Design of bistable arches by determining critical points in the force-displacement characteristic

Mechanism and Machine Theory 117 (2017) 175–188

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Research paper

Design of bistable arches by determining critical points in the force-displacement characteristic Safvan Palathingal, G.K. Ananthasuresh∗ Department of Mechanical Engineering, Indian Institute of Science, Bangalore, India

a r t i c l e

i n f o

Article history: Received 10 March 2017 Revised 8 July 2017 Accepted 10 July 2017

Keywords: Compliant bistable mechanisms Critical load Optimization Bistability Revolute flexures Buckling modes Asymmetric bistable arches

a b s t r a c t The boundary conditions, as-fabricated shape, and height-to-depth ratio of an arch characterize its bistability. We investigate this by modeling planar arches with torsion and translational springs at the pin joints anchored to the ground. A pinned-pinned arch, a special case of the model, has superior bistable characteristics as compared to a fixedfixed bistable arch, which is also a particular case of the model. Arches with revolute flexures at the ends retain bistable characteristics of the pinned-pinned arches while being amenable for easy fabrication. However, equilibrium equations for such arches become intractable for analytical solution unlike the extreme cases of fixed-fixed and pinned-pinned arches. Therefore, a semi-analytical method for analysis and shape-synthesis of bistable arches with general boundary conditions is developed in this work. This is done by numerically determining critical points in the force-displacement curve. These critical points correspond to switching and switch-back forces and travel between the two states thereby enabling synthesis for desired behavior. We present design and optimization examples of bistable arches with a variety of boundary conditions and as-fabricated shape without prestress. We also propose two approaches to design a new class of asymmetric bistable arches. © 2017 Published by Elsevier Ltd.

1. Introduction Arches, one of the commonly used structural elements, have been studied extensively in the literature. When subjected to a transverse load, they can undergo a nonlinear snap-through buckling resulting in two force-free equilibrium states. This behavior is a limiting factor while arches are designed for stiffness but desirable in bistable arches that are also known as curved-beam bistable mechanisms. Fig. 1 shows a bistable arch and its force-displacement characteristic. There are three force-free points on the curve: the first and the last points correspond to the stable State 1 and State 2 of the arch and the point in between corresponds to an unstable equilibrium. Bistable arch-based mechanisms are ideal for switches because power is required only for switching from one state to another but not for maintaining the states. Bistable arches also find application in micro-relays, electromagnetic actuators, micro-valves, mechanical memory components, retractable devices, consumer products, circuit breakers, and easy-chairs [1–7]. Further, multistable structures can be produced by combining bistable arches [8]. Bistability in arches can also be achieved by prestress: a buckled column with pinned-pinned boundary conditions is an example. However, precise prestress is hard to realize during bulk-manufacturing and in microfabrication. Therefore, it is ∗

Corresponding author. E-mail addresses: [email protected] (S. Palathingal), [email protected] (G.K. Ananthasuresh).

http://dx.doi.org/10.1016/j.mechmachtheory.2017.07.009 0094-114X/© 2017 Published by Elsevier Ltd.

S. Palathingal, G.K. Ananthasuresh / Mechanism and Machine Theory 117 (2017) 175–188

State 1

Force (F)

utr usb State 1

Fsb

F

Switching point

Fs

176

us

utr

State 2

State 2 (Second stable point)

Switch-back point Displacement (umid) Fig. 1. Typical force-displacement characteristics of a bistable arch.

Fig. 2. 3D-printed bistable arches with various boundary conditions and as-fabricated profiles in their two stable states. (a) Pinned-pinned sine arch (b) travel-optimized pinned-pinned arch (c) asymmetric pinned-pinned arch (d) asymmetric pinned-fixed arch (e) split-tube flexure-based arch, and (f) constrained double-cosine arch.

