Studies in the geometry of the discrete plane

This thesis is an attempt to initiate the development of a discrete geometry of the discrete plane H = {(qmxo,qnyo); m,n e Z - the set of integers}, where q s (0,1) is fixed and (xO,yO) is a fixed point in the first quadrant of the complex plane, xo,y0 ¢ 0. The discrete plane was first considered by Harman in 1972, to evolve a discrete analytic function theory for geometric difference functions. We shall mention briefly, through various sections, the principle of discretization, an outline of discrete a alytic function theory, the concept of geometry of space and also summary of work done in this thesis

Analytical Investigations 0n Collapse of Cylindrical Submarine Shells

Submarine hull structure is a watertight envelope, under hydrostatic pressure when in operation. Stiffened cylindrical shells constitute the major portion of these submarine hulls and these thin shells under compression are susceptible to buckling failure. Normally loss of stability occurs at the limit point rather than at the bifurcation point and the stability analysis has to consider the change in geometry at each load step. Hence geometric nonlinear analysis of the shell forms becomes. a necessity. External hydrostatic pressure will follow the deformed configuration of the shell and hence follower force effect has to be accounted for. Computer codes have been developed based on all-cubic axisymmetric cylindrical shell finite element and discrete ring stiffener element for linear elastic, linear buckling and geometric nonIinear analysis of stiffened cylindrical shells. These analysis programs have the capability to treat hydrostatic pressure as a radial load and as a follower force. Analytical investigations are carried out on two attack submarine cylindrical hull models besides standard benchmark problems. In each case, the analysis has been carried out for interstiffener, interdeepframe and interbulkhead configurations. The shell stiffener attachment in each of this configuration has been represented by the simply supported-simply supported, clamped-clamped and fixed-fixed boundary conditions in this study. The results of the analytical investigations have been discussed and the observations and conclusions are described. Rotation restraint at the ends is influential for interstiffener and interbulkhead configurations and the significance of axial restraint becomes predominant in the interbulkhead configuration. The follower force effect of hydrostatic pressure is not significant in interstiffener and interdeepframe configurations where as it has very high detrimental effect on buckling pressure on interbulkhead configuration. The geometric nonlinear interbulkhead analysis incorporating follower force effect gives the critical value of buckling pressure and this analysis is recommended for the determination of collapse pressure of stiffened cylindrical submarine shells.

Mathematical Modelling of the Dynamics of Granular Materials in a Rotating Cylinder

The current study is aimed at the development of a theoretical simulation tool based on Discrete Element Method (DEM) to 'interpret granular dynamics of solid bed in the cross section of the horizontal rotating cylinder at the microscopic level and subsequently apply this model to establish the transition behaviour, mixing and segregation.The simulation of the granular motion developed in this work is based on solving Newton's equation of motion for each particle in the granular bed subjected to the collisional forces, external forces and boundary forces. At every instant of time, the forces are tracked and the positions velocities and accelarations of each partcle is The software code for this simulation is written in VISUAL FORTRAN 90 After checking the validity of the code with special tests, it is used to investigate the transition behaviour of granular solids motion in the cross section of a rotating cylinder for various rotational speeds and fill fraction.This work is hence directed towards a theoretical investigation based on Discrete Element Method (DEM) of the motion of granular solids in the radial direction of the horizontal cylinder to elucidate the relationship between the operating parameters of the rotating cylinder geometry and physical properties ofthe granular solid.The operating parameters of the rotating cylinder include the various rotational velocities of the cylinder and volumetric fill. The physical properties of the granular solids include particle sizes, densities, stiffness coefficients, and coefficient of friction Further the work highlights the fundamental basis for the important phenomena of the system namely; (i) the different modes of solids motion observed in a transverse crosssection of the rotating cylinder for various rotational speeds, (ii) the radial mixing of the granular solid in terms of active layer depth (iii) rate coefficient of mixing as well as the transition behaviour in terms of the bed turnover time and rotational speed and (iv) the segregation mechanisms resulting from differences in the size and density of particles.The transition behaviour involving its six different modes of motion of the granular solid bed is quantified in terms of Froude number and the results obtained are validated with experimental and theoretical results reported in the literature The transition from slumping to rolling mode is quantified using the bed turnover time and a linear relationship is established between the bed turn over time and the inverse of the rotational speed of the cylinder as predicted by Davidson et al. [2000]. The effect of the rotational speed, fill fraction and coefficient of friction on the dynamic angle of repose are presented and discussed. The variation of active layer depth with respect to fill fraction and rotational speed have been investigated. The results obtained through simulation are compared with the experimental results reported by Van Puyvelde et. at. [2000] and Ding et at. [2002].The theoretical model has been further extended, to study the rmxmg and segregation in the transverse direction for different particle sizes and their size ratios. The effect of fill fraction and rotational speed on the transverse mixing behaviour is presented in the form of a mixing index and mixing kinetics curve. The segregation pattern obtained by the simulation of the granular solid bed with respect to the rotational speed of the cylinder is presented both in graphical and numerical forms. The segregation behaviour of the granular solid bed with respect to particle size, density and volume fraction of particle size has been investigated. Several important macro parameters characterising segregation such as mixing index, percolation index and segregation index have been derived from the simulation tool based on first principles developed in this work.

