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Advances in Nano Research Volume 14, Number 2, February 2023 , pages 127-139 DOI: https://doi.org/10.12989/.2023.14.2.127 |
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Proposing a dynamic stiffness method for the free vibration of bi-directional functionally-graded Timoshenko nanobeams |
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Mohammad Gholami, Mojtaba Gorji Azandariani, Ahmed Najat Ahmed and Hamid Abdolmaleki
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| Abstract | ||
| This paper studies the free vibration behavior of bi-dimensional functionally graded (BFG) nanobeams subjected to arbitrary boundary conditions. According to Eringen's nonlocal theory and Hamilton's principle, the underlying equations of motion have been obtained for BFG nanobeams. Moreover, the variable substitution method is utilized to establish the structure's state-space differential equations, followed by forming the dynamic stiffness matrix based on state-space differential equations. In order to compute the natural frequencies, the current study utilizes the Wittrick–Williams algorithm as a solution technique. Moreover, the nonlinear vibration frequencies calculated by employing the proposed method are compared to the frequencies obtained in previous studies to evaluate the proposed method's performance. Some illustrative numerical examples are also given in order to study the impacts of the nonlocal parameters, material property gradient indices, nanobeam length, and boundary conditions on the BFG nanobeam's frequency. It is found that reducing the nonlocal parameter will usually result in increased vibration frequencies. | ||
| Key Words | ||
| bidirectional functionally graded; dynamic stiffness method; Eringen's nonlocal theory, free vibration, nanobeams; state-space differential equations | ||
| Address | ||
| Mohammad Gholami: Department of Civil Engineering, Yasouj University, Yasouj, Iran Mojtaba Gorji Azandariani: Centre for Infrastructure Engineering, Western Sydney University, Sydney, Australia Ahmed Najat Ahmed: Department of Computer Engineering, College of Engineering and Computer Science, Lebanese French University, Kurdistan Region, Iraq Hamid Abdolmaleki: Department of Civil Engineering, Tuyserkan Branch, Islamic Azad University, Tuyserkan, Iran | ||