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.

A closed form solution for bending/stretching analysis of functionally graded circular plates under asymmetric loading using the Green function

2010 ◽  
Vol 224 (6) ◽  
pp. 1153-1163 ◽  
Author(s):  
A R Saidi ◽  
A Naderi ◽  
E Jomehzadeh
Keyword(s):  
Green Function ◽  
Closed Form ◽  
Volume Fraction ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Circular Plates ◽  
Equilibrium Equations

In this article, a closed-form solution for bending/stretching analysis of functionally graded (FG) circular plates under asymmetric loads is presented. It is assumed that the material properties of the FG plate are described by a power function of the thickness variable. The equilibrium equations are derived according to the classical plate theory using the principle of total potential energy. Two new functions are introduced to decouple the governing equilibrium equations. The three highly coupled partial differential equations are then converted into an independent equation in terms of transverse displacement. A closed-form solution for deflection of FG circular plates under arbitrary lateral eccentric concentrated force is obtained by defining a new coordinate system. This solution can be used as a Green function to obtain the closed-form solution of the FG plate under arbitrary loadings. Also, the solution is employed to solve some different asymmetric problems. Finally, the stress and displacement components are obtained exactly for each problem and the effect of volume fraction is also studied.

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.

Thermoelastic Fields in Boundary Layers of Isotropic Laminates

2005 ◽  
Vol 72 (1) ◽  
pp. 86-101 ◽  
Author(s):  
Christian Mittelstedt ◽  
Wilfried Becker
Keyword(s):  
Closed Form ◽  
Edge Effects ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Laminate Plate ◽  
Free Edge ◽  
Linear Displacement ◽  
Form Solution ◽  
State Variables ◽  
Layered Plates

An approximate approach to the calculation of displacements, strains, and stresses near edges and corners in symmetric rectangular layered plates of dissimilar isotropic materials under thermal load is presented. In the thickness direction the plate is discretized into an arbitrary number of sublayers/mathematical layers. A layerwise linear displacement field is formulated such that the terms according to classical laminate plate theory are upgraded with unknown in-plane functions and a linear interpolation scheme through the layer thickness in order to describe edge and corner perturbations. By virtue of the principle of minimum potential energy the governing coupled Euler–Lagrange differential equations are derived, which in the case of free-edge effects allow a closed-form solution for the unknown inplane functions. Free-corner effects are investigated by combining the displacement formulations of the two interacting free-edge effects. Hence, all state variables in the plate are obtained in a closed-form manner. Boundary conditions of traction free plate edges are satisfied in an integral sense. The present methodology is easily applied and requires only reasonable computational expenses.

Flexural stiffness of leaf springs for compliant micromechanisms

2008 ◽  
Vol 222 (12) ◽  
pp. 2505-2511 ◽  
Author(s):  
P Angeli ◽  
F De Bona ◽  
M G Munteanu
Keyword(s):  
Closed Form ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Constant Thickness ◽  
Flexural Stiffness ◽  
Beam Model ◽  
Flexural Behaviour ◽  
Leaf Springs ◽  
Linear Behaviour

Von Kármán equations have been used to evaluate the flexural behaviour of rectangular leaf springs with constant thickness. A closed form solution is obtained, showing that flexural stiffness varies continuously from that obtained by considering a beam model to the value given by the linear plate theory. This behaviour depends on section geometry, Poisson's ratio, and main curvature. A new characterizing parameter, whose relation with flexural stiffness allows a typical non-linear behaviour to be emphasized, is introduced in this work. In particular, for a given geometry and material, the flexural stiffness increases with the deflection and consequently with the load.

Closed Form Solutions for Vibration and Sound Radiation of Orthotropic Plates under Thermal Environment

2018 ◽  
Vol 18 (07) ◽  
pp. 1850098 ◽  
Author(s):  
Kai Zhou ◽  
Jinpeng Su ◽  
Hongxing Hua
Keyword(s):  
Closed Form ◽  
Orthotropic Plate ◽  
Acoustic Analysis ◽  
Thermal Environment ◽  
Separation Of Variables ◽  
Closed Form Solution ◽  
Form Solution ◽  
Thermal Loads ◽  
Orthotropic Plates ◽  
Heated Plates

This paper presents a closed form solution for the vibration and acoustic problem of orthotropic plates under a thermal environment. Hamilton’s principle is utilized to derive the governing equation of motion for the orthotropic plate with thermal loads, which is then solved by the method of separation of variables. The frequency equations and mode functions obtained for the orthotropic heated plates with at least two adjacent edges clamped are much simpler than those by the conventional methods. Several numerical examples are carried out for the modal, dynamic and acoustic analysis of orthotropic heated plates with different combinations of thermal loads and boundary conditions. The results of the parametric study for the orthotropic plate with different thermal loads are discussed in detail. The validity of the present formulation is confirmed by comparing the results obtained with the numerical ones. Due to its accuracy, efficiency and versatility, the present method offers an efficient tool for the structural and acoustic analysis of the orthotropic plate under the thermal environment.

Inverse-Ray Imaging of 2D Layered Structures from Seismic Reflection Data

2003 ◽  
Vol 19 (1) ◽  
pp. 191-196 ◽  
Author(s):  
Tan K. Wang ◽  
S. C. Tan
Keyword(s):  
Closed Form ◽  
Travel Time ◽  
Complex Structure ◽  
Closed Form Solution ◽  
Normal Incidence ◽  
Layered Structures ◽  
Form Solution ◽  
Seismic Reflection Data ◽  
Reflection Data ◽  
Uppermost Layer

ABSTRACTShooting angle of an inverse ray for imaging 2D multi-layered structures from reflected travel-times is derived in a closed form. By considering the normal incidence of two neighboring rays reflected at interfaces when sources are at the same locations as receivers, the traveling distance and direction of two inverse-rays are determined successively from the lowermost layer to the uppermost layer. This approach is similar to and also confirmed with the Huygens' principle that the equal travel-time along a wave front (perpendicular to the rays) is conserved. The closed-form solution of the inverse rays is further applied to image a complex structure of a ramp-flat fault with eleven layers. The results demonstrate that the inverse-ray imaging from travel-time picks of all layers is superior to that picked by a layer-stripping approach.

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.

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