A novel fully compliant tensural-compresural bistable mechanism

Sensors and Actuators A 268 (2017) 72–82

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

A novel fully compliant tensural-compresural bistable mechanism Qi Han, Kaifang Jin, Guimin Chen ∗ , Xiaodong Shao School of Electro-Mechanical Engineering, Xidian University, Xi’an, Shaanxi 710071, China

a r t i c l e

i n f o

Article history: Received 21 April 2017 Received in revised form 4 October 2017 Accepted 4 October 2017 Keywords: Bistable mechanism Tensural segment Compresural segment Beam constraint model Chained beam constraint model

a b s t r a c t Bistable mechanisms, which can maintain two distinct positions without power input, have been widely used for harvesting vibration energy, constructing metamaterials, sensing threshold acceleration, and achieving adaptable damping. In this work, we propose a novel configuration of fully compliant bistable mechanisms called tensural-compresural bistable mechanisms (TCBMs), in which both tensural segments (compliant segments that are subject to combined tensile and bending loads) and compresural segments (compliant segments that are subject to combined compressive and bending loads) are employed. The combination use of tensural and compresural segments makes TCBMs much easier to be tailored for different design requirements. Although compresural segments are incorporated, potential problems associated with buckling can always be avoided by using compresural segments with smaller slenderness as compared to the tensural segments. To facilitate the design of TCBMs, two kinetostatic models are developed by using the beam constraint model and the chained beam constraint model, respectively. Several TCBM designs accompanied with a prototype test are presented to demonstrate the feasibility of the new bistable configuration and the use of the two kinetostatic models. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Bistable mechanisms, which can maintain two distinct positions without power input [1], have been widely used for state switching [2–5], vibration energy harvesting [6,7], nonvolatile shock detecting [8], mechanical memory [9], metamaterials construction [10], tristable and quadristable mechanisms synthesis [11–14], and overload protection [15]. There are a variety types of compliant bistable mechanisms, for example, the compliant four-link bistable mechanisms [16,17], the Young bistable mechanisms [18,19], the compliant bistable Sarrus mechanisms [20], bistable mechanisms employing initially straight beams [21,23–26,22,27,28], and bistable mechanisms utilizing curved beams/plates [32,34,30,33,31,29]. In most of the bistable mechanisms, the compliant segments are designed to be subject primarily to combined compressive and bending loads [21,22,32,34,33,31,29]. These segments will be referred to as compresural (short for compressive flexural) segments hereafter. Interestingly, Masters and Howell [23] first used compliant segments loaded in tension and bending instead of compresural segments to create a fully compliant bistable mechanism called self-retracting fully compliant bistable mechanism (SRFBM). Compliant segments that are loaded in tension and bending are called tensural (short for tensile flexural) segments [23]. Because tensural segments are loaded in tension, buckling failure is no longer a concern [38,39], and non-symmetric switching modes can be avoided [34,36,37,35]. A tensile load always results in a moment that counteracts to the bending deflection of a compliant segment, thus tensural segments are suitable for designing small displacement bistable mechanisms (to lower power assumption for switching). Besides SRFBMs, another two types of bistable mechanisms employing tensural segments were proposed, which are double-tensural fully compliant bistable mechanisms (DTBMs) [24] and fully compliant tensural bistable mechanism (FTBMs) [25], respectively. To facilitate the design of DTBMs, two kinetostatic models were developed [40]. However, design of these bistable mechanisms is still nontrivial because the bistable behavior requires that the strain energy due to tension dominates over the strain energy due to bending in the tensural segments, which often leads to tensural segments designs that are too thin to be realized due to the practical limitations of stress and fabrication constraints [41]. In this work, we propose a novel configuration of fully compliant bistable mechanisms called tensural-compresural bistable mechanisms (TCBMs), in which both tensural segments and compresural segments are employed, as illustrated in Fig. 1. TCBMs significantly extend

∗ Corresponding author. E-mail address: [email protected] (G. Chen). https://doi.org/10.1016/j.sna.2017.10.012 0924-4247/© 2017 Elsevier B.V. All rights reserved.

