Free Vibration of Thin Spherical Shells

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
April Bryan
Keyword(s):  
Differential Equation ◽  
Spherical Shell ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Transverse Motion ◽  
Hemispherical Shell ◽  
Surface Displacements ◽  
Thin Spherical Shell ◽  
The One ◽  
The Relationship

This research introduces a new approach to analytically derive the differential equations of motion of a thin spherical shell. The approach presented is used to obtain an expression for the relationship between the transverse and surface displacements of the shell. This relationship, which is more explicit than the one that can be obtained through use of the Airy stress function, is used to uncouple the surface and normal displacements in the spatial differential equation for transverse motion. The associated Legendre polynomials are utilized to obtain analytical solutions for the resulting spatial differential equation. The spatial solutions are found to exactly satisfy the boundary conditions for the simply supported and the clamped hemispherical shell. The results to the equations of motion indicate that the eigenfrequencies of the thin spherical shell are independent of the azimuthal coordinate. As a result, there are several mode shapes for each eigenfrequency. The results also indicate that the effects of midsurface tensions are more significant than bending at low mode numbers but become negligible as the mode number increases.

Free Vibration of Thin Shallow Elliptical Shells

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
April Bryan
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Transverse Motion ◽  
Transverse Strain ◽  
Compatibility Equation ◽  
New Approach ◽  
Strain Compatibility ◽  
Mathieu Equations

This research presents a study of the free vibration of thin, shallow elliptical shells. The equations of motion for the elliptical shell, which are developed from Love's equations, are coupled and nonlinear. In this research, a new approach is introduced to uncouple the transverse motion of the shallow elliptical shell from the surface coordinates. Through the substitution of the strain-compatibility equation into the differential equations of motion in terms of strain, an explicit relationship between the curvilinear surface strains and transverse strain is determined. This latter relationship is then utilized to uncouple the spatial differential equation for transverse motion from that of the surface coordinates. The approach introduced provides a more explicit relationship between the surface and transverse coordinates than could be obtained through use of the Airy stress function. Angular and radial Mathieu equations are used to obtain solutions to the spatial differential equation of motion. Since the recursive relationships that are derived from the Mathieu equations lead to an infinite number of roots, not all of which are physically meaningful, the solution to the eigenvalue problem is used to determine the mode shapes and eigenfrequencies of the shallow elliptical shell. The results of examples demonstrate that the eigenfrequencies of the thin shallow elliptical shell are directly proportional to the curvature of the shell and inversely proportional to the shell's eccentricity.

Computer simulation of the hydraulic milking method

1992 ◽  
Vol 59 (3) ◽  
pp. 253-264
Author(s):  
M. Clare Butler ◽  
Robert J. Grindal
Keyword(s):  
Differential Equation ◽  
Flow Rate ◽  
Equations Of Motion ◽  
Flow Rates ◽  
Air Inlet ◽  
Wall Movement ◽  
Linear Differential ◽  
Milking Machine ◽  
The Relationship ◽  
Return Valve

SummaryA mathematical model of the interactions within a milking machine teatcup has been developed, which describes the relationship between liner wall movement, pressures and flow rate when milking without an air inlet. It is based on equations of motion for a column of incompressible fluid and requires a second-order, non-linear differential equation to be solved. Incorporating a non-return valve allows hydraulic milking to be modelled, and the comparison between predicted and measured pressures, flow rates and liner wall movement when milking hydraulically is shown. The model can be used to optimize milking conditions to reduce vacuum peaks, improve liner opening and thus maximize flow rate.

scholarly journals Free vibration characteristics of thick doubly curved variable stiffness composite laminated shells using higher-order shear deformation theory

2018 ◽  
Vol 172 ◽  
pp. 03010 ◽  
Author(s):  
Anand Venkatachari ◽  
K. Ramajeyathilagam
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Variable Stiffness ◽  
Mode Shapes ◽  
Laminated Shells ◽  
Surface Displacements ◽  
Edge Conditions ◽  
Geometric Variables

In the present work, the natural frequencies of cylindrical and spherical laminated shells with variable stiffness are numerically studied using a shear flexible isogeometric finite element. The kinematics relies on cubic shear deformation theory in which cubic variation is assumed for the surface displacements and a quadratic variation for the traverse displacement along the thickness. A zig-zag function, used for the in-plane displacements, accounts for the abrupt discontinuity at the boundaries of the laminae. The Lagrangian equations of motion is deployed to solve the frequencies of curved panels. A detailed parametric analysis examines the influence of fibre centre/edge angles, shell geometric variables, material anisotropy and edge conditions on frequencies and mode shapes.

Convergence of a thin spherical shell

2007 ◽  
Vol 5 ◽  
pp. 307-323
Author(s):  
S.V. Khabirov ◽  
A.R. Garifullin
Keyword(s):  
Spherical Shell ◽  
Dynamic Stability ◽  
Equations Of Motion ◽  
Asymptotic Formulas ◽  
Precise Solution ◽  
Solution Of Equations ◽  
Initial Stage ◽  
Thin Spherical Shell

The precise solution of equations of motion of a true liquid describing a convergence to the center of a thin spherical shell is considered. At the initial stage prime asymptotic formulas which are used for studying of a dynamic stability in relation to potential indignations are received.

