Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

Luongo, Angelo; Paolone, Achille; Egidio, Angelo Di

doi:10.1023/b:nody.0000013508.50435.39

Cited by 59 publications

(37 citation statements)

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“…approach, the authors have systematically applied the multiple scale method to analyze a number of bifurcations of linear codimension-one, two and three, to general finite dimensional systems [13][14][15][16][17]; the review paper [18] resumes their main results. More recently, they have extended the method to infinite dimensional system, to analyze divergence, Hopf and doublezero bifurcations [19][20][21][22].…”

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confidence: 99%

Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

Egidio

Luongo

Paolone

2007

International Journal of Non-Linear Mechanics

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. Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams. International Journal of Non-Linear Mechanics, Elsevier, 2007, 42 (1), pp.88-98. approach, the authors have systematically applied the multiple scale method to analyze a number of bifurcations of linear codimension-one, two and three, to general finite dimensional systems [13][14][15][16][17]; the review paper [18] resumes their main results. More recently, they have extended the method to infinite dimensional system, to analyze divergence, Hopf and doublezero bifurcations [19][20][21][22]. The method is based on the direct treatment of the original (integro)-differential equations, avoiding any a priori discretization (as that, e.g., performed in [3]), according to the so-called direct method, widely applied by Nayfeh and co-workers [23][24][25], and many other authors (see, e.g., [26,27]), to several problems of non-linear dynamics.In this paper the algorithm is applied to study a cantilever beam, constrained by a spring and two dashpot, loaded by a follower force. This system, for its simplicity, was already studied in [19,20] as an example of structure undergoing divergence, Hopf and double-zero bifurcation. A deeper parametric analysis performed on the system permitted to reveal a richer bifurcation scenario, including Hopf-divergence (HD) and resonant and non-resonant double-Hopf, not discovered in the previous analysis. Therefore, the beam viscous-elastically restrained could be taken as paradigmatic system undergoing all the low-codimension bifurcations of mechanical interest. Here, the analysis of the codimension-two bifurcations is completed, while the codimension-three one (resonant Hopf-Hopf (HH)) is left for future investigation.The paper is thus organized. In Section 2 the equations of motions are given, and detailed in Appendix A, where they are derived through a procedure alternative to that of Ref. [19]. Moreover the linear adjoint problem is defined. In Section 3 the critical scenario is depicted. Considerably emphasis is given to the influence of the relative magnitude of the two dashpots on bifurcations. The well-known destabilizing effect of damping is detected, and a new paradoxical result discovered. In Section 4 the post-critical analysis is carried-out. The bifurcation equations are first derived and then numerically studied to built-up equilibrium paths and bifurcation diagrams. Finally, in Section 5 some conclusions are drawn. ModelA planar beam is considered, fixed at end A and constrained by a linear visco-elastic device at end B, loaded at the tip by a follower force of intensity P (Fig. 1). The device consists of an extensional spring of stiffness k e and two dashpots of constants c e and c t , of extensional and torsional type, respectively. The beam is assumed to be inextensible and shear-undeformable, so that bending is the unique strain measure. The equations of motionThe equations of motion of the beam were derived in [19] through a variational approach by elimi...

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Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

Egidio

Luongo

Paolone

2007

International Journal of Non-Linear Mechanics

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“…In particular, the topics that can be recognized in these papers concern the investigation of: (a) divergenceHopf bifurcation [56][57][58], (b) double-zero and multiple-zero bifurcations [59,60], (c) simple Hopf bifurcation [61], multiple (non-defective) Hopf bifurcation [62,63] and multiple (defective) Hopf bifurcation, [64], (d) theoretical and technical [65,66], and algorithmic [67] aspects in perturbation methods.…”

Section: Methodsmentioning

confidence: 99%

“…3. The paper [63] deals with the study of the post-critical behavior of a general N-dimensional system around a 1:2 or 1:3 resonant double Hopf bifurcation. To this end, the MSM is employed in order to obtain, in closed form, the bifurcation equations in terms of the derivatives of the original vector field evaluated at the bifurcation point.…”

Section: In [62] a Two-time Version Of The Lindstedt-poincaré Methods mentioning

confidence: 99%

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

Piccardo

D’Annibale

Zulli

2014

Continuum Mech. Thermodyn.

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This paper intends to summarize the scientific production of Angelo Luongo on the occasion of his Sixtieth Birthday, focusing on his main contributions in the field of Mechanics. The task will not be easy because of the breadth of his scientific production, only apparently attributable to a restricted number of keywords. In fact, even when the work seems purely algorithmic, speculation on the physical and mechanical aspects of the problem is always present, providing new interpretations and innovative openings to a careful reader. Similarly, also the works, which apparently seem to be high-level applications, always reserve methodological aspects that are not negligible. The editorial choice to divide his papers through a small number of keywords is certainly simplistic, but offers the possibility to better highlight all the connections among his variegated production. The most original contributions of Angelo Luongo in the context of perturbation methods, linear and nonlinear dynamics and control, elastic buckling and structural analysis, bifurcation and stability of non-conservative systems, are discussed in detail. Finally, the Angelo Luongo's central role in the creation and development of activities of the international research center M&MoCS is pointed out.

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“…As a future advancement to this proposed equation, perturbation methods, such as the two-time Lindsted-Poincaré method [58] or the multiple scale method [12,13,[58][59][60] could be applied to tackle the nonlinear problem in an analytical way, thus completing and enriching the investigation. The results within this paper, even though, are worthy of further and deeper investigation, provide unique findings and alternative routes to chaos compared with the single vertically forced case.…”

Section: Final Remarksmentioning

confidence: 99%

Dynamics of a Parametrically Excited System with Two Forcing Terms

Sofroniou

Bishop

2014

Mathematics

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Motivated by the dynamics of a trimaran, an investigation of the dynamic behaviour of a double forcing parametrically excited system is carried out. Initially, we provide an outline of the stability regions, both numerically and analytically, for the undamped linear, extended version of the Mathieu equation. This paper then examines the anticipated form of response of our proposed nonlinear damped double forcing system, where periodic and quasiperiodic routes to chaos are graphically demonstrated and compared with the case of the single vertically-driven pendulum.

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Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

Cited by 59 publications

References 15 publications

Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

Dynamics of a Parametrically Excited System with Two Forcing Terms

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