A mathematical model for linear elastic systems with structural damping

Abstract

Abstract. We present a mathematical model exhibiting the empirically observed damping rates in elastic systems. The models studied are of the form (A the relevant elasticity operator)x + Bx + Ax = 0 with B related in various ways to the positive square root, A1'2, of A. Comparison with existing "ad hoc" models is made.

Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers

“…Let us position ten transducers constituted by piezoceramic patches, produced by PiezoSystem, made of lead zirconate titanate [PZT]. The characteristics of these piezoceramic * * * * We refer to Reference [31] for the (rather involved) characterization of this class of boundary conditions. This is a delicate issue: we simply remark that the square root operator of the fourth-order derivative in the case of boundary conditions relative to a cantilever beam, is not a di erential operator.…”

“…The mathematical properties of the square root of the fourth derivative operator have been extensively studied in References [31,32]. In Reference [31], such an operator is introduced to rigorously describe empirically observed damping rates in vibrating beams.…”

“…In Reference [31], such an operator is introduced to rigorously describe empirically observed damping rates in vibrating beams. While, in Reference [32], its mathematical properties are deeply analysed, and it is shown that this operator…”

Spectralizable Operators

“…We mention here only [8,12,19,23,28] The aim of our work is the investigation of the class of spectralizable operators. Spectralizable operators arise in many different problems, see, e.g., [7,14,15,16,31,32,33]. We mention that the term spectralizable was used first in [30] in a very special setting, where, in particular, one can find an example of a non-self-adjoint difference operator with a self-adjoint square.…”

Parabolicity of a class of higher order abstract differential equations

“…But under what condition is equation (1.2) parabolic? For second-order equations, this problem has been studied by many authors (see, e.g., [1][2][3][5][6][7][8][9] and references therein); here (1.2) amounts to (1.5) u"(t) + Axu'(t) + A2u(t) = 0, and (1.3) to…”