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

Steady states of elastically-coupled extensible double-beam systems

Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary model in dimensionless form $$\begin{cases} u"" - \Big(\beta+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" - \Big(\beta+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases}$$ describing the equilibria of an elastically-coupled extensible double-beam system subject to evenly compressive axial loads. Necessary and sufficient conditions in order to have nontrivial solutions are established, and their explicit closed-form expressions are found. In particular, the solutions are shown to exhibit at most three nonvanishing Fourier modes. In spite of the symmetry of the system, nonsymmetric solutions appear, as well as solutions for which the elastic energy fails to be evenly distributed. Such a feature turns out to be of some relevance in the analysis of the longterm dynamics, for it may lead up to nonsymmetric energy exchanges between the two beams, mimicking the transition from vertical to torsional oscillations

Long-Term Damped Dynamics of the Extensible Suspension Bridge

This work is focused on the doubly nonlinear equation, whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness k^2. When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load p and stiffness k^2. For a general external source f, we prove the existence of bounded absorbing sets.When f is timeindependent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.Comment: 19 pages, 1 figur

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

Steady-state solutions for a suspension bridge with intermediate supports

2This work is focused on a system of boundary value problems whose solutions represent the equilibria of a bridge suspended by continuously distributed cables and supported by M intermediate piers. The road bed is modeled as the junction of N=M+1 extensible elastic beams which are clamped each other and pinned at their ends to each pier. The suspending cables are modeled as one-sided springs with stiffness k. Stationary solutions of these doubly nonlinear problems are explicitly and analytically derived for arbitrary k and a general axial load p applied at the ends of the bridge. In particular, we scrutinize the occurrence of buckled solutions in connection with the length of each sub-span of the bridge.openopenGiorgi C.; Vuk E.Giorgi, Claudio; Vuk, Elen

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

On some nonlinear models for suspension bridges

In this paper we discuss some mathematical models describing the nonlinear vibrations of different kinds of single-span simply supported suspension bridges and we summarize some results about the longtime behavior of solutions to the related evolution problems. Finally, in connection with the static counterpart of a general string-beam nonlinear model, we present some original results concerning the existence of multiple buckled solutions