High power N2 laser with a modified gas flow system and discharge geometry

A high power Nz laser of the double-Blumlein type having a modified gas flow system, electrode configuration, and discharge geometry with minimum inductance is described. By incorporating a triggere’d-pressurized spark gap switch, arc-free operation was achieved for a wide E/P range. The device gives a peak power in excess of 700 kW with a FWHM of 3 ns and an efficiency of 0.51%, which is remarkably high for a pulsed nitrogen laser system. The dependence of output power on parameters such as operating pressure, voltage, and repetition rate are discussed.

Dynamics of the logistic map under discrete parametric perturbation

By introducing a periodic perturbation in the control parameter of the logistic map we have investigated the period locking properties of the map. The map then gets locked onto the periodicity of the perturbation for a wide range of values of the parameter and hence can lead to a control of the chaotic regime. This parametrically perturbed map exhibits many other interesting features like the presence of bubble structures, repeated reappearance of periodic cycles beyond the chaotic regime, dependence of the escape parameter on the seed value and also on the initial phase of the perturbation etc.

Numerical Investigations on Soil Structure Interaction of Multistorey Frames

Frames are the most widely used structural system for multistorey buildings. A building frame is a three dimensional discrete structure consisting of a number of high rise bays in two directions at right angles to each other in the vertical plane. Multistorey frames are a three dimensional lattice structure which are statically indeterminate. Frames sustain gravity loads and resist lateral forces acting on it. India lies at the north westem end of the Indo-Australian tectonic plate and is identified as an active tectonic area. Under horizontal shaking of the ground, horizontal inertial forces are generated at the floor levels of a multistorey frame. These lateral inertia forces are transferred by the floor slab to the beams, subsequently to the columns and finally to the soil through the foundation system. There are many parameters that affect the response of a structure to ground excitations such as, shape, size and geometry of the structure, type of foundation, soil characteristics etc. The Soil Structure Interaction (SS1) effects refer to the influence of the supporting soil medium on the behavior of the structure when it is subjected to different types of loads. Interaction between the structure and its supporting foundation and soil, which is a complete system, has been modeled with finite elements. Numerical investigations have been carried out on a four bay, twelve storeyed regular multistorey frame considering depth of fixity at ground level, at characteristic depth of pile and at full depth. Soil structure interaction effects have been studied by considering two models for soil viz., discrete and continuum. Linear static analysis has been conducted to study the interaction effects under static load. Free vibration analysis and further shock spectrum analysis has been conducted to study the interaction effects under time dependent loads. The study has been extended to four types of soil viz., laterite, sand, alluvium and layered.The structural responses evaluated in the finite element analysis are bending moment, shear force and axial force for columns, and bending moment and shear force for beams. These responses increase with increase in the founding depth; however these responses show minimal increase beyond the characteristic length of pile. When the soil structure interaction effects are incorporated in the analysis, the aforesaid responses of the frame increases upto the characteristic depth and decreases when the frame has been analysed for the full depth. It has been observed that shock spectrum analysis gives wide variation of responses in the frame compared to linear elastic analysis. Both increase and decrease in responses have been observed in the interior storeys. The good congruence shown by the two finite element models viz., discrete and continuum in linear static analysis has been absent in shock spectrum analysis.

