New relationships between spherical and spheroidal harmonics and applications

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

New relationships between spherical and spheroidal harmonics and applications

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

Influence of Roller-End Axial Profiling Type on Lubrication Performance in Thermal Elastohydrodynamic Finite Line Contacts

This study presents a finite-element-based numerical investigation of the influence of roller-end axial profiling type on the lubrication performance of thermal elastohydrodynamic finite line contacts. Performance is evaluated with respect to the reference case of straight rollers. The two most commonly used profiling types (i.e., dub-off and logarithmic) are compared under similar profiling length and height. It is found that a logarithmic profile outperforms a dub-off one by all accounts (i.e., frictional dissipation, lubricant film thickness, pressure buildup, and temperature rise), unless an extremely steep logarithmic shape is adopted. In the latter case, lubricant film thickness and pressure buildup may end up being negatively affected.

Improvement of thermal EHL by selecting bush-pin geometry based on an investigation for industrial chains

Purpose This study aims to investigate the influence of geometry of bush-pin pair from a perspective of optimal lubrication through a thermal elastohydrodynamic lubrication model for finite line contact. Design/methodology/approach A constitutive equation: Ree-Eyring fluid is used in the calculations. The real chain sizes, i.e. equivalent radius of curvature, bush length, length of the rounded corner area and rounded corner radius, are jointed investigated. Moreover, the effects of the length of the rounded corner area and the radius of rounded corner are investigated. Findings It is found that the current standard of the chain might not consider the importance of lubrication, and the lubrication state can be improved effectively by choosing an optimal radius of rounded corner and the length of the corner area. Originality/value By optimally selecting sizes, the occurrence of high pressure, high temperature rise and near zero film thickness at the ends of bush, especially under heavier load, can be effectively avoided. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-01-2020-0031/

Transient elastohydrodynamic analysis of finite line contact under load impulse

Numerical analysis is performed to study the transient behavior of EHL finite line contact of a cylindrical roller and flat plane under load impulse. In the present work, effect of pressure on the density and viscosity of lubricant is considered. Finite difference method is used to discretize the governing equations. Multilevel Multi-integration method is used to calculate the elastic deformation. Moreover, Multigrid method is implemented to accelerate the convergence process. Uniqueness of this finite line contact analysis is that it provides an ability to determine the transient behavior of lubricated contact even at the edges of roller. Results show that the load impulse causes squeezing and separation movement within the contact that develops film dimple and pressure ripples at the inlet region, which propagate towards the exit region due to the entrainment motion. It is noticed that the time taken by oil film [Formula: see text] to travel the Hertzian contact width and the time period [Formula: see text] of load impulse decides the behavior of lubricated contacts. Firstly, under a relatively heavy load when the contact width is large enough so that [Formula: see text], then a significant rise in central film thickness (CFT), central minimum film thickness (CMFT) and minimum film thickness (MFT) occurs after the execution of load impulse. Further, under the light load generating a relatively small contact width such that [Formula: see text], then comparatively a small rise in CFT occurs right during the load impulse. Lastly, for a given load if the time period of impulse [Formula: see text] is large enough satisfying the condition [Formula: see text], then a considerable reduction in CFT, CMFT and MFT takes place during the application of load impulse. Moreover, as compare to other cases, for [Formula: see text] the steady state condition is reestablished after a relatively more number of time cycles. It is observed that the maximum pressure and MFT occurs at the contact edges of roller which can be controlled by a proper choice of the radius of end profile [Formula: see text].

On the Negative Influence of Roller-End Axial Profiling on Friction in Thermal Elastohydrodynamic Lubricated Finite Line Contacts

This paper presents a finite element model for the solution of thermal elastohydrodynamic lubrication in finite line contacts, including edge effects. The model is used to investigate the influence of roller-end axial profiling on the frictional behavior of such contacts. Roller-end profiling in finite line contacts has always been used to enhance fatigue life by increasing lubricant minimum film thickness and reducing stress concentration at roller ends. The influence on friction on the other hand has often been overlooked in the literature. The current work reveals that roller-end profiling has a negative effect on friction. In fact, it turns out that the improvement in fatigue life comes at the expense of increased friction.