Hamiltonian modelling and buckling analysis of a nonlinear flexible beam with actuation at the bottom

Authors

Megha V. Trivedi, Ravi N. Banavar, Paul Kotyczka

Abstract

ABSTRACT The use of beams and similar structural elements is finding increasing application in many areas including micro and nanotechnology devices. For the purpose of buckling analysis and control, it is essential to account for nonlinear terms in the strains while modelling these flexible structures. Further, the Poisson’s effect can be accounted in modelling by the use of a two-dimensional stress–strain relationship. This paper studies the buckling effect for a slender, vertical beam (in the clamped-free configuration) with horizontal actuation at the fixed end and a tip-mass at the free end. Including also the inextensibility constraint of the beam, the equations of motion are derived. A preliminary modal analysis of the system has been carried out to describe candidate post-buckling configurations and study the stability properties of these equilibria. The vertical configuration of the beam under the action of gravity is without loss of generality, since the objective is to model a potential field that determines the equilibria. Neglecting the inextensibility constraint, the equations of motion are then casted in port-Hamiltonian form with appropriately defined flows and efforts as a basis for structure-preserving discretization and simulation. Finally, the finite-dimensional model is simulated to obtain the time response of the tip-mass for different loading conditions.

Citation

• Journal: Mathematical and Computer Modelling of Dynamical Systems
• Year: 2016
• Volume: 22
• Issue: 5
• Pages: 475–492
• Publisher: Informa UK Limited
• DOI: 10.1080/13873954.2016.1201517

BibTeX

@article{Trivedi_2016,
  title={{Hamiltonian modelling and buckling analysis of a nonlinear flexible beam with actuation at the bottom}},
  volume={22},
  ISSN={1744-5051},
  DOI={10.1080/13873954.2016.1201517},
  number={5},
  journal={Mathematical and Computer Modelling of Dynamical Systems},
  publisher={Informa UK Limited},
  author={Trivedi, Megha V. and Banavar, Ravi N. and Kotyczka, Paul},
  year={2016},
  pages={475--492}
}

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References

• Meressi, T. & Paden, B. Buckling control of a flexible beam using piezoelectric actuators. Journal of Guidance, Control, and Dynamics vol. 16 977–980 (1993) – 10.2514/3.21113
• Thompson, S. P. & Loughlan, J. The active buckling control of some composite column strips using piezoceramic actuators. Composite Structures vol. 32 59–67 (1995) – 10.1016/0263-8223(95)00048-8
• Wang, Q. S. Active buckling control of beams using piezoelectric actuators and strain gauge sensors. Smart Materials and Structures vol. 19 065022 (2010) – 10.1088/0964-1726/19/6/065022
• MA, H., GAO, X. & REDDY, J. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids vol. 56 3379–3391 (2008) – 10.1016/j.jmps.2008.09.007
• Reddy, J. N. & Mahaffey, P. Generalized beam theories accounting for von Kármán nonlinear strains with application to buckling. Journal of Coupled Systems and Multiscale Dynamics vol. 1 120–134 (2013) – 10.1166/jcsmd.2013.1006
• Patil, O. & Gandhi, P. On the Dynamics and Multiple Equilibria of an Inverted Flexible Pendulum With Tip Mass on a Cart. Journal of Dynamic Systems, Measurement, and Control vol. 136 (2014) – 10.1115/1.4026831
• Osita D.. The Mechanical Systems Design Handbook: modelling, Measurement, and Control (2001)
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
• Macchelli, A. & Melchiorri, C. Control by Interconnection and Energy Shaping of the Timoshenko Beam. Mathematical and Computer Modelling of Dynamical Systems vol. 10 231–251 (2004) – 10.1080/13873950412331335243
• Banavar, R. & Dey, B. Stabilizing a Flexible Beam on a Cart: A Distributed Port-Hamiltonian Approach. Journal of Nonlinear Science vol. 20 131–151 (2009) – 10.1007/s00332-009-9054-1
• Reddy J.N.. An Introduction to Nonlinear Finite Element Analysis: with Applications to Heat Transfer, Fluid Mechanics, and Solid Mechanics (2015)
• Advanced Materials Research vol. 745 (2013) – 10.4028/www.scientific.net/amr.745
• Craig R.R.. Fundamentals of Structural Dynamics (2006)
• Friswell, M. I. et al. Non-linear piezoelectric vibration energy harvesting from a vertical cantilever beam with tip mass. Journal of Intelligent Material Systems and Structures vol. 23 1505–1521 (2012) – 10.1177/1045389x12455722
• Khalil H.K.. Nonlinear Systems (1996)
• Nayfeh, A. H. & Emam, S. A. Exact solution and stability of postbuckling configurations of beams. Nonlinear Dynamics vol. 54 395–408 (2008) – 10.1007/s11071-008-9338-2
• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Moulla, R., Lefévre, L. & Maschke, B. Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics vol. 231 1272–1292 (2012) – 10.1016/j.jcp.2011.10.008
• Schöberl, M. & Siuka, A. Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators. Automatica vol. 50 607–613 (2014) – 10.1016/j.automatica.2013.11.035