Misreading Egypt

2018 ◽  
Author(s):  
Jesse Schotter
Keyword(s):  
New Media ◽  
Narrative Form ◽  
Experimental Film ◽  
Tower Of Babel ◽  
Avant Garde ◽  
Sergei Eisenstein ◽  
History Of ◽  
The One ◽  
The Relationship ◽  
Modern Technologies

The first chapter of Hieroglyphic Modernisms exposes the complex history of Western misconceptions of Egyptian writing from antiquity to the present. Hieroglyphs bridge the gap between modern technologies and the ancient past, looking forward to the rise of new media and backward to the dispersal of languages in the mythical moment of the Tower of Babel. The contradictory ways in which hieroglyphs were interpreted in the West come to shape the differing ways that modernist writers and filmmakers understood the relationship between writing, film, and other new media. On the one hand, poets like Ezra Pound and film theorists like Vachel Lindsay and Sergei Eisenstein use the visual languages of China and of Egypt as a more primal or direct alternative to written words. But Freud, Proust, and the later Eisenstein conversely emphasize the phonetic qualities of Egyptian writing, its similarity to alphabetical scripts. The chapter concludes by arguing that even avant-garde invocations of hieroglyphics depend on narrative form through an examination of Hollis Frampton’s experimental film Zorns Lemma.

Researching (with) Major Food Retailers

2015 ◽  
Vol 15 (3) ◽  
pp. 33-39 ◽  
Author(s):  
David Evans
Keyword(s):  
Food Systems ◽  
Academic Research ◽  
Social Scientists ◽  
Major Food ◽  
Food Retailers ◽  
Food Retailer ◽  
The One ◽  
The Uk ◽  
The Relationship ◽  
Opening Up

This paper considers the relationship between social science and the food industry, and it suggests that collaboration can be intellectually productive and morally rewarding. It explores the middle ground that exists between paid consultancy models of collaboration on the one hand and a principled stance of nonengagement on the other. Drawing on recent experiences of researching with a major food retailer in the UK, I discuss the ways in which collaborating with retailers can open up opportunities for accessing data that might not otherwise be available to social scientists. Additionally, I put forward the argument that researchers with an interest in the sustainability—ecological or otherwise—of food systems, especially those of a critical persuasion, ought to be empirically engaging with food businesses. I suggest that this is important in terms of generating better understandings of the objectionable arrangements that they seek to critique, and in terms of opening up conduits through which to affect positive changes. Cutting across these points is the claim that while resistance to commercial engagement might be misguided, it is nevertheless important to acknowledge the power-geometries of collaboration and to find ways of leveling and/or leveraging them. To conclude, I suggest that universities have an important institutional role to play in defining the terms of engagement as well as maintaining the boundaries between scholarship and consultancy—a line that can otherwise become quite fuzzy when the worlds of commerce and academic research collide.

Choice Of Technique And The Volume Of Saving (A Re-Examination Of The Cobb-Douglas Model)

1968 ◽  
Vol 8 (4) ◽  
pp. 606-617
Author(s):  
Mohammad Anisur Rahman
Keyword(s):  
Economic Development ◽  
Production Function ◽  
Development Policy ◽  
Social Goals ◽  
The Other ◽  
Economic Development Policy ◽  
Labour Intensity ◽  
The One ◽  
Choice Of Technique ◽  
The Relationship

The purpose of this paper is to re-examine the relationship between the degree of aggregate labour-intensity and the aggregate volume of saving in an economy where a Cobb-6ouglas production function in its traditional form can be assumed to give a good approximation to reality. The relationship in ques¬tion has an obviously important bearing on economic development policy in the area of choice of labour intensity. To the extent that and in the range where an increase in labour intensity would adversely affect the volume of savings, a con¬flict arises between two important social objectives, i.e., higher rate of capital formation on the one hand and greater employment and distributive equity on the other. If relative resource endowments in the economy are such that such a "competitive" range of labour-intensity falls within the nation's attainable range of choice, development planners will have to arrive at a compromise between these two social goals.

Bastard Feudalism and the Framing of Thirteenth-Century England

2018 ◽  
Author(s):  
Peter Coss
Keyword(s):  
Middle Ages ◽  
Thirteenth Century ◽  
Twelfth Century ◽  
The State ◽  
The Other ◽  
Great Work ◽  
Early Middle Ages ◽  
The One ◽  
The Relationship

In the introduction to his great work of 2005, Framing the Early Middle Ages, Chris Wickham urged not only the necessity of carefully framing our studies at the outset but also the importance of closely defining the words and concepts that we employ, the avoidance ‘cultural sollipsism’ wherever possible and the need to pay particular attention to continuities and discontinuities. Chris has, of course, followed these precepts on a vast scale. My aim in this chapter is a modest one. I aim to review the framing of thirteenth-century England in terms of two only of Chris’s themes: the aristocracy and the state—and even then primarily in terms of the relationship between the two. By the thirteenth century I mean a long thirteenth century stretching from the period of the Angevin reforms of the later twelfth century on the one hand to the early to mid-fourteenth on the other; the reasons for taking this span will, I hope, become clearer during the course of the chapter, but few would doubt that it has a validity.

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