An analytical method for shallow spherical shell free vibration on two-parameter foundation

Abstract

The free vibration control differential equation of shallow spherical shell on two-parameter foundation is a four order differential equation. Using the intermediate variable, the four order differential equation is reduced to two lower order differential equations. The first lower order differential equation is a Helmholtz equation. A new method of two-dimensional Helmholtz operator is proposed as shown in the paper in which the Bessel function included in Helmholtz equation needs to be treated appropriately to eliminate singularity. The first lower order differential equation is transformed into the integral equation using the proposed method in the paper. The second lower order differential equation which is a Laplace equation is transformed into the integral equation by existing methods. Then the two integral equations are discretized according to the middle rectangle formula, and the corresponding solutions can be obtained by MATLAB programming. In this paper, the R-function theory is used to select the appropriate boundary equation to eliminate the singularity. Based on the properties of Rfunction, the combined method of Helmholtz equation and Laplace equation can solve the free vibration problem of irregular shallow spherical shell on two-parameter foundation. Five examples are given to verify the feasibility of the method.

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“…The R -function is a series of real functions whose result is positive or negative only determined solely by the signs (positive or negative) of the variables, regardless of their values (Rvachev, 1982). In this way, if the implicit representation of the parts is known, any shape that is represented by set theory can be transformed into an implicit inequality by R -functions (Varvak, 2015; Gan et al., 2021). Owing to their mathematical properties and natural ability to express complex geometric objects, R -functions have been widely used in computer graphics and geometric modelling (Varvak, 2015; Lu and Wang, 2019; Gan et al., 2021; Sukumar and Srivastava, 2022).…”