Modeling Large Deflections of Initially Curved Beams

Guimin Chen1 State Key Laboratory for Manufacturing Systems Engineering, Shaanxi Key Lab of Intelligent Robots, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, China e-mail: [email protected]

Fulei Ma School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, Shaanxi, China e-mail: [email protected]

Guangbo Hao School of Engineering, University College Cork, Cork, Ireland e-mail: [email protected]

Weidong Zhu Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD 21250 e-mail: [email protected]

1

Modeling Large Deflections of Initially Curved Beams in Compliant Mechanisms Using Chained Beam Constraint Model Understanding and analyzing large and nonlinear deflections are the major challenges of designing compliant mechanisms. Initially, curved beams can offer potential advantages to designers of compliant mechanisms and provide useful alternatives to initially straight beams. However, the literature on analysis and design using such beams is rather limited. This paper presents a general and accurate method for modeling large planar deflections of initially curved beams of uniform cross section, which can be easily adapted to curved beams of various shapes. This method discretizes a curved beam into a few elements and models each element as a circular-arc beam using the beam constraint model (BCM), which is termed as the chained BCM (CBCM). Two different discretization schemes are provided for the method, among which the equal discretization is suitable for circulararc beams and the unequal discretization is for curved beams of other shapes. Compliant mechanisms utilizing initially curved beams of circular-arc, cosine and parabola shapes are modeled to demonstrate the effectiveness of CBCM for initially curved beams of various shapes. The method is also accurate enough to capture the relevant nonlinear loaddeflection characteristics. [DOI: 10.1115/1.4041585]

Introduction

Compliant mechanisms have gained increasing popularity in scientific instruments, minimally invasive surgeries, robotic systems, and microelectromechanical systems in recent years [1]. They offer many advantages over their rigid counterparts such as increased precision and reliability, and reduced wear and backlash. Because the flexible beams in compliant mechanisms usually undergo large and nonlinear deflections [2], one of the major challenges of designing compliant mechanisms lies in understanding and analyzing such deflections. Previous work has mainly focused on modeling initially straight beams [3–11]. The use of initially curved beams can provide several advantages over initially straight beams in compliant mechanisms: they exhibit more diverse mechanical behaviors than the initially straight beams upon loading [12], and they distribute stresses more evenly [13]. Therefore, this work will focus on modeling large deflections of initially curved beams. Initially curved beams have been extensively employed for designing compliant mechanisms. For example, curved beams of cosine shape were utilized for obtaining bistable and multistable behaviors [14–16]; a simple bistable equilibrium compliant mechanism utilizing a partially rigid and partially flexible curved beam with pinned-fixed boundary conditions was proposed, in which the initial curvature of the flexible part was determined by the shape of a buckled straight beam [17]; the snap-through characteristics of a semi-circular flexible arc were utilized to realize a long dwell behavior of a five-bar compliant mechanism [18]; a constant torque compliant mechanism was synthesized using a set of variable width spline-curve beams [19]; Wu and Lan [20] utilized parallel connected lateral curved beams preloaded by an axial spring to achieve linear variable stiffness; Ahuett-Garza et al. [21] explored the use of semi-circular beams as large displacement hinges in planar compliant mechanisms; Wang et al. [22] 1 Corresponding author. Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received March 24, 2018; final manuscript received September 22, 2018; published online November 12, 2018. Assoc. Editor: James J. Joo.

