arXiv subjects
Publications and source records attributed to Filippo Gazzola.
We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.
A full theory for hinged beams and degenerate plates with multiple intermediate piers is developed. The analysis starts with the variational setting and the study of the linear stationary problem in one dimension. Well-posedness results are provided and the possible loss of regularity, due to the presence of the piers, is underlined. A complete spectral theorem is then proved, explicitly determining the eigenvalues on varying of the position of the piers and exhibiting the fundamental modes of oscillation. The obtained eigenfunctions are used to tackle the study of the nonlinear evolution problem in presence of different nonlinearities, focusing on the appearance of linear and (a suitable notion of) nonlinear instability, with a twofold goal: finding the most reliable nonlinearity describing the oscillations of real structures and determining the position of the piers that maximizes the stability of the structure. The last part of the paper is devoted to the study of degenerate plate models, where the structure is composed by a central beam moving vertically and by a continuum of cross-sections free to rotate around the beam and allowed to deflect from the horizontal equilibrium position. The torsional instability of the structure is investigated, taking into account the impact of different nonlinear terms aiming at modeling the action of cables and hangers in a suspension bridge. Again, the optimal position of the piers in terms of stability is discussed. The stability analysis is carried out both by means of analytical tools, such as Floquet theory, and numerical experiments. Several open problems and possible future developments are presented.
A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be orthogonally split into two subspaces containing, respectively, the longitudinal and the torsional movements of the plate. Sufficient conditions for the stability of periodic solutions and of solutions having only a longitudinal component are given. A stability analysis of the so-called prevailing mode is also performed. Some numerical experiments show that instabilities may occur. This plate can be seen as a simplified and qualitative model for the deck of a suspension bridge, which does not take into account the complex interactions between all the components of a real bridge.
We use a gap function in order to compare the torsional performances of different reinforced plates under the action of external forces. Then, we address a shape optimization problem, whose target is to minimize the torsional displacements of the plate: this leads us to set up a minimaxmax problem, which includes a new kind of worst-case optimization. Two kinds of reinforcements are considered: one aims at strengthening the plate, the other aims at weakening the action of the external forces. For both of them, we study the existence of optima within suitable classes of external forces and reinforcements. Our results are complemented with numerical experiments and with a number of open problems and conjectures.
The Melan equation for suspension bridges is derived by assuming small displacements of the deck and inextensible hangers. We determine the thresholds for the validity of the Melan equation when the hangers slacken, thereby violating the inextensibility assumption. To this end, we preliminarily study the possible shortening of the cables: it turns out that there is a striking difference between even and odd vibrating modes since the former never shorten the cables. These problems are studied both on beams and plates.
Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation $$ u_{tt} + u_{xxxx} + f(u)= g(x, t) $$ in bounded space-time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities $f$ and for some forcing terms $g$, highlighting some of their structural properties and performing some numerical simulations.
We consider the nonlinear nonlocal beam evolution equation introduced by Woinowsky- Krieger. We study the existence and behavior of periodic solutions: these are called nonlinear modes. Some solutions only have two active modes and we investigate whether there is an energy transfer between them. The answer depends on the geometry of the energy function which, in turn, depends on the amount of compression compared to the spatial frequencies of the involved modes. Our results are complemented with numerical experiments, overall, they give a complete picture of the instabilities that may occur in the beam. We expect these results to hold also in more complicated dynamical system
We consider a class of Hill equations where the periodic coefficient is the squared solution of some Duffing equation plus a constant. We study the stability of the trivial solution of this Hill equation and we show that a criterion due to Burdina (V.I. Burdina, Boundedness of solutions of a system of differential equations) is very helpful for this analysis. In some cases, we are also able to determine exact solutions in terms of Jacobi elliptic functions. Overall, we obtain a fairly complete picture of the stability and instability regions. These results are then used to study the stability of nonlinear modes in some beam equations.
