Modified Mixed Variational Principle and the State-Vector Equation for Elastic Bodies and Shells of Revolution

1992 ◽  
Vol 59 (3) ◽  
pp. 587-595 ◽  
Author(s):  
Charles R. Steele ◽  
Yoon Young Kim
Keyword(s):  
Variational Principle ◽  
Equations Of State ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Shells Of Revolution ◽  
Elastic Bodies ◽  
Mixed Variational Principle ◽  
Stress Components ◽  
Independent Variable ◽  
Nonsymmetric Deformation

A modified mixed variational principle is established for a class of problems with one spatial variable as the independent variable. The specific applications are on the three-dimensional deformation of elastic bodies and the nonsymmetric deformation of shells of revolution. The possibly novel feature is the elimination in the variational formulation of the stress components which cannot be prescribed on the boundaries. The result is a form exactly analogous to classical mechanics of a dynamic system, with the equations of state exactly in the form of the canonical equations of Hamilton. With the present approach, the correct scale factors of the field variables to make the system self-adjoint are readily identified, and anisotropic materials including composites can be handled effectively. The analysis for shells of revolution is given with and without the transverse shear deformation considered.

scholarly journals On the quantization of a theory arising from a variational principle for multiple integrals with application to born's electrodynamics

1936 ◽  
Vol 156 (887) ◽  
pp. 192-220 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Classical Mechanics ◽  
Mathematical Aspect ◽  
Quantization Method ◽  
Generalized Coordinates ◽  
Starting Point ◽  
Independent Variable ◽  
The Common ◽  
Multiple Integrals

The transition from classical mechanics to quantum mechanics has been formulated in various ways. But all these ways have the common feature that they use as starting point mechanics not in its Newtonian form, but in the form which it was given by Lagrange and Hamilton. The essence of this form consists in the fact that the differential equations are represented as Euler equations of a variational principle for v functions, the generalized coordinates, of one variable, the time. In its purely mathematical aspect the quantization method may, therefore, be regarded as a mode of taking certain notions and certain relations arising from a variational principle for several dependent and one independent variable and of giving them a new meaning.

Research on Three-Dimensional Displacement Measurement Technology of the Beating Heart

2011 ◽  
Vol 403-408 ◽  
pp. 5182-5186
Author(s):  
Sheng Yi Yang ◽  
An Gu ◽  
Meng Li ◽  
Chang Jian Lu
Keyword(s):  
Heart Surgery ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Beating Heart ◽  
Displacement Measurement ◽  
Surgical Instruments ◽  
Analytical Models ◽  
Measurement Technology ◽  
Position Measurement ◽  
Mechanics Of Materials

In robotic-assisted heart surgery, the method of canceling the relative motion between the surgical site on the heart and the surgical instruments was introduced in this paper. A whisker sensor was designed for three dimensional position measurement in beating heart surgery. Analytical models were developed according to the classical mechanics of materials, and theoretical formulas were derived for displacement measurement. Feasibility and effectiveness of the method were verified by simulation experiments. We can obtain measurements by loading displacement to the whisker sensor, and draw conclusions by comparing the measurements.

Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2005 ◽  
Vol 72 (5) ◽  
pp. 797-800 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Shells Of Revolution ◽  
Dimensional Theory ◽  
Conical Shells ◽  
Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

Tip Leakage Flow Characteristics Downstream of an Axial Flow Fan

1996 ◽  
Author(s):  
P. Puddu
Keyword(s):  
Vortex Flow ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Axial Flow ◽  
Tip Leakage Flow ◽  
Single Wire ◽  
Vortex System ◽  
Tip Leakage ◽  
Stress Components ◽  
Computer Controlled

The three-dimensional viscous flow characteristics and the complex vortex system downstream of the rotor of an industrial exial fan have been determined by an experimental investigation using hot-wire anemometer. Single-wire slanted and straight type probes have been rotated about the probe axis using a computer controlled stepper motor. Measurements have been taken at four planes behind the blade trailing edge. The results show the characteristics of the relative flow as velocity components, secondary flow and kinetic energy defect. Turbulence intensity and Reynolds stress components in the leakage vortex area are also presented. The evolution of the leakage vortex flow during the decay process has also been evaluated in terms of dimension, position and intensity.

