Manuel Ferretti

Can a semi-simple eigenvalue admit fractional sensitivities?

We perform high-order sensitivity analysis of eigenvalues and eigenvectors of linear systems depending on parameters. Attention is focused on double not-semi-simple and semi-simple eigenvalues, undergoing perturbations, either of regular or singular type. The use of integer (Taylor) or fractional (Puiseux) series expansions is discussed, and the analysis carried out both on the characteristic polynomial and on the eigenvalue problem. It is shown that semi-simple eigenvalues can admit fractional sensitivities when the perturbations are singular, conversely to the not-semi-simple case. However, such occurrence only manifests itself when a second-order perturbation analysis is carried out. As a main result, it is found that such over-degenerate case spontaneously emerges in bifurcation analysis, when one looks for the boundaries of the stability domain of circulatory mechanical systems possessing symmetries. A four degree-of-freedom system under a follower force is studied as an illustrative example.

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.

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.

Bias Extension Test for In-Plane Shear Properties during Forming - Use at High Temperature and Limits of the Test

The results of in-plane shear tests performed on 5-hardness satin woven carbon/PPS thermoplastic prepregs are described. The experimental analyses are based on bias-extension tests performed in an environmental chamber. The results are given for different temperatures on both side of the melting point. This range of temperature is those of the part during a thermoforming process. In another hand it is shown that second-gradient energy terms allow for an effective prediction of the onset of internal shear boundary layers which are transition zones between two different shear deformation modes. The existence of these boundary layers cannot be described by a simple first-gradient model.