Analysis of a thermoelastic Timoshenko beam model
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O R I G I NA L PA P E R
Ivana Bochicchio · Marco Campo · José R. Fernández Maria Grazia Naso
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Analysis of a thermoelastic Timoshenko beam model
Received: 3 April 2020 / Revised: 30 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, we analyze a linear problem describing the vibrations of a coupled suspension bridge. The single-span roadbed is modeled as an extensible thermoelastic beam of Timoshenko type. The main cable is modeled as an elastic string and is connected to the roadbed by a distributed system of elastic springs. For this linear model, we obtain the existence and uniqueness of solutions by using the semigroup theory. The exponential decay property is also proved. Then, through a variational formulation, the model is numerically analyzed and some numerical experiments are performed to verify the behavior of the numerical method. Mathematics Subject Classification
74K10 · 74F05 · 74H20 · 74H40 · 65M60 · 65M15
1 Introduction In recent years, increasing attention has been turned to the analysis of some classical engineering structures describing the dynamics of nonlinear vibrations of suspension bridges. In problems concerning the dynamics of bridges, the stabilization of their oscillations plays a vital role. Hence, the study of their global dynamics is very important, especially if the coupling between the road bed and the main suspension cable is taken into account. The complex structure of such bridges requires the construction of realistic mathematical models that must be able to show dynamical properties of interest and that must be easy to handle. In the literature, the dynamic response of the bridge, coupled or not with that of the main cable, was mainly investigated in an isothermal environment and with models where the roadbed was modeled according The work of M. Campo and J. R. Fernández has been supported by the research Project PGC2018-096696-B-I00 (Ministerio de Ciencia, Innovación y Universidades, Spain) with the participation of FEDER I. Bochicchio Dipartimento di Ingegneria Civile, Universitá degli Studi di Salerno, Via Giovanni Paolo II, 84084 Fisciano, SA, Italy E-mail: [email protected] M. Campo Departamento de Matemáticas, Universidade da Coruña, Escola Politécnica Superior, Campus de Esteiro, 15403 Ferrol, Spain E-mail: [email protected] J. R. Fernández (B) Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain E-mail: [email protected] M. G. Naso Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy E-mail: [email protected]
to the Euler–Bernoulli beam theory (see, for instance, [1–7]). For such a model, doubly nonlinear elastic and viscoelastic coupled systems were analyzed and the longtime behavior of solutions deeply investigated (see [8–14]); however, the Euler–Bernoulli theory ignores the effect of transverse shear strain. This lim
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