Large free vibration of thin plates: Hierarchic finite Element Method and asymptotic linearization |
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| Authors: | M Taazount A Zinai A Bouazzouni |
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| Institution: | 1. Civil Engineering Laboratory “LGC”, Polytech-Blaise Pascal University, BP 206, 63174 Aubière, France;2. Mechanical, structures and Energetic Laboratory “LMSE”, Mouloud Mammeri University, Tizi-Ouzou, BP 17 RP, 15000 Tizi-Ouzou, Algeria;1. Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine, 14b, Metrolohichna Str., Kyiv 03680, Ukraine;2. Dumansky Institute of Colloid and Water Chemistry, Nat. Acad. of Sci. of Ukraine, 42, Acad. Vernadsky Boulevard, Kyiv 03680, Ukraine;1. State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, PR China;2. Earthquake Engineering Research and Test Center, Guangzhou University, Guangzhou 510405, PR China;3. Cardiff School of Engineering, Cardiff University, Cardiff CF 24 3AA, Wales, UK;1. School of Civil and Environmental Engineering, University of New South Wales, Australia;2. Universidade do Porto, Faculdade de Engenharia, Porto, Portugal;3. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia;4. Institute of Mechanics and Advanced Materials, Cardiff University, Wales, UK;5. Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, Italy;1. INEGI/LAETA, Faculty of Engineering, University of Porto, Porto, Portugal;2. INEGI/LAETA, Instituto Superior de Engenharia, IPP, Porto, Portugal |
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| Abstract: | The aim of this work is to study the free dynamic response of thin plates characterized by geometrical nonlinearities. To achieve this task, the equation of motion of the plate is first carried out through modeling by hierarchical finite element method whose interpolating shape functions are sinusoidal. Then, the study of the nonlinear vibrations was carried out by the development of asymptotic linearization and equivalent linearization methods in modal space. The nonlinear angular frequencies are successively deduced by exciting the corresponding vibrating mode of the structure. The confrontation of these results to those obtained by the iterative method in the physical space and to those found in the literature, showed a very good agreement between the various methods. From the elementary nonlinear frequencies we showed that there exists an equivalent linear dynamical system characterized by only one equivalent linear stiffness matrix. Numerical experiments were carried out on beams and thin plates of various dimensions ratios and boundary conditions. These numerical test simulations, whether in time space or frequency space, have showed that the nonlinear elastic energy is restored by the equivalent linear dynamical system. Nevertheless, we have to say that the dynamic effects of modes above the excited one are neglected. |
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