The Interaction between the Plane Wave and the Plate with Limited Height in Soil

N.A. Lokteva^a*, V.N. Paimushin^b,c*, D.O. Serdyuk^a***, D.V. Tarlakovskii^d****

^aMoscow Aviation Institute (National Research University), Moscow, 125993 Russia

^bKazan Federal University, Kazan, 420008 Russia

^cTupolev Kazan National Research Technological University, Kazan, 420111 Russia

^dMoscow State University, Moscow, 119991 Russia

E-mail: ^*nlok@rambler.ru, ^**vpajmushin@mail.ru, ^***d.serduk55@gmail.com, ^****tdvhome@mail.ru

Received December 28, 2016

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Abstract

A solution of the two-dimensional task on interaction between the harmonic wave and the plate with the limited height in soil has been provided. The plate surrounded on both sides with the half-spaces filled with soil medium has been used as a vibro-absorbing obstacle. The mechanical behavior of the plate has been described by S.P. Timoshenko's shift model and the mechanical behavior of soil by a linear elasticity theory equation. The main purpose of the paper is to determine the total acceleration vector field inducted by the penetrated and radiated waves in the second half-space. The mathematical formulation of the task includes a model of upcoming wave, soil medium and plate movement equation, infinity conditions, and conditions of soil contact with obstacle. Conditions of free slip have been taken as the contact conditions between the soil and the obstacle. We have considered a closed system of equations, which includes wave equations for scalar and vector potentials, elasticity theory equations for soil mediums, Koshi's relations, physical law, and plate movement equation. The boundary conditions for the plate correspond to a hinged support. To solve this task, all functions have been expanded in trigonometric series that allowed to obtain potential values in the coefficients of the series. To define the integrations constants, the contact conditions between the obstacle and soil have been used. On the basis of the revealed potentials, we have defined displacements on the boundary between the plate and soil and in other points of the second half-space. The vibro-absorbing properties of the plate have been investigated depending on the frequency of the harmonic wave falling on the plate. From the practical point of view, this task is related to protection of buildings from vibrations formed at a distance from underground railways.

Keywords: soil, plate, S.P. Timoshenko's model, harmonic wave, oscillation frequency, vibration absorption, vibration acceleration

Acknowledgments. This study was supported by the Russian Foundation for Basic Research (project no. 16-38-50025 mol_nr).

Figure Captions

Fig. 1. A model of interaction between the plate and soil.

Fig. 2. The dependence of the vibration acceleration field module: 1 – on the plate surface at z = 0 m; 2 – in the soil at z = 10 m.

Fig. 3. The dependence of the vibration acceleration field module at ω = 20 ÷ 300 Hz: 1 – on the plate surface at z = 0 m; 2 – a × 10³ in the soil at z = 10 m.

References

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For citation: Lokteva N.A., Paimushin V.N., Serdyuk D.O., Tarlakovskii D.V. The interaction between the plane wave and the plate with limited height in soil. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 1, pp. 64–74. (In Russian)

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