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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.

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In this paper, with the aim of calculating the Quasi-steady aerodynamic forces of articulated wing, a kinematic mechanism model for flapping wings is presented. First of all, the Kempf patent is used for simulating this mechanism, due to its simplicity and proper simulating of the flapping motion of articulated wings. This motion includes not-the-same-phase upstroke and downstroke motion of each part of wing. The angular position, angular velocity, angular acceleration and forces applied to both inner and outer part of the wing are analytically analyzed. Lifting line theory; that predicts the lift distribution of the three-dimensional wings based on the bounded vortex at the aerodynamic center, and is applied for single part wings of insects in the literature; is applied for articulated wings of birds, for the first time, in the present work. The average aerodynamic forces of articulated wing are obtained by calculating downwash and bounded vortex at each wing section and integrating on the wing surface. The results for an Ornithopter like 1kg gull with 5m/s of cruise speed indicate that both parts of the wing provide the lift. In addition, the outer wing has the main role to produce the thrust against the inner wing. The results show good agreement between the present work and the computational fluid dynamics method.