Design and fabrication of bistable arches
V S Varshni
School of Mechanical Engineering, Vellore Institute of Technology, Vellore 632001
G.K. Ananthasuresh
Professor, Department of Mechanical Engineering, Indian Institute of Science, Bengaluru 560012
Abstract
The purpose of this work is to design slender arches that can undergo snapthrough elastic deformation to switch to a second forcefree stable configuration. Actuation is only required to switch between the states and not to maintain either. An analytical bilateral relationship between archprofiles developed in my advisor's group was used to design arches with different initial and final profiles, crosssection dimensions; with fixedfixed and pinnedpinned boundary conditions. Various models of bistable arches are presented with a variety of switching forces, switchback forces and travel between the two states. Finite element analysis was performed for all arches to understand the nonlinear force displacement characteristic and prototypes were made using 3D printing. As 3D printed prototypes were fragile and did not last long, CNC machining was used for fabricating fixedfixed arches using polypropylene sheets. Experimental testing was done for an arch and compared with finite element analysis. The main deliverable in this project is a set of bistable arches that show interesting behaviours so that they can be used in a variety of applications.
Keywords: compliant mechanisms, CNC machining, Finite Element Analysis, bistability
Abbreviations
FEA  Finite Element Analysis 
INTRODUCTION
When a transverse load is applied on a slender arch, it undergoes snapthrough elastic deformation. Fig. 1 shows the forcedisplacement characteristics of a bistable arch ^{[1]}. It can be seen that the curve corresponds to three forcefree states: the initial stressfree stable state 1, the final stressed stable state 2 and a state of unstable equilibrium between the two stable states. As an arch deforms, its strain energy increases from zero to maximum and reaches to a minimum again^{[2]}. Stable equilibrium states occur at the minima of the strain energy of an arch.
The minimum force required for an arch to switch from stable state 1 to state 2 is called the switching force, F_{s}; the minimum force required to switch back from state 2 to state 1 is called the switchback force, F_{sb} and the distance moved by the midpoint of an arch between the two states is defined as travel, U_{tr}. Actuation is required only to switch between states and not to maintain either. Arches can have boundary conditions such as fixedfixed, pinnedpinned; different initial and final profiles and height to depth ratio. All these parameters are critical for designing bistable arches.
Bistable arches can be seen in various applications such as easy chairs ^{[3]}, energy harvesters ^{[4]}, switches, valves, mechanical OR gates, electromagnetic actuators, microrelays^{[5]}, clips, consumer products and so on.
In this report, literature on bistable arches is summarised in section 2. In section 3, an analytical method to design arches through bilateral relationships^{[6]}^{[2]} is stated. Focus is laid on arches with fixedfixed and pinnedpinned boundary conditions. Hessian matrix is used to check for bistability, as not all archprofiles are bistable. As 3D printed arches do not last long and lose their bistability, a new fabrication method is required to make them durable. This is done by CNC machining. Fabricating arches with pinnedpinned boundary condition is difficult; hence, only fixedfixed arches are CNC machined. Polypropylene is the material used for fabrication.
FEA analysis was performed for all arches and their forcedisplacement curve determined. Prototypes were made using 3D printing and compared with the archprofiles obtained through FEA simulations. This is presented in section 4. Working video of CNC machined arches is also attached. We present conclusions of this work in section 5.
LITERATURE REVIEW
Buckling is a phenomenon that is usually avoided through design. But research was carried out to make buckling more favourable, especially in the case of arches to attain bistability. This is similar to prestressed beams or columns with similar boundary conditions that can exist in two stable states.
Determining the amount of prestress required would be difficult during fabrication. Hence, Qiu ^{[7]} suggested making the asfabricated shape of an arch as the linear combination of buckling mode shapes of a beam, with required boundary conditions. This paved way for fabricating compliant bistable arches without prestress. Detailed analysis of fixedfixed arches can also be seen in ^{[7]}.
In ^{[1]}, the initial and final profile of an arch is represented as the combination of buckling mode shapes of a similar beam or column. The potential energy of the system is minimized to determine the equilibrium equations. These equations are used to find the critical points on the forcedisplacement curve of an arch, namely, the switching force, switchback force and travel. Arches are designed with different profiles and boundary conditions using these critical points.