beneficial to pursue fully-compliant monolithic bistable arches that do not rely on prestress for their bistability. One way of achieving bistability without prestress is to make the fundamental buckling mode shape as the as-fabricated arch-shape [9]. Analysis of arches under fixed-fixed boundary conditions has been treated with prestress [10] and without it [9]. Buckling and post-buckling analysis of sine-curved arches, which is also the fundamental mode shape of a straight column with pinned-pinned boundary conditions, has been well studied in two pioneering studies of shallow arches [11,12]; later, this was improved to find the exact snapping loads [13]. A bistable arch with pinned-pinned boundary conditions has advantages over fixed-fixed arches. First, a pinned-pinned arch does not need an interconnected double cosine curve (see Fig. 2(f)) to restrict the asymmetric mode of switching. Second, it has enhanced range of travel between its two stable states and reduced switching force. Third, it has provision for secondary lateral actuation [5]. However, pin joints lead to difficulties in manufacturing at the micro scale and problems in operation due to friction and wear. Therefore, we proposed [14] bistable arches with rotational flexures at the ends with a monolithic design retaining the aforementioned three advantages while easing manufacturing. As shown in Fig. 1, State 1 is the as-fabricated force-free equilibrium state and State 2 the stressed force-free equilibrium state. Switching force, Fs , is the minimum force required to switch from State 1 to State 2; switch-back force, Fsb , is the minimum force required to switch back to State 1 from State 2; and travel, utr , is the distance the midpoint of the arch moves between the two stable states. Generally, points on the force-displacement curve corresponding to Fs , Fsb and utr are sufficient for designing a bistable arch. Furthermore, bistability of the arch can be assessed by examining the values of Fs and Fsb . Hence, we refer to these three points as the critical points on the force-displacement curve. We design and optimize bistable arches based on these critical points. This approach simplifies the design methodology to find the optimal bistable design parameters pertaining to size, arch-shape, and cross-section dimensions.

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177

Mode 1 A κA

κB

B

P Mode 2 Mode 3

Fig. 3. Buckling of a straight beam with torsional springs at the ends.

In our previous work [15], we had proposed a computational approach based on critical points on the force-displacement curve for a bistable arch with torsional springs at the joints. In this paper, we extend the critical-point method to a generalized joint constrained by one rotational and two translational springs. An algorithm to obtain the critical-points for arches with general boundary conditions is presented. The critical-point method enables the design of bistable arches with a wide range of boundary conditions and initial shape by taking the as-fabricated shape as a linear combination of buckling mode shapes. Furthermore, the optimal arch profile for the given boundary conditions can also be obtained. This paper has four important contributions to the bistable arches literature. First, A general model of a bistable arch with revolute flexures at the ends is introduced. This can be used to design bistable arches with various boundary conditions. Second, a computationally efficient design method for planar bistable arches using three critical points on the forcedisplacement characteristic is described. This approach solves the computational intricacies while designing bistable arches, especially with flexures at the ends. Third, shape optimization for improving the travel and the switch-back force of the arch using the critical-point method is illustrated. Fourth, two design approaches for a new class of asymmetric bistable arches are presented. In Section 2, we explain our approach of approximating the initial stress-free shape and the deformed shape of an arch as a linear combination of the buckling mode shapes of their corresponding straight column with the torsional springs at the ends. The equilibrium equations are found by minimizing the potential energy of the system with respect to the assumed mode weights of the deformed shape and the displacements in the translational springs. In Section 3, a numerical strategy to find the critical points from the equilibrium equations is discussed. In Section 4, we use the critical-point method as a tool to design a variety of bistable arches shown in Fig. 2. Design and shape optimization of arches with split-tube flexures and the effect of the torsional stiffness and initial shape on them is presented. We show that the critical-point method is equally effective in the design of pinned-pinned and fixed-fixed bistable arches. Furthermore, we explore the design of asymmetric bistable arches using the same method. It may be noted that a detailed force-deflection study on the prototypes presented in the Fig. 2 is not part of this work. 2. Analysis The critical-point method is semi-analytical wherein we solve only for the critical points without obtaining the entire force-displacement curve. Equilibrium equations derived in this section will be used to find the critical points in Section 3. As mentioned before, a column that is buckled into its first fundamental buckling mode can act like a bistable arch if there is prestress. Moreover, arches with the fundamental buckling mode shape as their as-fabricated shape are found to be bistable for pinned-pinned boundary conditions. For fixed-fixed boundary conditions too, its corresponding fundamental buckling mode shape is bistable although restricting its asymmetric mode is necessary, as shown in Fig. 2(f). This prompts us to explore arches with as-fabricated shape, h(x), as a weighted combination of the buckling mode shapes of their corresponding straight columns. For the case of a bistable arch with revolute flexures, buckling mode shapes of a column with torsional springs with equivalent stiffness of the flexure at the ends is considered as shown in Fig. 3. It may be noted that we are not taking the translational stiffnesses of the revolute flexures into account for computing the buckling mode shapes as we only study arch profiles with ends at the same level (i.e., we consider only the loads that are perpendicular to the line joining the ends). Buckling mode shapes, wi , for a column with torsional springs at the ends can be obtained from linear buckling analysis [16]. 2.1. As-fabricated and deformed profiles Let h(x) denote the arch-profile. Let wi indicate the ith buckling mode shape of the straight beam of the same boundary conditions. Since it is likely to get a bistable behaviour from a linear combination of wi s, we write:

