Free Dry and Wet Vibration of 2-Way Tapered Hollow Marine Rudder With Non-Classical Pivot: Theoretical Study

Abstract

A theoretical analysis of free dry and wet vibration of a trapezoidal, 2-way tapered, marine spade rudder, is presented. The rudder is considered as a hollow Kirchhoff’s plate, with the chord section as a NACA profile. The chord length and the thickness taper from the top to the bottom, over the vertical span. The rudder is pivoted at the top, with the pivot behind the leading edge. The pivot is modeled as a combination of a translational and a rotational spring, in order to include the rigid body modes of the rudder vibration. The span-wise and chord-wise non-uniform beam vibration is first analyzed by the Rayleigh-Ritz method, in order to establish the non-uniform beam mode shapes. The span-wise beam is a linearly tapered vertical cantilever, with non-classical edge at the top and free at the bottom. The chord-wise section is a 2-span beam with the ends free, and four continuity conditions at the pivot. The non-uniform mode shapes, in either direction, are a weighted summation of the uniform beam mode shapes, which also satisfy the boundary/continuity conditions. They now act as admissible spatial functions to the plate vibration, which is analyzed by the Galerkin’s method. Eigenvalue analysis generates the plate natural frequencies. A weighted superposition, of the product of the beam mode shapes, in either direction, generates the plate mode shapes. Alternately, uniform beam mode shapes are used as admissible functions into the Galerkin’s method for the plate natural frequencies and mode shapes. The natural frequencies are generated for various positions of the rudder stock along the chord length. The pivot conditions (in both translational and rotational rigid body degree of freedom) influence the prominence of the rigid body mode shapes. The natural frequencies are analyzed for various pivot fixities, taper ratios, and aspect ratios of the plate. This is followed by the wet vibration analysis of the rudder. First, 2D strip theory is used to generate the added mass of each chord section. Constant strength source distribution technique is used to generate the added mass in sway and yaw of a 2D aerofoil. Each flexural and torsional mode is associated with its own added mass. Various empirical corrections are done to account for the 3D flow. Finally, 3D panel method is used to generate the modal added masses, and hence the wet natural frequencies. The added mass coefficient is generated for various aerofoil fineness ratios, pivot fixities, taper ratios, and aspect ratios of the plate.