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Exact solution of the free vibration of exponentially non-uniform functionlly graded Timoshenko beams |
Deng Hao, Cheng wei |
School of Aeronautic Science and Engineering,Beijing University of Aeronautics and Astronautics,Beijing 100191,China |
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Abstract Based on the state space variable replacement,the transfer matrix equation of a Timoshenko beam with axially exponential distributed functional gradation was derived.The exact solution of natural frequencies of the structure with multiple boundary conditions was obtained by the transfer matrix method and compared with the available analytical solution.The results show that the relation curve between the frequency and the gradient of the material is continuous and smooth,and there is no jumping phenomenon.Meanwhile the finite element method was used to verify the results.The effects of shear stiffness and moment of inertia on the natural frequencies of the structure were analyzed by comparing the results according to different beam theories.The calculation results show that the method presented is clear in physical concept and can reduce the computational complexity and the amount of computation.
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Received: 11 August 2015
Published: 15 March 2017
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