Vibrations of an Infinite Piece-homogeneous Two-layer Plate under the Influence of Normal Load

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In real structures, the destruction of their elements is usually accompanied by impact loads. In this work, a solution is constructed on the vibrations of an infinite two-layer plate under the action of a normal load applied to the surface of a two-layer plate. The problem is reduced to solving an approximate equation for the transverse displacement W of points of the contact plane of a two-layer plate of constant thickness, obtained in [1] and [2]. (1) where the coefficients Qj are determined by the formula obtained in [2]. Assuming the load F(x, y, t) to be even in (x, y), the transverse displacement W will be sought in the form of the Fourier integrals (2) Substituting (2) into equations (1), for W0 we obtain the ordinary differential equation (3) where the coefficients Aj and F0(k, q, t) are equal: and the coefficients Qjʹ are determined by the formulas and γ is determined by the formula γ = h20(k2 + q), and immeasurable parameters were introduced: For ξ from equation (3) we obtain the frequency equation ξ6 + A1ξ4 + A2ξ2 + A3 = 0. (5) frequency equation (5) has purely imaginary roots, i.e., frequencies of own fluctuations. Then, the common decision of the homogeneous differential equation (4) is equal (6) Applying a method of a variation of any constants, for Cjʹ we will receive: (7) private decision of the differential equation (3) we will write down in a kind (8) Satisfying with a zero initial condition, i.e., (9) We find that C1ʹ = C2ʹ = … = C6ʹ = 0. Then, the decision of a problem for displacement W looks like (10) Let, if F(x, y, t) = σ0δ(x)δ(y)δ(z), where σ0 - constant of dimension of pressure; δ(ζ) - Dirac delta function. Then, problem decisions will register in a kind (11) Conclusions From the analytical decision of a problem on influence of normal loading on a surface of a two-layer plate follows that the deflection depends on geometrical and mechanical characteristics of a material of a plate, and also allows to describe precisely enough tensely - the deformed status of a plate in its any point eventually.

About the authors

Mamatisa L. Djalilov

Fergana branch of the Tashkent University of Information Technologies named after Muhammad Al-Khorazmiy

Email: mamatiso2015@yandex.ru
Cand. Sci. (Eng.); Head at the Department “Computer Systems” Fergana, Republic of Uzbekistan

Rustam Kh. Rakhimov

Institute of Materials Science of the SPA “Physics-Sun” of the Academy of Science of Uzbekistan

Email: rustam-shsul@yandex.com
Dr. Sci. (Eng.); Head at the Laboratory No. 1 Tashkent, Republic of Uzbekistan

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