Abstract
The Multiple-Scale Method is applied directly to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded up to cubic terms in the transversal displacements and velocities of the beam. They are put in an operator form incorporating the mechanical boundary conditions, which account for the lumped viscoelastic device; the problem is thus governed by mixed algebraic-integro-differential operators. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence, Hopf and double-zero bifurcations. The spectral properties of the linear operator and its adjoint are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system is defective at the double-zero point, thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Preliminary numerical results are illustrated for the double-zero bifurcation.
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References
Steindl, A. and Troger, H., ‘Methods for dimension reduction and their application in nonlinear dynamics’, International Journal of Solids and Structures 38, 2001, 3131–2147.
Nayfeh, A. H., ‘Reduced-order models of weakly nonlinear spatially continuous systems’, Nonlinear Dynamics 16, 1998, 105–125.
Nayfeh, A. H. and Lacarbonara, W., ‘On the discretization of spatially continuous systems with quadratic and cubic nonlinearities’, JSME International Journal 41, 1998, 510–531.
Rega, G., Lacarbonara, W., Nayfeh, A. H., and Chin, C., ‘Multiple resonances in suspended cables: Direct versus reduced-order models’, International Journal of Non-Linear Mechanics 34, 1999, 901–924.
Rega, G., Lacarbonara, W., and Nayfeh, A. H., ‘Reduction methods for nonlinear vibrations of spatially continuous systems with initial curvature’, Solid Mechanics and Its Applications 77, 2000, 235–246.
Nayfeh, A. H., Arafat, H., Chin, C. M., and Lacarbonara, W., ‘Multimode interactions in suspended cables’, Journal of Vibration and Control 8, 2002, 337–387.
Luongo, A. and Paolone, A., ‘Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues’, Nonlinear Dynamics 14, 1997, 193–210.
Luongo, A. and Paolone, A., ‘Multiple scale analysis for divergence-Hopf bifurcation of imperfect symmetric systems’, Journal Sound and Vibration 218, 1998, 527–539.
Luongo, A., Paolone, A., and Di Egidio, A., ‘Multiple time scales analysis for 1:2 and 1:3 resonant Hopf bifurcations’, Nonlinear Dynamics 34(3–4), 2003, 269–281.
Luongo, A., Paolone, A., and Di Egidio, A., ‘Sensitivity and linear stability analysis around a double zero eigenvalues’, AIAA Journal 38(4), 2000, 702–710.
Luongo, A., Di Egidio, A., and Paolone, A., ‘Multiple time scale analysis for bifurcation from a multiple-zero eigenvalue’, AIAA Journal 41(6), 2003, 1143–1150.
Luongo, A., Di Egidio, A., and Paolone, A., ‘Multiple scale bifurcation analysis for finite-dimensional autonomous systems’, in Recent Research Developments in Sound & Vibration, Transworld Research Network, Kerala, India, Vol. 1, 2002, pp. 161–201.
Nayfeh, A. H., Problems in Perturbation, Wiley, New York, 1985.
Kirillov, O. N. and Seyranian, A. P., ‘Collapse of Keldysh chain and stability of nonconservative systems’, Doklady Mathematics 66(1), 2002, 127–131 (Translated from Doklady Mathematics Nauk 385(2), 2002, 172–176).
Kirillov, O. N. and Seyranian, A. P., ‘Solution to the Herrmann–Smith problem’, Doklady Physics 47(10), 2002, 767–771 (from Doklady Akademii Nauk 386(6), 2002, 761–766).
Keldysh, M. V., ‘On eigenvalues and eigenfunctions of certain classes of not self-adjoint equations’, Doklady Akademii Nauk SSRR 77(1), 1951, 11–14.
Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981.
Nayfeh, A. H., ‘Topical Course on Nonlinear Dynamics’, Società Italiana di Fisica, Santa Margherita di Pula, Sardinia, Perturbation Methods in Nonlinear Dynamics, 1985.
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LUONGO, A., EGIDIO, A.D. Bifurcation Equations Through Multiple-Scales Analysis for a Continuous Model of a Planar Beam. Nonlinear Dyn 41, 171–190 (2005). https://doi.org/10.1007/s11071-005-2804-1
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DOI: https://doi.org/10.1007/s11071-005-2804-1