On the Proper Boundary Conditions for a Beam

1992 ◽  
Vol 59 (4) ◽  
pp. 915-922 ◽  
Author(s):  
H. Fan ◽  
G. E. O. Widera
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Cross Section ◽  
Boundary Data ◽  
Circular Cross Section ◽  
Numerical Calculations ◽  
Analysis Technique ◽  
Outer Expansion ◽  
Beam Theories

Employing the asymptotic expansion approach, the boundary conditions of a beam are reconsidered in the present paper. Gregory and Wan’s (1984) decay analysis technique is extended here to formulate the boundary conditions for the outer expansion. Among the various prescribed boundary data, most of the attention is focused on the displacement case because engineering beam theories employ incorrect conditions for these data. Numerical calculations are carried out for the displacement prescribed beam having a circular cross-section.

Natural Frequencies of In-Plane Vibration of Arcs

1983 ◽  
Vol 50 (2) ◽  
pp. 449-452 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
K. Tanaka
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Plane Vibration

The natural frequencies of in-plane vibration are presented for uniform arcs with circular cross section under all combinations of boundary conditions.

Finite amplitude convection cells

1967 ◽  
Vol 30 (3) ◽  
pp. 577-600 ◽  
Author(s):  
J. L. Robinson
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Cross Section ◽  
Circular Cross Section ◽  
Maximum Heat ◽  
Mean Field Equations

In this paper we consider two-dimensional steady cellular motion in a fluid heated from below at large Rayleigh number and Prandtl number of order unity. This is a boundary-layer problem and has been considered by Weinbaum (1964) for the case of rigid boundaries and circular cross-section. Here we consider cells of rectangular cross-section with three sets of velocity boundary conditions: all boundaries free, rigid horizontal boundaries and free vertical boundaries (referred to here as periodic rigid boundary conditions), and all boundaries rigid; the vertical boundaries of the cells are insulated. It is shown that the geometry of the cell cross-section is important, such steady motion being not possible in the case of free boundaries and circular cross-section; also that the dependence of the variables of the problem on the Rayleigh number is determined by the balances in the vertical boundary layers.We assume only those boundary layers necessary to satisfy the boundary conditions and obtain a Nusselt number dependence $N \sim R^{\frac{1}{3}}$ for free vertical boundaries. For the periodic rigid case, Pillow (1952) has assumed that the buoyancy torque is balanced by the shear stress on the horizontal boundaries; this is equivalent to assuming velocity boundary layers beside the vertical boundaries (rather than the vorticity boundary layers demanded by the boundary conditions) and leads to a Nusselt number dependence N ∼ R¼. If it is assumed that the flow will adjust itself to give the maximum heat flux possible the two models are found to be appropriate for different ranges of the Rayleigh number and there is good agreement with experiment.An error in the application of Rayleigh's method in this paper is noted and the correct method for carrying the boundary-layer solutions round the corners is given. Estimates of the Nusselt numbers for the various boundary conditions are obtained, and these are compared with the computed results of Fromm (1965). The relevance of the present work to the theory of turbulent convection is discussed and it is suggested that neglect of the momentum convection term, as in the mean field equations, leads to a decrease in the heat flux at very high Rayleigh numbers. A physical argument is given to derive Gill's model for convection in a vertical slot from the Batchelor model, which is appropriate in the present work.

Beam Theory and Its Suitability for the Analysis of Pipes

1991 ◽  
Vol 113 (2) ◽  
pp. 291-296
Author(s):  
H. Fan ◽  
G. E. O. Widera ◽  
P. Afshari
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Cross Section ◽  
Three Dimensional ◽  
Beam Theory ◽  
Arbitrary Cross Section ◽  
Elasticity Equations ◽  
Expansion Technique ◽  
Three Dimensional Elasticity ◽  
Benchmark Solutions

The use of the asymptotic expansion technique when applied to the three-dimensional elasticity equations is outlined and used to demonstrate the development of an asymptotic beam theory and associated boundary conditions. The formulation thus obtained holds for arbitrary cross section shapes and is applied here to pipes. It can be used to provide benchmark solutions to test the suitability of engineering beam and shell theories.