Q. Han et al. / Sensors and Actuators A 268 (2017) 72–82

73

Fig. 1. A fully compliant tensural-compresural bistable mechanism.

Fig. 2. Geometrical parameters of a TCBM (Ot A, Oc B and AB are corresponding to the tensural segment, the compresural segment and the rigid coupler, respectively).

the available design space of bistable mechanisms. The combination use of tensural and compresural segments makes TCBMs much easier to be tailored for different design requirements. Although compresural segments are incorporated, potential problems associated with buckling can always be avoided by using compresural segments with smaller slenderness (the ratio of length to thickness) as compared to the tensural segments. To accurately capture the nonlinear deflections of the tensural and the compresural segments in TCBMs, the Beam Constraint Model (BCM) [42–44] and the Chained Beam Constraint Model (CBCM) [45,46] are employed (which are more robust than the elliptic integral solutions [47,48]), based on which two kinetostatic models are developed to facilitate the design of TCBMs. In the following, Section 2 briefly introduces the main parameters of a TCBM, Section 3 formulates the two kinetostatic models for it, and Section 2 provides several TCBM designs accompanied with a prototype test to demonstrate the feasibility of the new bistable configuration and the use of the two kinetostatic models. 2. Tensural-compresural bistable mechanisms (TCBMs) Because a TCBM is symmetric with respect to its vertical centerline, only half of it is considered. Fig. 2 shows the right half for a TCBM, in which three rigid links (the coupler link, the grounded link and the mobile link) are connected by two compliant segments and form a closed loop. This design configuration ensures that the right compliant segment to tensile-force dominated loads (tensural segment), while the left compliant segment is subject to compressive-force dominated loads (compresural segment). The lengths of the tensural segment and the compresural segment are denoted as Lt and Lc , respectively. The slenderness (L/t) of the compresural segment should be smaller than that of the tensural segment to reduce the risk of buckling. Their corresponding in-plane and out-of-plane thicknesses are Ht , Wt , Hc and Wc . It and Ic represent the flexural rigidities, which are defined as It = Wt Ht3 /12, Ic = Wc Hc3 /12

(1)

The length of the rigid coupler is Lr . By placing the global coordinate frame (XOY) with its Y-axis along the opposite travel direction of the shuttle, the initial angles of the tensural segment, the compresural segment and the coupler measured with respect to the X-axis are  t ,  c and  r , respectively. Then we have the following expressions for Lm and Ln : Lm = −Lt cos t − Lr cos r − Lc cos c

(2)

Ln = Lt sin t + Lr sin r + Lc sin c

(3)

It should be noted that Lm is always positive but Ln can be either positive or negative. A negative Ln indicates that point Oc is below point Ot with respect to the XOY coordinate frame at the as-fabricated (undeflected) position. The dominating parameters of TCBMs include Lm , Ln , Lt , Lc , Ht , Hc ,  t and  c . For the purpose of simplifying the kinetostatic modeling, the local coordinate frame for the tensural segment Lt is established with the origin placed at its fixed end Ot and the Xt -axis along the length direction of its undeflected configuration (Xt Ot Yt ). When deflected, this segment is subject to a horizontal force Fxt , a vertical force Fyt and a moment Mzt at its tip (point A), and the corresponding tip deflections are denoted by axial deflection xt , transverse deflection yt and tip slope  t , all measured with respect to Xt Ot Yt . Similarly, for compresural segment Lc , the local coordinate frame Xc Oc Yc is established with the origin placed at its end attached to the shuttle (point Oc ), and the Xc -axis along the length direction of its undeflected configuration. This segment is subject to tip loads Fyc , Fxc and Mzc at point B, and the corresponding tip deflections are denoted as xc , yc , and  c , all of which are measured with respect to Xc Oc Yc (Fig. 3). 3. Kinetostatic modeling Modeling the nonlinear deflections of the tensural and the compresural segments is critical to the development of the kinetostatic model for TCBMs. Model I employs the beam constraint model (BCM) [42] for these segments because it accurately captures the relevant geometric nonlinearities in the intermediate deflection range (10% of the segment length) and the moderate axial-force range (±10 for

