Abstract
Micro Electro Mechanical Systems (MEMS) have found a large range of applications over the recent years. One of the prodigious application of micro-cantilever beams that is in use is represented by AFM probes (Atomic Force Microscopy). The AFM principle is based on the real-time measurement of the deflection of a micro-beam while following a surface profile. Hence, the prior knowledge of the deflection of beams has been of great interest to designers. Although both analytical and numerical solutions have been found for specific type of loads, there is no general solution specifically formulated for micro-cantilever beams that are not geometrically perfectly straight. Hence, the problem has not been specifically considered so far. The current work presents an analytical method based on Lie symmetry groups. The presented method produces an exact analytical solution for the deflection of Ludwick type beams subjected to any point load for non-straight beams. The Lie symmetry method is used to reduce the order of the Ordinary Differential Equation (ODE) and formulate an analytical solution of the deflection function. The result is compared with an analytical solution for a particular case that is available in the open literature. It was found that the two results coincide.
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References
Brojan, M., Videnic, T.: Large deflections of nonlinearly elastic non-prismatic cantilever beams made from materials obeying the generalized Ludwick constitutive law. Meccanica 44, 733–739 (2009)
Bisshopp, K.E., Drucker, D.C.: Large deflection of cantilever beams. Q. Appl. Math. 3(3), 272–275 (1945)
Wang, T.M., Lee, S.L., Zienkiewicz, O.C.: A numerical analysis of large deflections of beams. Int. J. Mech. Sci. 3(3), 219–228 (1961)
Kemper, J.D.: Large deflections of tapered cantilever beams. Int. J. Mech. Sci. 10(6), 469–478 (1968)
Lewis, G., Monasa, F.: Large deflections of cantilever beams of non-linear materials of the Ludwick type subjected to an end moment. Int. J. Non-Linear Mech. 17(1), 1–6 (1982)
Monasa, F., Lewis, G.: Large deflections of point loaded cantilevers with nonlinear behaviour. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 34(1), 124–130 (1983)
Ang, M.H., Wei, W., Teck-Seng, L.: On the estimation of the large deflection of a cantilever beam. In: Proceedings of the International Conference on Industrial Electronics, Control, and Instrumentation, IEEEIECON 1993 (1993)
Lee, K.: Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int. J. Non-Linear Mech. 37(3), 439–443 (2002)
Beléndez, T., Neipp, C., Beléndez, A.: Large and small deflections of a cantilever beam. Eur. J. Phys. 23, 371 (2002)
Dado, M., Al-Sadder, S.: A new technique for large deflection analysis of non-prismatic cantilever beams. Mech. Res. Commun. 32(6), 692–703 (2005)
Baykara, C., Guven, U., Bayer, I.: Large deflections of a cantilever beam of nonlinear bimodulus material subjected to an end moment. J. Reinf. Plast. Compos. 24(12), 1321–1326 (2005)
Belendez, T., et al.: Numerical and experimental analysis of large deflections of cantilever beams under a combined load. Phys. Scr. 2005, 61 (2005)
Al-Sadder, S., Al-Rawi, R.A.O.: Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings. Arch. Appl. Mech. 75(8), 459–473 (2006)
Wang, J., Chen, J.K., Liao, S.: An explicit solution of the large deformation of a cantilever beam under point load at the free tip. J. Comput. Appl. Math. 212(2), 320–330 (2008)
Shvartsman, B.S.: Large deflections of a cantilever beam subjected to a follower force. J. Sound Vib. 304(3–5), 969–973 (2007)
Changizi, M.A., Stiharu, I.: Sensitivity analysis of the non-linear deflection of non-straight AFM micro-cantilever. J. Adv. Microsc. Res. 7(1), 51–63 (2012)
Athisakul, C., et al.: Effect of material nonlinearity on large deflection of variable-arc-length beams subjected to uniform self-weight. Math. Probl. Eng. (2012). https://doi.org/10.1155/2012/345461
Changizi, M.A., Stiharu, I., Sahin, D.E.: A new approach for the non-linear analysis of the deflection of beams using lie symmetry groups. In: Proceedings of the International Conference of Mechatronics and Cyber-MixMechatronics, Bucharest, Romania, 9–11 September 2017. https://doi.org/10.1007/978-3-319-63091-5_17
Changizi, M.A.: Geometry and material nonlinearity effects on static and dynamics performance of MEMS. Ph.D. Thesis. Concordia University, Canada (2011)
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Changizi, M.A., Sahin, D.E., Stiharu, I. (2019). A Closed Form Solution for Non-linear Deflection of Non-straight Ludwick Type Beams Using Lie Symmetry Groups. In: Gheorghe, G. (eds) Proceedings of the International Conference of Mechatronics and Cyber-MixMechatronics – 2018. ICOMECYME 2018. Lecture Notes in Networks and Systems, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-319-96358-7_12
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