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A formula for the frequency counting function of a thin shell of revolution for the case of a simple turning point. (English. Russian original) Zbl 0848.73036
Funct. Anal. Appl. 28, No. 4, 282-284 (1994); translation from Funkts. Anal. Prilozh. 28, No. 4, 74-77 (1994).
The authors investigate free vibrations of a thin elastic shell of revolution with a fixed number $$m$$ of waves along a parallel. The shell is formed by revolving a curve $$x= B(s)$$, $$y= \int^s_0 \sqrt{1-B'{}^2 (s)} dx$$, $$0\leq s\leq 1$$, about the $$y$$-axis. Here $$s$$ is the arc length, and $$B(s)$$ is the distance from the meridian to the axis of revolution. The vibrations are described by a system of three ordinary differential equations in terms of displacements: $$(\mu^4 N+ L)f= \lambda f$$, where $$N$$ and $$L$$ are matrix differential operators, $$\mu^4= h^2/ 12$$, and $$h$$ is the thickness of the shell. The spectrum of degenerate $$(h=0)$$ boundary value problem contains a continuous part $$\Phi_1= [\alpha_1, \beta_1]$$ that is the range of values of the function $$\varphi_1 (s)= (1- \sigma^2)/ R^2_2 (s)$$, where $$R_2^{-1} (s)= \sqrt {1- B'{}^2 (s)}/B(s)$$ is one of the principal curvatures of the shell, and $$\sigma$$ is the Poisson ratio. An asymptotic formula for eigenvalues and an asymptotic formula for the counting function are proved for $$\lambda\in \Phi_1$$ under minimal smoothness assumptions imposed on the function $$B(s)$$ and for arbitrary boundary conditions at the edges.
##### MSC:
 74H45 Vibrations in dynamical problems in solid mechanics 74K15 Membranes
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##### References:
 [1] A. G. Aslanyan and V. B. Lidskii, Distribution of Fundamental Frequencies of Thin Elastic Shells [in Russian], Nauka, Moscow (1974). [2] A. L. Goldenveizer, V. B. Lidskii, and P. E. Tovstik, Free Oscillations of Thin Elastic Shells [in Russian], Nauka, Moscow (1979). [3] A. G. Aslanyan and V. B. Lidskii, Dokl. Akad. Nauk SSSR,222, No. 4, 790-792 (1975). [4] A. G. Aslanyan, Funkts. Anal. Prilozhen.,10, No. 2, 63-64 (1976). · Zbl 0328.28006 [5] A. G. Aslanyan, Funkts. Anal. Prilozhen.,12, No. 3, 61-63 (1978).
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