New solution of the Euler problem about a flexible body sliding over a fixed block

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Authors:

N.A. Lubenets, Candidate of Technical Sciences, Associate Professor, State Higher Educational Institution “National Mining University”, Senior Lecturer of the Department of Transport Systems and Technology, Dnipropetrovsk, Ukraine

Abstract:

Purpose. To solve the problem about a flexible body sliding over a fixed block taking into account the principle of conservation of energy and present knowledge about friction.

Methodology. We substantiate the system of differential equilibrium equations of a mechanical system describing friction of a flexible body surface element against a fixed block, friction law, and law of conservation of energy. The solution shows the dependence of the tension of the flexible body on the line of contact with the block under friction, which corresponds to the law of conservation of energy.

Findings. The solution describes the interrelations between the wrap angle, friction coefficient, flexibility at the body unit, the forces applied to the ends of the flexible body, its linear velocity and mass. The solution is applicable to sliding and cohesion of bodies.

Originality. We have justified the new system of differential equations of equilibrium of a mechanical system of a flexible body friction against a fixed block that contains the two-parameter friction law, condition of equilibrium tension of the flexible body on the line of contact with the block under friction and provides an analytic solution. The article presents the linear dependence the dependence of the tension of the flexible body on the line of contact with the block under friction; the direct expressions for calculating the friction coefficient and the normal reaction between the bodies unaffected by the friction properties of bodies; describes the full range of possible forces that can be applied to the ends of the flexible body tested.

Practical value. The new solution overcomes the contradictions between the accumulated experimental data and earlier solutions. It allows us to specify the test normal reaction between the bodies and determine their coefficient of friction directly. The knowledge obtained develops mathematical methods for solving systems of differential equations of mechanics and physics; it enriches representation of friction flexible bodies; and contributes to progress in research, education, and engineering. 

References:

1. Кирия Р.В. Применение метода возмущений Л. Прандтля к разрешению парадокса Н.Е Жуковского / Р.В.Кирия, Е.А. Стаховский // Системні технології. –2002. – № 4 (21). С. 33–46

Kiriya,R.V. and StakhovskyE.A. (2002),The use of the perturbation method Prandtl to resolve th paradox N.EZhukovsky”, Sistemnі tehnologіїDnepropetrovsk, no. 4(21) pp. 33–46.

2. Лубенец Н.А. Влияние центробежных сил гибкого тела на реализацию тягового усилия трением / Н.А.Лубенец, Т.Н.Лубенец // Науковий вісник НГУ. – 2012. – № 5. – С. 28–33.

Lubenets, N.A. and Lubenets, T.N. (2012), The influence of the centrifugal forces of a flexible body for the implementation of traction friction, Naukovyi Visnyk Natsіonalnoho Hіrnychoho Unіversitetu, no.5, pp. 28–33.

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Date 2014-07-08 Filesize 523.86 KB Download 582

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ISSN (print) 2071-2227,
ISSN (online) 2223-2362.
Journal was registered by Ministry of Justice of Ukraine.
Registration number КВ No.17742-6592PR dated April 27, 2011.

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