Journal of Mechanisms and Robotics

presented a 2DOF micropositioning stage using periodically corrugated flexural units; a novel constant-force bistable mechanism allowing constant contact force and overload protection was developed utilizing curved flexible beams of cubic Bezier curve and cosine curve [23]; curved flexible elements had also been employed in topology optimization of compliant mechanisms [12,24]. Although initially curved beams are commonly used in compliant mechanisms, the literature on analysis and design using such beams is rather limited. Several pseudo-rigid-body models were developed to approximate the large deflections of initially curved beams of circular-arc shapes: Howell and Midha [25] developed a pseudo-rigid-body model consisting of two rigid links joined by a torsional spring for initially curved cantilever beams subject to pure end-force loads, which was extended by Edwards et al. [26] for initially curved pinned-pinned beams to allow for moment loads at the beam tip; Venkiteswaran and Su [13] further added a torsional spring to the pseudo-rigid-body model at the tip; Wang et al. [27] presented a pseudo-rigid-body model for semi-circular beams subject to pure end-moment loads, which was employed to model periodically corrugated beams; Venkiteswaran and Su [28] also studied a pseudo-rigid-body model for curved beams consisting of four rigid links joined by three torsional springs. Besides, Belfiore and Simeone [29] derived the closed-form expressions for the compliance matrix of a curved beam joint which has initially a circular axis, Awtar and Sen [30,31] developed a beam constraint model (BCM) for circular-arc beams which can accurately capture the relevant nonlinearities when the deflections are within 10% of the beams’ lengths, Lan and Lee [32] proposed a generalized shooting method for analyzing compliant mechanisms with curved members, and Shvartsman [33] transformed the large deflection problem of a circular-arc cantilever beam into an initial-value problem and solved numerically using a modified Numerov’s method. Generally speaking, most of the available methods are limited to initially curved beams of circular-arc shapes [25–27,30,33] or a certain range of deflections [29,30]. In this work, we propose a method called chained beam constraint model (CBCM) for modeling initially curved beams of various shapes in compliant

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mechanisms (noting that initially curved beams of noncircular shapes are favored in many designs [16,17]). CBCM discretizes an initially curved beam into a few elements (equally or unequally) and models each element using BCM. Through the discretization, CBCM eliminates the limitations for small and constant curvature, small deflection, and maximum allowable tangential force of BCM. CBCM provides designers a simple, fast and practically applicable tool for study, design and optimization of compliant mechanisms utilizing various initially curved beams. The rest of this paper is organized as follows. BCM for initially curved beams developed by Awtar and Sen [30] is briefly introduced in Sec. 2. Section 3 presents two discretization schemes of CBCM (equal discretization and unequal discretization) and the corresponding formulations. Compliant mechanisms utilizing initially curved beams of circular-arc, cosine and parabola shapes are modeled to demonstrate the effectiveness of CBCM for beams of various shapes in this section. Section 4 includes the concluding remarks.

2

Beam Constraint Model for Initially Curved Beams

Figure 1 shows a curved beam with an initial constant radius of R. The beam is subject to radial force F, tangential force P and moment M at its free end, resulting in tangential and radial deflections K and D and end slope a. The parameters of the beam include: the length L along the X-axis, the in-plane thickness T, the out-of-plane thickness W, and Young’s modulus of the material E. I ¼ WT3/12 represents the area moment of inertia of the beam cross section. The nondimensionalized curvature is defined as j¼6

1 R=L

(1)

in which R/L corresponds to the nondimensionalized radius of curvature. For jjj  0:1, the curve of the undeformed beam configuration can be approximately expressed as [30]: j yð xÞ ¼ x2 2

ð0  x  LÞ

(2)

The sign of j indicates the direction in which the unit tangent vector rotates as a function of the parameter along the curve. If the unit tangent rotates counterclockwise, then j > 0. If it rotates clockwise, then j < 0. Beam constraint model accurately captures the nonlinear loaddeflection relations of the beam using the following closed-form equations for jjj  0:1 and jpj  5:0 [30]: 2 3 2 32 3 2 32 3 f 12 6 d 6=5 1=10 d 4 5¼4 54 5 þ p4 54 5 m 6 4 a 1=10 2=15 a 2 32 3 2 3 1=700 1=1400 d j=2 54 5 þ p4 5 þp2 4 (3) 1=1400 11=6300 a j=12

Fig. 1 An initially curved beam subject to combined force and moment loads at its free end

