Frictional Wear of a Thermoelastic Beam

Gu, Randy J.; Kuttler, Kenneth; Shillor, Meir

doi:10.1006/jmaa.1999.6652

Cited by 29 publications

(24 citation statements)

References 10 publications

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“…We study the nonlinear evolution fifth-order equation with second-order temporal derivative which is a multidimensional nonlinear generalization of the well known one-dimensional linear equation of beam vibrations in the Timoshenko model [7]. Equations of such a type describe propagation of perturbations in a viscoelastic material under action of external ultrasonic aerodynamical forces [8].…”

Section: Introductionmentioning

confidence: 99%

“…Equations of such a type describe propagation of perturbations in a viscoelastic material under action of external ultrasonic aerodynamical forces [8]. Investigation of mixed problems for these equations and systems can be explained by the worn-out contact surfaces [7]. In paper [7] there is investigated the existence of weak solutions c Wydawnictwa AGH, Krakow 2017 for the mixed problems in the bounded domain D for a system of two linear evolution equations with partial one-and second-order temporal derivatives, where one of unknown functions describes a vertical displacement of a beam.…”

Section: Introductionmentioning

confidence: 99%

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On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model

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Nytrebych

et al. 2017

OpMath

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model

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Ільків

Nytrebych

et al. 2017

OpMath

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“…Moreover, the efficient wear modeling and simulation of thin elastic structures submitted to sharp contact, is in an even younger stage of development (see e.g. [37][38][39][40]). …”

Section: Introductionmentioning

confidence: 99%

Modeling and computation of fretting wear of structures under sharp contact

Dhia

Torkhani

2010

Numerical Meth Engineering

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

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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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Frictional Wear of a Thermoelastic Beam

Cited by 29 publications

References 10 publications

On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model

On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model

Modeling and computation of fretting wear of structures under sharp contact

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

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