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Fig. 3. Free body diagram of a TCBM (A’B’ represents the rigid coupler at a deflected position).

the normalized axial-force) through a set of simple and parametric equations. For TCBMs where stubby segments are incorporated and shear effects become significant, a similar model called Timoshenko beam constraint model [26] can be used instead. In Model II, the chained beam constraint model (CBCM) [45] is utilized to formulate the load-deflection relations of the compliant segments. CBCM breaks aforementioned limitations of BCM through segmentation, thus Model II is more accurate and more broadly applicable. 3.1. Model I Fig. 4 shows the layout of the half model for a TCBM with its associated design variables. At a deflected position with a given shuttle displacement y along the Y axis, the loop-closure equations for the half model can be expressed as: Lm = −[(Lt + xt ) cos t − yt sin t ] − Lr cos(r + ˇ) − [(Lc + xc ) cos c − yc sin c ]

(4)

Ln − y = [(Lt + xt ) sin t + yt cos t ] + Lr sin(r + ˇ) + [(Lc + xc ) sin c + yc cos c ]

(5)

and

where ˇ represents the rotation of the rigid coupler. Because the coupler is rigidly attached to the tensural and the compresural segments, we have ˇ = t = c In the following,  t and  c will be replaced by ˇ for the purpose of reducing the number of the parameters. Applying the beam constraint model (BCM) [43] to the tensural segment with respect to its own local coordinate frame (Xt Ot Yt ) yields

⎡ ⎢ ⎣

⎤

Fyt Lt2 EI t Mzt Lt EI t





⎡

12 −6 ⎥ ⎦ = −6 4 ⎣

yt Lt

⎤ ⎦+

Fxt Lt2

ˇ



1.2

−0.1 2/15

EI t

and H 2 Fxt xt = t − 0.5 Lt 12EI t



yt Lt

 1.2

ˇ

−0.1



−0.1



−0.1 2/15

⎡ ⎣

yt Lt ˇ

⎡

⎤

yt Lt

⎣

⎦+



Fxt Lt2 EI t

ˇ

⎤ 2 ⎦ − Fxt Lt EI t



2 

yt Lt

ˇ

−1/700

1/1400

1/1400

−11/6300

 −1/700 1/1400

1/1400



−11/6300

⎡ ⎣



⎡

yt Lt

⎣

⎤ ⎦

(6)

ˇ

yt Lt

⎤ ⎦

(7)

ˇ

where E is the Young’s modulus (plane stress is assumed [49]) of the material of the segment. Similarly, formulating the load–displacement relations of the compresural segment using BCM with respect to Xc Oc Yc yields

⎡ ⎢ ⎣

Fyc Lc2 EI c Mzc Lc EI c

⎤



⎥ ⎦ = −6 12

−6



4

⎡ ⎣

yc Lc ˇ

⎤ 2 ⎦ + Fxc Lc EI c



1.2

−0.1

−0.1

2/15

and Hc2 Fxc

xc = − 0.5 Lc 12EI c



yc Lc

ˇ

 1.2

−0.1

−0.1 2/15



⎡ ⎣

yc Lc ˇ



⎡ ⎣

yc Lc

⎦+



ˇ

⎤ ⎦−

⎤

Fxc Lc2 EI c



yc Lc

Fxc Lc2 EI c

ˇ

2 

−1/700

1/1400

1/1400

−11/6300

 −1/700 1/1400

1/1400 −11/6300



⎡ ⎣



⎡ ⎣

yc Lc

⎤ ⎦

(8)

ˇ

yc Lc ˇ

⎤ ⎦

(9)

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75

Fig. 4. The tensural and the compresural segments in Model II.

