# 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 snap-through elastic deformation to switch to a second force-free stable configuration. Actuation is only required to switch between the states and not to maintain either. An analytical bilateral relationship between arch-profiles developed in my advisor's group was used to design arches with different initial and final profiles, cross-section dimensions; with fixed-fixed and pinned-pinned boundary conditions. Various models of bistable arches are presented with a variety of switching forces, switch-back forces and travel between the two states. Finite element analysis was performed for all arches to understand the non-linear 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 fixed-fixed 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

Abbreviations
 FEA Finite Element Analysis

## INTRODUCTION

When a transverse load is applied on a slender arch, it undergoes snap-through elastic deformation. Fig. 1 shows the force-displacement characteristics of a bistable arch [1]. It can be seen that the curve corresponds to three force-free states: the initial stress-free 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.

Force-displacement characteristics of bistable arch​[1]

The minimum force required for an arch to switch from stable state 1 to state 2 is called the switching force, Fs; the minimum force required to switch back from state 2 to state 1 is called the switch-back force, Fsb and the distance moved by the midpoint of an arch between the two states is defined as travel, Utr. Actuation is required only to switch between states and not to maintain either. Arches can have boundary conditions such as fixed-fixed, pinned-pinned; 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, micro-relays[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 fixed-fixed and pinned-pinned boundary conditions. Hessian matrix is used to check for bistability, as not all arch-profiles 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 pinned-pinned boundary condition is difficult; hence, only fixed-fixed arches are CNC machined. Polypropylene is the material used for fabrication.

FEA analysis was performed for all arches and their force-displacement curve determined. Prototypes were made using 3D printing and compared with the arch-profiles 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 pre-stressed beams or columns with similar boundary conditions that can exist in two stable states.

Determining the amount of pre-stress required would be difficult during fabrication. Hence, Qiu [7] suggested making the as-fabricated 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 pre-stress. Detailed analysis of fixed-fixed 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 force-displacement curve of an arch, namely, the switching force, switch-back force and travel. Arches are designed with different profiles and boundary conditions using these critical points.

In ​[2]​, arches are designed with pinned-pinned boundary condition. The arch-profiles are taken as weighted combination of corresponding buckling mode shapes. A two-way relationship between the mode weights of the arch profiles was derived in their force-free 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 arch-profiles with fixed-fixed boundary condition in ​[6]​. We use these relationships to design arches for fabrication.

## Analytical Method

Bistability can be achieved by taking the as-fabricated 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 fixed-fixed boundary conditions, we first define the geometric parameters as listed in Table 1. We first focus on designing double-curved fixed-fixed arches.

Geometric parameters of fixed-fixed bistable arches
Geometric parameters (in mm)
 Arch span or length L Arch height hmid In-plane depth t Out-of-plane 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 as-fabricated profile H(x) and the deformed profile W(x) of a fixed-fixed arch can be expressed as the combination of buckling mode weights of straight fixed-fixed column. In equation 1 and 2, ai and Ai represent unknown weights corresponding to ith buckling mode shape of the respective initial and final profiles. The normalized parameters in the equation are shown in Table 2.

$\text{H(x)=}\sum _{\mathrm{i}=1}^{\infty }{\mathrm{a}}_{\mathrm{i}}{\mathrm{W}}_{\mathrm{i}}\left(\mathrm{x}\right)$

where,

Normalized parameters
 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=hmid/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 arch-profiles are bistable.

$\frac{{\partial }^{2}SE}{\partial {{A}_{i}}^{2}}=0$

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.

${\mathrm{A}}_{\mathrm{i}}=\frac{{\mathrm{a}}_{\mathrm{i}}}{1-\frac{3{\mathrm{Q}}^{2}\mathrm{C}}{{{\mathrm{M}}_{\mathrm{i}}}^{2}}}$

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, A1. Using A1, 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, pinned-pinned 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 fixed-fixed boundary condition. Polypropylene is used for fabrication whose material properties are given in Table 1.

Material properties of polypropylene
 Mass density 946 kg/m3 Young's modulus 1.325x109 N/m2 Poisson's ratio 0.35

## Planar Bi-axial Test

Experimental testing of bistable arch in Fig 7

The planar biaxial test was performed for the fixed-fixed 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 force-displacement curve for the arch.

## RESULTS

For the double cosine fixed-fixed arch, the as-fabricated shape is the same as the first fundamental buckling mode shape of straight fixed-fixed 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 hmid=20mm and t=1mm. Force-displacement curve obtained through FEA results given in Fig 3b shows bistability. The arch-profiles obtained through FEA analysis matches with the 3D printed prototypes as shown in Fig 4. Snap-through bistability can be observed in Video 2 for the arch.