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" #" #  6=5 1=10 d t2 p j j 1  d a d a k¼ 12 2 12 2 1=10 2=15 a " #" #   1=700 1=1400 d j j2 aþp p d a þp 360 720 1=1400 11=6300 a (4) in which the nondimensionalized thickness t ¼ T/L, and all the load and deflection parameters are normalized with respect to the beam parameters as (note that the lower-case symbols denote normalized parameters if not indicated otherwise) p¼

PL2 ; EI

f ¼

FL2 ; EI

m¼

ML ; EI

k¼

K ; L

d¼

D ; L

a¼a (5)

3 Chained Beam Constraint Model for Initially Curved Beams Figure 2 shows a curved beam (thickness of T and width of W) at its undeflected position. The beam is discretized into N elements, and for the ith element shown in Fig. 3, its local coordinate frame (OiXiYi) is attached to and moves along with the free end of (i  1)th element, i.e., Oi. The length of each curved element along its Xi-axis is denoted as Li. The slope at each node Oi is denoted as bi, and the coordinates of each node Oi are denoted as xi and yi, respectively, all measured with respect to the global coordinate frame (noted at x1 ¼ 0, y1 ¼ 0 and b1 ¼ 0). When deflected, the slope at each node Oi is denoted as hi with respect to the global coordinate frame. 3.1 Chained Beam Constraint Model: Equal Discretization. For a flexible beam of circular-arc shape, the curvature is constant; thus, an equal discretization can be adopted when modeled using CBCM. Assume that the beam is equally discretized into N elements. Because BCM requires j  0.1, it is required that N  d10we

(6)

where w is the central angle of the circular arc and w ¼ ho at the undeflected position. For example, to be accurately modeled using

Fig. 2 Discretization of a circular-arc beam at its undeflected position

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" #" #  6=5 1=10 di t2 pi j j 1  di  ai  di ai ki ¼ 12 2 12 2 1=10 2=15 ai " #" #   1=700 1=1400 di j j2 ai þ pi þ pi pi di ai 360 720 1=1400 11=6300 ai (13) (2) Static equilibrium equations (3N equations) Static equilibrium between beam tip and first element: p1 ¼ po ; and static equilibrium (i ¼ 2;   ; N): 2

cos hi 4 sin hi ð1 þ ki Þ

Fig. 3 The ith element at its deflection position

CBCM, at least 16 elements are required for a quarter-circle (w ¼ p/2) shaped beam, and at least 32 elements are required for a half-circle (w ¼ p) shaped beam. The slope, bi, and the coordinates of each node Oi, xi, and yi, are calculated as bi ¼

ði  1Þp ; 2N

xi ¼ R sin bi ;

yi ¼ Rð1  cos bi Þð1 < i  N Þ (7)

p 2N

(8)

(9)

Pi L2e Fi L2e Mi Le Ki Di pi ¼ ; fi ¼ ; mi ¼ ; ki ¼ ; di ¼ ; ai ¼ ai EI EI EI Le Le (10) Similarly, the tip loads and the tip coordinates are normalized as Po L2e Fo L2e Mo Le Xo Yo ; fo ¼ ; mo ¼ ; xo ¼ ; yo ¼ ; h o ¼ h o EI EI EI Le Le (11)

(1) Beam constraint model_equations (3N equations) [30] 2 3 2 2 32 3 32 3 6=5 1=10 12 6 fi di di 4 5¼4 54 5 54 5 þ pi 4 mi ai ai 1=10 2=15 6 4 2 32 3 2 3 1=700 1=1400 j=2 di 24 54 5 þ pi 4 5 þ pi ai 1=1400 11=6300 j=12 (12) Journal of Mechanisms and Robotics

and

ith

3 32 3 2 fi 0 f1 0 54 pi 5 ¼ 4 p1 5 1 mi mi1

elements

(15)

where (0.5j þ di) is the normalized tip coordinate of the deflected ith element along the Yi-axis (note that 0.5j is determined by the curve expression given in Eq. (2)), and hi (considered as the rotation angle of the ith element’s coordinate frame with respect to the global coordinate frame) can be directly expressed by the unknowns h1 ¼ 0;

hi ¼ bi þ

i1 X

ak ði ¼ 2; 3; …; NÞ

(16)

