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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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