Can a semi-simple eigenvalue admit fractional sensitivities?

Luongo, Angelo; Ferretti, Manuel

doi:10.1016/j.amc.2014.01.178

Cited by 13 publications

(10 citation statements)

References 16 publications

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“…Since λ • must be non-defective (semi-simple), it can be proved that C µ = 0 [48]. Consequently, the 1 -order equation (31) is trivially satisfied, but leaves λ undetermined.…”

Section: Accepted Manuscriptmentioning

confidence: 95%

High-frequency parametric approximation of the Floquet-Bloch spectrum for anti-tetrachiral materials

Bacigalupo

Lepidi

2016

International Journal of Solids and Structures

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The class of anti-tetrachiral cellular materials is phenomenologically characterized by a strong auxeticity of the elastic macroscopic response. The auxetic behavior is activated by rolling-up deformation mechanisms developed by the material microstructure, composed by a periodic pattern of stiff rings connected by flexible ligaments. A linear beam lattice model is formulated to describe the free dynamic response of the periodic cell, in the absence of a soft matrix. After a static condensation of the passive degrees-of-freedom, a general procedure is applied to analyze the wave propagation in the low-dimensional space of the active degrees-of-freedom. The exact dispersion functions are compared with explicit – although approximate – dispersion relations, obtained from asymptotic perturbation solutions of the eigenproblem governing the Floquet–Bloch theory. A general hierarchical scheme is outlined to formulate and solve the perturbation equations, taking into account the dimension of the perturbation vector. Original recursive formulas are presented to achieve any desired order of asymptotic approximation. For the anti-tetrachiral material, the fourth-order asymptotic solutions are found to approximate the dispersion curves with fine agreement over wide regions of the parameter space. The asymptotic eigensolutions allow an accurate sensitivity analysis of the material spectrum under variation of the key physical parameters, including the cell aspect ratio, the ligament slenderness and the spatial ring density. Finally, the explicit dependence of the dispersion functions on the mechanical parameters may facilitate the custom design of specific spectral properties, such as the wave velocities and band gap amplitudes

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“…Since λ • must be non-defective (semi-simple), it can be proved that C µ = 0 [48]. Consequently, the 1 -order equation (31) is trivially satisfied, but leaves λ undetermined.…”

Section: Accepted Manuscriptmentioning

confidence: 95%

High-frequency parametric approximation of the Floquet-Bloch spectrum for anti-tetrachiral materials

Bacigalupo

Lepidi

2016

International Journal of Solids and Structures

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“…Said in other words, passive control strategies were designed in such a way that the Hopf critical load of the controlled system could be greater than that of the uncontrolled one. By studying the sensitivity analysis of the eigenvalues, carried out via ad hoc perturbation methods [4,[57][58][59], the following three controllers, differing each other with respect the demand or not of the resonance condition, and for the order of magnitude of the electrical coefficients, were detected (ε small parameter): …”

Section: Controlled Linearly Damped Systemsmentioning

confidence: 99%

Piezoelectric control of the Hopf bifurcation of Ziegler’s column with nonlinear damping

D’Annibale

2016

Nonlinear Dyn

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Linear and nonlinear analyses of a piezoelectric-controlled Ziegler's column, endowed with a Van der Pol-like nonlinear damping, are carried out in the present paper. The effects of three linear and passive piezoelectric controllers on the position and on the amplitude of the limit-cycle at the Hopf bifurcation, triggered by the follower force, are investigated. The controllers, previously introduced in the literature and here referred to as Non-Singular Resonant, Singular Non-Resonant, and Tuned Piezoelectric Damper, resemble well-known devices, adopted for controlling both linear and nonlinear oscillations, namely, a largemass Tuned Mass Damper, a singular Energy Sink, and a classical small-mass Tuned Mass Damper. Numerical simulations on a linearly and uniformly damped column, under different nonlinear damping conditions, are carried out to show the effectiveness of the controllers in reducing the amplitude of the limit-cycle and their behavior with respect to the possible occurrence of the hard loss bifurcation.

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“…It is shown that for a double semi-simple imaginary eigenvalue the region of flutter instability lies inside a cone with the apex at the singular point [14]. More recently, it has been shown that when the parameters move along the stability boundary the second approximation must also be considered [15].…”

Section: Introductionmentioning

confidence: 99%

On the stability and instability of linear potential systems subjected to infinitesimal circulatory forces

Bulatović

Udwadia

2021

Proc. R. Soc. A.

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The stability of linear multi-degree-of-freedom stable potential systems with multiple natural frequencies under the action of infinitesimal circulatory forces is considered. Contrary to the received view that such systems are inherently unstable, a careful study shows that such systems have a much more complex behaviour than previously recognized and could exhibit an alternation of stability and instability that depends on the structure of the potential system and its interaction with the circulatory forces. The conditions under which stability or instability ensues and the nature of this alternation in stability are explicitly obtained. In low-dimensional stable potential systems, when the coefficients of the circulatory forces are proportional to an arbitrarily small scalar parameter, all the circulatory forces that cause flutter instability are described.

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Can a semi-simple eigenvalue admit fractional sensitivities?

Cited by 13 publications

References 16 publications

High-frequency parametric approximation of the Floquet-Bloch spectrum for anti-tetrachiral materials

High-frequency parametric approximation of the Floquet-Bloch spectrum for anti-tetrachiral materials

Piezoelectric control of the Hopf bifurcation of Ziegler’s column with nonlinear damping

On the stability and instability of linear potential systems subjected to infinitesimal circulatory forces

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