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On perturbation of the Kirchhoff operator – analysis and numerical simulation. (English) Zbl 1254.74068

Summary: We consider a new model for vertical vibrations of an elastic string fixed at the ends. When the tension on the string is not constant, Kirchhoff obtained the model \[ \frac{\partial^2u}{\partial t^2}-(a(x)+b(x)|\nabla u|^2)\frac{\partial^2u}{\partial x^2}=0, \] with the nonlinear perturbations \(b(x)|\nabla u|^2\), which represents the additional tension due to the length change of the string. The Kirchhoff model is extensively investigated in the literature.
In the present paper, for strings with variable density and cross section, we obtain a model which is a perturbation of the Kirchhoff equation by an additional term: \[ -c(x,t)|\nabla u|^2\frac{\partial u}{\partial x}. \] We prove that for every \(T>0\) a mixed problem for this new model is well-posed in the interval \(0\leq t<T\), with a restriction on the initial data \(\phi_0\) and \(\phi_1\) that depends on \(T\). We apply the Galerkin method, multiplier techniques and compactness results to obtain the existence and uniqueness of solutions. For the numerical solution, we employ the finite element method and also introduce an implicit time discretization. Some numerical examples are presented to validate the numerical method and numerical experiments are presented to compare with the Kirchhoff model and to investigate the effects of coefficients in the string vibration.

MSC:

74K05 Strings
35L70 Second-order nonlinear hyperbolic equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
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