Some concepts and models usefulninthe analysis of discrete life time data

This thesis entitled Reliability Modelling and Analysis in Discrete time Some Concepts and Models Useful in the Analysis of discrete life time data.The present study consists of five chapters. In Chapter II we take up the derivation of some general results useful in reliability modelling that involves two component mixtures. Expression for the failure rate, mean residual life and second moment of residual life of the mixture distributions in terms of the corresponding quantities in the component distributions are investigated. Some applications of these results are also pointed out. The role of the geometric,Waring and negative hypergeometric distributions as models of life lengths in the discrete time domain has been discussed already. While describing various reliability characteristics, it was found that they can be often considered as a class. The applicability of these models in single populations naturally extends to the case of populations composed of sub-populations making mixtures of these distributions worth investigating. Accordingly the general properties, various reliability characteristics and characterizations of these models are discussed in chapter III. Inference of parameters in mixture distribution is usually a difficult problem because the mass function of the mixture is a linear function of the component masses that makes manipulation of the likelihood equations, leastsquare function etc and the resulting computations.very difficult. We show that one of our characterizations help in inferring the parameters of the geometric mixture without involving computational hazards. As mentioned in the review of results in the previous sections, partial moments were not studied extensively in literature especially in the case of discrete distributions. Chapters IV and V deal with descending and ascending partial factorial moments. Apart from studying their properties, we prove characterizations of distributions by functional forms of partial moments and establish recurrence relations between successive moments for some well known families. It is further demonstrated that partial moments are equally efficient and convenient compared to many of the conventional tools to resolve practical problems in reliability modelling and analysis. The study concludes by indicating some new problems that surfaced during the course of the present investigation which could be the subject for a future work in this area.

Discrete Time Inventory Models with/without Positive Service Time

In everyday life different flows of customers to avail some service facility or other at some service station are experienced. In some of these situations, congestion of items arriving for service, because an item cannot be serviced Immediately on arrival, is unavoidable. A queuing system can be described as customers arriving for service, waiting for service if it is not immediate, and if having waited for service, leaving the system after being served. Examples Include shoppers waiting in front of checkout stands in a supermarket, Programs waiting to be processed by a digital computer, ships in the harbor Waiting to be unloaded, persons waiting at railway booking office etc. A queuing system is specified completely by the following characteristics: input or arrival pattern, service pattern, number of service channels, System capacity, queue discipline and number of service stages. The ultimate objective of solving queuing models is to determine the characteristics that measure the performance of the system

Discrete commutative difference operator theory

The object of this thesis is to formulate a basic commutative difference operator theory for functions defined on a basic sequence, and a bibasic commutative difference operator theory for functions defined on a bibasic sequence of points, which can be applied to the solution of basic and bibasic difference equations. in this thesis a brief survey of the work done in this field in the classical case, as well as a review of the development of q~difference equations, q—analytic function theory, bibasic analytic function theory, bianalytic function theory, discrete pseudoanalytic function theory and finally a summary of results of this thesis

Some problems of discrete function theory

An attempt is made by the researcher to establish a theory of discrete functions in the complex plane. Classical analysis q-basic theory, monodiffric theory, preholomorphic theory and q-analytic theory have been utilised to develop concepts like differentiation, integration and special functions.