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Analytical and numerical investigation into the longitudinal vibration of uniform nanotubes
Masoud MASOUMI,Mehdi MASOUMI
Front. Mech. Eng.. 2014, 9 (2): 142-149.
https://doi.org/10.1007/s11465-014-0292-z
In recent years, prediction of the behaviors of micro and nanostructures is going to be a matter of increasing concern considering their developments and uses in various engineering fields. Since carbon nanotubes show the specific properties such as strength and special electrical behaviors, they have become the main subject in nanotechnology researches. On the grounds that the classical continuum theory cannot accurately predict the mechanical behavior of nanostructures, nonlocal elasticity theory is used to model the nanoscaled systems. In this paper, a nonlocal model for nanorods is developed, and it is used to model the carbon nanotubes with the aim of the investigating into their longitudinal vibration. Following the derivation of governing equation of nanorods and estimation of nondimensional frequencies, the effect of nonlocal parameter and the length of the nanotube on the obtained frequencies are studied. Furthermore, differential quadrature method, as a numerical solution technique, is used to study the effect of these parameters on estimated frequencies for both classical and nonlocal theories.
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A fast compound direct iterative algorithm for solving transient line contact elastohydrodynamic lubrication problems
Jian LIU,Yuxue CHEN,Zhenzhi HE,Shunian YANG
Front. Mech. Eng.. 2014, 9 (2): 156-167.
https://doi.org/10.1007/s11465-014-0297-7
A fast compound direct iterative algorithm for solving transient line contact elastohydrodynamic lubrication (EHL) problems is presented. First, by introducing a special matrix splitting iteration method into the traditional compound direct iterative method, the full matrices for the linear systems of equations are transformed into sparse banded ones with any half-bandwidth; then, an extended Thomas method which can solve banded linear systems with any half-bandwidth is derived to accelerate the computing speed. Through the above two steps, the computational complexity of each iteration is reduced approximately from O(N3/3) to O(β2N), where N is the total number of nodes, and β is the half-bandwidth. Two kinds of numerical results of transient EHL line contact problems under sinusoidal excitation or pure normal approach process are obtained. The results demonstrate that the new algorithm increases computing speed several times more than the traditional compound direct iterative method with the same numerical precision. Also the results show that the new algorithm can get the best computing speed and robustness when the ratio, half-bandwidth to total number of nodes, is about 7.5%–10.0% in moderate load cases.
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Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM
M. R. AKBARI,D. D. GANJI,A. MAJIDIAN,A. R. AHMADI
Front. Mech. Eng.. 2014, 9 (2): 177-190.
https://doi.org/10.1007/s11465-014-0288-8
In the present paper, three complicated nonlinear differential equations in the field of vibration, which are Vanderpol, Rayleigh and Duffing equations, have been analyzed and solved completely by Algebraic Method (AGM). Investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by numerical method (Runge-Kutte 4th). Based on the comparisons which have been made between the gained solutions by AGM and numerical method, it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. The results reveal that this method is not only very effective and simple, but also reliable, and can be applied for other complicated nonlinear problems.
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