Generalized Hypergeometric Function Solutions on the Transverse Vibration of a Class of Nonuniform Beams

1967 ◽  
Vol 34 (3) ◽  
pp. 702-708 ◽  
Author(s):  
Han-Chung Wang
Keyword(s):  
Transverse Vibration ◽  
Hypergeometric Functions ◽  
Beam Theory ◽  
Frequency Equation ◽  
Cantilever Beams ◽  
Sectional Area ◽  
Cross Sectional ◽  
Generalized Hypergeometric Functions ◽  
Normal Functions ◽  
Longitudinal Coordinate

With simple beam theory, solutions of normal functions for transverse vibration of a tapered beam are obtained in terms of generalized hypergeometric functions by the method of Frobenius. For the beams considered, the cross-sectional area and the area moment of inertia vary along the beam according to any two arbitrary powers of the longitudinal coordinate. The frequency equation is formulated, and the numerical results for many different tapered cantilever beams are presented.

Concerning the Transverse and Longitudinal Vibrations of a Fluid Conveying Beam and the Pipe Walking Phenomenon

2007 ◽  
Author(s):  
Vincent O. S. Olunloyo ◽  
Ayo A. Oyediran ◽  
Ajayi Adewale ◽  
Adekunle O. Adelaja ◽  
Charles A. Osheku
Keyword(s):  
Transverse Vibration ◽  
Oil And Gas ◽  
Cross Sectional Area ◽  
Longitudinal Motion ◽  
Sectional Area ◽  
Cross Sectional ◽  
External Temperature ◽  
Area Change ◽  
Start Up ◽  
Longitudinal Extension

In oil and gas installations, whether on-shore or off-shore, pipes are the primary vessel for the conveyance of either crude or products from one location to another. Under use, the pipes are subjected to both internal and external temperature fluctuations while repeated operational start-up and shut-down procedures triggers vibrations of these pipes, propagates internal waves and results in finite and irreversible longitudinal extension of the pipe over time. This longitudinal extension which is sometimes accompanied by pipe buckling is known as ratcheting and has also been described by some as pipe walking. In view of the complicated and intractable nature of the problem, most attempts to study the behavior of these pipes have been limited to the analysis of some reduced problem based on heuristic arguments and idealizations. Within this context, the transverse vibration and stability of such pipes have been studied while the problem of undamped clamped-pinned pipe conveying fluid has also been tackled numerically. Keiper and Metrikine [2004] however pointed out that such numerical schemes sometimes lead to disputed or controversial results. More importantly, the coupling between the transverse vibration and longitudinal motion has been largely ignored or neglected altogether by most writers. The objective of this paper is to formally derive the governing equations of Euler-Bernoulli beam capturing various effects including temperature variations (within and without), Coriolis acceleration, transverse acceleration, pre-stress, pressurization, rotatory inertia, and cross-sectional area change. In particular, it is shown that the latter effect is what causes pipe walking phenomenon. Most of the other effects were either earlier accounted for by Semler, et al [1994] or recently captured by Reddy and Wang [2004]. Nonetheless earlier contributions neglected the effect of the cross-sectional area change completely, thereby omitting the pipe walking phenomenon. Simple examples are considered to demonstrate the importance of these terms.

Frequencies of Longitudinal Vibration for a Slender Rod of Variable Section

1953 ◽  
Vol 20 (2) ◽  
pp. 173-177
Author(s):  
James L. Lubkin ◽  
Yudell L. Luke
Keyword(s):  
Asymptotic Expansions ◽  
Natural Frequencies ◽  
Longitudinal Vibration ◽  
Frequency Equation ◽  
Sectional Area ◽  
Dimensional Theory ◽  
Cross Sectional ◽  
One Dimensional ◽  
Variable Section ◽  
Fixed End

Abstract The natural frequencies of a slender, homogeneous, fixed-free rod of variable section are studied in a one-dimensional theory. The rod is uniform at the fixed end for a certain distance L1, and then tapers for a distance L2 so that the cross-sectional area varies linearly. The frequency equation is derived and its first five zeros are tabulated for several ratios of L1 to L2, and various tapers. The first few terms of asymptotic expansions for the roots of the frequency equation are presented and discussed.

