Unsteady motion of a concentrated load, applied to the boundary of the half-space

Аuthors

Okonechnikov A. S.^1*, Tarlakovsky D. V.^2**, Fedotenkov G. V.^3***

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia
3. ,

*e-mail: leon_lionheart@mail.ru
**e-mail: tdvhome@mail.ru
***e-mail: greghome@mail.ru

Abstract

Abstract

Purpose

In some aircraft engineering problems it is necessary to take into account different unsteady state effects. Such effects can occur, for example, when the aircraft contacts with small solid particles from the atmosphere. In such problems sometimes one has to take into account the movement of the load, produced by such particles. Also such problem can be interested in high-speed transport engineering. In general formulation of the problem, it is extremely difficult to analyze such effects.

In this paper the unsteady state response of a half-space due to a concentrated moving load is proposed. The half-space material is assumed isotropic, elastic and homogeneous. By means of the proposed method we found the solution of the problem and analyzed all possible singularities of the solution for different speed modes.

Approach

This problem is formulated in terms of elastic potentials. The load is modeled by means of Dirac’s function. In this formulation, the load has a random law of motion. After the problem is formulated, the superposition principle is applied. In this principle the Lamb’s problem solution is used as a Green’s function. After the general analysis, it is necessary to assume some definite law of motion of the load.

Findings

As an example, the case of load movement with constant velocity has been studied. The solution has been obtained and the result is analytical. A part of great interest play singularities of the solution on different speed modes and on some critical velocities. Such critical velocities like- speed of Rayleigh wave, and speed of elastic waves.

Practical implications

Such results can be used in transport engineering. Effect of Rayleigh wave as a result of train wheels and rails contact. Our results can help to find avoid such undesirable effects.

Originality

This is first time this problem was fully analyzed. In different papers only quasi-stationary or steady response was researched. Some authors obtained approximate solutions, our result is analytical and accurate.

Keywords:

moving load, unsteady problem, superposition principle, singular integrals, regularization, solution singularities

References

1. Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Volny v sploshnykh sredakh (Waves in continuous media), Moscow, FIZMATLIT, 2004, 472 p.

2. Tarlakovskii D.V., Fedotenkvov G.V. Izvestiya RAN. Mekhanika tverdogo tela, 2015, no. 2. pp. 118-128.

3. Gorshkov A.G., Tarlakovskii D.V. Dinamicheskie kontaktnye zadachi s podvizhnymi granitsami (Dynamic contact problems with moving boundaries), Moscow, Nauka. Fizmatlit, 1995, 352 p.

4. Medvedskii A.L., Tarlakovskii D.V. Elektronnyi zhurnal «Trudy MAI», 2011, no. 48. available at: https://www.mai.ru/science/trudy/eng/published.php?ID=27499 (accessed 22.11.2011)

5. Medvedskii A.L., Tarlakovskii D.V. Vestnik Moskovskogo aviatsionnogo instituta, 2011, Vol. 18, no. 6, pp. 125–132.

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