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

Area of Science:

• Mechanical Engineering
• Structural Dynamics
• Applied Mathematics

Background:

• Free vibration analysis of shallow spherical shells on foundations is crucial for structural integrity.
• Existing methods face challenges with irregular geometries and complex foundation interactions.
• The governing differential equation is a high-order equation requiring simplification.

Purpose of the Study:

• To develop a robust numerical method for analyzing the free vibration of shallow spherical shells on two-parameter foundations.
• To address the complexities introduced by irregular shell shapes and foundation properties.
• To provide a computationally efficient solution using integral equations.

Main Methods:

• Reducing a fourth-order differential equation to two lower-order equations (Helmholtz and Laplace).
• Developing a novel two-dimensional Helmholtz operator to handle Bessel function singularities.
• Transforming differential equations into integral equations using proposed and existing methods.
• Discretizing integral equations via the middle rectangle formula and solving with MATLAB.
• Employing R-function theory to select boundary equations and eliminate singularities.

Main Results:

• The proposed combined method effectively solves the free vibration problem for irregular shallow spherical shells.
• Singularity issues in the Helmholtz equation are successfully managed.
• The method demonstrates feasibility through five verification examples.
• MATLAB programming facilitates the numerical solution process.

Conclusions:

• The integrated approach combining Helmholtz and Laplace integral equations offers a viable solution for free vibration analysis.
• R-function theory is instrumental in managing boundary conditions and singularities.
• The method is applicable to complex, irregular shallow spherical shell structures on two-parameter foundations.