Analysis of a Thin Elastic Ring Under Arbitrary Loading

1974 ◽  
Vol 96 (3) ◽  
pp. 870-876 ◽  
Author(s):  
J. Y. Liu ◽  
Y. P. Chiu
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Elastic Ring ◽  
Closed Form Solutions ◽  
Distributed Load ◽  
Arbitrary Loading ◽  
Thin Ring ◽  
Special Cases ◽  
Loading System ◽  
Thin Elastic Ring

This paper presents a general solution for a thin ring under a self-equilibrating loading system comprising any combination of radial, tangential, and moment loads. The formulations are applicable to concentrated loads as well as to distributed load functions. Closed-form solutions are obtained for each case for engineering applications. Comparisons with recent published results for some special cases are demonstrated in some of the sample problems.

Closed Form Solutions For Mixed Convection With Magnetohydrodynamic Effect in a Vertical Porous Annulus Surrounding an Electric Cable

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
A. Barletta ◽  
E. Magyari ◽  
S. Lazzari ◽  
I. Pop
Keyword(s):  
Mixed Convection ◽  
Closed Form ◽  
Heat Generation ◽  
Internal Heat ◽  
Boussinesq Approximation ◽  
Closed Form Solutions ◽  
Electric Cable ◽  
Special Cases ◽  
Magnetohydrodynamic Effect ◽  
The Magnetic Field

Mixed convection Darcy flow in a vertical porous annulus around a straight electric cable is investigated. It is assumed that the flow is fully developed and parallel. Moreover, the Boussinesq approximation is used. The magnetic field with a steady electric current in the cable is radially varying according to the Biot–Savart law. Two flow regimes are investigated. The first is mixed convection with negligible effects of internal heat generation due to Joule heating and viscous dissipation. The second is forced convection with important effects of heat generation. In these two special cases, closed form expressions of the velocity profile and of the temperature profile, as well as of the flow rate and the Nusselt number, are obtained. The main features of these solutions are discussed.

scholarly journals Influence of Flexible Foundation on Isolator Wave Effects

1996 ◽  
Vol 3 (1) ◽  
pp. 61-67 ◽  
Author(s):  
Ye-Ping Xiong ◽  
Kong-Jie Song ◽  
Xing Ai
Keyword(s):  
Closed Form ◽  
Vibration Isolation ◽  
Response Ratio ◽  
New Model ◽  
Closed Form Solutions ◽  
Special Cases ◽  
Wave Effects ◽  
Rigid Mass ◽  
Isolation Systems ◽  
Flexible Foundation

This article deals with the interaction between wave effects in mounts and resonances of foundations inflexible vibration isolation systems. A new model is proposed that is represented as a rigid mass supported by two linear unidirectional isolators on a flexible foundation beam, whose closed-form solutions for transmissibility and response ratio are then obtained, with which the influence of wave effects coupled with the flexibility of the foundation on the effectiveness of isolation is discussed. The wave effects on flexible isolation systems are analyzed under various parametric conditions and compared with those in rigid systems. In addition, several special cases are presented to show the transition between various limiting cases. Some approaches to control wave effects are also proposed.

Stress-Free Strains in Martensitic Microstructures

2013 ◽  
Vol 738-739 ◽  
pp. 10-14
Author(s):  
Michaël Peigney
Keyword(s):  
Phase Transformation ◽  
Theoretical Prediction ◽  
Shape Memory ◽  
Closed Form ◽  
Solid Phase ◽  
Closed Form Solutions ◽  
Special Cases ◽  
Crystallographic Structures ◽  
Solid Phase Transformation ◽  
Free Strains

The peculiar properties of shape-memory alloys are the result of a solid/solid phase transformation between different crystallographic structures (austenite and martensite). This paper is concerned with the theoretical prediction of the set of strains that minimize the effective (or macroscopic) energy. Those strains, classically refered to as recoverable strains, play a central role in shape memory effect displayed by alloys such as NiTi or CuAlNi. They correspond to macroscopic strains that can be achieved in stress-free states. Adopting the framework of nonlinear elasticity, the theoretical prediction of stress-free strains amounts to find the austenite/martensite microstructures which minimize the global energy. Closed-form solutions to that problem have been obtained only in few special cases. This paper aims at complementing existing results on that problem, essentially by deriving bounds on the set of stress-free strains.

scholarly journals Entropy-regularized 2-Wasserstein distance between Gaussian measures

2021 ◽  
Author(s):  
Anton Mallasto ◽  
Augusto Gerolin ◽  
Hà Quang Minh
Keyword(s):  
Fixed Point ◽  
Closed Form ◽  
Numerical Study ◽  
Wasserstein Distance ◽  
Iteration Algorithm ◽  
Probability Measures ◽  
Gaussian Distributions ◽  
Closed Form Solutions ◽  
Special Cases

AbstractGaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. They furthermore stand as important special cases for frameworks providing geometries for probability measures, as the resulting geometry on Gaussians is often expressible in closed-form under the frameworks. In this work, we study the Gaussian geometry under the entropy-regularized 2-Wasserstein distance, by providing closed-form solutions for the distance and interpolations between elements. Furthermore, we provide a fixed-point characterization of a population barycenter when restricted to the manifold of Gaussians, which allows computations through the fixed-point iteration algorithm. As a consequence, the results yield closed-form expressions for the 2-Sinkhorn divergence. As the geometries change by varying the regularization magnitude, we study the limiting cases of vanishing and infinite magnitudes, reconfirming well-known results on the limits of the Sinkhorn divergence. Finally, we illustrate the resulting geometries with a numerical study.

Joint and Cross Acceptances for Cross-Flow-Induced Vibration—Part I: Theoretical and Finite Element Formulations

2000 ◽  
Vol 122 (3) ◽  
pp. 349-354 ◽  
Author(s):  
M. K. Au-Yang
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Cross Flow ◽  
Theoretical Development ◽  
Element Formulation ◽  
Flow Induced Vibration ◽  
Arbitrary Boundary ◽  
One Dimensional ◽  
Closed Form Solutions ◽  
Special Cases

The theoretical development of the acceptance integral method to estimate the random vibration of structures subject to turbulent flow is critically reviewed and put onto a firm mathematical basis. Closed-form solutions for the joint acceptances for cross-flow-induced vibration of one-dimensional structures are derived for two special cases of spring-supported and simply supported beams. These are used to check results from a finite element formulation of the acceptance integrals for one-dimensional structures with arbitrary boundary conditions, and for arbitrary correlation lengths. Agreements between the finite element and closed-form solutions are excellent. [S0094-9930(00)02303-9]

Coincidence of Boobnov-Galerkin and Closed-Form Solutions in an Applied Mechanics Problem

2003 ◽  
Vol 70 (5) ◽  
pp. 777-779 ◽  
Author(s):  
I. Elishakoff ◽  
M. Zingales
Keyword(s):  
Galerkin Method ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mechanics Of Solids ◽  
Applied Mechanics ◽  
Graduate Courses ◽  
Closed Form Solutions ◽  
Distributed Load ◽  
Comparison Functions

It is shown that the utilization of the Filonenko-Borodich set of functions, as the comparison functions in the Boobnov-Galerkin method leads to the result that coincides with the closed-form solution for the clamped-clamped uniform beam under uniformly distributed load. It is hoped that this remarkable, direct coincidence could be used in graduate courses and books on mechanics of solids.

Modeling Finite Squeeze Film Dampers

2002 ◽  
Author(s):  
A. El-Shafei
Keyword(s):  
Closed Form ◽  
Reynolds Equation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Perturbation Solution ◽  
Squeeze Film ◽  
Closed Form Solutions ◽  
Special Cases ◽  
Squeeze Film Dampers ◽  
Finite Difference Solution

Most closed form solutions of Reynolds’ equation assume either a short bearing approximation or a long bearing approximation. These closed form approximations are used in rotordynamic simulation applications, otherwise a Finite Difference solution of Reynolds’ equation would be prohibitively time consuming. Recently, there have been proposed series solutions for Reynolds’ equation for special cases. In this paper, a perturbation solution to the governing equations is proposed to obtain a closed form solution of Reynolds’ equation for a finite squeeze film damper executing a circular centered orbit. The pressure field and velocity profiles are obtained. It is shown that in the limit the finite damper solution approaches either the appropriate short or long damper. This perturbation solution can be used with appropriate boundary conditions, for various damper sealing configurations, and provides insight into the damper performance.

General Solutions of Anisotropic Laminated Plates

2003 ◽  
Vol 70 (4) ◽  
pp. 496-504 ◽  
Author(s):  
W.-L. Yin
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Analytic Functions ◽  
Infinite Plate ◽  
Elliptical Hole ◽  
Complex Variables ◽  
Laminated Plates ◽  
Eigenvalues And Eigenvectors ◽  
Closed Form Solutions ◽  
Repeated Eigenvalues

Anisotropic laminates with bending-stretching coupling possess eigensolutions that are analytic functions of the complex variables x+μky, where the eigenvalues μk and the corresponding eigenvectors are determined in the present analysis, along with the higher-order eigenvectors associated with repeated eigenvalues of degenerate laminates. The analysis and the resulting expressions are greatly simplified by using a mixed formulation involving a new set of elasticity matrices A*, B*, and D*. There are 11 distinct types of laminates, each with a different expression of the general solution. For an infinite plate with an elliptical hole subjected to uniform in-plane forces and moments at infinity, closed-form solutions are obtained for all types of anisotropic laminates in terms of the eigenvalues and eigenvectors.

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