0
Research Papers

Analysis of a Compressed Bistable Buckled Beam on a Flexible Support

[+] Author and Article Information
J. Beharic, T. M. Lucas

Electrical and Computer Engineering Department,
University of Louisville,
Louisville, KY 40208

C. K. Harnett

Electrical and Computer Engineering Department,
University of Louisville,
248 Shumaker Research Building,
2210 S. Brook Street,
Louisville, KY 40208
e-mail: c0harn01@louisville.edu

1Corresponding author.

Manuscript received November 21, 2013; final manuscript received April 17, 2014; accepted manuscript posted April 22, 2014; published online May 5, 2014. Assoc. Editor: Taher Saif.

J. Appl. Mech 81(8), 081011 (Jun 05, 2014) (5 pages) Paper No: JAM-13-1481; doi: 10.1115/1.4027463 History: Received November 21, 2013; Revised April 17, 2014; Accepted April 22, 2014

The bistable snap-through behavior of a compressed beam is modeled and measured experimentally as its supporting surface is bent through positive and negative curvatures. When the supporting angle of the beam exceeds a critical angle, bistability is lost and only one stable state is supported. The critical angle is controlled only by the initial compressive stress in the beam, and we report a nondimensionalized calculation method for this angle. This large-deflection nonlinear model provides design rules for low-power sensors and actuators that can measure and control surface curvature from the micro- to macroscale.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Krylov, S., Ilic, B. R., and Lulinsky, S., 2011, “Bistability of Curved Microbeams Actuated by Fringing Electrostatic Fields,” Nonlinear Dyn., 66(3), pp. 403–426. [CrossRef]
Goessling, B. A., Lucas, T. M., Moiseeva, E.V., Aebersold, J. W., and Harnett, C. K., 2011, “Bistable Out-of-Plane Stress-Mismatched Thermally Actuated Bilayer Devices With Large Deflection,” J. Micromech. Microeng., 21(6), p. 065030. [CrossRef]
Sachkov, Y. L., and Levyakov, S. V., 2010, “Stability of Inflectional Elasticae Centered at Vertices or Inflection Points,” Proc. Steklov Inst. Math., 271(1), pp. 177–192. [CrossRef]
Plaut, R. H., and Virgin, L. N., 2009, “Vibration and Snap-Through of Bent Elastica Strips Subjected to End Rotations,” ASME J. Appl. Mech., 76(4), p. 041011. [CrossRef]
Zhao, J., Jia, J., He, X., and Wang, H., 2008, “Post-Buckling and Snap-Through Behavior of Inclined Slender Beams,” ASME J. Appl. Mech.75(4), p. 041020. [CrossRef]
Kim, C., and Ebenstein, D., 2012, “Curve Decomposition for Large Deflection Analysis of Fixed-Guided Beams With Application to Statically Balanced Compliant Mechanisms,” ASME J. Mech. Rob.4(4), p. 041009. [CrossRef]
Kim, D.-H., Song, J., Choi, W. M., Kim, H.-S., Kim, R.-H., Liu, Z., Huang, Y. Y., Hwang, K.-C., Zhang, Y.-W., and Rogers, J. A., 2008, “Materials and Noncoplanar Mesh Designs for Integrated Circuits With Linear Elastic Responses to Extreme Mechanical Deformations,” Proc. Natl. Acad. Sci. U.S.A., 105(48), pp. 18675–18680. [CrossRef] [PubMed]
Kramer, R. K., Majidi, C., Sahai, R., and Wood, R. J., 2011, “Soft Curvature Sensors for Joint Angle Proprioception,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Francisco, CA, September 25–30, pp. 1919–1926. [CrossRef]
Abedi, A., Razi, A., and Afghan, F., 2011, “Small Battery-Free Wireless Sensor Networks for Structural Health Monitoring,” 8th International Workshop on Structural Health Monitoring, Stanford, CA, September 13–15, pp. 2107–2114.
Kornbluh, R. D., Prahlad, H., Pelrine, R., Stanford, S., Rosenthal, M. A., and von Guggenburg, P. A., 2004, “Rubber to Rigid, Clamped to Undamped: Toward Composite Materials With Wide-Range Controllable Stiffness and Damping,” Proc. SPIE, 5388 pp. 372–386. [CrossRef]
Seffen, K. A., 2006, “Mechanical Memory Metal: A Novel Material for Developing Morphing Engineering Structures,” Scr. Mater., 55(4), pp. 411–414. [CrossRef]
Ericson, P., 2002, “Bistable Domes Can Control Springback, Create Shape Memory,” Metalforming Magazine, May 2002, available at: http://home.earthlink.net/~barkingpo/metalform.html
Timoshenko, S. P., and Gere, J. M., 1961, Theory of Elastic Stability, McGraw-Hill, New York.
GitHub, 2014, “For a Suspended Compressed Beam on a Flexible Substrate, Calculate Curvature Energy vs Substrate Angle and Compression Ratio” (Repository for matlab code Optimized for High Performance Computing), https://github.com/harnettlab/hpcflip
GitHub, 2014, “CAD Files for Geared Bender and Test Strips for Experiments to Go Along With hpcflip,” (Repository for These Laser Cutting Files), https://github.com/harnettlab/bender
Vangbo, M., 1998, “An Analytical Analysis of a Compressed Bistable Buckled Beam,” Sens. Actuators A, 69(3), pp. 212–216. [CrossRef]
Rogers, J. A., Someya, T., and Huang, Y., 2010, “Materials and Mechanics for Stretchable Electronics,” Science, 327(5973), p. 1603–1607. [CrossRef] [PubMed]

Figures

Grahic Jump Location
Fig. 1

Clamped, compressed inclined slender beam and coordinate system for calculations. Half of the symmetric beam is shown. The initial angle θ at (x = 0, y = 0) is α and the final angle θ at x = w is 0. The final y-value, or height of the middle of the beam, is determined by integrating dy/ds from s = 0 to s = L. The two stable solutions are here illustrated for α = 40 deg and β = L/w = 1.15.

Grahic Jump Location
Fig. 2

Relationship between α, substrate curvature radius r, and beam support half-width w. sin(α) = w/r.

Grahic Jump Location
Fig. 3

Stable (zero-applied-force) beam shapes are shown by thick lines. Moving shapes (requiring a vertical force to stay in position) are illustrated by thin lines, for a beam having an arclength-to-gap width ratio L/w of 1.15, and having an inclination angle α of 30 deg.

Grahic Jump Location
Fig. 4

(a) Energy landscape of bistable compressed beam with compression ratio 1.14 for varying center heights and inclination angles. The displacement and inclination axes are nondimensionalized for any thin beam, but the curvature energy on the vertical axis is given in mJ for a three-mil thick, 1 cm wide spring steel beam (E ∼ 200 GPa) spanning a gap width w of 5 cm. (The arclength L for this beam is therefore 1.14 × w = 5.7 cm.) (b) Energy versus displacement trace for this beam with an intermediate inclination angle (30 deg). (c) The vertical force Fv versus displacement for this beam is the negative gradient of plot (b) (“downward force” means to push the center of the beam toward more negative y-values).

Grahic Jump Location
Fig. 5

(a) Setup for experiments with beams having a support width (2w) of 5 cm and symmetric end angles α. (b) Experimental results and prediction for thin plastic and metal strips. Snap-through angle is a function of the compression ratio β, and does not depend on absolute scale or material properties as long as assumptions are met.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In