Exact Eigensolutions for a Family of Nonuniform Rods With End Point Masses

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Lourdes Rubio ◽  
José Fernández-Sáez ◽  
Antonino Morassi
Keyword(s):  
Closed Form ◽  
Canonical Form ◽  
Longitudinal Vibration ◽  
Countable Family ◽  
Darboux Transformations ◽  
Closed Form Solutions ◽  
End Point ◽  
Sturm Liouville ◽  
Tip Mass

In this paper, new exact closed-form solutions for free longitudinal vibration of a one-parameter countable family of cantilever rods with one end tip mass are obtained. The analysis is based on the reduction of the equation governing the longitudinal vibration to the Sturm–Liouville canonical form and on the use of double Darboux transformations. The rods for which exact eigensolutions are provided are explicitly determined in terms of an initial rod with known closed-form eigensolutions. The method can be also extended to include longitudinally vibrating rods with tip mass at both ends.

Closed-form solutions for forced vibrations of piezoelectric energy harvesters by means of Green’s functions

2017 ◽  
Vol 28 (17) ◽  
pp. 2372-2387 ◽  
Author(s):  
X Zhao ◽  
EC Yang ◽  
YH Li ◽  
W Crossley
Keyword(s):  
Closed Form ◽  
Piezoelectric Materials ◽  
Forced Vibrations ◽  
Rotational Inertia ◽  
Closed Form Solutions ◽  
Energy Harvesters ◽  
Piezoelectric Energy ◽  
Laplace Transform Technique ◽  
Damping Effects ◽  
Tip Mass

In this article, the closed-form solutions are obtained for the forced vibrations of cantilevered unimorph piezoelectric energy harvesters. A tip mass is attached at the free end, and the moment of its inertia to the fixed end is considered. Timoshenko beam assumptions are used to establish a coupled electromechanical model for the harvester. Two damping effects, transverse and rotational damping effects, are taken into account. Green’s function method and Laplace transform technique are used to solve the coupled electromechanical vibration system. The conventional case of a harmonic base excitation is considered, and numerical calculations are performed. The present model is validated by comparing its predictions with the existing data, the experimental results, and the finite element method solutions. The influences of shear deformation and rotational inertia on the predictions are discussed. The effect of load resistance on the electrical power is studied, and the optimal load resistances are obtained. Ultimately, the optimal schemes are proposed to improve electricity generation performance for the soft piezoelectric materials: PZT-5A/5H.

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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