Exact Solution of the Elastica with Transverse Shear Effects

2014 ◽  
Vol 14 (08) ◽  
pp. 1440019
Author(s):  
Adrian Marcel Zalmanovici ◽  
Reyolando Manoel Lopes Rebello da Fonseca Brasil ◽  
José Manoel Balthazar
Keyword(s):  
Exact Solution ◽  
Transverse Shear ◽  
Closed Form Solution ◽  
Numerical Procedure ◽  
Form Solution ◽  
Total Potential ◽  
Euler’S Elastica ◽  
Shear Effects ◽  
Set Up ◽  
Transverse Shear Effects

A new derivation of Euler's Elastica with transverse shear effects included is presented. The elastic potential energy of bending and transverse shear is set up. The work of the axial compression force is determined. The equation of equilibrium is derived using the variation of the total potential. Using substitution of variables an exact solution is derived. The equation is transcendental and does not have a closed form solution. It is evaluated in a dimensionless form by using a numerical procedure. Finally, numerical examples of laminates made of composite material (fiber reinforced) and sandwich panels are provided.

Free-Vibration of Rotating Composite Beams Incorporating Higher-Order Transverse Shear Effects

2001 ◽  
Author(s):  
N. K. Chandiramani ◽  
L. I. Librescu ◽  
C. D. Shete
Keyword(s):  
Free Vibration ◽  
Transverse Shear ◽  
Present Model ◽  
Structural Model ◽  
Laminated Composite ◽  
Higher Order ◽  
Rotating Blade ◽  
Box Beam ◽  
Shear Effects ◽  
Transverse Shear Effects

Abstract The free vibration behavior of a rotating blade modeled as a laminated composite hollow (single celled) box beam is studied. The geometrically nonlinear structural model developed herein incorporates a number of non-classical effects such as anisotropy, heterogeneity, transverse shear flexibility, and warping inhibition. The centrifugal and Coriolis force field effects are also included. The main focus here being the refinement of the existing model, the traction-free boundary conditions are satisfied here in contrast to the existing model. The resulting linearized equations and numerical results based on them are presented. Results obtained for the present higher-order shearable model are compared with those of the existing first-order shearable and the non-shearable models. Tailoring studies using the present model reveal an enhancement of eigenfrequency characteristics.

scholarly journals Analytic Solution for Systems of Two-Dimensional Time Conformable Fractional PDEs by Using CFRDTM

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Abir Chaouk ◽  
Maher Jneid
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Analytic Solution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Two Dimensional ◽  
Fractional Pdes ◽  
Differential Transform ◽  
Computational Work ◽  
Similar Solutions

In this study we use the conformable fractional reduced differential transform (CFRDTM) method to compute solutions for systems of nonlinear conformable fractional PDEs. The proposed method yields a numerical approximate solution in the form of an infinite series that converges to a closed form solution, which is in many cases the exact solution. We inspect its efficiency in solving systems of CFPDEs by working on four different nonlinear systems. The results show that CFRDTM gave similar solutions to exact solutions, confirming its proficiency as a competent technique for solving CFPDEs systems. It required very little computational work and hence consumed much less time compared to other numerical methods.

Derivation of Position-Probability Density for the Transient Nano-Tunnel Problem in SET

2010 ◽  
Author(s):  
Sheam-Chyun Lin ◽  
Hsien-Chang Shih ◽  
Fu-Sheng Chuang ◽  
Ming-Lun Tsai ◽  
Harki Apri Yanto ◽  
...  
Keyword(s):  
Probability Density ◽  
Error Function ◽  
Closed Form Solution ◽  
Form Solution ◽  
Single Electron ◽  
Mathematical Operation ◽  
Similarity Parameter ◽  
Set Up ◽  
Position Probability ◽  
Good Agreement

This theoretical investigation intends to study the nano-tunnel problem of the single electron transistor (SET), which is one of the most important components in the nano-electronics industry. With a combined effort of quantum mechanics and similarity parameter, the partial differential equation of transient position-probability density is attained and can be applied to predict the electron’s position inside the nano tunnel. Also, an appropriate set of the initial and the boundary conditions is set up in accordance to the actual electron behavior for solving this PDE of probability density function. Thereafter, a simple, closed-form solution for the probability density is obtained and expressed in terms of the error function for a new similarity variable η. Note that this analytic similarity solution is easy to perform the calculation and suitable for any further mathematical operation, such as the optimization applications. In addition, it is shown that these predications are reasonable and in good agreement to the physical meanings, which are evaluated from both microscopic and macroscopic viewpoints. In conclusions, this is an innovative approach by using the Schro¨dinger equation directly to solve the nano-tunnel problem. Moreover, with the aids of this analytic position-probability-density solution, it is illustrated that the free single electron in the SET’s tunnel can only appear at some specified regions, which are defined by a dimensionless parameter η within a range of 0 ≤ η ≤ 2. This result can be served as a valuable design reference for setting the practical manufacture requirement.

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