ZHOU Zhenhuan, LI Yuejie, FAN Junhai, SUI Guohao, ZHANG Junlin, XU Xinsheng. A Symplectic Approach for Free Vibration of Functionally Graded Double-Nanobeam Systems Embedded in Viscoelastic Medium[J]. Applied Mathematics and Mechanics, 2018, 39(10): 1159-1171. doi: 10.21656/1000-0887.390130
Citation: ZHOU Zhenhuan, LI Yuejie, FAN Junhai, SUI Guohao, ZHANG Junlin, XU Xinsheng. A Symplectic Approach for Free Vibration of Functionally Graded Double-Nanobeam Systems Embedded in Viscoelastic Medium[J]. Applied Mathematics and Mechanics, 2018, 39(10): 1159-1171. doi: 10.21656/1000-0887.390130

A Symplectic Approach for Free Vibration of Functionally Graded Double-Nanobeam Systems Embedded in Viscoelastic Medium

doi: 10.21656/1000-0887.390130
Funds:  The National Natural Science Foundation of China(11672054); The National Basic Research Program of China(973 Program)(2014CB046803); The National Key R&D Program of China(2016YFB0201600)
  • Received Date: 2018-04-23
  • Rev Recd Date: 2018-05-18
  • Publish Date: 2018-10-01
  • A new analytical approach was proposed for free vibration of functionally graded (FG) double-nanobeam systems (DNBSs) embedded in viscoelastic medium under the framework of symplectic mechanics and the nonlocal Timoshenko beam theory. In the Hamiltonian system, the dual variables of the displacement and the rotation angle are the generalized shear force and bending moment, respectively. The high-order governing partial differential equations in the classical Lagrangian system were simplified into a set of ordinary differential equations through introduction of an unknown vector composed of the fundamental variables and their dual variables. The free vibration of DNBSs was finally reduced to an eigenproblem in the symplectic space. Analytical frequency equations and vibration mode functions were directly obtained with the symplectic eigensolutions and boundary conditions. Numerical results verify the accuracy and efficiency of the presented method. A systematic parametric study on the small size effect, the interaction between the double nanobeams and the viscoelastic foundation influence, was also provided.
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