h (x ) =

n 

ai wi ( x )

(1)

i=1

where ai is the weight for wi (x) and n is the number of buckling modes considered. Let a point transverse load applied at the mid-point of the arch deforms it to a shape given by w(x). Since, buckling mode shapes form a basis set, we approximate

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t κA

kH

h(x) κB

hmid

kAV

kBV

x=0

x=L

Fig. 4. Model of a bistable arch with flexures at the ends.

Table 1 Normalized parameters used in the spring-restrained pinned-pinned arch. Parameter

Normalized quantity

Normalizing factor

x h(x), w(x), wA , wB

X H(X), W(X), WA , WB

1 L

κA, κB

KA , KB

L EI

kAV , kBV

KAV , KBV

L3 EI

kH

KH

Lh2mid EI

u

U

L h2mid

f

F

L3 EIhmid

–

Q

hmid t

1 hmid

w(x) as follows by accounting for end-displacements:

w (x ) =

m 

Ai wi ( x ) +

i=1

( wB − wA )x L

+ wA

(2)

where Ai is the weight for wi (x) for each buckling mode shape in the deformed shape, wA and wB are the displacements in transverse springs at A and B respectively, and m is number of mode shapes used for approximating the deformed profile. It may be noted that m should not be smaller than n because to capture the as-fabricated shape itself m should be equal to n.

2.2. Equilibrium equations We continue our analysis of the generalized bistable arch shown in Fig. 4 in a normalized framework to identify the decisive geometric parameters that define bistability. All normalized quantities are denoted by uppercase symbols. Table 1 lists the normalized parameters, which are defined later. The potential energy of the system (PE) includes strain energy due to bending in the beam (SEb ), strain energy due to compression along the length of the arch (SEc ), strain energy in the elastic flexures accounted by the torsional and translational springs (SEs ), and the work potential (WP) due to the external transverse force F applied at the mid-point. As shown in Fig. 4, revolute flexures at the ends of the bistable arch have been modeled by two equivalent torsional springs, κ A and κ B , two transverse springs, kAV and kBV , and an axial spring, kH . Modeling with a single axial spring instead of two, one for each revolute flexure, does not result in loss of generality as two springs in series can always be replaced with a single spring of equivalent stiffness. This analysis assumes that the translational spring stiffness is much higher compared to the torsional stiffness, which is justifiable in the case of compliant revolute flexures and, in particular, the split-tube flexure [17] used in this work.

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179

The normalized potential energy for a bistable arch of length L, depth t, width b, area moment of inertia I, height at the mid-point hmid , and Young’s modulus E, is given by

P E =SEb + SEc + SEs + W P 2   d 2W 1 1 d2 H = − dX 2 0 dX 2 dX 2





+ 6Q

1

2

 

0

1 2

dH dX

 2

dH

1 dW KA −

2 dX dX

2

X=0

1 − 2



dW dX

2 

2 dX + U

(3)

 2

dH

1 dW + KB −

2 dX dX

X=1

1 1 1 + KH U 2 + KAV WA 2 + KBV WB 2 − F (Hmid − Wmid ) 2 2 2 h

where H(X) is the normalized as-fabricated shape, W(X) the normalized deformed shape, Q = mid t , the geometry constant, WA , WB , and U are the normalized displacements in the springs KAV , KBV , and KH . Table 1 summarizes the normalizing factors 2

for the these parameters. To obtain Eq. (3), the arch is assumed to be shallow (i.e., dh  1) while approximating the change dx in curvature of the beam. The potential energy expression in Eq. (3) shows the dependence of bistability on the geometry and the boundary conditions of the arch. Strain energy in the springs, SEs , is dependent on the boundary conditions. The terms involving the normalized as-fabricated shape, H(X), and the geometric quantity, Q, indicate the dependence of geometry on bistability. The role of the parameter Q is significant as 6Q2 times change in length of the arch gives the compression energy term, which is crucial to any bistable behaviour. The initial shape, Q, and the boundary conditions determine bistability, whereas geometric parameters hmid , L, t and b, and Young’s modulus E decide the magnitude of critical points in the bistable characteristic. The equilibrium equations obtained from the principle of minimum potential energy are:

