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

ZHOU Zhenhuan; LI Yuejie; FAN Junhai; SUI Guohao; ZHANG Junlin; XU Xinsheng

doi:10.21656/1000-0887.390130

Volume 39 Issue 10

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Article Navigation > Applied Mathematics and Mechanics > 2018 > 39(10): 1159-1171

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

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

doi: 10.21656/1000-0887.390130

State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology); Department of Engineering Mechanics, International Research Center for Computational Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

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

Abstract

Abstract

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.
- Hamiltonian system,
- symplectic method,
- functionally graded double-nanobeam system,
- free vibration,
- analytical solution

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References

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