Investigation of Fixed-Free Annular Plate Resonant Frequency Predictions

1988 ◽  
Vol 110 (3) ◽  
pp. 408-410 ◽  
Author(s):  
Alison Flatau ◽  
G. A. Flandro ◽  
W. K. Van Moorhem
Keyword(s):  
Free Boundary ◽  
Resonant Frequency ◽  
Elastic Plate ◽  
Annular Plate ◽  
Plate Theory ◽  
Outer Radius ◽  
Resonant Frequencies ◽  
Free Boundary Conditions ◽  
Annular Plates ◽  
Analytical Frequency

Nondimensional frequency parameters for predicting the resonant frequencies of annular plates with fixed-free boundary conditions as the plate inner to outer radius ratio approaches unity have been investigated experimentally. Frequency parameters have been determined by using modal analysis to measure resonant frequencies for annular plates of varied materials, thicknesses, and with radius ratios of 0.5 to 0.9. The data are compared to two different analytical frequency predictions which have been presented as solutions for resonance of fixed-free annular plates based on classical elastic plate theory.

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.

Buckling of Functionally Graded Circular/Annular Plates Based on the First-Order Shear Deformation Plate Theory

2004 ◽  
Vol 261-263 ◽  
pp. 609-614 ◽  
Author(s):  
L.S. Ma ◽  
Tie Jun Wang
Keyword(s):  
Shear Deformation ◽  
Material Properties ◽  
Volume Fraction ◽  
Plate Theory ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Outer Radius ◽  
Functionally Graded ◽  
First Order ◽  
Annular Plates

Based on the first-order shear deformation theory of plate, governing equations for the axisymmetric buckling of functionally graded circular/annular plates are derived. The coupled deflections and rotations in the pre-buckling state of the plates are neglected in analysis. The material properties vary continuously through the thickness of the plate, and obey a power law distribution of the volume fraction of the constituents. The resulting differential equations are numerically solved by using a shooting method. The critical buckling loads of circular and annular plates are obtained, which are compared with those obtained from the classical plate theory. Effects of material properties, ratio of inter to outer radius, ratio of plate thickness to outer radius, and boundary conditions on the buckling behavior of FGM plates are discussed.

PLASTIC BUCKLING OF MODERATELY THICK ANNULAR PLATES

2005 ◽  
Vol 05 (03) ◽  
pp. 337-357 ◽  
Author(s):  
TUN MYINT AUNG ◽  
C. M. WANG ◽  
J. CHAKRABARTY
Keyword(s):  
Shear Deformation ◽  
Transverse Shear ◽  
Plate Theory ◽  
Outer Radius ◽  
Stress Factors ◽  
Mindlin Plate ◽  
Transverse Shear Deformation ◽  
Plastic Buckling ◽  
Annular Plates ◽  
Buckling Stress

This paper is concerned with the plastic buckling of moderately thick annular plates under a uniform compressive stress state. The analysis is based on the incremental theory of plasticity which employs the Prandtl–Reuss equations and the plate material is assumed to obey the Ramberg–Osgood stress–strain relation. The effect of transverse shear deformation is taken into consideration by adopting the Mindlin plate theory. The governing differential equations for the plastic buckling problem are solved analytically and the plastic buckling stress factors for annular plates with the allowance of transverse shear deformation are presented for the first time. The influences of the boundary conditions, thickness to outer radius ratios, and inner to outer radius ratios on the buckling stress factors are also examined.

BUCKLING DEFORMATION OF ANNULAR PLATES DESCRIBING NATURAL FORMS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350054 ◽  
Author(s):  
YANG ZHANG ◽  
LU CHEN ◽  
SOMSAK SWADDIWUDHIPONG ◽  
ZISHUN LIU
Keyword(s):  
Solid Mechanics ◽  
Annular Plate ◽  
Soft Materials ◽  
Outer Radius ◽  
Inhomogeneous Deformation ◽  
Periodic Patterns ◽  
Energy Principle ◽  
Hyperelastic Materials ◽  
Annular Plates ◽  
Physical Response

It has been observed that undulating periodic patterns formed on an initial flat annular plate that model the leaves of plants are a physical response to the expansion of the surface under a lateral restraint. They are not visible at the beginning but become apparent only when the plants continue to grow. The behavior can be explained via the inhomogeneous deformation of gel materials that behave in a similar manner to hyperelastic materials in solid mechanics.This paper compares the stability of thin annular plates clamped along the inner edge and free along the outer periphery using numerical simulations of the swelling of thin gel annular plate held along the inner edge as well as analysis of a similar class of structures by solid mechanics concept via energy principle. The trends of results from both approaches compare favorably. The buckling patterns of annular plates with various values of inner radius to outer radius ratio illustrate the relationship between the geometry of the annular plate and the inhomogeneous deformation of gels or buckling patterns of solid mechanics materials. The undulating patterns on leaves such as those of flowering cabbage can thus be explained via the buckling behavior of annular plates, which can be regarded as thin soft materials adhered to a stiffer core. The study can also be extended to cover other stimuli under different environmental conditions and the outcome may bring further insights into the evolution of plants.

Axisymmetric Banding of Annular Plates

1978 ◽  
Vol 45 (4) ◽  
pp. 834-838 ◽  
Author(s):  
J. T. Tielking
Keyword(s):  
Annular Plate ◽  
Plate Theory ◽  
Ritz Method ◽  
Plate Thickness ◽  
Computational Procedure ◽  
Stress Distributions ◽  
The Finite Element Method ◽  
Annular Plates ◽  
Castigliano’S Theorem ◽  
Energy Formulation

The potential energy formulation of von Karman plate theory and the Ritz method are utilized to obtain solutions for deformation and stress in a variable thickness annular plate with built-in edges. The plate is loaded by prescribing an axisymmetric offset of the edges; the external axial force is calculated by Castigliano’s theorem. Solutions are presented to illustrate the influence of the plate thickness profile on the surface stress distribution and to show the effect of large deflections on the surface and membrane stress distributions in a plate with hyperbolic taper. The computational procedure is competitive with other procedures, such as the finite-element method.

Buckling of Polar Orthotropic Annular Plates Under Uniform Inplane Compressive Forces

1981 ◽  
Vol 48 (3) ◽  
pp. 643-653 ◽  
Author(s):  
G. K. Ramaiah
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Buckling Loads ◽  
Simply Supported ◽  
Free Boundary Conditions ◽  
Annular Plates ◽  
Design Engineers ◽  
Direct Use ◽  
Polar Orthotropic

The problem of buckling of polar orthotropic annular plates under various types of inplane compressive forces along the radial edges has been analyzed in detail by the Rayleigh-Ritz method for eight different combinations of clamped, simply supported, and free boundary conditions. Accurate estimates of critical buckling loads have been obtained for various values of hole ratios and for various values of rigidity ratios. The numerical results are presented in the form of data sheets for direct use by the design engineers.

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

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