An advanced ascent guidance algorithm for rocket-powered launch vehicles is developed. The ascent guidance function is responsible for commanding attitude, throttle and setting during the powered ascent phase of flight so that the vehicle attains target cutoff conditions in a near-optimal manner while satisfying path constraints such as maximum allowed bending moment and maximum allowed axial acceleration. This algorithm cyclically solves the calculus-of-variations two-point boundary-value problem starting at vertical rise completion through orbit insertion. This is different from traditional ascent guidance algorithms which operate in an open-loop mode until the high dynamic pressure portion of the trajectory is over, at which time there is a switch to a closed loop guidance mode that operates under the assumption of negligible aerodynamic forces. The main contribution of this research is an algorithm of the predictor-corrector type wherein the state/costate system is propagated with known (navigated) initial state and guessed initial costate to predict the state/costate at engine cutoff. The initial costate guess is corrected, using a multi-dimensional Newtons method, based on errors in the terminal state constraints and the transversality conditions. Path constraints are enforced within the propagation process. A modified multiple shooting method is shown to be a very effective numerical technique for this application. Results for a single stage to orbit launch vehicle are given. In addition, the formulation for the free final time multi-arc trajectory optimization problem is given. Results for a two-stage launch vehicle burn-coast-burn ascent to orbit in a closed-loop guidance mode are shown. An abort to landing site formulation of the algorithm and numerical results are presented. A technique for numerically treating the transversality conditions is discussed that eliminates part of the analytical and coding burden associated with optimal control theory.

The purpose of this paper is to develop the shooting method as a technique for approximating the solution to the two-point boundary value problem on the interval [a,b] with the even order differential equation {i.e. n is even) u(n)(t) + f(t, u(t), u(i)(t, ),..., u(n-1)(t)) = 0 and boundary conditions u(a) = A u(b) = B and with at most n-2 other boundary conditions specified at either a or b. The basic proceedure will be illustrated by the following example. Consider the two-point boundary value problem (0.1) (0.2) (0.3) with the additional boundary conditions u(i)(a) = mi for i = 1, ... ,k-1,k+l, ... ,n-1. The first step is to find values m1 and m2 such that the solutions or "shots", u1(t) and u2(t), to (0.1) that satisfy the initial conditions u(a) = A u(k-1)(a)= mk-1 u(k)(a) = m1 u(k+1)(a) = mk+l u(n-1)(a) = mn-1 with 1 = 1,2, respectively, with the property that u1(b) < B < u2(b). The interval [m1,m2] is then searched by seccussive bisection to find the value, m, such that the solution or "shot", u(t), to the initial value problem with (0.1) and initial conditions u(a) = A u(k-1)(a)= mk-1 u(k)(a) = m1 u(k+1)(a) = mk+l u(n-1)(a) = mn-1 has the property that u(b) = B.

<p>Routing techniques for plain ’flat’ microchips have been developed extensively and will soon reach its limitations. One natural step would be to develop chips which are manufactured in a more cubic type of volume, as oppose to the classical flat design. </p><p>This thesis proposes a method for routing two-point nets across a three- dimensional channel. The height required by this algorithm is of the order <i>O(n</i> ^(3/2)), where n is the number of terminals on a square top-layer with the side <i>2 (n</i>^(1/2)). </p><p>The algorithm proposed here is based on"On Routing Two-Point Nets Across a Channel", by Ron Y. Pinter [9], and the concepts from this paper are explainedin this thesis to familiarise the reader these. </p><p>It is also shown that the proposed algorithm is more effective in its volume than the two-dimensional counterpart. The algorithm here is of the order <i>O(n</i>^(3/2)) with the two-dimensional algorithm of the order <i>O</i>(<i>n</i>²).</p>

Routing techniques for plain ’flat’ microchips have been developed extensively and will soon reach its limitations. One natural step would be to develop chips which are manufactured in a more cubic type of volume, as oppose to the classical flat design. This thesis proposes a method for routing two-point nets across a three- dimensional channel. The height required by this algorithm is of the order O(n (3/2)), where n is the number of terminals on a square top-layer with the side 2 (n(1/2)). The algorithm proposed here is based on"On Routing Two-Point Nets Across a Channel", by Ron Y. Pinter [9], and the concepts from this paper are explainedin this thesis to familiarise the reader these. It is also shown that the proposed algorithm is more effective in its volume than the two-dimensional counterpart. The algorithm here is of the order O(n(3/2)) with the two-dimensional algorithm of the order O(n2).

