. 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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SUMMARYIn this paper the influence of base isolation on the behaviour of a work of art has been analysed. To make things more realistic, the work of art has been modelled with a non-symmetrical rigid body, sitting on a base that is connected to a visco-elastic device, which represents the passive control system. To prevent the breaking of the isolation device, security stops have been introduced to limit the displacement of the oscillating base to a maximum safety value. All analyses have been carried out comparing the behaviour of the non-isolated and the isolated non-symmetric rigid body subject to impulsive and seismic excitations. The analysis is particularly focused on the effects of the eccentricity of the rigid body and on the presence of the security stops. Generally, base isolation improves the behaviour of the system while the presence of an eccentricity makes the performance of the system worse with respect to the symmetric rigid body. Moreover the security stops, although they preserve the isolator devices, cause a worsening in the performance of the systems.

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.

An adapted version of the Multiple Scale Method is formulated to analyze 1:1 resonant multiple Hopf bifurcations of discrete autonomous dynamical systems, in which, for quasi-static variations of the parameters, an arbitrary number m of critical eigenvalues simultaneously crosses the imaginary axis. The algorithm therefore requires discretizing continuous systems in advance. The method employs fractional power expansion of a perturbation parameter, both in the state variables and in time, as suggested by a formal analogy with the eigenvalue sensitivity analysis of nilpotent (defective) matrices, also illustrated in detail. The procedure leads to an order-m differential bifurcation equation in the complex amplitude of the unique critical eigenvector, which is able to capture the dynamics of the system around the bifurcation point. The procedure is then adapted to the specific case of a double Hopf bifurcation (m = 2), for which a step-by-step, computationally-oriented version of the method is furnished that is directly applicable to solve practical problems. To illustrate the algorithm, a family of mechanical systems, subjected to aerodynamic forces triggering 1:1 resonant double Hopf bifurcations is considered. By analyzing the relevant bifurcation equation, the whole scenario is described in a three-dimensional parameter space, displaying rich dynamics. (C) 2004 Elsevier Ltd. All rights reserved

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