.2308 3.0259 4.0 0.3077 4.0434 5.0 0.3846 5.0681 6.0 0.4615 6.1017 7.0 0.5385 7.1455

0.6700 0.7000 0.8633 1.0850 1.3620 1.6950 2.0786

0.9962 1.9916 2.9886 3.9868 4.9866 5.9879 6.9905

0.3790 0.4200 0.3800 0.3300 0.2680 0.2017 0.1357

Rigid-link length, L = 20 mm and flexure joint length, l = 3 mm 1.0 0.0435 1.0019 2.0 0.0870 2.0054 3.0 0.1304 3.0064 4.0 0.1739 4.0100 5.0 0.2174 5.0147 6.0 0.2609 6.0208 7.0 0.3043 7.0286

0.1900 0.2700 0.2133 0.2500 0.2940 0.3467 0.4086

0.9988 1.9989 2.9960 3.9948 4.9939 5.9930 6.9924

0.1200 0.0550 0.1333 0.1300 0.1220 0.1167 0.1086

Table 2. In this evaluation, the PRB model for small-length flexure pivots is employed because it is the only PRB model which analyzes beam-based flexure configurations. From Table 2, when the rigid-link is much shorter than the beam-based flexure joint, i.e., L  l, the deflections obtained from the FEA (or FEA-based deflections) deviate less than 3% when compared with the desired deflections input into the semi-analytic model, while the FEA-based deflections have a maximum deviation of 25% when compared with the desired deflections input into the PRB model. In addition, when L ≤ l, the FEA-based deflections deviate less than 2% when compared with the desired deflections input into the semi-analytic model, while the FEA-based deflections have a deviation up to 8% when compared with the desired deflections input into the PRB model. As L becomes much longer than l, the deviations between the FEA-based deflections and the desired deflections input into the PRB model slowly reduce to less than 0.5%. Thus, it shows that the PRB model is indeed very accurate in such a beam-based flexure configuration. The deviations between the FEA-based deflections and the desired deflections input into the semi-analytic model not only just reduced to 0.1% but

also show constant differences throughout the incremental deflections. Results from all the five FEA-simulated beam-based flexure configurations have suggested that the stiffness predicted by the semi-analytic model are smaller than those predicted by the PRB model. Consequently, the FEA-based deflections obtained through the forces derived from the semi-analytic model have less deviation when compared to the desired deflections. In addition, the deviations between the FEA-based deflections and the desired deflections input into the PRB model increase when the deflection ratio increases. Such a trend is observed throughout all five simulated flexure configurations. This shows that the accuracy of the PRB model decreases when the deflection increases. On the other hand, the deviations between the FEA-based deflections and the desired deflections input into the semi-analytic model reduce when the deflection ratio increases throughout all simulations. Such observations suggest that the accuracy of the semi-analytic model increases along with the increase in deflection. One reason is because an increase in force at the free end of the rigid-link (to achieve larger deflection) will increase the domination of moment end loading on the flexure joint. As the semi-analytic model is derived based on a beam-based flexure joint subjected to a moment end load condition, such domination of moment suit its assumed loading condition. Assuming that the FEA simulations produce accurate nonlinear deflection analyses, these results have validated that the semi-analytic model offers a simple and robust approach with high-degree of accuracy in approximating the deflection of any form of beam-based flexure configuration. 4. Experimental investigations

Fig. 10. Steps for obtaining a set of FEA-based deflections from desired deflections along the y-axis input into the semi-analytical and PRB models.

