Analysis and simulations of a nonlinear elastic dynamic beam

Andrews, Kevin T.; Dumont, Yves; Mbengue, Mbagne Faly; Purcell, J.; Shillor, Meir

doi:10.1007/s00033-012-0233-9

Cited by 32 publications

(29 citation statements)

References 12 publications

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“…Not that, as follows from the estimates established in what follows, τ 0 = T . We multiply each equation of system (6) by ∂C k N ∂t , sum all equations with respect to k from 1 to N and integrate the result with respect to t over the interval [0, τ] , τ ≤ T . Reasoning as in [2], we conclude, in particular, that, for any τ ∈(0,T ] ,…”

Section: Procedures Of Getting the Main Resultsmentioning

confidence: 99%

“…The outlined problem is quite urgent for engineering [5]. However, in the general case, it was solved only for a fairly narrow class of vibration systems [2,6,7]. In [7], the problem of existence of weak solutions of mixed problems was studied in a bounded domain for a certain system of linear partial differential equations in which one of unknown functions describes the vertical displacements of the beam.…”

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confidence: 99%

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Investigation of Bending Vibrations in Voigt–Kelvin Bars with Regard for Nonlinear Resistance Forces

Пукач¹

2016

J Math Sci

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We present a procedure aimed at the qualitative investigation of the solution of a mathematical model of bending vibrations of viscoelastic bodies under the action of dissipative forces and nonlinear resistance forces in a bounded domain. This procedure is based of the general approaches of the theory of nonlinear boundary-value problems and the application of the Galerkin method and enables one to substantiate the correctness of the solution of the model and use approximate methods for its investigation. Urgency of the Problem and Survey of the Main ResultsThe mathematical models of vibration systems that describe actual technological processes are mainly based on nonlinear differential equations. The principal difficulties encountered in the analytic investigation of the mathematical models of dynamic processes are connected with their nonlinearity because, only on the basis of the analysis of solutions (exact or approximate), one can estimate the influence of the parameters of systems on the dynamic phenomena, predict resonances even in the stage of design, and choose the optimal strength characteristics of machine parts and units. As the most efficient analytic methods for the investigation of quasilinear vibration systems, we can mention the Poincaré method, the van der Pol method, and the KrylovBogolyubov-Mitropol'skii asymptotic method (see, e.g., [3,4]). At the same time, the advances of engineering and the implementation of fast machine building require the formulation and solution of new problems for which the corresponding mathematical models cannot be investigated by the asymptotic methods of nonlinear mechanics. These class of problems includes the problems of vibration of flexible elements of belt and chain gears, tape systems for data recording and reproduction, conveyer lines, various types of rope tows, paperrolling machines, equipment for winding metallic strips, wires, and threads, equipment for drilling oil and gas wells, pipelines, etc. For the major part of applied problems, the action of generalized forces of internal dissipation in vibration systems is obvious. Thus, in particular, bending waves in Voigt-Kelvin bars are described by a linear equation of the fifth order [1, p. 60] taking into account the influence of dissipative forces on the dynamic process.The present paper is devoted to the qualitative investigation of the mathematical model of bending vibrations in Voigt-Kelvin bars under the action of nonlinear resistance forces. In the case of a strongly nonlinear dependence of the amplitude of vibrations on the resistance forces, it is impossible to get the analytic solution of the problem and, hence, there are no general procedures aimed at the determination of the dynamic characteristics of the process of vibration. The outlined problem is quite urgent for engineering [5]. However, in the general case, it was solved only for a fairly narrow class of vibration systems [2,6,7]. In [7], the problem of existence of weak solutions of mixed problems was studied in a bounded domain for a ...

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Section: Procedures Of Getting the Main Resultsmentioning

confidence: 99%

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confidence: 99%

Investigation of Bending Vibrations in Voigt–Kelvin Bars with Regard for Nonlinear Resistance Forces

Пукач¹

2016

J Math Sci

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“…(10) and (12) we can identify the response terms as the so-called normal compliance terms describing a normal compliance contact between the beam and the foundation, see [1]. For this reason we will use for these cases the name problem with normal compliance.…”

Section: Analysis Of the Problem Formulationsmentioning

confidence: 99%

“…In this paper we will, regarding the simplicity, consider only the case F = 0. In literature, this model is the so-called nonlinear Gao beam, which has used for studying many engineering problems [9][10][11]. If the beam is subjected to an axial compressive load, the total potential energy of this beam is nonconvex whence the axial load reaches the Euler buckling load.…”

Section: Introductionmentioning

confidence: 99%

Mixed finite element solutions to contact problems of nonlinear Gao beam on elastic foundation

Gao

Machalová

Netuka

2015

Nonlinear Analysis: Real World Applications

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“…Moreover, one may use such models as tests and benchmarks for computer schemes meant for simulations of more complicated multidimensional contact problems. Models, analysis and computer simulations of various contact problems for rods and beams can be found in (Ahn et al, 2012;Andrews et al, 2012;Kuttler et al, 2001;Shillor et al, 2001) and the references therein. A mathematical model that describes unilateral contact of a beam between two deformable obstacles was studied in (Barboteu et al, 2012b).…”

Section: Introductionmentioning

confidence: 99%

Modeling and simulations for quasistatic frictional contact of a linear 2D bar

Barboteu¹,

Djehaf²,

Shillor³

et al. 2017

jtam

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This work considers a mathematical model that describes quasistatic evolution of an elastic 2D bar that may come in frictional contact with a deformable foundation. We present the model and some of its underlying assumptions. In particular, the novelty in the model is that both vertical and horizontal motions are taken into account, which makes it especially useful when frictional contact is concerned. Contact is described with the normal compliance condition and friction with the Coulomb law of dry friction. We introduce a hybrid variational formulation of the problem and a numerical discretization based on a uniform time step and the finite element method in space. The numerical algorithm has been implemented, and we present computer simulations that illustrate the mechanical behavior of the system with emphasis on frictional aspects of the problem.

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Analysis and simulations of a nonlinear elastic dynamic beam

Cited by 32 publications

References 12 publications

Investigation of Bending Vibrations in Voigt–Kelvin Bars with Regard for Nonlinear Resistance Forces

Investigation of Bending Vibrations in Voigt–Kelvin Bars with Regard for Nonlinear Resistance Forces

Mixed finite element solutions to contact problems of nonlinear Gao beam on elastic foundation

Modeling and simulations for quasistatic frictional contact of a linear 2D bar

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