For the rigid coupler (Lr ), its static balancing can be expressed by the following 3 equations:  cos  − F  sin  ) + [F  cos( − ) − F  sin( − )] = 0 (Fxt t t c c xc yc yt  sin  + F  cos  ) + [F  sin( − ) + F  cos( − )] = 0 (Fxt t t c c xc yc yt  + M  + F  L cos( + ˇ −  ) + F  L sin( + ˇ −  ) = 0 Mzt r t r t zc yt r xt r

⎫ ⎪ ⎬ ⎪ ⎭

in which



 , F = −F  , M = −M  Fxt = −Fxt yt zt yt zt  , F   Fxc = −Fxc yc = −Fyc , Mzc = −Mzc

These static balancing equations can be rewritten as (−Fxt cos t + Fyt sin t ) + (Fxc cos c − Fyc sin c ) = 0 (−Fxt sin t − Fyt cos t ) + (Fxc sin c + Fyc cos c ) = 0 −Mzt − Mzc − Fyt Lr cos(r + ˇ − t ) − Fxt Lr sin(r + ˇ − t ) = 0

⎫ ⎪ ⎬ ⎪ ⎭

(10)

The input force Fi of this half model (its positive direction is defined along the negative Y-axis) can be calculated as (the static balancing of Lc along the Y-axis) Fi = Fxc sin(c − ) + Fyc cos(c − )

(11)

For a TCBM design, given an input displacement y , the 12 unknowns (including Fi , ˇ, Fxt , Fyt , Mzt , xt , yt , Fxc , Fyc , Mzc , xc , and yc ) can be obtained by numerically solving the 12 equations in Eqs. (4)–(11). For the whole TCBM, the total input force is Fy = 2Fi = 2Fxc sin(c − ) + 2Fyc cos(c − )

(12)

If a bistable mechanism is composed of n TCBM structures in parallel, the total input force is calculated by Fy = 2nF i

(13)

3.2. Model II It should be noted that the compliant segments (Lt ) may be subject to axial forces that exceed the prediction range of BCM (the Fxt L2

F L2

c , exceeds ±10 [42], which could induce large modeling errors of Model I. In order to improve normalized axial force, fxt = EI t or fxc = xc EI c t the modeling accuracy, Model II models both compliant segments using the Chained Beam Constraint Model (CBCM) [45] by dividing them into two equal elements, as shown in Fig. 4(a).

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Q. Han et al. / Sensors and Actuators A 268 (2017) 72–82

3.2.1. Tensural segment The tensural segment Lt is divided into two equal elements at point Q. This division introduces 12 more intermediate parameters into the system, including 3 loading parameters and 3 deflection parameters for each of the elements. The lengths of these elements are: Lt1 = Lt2 = Lt /2. According to CBCM, we have the following relations for Lt1 :

⎡ ⎢ ⎣

⎤

⎤ ⎤ ⎤ ⎡ ⎡ ⎡  2yt1  2yt1

2  2yt1 12 −6 1.2 −0.1 −1/700 1/1400 Fxt1 Lt2 Fxt1 Lt2 ⎥ ⎣ Lt ⎦ + ⎣ Lt ⎦ ⎦ = −6 4 ⎣ Lt ⎦ + 4EI 4EI t −0.1 2/15 1/1400 −11/6300 t Mzt1 Lt ˛t1 ˛t1 ˛t1 Fyt1 Lt2 4EI t

(14)

2EI t

and H 2 Fxt1 2xt1 = t − 0.5 Lt 12EI t



2yt1 ˛t1 Lt

 1.2

−0.1



−0.1 2/15

⎡ ⎣

⎤

2yt1 Lt



2 ⎦ − Fxt1 Lt 4EI t

˛t1

2yt1 Lt

˛t1

 −1/700 1/1400

1/1400



−11/6300

⎡

⎤

2yt1 Lt

⎣

⎦

(15)

˛t1

For Lt2 (note that Lt2 rotates along with the deflection of Lt1 by ˛t1 at point Q, thus Fxt2 , Fyt2 , xt2 , and yt2 are measured with respect to its local coordinate frame obtained by rotating Xt Ot Yt by ˛t1 ):