An experimental investigation was conducted to evaluate the accuracy of the semi-analytic model in analyzing the large nonlinear deflection of the beam-based flexure configurations. For each

614

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

Fig. 13. (a) Two spherical balls attached to the rigid-link for the measuring device to capture (b) the image of the actual coordinates of deflection where triangular coordinates determined through the center of the spherical balls from each deflection were used to (c) calculate the actual deflections along the y- and x-axes. Fig. 11. Specimens of the beam-based flexure joint in various form of flexure configurations.

beam-based flexure configuration, weights are added at the rigidlink free end to obtain the deflections along the y- and x-axes that are captured by a high-resolution measurement device simultaneously. 4.1. Experimental setup Four beam-based flexure configuration specimens are prepared for the experiment. The dimensions of each specimen are calculated to ensure that a large non-dimensional deflection ratio of 0.5 along the y-axis (ıy /l) is achievable within the elastic limit of the material used to fabricate each specimen. All four specimens and their respective dimensions are shown in Fig. 11. Specimen 1 is designed with a long beam-based flexure joint coupled to a shorter rigid-link. Specimens 2–4 have the same length for their respective beam-based flexure joint but have different lengths for their respective rigid-links. For all the specimens, the thickness of the beam-based flexure joints is 0.2 mm. Spring steel (SUS301) was used to fabricate Specimen 1 and aluminum (7075-T6) was used to fabricate the remaining specimens. All specimens are fabricated in larger dimensions so that noticeable and detectable deflections can be captured by the measurement device effortlessly. An experimental platform is constructed for this investigation with various components labelled from (1) to (6) as shown in Fig. 12. Here, each specimen is fixed on a mounting (1), which is adjustable along the vertical direction (or z-axis). Specimens are mounted

in this arrangement to avoid the effect of gravitational pull from causing any undesired deflection. A nylon string (2), which can withstand 5 kg load, connects the end of the rigid-link to a container (3). This container is used to put the pre-calculated weights, which act as the load for deflecting the specimens. The nylon string is rested on a bearing (4), which rotates and slides along a rotating shaft (5), to ensure that it is always kept parallel to the y-axis. Creeping effects caused by friction is addressed by lubricating the contact surface between the bearing and the nylon string with oil. This will ensure repeatability of the deflections. Subsequently, the rotating shaft is connected to an adjustable mounting (6). This mounting can orientate about the x- and y-axes to ensure the rotating shaft is kept in parallel to the specimens. In this experiment, a 3-dimensional (3D) digitized measuring device from ATOS is used to capture the deflections of those specimens as shown in Fig. 12. This device has a resolution of 3 ␮m, an accuracy of 30 ␮m and a measurement volume of 150 mm × 250 mm × 200 mm. As all specimens are fabricated in larger dimensions, the differences between the experimental data and the results obtained from both models are in few hundreds of micrometers. Hence, an accuracy of 30 ␮m is sufficient for this evaluation. To capture the precise coordinates of the deflection, two spherical stainless steel balls were attached onto the rigid-link of each specimen as shown in Fig. 13a. One of the spherical balls is located in line with the nylon string while the other acts as a reference. From the initial undeflected position, five pre-calculated weights will be loaded into the container and cause the specimen to deflect. Each deflection caused by each loading is captured by the measuring device as an image. Subsequently, the images are combined together to form a complete image of the deflections obtained from each specimen as shown in Fig. 13b. Consequently, the center coordinates of each spherical ball is processed and given by the measurement device. For each deflection, the distance from the center of the spherical balls with respect to the undeflected position were measured (Fig. 13b). These distances formed a triangle (Fig. 13c) where an angle, ˇ, can be determined from

 ˇ = cos−1

A2 + C 2 − B2 2AC

 (34)

Subsequently, this angle is used to determine the deflection in the y-axis Fig. 12. Experimental platform and setup.

y = C sin ˇ

(35)

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

615

Table 3 Deviations in deflection along the x- and y-axes obtained from Specimen 1 through experiment, semi-analytic model and conventional PRB model. Specimen 1: L = 75 mm, l = 150 mm Experimental vs. semi-analytic model (%)

Fig. 14. Experimental and semi-analytic results for Specimen 1.

