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Vibrations of truncated shallow and deep conical shells with non-uniform thickness

机译：厚度不均匀的圆锥形浅壳和深壳的振动

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摘要

A three-dimensional (3-D) method of analysis is presented for determining the natural frequencies of a truncated shallow and deep conical shell with linearly varying thickness along the meridional direction free at its top edge and clamped at its bottom edge. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u(r), u(theta),and u(z) in the radial, circumferential, and axial directions, respectively, are taken to be periodic in. and in time, and algebraic polynomials in the r and z directions. Strain and kinetic energies of the truncated conical shell with variable thickness are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated. The frequencies from the present 3-D method are compared with those from other 3-D finite element method and 2-D shell theories.
机译：提出了一种三维（3-D）分析方法，用于确定截短的浅圆锥壳和深圆锥壳的固有频率，该圆锥壳沿子午方向的厚度线性变化，且顶部边缘自由，并在底部边缘被夹紧。与数学上是二维（2-D）的常规壳理论不同，本方法基于3-D动态弹性方程。分别在径向，圆周和轴向上的位移分量u（r），u（theta）和u（z）在时间上和时间上都是周期性的，在r和z方向上的代数多项式被认为是周期性的。拟定了厚度可变的截顶圆锥壳的应变和动能，并使用Ritz方法解决了特征值问题，从而通过使频率最小化来产生频率的上限值。随着多项式次数的增加，频率收敛到精确值。证明了收敛到四位精度。将本3-D方法的频率与其他3-D有限元方法和2-D壳理论的频率进行比较。

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