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Dynamic stiffness of nonlocal damped nano-beams on elastic foundation
European Journal of Mechanics - A/Solids, Volume: 86, Start page: 104144
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Free and forced bending vibration of damped nonlocal nano-beams resting on an elastic foundation is investigated. Two types of nonlocal damping models, namely, strain-rate-dependent viscous damping and velocity-dependent viscous damping are considered. A frequency-dependent dynamic finite element me...
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Free and forced bending vibration of damped nonlocal nano-beams resting on an elastic foundation is investigated. Two types of nonlocal damping models, namely, strain-rate-dependent viscous damping and velocity-dependent viscous damping are considered. A frequency-dependent dynamic finite element method is developed to obtain the forced vibration response. Frequency-adaptive complex-valued shape functions are used for the derivation of the dynamic stiffness matrix. It is shown that there are six unique coefficients which define the general dynamic stiffness matrix. These complex-valued coefficients are obtained exactly in closed-form and illustrated numerically as functions of the frequency. It is proved that the general dynamic stiffness matrix reduces to the well known special cases under different conditions. The stiffness and mass matrices of the nonlocal beam are also obtained using the conventional finite element method. A numerical algorithm to extract the eigenvalues from the dynamic stiffness matrix with a transcendental element for the special case when the system is undamped is suggested. Results from the dynamic finite element method and the conventional finite element method are compared. The application of the dynamic stiffness approach is shown through forced response analysis of a double-walled carbon nanotube in pinned-pinned and cantilever configurations. Explicit closed-form expressions of the dynamic response for both the cases have been obtained and the role of crucial system parameters such as, the damping factors, the nonlocal parameter and the foundation stiffness have been investigated.
Bending vibration; Nonlocal mechanics; Dynamic stiffness; Asymptotic analysis; Frequency response
College of Engineering