Experimental vs. PRB model (%)

Under 7.65 g loading 1.8969 y 1.6992 x

11.9109 4.8446

Under 16.55 g loading 3.9660 y 0.1808 x

6.5919 3.0853

Under 25.52 g loading 4.3444 y 4.1683 x

6.1787 1.6115

Under 35.49 g loading 2.9878 y 6.5670 x

12.7056 8.2817

Under 47.45 g loading 2.0379 y 2.8347 x

8.1526 3.8520

From Eq. (35), the deflection in the x-axis expressed as



x =

B2 − 2y

(36)

4.2. Results In this experiment, the deflection along the y- and x-axes are calculated using Eqs. (35) and (36), whereby the values of A, B and C are obtained through the center coordinates of the spherical balls via the captured images (Fig. 13b). For each specimen, every precalculated weight was loaded for five separate occasions to obtain five sets of deflection values along the y- and x-axes. Hence, averaging these five sets of values gave the deflections along the yand x-axes caused by each weight. For the semi-analytic modeling results, all pre-calculated weights are converted to input forces, which are used to obtain the deflection angles based on Eq. (22). The obtained deflection angles are subsequently used to determine the deflections along the y- and x-axes based on Eqs. (16) and (17) respectively. Here, the experimental deflections and the deflections obtained from the semi-analytic model for Specimens 1–4 are plotted in Figs. 14–17 respectively. In addition, the deflections along the y- and x-axes predicted by the semi-analytic model for each specimen are compared against the approximated deflections obtained through the PRB model using Eqs. (47) and (48). The variation between the experimental results, the semi-analytic model, and the PRB model for all four specimens are listed in Tables 3–6. In this work, Specimens 1 and 2 represent cases where the beambased flexure length is equal or longer than the rigid-link length. Fig. 14 shows that the semi-analytic model results in a good prediction of the deflections along the x- and y-axes of Specimen 1, where the beam-based flexure length is twice as large as that of the rigidlink. Table 3 also indicates that the semi-analytic model results in a higher accuracy in predicting the deflections along the y- and xaxes when compared to the PRB model. The deviation between the results obtained via experiments and those obtained by using the semi-analytic model for the deflection along the y-axis range from 0.4% to 4%, while the deviation between the results obtained experimentally and by using the PRB model is found to be ranging from 6% to 13%. In addition, the deviation between the results obtained via experiments and those obtained by the semi-analytic model for deflection along the x-axis range from 0.2% to 6.6%, while the deviation between the results obtained experimentally and by using the PRB model is found to be ranging from 1.6% to 8.3%. Fig. 15 shows that the semi-analytic model has a good prediction of the deflection along the x- and y-axes of Specimen 2, where both

beam-based flexure and the rigid-link have equal length. Table 4 indicates the deviation between the results obtained via experiments and those obtained by using the semi-analytic model for the deflection along the y-axis range from 1.4% to 5.6%, while the deviation between the results obtained experimentally and by using the PRB model is found to be ranging from 6.8% to 10.7%. For deflection along the x-axis, the deviation between the results obtained via experiments and those obtained by the semi-analytic model range from 2.3% to 6.6%, while the deviation between the results obtained experimentally and by using the PRB model is found to be ranging from 3.1% to 8.6%. For other general cases where the beam-based flexure length is shorter than the rigid-link length, Figs. 16 and 17 show that the semi-analytic model has made accurate predictions on the deflection when compared against the actual deflection obtained from Specimens 3 and 4. For Specimen 3, Table 5 also shows that the semi-analytic model results in a higher accuracy in predicting the deflections along the y- and x-axes when compared to the PRB model. The deviation between the results obtained via experiments and those obtained by using the semi-analytic model for the deflection along the y-axis range from 0.4% to 4.3%, while the deviation between the results obtained experimentally and by using the PRB model is found to be ranging from 1.2% to 6.4%. For deflection

Fig. 15. Experimental and semi-analytic results for Specimen 2.

