Nonlinear Stability and Bifurcation Theory

Tröger, H.; Steindl, Alois

doi:10.1007/978-3-7091-9168-2

Cited by 241 publications

(111 citation statements)

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“…However, the numerical values that they assume at the bifurcation points studied ahead (Section 4.2), are given in the Appendix C. Eqs. (17) constitute the bifurcation equations, in standard normal form and specialized to a symmetric system, for two pairs of non-resonant purely imaginary critical eigenvalues, or for a one zero and a purely imaginary pair of critical eigenvalues [11]. They are invariant under the transformations a 1 → −a 1 and/or a 2 → −a 2 .…”

Section: Bifurcation Equationsmentioning

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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Section: Bifurcation Equationsmentioning

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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“…The non-passive systems that will be studied are linear mechanical systems with a non-positive definite damping matrix with additional dry friction elements. The non-positivedefiniteness of the damping matrix of linearised systems can be caused by fluid, aeroelastic, control and gyroscopical forces, which can cause instabilities such as flutter vibrations of airfoils [14], shimmying of wheels in vehicle systems [15] or flutter instabilities of fluidconveying tubes [16]. It will be demonstrated in this paper that the presence of dry friction in such an unstable linear system can (conditionally) ensure the local attractivity of the equilibrium set of the resulting system with dry friction.…”

Section: Introductionmentioning

confidence: 99%

Attractivity of Equilibrium Sets of Systems with Dry Friction

Wouw

Leine²

2004

Nonlinear Dynamics

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Abstract. The dynamics of mechanical systems with dry friction elements, modelled by set-valued force laws, can be described by differential inclusions. An equilibrium set of such a differential inclusion corresponds to a stationary mode for which the friction elements are sticking. The attractivity properties of the equilibrium set are of major importance for the overall dynamic behaviour of this type of systems. Conditions for the attractivity of the equilibrium set of MDOF mechanical systems with multiple friction elements are presented. These results are obtained by application of a generalisation of LaSalle's principle for differential inclusions of Filippov-type. Besides passive systems, also systems with negative viscous damping are considered. For such systems, only local attractivity of the equilibrium set can be assured under certain conditions. Moreover, an estimate for the region of attraction is given for these cases. The effectiveness of the results is illustrated by means of both 1DOF and MDOF examples.

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“…This result contrasts with some statements existing in the literature about this question. 11,12 6) A mechanical two-DOF system was studied as an example. It was found that the non-singular perturbation expansion furnishes a good approximation almost everywhere in the parameter space, except in the neighborhood of the subspace tangent to the divergence manifold.…”

Section: Discussionmentioning

confidence: 99%