⎡ ⎢ ⎣

Fyt2 Lt2 4EI t Mzt2 Lt 2EI t

⎤





⎡

12 −6 ⎥ ⎦ = −6 4 ⎣

⎤

2yt2 Lt

⎦+

˛t2

Fxt2 Lt2



−0.1 2/15

4EI t

and H 2 Fxt2 2xt2 = t − 0.5 Lt 12EI t



2yt2 ˛t2 Lt

 1.2

−0.1

1.2



−0.1



−0.1 2/15

⎡ ⎣

2yt2 Lt ˛t2

⎡ ⎣

2yt2 Lt

⎤ ⎦+



Fxt2 Lt2 4EI t

˛t2

⎤



2 ⎦ − Fxt2 Lt 4EI t

2 

2yt2 Lt

˛t2

−1/700

1/1400

1/1400

−11/6300

 −1/700 1/1400

1/1400



⎡ ⎣

2yt2 Lt

⎤ ⎦

(16)

˛t2



−11/6300

⎡ ⎣

2yt2 Lt

⎤ ⎦

(17)

˛t2

The total deflection of segment Lt with respect to Xt Ot Yt can be calculated as

⎫ ⎪ ⎬

ˇ = t = ˛t1 + ˛t2 xt = xt1 + (Lt /2 + xt2 ) cos ˛t1 − yt2 sin ˛t1 − Lt /2 yt = yt1 + (Lt /2 + xt2 ) sin ˛t1 + yt2 cos ˛t1

(18)

⎪ ⎭

Besides, the following load relations are used to eliminate the 6 newly introduced intermediate load parameters (i.e., Fxt1 , Fyt1 , Mzt1 , Fxt2 , Fyt2 and Mzt2 ):

⎧ ⎪ ⎨ ⎪ ⎩

and

Fxt1 = Fxt Fyt1 = Fyt Mzt1 = Mzt − (Fyt sin ˛t1 + Fxt cos ˛t1 )yt1 + (Fyt cos ˛t1 − Fxt sin ˛t1 )(xt1 + Lt /2)

⎧ F = Fyt sin ˛t1 + Fxt cos ˛t1 ⎪ ⎨ xt2 ⎪ ⎩

Fyt2 = Fyt cos ˛t1 − Fxt sin ˛t1 Mzt2 = Mzt

In model II, the 9 equations in Eqs. (14)–(18) will be used instead of Eqs. (6) and (7) in model I for the tensural segment. That is to say, the remaining 6 intermediate parameters, including xt1 , yt1 , ˛t1 , xt2 , yt2 and ˛t2 , are dealt with by 6 more equations when modeling with CBCM. 3.2.2. Compresural segment Similarly, the tensural segment Lc is divided into two equal elements at point R. The lengths of these elements are: Lc1 = Lc2 = Lc /2. For Lc1 , we have

⎡ ⎢ ⎣

Fyc1 Lc2 4EI c Mzc1 Lc 2EI c

⎤



⎥ ⎦ = −6 12

−6 4



⎡ ⎣

2yc1 Lc ˛c1

⎤ 2 ⎦ + Fxc1 Lc 4EI c



1.2

−0.1 2/15

and Hc2 Fxc1

2xc1 = − 0.5 Lc 12EI c



2yc1 ˛c1 Lc

 1.2

−0.1

−0.1

−0.1 2/15



⎡ ⎣

2yc1 Lc ˛c1



⎡ ⎣

2yc1 Lc

⎤ ⎦+

˛c1

⎤ ⎦−

Fxc1 Lc2 4EI c





Fxc1 Lc2 4EI c

2yc1 Lc

2 

˛c1

−1/700

1/1400

1/1400

−11/6300

 −1/700 1/1400



1/1400 −11/6300

⎡ ⎣

2yc1 Lc

⎤ ⎦

(19)

˛c1



⎡ ⎣

2yc1 Lc ˛c1

⎤ ⎦

(20)

