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Analysis of the bar's free vibrations with considering lateral shear strain by the finite element method

In this paper, the author uses the forced displacement method combined with the finite element method to study the free vibration of the bar with different boundary conditions with considering the influence of the shear strain, the theory used here is the full beam theory. | NGHIÊN CỨU KHOA HỌC nNgày nhận bài 01 4 2022 nNgày sửa bài 15 4 2022 nNgày chấp nhận đăng 16 5 2022 Analysis of the bar s free vibrations with considering lateral shear strain by the finite element method Phân tích dao động tự do của thanh có xét đến biến dạng trượt ngang bằng phương pháp phần tử hữu hạn gt A. Prof. Phd DOAN VAN DUAN Faculty of Engineering - Vietnam Maritime University. Email duandv.ct@vimaru.edu.vn ABSTRACT shear force this is only true for beams with a small cross-sectional The beam structure with a large cross-sectional height compared height compared to the beam length h Ldisplacement method combined with the finite element method Putting expressions 1 and 3 in 2 get to build and solve the problem of free oscillation of the bar with 4W 3V 2W considering the influence of the lateral shear strain according to EJ 3 m 2 0 a x 4 GF x t the numerical solution. 4 3W 2V EJ V 0 b 2. THE PROBLEM OF FREE VIBRATION OF THE BAR WITH x 3 GF x 2 CONSIDERING THE LATERAL SHEAR STRAIN The solution of system 4 can be written in the form Consider a straight bar of constant cross-section with mass m uniformly distributed over the bar. When there is a lateral W x t y x cos t y cos t 5 displacement then in addition to the internal forces M and Q the V x t Q x cos t Q cos t inertia force fm must also be considered. The force of inertia fm is Then system 4 has the form the product of the mass and the acceleration of motion and whose d4y d 3Q direction of action is the direction of motion the direction of EJ 4 m 2 y cos t 0 dx GF dx 3 deflection of the bar. Thus the inertial force has the same effect as 6 the lateral force in this case is the distributed lateral force applied d y 3 d Q 2 EJ 3 Q cos t 0 at the bar axis. If the mass m is distributed over the height of the dx GF dx 2 bar section then due to the rotation of the bar cross section there is also a rotational inertia force of the bar cross section. For Since the component in brackets does not depend on t system