Microscale Flow Visualization

2002 ◽  
Author(s):  
David Sinton ◽  
Dongqing Li
Keyword(s):  
Cross Section ◽  
Fluorescent Dyes ◽  
Numerical Technique ◽  
Circular Cross Section ◽  
Fluorescent Dye ◽  
Double Layers ◽  
Wall Region ◽  
Velocity Data ◽  
Analysis Technique ◽  
Near Wall

A combined experimental and numerical analysis technique for velocity profiles in liquid microchannel flows is described. The working fluids employed are aqueous solutions with caged fluorescent dyes. A sheet of fluorescent dye is photo-injected by briefly exposing a cross-section of the fluid to ultraviolet light. The transport of the resulting ‘band’ of fluorescent dye is imaged onto a CCD camera using an epi-illumination fluorescent microscope system. The velocity profile is calculated from images acquired and processed after each uncaging event. The analysis technique utilizes several images, to provide velocity data with an increased signal-to-noise ratio. This combined experimental/numerical technique can provide velocity data in the near-wall region as well as in the bulk. Results are shown to compare favorably to analytical solutions in both pressure-driven and electroosmotic flow in circular cross-section capillaries of 205μm and 102μm inner diameters, respectively. Near-wall resolution is verified through application to electroosmotic flows with thin electrical double layers.

Natural Frequencies of Out-of-Plane Vibration of Arcs

1982 ◽  
Vol 49 (4) ◽  
pp. 910-913 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
K. Tanaka
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Plane Vibration ◽  
Out Of Plane

The natural frequencies of out-of-plane vibration based on the Timoshenko beam theory are calculated numerically for uniform arcs of circular cross section under all combination of boundary conditions, and the results are presented in some figures.

scholarly journals Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Zhongmin Wang ◽  
Rongrong Li
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Cantilever Beam ◽  
Cross Section ◽  
Transverse Vibration ◽  
Slenderness Ratio ◽  
Circular Cross Section ◽  
Dimensionless Parameters ◽  
Inner Radius ◽  
Tapered Cantilever Beam

Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross-section are addressed, in which the inner radius of cross-section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross-sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen-equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

scholarly journals Electrical Conductivity of Thin Wires

1986 ◽  
Vol 12 (2) ◽  
pp. 103-109 ◽  
Author(s):  
R. Dimmich ◽  
F. Warkusz
Keyword(s):  
Electrical Conductivity ◽  
Boundary Conditions ◽  
Cross Section ◽  
Electron Scattering ◽  
Circular Cross Section ◽  
Boltzmann Transport ◽  
Wire Surface ◽  
Thin Wires ◽  
The Wire ◽  
Metal Wires

An expression for the electrical conductivity of thin metal wires of a circular cross-section has been derived taking account of conduction-electron scattering at grain boundaries and the wire surface. An angular-dependent specularity parameter has been introduced and possible fluctuations of the wire diameter along its length have been taken into account. Boltzmann transport equations have been solved for the boundary conditions proposed by Dingle.

SAF file: It's really three dimensional file

2018 ◽  
Vol 14 (1) ◽  
pp. 1
Author(s):  
Prof. Dr. Jamal Aziz Mehdi
Keyword(s):  
Cross Section ◽  
Root Canal ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Circular Motion ◽  
Root Canal Treatment ◽  
Metal Core ◽  
Rotating Blade

The biological objectives of root canal treatment have not changed over the recentdecades, but the methods to attain these goals have been greatly modified. Theintroduction of NiTi rotary files represents a major leap in the development ofendodontic instruments, with a wide variety of sophisticated instruments presentlyavailable (1, 2).Whatever their modification or improvement, all of these instruments have onething in common: they consist of a metal core with some type of rotating blade thatmachines the canal with a circular motion using flutes to carry the dentin chips anddebris coronally. Consequently, all rotary NiTi files will machine the root canal to acylindrical bore with a circular cross-section if the clinician applies them in a strictboring manner

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