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77

Table 1 Parameters of 3 TCBM designs. Parameters

Design 1

Design 2

Design 3

E (PLA) Wt = Wc Lt t Ht Lc c Hc Lm Ln

3.5 × 109 Pa 5 mm 30 mm 348◦ 0.4 mm 16 mm 13◦ 0.7 mm 36 mm 16 mm

3.5 × 109 Pa 5 mm 40 mm 348◦ 0.5 mm 21 mm 13◦ 0.7 mm 30 mm 13 mm

3.5 × 109 Pa 5 mm 25 mm 350◦ 0.7 mm 14 mm 10◦ 0.7 mm 40 mm 15 mm

While for Lc2 (note that Lc2 rotates along with the deflection of Lc1 by ˛c1 at point R, thus Fxc2 , Fyc2 , xc2 , and yc2 are measured with respect to its local coordinate frame obtained by rotating Xc Oc Yc by ˛c1 ), we have

⎡ ⎢ ⎣

Fyc2 Lc2 4EI c Mzc2 Lc 2EI c

⎤



⎥ ⎦ = −6 12

−6 4



⎡ ⎣

2yc2 Lc ˛c2

⎤ 2 ⎦ + Fxc2 Lc 4EI c



1.2

2xc2 Lc



2yc2 ˛c2 Lc

 1.2



−0.1 2/15

and H 2 Fxc2 = t − 0.5 12EI c

−0.1

−0.1

−0.1 2/15



⎡ ⎣

2yc2 Lc ˛c2

⎡ ⎣

2yc2 Lc

⎤ ⎦+

˛c2

⎤ ⎦−

Fxc2 Lc2



4EI c



Fxc2 Lc2 4EI c

2yc2 Lc

2 

˛c2

−1/700

1/1400

1/1400

−11/6300

 −1/700 1/1400



1/1400 −11/6300

⎡ ⎣

2yc2 Lc

⎤ ⎦

(21)

˛c2



⎡ ⎣

2yc2 Lc

⎤ ⎦

(22)

˛c2

The total deflection of segment Lc with respect to Xc Oc Yc can be calculated as c = ˛c1 + ˛c2 xc = xc1 + (Lc /2 + xc2 ) cos ˛c1 − yc2 sin ˛c1 − Lc /2 yc = yc1 + (Lc /2 + xc2 ) sin ˛c1 + yc2 cos ˛c1

⎫ ⎪ ⎬

(23)

⎪ ⎭

Besides, the following load relations are used to eliminate the 6 intermediate load parameters (i.e., Fxc1 , Fyc1 , Mzc1 , Fxc2 , Fyc2 , and Mzc2 ):

⎧ ⎪ ⎨ ⎪ ⎩

and

Fxc1 = Fxc Fyc1 = Fyc Mzc1 = Mzc − (Fyc sin ˛c1 + Fxc cos ˛c1 )yc1 + (Fyc cos ˛c1 − Fxc sin ˛c1 )(xc1 + Lc /2)

⎧ F = Fyc sin ˛c1 + Fxc cos ˛c1 ⎪ ⎨ xc2 ⎪ ⎩

Fyc2 = Fyc cos ˛c1 − Fxc sin ˛c1 Mzc2 = Mzc

Likewise, CBCM uses Eqs. (19)–(23) for the tensural segment in model II. In general, Model II is constructed from Model I by replacing Eqs. (6) and (7) and Eqs. (8) and (9) with Eqs. (14)–(18) and (19)–(23)), respectively. Model II involves more equations thus is more complex and less efficient than Model I. However, it always yields more accurate predictions of the kinetostatic behaviors of TCBMs. 4. Example Three TCBM designs are presented in this section to demonstrate the feasibility of this novel bistable configuration and the use of the two models for predicting the bistable behaviors of TCBMs. Table 1 lists the parameters of the designs which are assumed to be 3D-printed using Polylactic Acid (PLA). The force–deflection curves for 3 designs obtained by Model I and Model II are plotted in Figs. 5–7. The figures show that both Model I and Model II successfully predict the bistable behaviors of the TCBM designs. Both the unstable equilibrium position and the second stable equilibrium position predicted by Model I agree well with those obtained by Model II. However, the magnitudes of the two critical forces (the minimum forces required to switch the mechanism between its two stable positions) obtained Model I are smaller than those of Model II. Nonlinear finite element analysis (FEA) models were built for these TCBM designs in ANSYS (with the geometric nonlinearity option turned on), in which each of the compliant segments was meshed into 50 elements using Beam188. Our tests show that 30 elements for each segment can yield reasonable predictions of TCBMs. However, 50 elements were used in this work to guarantee the accuracy of the results so that they can be used as the exact solutions for the purpose of comparison. The FEA results were obtained by applying a series of displacement load steps (with the displacement step set to 0.1 mm) and recording the reaction force at each step. The FEA results are overlaid on the results of Model I and Model II in Figs. 5–7. As compared to FEA results, Model II obtained more accurate predictions of the kinetostatic behaviors for TCBMs than Model I for all the three designs.

