Natural Vibrations of Truncated Conical Shells of Variable Thickness

• Published in: Journal of applied mechanics and technical physics Vol. 62; no. 7; pp. 1222 - 1233
• Main Author: Bochkarev, S. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.12.2021; Springer Nature B.V
• Subjects: Angles (geometry); Applications of Mathematics; Boundary conditions; Boundary value problems; Classical and Continuum Physics; Classical Mechanics; Conical shells; Differential equations; Equations of motion; Exact solutions; Fluid- and Aerodynamics; Frequency spectrum; Mathematical Modeling and Industrial Mathematics; Mechanical Engineering; Numerical integration; Ordinary differential equations; Physics; Physics and Astronomy; Resonant frequencies; Runge-Kutta method; Variable thickness; Godunov orthogonal sweep method; classical shell theory; natural vibrations; variable thickness; conical shell
• ISSN: 0021-8944; 1573-8620
• DOI: 10.1134/S0021894421070038

— The paper presents the results of studying the natural frequencies of circular truncated conical shells, the thickness of which varies according to different laws. The behavior of the elastic structure is described in the framework of the classical theory of shells based onthe Kirchhoff–Love hypotheses. The corresponding geometric and physical relations together with the equations of motion are reduced toasystem of ordinary differential equations for new variables. The solution to the formulated boundary value problem is found using Godunov orthogonal sweep method involving the numerical integration of differential equations by the Runge–Kutta fourth order method. The natural frequencies of vibrations are evaluated using a combination of a step-wise procedure and subsequent refinement by the interval bisection method. The reliability of the results is verified by comparison with the known numerical-analytical solutions. Thedependences of the minimum vibration frequencies obtained at shell thicknesses subject to a power-law variation (linear and quadratic, with symmetric and asymmetric shapes) and harmonic variation (with positive and negative curvature) are investigated for shells withdifferent combinations of boundary conditions (simply supported, rigidly clamped, and cantilevered support), cone angles and linear sizes. The results of the study confirm the existence of configurations that provide a significant increase in the frequency spectrum compared to shells of constant thickness under the same limitations on the structure weight.