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Euler‐Bernoulli Beam with Boundary Control: Stability and FEM
Author(s) -
Miletic Maja,
Arnold Anton
Publication year - 2011
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201110330
Subject(s) - dissipative system , exponential stability , cantilever , passivity , finite element method , euler's formula , controller (irrigation) , beam (structure) , piecewise , boundary (topology) , mathematical analysis , mathematics , control theory (sociology) , ode , boundary value problem , timoshenko beam theory , physics , computer science , engineering , control (management) , structural engineering , agronomy , electrical engineering , optics , nonlinear system , quantum mechanics , artificial intelligence , biology , thermodynamics
We consider a model for the time evolution of a piezoelectric cantilever with tip mass. With appropriately shaped actuator and sensor electrodes, boundary control is applied and a passivity based feedback controller is designed to include damping into the system. Assuming that the cantilever can be modeled by the Euler‐Bernoulli beam equation, we obtain a coupled PDE‐ODE system. First we discuss its dissipativity, and its asymptotic but non ‐exponential stability. Next we derive a FEM using piecewise cubic Hermitian shape functions that is still dissipative. This is illustrated on a numerical simulation. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)