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Q. Han et al. / Sensors and Actuators A 268 (2017) 72–82

Fig. 5. The force–deflection curves of Design 1 predicted by Model I, Model II and the FEA model (FCF, forward critical force; BCF, backward critical force; USP, unstable equilibrium position, and SSP, stable equilibrium position).

Fig. 6. The force–deflection curves of Design 2 predicted by Model I, Model II and the FEA model.

Fig. 7. The force–deflection curves of Design 3 predicted by Model I, Model II and the FEA model.

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79

Table 2 Results comparison of the forward critical force (FCF), the backward critical force (BCF), unstable equilibrium position (USP), and stable equilibrium position (SSP). USP

SSP

FCF

BCF

Value

Error

Value

Error

Value

Error

Value

Error

Design 1

FEA Model I Model II

23.0 mm 21.9 mm 22.9 mm

– 4.8% 0.4%

40.8 mm 40.5 mm 40.7 mm

– 0.7% 0.2%

6.506 N 6.119 N 6.32 N

– 5.9% 2.9%

−6.686 N −7.217 N −6.578 N

– 7.9% 1.6%

Design 2

FEA Model I Model II

22.1 mm 21.4 mm 22.1 mm

– 3.2% 0%

37.2 mm 36.8 mm 37.0 mm

– 1.1% 0.5%

3.202 N 3.034 N 3.132 N

– 5.2% 2.2%

−2.57 N −2.712 N −2.489 N

– 5.5% 3.2%

Design 3

FEA Model I Model II

21.1 mm 21.0 mm 21.1 mm

– 0.5% 0%

35.4 mm 35.1 mm 35.4 mm

– 0.8% 0%

7.496 N 6.983 N 7.342 N

– 6.8% 2.1%

−7.743 N −7.18 N −7.631 N

– 7.3% 1.4%

Fig. 8. The predicted axial forces exerted on the tensural and the compresural segments in Design 3.

Table 2 compares the results of Model I, Model II and the FEA models for the forward critical force (FCF), the backward critical force (BCF), unstable equilibrium position (USP), and stable equilibrium position (SSP) of 3 designs. Both Model I and Model II successfully predicted these critical forces and equilibrium positions, with the largest error less than 8%. Model II are more accurate than Model I, with the largest error 3.2% as compared to the FEA results. Hao [50] showed that the modeling accuracy of BCM can be improved by including the third-order terms. Model I is modified to incorporate the third-order BCM by replacing Eqs. (6)–(7)) with the following equations:

⎡

⎤

  ⎡ yt ⎤   ⎡ yt ⎤ 12 −6 1.2 −0.1 2 ⎢ ⎥ ⎢ ⎥ ⎣ Lt ⎦ + Fxt Lt ⎣ Lt ⎦ + ⎣M L ⎦ = EI t zt t −6 4 −0.1 2/15 ˇ ˇ EI t ⎤ ⎡

2  −1/700 1/1400  yt

3  1/63000 2 Fxt Lt2 ⎣ Lt ⎦ + Fxt Lt Fyt Lt2 EI t

EI t

H 2 Fxt xt = t − 0.5 Lt 12EI t

1.5

F

2 xt Lt

EI t

2



yt Lt

1/1400

yt Lt





1.2

−11/6300 −0.1

ˇ



−0.1 2/15 1/63000

  yt  Lt ˇ

−1/126000

ˇ −1/126000

1/27000

EI t

ˇ

−

Fxt Lt2



EI t

  yt  Lt ˇ

+

yt Lt

F

−1/700

1/1400

1/1400

−11/6300

3

yt Lt

1/27000



(24)