The present work investigates the phenomena of whip and whirl for a rigid rotor contacting at two bearing locations. The idea originated from an anemometer consisting of a rotor with an elastically supported stator undergoing the phenomena of dry friction whip and whirl at the two bushing contacts. To analyze the behavior, a mathematical model similar to the anemometer is developed and analyzed assuming two possible solutions, Mode1 (normal reaction forces in phase at two contacts) and Mode 2 (normal reaction forces out of phase at two contacts). Analytical solutions are only possible for the models with same RCl (Radius to Clearance ratio) at the two rub locations. A simulation model is constructed using the Texas A&M University (TAMU) Turbomachinery Laboratory rotordynamics software suite XLTRC² comprised of Timoshenko beam finite elements to form multiple degrees of freedom rotor and stator models. The nonlinear connections at the rub surface are modeled using Hunt and Crossley‘s contact model with coulomb friction. Dry friction simulations are performed for three separate models depending on the rotor‘s mass disk location with respect to the contact locations. The three models used have (1) Disk at center location (2) Disk at 3/4 location (3) Disk at overhang location. The adequacy of the analytical solution is investigated using the above simulations. Also, cases are explored where the general assumed solution would not solve the mathematical model, e.g. different RCl ratios at the two contacts. Simulations are performed for increasing as well as decreasing running speeds. There is partial agreement between simulation predictions and the analytical solutions for the cases with the mass center at centered and at 3/4 location. First, whirl-to-whip transitions occur at near the combine rotor-stator bounce frequency for both disk at center and disk at 3/4 location. The case with overhang mass disk predicts the two contacts to whip and at different frequencies simultaneously. Neither of the analytical solutions predicts a case where precession occurs at two different frequencies at the two contact points. Predictions for models with different RCl on the Backward Precessional (BP) graph imitate whirling. The BP graph predicts increasing BP frequency with increasing rotor speeds which is a characteristic of whirling, whereas investigation of individual contact velocities suggest that they are slipping at all conditions, one of them slipping more than the other netting a whirling like motion. For the overhang model with different RCl, apart from whipping at different frequency the two contacts also whirl at different frequencies corresponding to the RCl at the respective contacts. Simulations for decreasing rotor speed predict jump down from whirl- to-whip different BP frequency as compared to the jump up from whip-to-whirl for the speed up.

In the first, a time-variant modulus phase lock loop(PLL) model is established. Applying the model, Theorems of fractional-N synthesizers are introduced. We also explain theorems and simulations of Closed-Loop Modulation and Two-Point Delta Sigma Modulation with the model. In the end, a 2.4GHz FHSS transmitter using Two-Point Delta Sigma Modulation which meets Bluetooth specifications is demonstrated.

This dissertation presents a 2.4 GHz wideband GFSK-modulated frequency synthesizer using two-point delta-sigma modulation (TPDSM). The two bottlenecks in this design have been rigorously investigated. One bottleneck is the nonlinear performance of the phase-locked loop (PLL). The other one is the inherent gain and delay mismatch between the two modulation points. Both nonlinear and mismatch factors dominate the modulation accuracy in the closed PLL. The proposed formulation can successfully predict the dependencies of the modulation accuracy on both factors. The comparison of the averaged frequency deviation and frequency-shift -keying (FSK) error between theory and measurement shows excellent agreement. The modulated frequency synthesizer implemented in this study can achieve a 2.5 Mbps data rate as well as a 15 £gs PLL stable time with only 2.2 % FSK error under good design and operating conditions.

Area averaging is one of the techniques used to solve problems encountered in the transport of momentum, heat, and mass. The application of this technique simplifies the mathematical solution of the problem. However, it necessitates expressing the local value of the dependent variable and/or its derivative(s) on the system boundaries in terms of the averaged variable. In this study, these expressions are obtained by the two-point Hermite expansion and this approximate method is applied to some specific problems, such as, unsteady flow in a concentric annulus, unequal cooling of a long slab, unsteady conduction in a cylindrical rod with internal heat generation, diffusion of a solute into a slab from limited volume of a well-mixed solution, convective mass transport between two parallel plates with a wall reaction, convective mass transport in a cylindrical tube with a wall reaction, and unsteady conduction in a two -layer composite slab. Comparison of the analytical and approximate solutions is shown to be in good agreement for a wide range of dimensionless parameters characterizing each system.

In the first, we design and implement a Two -Point Voltage-Controlled Oscillator which applied in Open-Loop and Two-Point Close-Loop Modulation Bluetooth transmitter and include Hybrid and CMOS RFIC design. Second, we design six LTCC embedded components including inductors and capacitors. The extraction result via traditional equivalent £kmodel match the simulation and the frequency response of adopted model can accurate to device¡¦s Self-Resonant-Frequency nearby compare with measurement. In the end, we design and implement a LTCC Two-Point Voltage-Controlled Oscillator module and effective ly reduce the module size.

Philosophiae Doctor - PhD / Singularly perturbed problems have been studied extensively over the past few years from different perspectives. The recent research has focussed on the problems whose solutions possess interior layers. These interior layers appear in the interior of the domain, location of which is difficult to determine a-priori and hence making it difficult to investigate these problems analytically. This explains the need for approximation methods to gain some insight into the behaviour of the solution of such problems. Keeping this in mind, in this thesis we would like to explore a special class of numerical methods, namely, fitted finite difference methods to determine reliable solutions. As far as the fitted finite difference methods are concerned, they are grouped into two categories: fitted mesh finite difference methods (FMFDMs) and the fitted operator finite difference methods (FOFDMs). The aim of this thesis is to focus on the former. To this end, we note that FMFDMs have extensively been used for singularly perturbed two-point boundary value problems (TPBVPs) whose solutions possess boundary layers. However, they are not fully explored for problems whose solutions have interior layers. Hence, in this thesis, we intend firstly to design robust FMFDMs for singularly perturbed TPBVPs whose solutions possess interior layers and to improve accuracy of these approximation methods via methods like Richardson extrapolation. Then we extend these two ideas to solve such singularly perturbed TPBVPs with variable diffusion coefficients. The overall approach is further extended to parabolic singularly perturbed problems having constant as well as variable diffusion coefficients. / 2023-08-31