Besides, we have the following relation: (17)

(3) Geometric constraint equations (three equations)

1 j ¼ ji ¼ R=Le

The discretization introduces 6N intermediate parameters: 3 deflection parameters and 3 load parameters of each element. For the ith element, we use Fi, Pi, and Mi to denote the radial force, the tangential force, and the end moment applied on the free end and Ki, Di, and ai to denote the corresponding tangential and radial deflections and the end slope—all are measured with respect to its local coordinate frame OiXiYi. All of the parameters are normalized with respect to the dimensions of an element as:

po ¼

sin hi cos hi ð0:5j þ di Þ

first

m N ¼ mo

and the normalized thickness and curvature are T t ¼ ti ¼ ; Le

between

(14)

k¼1

The length of each curved element along its Xi-axis is Le ¼ Li ¼ R sin

f1 ¼ fo

8 N > > > X½ð1 þ k Þcos h  ð0:5j þ d Þsin h  ¼ x > > i i i i o > > > i¼1 > > > > i¼1 > > > > N X > > > > bN þ ai ¼ ho > :

(18)

i¼1

Equations (12)–(18) constitute the CBCM equations (totally (6N þ 3) equations) for the beam. Among six parameters po, fo, mo, xo, yo and ho, given any three parameters, the other three can be obtained by numerically solving the CBCM equations. In the following, three examples are presented to demonstrate the effectiveness of CBCM in dealing with initially curved beams of circular-arc shape. 3.1.1 Quarter Circle. We assume a combined load of Po ¼ 0.07 N, Fo ¼ 0.01 N, and Mo ¼ 0.0002 Nm is applied at the free end of a beam of a quarter circle (shown in Fig. 4). The parameters of the beam include: E ¼ 1.4  109 Pa, R ¼ 0.1 m, T ¼ 0.001 m, and W ¼ 0.01 m. The beam is evenly divided into 16 elements (N ¼ 16) and modeled using CBCM. The deflected configuration achieved by CBCM is shown in Fig. 4. A nonlinear finite element model (NLFEM) for the beam was built in ANSYSTM, with the beam being meshed into 200 elements using BEAM189 element and the geometric nonlinearity option turned on. The NLFEM results are also plotted in this figure for the purpose of comparison. The results of CBCM show a good agreement with the NLFEM results. The errors between the results obtained by CBCM and NLFEM for the tip coordinates and the tip slope are less than 0.07%. FEBRUARY 2019, Vol. 11 / 011002-3

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issue. As envisaged that the more elements employed, the more accurate the model is and the more computational time is required. 3.1.2 Half Circle. A combined load of Po ¼ 0.07 N, Fo ¼ 0.01 N and Mo ¼ 0.0002 Nm is applied at the free end of a beam of a half circle, as shown in Fig. 6. The parameters of the beam are: E ¼ 1.4  109 Pa, R ¼ 0.1 m, T ¼ 0.001 m, and W ¼ 0.01 m. CBCM evenly divided the beam into 32 elements (N ¼ 32), and the predicted deflection is plotted in Fig. 6. Figure 6 also plots the deflected configuration of the corresponding NLFEM model. The CBCM and NLFEM results agree very well. The errors between the results obtained by CBCM and NLFEM for the tip coordinates and the tip slope are less than 0.11%.

Fig. 4

Deflections of a quarter circle

Table 1 Predicted tip coordinates of the quarter circle using CBCM with different numbers of elements Tip coordinates N ¼ 16 N ¼ 10 N¼8 N¼6 N¼4 N¼3 N¼2

Xo

Yo

0.04403 mm 0.04417 mm 0.04431 mm 0.04461 mm 0.04557 mm 0.04715 mm 0.05290 mm

0.11901 mm 0.11899 mm 0.11898 mm 0.11893 mm 0.11873 mm 0.11827 mm 0.11574 mm

Table 1 lists the deflected tip coordinates of the quarter circle predicted by CBCM with different numbers of elements under the same loading scheme. The results are also plotted in Fig. 5. It has been observed that all the curves tend to be stable with the increase in the number of elements, indicating no convergence