616

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

Table 4 Deviations in deflection along the x- and y-axes obtained from Specimen 2 through experiment, semi-analytic model and conventional PRB model. Specimen 2: L = 20 mm, l = 20 mm Experimental vs. semi-analytic model (%)

Table 5 Deviations in deflection along the x- and y-axes obtained from Specimen 3 through experiment, semi-analytic model and conventional PRB model. Specimen 3: L = 40 mm, l = 20 mm

Experimental vs. PRB model (%)

Experimental vs. semi-analytic model (%)

Experimental vs. PRB model (%)

Under 122.42 g loading 5.5918 y 2.3161 x

10.6759 3.0913

Under 76.15 g loading 4.2926 y 0.5655 x

6.4320 3.0211

Under 248.32 g loading 2.8371 y 5.8210 x

8.0670 6.7846

Under 155.15 g loading 1.0810 y 6.8485 x

1.2066 9.5121

Under 385.14 g loading 2.2337 y 6.4953 x

7.5042 7.7278

Under 240.15 g loading 0.4284 y 4.4898 x

1.8987 7.1755

Under 535.34 g loading 4.4822 y 4.8409 x

9.6680 6.3859

Under 335.15 g loading 0.8295 y 1.6270 x

3.2193 4.3331

Under 709.02 g loading 1.3780 y 6.6358 x

6.8271 8.5940

Under 445.95 g loading 0.9020 y 5.4985 x

3.4373 2.8955

Fig. 16. Experimental and semi-analytic results for Specimen 3.

along the x-axis, the deviation between the results obtained via experiments and those obtained by the semi-analytic model range from 0.6% to 6.8%, while the deviation between the results obtained experimentally and by using the PRB model is found to be ranging from 2.9% to 9.5%. Here, it is observed that the PRB model accuracy for predicting the deflection along y-axis is improving as compared to the previous two cases (Specimens 1 and 2). For Specimen 4, Table 6 indicates that the semi-analytic model results in a higher accuracy in predicting the deflections along the y-axis when compared to the PRB model. The deviation between the results obtained via experiments and those obtained by using the semi-analytic model for the deflection along the y-axis range from 0.1% to 4.2%, while the deviation between the results obtained experimentally and by using the PRB model is found to be ranging from 0.5% to 3.2%. Here, it is also observed that the accuracy of the PRB model is improving for predicting the deflection along yaxis. The deviation between the results obtained via experiments and those obtained by the semi-analytic model for deflection along the x-axis range from 2.0% to 20.7%, while the deviation between the results obtained experimentally and by using the PRB model is found to be ranging only from 0.8% to 19.1%. These observations Table 6 Deviations in deflection along the x- and y-axes obtained from Specimen 4 through experiment, semi-analytic model and conventional PRB model. Specimen 4: L = 100 mm, l = 20 mm Experimental vs. semi-analytic model (%)

Fig. 17. Experimental and semi-analytic results for Specimen 4.

Experimental vs. PRB model (%)

Under 35.35 g loading 2.6969 y 20.6994 x

3.2072 19.0615

Under 72.15 g loading 4.1918 y 12.9521 x

4.7221 11.1554

Under 111.95 g loading 0.0593 y 1.9713 x

0.5467 0.0470

Under 156.05 g loading 1.5263 y 6.7368 x

2.2002 4.8286

Under 208.95 g loading 0.7859 y 2.7457 x

1.5797 0.7820

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

suggest that the predictions of the PRB model on the deflection along the x-axis are closer to the actual deflection as compared to the semi-analytic model. Hence, these observations show that the PRB model for small-length flexure pivot is accurate for predicting the deflections along x- and y-axes when the rigid-link length is much longer the length of the beam-based flexure joint. Nevertheless, all experimental data have shown that the approximated deflections made by the semi-analytic model are much closer and consistent with the actual measured deflections for all four specimens. Consequently, this experiment has verified that the semi-analytic model has both the accuracy and robustness in predicting large nonlinear deflections, and deflection stiffness of any form of beam-based flexure configuration.

Using the chain rule on Eq. (37) with d y, dx and ds being infinitesimal yields d dy d M · = = · sin  EI dy ds dy

M d dx d = · = · cos  EI dx ds dx

(40)