∂ PE = 0 where i = 1, 2, 3, . . . m ∂ Ai

(4)

∂ PE =0 ∂U

(5)

∂ PE =0 ∂ WA

∂ PE =0 ∂ WB

Taking only the fundamental mode shape as the as-fabricated shape, the preceding equations can be solved analytically for the cases of pinned-pinned [11,13] and fixed-fixed [9] boundary conditions. On the other hand, getting analytical solutions for bistable arches with flexures at their ends is difficult. In the next section, we explain a semi-analytical method that can be used to design flexure-based bistable arches as well as pinned-pinned and fixed-fixed arches for a wide range of as-fabricated shapes. 3. Critical-point method As the number of mode shapes used to approximate the deformed shape increases, i.e., as m increases, the accuracy of the solution improves at the expense of solving a larger system of equations. It will be shown later that if we take only the first three mode shapes, i.e. m = 3, results will be closer to those obtained using the nonlinear finite element analysis using continuum elements. This assumption leaves us with a system of six equations (Eq. (4) with m = 3 and Eq. (5)) and seven unknowns: A1 , A2 , A3 , U, WA , WB , and F. Using these, the parameters corresponding to the critical points mentioned earlier (see Fig. 1) are discussed next. 3.1. Switching point The force at the switching point (Fs ) is the minimum force required to switch the bistable arch to the second stable state. It is a critical factor in design. Switching from the first stable state to the second can either happen with a symmetrydominant deformation (symmetric switching) or with an asymmetry-dominant deformation (asymmetric switching). A2 will have larger magnitude in asymmetric switching than in symmetric switching. The solid curve in Fig. 5 shows a typical forcedisplacement curve for symmetric switching and the dashed curve for asymmetric switching. Fs is dependent on the mode of switching of the bistable arch. Asymmetric switching lowers the switching force significantly by taking a path with reduced compressive strain energy during the deformation. Eqs. (4) and (5) satisfy both symmetric switching and asymmetric switching conditions. An arch would prefer one mode of switching over the other depending on its as-fabricated shape, Q, and boundary conditions. So, it is important to identify the mode of switching between the two for a given arch. As noted earlier, for the cases of pinned-pinned and fixed-fixed boundary conditions, these two modes of switching can be

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Force (F)

Fss Symmetric switching Fas

0

Assymmetric switching uas

uss

Displacement (u) Fig. 5. Asymmetric mode of switching lowers the switching force.

described analytically [9,11]. Analytical solution has not been possible so far when springs are included. Therefore, it is done numerically by solving for two different switching forces: one assuming symmetric switching (Fss at uss ) and other assuming asymmetric switching (Fas at uas ). Fs will be equal to Fas if uas is less than uss as shown in Fig. 5. In this case the arch prefers asymmetric switching over symmetric switching. It can happen that Fs = Fss if uss < uas . In some other cases, when only one mode of switching is present, Fs would be the switching force corresponding to the only existing switching-mode. Asymmetric arches in Fig. 2(c) and (d) have only asymmetric mode of switching, and hence Fs is equal Fas . The double-cosine arch with a restraint in the middle as in Fig. 2(f) switches only symmetrically as the asymmetric mode of switching is physically constrained. In this arch, Fs is equal to Fss . Fss is the point where the force is maximum with respect to the displacement on the force-displacement curve such that the equilibrium equations are satisfied. When both Fss and Fas exist, this maximum will always be symmetric switching force since Fas cannot be greater than Fss . The finite variable optimization problem to obtain Fss can be written as:

Maximize

A1 ,A2 ,A3 ,U,WA ,WB

F

Sub ject to

i : 4 : 5 : 6 :

