A rod‐like piezoelectric controller for the improvement of the visco‐elastic Beck's beam linear stability

Abstract

The linear stability of a cantilever beam subject to a follower force is investigated. A system of distributed piezoelectric devices is attached to the beam with the purpose of improving the dynamic stability of the structure. They are connected to a second-order network and second-order analog electrical circuit. The assembly of the mechanical and piezoelectric subsystems constitutes a coupled Piezo-Electro-Mechanical (PEM) system, whose equations of motion, derived within a variational approach, are discretized via the Galerkin weighted residual method and the stability of the trivial equilibrium is addressed by numerically solving the associated eigenvalue problem. A sensitivity analysis, carried out on a numerical ground, is thus conducted on a wide range of the electrical parameters to investigate the effectiveness of the proposed controller.

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

“…( 3) and applying the fundamental lemma of the calculus of variations, the necessary condition related to the thermal variable, namely, the entropy displacement s provides the Cattaneo law in the following form: alongside the thermal natural boundary conditions related to the portion of the boundary where the temperature is applied B , and, similarly to the mechanical displacement, we can also give the entropy displacement as a thermal essential boundary condition Equation ( 35) is significant since it tells us that the dual of the displacement, i.e., f i , defined in (24), is connected not only with the stress tensor, as in the non-dissipative case, but also to the viscous tensor D and the strain rate. Equation ( 24) defines also the dual of the entropy displacement Θ ext n , that is not only assumed to be parallel to the boundary normal (36)…”

“…The use of a variational principle is a highly effective and safe conceptual tool for developing complex theories regarding generalized continua [21][22][23][24][25][26][27][28], or approximate theories of structural components, alongside natural boundary conditions [29]. Additionally, it may serve as a preparatory step towards the establishment of a comprehensive thermo-mechanical formulation, a synthetic viewpoint to facilitate the proof of many mathematical theorems, and the theoretical basis for the application of modern numerical techniques like, for instance, the finite element method [30][31][32][33][34][35][36][37][38]. Variational principles are a widely accepted method of organizing information about dynamic systems and developing their evolution equations [7,[39][40][41][42][43][44][45][46].…”