Integrating both sides of Eq. (39) yields



ıy

M dy = EI

ıy,nonlinear

A generic solution for predicting large nonlinear deflections of a beam-based flexure joint is introduced. Termed a semi-analytic model, its accuracy is not limited by any form of beam-based flexure configuration. A shifting factor, , is introduced and adopted by the semi-analytic model to determine the parasitic shifting of ‘pivot’ point for a beam-based flexure joint during large deflection. Comparison with past theoretical models and experimental results show that  allows the semi-analytic model to approximate the parasitic shift of the ‘pivot’ point accurately. This allows the semi-analytic model to provide an accurate approximation on the nonlinear deflection of a beam-based flexure joint subjected to a moment end load. A comparison with the classical nonlinear model has proven the semi-analytic model is accurate in approximating such nonlinear deflections. This semi-analytic model is subsequently extended to predict the deflection of any beam-based flexure configuration. For analyzing the stiffness of any beam-based flexure configuration, another factor, , is adopted by the semianalytic model to determine the moment arm of the torsional spring, which is used to represent the stiffness of the compliant joint. Comparison with past theoretical models and observations have shown that  can effectively predict the moment arm via the changing-arm with respect to any form of beam-based flexure configuration. Lastly, an experiment has been conducted to evaluate the accuracy of this generic solution. Experimental results show that the semi-analytic model has higher accuracy in approximating the large deflection of any beam-based flexure configuration as compared to the PRB model. Based on all theoretical comparisons and experimental results, the semi-analytic model has proven to be a generic, simple and accurate solution for quick parametric studies and analyses of any compliant mechanism.

(39)

and

0

5. Conclusions

617

l

=



f

sin  d 0

EI (1 − cos f ) Ml

(41)

Deflection along y-axis is derived by substituting Eq. (38) into Eq. (41) and expressed as ıy,nonlinear l

=

1 − cos f

(42)

f

Integrating both sides of Eq. (40) yields



ıx

0

M dx = EI



f

cos  d 0

M ı = sin f EI x,nonlinear

(43)

Deflection along x-axis is derived by substituting Eq. (38) into Eq. (43) and expressed as sin f l − ıx,nonlinear = l f

(44)

From Eqs. (42) and (44), the non-dimensional deflection ratio in the y- and x-axes are given as ıy,nonlinear l

 =

ıx,nonlinear = l

1 − cos f

 (45)

f



f − sin f



f

(46)

Appendix B. PRB model for small-length flexure pivots Appendix A. Classical nonlinear solution for a cantilever beam subjected to moment end load The Bernoulli–Euler’s law states that the bending moment at any point of the bar is proportional to the change in the curvature, which gives M d = EI ds



b=



l

ds = 0

Ml f = EI

a=

(37)

Hence, the deflection angle at the deflection beam end, f , can be determined by integrating both sides of Eq. (37) and yields M EI

In 1994, Howell and Midha [24] introduced a PRB model that approximates the deflection of a small-length flexure joint. Coupled with a rigid-link, the deflection of y- and x-axes will be expressed as

f

0

(38)



L+

l 2



cos 

(47)

 sin 

(48)

While the force–deflection relationship is given as F=

d



l l + L+ 2 2

K F(L + 2l ) sin( − )

(49)

where K = (EI)l /l is the stiffness of the torsional spring (or the flexure joint), is the initial deflection angle, and  is the PRB approximated deflection angle.

618

T.J. Teo et al. / Precision Engineering 34 (2010) 607–618

Fig. 18. Comparison between (1 − cos )/ and sin /2.

Fig. 19. Comparison between (1 − cos )/ and sin /2.

Appendix C. Approximations and assumptions Fig. 18 shows that (1 − cos )/ can be initially approximated by sin /2 within a deflection angle ranging from 0◦ to 45◦ . With , the deflection angle range is increased to 60◦ for this approximation as shown in Fig. 19. References [1] Ryu JW, Gweon D-G, Moon KS. Optimal design of a flexure hinge based XY␪ wafer stage. Precision Engineering 1997;21:18–28. [2] Choi BJ, Sreenivasan SV, Johnson S, Colburn M, Wilson CG. Design of orientation stages for step and flash imprint lithography. Precision Engineering 2001;25:192–9.