∂ PE = 0 where i = 1, 2, 3 ∂ Ai ∂ PE =0 ∂U ∂ PE =0 ∂ WA ∂ PE =0 ∂ WB

(6)

with the necessary conditions:

∂L = 0 where i = 1, 2, 3 ∂ Ai ∂L ∂L ∂L =0 =0 =0 ∂U ∂ WA ∂ WB ∂L = 0 where i = 1, 2, 3, 4, 5, 6 ∂ i

(7)

where L is the Lagrangian of the optimization problem Eq. (6). These equations can be solved numerically using the Newton– Raphson method as

xn+1 = xn − [H f (xn )]−1 ∇ f (xn )

(8)

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181

where

xn = [A1 , A2 , A3 , U, WA , WB , 1 , 2 , 3 , 4 , 5 , 6 ]T



f =

dL dL dL dL dL dL ∂L ∂L ∂L ∂L ∂L ∂L , , , , , , , , , , , dA1 dA2 dA3 dU dWA dWB ∂ 1 ∂ 2 ∂ 3 ∂ 4 ∂ 5 ∂ 6

T

Hf(xn ) the Hessian of f(xn ), and ∇ f(xn ) the gradient of f(xn ). Asymmetric switching force is the minimum amount of force required for the arch to switch from State 1 to State 2 with an asymmetric switching path. In the case of symmetric arches, this is also the point at which the arch begins to deflect asymmetrically. Starting from this point, two solutions can exist for the equilibrium equations; symmetric solution, where the change, |A2 − a2 | remains negligible and the asymmetric solution with a higher A2 along the deformation. This is one of the points where the equilibrium equations have repeated roots with respect to A2 , i.e., at this point, we have

d2 PE =0 d 2 A2

(9)

Fas for symmetric arches is found by solving Eqs. (9), (4) and (5) numerically using the Newton–Raphson method as

xn+1 = xn − [∇ f (xn )]−1 f (xn )

(10)

where

xn = [A1 , A2 , A3 , U, WA , WB , F ]T



f =

dP E dP E dP E d2 P E dP E dP E dP E , , , , , , dA1 dA2 dA3 d2 A2 dU dWA dWB

T

Initial values of the unknowns are taken to be their corresponding values in the State 1 of the arch. In the cases of asymmetric arches, where only asymmetric-switching is present, Fas is found by solving Eq. (7). 3.2. Switch-back point Switch-back force, Fsb is the minimum force required to switch a bistable arch from State 2 to State 1. Fsb can also be thought as a measure of bistability. The larger the switch-back force, the higher the stability of the arch in the second stable state. Similar to Fs , Fsb also depends on the nature of switching. Existence of asymmetric switching from State 2 to State 1 reduces the stability. As in the case of the switching point, we proceed by solving for two different Fsb s, corresponding to symmetric and asymmetric switching-back. The symmetric and the asymmetric switch-back points are found as described in the previous section using Eqs. (8) and (10). To get the switch-back point, the initial guess corresponding to State 2 is to be taken. While implementing the critical-point method, the second stable point (State 2) is determined before the switch-back point so that it can be used as the initial guess in the algorithm. 3.3. Second stable point The second stable point corresponds to the second force-free equilibrium state of the arch. Since F = 0 at this point, we have six equations and six unknowns, which can be solved numerically as

xn+1 = xn − [∇ f (xn )]−1 f (xn )

(11)

where

xn = [A1 , A2 , A3 , U, WA , WB ]T



f =

dP E dP E dP E dP E dP E dP E , , , , , dA1 dA2 dA3 dU dWA dWB

T F =0

By taking x0 = [−a1 , a2 , a3 , 0, 0, 0], it was observed that the numerical method does not converge to the point corresponding to unstable force-free equilibrium state, which also satisfies the same equations. This is because of the broad basin of attraction at the second stable point. 4. Design and optimization using the critical-point method In this section, we first discuss design and profile-optimization of flexure-based bistable arches. Then we illustrate the generality of the method by applying it to four special cases of flexure-based bistable arches: (1) bistable arches with pinned-pinned boundary conditions, (2) design of asymmetric bistable arches, (3) bistable arches with fixed-fixed boundary conditions, and (4) limiting axial spring stiffness design for a bistable arch.

