Suspension bridges with non-constant stiffness: bifurcation of periodic solutions
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Suspension bridges with non-constant stiffness: bifurcation of periodic solutions Gabriela Holubov´ a
and Jakub Janouˇsek
Abstract. We consider a modified version of a suspension bridge model with a spatially variable stiffness parameter to reflect the discrete nature of the placement of the bridge hangers. We study the qualitative and quantitative properties of this model and compare the cases of constant and non-constant coefficients. In particular, we show that for certain values of the stiffness parameter, the bifurcation occurs. Moreover, we can expect also the appearance of blowups, whose existence is closely connected with the so-called Fuˇc´ık spectrum of the corresponding linear operator. Mathematics Subject Classification. 35B10, 35B32, 70K30. Keywords. Suspension bridge, Jumping nonlinearity, Variable coefficient, Bifurcation.
1. Introduction We study a modified version of a standard [12,13] one-dimensional nonlinear beam model of a suspension bridge utt + uxxxx + b r(x)u+ = h(x, t) in (0, 1) × R, u(0, t) = u(1, t) = uxx (0, t) = uxx (1, t) = 0, u(x, t) = u(x, t + 2π) = u(x, −t).
(1)
Here, the term b r(x)u+ represents the nonlinear restoring force due to the bridge hangers (sometimes called suspender cables in the literature) with the constant stiffness b and a variable hanger placement density r(x). Unless stated otherwise, we consider r(x) to be a continuous function on (0, 1) such that 0 < r(x) ≤ 1 almost everywhere in (0, 1). For r(x) ≡ 1, there are several results concerning multiplicity of periodic solutions: see [12,13] and also [2,10] as an example of the problem setting and the application of various tools leading to a conclusion, that when the hanger stiffness b crosses eigenvalues of the corresponding linear beam operator, more solutions appear. These works were followed by [4] and [7], which approached this problem from a different perspective through utilizing a global bifurcation framework. In this paper, we use some of the basic ideas appearing in [4] and implement them in (1) with a generally non-constant function r(x) ≡ 1. The reason for introducing the density (or weight) function r(x) is to interpret more realistically the fact that the hangers are actually not a uniformly distributed force acting on the roadbed, but they are “distinctly distributed.” That is, the restoring force should attain its maximum where the hangers are connected to the roadbed, whereas being considerably weaker in between. Such phenomenon can be described, e.g., with r(x) being a high even power of the cosine function. We show that making the stiffness parameter spatially variable actually improves the behavior of the considered model while not changing its qualitative properties (see the results in Sects. 3 and 5). Bifurcation phenomena (from the stationary solution as well as from infinity) are still observable. But 0123456789().: V,-vol
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G. Holubov´ a and J. Janouˇsek
ZAMP
since the variable coefficient shifts the eigenvalues of the correspon
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