Geometric parameters
Force-displacement curve obtained through FEA
Double cosine fixed-fixed arch
Initial and final profile obtained through FEA
3D printed prototype
Double cosine fixed-fixed arch
CNC machined double cosine fixed-fixed arch showing bistability

The fixed-fixed arch shown in Fig 5 has been obtained through analysis method. For hmid=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 A1=-0.5388 using Eq.8. The remaining Ai values are found using Eq.9 and the arch is then modeled. The arch-profiles along with the 3D printed prototypes can be seen in Fig 6. Video 3 shows the snap-through bistability of the arch.

Geometric parameters
Force-displacement curve obtained through FEA
Fixed-Fixed arch
Initial and final profile obtained through FEA
3D printed prototype
Fixed-Fixed arch
Working video of CNC machined arch in Fig 5

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, Ai(i=1:20) values are used to compute a1 using Eq.10. The remaining ai values are found using Eq. 9 and the arch was modeled using Eq.1. The force-displacement 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 switch-back force in Fig.8b is small as compared to switch-back 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 arch-profiles and the 3D printed prototype is given in Fig. 9a and 9b respectively.

Geometric parameters of fixed-fixed arch
Force-displacement curve obtained through FEA
Force-displacement curve obtained through experimental testing
Fixed-Fixed arch
Initial and final profile obtained through FEA
3D printed prototype
Fixed-Fixed arch
Working video of CNC machined arch in Fig.7

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 a1. Remaining values in ai are found using Eq.9. This is then modeled by representing ai values in the form of Eq. 1. The geometric parameters and force-displacement curve is shown in Fig. 10. Video 5 presents the working of the arch with the 3D printed prototypes seen in Fig 11b.

Geometric parameters of arch 4
Force-displacement curve obtained through FEA
Fixed-Fixed arch
Initial and final profile obtained through FEA
3D printed prototype
Fixed-Fixed arch
Working video of CNC machined arch in Fig.10a

Similarly, for getting the desired arch-profiles in Fig.13a, the geometric parameters with hmid= 10mm and t=1mm were used as per Fig. 12a. The force-displacement 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 force-displacement curve are presented in Fig 14. Design for single-curved fixed-fixed arch was explored. The arch-profiles are presented in Fig 16a with the force-displacement curve in Fig 16b. Design examples of pinned-pinned arches with their force-displacement curves have been presented in Figures 17 and 18. They can be modeled using the relations in ​[2]​.

Geometric parameters of arch 5
Force-displacement curve obtained through FEA
Fixed-Fixed arch
Initial and final profile obtained through FEA
3D printed prototype
Fixed-Fixed arch
Working video of CNC machined arch in Fig.12a
Geometric parameters
Force-displacement curve obtained through FEA
Fixed-Fixed arch
Initial and final profile obtained through FEA
3D printed prototype
Fixed-Fixed arch
Single curved fixed-fixed arch
Force-displacement curve of arch in Fig.16a
Single-curved fixed-fixed arch

Sine curved pinned-pinned arches
Force-displacement curve of arch in Fig.17a
Elliptical curved pinned-pinned arches
Force-displacement curve of arch in Fig.17c
Pinned-pinned arches
Pinned-pinned arch
Force-displacement curve of arch in Fig.18a
Pinned-Pinned arch

## CONCLUSION

In this work, bistable arches were designed using bilateral relationships and prototyped using 3D printing. Fixed-Fixed 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 switch-back. Hence, as-fabricated 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 fixed-fixed bistable arches and make durable pinned-pinned 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.

#### References

• Safvan Palathingal, G.K. Ananthasuresh, 2017, Design of bistable arches by determining critical points in the force-displacement characteristic, Mechanism and Machine Theory, vol. 117, pp. 175-188

• 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. 183-193

• Darshan Sarojini, T.J. Lassche, J.L. Herder, G.K. Ananthasuresh, 2016, Statically balanced compliant two-port bistable mechanism, Mechanism and Machine Theory, vol. 102, pp. 1-13

• M H Ansari, M Amin Karami, 2015, Energy harvesting from controlled buckling of piezoelectric beams, Smart Materials and Structures, vol. 24, no. 11, pp. 115005

• Troy Gomm, Larry L Howell, Richard H Selfridge, 2002, In-plane linear displacement bistable microrelay, Journal of Micromechanics and Microengineering, vol. 12, no. 3, pp. 257-264

• Safvan Palathingal, G. K. Ananthasuresh, 2019, Analysis and Design of Fixed–Fixed Bistable Arch-Profiles Using a Bilateral Relationship, Journal of Mechanisms and Robotics, vol. 11, no. 3, pp. 031002

• J. Qiu, J.H. Lang, A.H. Slocum, 2004, A Curved-Beam Bistable Mechanism, Journal of Microelectromechanical Systems, vol. 13, no. 2, pp. 137-146

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