Vibration analysis of partially cracked thin orthotropic cylindrical shells with linearly and quadratically varying thickness: An analytical approach

This study presents an analytical model for the free vibration analysis of thin orthotropic cylindrical shells featuring linearly and quadratically varying thickness. The shell is assumed to have a length significantly greater than its other dimensions, and a small mid-span surface crack is considered to evaluate its impact on dynamic behaviour. The governing equations of motion are derived using classical shell theory and simplified using the Donnell–Mushtari–Vlasov (DMV) formulation to incorporate the effects of thickness variation. Crack-induced stiffness reduction is modelled using crack compliance coefficients based on the Line Spring Model (LSM), accounting for both axial and circumferential crack orientations. The influence of variable thickness on structural stiffness and vibration characteristics is explicitly addressed. A closed-form solution for simply supported and clamped boundary conditions is obtained via Hamilton’s principle and separation of variables. The analytical results are validated through comparison with existing literature. A comprehensive parametric study is conducted to assess the effects of thickness gradients, crack size and position, and orthotropic material properties on the fundamental natural frequency.