In ^{[2]}, arches are designed with pinnedpinned boundary condition. The archprofiles are taken as weighted combination of corresponding buckling mode shapes. A twoway relationship between the mode weights of the arch profiles was derived in their forcefree equilibrium states. These relationships are used to design the arches and bistability is checked using Hessian matrix. The matrix is obtained by differentiating the strain energy equation twice. FEA analysis was done and prototypes were made by 3D printing. Similar bilateral relationships were derived between the archprofiles with fixedfixed boundary condition in ^{[6]}. We use these relationships to design arches for fabrication.
METHODOLOGY
Analytical Method
Bistability can be achieved by taking the asfabricated profile of an arch as the linear combination of buckling modes of corresponding column with similar boundary conditions. The analytical equations needed to design the arches are stated here. These have been dealt with extensive mathematical detail in ^{[6]} and ^{[2]}. For arches with fixedfixed boundary conditions, we first define the geometric parameters as listed in Table 1. We first focus on designing doublecurved fixedfixed arches.
Arch span or length  L 
Arch height  hmid 
Inplane depth  t 
Outofplane thickness  b 
Connector width  3 mm 
Gap between two arches  4 mm 
Fixed support  Square ends 20mmx20mm 
Hole diameter in support  d=3mm 
The normalised asfabricated profile H(x) and the deformed profile W(x) of a fixedfixed arch can be expressed as the combination of buckling mode weights of straight fixedfixed column. In equation 1 and 2, a_{i} and A_{i} represent unknown weights corresponding to i^{th} buckling mode shape of the respective initial and final profiles. The normalized parameters in the equation are shown in Table 2.
where,
Parameter  Normalizing factor  Normalized quantity 
x  1/L  X 
w(x),h(x)  1/hmid  W(X), H(X) 
  hmid/t  Q 
The strain energy equation of the arch is taken as the combination of bending and compression energies. This equation is minimized with respect to the unknown mode weights of deformed profile ^{[6]}. This leads us to equations 5 and 6 where Q=h_{mid}/t.
The Hessian matrix is then computed by finding the second derivative of strain energy equation SE, with respect to unknown mode weights of deformed profile. If the matrix is positive definite, then archprofiles are bistable.
Based on the above equations, relationships between the initial and final profiles of the arch have been obtained in ^{[6]}. These are stated in equations 8, 9 and 10. These equations can be used to model the arches through design and analysis method.
In the analysis method, the mode weights of the initial profile are determined. They are then substituted in equation 8 to find the fundamental modal value of the final profile, A_{1}. Using A_{1}, the remaining mode weights are found using equation 9. The arch is then modeled. The design method can be used to determine the initial profile from the final profile. The mode weights of the initial profile, ais can be determined from A is using equation 10. Similarly, pinnedpinned arches can be designed using equations in ^{[2]}.
Fabrication Process
The modeled arches are 3D printed using Vero White and Vero Black material. CNC machining is done for five arches with fixedfixed boundary condition. Polypropylene is used for fabrication whose material properties are given in Table 1.
Mass density  946 kg/m^{3} 
Young's modulus  1.325x10^{9} N/m^{2} 
Poisson's ratio  0.35 
Planar Biaxial Test
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The planar biaxial test was performed for the fixedfixed arch given in Fig 7. Grips are used to attach the unstressed arch on both sides. A displacement of 27 mm is given from the centre of the arch by the actuator as seen in Video 1. The corresponding load is measured when the arch switches from initial to final state and from the final to initial state. This is used to plot the forcedisplacement curve for the arch.
RESULTS
For the double cosine fixedfixed arch, the asfabricated shape is the same as the first fundamental buckling mode shape of straight fixedfixed column. A connector connects two cosine arches at the centre to restrict the asymmetric mode of switching as suggested by ^{[7]}. Fig 3a shows the geometric parameters of the arch. Q is 20 for h_{mid}=20mm and t=1mm. Forcedisplacement curve obtained through FEA results given in Fig 3b shows bistability. The archprofiles obtained through FEA analysis matches with the 3D printed prototypes as shown in Fig 4. Snapthrough bistability can be observed in Video 2 for the arch.
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The fixedfixed arch shown in Fig 5 has been obtained through analysis method. For h_{mid}=10mm and t=1mm, the value of Q is 10. Taking the first 20 buckling mode shapes (for i=1, 2...20), the desired initial profile is represented in form of Eq.1. The Hessian matrix is positive definite and A_{1}=0.5388 using Eq.8. The remaining Ai values are found using Eq.9 and the arch is then modeled. The archprofiles along with the 3D printed prototypes can be seen in Fig 6. Video 3 shows the snapthrough bistability of the arch.
 0