We introduce a new functional, named gap function, which measures the torsional instability of a partially hinged rectangular plate, aiming to model the deck of a bridge. Then we test the performances of the gap function on a plate subject to two kinds of loads: one modeled by a force concentrating on one of the free edges and one in resonance with the eigenmodes.
We consider a partially hinged rectangular plate and its normal modes. There are two families of modes, longitudinal and torsional. We study the variation of the corresponding eigenvalues under domain deformations. We investigate the possibility of finding a shape functional able to quantify the torsional instability of the plate, namely how prone is the plate to transform longitudinal oscillations into torsional ones. This functional should obey several rules coming from both theoretical and practical evidences. We show that a simple functional obeying all the required rules does not exist and that the functionals available in literature are not reliable.
We consider a thin and narrow rectangular plate where the two short edges are hinged whereas the two long edges are free. This plate aims to represent the deck of a bridge, either a footbridge or a suspension bridge. We study a nonlocal evolution equation modeling the deformation of the plate and we prove existence, uniqueness and asymptotic behavior for the solutions for all initial data in suitable functional spaces. Then we prove results on the stability/instability of simple modes motivated by a phenomenon which is visible in actual bridges and we complement these theorems with some numerical experiments.
This paper studies the existence of solitons, namely stable solitary waves, in a suspension bridge. The bridge is modeled as a degenerate plate, that is, a central beam with cross sections, and displays two degrees of freedom: the vertical displacement of the beam and the torsional angles of the cross sections. Under fairy general assumptions, we prove the existence of solitons. Under the additional assumption of large tension in the sustaining cables, we prove that these solitons have a nontrivial torsional component. This appears relevant for the security since several suspension bridges collapsed due to torsional oscillations.
The Melan beam equation modeling suspension bridges is considered. A slightly modified equation is derived by applying variational principles and by minimising the total energy of the bridge. The equation is nonlinear and nonlocal, while the beam is hinged at the endpoints. We show that the problem always admits at least one solution whereas the uniqueness remains open although some numerical results suggest that it should hold. We also emphasize the qualitative difference with some simplified models.
All attempts of aeroelastic explanations for the torsional instability of suspension bridges have been somehow criticised and none of them is unanimously accepted by the scientific community. We suggest a new nonlinear model for a suspension bridge and we perform numerical experiments with the parameters corresponding to the collapsed Tacoma Narrows Bridge. We show that the thresholds of instability are in line with those observed the day of the collapse. Our analysis enables us to give a new explanation for the torsional instability, only based on the nonlinear behavior of the structure.
We consider a partial differential equation that arises in the coarse-grained description of epitaxial growth processes. This is a parabolic equation whose evolution is governed by the competition between the determinant of the Hessian matrix of the solution and the biharmonic operator. This model might present a gradient flow structure depending on the boundary conditions. We first extend previous results on the existence of stationary solutions to this model for Dirichlet boundary conditions. For the evolution problem we prove local existence of solutions for arbitrary data and global existence of solutions for small data. By exploiting the boundary conditions and the variational structure of the equation, according to the size of the data we prove finite time blow-up of the solution and/or convergence to a stationary solution for global solutions.
We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation whose nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of self-similar solutions.
We suggest a new model for the dynamics of a suspension bridge through a system of nonlinear nonlocal hyperbolic differential equations. The equations are of second and fourth order in space and describe the behavior of the main components of the bridge: the deck, the sustaining cables and the connecting hangers. We perform a careful energy balance and we derive the equations from a variational principle. We then prove existence and uniqueness for the resulting problem.
A rectangular plate modeling the deck of a suspension bridge is considered. The plate may widely oscillate, which suggests to consider models from nonlinear elasticity. The von Kármán plate model is studied, complemented with the action of the hangers and with suitable boundary conditions describing the behavior of the deck. The oscillating modes are determined in full detail. Existence and multiplicity of static equilibria are then obtained under different assumptions on the strength of the buckling load.