Concentrated Forces on Shallow Cylindrical Shells

Solutions for the normal displacement w and tangential displacements ux and uy for a shallow cylindrical shell subjected to concentrated forces are obtained in this paper. The normal force and the two tangential force cases are treated. The results for the displacements in all cases are expressible in terms of elementary functions, modified Bessel functions, and one new function of two variables. A reasonably complete investigation of this function is included.

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Error bounds for the Liouville–Green (or WKB) approximation

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100035945 ◽

1961 ◽

Vol 57 (4) ◽

pp. 790-810 ◽

Cited By ~ 57

Author(s):

F. W. J. Olver

Keyword(s):

Error Bounds ◽

Bessel Functions ◽

Approximate Solutions ◽

Parabolic Cylinder ◽

Real Interval ◽

Positive Parameter ◽

Modified Bessel Functions ◽

Elementary Functions ◽

Parabolic Cylinder Functions ◽

Image Position

ABSTRACTError bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equationsin which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.

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Response Due to Concentrated Force in Micropolar Elastic Solid with Voids

International Journal of Applied Mechanics and Engineering ◽

10.2478/ijame-2014-0052 ◽

2014 ◽

Vol 19 (4) ◽

pp. 755-769

Author(s):

R. Singh ◽

K. Singh

Keyword(s):

Fourier Transforms ◽

Normal Force ◽

Volume Fraction ◽

Tangential Force ◽

Elastic Solid ◽

Normal Displacement ◽

Laplace And Fourier Transforms ◽

Concentrated Force ◽

Eigen Value ◽

Stress Components

Abstract The eigen value approach, following Laplace and Fourier transforms has been employed to find the general solution of the field equation in a micropolar elastic solid with voids for the plane strain problem. An application of an infinite space with impulsive force has been taken to illustrate the utility of the approach. The integral transformations have been inverted by using a numerical inversion technique to get result in physical domain. The result in the form of normal displacement, volume fraction, normal force stress, tangential force stress and tangential couple stress components has been obtained numerically and illustrated graphically to depict the effect of micropolarity and voids.

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An Elastic-Plastic Spherical Contact Model Under Combined Normal and Tangential Loading

ASME/STLE 2011 Joint Tribology Conference ◽

10.1115/ijtc2011-61126 ◽

2011 ◽

Cited By ~ 1

Author(s):

Aizhong Wu ◽

Xi Shi ◽

Andreas A. Polycarpou

Keyword(s):

Coulomb Friction ◽

Normal Force ◽

Tangential Force ◽

Element Model ◽

Friction Model ◽

Normal Displacement ◽

Tangential Displacement ◽

Spherical Contact ◽

Elastic Plastic ◽

Proposed Model

In this work, by utilizing the shear strength criterion for the sliding inception, a finite element model for obliquely loaded spherical contact has been developed, which realized a friction transition from perfect slip case to full stick case with increasing normal approach. Both tangential force and normal force during tangential loading were investigated using different models. It was found that with elastic-plastic normal displacement preload, there is an obvious normal force release during tangential loading. Furthermore, both Coulomb friction model and the proposed model predict a lower tangential force at the same tangential displacement compared to the full stick model. However, the Coulomb friction is more empirically determined with some arbitrary friction coefficient whereas the proposed model is based on physics parameters.

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Eigen Value Approach in Micropolar Elastic Medium with Voids

International Journal of Applied Mechanics and Engineering ◽

10.2478/ijame-2013-0031 ◽

2013 ◽

Vol 18 (2) ◽

pp. 521-536

Author(s):

R. Singh ◽

K. Singh

Keyword(s):

Elastic Medium ◽

Normal Force ◽

Volume Fraction ◽

Tangential Force ◽

Normal Displacement ◽

Physical Domain ◽

Field Equations ◽

Hankel Transformation ◽

Eigen Value ◽

Stress Components

The eigen value approach, following the Laplace and Hankel transformation has been employed to find a general solution of the field equations in a micropolar elastic medium with voids for an axisymmetric problem. An infinite space with the mechanical source has been applied to illustrate the utility of the approach. The integral transformations has been inverted by using a numerical inversion technique to get the result in physical domain. The results in the form of normal displacement, volume fraction, normal force stress, tangential force stress and tangential couple stress components have been obtained numerically and illustrated graphically.

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Effect of Water-Based Nanolubricants in Ultrasonic Vibration Assisted Grinding

Journal of Manufacturing and Materials Processing ◽

10.3390/jmmp2040080 ◽

2018 ◽

Vol 2 (4) ◽

pp. 80 ◽

Cited By ~ 1

Author(s):

Mir Molaie ◽

Ali Zahedi ◽

Javad Akbari

Keyword(s):

Surface Roughness ◽

Specific Energy ◽

Material Removal ◽

Normal Force ◽

Tangential Force ◽

Process Efficiency ◽

Multiwall Carbon Nanotube ◽

Ultrasonic Vibrations ◽

Mql Grinding ◽

Water Based

Currently, because of stricter environmental standards and highly competitive markets, machining operations, as the main part of the manufacturing cycle, need to be rigorously optimized. In order to simultaneously maximize the production quality and minimize the environmental issues related to the grinding process, this research study evaluates the performance of minimum quantity lubrication (MQL) grinding using water-based nanofluids in the presence of horizontal ultrasonic vibrations (UV). In spite of the positive impacts of MQL using nanofluids and UV which are extensively reported in the literature, there is only a handful of studies on concurrent utilization of these two techniques. To this end, for this paper, five kinds of water-based nanofluids including multiwall carbon nanotube (MWCNT), graphite, Al2O3, graphene oxide (GO) nanoparticles, and hybrid Al2O3/graphite were employed as MQL coolants, and the workpiece was oscillated along the feed direction with 21.9 kHz frequency and 10 µm amplitude. Machining forces, specific energy, and surface quality were measured for determining the process efficiency. As specified by experimental results, the variation in the material removal nature made by ultrasonic vibrations resulted in a drastic reduction of the grinding normal force and surface roughness. In addition, the type of nanoparticles dispersed in water had a strong effect on the grinding tangential force. Hybrid Al2O3/graphite nanofluid through two different kinds of lubrication mechanisms—third body and slider layers—generated better lubrication than the other coolants, thereby having the lowest grinding forces and specific energy (40.13 J/mm3). It was also found that chemically exfoliating the graphene layers via oxidation and then purification prior to dispersion in water promoted their effectiveness. In conclusion, UV assisted MQL grinding increases operation efficiency by facilitating the material removal and reducing the use of coolants, frictional losses, and energy consumption in the grinding zone. Improvements up to 52%, 47%, and 61%, respectively, can be achieved in grinding normal force, specific energy, and surface roughness compared with conventional dry grinding.

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Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1959.0023 ◽

1959 ◽

Vol 249 (1257) ◽

pp. 284-292 ◽

Cited By ~ 1

Keyword(s):

Asymptotic Expansions ◽

Bessel Functions ◽

Modified Bessel Functions ◽

Lommel Functions ◽

Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

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