Free vibration analysis of elastic elliptic cylinders with an eccentric elliptic cavity

• First exact solution for an eccentric elliptic cylinder with shear diaphragm ends.
• Assessment of geometric effects on the key free vibrational characteristics.
• Presentation of 3D deformation mode shapes in vivid graphical form.
• Insights via benchmark simulations and comparisons with FEM/literature data.

An exact elastodynamic model based on Navier equations of linear elasticity is formulated to describe the three-dimensional natural oscillations of an elliptic cylinder of finite length with shear diaphragm end conditions, and containing an inner (coaxial) elliptical cavity of arbitrary size, location, and orientation. The formulation is based on Helmholtz decomposition theorem, the method of separation of variables in elliptical coordinates, and the translational addition theorems for Mathieu functions. The first three natural frequencies are calculated for selected cylinder lengths, cross-sectional aspect ratios, and cavity location/orientation parameters. Also, some representative 3D deformation mode shapes are depicted in vivid graphical form. The precision of solutions is checked through proper convergence studies, and the validity of results is verified with the aid of a commercial finite element package as well as by comparison with the available literature data. The presented exact Mathieu series solution is believed to be the first attempt on the vibrational characteristics of finite-length eccentric elliptical cylinders.