Closed form solutions for nonlinear static response of curled cantilever micro-/nanobeams including both the fringing field and van der Waals force effect

2016 ◽  
Vol 23 (1) ◽  
pp. 163-174 ◽  
Author(s):  
Youhong Sun ◽  
Yongping Yu ◽  
Baisheng Wu ◽  
Baochang Liu
Keyword(s):  
Closed Form ◽  
Van Der Waals ◽  
Van Der Waals Force ◽  
Static Response ◽  
Closed Form Solutions ◽  
Force Effect ◽  
Fringing Field ◽  
Nonlinear Static

Closed Form Solutions for the Problem of Statical Behavior of Nano/Micromirrors Under the Effect of Capillary Force and Van Der Waals Force

2011 ◽  
Author(s):  
Ali Darvishian ◽  
Hamid Moeenfard ◽  
Hasan Zohoor ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Capillary Force ◽  
Rotation Angle ◽  
Van Der Waals ◽  
Van Der Waals Force ◽  
Combined Effect ◽  
Geometrical Parameters ◽  
Closed Form Solutions ◽  
The Stability ◽  
Vdw Force ◽  
Stability Limits

The current paper deals with the problem of static instability of Micro/Nano mirrors under the combined effect of capillary force and van der Waals force. First the governing equations of the statical behavior of Micro/Nano mirrors under the combined effect of capillary force and casimir force is obtained using the newtons first law of motion. The dependence of the critical tilting angle on the physical and geometrical parameters of the nano/micromirror and its supporting torsional beams is investigated. It is found that existence of vdW torque can considerably reduce the stability limits of the nano/micromirror. It is also found that rotation angle of the mirror due to capillary force highly depends on the vdW toque applied to the mirror. Finally analytical tool Homotopy Perturbation Mehtod (HPM) is utilized for prediction of the nano/micromirror behaviour under combined capillary and vdW force. It is observed that a sixth order perturbation approximation accurately predicts the rotation angle and stability limits of the mirror. Results of this paper can be used for successful fabrication of nano/micromirrors using wet etching process where capillary force plays a major role in the system.

XXXVII.—The Van der Waals Force between a Proton and a Hydrogen Atom. II. Excited States

1948 ◽  
Vol 62 (3) ◽  
pp. 360-368 ◽  
Author(s):  
C. A. Coulson ◽  
C. M. Gillam
Keyword(s):  
Hydrogen Atom ◽  
Closed Form ◽  
Interaction Energy ◽  
Excited States ◽  
Internuclear Distance ◽  
Van Der Waals ◽  
Quantum States ◽  
Van Der Waals Force ◽  
Image Position

SummaryThe interaction energy, or Van der Waals force, between a proton and a hydrogen atom in any one of its allowed quantum states is calculated in terms of the internuclear distance R by an expansion of the formAll the coefficients up to and including E5 are obtained in closed form. For values of R for which the expansion is valid, the coefficients are determined absolutely, no approximations being introduced.

Closed Form Solutions of Joint Water-Filling for Coordinated Transmission

2010 ◽  
Vol E93-B (12) ◽  
pp. 3461-3468 ◽  
Author(s):  
Bing LUO ◽  
Qimei CUI ◽  
Hui WANG ◽  
Xiaofeng TAO ◽  
Ping ZHANG
Keyword(s):  
Closed Form ◽  
Closed Form Solutions ◽  
Water Filling
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