3.1.3 Buckling-Arc Follower in Long Dwell Mechanism. Figure 7 shows a buckling-arc follower, whose snap-through characteristics were utilized to realize a long dwell behavior of a five-bar compliant mechanism [18]. The follower is a semicircular flexible arc made of spring steel whose Young’s modulus is E ¼ 2.07  1011 N/m2. The radius of the arc is R ¼ 50 mm, and the cross-sectional dimensions are w ¼ 16 mm (width) and h ¼ 0.1 mm (thickness). Considering the symmetry of the follower, the middle point of the arc (point Q) undergoes a rectilinear motion and only the lower half is analyzed using CBCM. For a given displacement DA of middle point Q, the corresponding tip coordinates and tip slope are given as Xo ¼ 50  DA ;

Yo ¼ 50;

ho ¼ p=2

(19)

Then Po can be obtained by solving the BCM equations. The force required to actuate the whole buckling-arc follower is calculated as FA ¼ 2Po

(20)

Figure 8 shows the force-deflection curve of the buckling-arc follower for the input displacement DA varies from 0 to 70 mm. CBCM successfully predicts the snap-through behavior and the negative stiffness of the follower. The snap-through force and the snap-back force reported in Ref. [18] were 1.15 N and 0.49 N, respectively, which agree well with the CBCM results shown in Fig. 8.

Fig. 5 Convergence analysis results

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Fig. 6 Deflection of a half circle

Fig. 8 The force-deflection curve of the buckling-arc follower

lengths. After discretization, each element can be approximately treated as a flexural beam having an initial constant curvature. Assume that the curve of a beam is given as y ¼ f ðxÞ;

0xL

(21)

0

which satisfies f ð0Þ ¼ 0. The beam is divided into N elements at nodes Oi(i ¼ 2, 3, …, N) whose coordinates along the X-axis are xi. Besides, we define two nodes at the fixed end and the tip of the beam denoted as O1 (x1 ¼ 0 mm) and ONþ1 (xNþ1 ¼ L), respectively. Then the coordinates along the Y-axis, the slopes, and the curvatures of these (N þ 1) nodes are

Fig. 7

Buckling-arc follower in long dwell mechanism

yi ¼ f ðxi Þ

(22)

bi ¼ arctanf 0 ðxi Þ

(23)

f 00 ðxi Þ Ki ¼  3=2 1 þ f 02 ðxi Þ

(24)

and 3.2 Chained Beam Constraint Model: Unequal Discretization. For a beam of noncircular shape, it is inconvenient to implement equal discretization because the curvature changes along the beam. A practical way is to select a discretization strategy that divide the beam into N elements with approximately equal Journal of Mechanisms and Robotics

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Given the length components of each element along the X- and Y-axes (as shown in Fig. 2) Lxi ¼ xiþ1  xi ;

Lyi ¼ yiþ1  yi

Li ¼ Lxi cos bi þ Lyi sin bi

(26)

The thickness and the approximate curvature of the ith element are normalized with respect to its own length as T ; Li

ji 

Ki þ Kiþ1 Li 2

(27)

The discretization introduces 6N intermediate parameters, i.e., the deflection and the load parameters of each element. The parameters of the ith element are normalized with respect to Li (note that the lengths of the elements are different) as Pi L2i ; EI ai ¼ ai pi ¼

fi ¼

Fi L2i ; EI

mi ¼

Mi Li ; EI

ki ¼

Ki ; Li

di ¼

Di ; Li (28)

This treatment allows us to use the BCM equations for each element without change. (1) Beam constraint model equations (3N equations) [30] The BCM equations for each element are still the same " # " #" # " #" # 12 6 di 6=5 1=10 di fi ¼ þ pi mi ai ai 6 4 1=10 2=15 " # " #" # 1=700 1=1400 di ji =2 þ p2i (29) þ pi 1=1400 11=6300 ai ji =12 " #" #  6=5 1=10 di t2i pi ji ji 1  di  ai  di ai ki ¼ 2 12 2 12 1=10 2=15 ai " #" #   1=700 1=1400 di ji j2 þ pi ai þ pi i pi di ai 360 720 1=1400 11=6300 ai (30) As compared to Eqs. (12) and (13), j becomes ji due to the noncircular shape and t becomes ti due to the unequal discretization in the BCM equations.