scholarly journals Anharmonic Self-Consistent Theory of Crystals. II. The 2d and 3d Quartic Crystal Models

1994 ◽  
Vol 49 (6) ◽  
pp. 663-670
Author(s):  
S. Sh. Soulayman ◽  
C. Ch. Marti ◽  
Ch. Ch. Guilpin
Keyword(s):  
Helmholtz Free Energy ◽  
Equations Of State ◽  
Three Dimensional ◽  
Inert Gases ◽  
Thermoelastic Properties ◽  
Consistent Theory ◽  
Jones Potential ◽  
Lennard Jones ◽  
2D And 3D ◽  
Strong Anharmonicity

Abstract In this paper we apply the method developed in part I for describing the crystalline state of two and three dimensional inert gases. For strong anharmonicity of fourth order, the equations of state of these gases are obtained. This way we calculate the thermoelastic properties of two and three dimensional argon, krypton and xenon using the Lennard-Jones potential. The corrections to the Helmholtz free energy and thermodynamic properties due to quantum effects are considered. The results are compared with the available experimental data.

Studies in Three-Dimensional Photoelasticity Torsional Stresses by Oblique Incidence

1944 ◽  
Vol 11 (4) ◽  
pp. A229-A234
Author(s):  
Max M. Frocht
Keyword(s):  
Oblique Incidence ◽  
Three Dimensional ◽  
Complete Agreement ◽  
Stress System ◽  
Pure Torsion ◽  
Photoelastic Method ◽  
Stress Components ◽  
Single Stress ◽  
Stress Patterns ◽  
Maximum Stresses

Abstract In this paper a photoelastic method is described for the study of stresses in cylindrical shafts due to pure torsion. The basic photoelastic equation governing this case is derived. It is shown that the maximum stresses, i.e., the boundary stresses, can be determined from a single stress pattern, and that the two stress components which define the complete stress system in pure torsion at an interior point can be determined from two stress patterns obtained from one section of a shaft into which the pure-torsion system has been frozen, employing to this end the Drucker-Mindlin suggestion of oblique incidence. The method is applied to a circular shaft, and the experimental results are found to be in complete agreement with the theory.

Management of Complex Loading Histories for Use in Probabilistic Creep-Fatigue Damage Assessments

2018 ◽  
Author(s):  
N. A. Zentuti ◽  
J. D. Booker ◽  
R. A. W. Bradford ◽  
C. E. Truman
Keyword(s):  
Fatigue Damage ◽  
Input Parameter ◽  
Three Dimensional ◽  
Complex Loading ◽  
Fe Model ◽  
Plant Component ◽  
Fatigue Lifetime ◽  
Wide Range ◽  
Stress Components ◽  
Creep Fatigue

An approach is outlined for the treatment of stresses in complex three-dimensional components for the purpose of conducting probabilistic creep-fatigue lifetime assessments. For conventional deterministic assessments, the stress state in a plant component is found using thermal and mechanical (elastic) finite element (FE) models. Key inputs are typically steam temperatures and pressures, with the three principal stress components (PSCs) at the assessment location(s) being the outputs. This paper presents an approach which was developed based on application experience with a tube-plate ligament (TPL) component, for which historical data was available. Though both transient as well as steady-state conditions can have large contributions towards the creep-fatigue damage, this work is mainly concerned with the latter. In a probabilistic assessment, the aim of this approach is to replace time intensive FE runs with a predictive model to approximate stresses at various assessment locations. This is achieved by firstly modelling a wide range of typical loading conditions using FE models to obtain the desire stresses. Based on the results from these FE runs, a probability map is produced and input(s)-output(s) functions are fitted (either using a Response Surface Method or Linear Regression). These models are thereafter used to predict stresses as functions of the input parameter(s) directly. This mitigates running an FE model for every probabilistic trial (of which there typically may be more than 104), an approach which would be computationally prohibitive.

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