Forced wet flexural dynamics of spade rudder in propeller slipstream: semi-analytical approach of pivoted FFFF plate

Abstract

A semi-analytical approach of forced vibration of a trapezoidal, two-way tapered, hollow model spade rudder, is presented. The pivoted rudder is in the propeller slipstream; which causes harmonic hydrodynamic load. The excitation frequency of propeller-induced vibration (PIV) equals the propeller blade passing frequency. The rudder operates at a Reynolds’s number of the order of 108. It has a span of 1 m, root chord of 0.6 m, and tip chord of 0.5 m. It is considered as a hollow Free–Free–Free–Free Kirchhoff’s plate, with the chord section as an NACA0018 profile. The dry vibration is analyzed by the energy-based Galerkin’s method, using closed-form admissible functions in the two perpendicular directions. 3D constant-strength panel method is used to generate the modal added masses, and hence the wet natural frequencies. The drag and the lift coefficient of the rudder, at various angles of attack, are estimated to find the spatial distribution of the total PIV-causing normal force. The mean flow velocity about the rudder is the velocity of advance behind the propeller. The fluid velocity past the aerofoil section depends on the profile thickness and the angle of attack, thereby influencing the local lift. The hydrodynamic pressure varies parabolically along the span. The effect of the wake developed by the propeller, the stern shape, and the rudder angle on the hydrodynamic loading on rudder is included to express the transverse time-varying loading. The wet, forced vibration of the Kirchhoff’s plate is analyzed in MATLAB. The maximum bending stress and twisting sheer at the rudder stock is calculated for various angles of attack.