Fig. 9

011002-6 / Vol. 11, FEBRUARY 2019

Static equilibrium between beam tip and first element

(25)

the nominal length (defined as the length along the Xi-axis, i.e., the element’s coordinate frame) of each element can be calculated as

ti ¼

(2) Static equilibrium equations (3N equations)

Po ¼

p1 EI ; L21

Fo ¼

f1 EI ; L21

Mo ¼

mN EI LN

(31)

and static equilibrium between first and ith elements ði ¼ 2;   ; NÞ 3 2 32 3 2 sin hi 0 fi cos hi f1 L2i =L21 4 sin hi cos hi 0 54 pi 5 ¼ 4 p1 L2i =L21 5 (32) mi ð1 þ ki Þ ð0:5ji þ di Þ 1 mi1 Li =Li1 where hi is the rotation angle of ith element’s coordinate frame to the global coordinate frame that can be directly expressed by the unknowns h1 ¼ 0; hi ¼ bi þ

i1 X

ak ði ¼ 2; 3; …; NÞ

(33)

k¼1

Geometric constraint equations (3 equations) 8 N > X > > > ½ð1 þ ki ÞLi cos hi  ð0:5ji þ di ÞLi sin hi  ¼ Xo > > > > i¼1 > > > i¼1 > > > N X > > > > bNþ1 þ ai ¼ ho > :

(34)

i¼1

It should be noted that, as compared to the equations of CBCM with equal discretization, Li is newly included in the expressions for the static equilibrium equations (Eqs. (31) and (32)) and the geometric constraint equations (Eq. (34)). These equations can be simplified to those of CBCM with equal discretization when Li ¼ L/N. In the following, two examples are presented to demonstrate the effectiveness of CBCM in dealing with initially curved beams of noncircular shapes. 3.2.1 Curved-Beam Bistable Mechanism. Figure 9 shows a bistable mechanism utilizing initially curved beams of cosine shape [14,16]. The mechanism is assumed to be made of brass whose Young’s modulus is 100 GPa and Poisson’s ratio is 0.279. Considering the symmetry of the mechanism, only the left lower beam (whose length along the X-axis is 30 mm, denoted as L) of the bistable mechanism is modeled, as illustrated in Fig. 9. The

Bistable mechanism employing initially curved beams

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width and the thickness of the beam are W ¼ 2 mm and T ¼ 0.1 mm, respectively. The shape of the beam is given as y¼

1 px 1 cos þ 4000 30 4000

(35)

in which x ranges from 0 to 30 mm. For the purpose of using CBCM, the beam is divided into 12 elements (N ¼ 12) at nodes Oi (i ¼ 2, 3, …, 12) whose coordinates along the X-axis are xi ¼ 2:5; 5; 7:5; 10; 12:5; 15; 17:5; 20; 22:5; 25; and 27:5 mm (36) Although these elements are of equal length along the X-axis, the lengths along their own Xi-axes are different. By defining two nodes at the fixed end and the tip of the beam denoted as O1 (x1 ¼ 0 mm) and O13 (x13 ¼ 30 mm), respectively, the coordinates of all the nodes on the beam along the Y-axis are calculated as yi ¼ 

1 pxi 1 cos þ 4000 30 4000

and the corresponding slopes and curvatures are   p pxi sin bi ¼ arctan 120 30

(37)

Fig. 10 The force-deflection curve of the bistable mechanism

(38)

and p2 cos Ki ¼ 

pxi 30

216000 þ 15p2 sin2

pxi 30

3=2

(39)

For a given shuttle displacement DA, the corresponding tip coordinates and tip slope are expressed as Xo ¼ 30;

