. 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...

No abstract

SUMMARYThe governing equations for dynamic transverse motion of a cable-stayed beam are obtained by means of a classical variational formulation. The analytical model permits a parametric investigation of linear and non-linear behaviour in a family of cable-stayed beam systems. Analytical eigensolutions of the linearized equations are used to investigate how the mechanical characteristics in uence the occurrence of global, local and coupled modes. The exact eigenfunctions are assumed to describe the forced harmonic motion in the neighbourhood of a selected frequency. The frequency-amplitude relationship, obtained by the use of the multiple scale method, permits the description of softening and hardening behaviour in the global, local and coupled classes of motion.

Abstract. The postcritical behavior of a general n-dimensional system around a resonant double Hopf bifurcation is analyzed. Both cases in which the critical eigenvalues are in ratios of 1:2 and 1:3 are investigated. The Multiple Scale Method is employed to derive the bifurcation equations systematically in terms of the derivatives of the original vector field evaluated at the critical state. Expansions of the n-dimensional vector of state variables and of a three-dimensional vector of control parameters are performed in terms of a unique perturbation parameter ε, of the order of the amplitude of motion. However, while resonant terms only appear at the ε 3 -order in the 1:3 case, they already arise at the ε 2 -order in the 1:2 case. Thus, by truncating the analysis at the ε 3 -order in both cases, first or second-order bifurcation equations are respectively drawn, the latter requiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations is studied. The complete postcritical scenario is analyzed in terms of the three control parameters and the asymptotic results are compared with exact numerical integrations for both resonances. Branches of periodic as well as periodically modulated solutions are found and their stability analyzed.

The classical Lindstedt-Poincare method is adapted to analyze the nonlinear normal modes of a piecewise linear system. A simple two degrees-of-freedom, representing a beam with a breathing crack is considered. The fundamental branches of the two modes and their stability are drawn by varying the severity of the crack, i.e., the level of nonlinearity. Results furnished by the asymptotic method give insight into the mechanical behavior of the system and agree well with numerical results; the existence of superabundant modes is proven. The unstable regions and the bifurcated branches are followed by a numerical procedure based on the Poincare map

No abstract

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

No abstract