Pelvic Canal Narrowing Caused by Triple Pelvic Osteotomy in the Dog

1994 ◽  
Vol 07 (03) ◽  
pp. 110-113 ◽  
Author(s):  
D. L. Holmberg ◽  
M. B. Hurtig ◽  
H. R. Sukhiani
Keyword(s):  
Postoperative Complications ◽  
Pelvic Osteotomy ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Cross Sectional ◽  
Canal Width ◽  
Operative Complications ◽  
Post Operative Complications

SummaryDuring a triple pelvic osteotomy, rotation of the free acetabular segment causes the pubic remnant on the acetabulum to rotate into the pelvic canal. The resulting narrowing may cause complications by impingement on the organs within the pelvic canal. Triple pelvic osteotomies were performed on ten cadaver pelves with pubic remnants equal to 0, 25, and 50% of the hemi-pubic length and angles of acetabular rotation of 20, 30, and 40 degrees. All combinations of pubic remnant lengths and angles of acetabular rotation caused a significant reduction in pelvic canal-width and cross-sectional area, when compared to the inact pelvis. Zero, 25, and 50% pubic remnants result in 15, 35, and 50% reductions in pelvic canal width respectively. Overrotation of the acetabulum should be avoided and the pubic remnant on the acetabular segment should be minimized to reduce postoperative complications due to pelvic canal narrowing.When performing triple pelvic osteotomies, the length of the pubic remnant on the acetabular segment and the angle of acetabular rotation both significantly narrow the pelvic canal. To reduce post-operative complications, due to narrowing of the pelvic canal, overrotation of the acetabulum should be avoided and the length of the pubic remnant should be minimized.

DETERMINATION OF THE FUNCTION OF THE ROD’S CROSS-SECTIONAL AREA USING VIBRATION EIGENFREQUENCIES

2020 ◽  
Vol 0 (4) ◽  
pp. 19-24
Author(s):  
I.M. UTYASHEV ◽  
◽  
A.A. AITBAEVA ◽  
A.A. YULMUKHAMETOV ◽  
◽  
...  
Keyword(s):  
Cross Sectional Area ◽  
Variable Cross ◽  
Sectional Area ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Method Error ◽  
Various Boundary Conditions ◽  
Elastic Fixation ◽  
Crosssectional Area

The paper presents solutions to the direct and inverse problems on longitudinal vibrations of a rod with a variable cross-sectional area. The law of variation of the cross-sectional area is modeled as an exponential function of a polynomial of degree n . The method for reconstructing this function is based on representing the fundamental system of solutions of the direct problem in the form of a Maclaurin series in the variables x and λ. Examples of solutions for various section functions and various boundary conditions are given. It is shown that to recover n unknown coefficients of a polynomial, n eigenvalues are required, and the solution is dual. An unambiguous solution was obtained only for the case of elastic fixation at one of the rod’s ends. The numerical estimation of the method error was made using input data noise. It is shown that the error in finding the variable crosssectional area is less than 1% with the error in the eigenvalues of longitudinal vibrations not exceeding 0.0001.

Changes in the parameters of nasal breathing and parameters of the upper respiratory tract during orthognathic operations in patients with II and III skeletal class of jaw anomalies

2020 ◽  
pp. 22-27
Author(s):  
S.Sh. Gammadaeva ◽  
M.I. Misirkhanova ◽  
A.Yu. Drobyshev
Keyword(s):  
Respiratory Tract ◽  
Nasal Cavity ◽  
Nasal Septum ◽  
Cross Sectional Area ◽  
Upper Respiratory Tract ◽  
Sectional Area ◽  
Nasal Breathing ◽  
Cross Sectional ◽  
Skeletal Class ◽  
Upper Respiratory

The study analyzed the functional parameters of nasal breathing, linear parameters of the nasal aperture, nasal cavity and nasopharynx, volumetric parameters of the upper airways in patients with II and III skeletal class of jaw anomalies before and after orthognathic surgery. The respiratory function of the nose was assessed using a rhinomanometric complex. According to rhinoresistometry data, nasal resistance and hydraulic diameter were assessed. According to the data of acoustic rhinometry, the minimum cross-sectional area along the internal valve, the minimum cross-sectional area on the head of the inferior turbinate and nasal septum and related parameters were estimated. According to the CBCT data, the state of the nasal septum, the inferior turbinates, the nasal aperture, the state of the nasal cavity, and the linear values of the upper respiratory tract (nasopharynx) were analyzed. The patients were divided into 4 groups according to the classification of the patency of the nasal passages by

Application of Image Analysis System for Study of Activated Sludge Flocs

1986 ◽  
Vol 21 (1) ◽  
pp. 130-140 ◽  
Author(s):  
Da-hong Li ◽  
J. J. Ganczarczyk
Keyword(s):  
Image Analysis ◽  
Activated Sludge ◽  
Statistical Processing ◽  
Equivalent Diameter ◽  
Image Analysis System ◽  
Sectional Area ◽  
Cross Sectional ◽  
Computerized Image Analysis ◽  
Sludge Flocs ◽  
Analysis System

Abstract The computerized image analysis system has been successfully used for determination and statistical processing of the following geometric characteristics of activated sludge flocs: longest dimension, breadth, equivalent diameter, cross-sectional area, perimeter, elongation, and circularity. These parameters could be effectively and precisely determined by the system applied. In addition, the studied method, as compared to direct microscope observation and photography floc-sizing methods, was found to be more accurate, less time-consuming, and less dependent on the investigators.

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