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400 350 Q=5 Q=10 Q=20

300 250 200 150 0

10

20

30

40

50

(a) Switching force 100 Q=5 Q=10 Q=20

80 60 40 20 0 -20

0

10

20

30

40

50

(b) Switch-back force 2

Q=5 Q=10 Q=20

1.98

1.96

1.94

0

10

20

30

40

50

(c) Travel Fig. 6. Effect of torsion stiffness on bistable characteristics.

4.1. Bistable arches with revolute flexures at the joints An ideal revolute flexure has minimal torsional stiffness, and high axial and transverse stiffnesses. It is important to understand how the torsion stiffness affects bistability. In Fig. 6 bistable characteristics are plotted against non-dimensional torsional stiffness for a range of values of Q by taking the first buckling mode shape to be the as-fabricated shape with equal torsion stiffnesses at the ends. Higher torsional stiffness implies that the arch has to overcome more resistance at the boundaries. This would cause an increase in switching force and reduction of switch-back force and travel. We see a decrease in desirable bistable characteristics as K increases, i.e., as we move from pinned-pinned to fixed-fixed boundary conditions. When we design revolute flexure-based bistable arches, it is advisable to restrict the normalized torsion stiffness K to be less than 5 to take advantage of the rotational compliance at the joints. As shown in Fig. 6a and b, as Q increases,

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183

Table 2 Geometric and material parameters. E 2.1 × 10 N/m kH 7.3 × 105 N/m 9

2

κA

κB

0.00525 Nm/rad kA V 1.45 × 106 N/m

0.00525 Nm/rad kBV 1.45 × 106 N/m

Fig. 7. Critical-point method compared to FEA in Comsol using continuum elements.

switching force increases and switch-back force reduces. Travel is reduced as K increases only if Q is small as in Fig. 6c. The increase in the torsional stiffness at the ends will restrict the travel. If the arch is compliant enough to deform by overcoming the stiffness experienced at the ends, the travel would be independent of K. We observe this independence of travel on torsion stiffness when Q is 20 or larger. We now discuss the design of a bistable arch with split-tube flexures [17] at the ends for the data given in Table 2 for maximum stability for a given hmid of 10 mm. Taking t = 1 mm, b = 10 mm, and L = 200 mm, the non-dimensional rotational stiffness is K = κEIL = 0.6; this retains characteristics comparable to that of a pinned-pinned bistable arch. One may note that the linear and axial stiffness values are large enough to justify the assumptions made in our analysis. Taking the first mode shape as the as-fabricated shape, we find the critical points: Fs = 197.30, Fsb = −83.23, Utr = 2.00. In Fig. 7 the critical points obtained from critical-point method is plotted on the force-displacement curve obtained from the finite element analysis (FEA) using continuum elements in COMSOL software. For low K values, since the switch-back force is independent of Q, we can optimize the shape of the profile to achieve maximum stability. The bistable characteristics can be optimized with respect to a1 , a2 and a3 , the unknown weights of the first three buckling modes, using a numerical approach. Utilizing the fact that the critical-point method is computationally a a fast, contour plots are constructed for a range of a2 and a3 ratios taking a1 = 1. An arch is not bistable for any combination 1 1 of a1 , a2 and a3 . Each point inside the dashed contour in Fig. 8 gives the combinations of a1 , a2 and a3 that are bistable. a3 a Arches are bistable for a broader range of a ratios as compared to a2 . The loss of bistability beyond certain combinations a

1

a

1

of a2 and a3 can be understood from Fig. 8b. The switch-back force is a measure of stability of the bistable arch in the 1 1 second stable state. As we approach towards the dashed contour, the Fsb decreases to zero indicating an unstable State 2. Physically, this implies that even a slight disturbance can bring the arch back from the second to the first equilibrium state. a a We also observe that the ratios a3 = 0.049 and a2 = 0 (Profile A in Fig. 9) maximize the switch-back force, the arch profile 1 1 with highest stability for the selected revolute flexure. Arch (e) in Fig. 2 shows the optimized 3D printed bistable arch with compound split-tube flexures and its two stable force-free equilibrium states. a a We see in Fig. 8a that the switching force reduces as a3 and a2 ratios increase. But it does not have a minimum in the a

