We analyse a continuous Cosserat model of a visco-elastic rod subjected to a combination of a conservative load and a follower term in one of the ends. The formalism takes into account the geometric non linearities that appear for large deformations from the straight solution. The resulting equation is a PDE whose solution can be analysed for special cases and the eigenvalues of the linearisation can be computed by a combination of numerical continuation and bifurcation results. We illustrate this method analysing the bifurcations of the straight solution of a rod subjected to a follower force. The dynamics and the bifurcation behaviour of the model are explored as the intensity of force is varied and as the mixture of conservative-follower terms varies continuously from the standard conservative case to the purely follower one. Special attention is paid to the corresponding transition from symmetry breaking pitchfork bifurcation (falling over mode) to the appearance of oscillations in a Hopf like bifurcation (if some material damping is included) or pure reversible Hamiltonian Hopf bifurcation in the absence of damping. After the onset of oscillations a complex dynamical behaviour frequently called flutter instability appears. The study is supplemented with the bifurcation analysis of the two elastically jointed follower pendula model as a simplified model of the continuous problem.
- Design Engineering Division and Computers and Information in Engineering Division
Stability and Bifurcations in a Model of a Follower Loaded Rod
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Gala´n-Vioque, J, Lago, MA, & Valverde, J. "Stability and Bifurcations in a Model of a Follower Loaded Rod." Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 4: 8th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A and B. Washington, DC, USA. August 28–31, 2011. pp. 51-58. ASME. https://doi.org/10.1115/DETC2011-47441
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