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

Abstract

. 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. <hal-00790054> 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...

Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams

“…The quantity F c is known to be the critical load for a clamped-free Euler beam (the historical reference is [1]; for interesting recent developments see also [34][35][36][37][38]). The previous formula provides the correction to be applied for a Timoshenko beam.…”

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

“…In [78] the main steps of the multiple scale analysis, detailed in [77], are resumed in a more problem-oriented form; moreover, a deep numerical investigation, which leads to several new results, concerning both the linear stability and the post-critical scenario around the three different bifurcation points, is developed. Finally, a deeper parametric analysis, performed in [79], revealed a richer bifurcation scenario, thus including Hopf-divergence and both resonant and non-resonant double Hopf, not discovered in the two previous papers. Then, the bifurcation diagrams, illustrating the system behavior around these critical points of the parameter space, are discussed.…”

“…In this Section, some results, concerning the papers [77][78][79][80][81][82][83][84], which are devoted to apply on beam models the perturbation methods discussed above, are briefly synthesized.…”

“…In a group of three papers, namely [77][78][79], the MSM is applied to a one-dimensional continuous cantilevered planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. It has been shown that the beam model here addressed can be considered as a paradigmatic system undergoing all the low codimension bifurcations of mechanical interest.…”

Linear Stability Analysis of Multiparameter Dynamical Systems via a Numerical-Perturbation Approach

“…5b (system 4), obtained by the previous one by removing all distributed damping and lumping it at the tip of the beam by means of two linear dashpots of constants C e and C t of extensional and torsional types, respectively. System 4 was chosen to test the algorithm in a difficult case, since it was shown by analytical methods in [11] that this beam exhibits a rich bifurcation scenario containing several bifurcation points of different natures close to each other.…”