a

1

1

feasible space. The ratios a3 = 0.192 and a2 = 0 (Profile B in Fig. 9) maximize the travel as seen in Fig. 8c. But this increased 1 1 travel comes at the expense of reduced switch-back force. So, it advisable to keep a lower limit on the switch-back force while maximizing the travel in bistable arches. 4.2. Pinned-pinned bistable arches Analysis for pinned-pinned arches is a special case of flexure-based arches with κA = κB = 0 and very high translational and axial stiffness. The reduced number of unknown variables, as WA = WB = U = 0, makes analysis for pinned-pinned arches even faster than the case with a flexure joint. Following a similar procedure as in the previous problem, geometric parameters can be designed from the non-dimensional critical points obtained. Fig. 10 shows a good agreement between critical-point method and FEA for t = 1 mm, b = 5 mm, E = 2.1 GPa and L = 100 mm. Furthermore, geometric profile opti-

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Fig. 8. Dependence of Fs , Fsb , and Utr on the as-fabricated shape.

Fig. 9. Optimal profiles of a bistable arch with split-tube flexure at the ends.

S. Palathingal, G.K. Ananthasuresh / Mechanism and Machine Theory 117 (2017) 175–188

a2 a1 = 0

185

a3 a1 = 0

Fig. 10. Critical-point method compared to FEA in Comsol using continuum elements for pinned-pinned boundary conditions.

a2 a3 = 0.12 a1 a1 = 0.1 (a)

a2 a1 = 0

a3 a1 = 0 (b)

Fig. 11. Critical-point method compared to FEA in Comsol using continuum elements for asymmetric bistable arches with (a) asymmetric initial shape and (b) asymmetric boundary conditions.

mization can also be carried out to obtain the optimal bistable arches with pinned-pinned boundary conditions. For example, arch (b) in Fig. 2 maximizes the travel between the two states for a given initial height. 4.3. Asymmetric bistable arches Asymmetric bistable arches (Fig. 2(c) and (d)) have not been explored in the literature. One can observe that the point at the maximum height shifts along the span as an asymmetric arch switches between the states. This shift can be utilized in switch embodiments requiring offset contact points. They can be good candidates to utilize bimodality in the bistable arches for two-port actuation [5] as well. Asymmetric bistable arches can be designed in two ways: (1) by having a non-zero a2 in the as-fabricated profile of a bistable arch, (2) by having two different torsional stiffnesses at the ends of the arch. We already considered asymmetric a arches with non-zero a2 as we optimized the as-fabricated profile for the case of split-tube flexure based bistable arch 1 using the critical-point method. In Fig. 8, we observe that the asymmetry in the initial profile decreases switching force and switch-back force. Asymmetry in the as-fabricated shape makes the choice between symmetric and asymmetric switching easy for the arch, making it directly go into asymmetric deformation causing a decrease in the switching force. As the arch is already asymmetric, symmetric solution is absent, i.e. Fs = Fas . Fig. 11(a) compares critical-point method to FEA for an asymmetric bistable arch of t = 1 mm, b = 5 mm, E = 2.1 GPa and L = 100 mm with pinned-pinned boundary conditions. The smooth FEA curve that is unlike the curves for symmetric designs with two sharp points indicates the absence of multiple switching solutions. The second way to design asymmetric bistable arches is to take unequal torsional stiffnesses at the two ends. Buckling mode shapes including the fundamental mode shape will be asymmetric for the straight column with unequal torsion stiffness at the ends. For example, taking pinned-fixed boundary conditions with the fundamental

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buckling mode as the as-fabricated shape, an asymmetric bistable arch in Fig. 2(d) is obtained. Critical-point method for this case with t = 1 mm, b = 5 mm, E = 2.1 GPa and L = 100 mm also agrees quite well with FEA as shown in Fig. 11(b). The sharp region at the switch-back force is due to an instantaneous deformation of the arch into a symmetric profile. The stability of the asymmetric bistable arches that belong to the second design method is found to be better than that of the first. 4.4. Fixed-fixed double cosine bistable arches with restricted asymmetric mode For fixed-fixed boundary conditions, it can be shown [9] that the first fundamental buckling mode shape is not bistable unless its asymmetric mode of switching is restricted. This can be physically realized by taking two cosine shaped arches parallel to each other and joining them at the center. Analysis described in the last two sections needs to be modified by taking into consideration two arches instead of one with the second mode of deformation restricted. Bending energy and compression energy components in Eq. (3) will become double and strain energy in the springs will be zero as the displacements at the boundary will be zero for fixed-fixed boundary conditions. The total potential energy can be written as

