Francesco D’Annibale

Nonlinear hysteretic damping effects on the post-critical behaviour of the visco-elastic Beck’s beam:

The effects of nonlinear hysteretic damping on the post-critical behaviour of the visco-elastic Beck’s beam are discussed in this paper. The model consists of an inextensible and shear-undeformable cantilever beam, internally and externally damped, loaded at the free end by a follower force. Equations are derived in finite kinematics by taking as the configuration variable the rotation θ of the cross-section, instead of the usual deflection v of the beam axis, thus making the description of the contact internal actions simpler. The linear stability analysis is carried out and results of the literature are re-obtained. Then, a multiple-scale approach is developed in order to evaluate the nature, super- or subcritical, and the amplitude of the limit-cycle, occurring close to the Hopf critical load. The effects of the nonlinear damping on the amplitude of the limit-cycle are finally discussed for different linear damping coefficients.

Paradoxes in dynamic stability of mechanical systems: investigating the causes and detecting the nonlinear behaviors.

AbstractA critical review of three paradoxical phenomena, occurring in the dynamic stability of finite-dimensional autonomous mechanical systems, is carried out. In particular, the well-known destabilization paradoxes of Ziegler, due to damping, and Nicolai, due to follower torque, and the less well known failure of the so-called ‘principle of similarity’, as a control strategy in piezo-electro-mechanical systems, are discussed. Some examples concerning the uncontrolled and controlled Ziegler column and the Nicolai beam are discussed, both in linear and nonlinear regimes. The paper aims to discuss in depth the reasons of paradoxes in the linear behavior, sometimes by looking at these problems in a new perspective with respect to the existing literature. Moreover, it represents a first attempt to investigate also the post-critical regime.

Linear and Nonlinear Damping Effects on the Stability of the Ziegler Column

The destabilizing effect of damping on both linear and nonlinear behavior of the Ziegler column is discussed. The paper addresses classical and non-classical aspects related to the ‘Ziegler paradox’. First, the linear problem is illustrated in a new perspective, according to which no discontinuities in the critical load exist between undamped and damped systems. Second, it furnishes a first overview of the mechanical behavior of the system in the post-critical range. The equations of motion for the system are derived via the extended Hamilton’s principle. Then a linear stability analysis is performed via a perturbation approach, in which, however, simple and not double eigenvalues are perturbed, in contrast with a commonly pursued strategy in the literature. According to this idea, a series expansion around the distinct purely imaginary eigenvalues of the undamped and under-critically loaded system is carried out, with the load kept as a fixed, although unknown, parameter. By pursuing the same idea, an algorithm based on the Multiple Scale Method is developed to investigate the post-critical behavior of the system. The role played by the nonlinear damping on the existence of limit-cycles is discussed.

Controlling the Limit-Cycle of the Ziegler Column via a Tuned Piezoelectric Damper

This paper is about the nonlinear analysis of a piezoelectric controlled Ziegler column. The piezoelectric controller, here referred to as Tuned Piezoelectric Damper (TPD), possesses evanescent characteristics and, moreover, it is tuned to the first natural frequency of the mechanical system, thus resembling the well-known Tuned Mass Damper. This means that the flow of energy between mechanical and electrical subsystems is driven by the resonance (Den Hartog principle) and magnified by the singularity of the evanescent electrical characteristics. Numerical simulations, showing how the proposed control strategy is effective in increasing the linear stability domain and decreasing the amplitude of the limit-cycles in the postcritical range, are presented.

Flexural-Torsional Flutter and Buckling of Braced Foil Beams under a Follower Force

The flutter and buckling behavior of a cantilever foil beam, loaded at the tip by a follower force, are addressed in this paper. The beam is internally and externally damped and braced at the tip by a linear spring-damper device, which is located in an eccentric position with respect to beam axis, thus coupling the flexural and torsional behaviors. An exact linear stability analysis is carried out, and the linear stability diagram of the trivial rectilinear configuration is built up in the space of the follower load and spring’s stiffness parameters. The effects of the flexural-torsional coupling, as well as of the damping, on the flutter and buckling critical loads are discussed.