1,720,970 research outputs found
Torsional instability and sensitivity analysis in a suspension bridge model related to the Melan equation
No full textInspired by the Melan equation we propose a model for suspension bridges with two cables linked to a deck, through inextensible hangers. We write the energy of the system and we derive from variational principles two nonlinear and nonlocal hyperbolic partial differential equations, involving the vertical displacement and the torsional rotation of the deck. We prove existence and uniqueness of a weak solution and we perform some numerical experiments on the isolated system; moreover we propose a sensitivity analysis of the system by mechanical parameters in terms of torsional instability. Our results display that there are specific thresholds of torsional instability with respect to the initial amplitude of the longitudinal mode excited
About Symmetry in Partially Hinged Composite Plates
Full text linkWe consider a partially hinged composite plate problem and we investigate qualitative properties, e.g. symmetry and monotonicity, of the eigenfunction corresponding to the density minimizing the first eigenvalue. The analysis is performed by showing related properties of the Green function of the operator and by applying polarization with respect to a fixed plane. As a by-product of the study, we obtain a Hopf type boundary lemma for the operator having its own theoretical interest. The statements are complemented by numerical results
Remarks on the 3D Stokes eigenvalue problem under Navier boundary conditions
Full text linkWe study the Stokes eigenvalue problem under Navier boundary conditions in C1,1-domains Ω ⊂ R3. Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple, it has multiplicity three. We apply these results to show the validity/failure of a suitable Poincaré-type inequality. The proofs are obtained by combining analytic and geometric arguments
Maximizing the ratio of eigenvalues of non-homogeneous partially hinged plates
Full text linkWe study the spectrum of non-homogeneous partially hinged plates having structural engineering applications. A possible way to prevent instability phenomena is to maximize the ratio between the frequencies of certain oscillating modes with respect to the density function of the plate; we prove existence of optimal densities and we investigate their analytic expression. This analysis suggests where to locate reinforcing material within the plate; some numerical experiments give further information and support the theoretical results
A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions
Full text linkIt is well known that for higher order elliptic equations, the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e., nonnegative loads yield positive solutions. The result follows by fine estimates of the Fourier expansion of the corresponding Green function
A new model for suspension bridges involving the convexification of the cables
Full text linkThe final purpose of this paper is to show that, by inserting a convexity constraint on the cables of a suspension bridge, the torsional instability of the deck appears at lower energy thresholds. Since this constraint is suggested by the behavior of real cables, this model appears more reliable than the classical ones. Moreover, it has the advantage to reduce to two the number of degrees of freedom, avoiding to introduce the slackening mechanism of the hangers. The drawback is that the resulting energy functional is extremely complicated, involving the convexification of unknown functions. This paper is divided in two main parts. The first part is devoted to the study of these functionals, through classical methods of calculus of variations. The second part applies this study to the suspension bridge model with convexified cables
On the stability of a nonlinear nonhomogeneous multiply hinged beam
Full text linkThe paper deals with a nonlinear evolution equation describing the dynamics of a nonhomogeneous multiply hinged beam, subject to a nonlocal restoring force of displacement type. First, a spectral analysis for the associated weighted stationary problem is performed, providing a complete system of eigenfunctions. Then, a linear stability analysis for bimodal solutions of the evolution problem is carried out, with the final goal of suggesting optimal choices of the density and of the position of the internal hinged points in order to improve the stability of the beam. The analysis exploits both analytical and numerical methods; the main conclusion of the investigation is that nonhomogeneous density functions improve the stability of the structure
On the long-time behaviour of solutions to unforced evolution Navier–Stokes equations under Navier boundary conditions
Full text linkWe study the asymptotic behaviour of the solutions to Navier-Stokes unforced equations under Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest. The paper draws its main motivation from celebrated results by Foias and Saut (1984) under Dirichlet conditions; here the choice of the boundary conditions requires carefully considering the geometry of the domain Ω, due to the possible lack of the Poincaré inequality in presence of symmetries. In non-axially symmetric domains we show the validity of the Foias-Saut result about the limit at infinity of the Dirichlet quotient, in axially symmetric domains we provide two invariants of the flow which completely characterize the motion and we prove that the Foias-Saut result holds for initial data belonging to one of the invariants
The evolution Navier–Stokes equations in a cube under Navier boundary conditions: rarefaction and uniqueness of global solutions
Full text linkWe study the evolution Navier-Stokes equations in a cube under Navier boundary conditions. For the related stationary Stokes problem, we determine explicitly all the eigenvectors, eigenvalues and the corresponding Weyl asymptotic. We introduce the notion of rarefaction, namely families of eigenvectors that weakly interact with each other through the nonlinearity. By combining the spectral analysis with rarefaction, we expand the solutions in Fourier series, making explicit some of their properties. We then suggest several new points of view in order to explain the striking difference in uniqueness results between 2D and 3D. First, we construct examples of solutions for which the nonlinearity plays a minor role, both in 2D and 3D. Second, we show that, if a solution is rarefied, then its energy is decreasing: hence, rarefaction may be seen as an almost two dimensional assumption. Finally, by exploiting the explicit form of the eigenvectors we provide a numerical explanation of the difficulty in using energy methods for general solutions of the 3D equations
Elasticity solution for a hollow cylinder under axial end loads: Application to a blister of a stayed bridge
No full textStarting from an applied problem related to the modeling of an element of a cable-stayed bridge, we compute the elasticity solution for a hollow circular cylinder under axial end loads. We prove results of symmetry for the solution and we expand it in proper Fourier series; computing the Fourier coefficients in adapted power series, we provide the explicit solution. We consider an engineering case of study, applying the corresponding approximate formula and giving some estimates on the error committed with respect to the truncation of the series
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