P E = 2SEb + 2SEc + W P

(12)

The equilibrium equations of the double cosine bistable arch with fixed-fixed boundary conditions and restricted asymmetric mode have multiple solutions w.r.t. A3 [9]. As the asymmetric deformation is restricted, A2 will be always zero. Moreover, at the point where multiple solutions exist, one of the solution corresponds to the case when A3 = 0. So, instead of Eq. (9), we have

A2 = 0 A3 = 0 2

(13)

d PE =0 d 2 A3 Let us consider a design example of a bistable arch with a minimum travel, utr of 18 mm, a maximum switching force, fs of 15 N, a span not exceeding 100 mm and a depth not less than 1 mm. Let us assume the height, hmid to be equal to 10 mm to achieve a travel greater than 18 mm. By taking the depth to be minimum we have

Q=

hmid = 10 t

By substituting this into the equilibrium equations and taking the as-fabricated to be the first buckling mode shape, we obtain from the critical-point method:

fs l 3 = 1486.57 EIhmid utr Utr = = 1.98 hmid Fs =

which implies that the travel, utr = 19.8 mm, which is greater than 18 mm. Solving for width, we get b =

100Et 3 hmid 12 f l 3

= 11.67 mm. By taking b = 10 mm the switching force can be reduced below 15 N. The FEA results juxtaposed with the critical points is shown in Fig. 12(a) for the final fixed-fixed design. Note that in the case of fixed-fixed bistable arches, the first five mode shapes were used for approximation, i.e., m = 5, for improved accuracy. Furthermore, let us design for an axial spring attached to one of the ends of the designed bistable arch. As we reduce the axial stiffness, the compression energy reduces at the intermediate unstable equilibrium that helps the arch to have two stable states; eventually resulting only in snap-through but no bistability. Results from critical-point method for an axial stiffness KH = 3 × 105 N/m attached to the fixed-fixed double-cosine designed previously is shown in Fig. 12(b). The deviation of the critical points from FEA is mainly because of the displacement in the axial spring. The reduction of bending energy of the arch due to this displacement is not accounted for in Eq. (3) to simplify the model. As a result, the critical-point method over-estimates switching and switch-back forces when axial displacement is significant. The limiting stiffness, i.e., the minimum stiffness of the lumped axial spring required so that the arch is bistable is found out to be 1.19 × 104 N/m (KH ) as shown in Fig. 13. We can observe how the switch-back force reduces and eventually becomes zero as the limiting stiffness decreases. Note that the critical points are joined with lines only for ease in visualization. They are neither considered to be approximate force-displacement curve nor are they used for obtaining the limiting stiffness. 5. Conclusions The motivation for developing bistable arches with revolute flexures was to make them amenable for assembly-free fabrication. Such a bistable arch with compound split-tube flexures at the pinned-pinned ends is fabricated as an example. A semi-analytical yet powerful, critical-point design method has been presented to design bistable arches with revolute flexures. Effectiveness of the method was illustrated for solving various bistable arch design problems. If needed, the accuracy

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187

(a)

KH

(b) Fig. 12. Critical-point method compared to FEA in Comsol using continuum elements for fixed-fixed boundary conditions with (a) KH = ∞ (b) KH = 3 × 105 N/m.

Fig. 13. Limiting horizontal stiffness of fixed-fixed double-cosine arch.

of the results can be improved by adding more terms in the number of mode shapes used in approximating the deformed profile. With the aid of the critical-point method, we studied the effect of the torsional stiffness and the as-fabricated shape on the bistable characteristics. It was found that including higher mode shapes in the as-fabricated geometric profile can lead to improved designs with reduced switching force and larger switch-back force and travel. One of the important observations a from the contours of feasible space is the possibility of asymmetric bistable arches, i.e., arches with non-zero a2 , are rarely 1 seen and utilized in the literature. Two different approaches to design these arches were discussed. One limitation of the analysis we followed here is that the choice of initial profile is limited to the combinations of the buckling mode shapes of the corresponding column. But it is not true that these are the only arch profiles that are bistable for the given boundary conditions. Using a generalized approach that considers any arbitrary as-fabricated profile might provide further insights into the nature of bistability of arches.

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