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Variable pitch propeller (VPP) are used in advanced helicopters, in order to achieve greater efficiency, better stability and achieve higher altitudes. This study is going to assess the behavior of VPP propeller with coupled non-linear displacement in three degrees of freedom. Accordingly, the behavior of this type of propeller with Changes of elastic axis distance, Length, mass, speed, polar radius of gyration, Stiffness in three degrees of freedom, and pitch have been investigated. In this paper, Gallerkin method is used to extract natural frequencies and the results were evaluated with the results reported by other researchers. The results show convergence and accuracy of the used method. In this study, it was found that parameters of mass, length and rotational speed of the propeller have effect on the natural frequencies, and all modes of vibration. However, other parameters except for the pitch angle effect on the odd or even number of frequency modes. It was also found that the pitch angle in the static mode does not effect on natural frequencies but in the case of rotation of propeller, affect on natural frequency of vibration modes as sine or cosine form.

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Free hydroelastic coupled vibration analysis of frictionless liquids with a free surface in spherical tanks with a flexible bottom has been performed. The side wall has been considered to treate as a rigid body. The flexible bottom treats as a membrane at a certain distance bellow the center point, and the free surface is considered as a cross cutting at the top of the center point. The spherical coordinate system is adopted to derive the governing coupled equations, and finally a vibration analysis is carried out, using the traditional Galerkin's method, leading to closed-form solutions. Effects of various system parameters, i.e., membrane tension, liquid density, geometric parameters of the system such as the container radius, free surface distance discriminate parameter, and bottom distance discriminate parameter on the vibration behavior are investigated. The novelty of the present work is to obtain direct formulas for hydroelastic coupled vibration analysis of the mentioned system, which can be easily used in engineering design applications. Coupling between two mode numbers can be clearly seen in results, in other words, there is a coupling between vibration modes as interaction in spherical geometry.