The arch in Fig 7 was obtained through design method. For the geometric parameters given in Fig.7, the desired final profile is represented in the form of Eq. 2. For the first 20 buckling mode shapes, A_{i}(i=1:20) values are used to compute a1 using Eq.10. The remaining a_{i} values are found using Eq. 9 and the arch was modeled using Eq.1. The forcedisplacement curves obtained through FEA and experimental testing are shown in Fig. 8a and 8b respectively. Small variations were found in both the results. Although switching forces in both the curves are almost equal, the switchback force in Fig.8b is small as compared to switchback force in Fig.8a. This means that, the final profile is less stable then what was predicted by the FEA result. Fig 8b does not appear smooth as compared to the FEA result. This is because of vibrations in the arch during measurement. The shape of the archprofiles and the 3D printed prototype is given in Fig. 9a and 9b respectively.
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The design method was used to design arch 4 given in Fig.10. More examples have been provided with the design method. This is because; the method ensures the desired final profile without many trial and errors. The final profile shown in Fig 11a is represented in the form of Eq.2 for first 20 buckling modes. Eq. 10 is used to find the value of a_{1}. Remaining values in a_{i} are found using Eq.9. This is then modeled by representing a_{i} values in the form of Eq. 1. The geometric parameters and forcedisplacement curve is shown in Fig. 10. Video 5 presents the working of the arch with the 3D printed prototypes seen in Fig 11b.
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Similarly, for getting the desired archprofiles in Fig.13a, the geometric parameters with h_{mid}= 10mm and t=1mm were used as per Fig. 12a. The forcedisplacement curve is shown in Fig. 12b with 3D printed prototypes in Fig.13b. Video 6 shows the bistable nature of the CNC machined arch. The arch shown in Fig 15 has been modeled using design method. The geometric parameters and forcedisplacement curve are presented in Fig 14. Design for singlecurved fixedfixed arch was explored. The archprofiles are presented in Fig 16a with the forcedisplacement curve in Fig 16b. Design examples of pinnedpinned arches with their forcedisplacement curves have been presented in Figures 17 and 18. They can be modeled using the relations in ^{[2]}.
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CONCLUSION
In this work, bistable arches were designed using bilateral relationships and prototyped using 3D printing. FixedFixed arches have been CNC machined to make them durable. We note that the parameter Q, the initial shape and the boundary conditions of an arch influence bistability. More energy is required for switching from state 1 to state 2 as compared to switchback. Hence, asfabricated initial profile of an arch is more stable than the final profile. The design method provides better results as compared with analysis method. Although CNC machining produces sturdy arches, they impose design constraints as compared to 3D printing. Further, this work can be extended to design more single curved fixedfixed bistable arches and make durable pinnedpinned arches.
ACKNOWLEDGEMENTS
I wish to thank my advisor, Professor G.K. Ananthasuresh for his guidance and support. I would also like to thank Mr Safvan and members of the M2D2 lab for helping me with my project. I am grateful to the Indian Academy of Sciences for providing me with the summer research fellowship.
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Safvan Palathingal, G.K. Ananthasuresh, 2017, Design of bistable arches by determining critical points in the forcedisplacement characteristic, Mechanism and Machine Theory, vol. 117, pp. 1751881

Safvan Palathingal, G.K. Ananthasuresh, 2018, A bilateral relationship between stable profiles of pinned–pinned bistable shallow arches, International Journal of Solids and Structures, vol. 143, pp. 1831931

Darshan Sarojini, T.J. Lassche, J.L. Herder, G.K. Ananthasuresh, 2016, Statically balanced compliant twoport bistable mechanism, Mechanism and Machine Theory, vol. 102, pp. 1131

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Troy Gomm, Larry L Howell, Richard H Selfridge, 2002, Inplane linear displacement bistable microrelay, Journal of Micromechanics and Microengineering, vol. 12, no. 3, pp. 2572641

Safvan Palathingal, G. K. Ananthasuresh, 2019, Analysis and Design of Fixed–Fixed Bistable ArchProfiles Using a Bilateral Relationship, Journal of Mechanisms and Robotics, vol. 11, no. 3, pp. 0310021

J. Qiu, J.H. Lang, A.H. Slocum, 2004, A CurvedBeam Bistable Mechanism, Journal of Microelectromechanical Systems, vol. 13, no. 2, pp. 1371461
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Fig 1a: https://medium.com/search?q=design
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