An approach to determining the natural frequencies and modes of compound systems of shells of revolution of different geometry and relative thickness, continuously and/or discretely inhomogeneous across the thickness is proposed. The shells are made of isotropic, orthotropic, and anisotropic materials with a single plane of elastic symmetry. The approach involves construction of a mathematical model based on the classical Kirchhoff–Love theory, Timoshenko-type refined theory, spatial elasticity theory (particular case), and numerical-analytical technique of solving associated two- and three-dimensional problems by reducing their dimension and using the successive approximation and step-by-step search methods in combination with the orthogonal sweep method. Examples of solving various problems in different fields of engineering are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. E. Bellman and R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems, Elsevier, New York (1965).
S. K. Godunov, “Numerical solution of boundary-value problems for systems of linear ordinary differential equations,” Usp. Mat. Nauk, 16, No. 3, 171–174 (1961).
Ya. M. Grigorenko, E. I. Bespalova, A. B. Kitaigorodskii, and A. I. Shinkar, Free Vibrations of Elements of Shell Structures [in Russian], Naukova Dumka, Kyiv (1986).
Ya. M. Grigorenko and A. T. Vasilenko, Theory of Shells of Varying Stiffness, Vol. 4 of the five-volume series Methods of Shell Design [in Russian], Naukova Dumka, Kyiv (1981).
Von L. Collatz, Eigenvalue Problems with Engineering Applications [in German], Akad. Verlagsges., Leipzig (1963).
S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body [in Russian], Mir, Moscow (1977).
V. D. Budak, A. Ya. Grigorenko, M. Yu. Borisenko, and E. V. Boichuk, “Frequencies and modes of natural vibrations of noncircular cylindrical shells of variable thickness,” Int. Appl. Mech., 53, No. 2, 164–172 (2017).
M. Caresta and N. J. Kessissoglou, “Free vibrational characteristics of isotropic coupled cylindrical-conical shells,” J. ound Vibr., 329, 733–784 (2010).
L. Cheng and J. Nicolas, “Free vibration analysis of a cylindrical shell-circular plate system with general coupling and various boundary conditions,” J. Sound Vibr., 155, 231–247 (1992).
D. Chronopoulos, M. Ichchou, B. Troclet, and O. Bareille, “Predicting the broadband response of a layered cone-cylinder-cone shell,” Compos. Struct., 107, 149–159 (2014).
A. Ya. Grigorenko, T. L. Efimova, and Yu. A. Korotkikh, “Free vibrations of non-thin cylindrical shells of a variable thickness with elliptic cross-section,” Int. Appl. Mech., 53, No. 6, 668–679 (2017).
W. C. L. Hu and J. P. Raney, “Experimental and analytical study of vibrations of joined shells,” AIAA J., 5, No. 5, 976–981 (1967).
E. Kamke, Differentialgleichungen. Losungmethoden und Losungen. I Gewonliche Differentialgleichungen, 6th Verbesserte-Auflage, Leipzig (1959).
Y. S. Lee, M. S. Yang, Y. S. Kim, and J. H. Kim, “A study on the free vibration of the joined cylindrical-spherical shell structures,” Compos. Struct., 80, No. 27–30, 2405–2414 (2002).
S. Liang and H. L. Chen, “The natural vibration of a conical shell with an annular end plate,” J. Sound Vibr., 294, 927–943 (2006).
A. V. Marchuk, S. V. Gniedash, and S. A. Levkovsky, “Free and forced vibrations of thick-walled anisotropic cylindrical shells,” Int. Appl. Mech., 53, No. 2, 181–195 (2017).
Y. Qu, S. Wu, Y. Chen, and Y. Hua, “Vibration analysis of ring-stiffened conical-cylindrical-spherical shells based on a modified variational approach,” Int. J. Mech. Sci., 69, 72–84 (2013).
B. P. Patel, M. Ganapathi, and S. Kamat, “Free vibration characteristics of laminated composite joined conical-cylindrical shells,” J. Sound Vibr., 237, 920–930 (2000).
M. Shakouri and M. A. Kouchakzadeh, “Free vibration analysis of joined conical shells: analytical and experimental tudy,” J. Thin-Walled Struct., 85, 350–358 (2014).
X. C. Shang, “Exact analysis for three vibration of a composite shell structure-hermetic capsule,” Appl. Math. Mech., 22, 1035–1045 (2001).
Z. Su and G. Jin, “Vibration analysis of coupled conical-cylindrical-spherical shells using a Fourier spectral element method,” J. Acoust. Soc. Am., 140, No. 5, 3925–3940 (2016).
C. K. Susheel, T. K. Rajeev Kumar, and Vishal Singh Chauhan, “Nonlinear vibration analysis of piezolaminated functionally graded cylindrical shells,” Int. J. Nonlin. Dynam. Cont., 1, No. 1, 27–50 (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Mekhanika, Vol. 55, No. 1, pp. 44–59, January–February, 2019.
Rights and permissions
About this article
Cite this article
Bespalova, E.I., Boreiko, N.P. Determination of the Natural Frequencies of Compound Anisotropic Shell Systems Using Various Deformation Models. Int Appl Mech 55, 41–54 (2019). https://doi.org/10.1007/s10778-019-00932-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10778-019-00932-8