Closed form solution for free vibrations analysis of FGPM thick cylinders employing FSDT under various boundary conditions

2019 ◽  
Vol 229 ◽  
pp. 111403
Author(s):  
Hossein Daemi ◽  
Hamidreza Eipakchi
Keyword(s):  
Boundary Conditions ◽  
Closed Form ◽  
Closed Form Solution ◽  
Free Vibrations ◽  
Form Solution ◽  
Various Boundary Conditions

A Closed Form Solution of Dual-Phase Lag Heat Conduction Problem With Time Periodic Boundary Conditions

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Pranay Biswas ◽  
Suneet Singh ◽  
Hitesh Bindra
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Closed Form ◽  
Computational Cost ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fourier Series Expansion ◽  
Phase Lag ◽  
Dual Phase ◽  
Time Periodic

The Laplace transform (LT) is a widely used methodology for analytical solutions of dual phase lag (DPL) heat conduction problems with consistent DPL boundary conditions (BCs). However, the inversion of LT requires a series summation with large number of terms for reasonably converged solution, thereby, increasing computational cost. In this work, an alternative approach is proposed for this inversion which is valid only for time-periodic BCs. In this approach, an approximate convolution integral is used to get an analytical closed-form solution for sinusoidal BCs (which is obviously free of numerical inversion or series summation). The ease of implementation and simplicity of the proposed alternative LT approach is demonstrated through illustrative examples for different kind of sinusoidal BCs. It is noted that the solution has very small error only during the very short initial transient and is (almost) exact for longer time. Moreover, it is seen from the illustrative examples that for high frequency periodic BCs the Fourier and DPL model give quite different results; however, for low frequency BCs the results are almost identical. For nonsinusoidal periodic function as BCs, Fourier series expansion of the function in time can be obtained and then present approach can be used for each term of the series. An illustrative example with a triangular periodic wave as one of the BC is solved and the error with different number of terms in the expansion is shown. It is observed that quite accurate solutions can be obtained with a fewer number of terms.

Application of Orthotropic Plate Theory to Windmill Blade Design

1981 ◽  
Vol 103 (4) ◽  
pp. 892-894 ◽  
Author(s):  
C. Rubin
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Closed Form ◽  
Orthotropic Plate ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Layered Structures ◽  
Form Solution ◽  
Blade Design ◽  
Free Boundary Conditions

The windmill blade is treated as a semi-infinite orthotropic wedge with free-free boundary conditions. A closed form solution for the deflections and stresses is obtained as a function of the loading. The loading may be quite general. Results for three different materials which are commonly used for windmill blades (aluminum, sitka spruce, and fiberglass) are obtained. Applications also include ribbed, corrugated, and layered structures. In addition, other types of boundary conditions may be used to obtain solutions to a wide variety of other orthotropic plate problems.

scholarly journals Analytic Simulations of Packaging Cardboards with Non-Rigid Adhesives

2019 ◽  
Vol 7 (6) ◽  
pp. 01-15
Author(s):  
R. Hussein
Keyword(s):  
Boundary Conditions ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Equilibrium Equations ◽  
The Real ◽  
Usual Assumption ◽  
Perfect Bonding

The understanding of the cardboard performance is necessary to the design of packaging containers and the protection of their contents for safe deliveries. The use of adhesives is unavoidable in the manufacturing of the cardboards. Like all materials, the adhesives have finite stiffness but when used in the literature, they are assumed perfectly rigid. This study changes this assumption by using the real properties of adhesives. A closed-form solution for cardboard panelsassembled withnon-rigid adhesives, and subjected to edgewise loading is presented. The solution satisfies the equilibrium equations of the layers, the compatibility equations of stresses and strains at the interfaces, and the boundary conditions. To investigate the effects of the finite values of adhesivestiffness on the responses, numerical evaluations are conducted. The results obtained have shown that the adhesive stiffness has a strong effect on the performance. Beyond a certain level of stiffness, the usual assumption of perfect bonding used in classical theories is acceptable. This could provide an answer to what constitutes perfect bonding in terms of the ratio of the fluted layer, or simply flute, stiffness to the bonding stiffness.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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