On the extensible viscoelastic beam

This work is focused on the equation \ptt u+\partial_{xxxx}u +\int_0^\infty \mu(s) \partial_{xxxx}[u(t)-u(t-s)]ds- \big(\beta+\|\partial_x u\|_{L^2(0,1)}^2\big)\partial_{xx}u= f describing the motion of an extensible viscoelastic beam. Under suitable boundary conditions, the related dynamical system in the history space framework is shown to possess a global attractor of optimal regularity. The result is obtained by exploiting an appropriate decomposition of the solution semigroup, together with the existence of a Lyapunov functional

Longtime behavior for oscillations of an extensible viscoelastic beam with elastic external supply

This work is focused on a nonlinear equation describing the oscillations of an extensible viscoelastic beam with fixed ends, subject to distributed elastic external force. For a general axial load $\beta$, the existence of a finite/infinite set of stationary solutions and buckling occurrence are scrutinized. The exponential stability of the straight position is discussed. Finally, the related dynamical system in the history space framework is shown to possess a regular global attractor.Comment: 16 pages, 2 figure

Buckling and longterm dynamics of a nonlinear model for the extensible beam

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Buckling and longterm dynamics of a nonlinear model for the extensible beam

This work is focused on the longtime behavior of a non linear evolution problem describing the vibrations of an extensible elastic homogeneous beam resting on a viscoelastic foundation with stiffness k>0 and positive damping constant. Buckling of solutions occurs as the axial load exceeds the first critical value, \beta_c, which turns out to increase piecewise-linearly with k. Under hinged boundary conditions and for a general axial load P, the existence of a global attractor, along with its characterization, is proved by exploiting a previous result on the extensible viscoelastic beam. As P<\beta_c, the stability of the straight position is shown for all values of k. But, unlike the case with null stiffness, the exponential decay of the related energy is proved if P<\bar\beta(k), where \bar\beta(k) < \beta_c(k) and the equality holds only for small values of k.Comment: 14 pages, 2 figure

Perturbation methods and proper orthogonal decomposition analysis for nonlinear aeroelastic systems

The modern engineering deals with applications of high complexity. From a mathematical point of view such a complexity means a large number of degrees of freedom and nonlinearities in the equations describing the process. To approach this difficult problem there are two levels of simplification. The first level is a physical reduction: the real problem is represented by mathematical models that are treated in order to be studied and their solution computed. At this level we can find all the discretization techniques like Galerkin projection or Finite Element Methods. The second level is a simplification of the original problem in order to study it in an easier way: a reduced order model is advocated. Simplification means to determine a dominant dynamics which drives the whole problem: not all the unknowns are considered independent being some of them functions of the remaining others. Two methodologies are considered in this a Thesis. The first is the Lie Transform Method based on the results of Normal Form Theory and the Center Manifold Theorem. For some conditions, called resonance or zero divisors, depending on combinations of the eigenvalues of the linearized system, the nonlinearity of the problem is reduced and a driving dynamics determined. The second is the Proper Orthogonal Decomposition (POD), which from the analysis of representative time responses of the original problem determines a subspace of state variables energetically significant spanned by the Proper Orthogonal Modes (POMs). The main issues related to the Lie Transform Method, as for all the Normal Form based Method, is the presence of small divisors for which there is no general rules to be determined when considering nonconservative systems. In the present work, this problem is considered and some physical parameters are related to such conditions determining qualitatively what small means for a divisor relatively to a perturbation parameter. Moreover, starting from the analytical results obtained the POD behavior in the neighborhood of a bifurcation point has been studied. In particular, POMs has been related to the linearized modes of the studied systems and it has been demonstrated their equivalence for systems experiencing a Hopf bifurcation. Moreover, some conditions of equivalence are addressed also in presence of static bifurcations with forcing loads. Finally, the relation i between modal activation and energy distribution has been studied and the possibility to relate POD behavior and nonlinearity (small divisors) of the response has been addressed

Nonlinear damping model for flexible structures

The study of nonlinear damping problem of flexible structures is addressed. Both passive and active damping, both finite dimensional and infinite dimensional models are studied. In the first part, the spectral density and the correlation function of a single DOF nonlinear damping model is investigated. A formula for the spectral density is established with O(Gamma(sub 2)) accuracy based upon Fokker-Planck technique and perturbation. The spectral density depends upon certain first order statistics which could be obtained if the stationary density is known. A method is proposed to find the approximate stationary density explicitly. In the second part, the spectral density of a multi-DOF nonlinear damping model is investigated. In the third part, energy type nonlinear damping model in an infinite dimensional setting is studied