Yo ¼ 0:5  DA ;

ho ¼ 0

(40)

and Fo can be obtained by solving the CBCM equations. The force required to actuate the whole bistable mechanism is calculated as FA ¼ 4Fo

Fig. 11 Circular-path guided compliant mechanism employing an parabolic-shape beam

(41) and the rotation center of the crank R as

Figure 10 shows the force-deflection curve of the mechanism when the shuttle is moved 1 mm downward from its undeflected position. The curve clearly shows the bistable behavior of the mechanism, kinetostatic curve, and strain energy, which indicate that the mechanism is a bistable mechanism, with the second stable equilibrium position occurring at DA ¼ 0.97 mm. The mechanism also exhibits a very straight negative-stiffness behavior, which could be useful for obtaining constant-force and staticbalancing in compliant mechanisms. The actuation force and the returning force reported in Ref. [16] were 53.5 mN and 24.2 mN, respectively, which agree with the CBCM results shown in Fig. 10. 3.2.2 Circular-Path Guided Compliant Mechanism. Figure 11 shows a circular-path guided compliant mechanism [8] utilizing an initially curved flexible beam. The flexible beam (W ¼ 2 mm, T ¼ 0.5 mm) is assumed to be made of brass whose Young’s modulus is 100 GPa and Poisson’s ratio is 0.279. The shape of the flexible beam is a parabola given as y ¼ 2x2

ð0  x  0:1mÞ

yQ ¼ 0:02 m

Journal of Mechanisms and Robotics

(43)

yR ¼ yQ  Lr sin /0

(44)

For the purpose of using CBCM, the beam is divided into ten elements (N ¼ 10) at nodes Oi (i ¼ 2, 3, …, 10) whose coordinates along the X-axis are xi ¼ 0:01; 0:02; 0:03; 0:04; 0:05; 0:06; 0:07; 0:08 and 0:09 m (45) By defining two nodes at the fixed end and the tip of the beam denoted as O1 (x1 ¼ 0 mm) and O11 (i.e., point Q, and x11 ¼ 0.1 m), respectively, the coordinates of all the nodes on the beam along the Y-axis are yi ¼ 2x2i

(46)

and the corresponding slopes and curvatures are bi ¼ arctanð4xi ÞðbNþ1 ¼ b0 Þ

(42)

The length of the crank Lr ¼ 0.07 m and the initial crank angle /0 ¼ 20 deg. The coordinates of the tip point Q are given as xQ ¼ 0:1m;

xR ¼ xQ  Lr cos /0 ;

(47)

and Ki ¼ 

4 1 þ 16x2i

3=2

(48)

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Fig. 12 Deflected configurations of the parabolic-shape beam

We tried different element numbers for the NLFEM and found out that at least 40 elements are required to accurately capture the kinetostatic behaviors of this mechanism. On a personal laptop with 3.1 GHz Intel Core i5-3350P processor and memory of 8 GB RAM, it took the NLFEM (40 elements) 14.76 s to obtain the results on average, while it took CBCM programed in MATLAB 12.79 s to complete the same calculation. Generally speaking, the efficiency of CBCM could be further improved by using programming languages such as Cþþ and FORTRAN.

Fig. 13 The input moment Mr versus the crank angle /

For a given crank angle /, the tip coordinates and the tip slope are ho ¼ b0 þ ð/  /0 Þ;

Xo ¼ xR þ Lr cos /;

Yo ¼ yR þ Lr sin / (49)

and the required input moment on the rigid crank is given as Mr ¼ MQ þ FQ Lr cos /  PQ Lr sin /

(50)

The deflected configurations of the parabolic beam obtained by CBCM are shown in Fig. 12. The input moment is plotted in Fig. 13, which shows that the NLFEM results agree well with those obtained by CBCM, with the maximum error less than 0.15%. An interesting phenomenon is found on the moment curve that the stiffness of the mechanism changes at / ¼ 25 deg, which corresponds to the change of the deflection of the curved beam from flattening out to curving up (strain energy releasing). This could be useful for achieving variable stiffness behaviors in robotic systems. 011002-8 / Vol. 11, FEBRUARY 2019