[3] Chen K-S, Trumper DL, Smith ST. Design and control for an electromagnetically driven X–Y– stage. Precision Engineering 2002;26:355–69. [4] Culpepper ML, Anderson G. Design of a low-cost nano-manipulator which utilizes a monolithic, spatial compliant mechanism. Precision Engineering 2004;28:469–82. [5] Slocum AH. Precision machine design. Englewood Cliffs, NJ: Prentice Hall; 1992. [6] Paros J, Weisbord L. How to design flexure hinges. Journal of Machine Design 1965;37:151–7. [7] Smith ST, Chetwynd DG. Foundations of ultra-precision mechanism design; 1991. [8] Moon Y-M, Kota S. Design of compliant parallel kinematic machines. In: 2002 ASME design engineering technical conferences and computer and information in engineering conference. October 2002. [9] Teo TJ, Chen I-M, Yang GL, Lin W. A flexure-based electromagnetic linear actuator. Nanotechnology 2008;19:315501. [10] Bisshopp KE, Drucker DC. Large deflection of cantilever beams. Quarterly of Applied Mathematics 1945;3:272–5. [11] Frisch-Fay R. Flexible bars. Washington, DC: Butterworth; 1962. [12] Mattiasson K. Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals. International Journal for Numerical Methods in Engineering 1981;17:145–53. [13] Wang CM, Lam KY, He XQ, Chucheepsakul S. Large deflections of an end supported beam subjected to a point load. International Journal of Non-Linear Mechanics 1997;32:63–72. [14] De Bona F, Zelenika S. A generalized elastica-type approach to the analysis of large displacements of spring-strips. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 1997;211:509–17. [15] Lee KM. On the development of a compliant grasping mechanism for on-line handling of live objects. Part 1. Analytical model. In: International conference on advance intelligent mechatronics. 1999. [16] Awtar S, Slocum AH. Closed-form nonlinear analysis of beam-based flexure modules. In: ASME 2005 international design engineering technical conferences & computers and information in engineering conference. 2005. [17] Wriggers P, Reese S. A note on enhanced strain methods for large deformations. Computer Methods in Applied Mechanics and Engineering 1996;135:201–9. [18] Pai PF, Palazotto AN. Large-deformation analysis of flexible beams. International Journal of Solids and Structures 1996;33:1335–53. [19] Gummadi LNB, Palazotto AN. Large strain analysis of beams and arches undergoing large rotations. International Journal of Non-Linear Mechanics 1998;33:615–45. [20] Zelenina AA, Zubov LM. The non-linear theory of the pure bending of prismatic elastic solids. Journal of Applied Mathematics and Mechanics 2000;64:399–406. [21] 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;26:183–352. [22] Smith ST. Elements of elastic mechanisms. Gordon and Breach Science Publishers; 2000. [23] Howell LL. Compliant mechanism. John Wiley & Sons Publisher; 2001. [24] Howell LL, Midha A. A method for the design of compliant mechanisms with small-length flexure pivots. ASME Journal of Mechanical Design 1994;116:280–90. [25] Howell LL, Midha A, Norton TW. Evaluation of equivalent spring stiffness for use in a pseudo-rigid-body model of large-deflection compliant mechanisms. ASME Journal of Mechanical Design 1996;118:126–31. [26] Edwards BT, Jensen BD, Howell LL. A pseudo-rigid-body model for initiallycurved pinned-pinned segments used in compliant mechanisms. Journal of Mechanical Design 2001;123:464–8. [27] Dado MH. Variable parametric pseudo-rigid-body model for large-deflection beams with end loads. International Journal of Non-Linear Mechanics 2001;36:1123–33. [28] Sönmez Ü, Tutum CC. A compliant bistable mechanism design incorporating elastica buckling beam theory and pseudo-rigid-body model. Journal of Mechanical Design 2008;130:042304. [29] Haringx JA. The cross strip pivot as a constructional element. Applied Scientific Research 1949;A1:5–6. [30] Young WE. An investigation of the cross strip pivot. Journal of Applied Mechanics 1944;11:A113–20. [31] Zelenika S, DeBona F, Analytical. Experimental characterization of highprecision flexural pivots subjected to lateral loads. Precision Engineering 2002;26:381–8. [32] Thorp II AG. Flexure pivots—design formulas and charts. Product Engineering 1953:192–200.