Dynamics of Gyroelastic Continua

1984 ◽  
Vol 51 (2) ◽  
pp. 415-422 ◽  
Author(s):  
G. M. T. D’Eleuterio ◽  
P. C. Hughes
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Angular Momentum ◽  
Volume Element ◽  
Basis Functions ◽  
Vibration Modes ◽  
Linear Elastic ◽  
Operator Notation ◽  
General Motion ◽  
The Continuum

This paper introduces the idea of distributed gyricity, in which each volume element of a continuum possesses an infinitesimal quantity of stored angular momentum. The continuum is also assumed to be linear-elastic. Using operator notation, a partial differential equation is derived that governs the small displacements of this gyroelastic continuum. Gyroelastic vibration modes are derived and used as basis functions in terms of which the general motion can be expressed. A discretized approximation is also developed using the method of Rayleigh-Ritz. The paper concludes with a numerical example of gyroelastic modes.

A Meshless Method for Solving the Mathematical Model Associated the Leakage Problem in Gas Pipeline

2014 ◽  
Vol 986-987 ◽  
pp. 1418-1421
Author(s):  
Jun Shan Li
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Basis Function ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Gas Pipeline ◽  
The Mathematical Model

In this paper, we propose a meshless method for solving the mathematical model concerning the leakage problem when the pressure is tested in the gas pipeline. The method of radial basis function (RBF) can be used for solving partial differential equation by writing the solution in the form of linear combination of radius basis functions, that is, when integrating the definite conditions, one can find the combination coefficients and then the numerical solution. The leak problem is a kind of inverse problem that is focused by many engineers or mathematical researchers. The strength of the leak can find easily by the additional conditions and the numerical solutions.

The Hamilton-Jacobi Equation Applied to Continuum

1997 ◽  
Vol 64 (3) ◽  
pp. 658-663 ◽  
Author(s):  
C. M. Leech
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Principal Function ◽  
Partial Differential ◽  
One Dimensional ◽  
Linear Elastic ◽  
Elastic Bar ◽  
Continuum Systems ◽  
Hamilton Jacobi Equation

The Hamilton-Jacobi partial differential equation is established for continuum systems; to do this a new concept in material distributions is introduced. The Lagrangian and Hamiltonian are developed, so that the Hamilton-Jacobi equation can be formulated and the principal function defined. Finally the principal function is constructed for the dynamics of a one-dimensional linear elastic bar; the solution for its’ vibrations is then established following the differentiation of the principal function.

Impact Mechanics of Elastic Structures With Point Contact

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Róbert Szalai
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Point Contact ◽  
Simple Criterion ◽  
Modeling Framework ◽  
Regular Behavior ◽  
Impact Phenomena ◽  
New Form ◽  
Delay Differential ◽  
The Continuum

This paper introduces a modeling framework that is suitable to resolve singularities of impact phenomena encountered in applications. The method involves an exact transformation that turns the continuum, often partial differential equation description of the contact problem into a delay differential equation. The new form of the physical model highlights the source of singularities and suggests a simple criterion for regularity. To contrast singular and regular behavior the impacting Euler–Bernoulli and Timoshenko beam models are compared.

Note on the physical interpretation of Chandrasekhar’s separation constants in his solution of Dirac’s equation in Kerr geometry

1989 ◽  
Vol 424 (1867) ◽  
pp. 323-326
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Angular Momentum ◽  
Differential Equations ◽  
Physical Interpretation ◽  
Total Angular Momentum ◽  
Flat Space ◽  
Single Factor ◽  
Partial Differential ◽  
Kerr Geometry

It is shown that in the limit of vanishing Kerr length-parameter a, Chandrasekhar’s separation constant λ is equal to ± ( j + ½ )/√2, where j is the total angular momentum. This result is derived from a comparison of the form of the simultaneous partial differential equation obeyed by Chandrasekhar’s angular functions, S +½ (θ, φ) and S -½ (θ, φ), with the differential equations obeyed by the corresponding pair of angular functions in flat space. The latter are taken from the solution of Dirac’s equation given by Schrödinger and by Pauli, in which the dependence of all four spinors on the azimuthal variable φ is given by the single factor of e imp , as in Chandrasekhar’s solution. For finite values of a, one can use the analytic expansion of λ in powers of a given by Pekeris & Frankowski.

scholarly journals A free boundary model of epithelial dynamics

2018 ◽  
Author(s):  
Ruth E Baker ◽  
Andrew Parker ◽  
Matthew J Simpson
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Partial Differential Equation ◽  
Free Boundary ◽  
Continuum Limit ◽  
Equation Model ◽  
Free Boundary Condition ◽  
Partial Differential ◽  
Cell Dynamics ◽  
The Continuum

AbstractIn this work we analyse a one-dimensional, cell-based model of an epithelial sheet. In this model, cells interact with their nearest neighbouring cells and move deterministically. Cells also proliferate stochastically, with the rate of proliferation specified as a function of the cell length. This mechanical model of cell dynamics gives rise to a free boundary problem. We construct a corresponding continuum-limit description where the variables in the continuum limit description are expanded in powers of the small parameter 1/N, where N is the number of cells in the population. By carefully constructing the continuum limit description we obtain a free boundary partial differential equation description governing the density of the cells within the evolving domain, as well as a free boundary condition that governs the evolution of the domain. We show that care must be taken to arrive at a free boundary condition that conserves mass. By comparing averaged realisations of the cell-based model with the numerical solution of the free boundary partial differential equation, we show that the new mass-conserving boundary condition enables the coarsegrained partial differential equation model to provide very accurate predictions of the behaviour of the cell-based model, including both evolution of the cell density, and the position of the free boundary, across a range of interaction potentials and proliferation functions in the cell based model.

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