3.2.3 Flexural Pivot With Two Flexible Parabola Beams. Figure 14 shows a flexural pivot with two initially curved beams. Each of the beams is fixed on the one end and rigidly attached to rigid link AB (whose effective length Lc ¼ 0.1 m) on the other end. The geometric and physical parameters of the flexible beams are identical to the one employed in the circular-path guided mechanism. The global coordinate frame XOY is established with its X-axis oriented along the line connecting the two fixed ends and its origin placed at the center of the line. We assume a combined load composed of F, P, and M is applied at the center of the rigid link, resulting in displacements DX, DY, and D/ of the rigid link, as illustrated in Fig. 14. A local coordinate frame is established for each flexible beam (X1O1Y1 for the left beam and X2O2Y2 for the right beam). The tip coordinates and the tip slope of the two beams are denoted as DA, KA, /A, DB, KB, and /B with respect to their own local coordinate frames, respectively. The two flexible beams are divided into ten elements and modeled in a similar way as the beam in the circular-path guided mechanism. Besides, the static equilibrium relations of the rigid link can be written as FA  FB  F ¼ 0 PA  PB  P ¼ 0 MA þ MB þ M  FB Lc cosD/ þ PB Lc cosD/ þ 

PLc cos D/ ¼ 0 2

FLc cosD/ 2 (51)

and the loop closure equations as XB  XA ¼ Lc cos D/ YB  YA ¼ Lc sin D/

(52)

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Fig. 14 Diagram of flexural pivot with two parabola beams

Assuming F ¼ 0, P ¼ 0 and M is gradually increased from 0 to 0.15 Nm, the kinetostatic behaviors of the flexural pivot can be obtained by simultaneously solving the above-mentioned equations and the CBCM equations for both flexible beams. Figure 15 plots the rotation angle of the rigid link as a function of the applied moment, and Fig. 16 plots the deflected configurations of the flexural pivot. The NLFEM results are also provided in both figures for the purpose of comparison. The NLFEM results agree well with those obtained by CBCM.

4

Conclusions

This paper presented a general and accurate method for modeling large planar deflections of initially curved beams of uniform cross section, which can be easily adapted to curved beams of various shapes. The method discretizes a curved beam into a few elements and models each element as a circular-arc beam using BCM. Two discretization schemes were discussed, in which the equal discretization is suitable for beams of circular-arc shape, while the unequal discretization is for beams of noncircular shapes. Compliant mechanisms utilizing initially curved beams of circular-arc, cosine and parabola shapes were modeled to demonstrate the effectiveness of CBCM for initially curved beams of various shapes. The method is also accurate enough to capture the relevant nonlinear characteristics.

Fig. 15 Kinetostatic behaviors of the flexural pivot

Funding Data The National Natural Science Foundation of China (Grant No. 51675396). The Fundamental Research Funds for the Central Universities (Grant No. K5051204021). The Joint Fund of Ministry of Education for Equipment PreResearch (Grant No. 6141A020226).

Appendix: Curvature For a plane curve given explicitly in Cartesian coordinates as Fig. 16 Deflected configurations of the flexural pivot for different moments

where XA, YA, XB, and YB are the coordinates of points A and B with respect to XOY, which can be expressed XA YA XB YB

Lc ¼ KA  L  2 ¼ DA Lc ¼ KB þ L þ 2 ¼ DB

Journal of Mechanisms and Robotics

y ¼ f ðxÞ the curvature can be determined as K ð xÞ ¼

(53)

(A1)

y00 ð1 þ y02 Þ3=2

(A2)

where the prime refers to differentiation with respect to x. For a curve defined in polar coordinates as its curvature is

r ¼ f ðhÞ

(A3)

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K ðhÞ ¼

r 2 þ 2r 02  rr00 ðr 2 þ r 02 Þ3=2

(A4)

where the prime refers to differentiation with respect to h.

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