 ⎡ yt ⎤ ⎣ Lt ⎦ ˇ

  yt  Lt

ˇ

2 xt Lt

EI t

−1/126000



−1/126000

−

ˇ

37/97020000

−37/194040000

ˇ −37/194040000

509/291060000

  yt 

(25)

Lt ˇ

and the corresponding results for Design 3 are plotted in Fig. 7. The results show that Model I with the third-order BCM significantly improves the prediction accuracy of the kinetostatic curve between the unstable equilibrium position and the second stable equilibrium position, but it is still less accurate than Model II. To further observe the cause of the discrepancy of Model I, Fig. 8 compares the axial forces applied on the compliant segments at different deflected positions. Again, Model I yielded large errors as compared to the FEA predictions. According to the FEA results, the maximum normalized axial forces applied on the compresural and the tensural segments are determined as fxc =−12.01 and fxt = 37.35, respectively. That is to say, the axial forces exceed the allowed axial-force range of a single BCM (±10), which induces the large errors of Model I. For Model II, the maximum normalized axial forces applied on elements Lc1 and Lt1 are reduced to fxc1 =−3 and fxt1 = 9.34 [45], respectively,

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Fig. 9. Experimental setup for measuring the force–deflection characteristics.

Fig. 10. The measured force–deflection curve compared to that of Model II.

which are within the allowed range. Generally speaking, Model II is more accurate than Model I in capturing the kinetostatic behaviors of TCBMs because CBCM extends the allowed axial-force range of BCM. A PLA prototype of the TCBM design, together with the measuring devices, is shown in Fig. 9. The shuttle of the TCBM is attached to a ball screw. By manually operating the ball screw, the shuttle was moved approximately 0.1 mm each displacement step along its travel direction. At each step, the deflections were measured by a diffraction grating scale with 20 ␮m resolution, and the corresponding forces were recorded by a force gauge with 0.05 N resolution. The recorded force–deflection data were plotted in Fig. 10 and compared to the results of Model II. The prototype successfully demonstrated the bistability of the TCBM design. The experimental curve follows the predicted one of Model II well, and passes through the predicted unstable equilibrium position and the second stable equilibrium position. However, the measured return critical force (the minimum force to switch the TCBM from the second stable position to the first) is smaller than the predicted one. This discrepancy could be attributed to the creep and stress relaxation of PLA and the imprecise geometric parameters of the prototype resulted from the 3D printing process.

5. Conclusions This work presented a novel configuration of fully compliant bistable mechanisms called tensural-compresural bistable mechanisms, in which both tensural segments and compresural segments are employed. Two kinetostatic models were developed to facilitate the design of TCBMs. Several TCBM designs were presented to demonstrate the feasibility of the new bistable configuration and the use of the two kinetostatic models. Model I is more tractable and can be used in the early stage of design; in contrast, Model II is more accurate in capturing the kinetostatic behaviors of TCBMs but a little more complex because more equations are involved. Finally, the bistability of one TCBM design was successfully demonstrated by a 3D-printed prototype.

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Acknowledgment The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China under Grant no. 51675396, the Science and Technology Research Funds from Shaanxi Province under no. 2013KJXX-65/2014KTCQ01-27, and the Fundamental Research Funds for the Central Universities under no. K5051204021/SPSZ011407. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

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Biographies Qi Han is currently working towards her Ph.D degree in Mechanical Engineering at Xidian University. Her main research interests include compliant multistable mechanisms and their applications. Kaifang Jin is a Master student at Xidian University. His main research interests include compliant mechanisms and their applications.

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Guimin Chen, PhD, is a Full Professor at Xidian University. His main research interests include compliant mechanisms and their applications. Currently he serves as an Associate Editor of ASME Journal of Mechanisms and Robotics.

Xiaodong Shao, PhD, is a Full Professor at Xidian University. His main research interests include computer-aided design and manufacturing. He has authored more than 30 papers in referred journals and peer-reviewed international conferences.