Determination of optimal trajectory under design constraints for a satellite launch vehicle

An optimal pitch steering programme of a solid-fuel satellite launch vehicle to maximize either (1) the injection velocity at a given altitude, or (2) the size of circular orbit, for a given payload is presented. The two-dimensional model includes the rotation of atmosphere with the Earth, the vehicle's lift and drag, variation of thrust with time and altitude, inverse-square gravitational field, and the specified initial vertical take-off. The inequality constraints on the aerodynamic load, control force, and turning rates are also imposed. Using the properties of the central force motion the terminal constraint conditions at coast apogee are transferred to the penultimate stage burnout. Such a transformation converts a time-free problem into a time-fixed one, reduces the number of terminal constraints, improves accuracy, besides demanding less computer memory and time. The adjoint equations are developed in a compact matrix form. The problem is solved on an IBM 360/44 computer using a steepest ascent algorithm. An illustrative analysis of a typical launch vehicle establishes the speed of convergence, and accuracy and applicability of the algorithm.

The Noether-Lefschetz theorem for the divisor class group

Let X be a normal projective threefold over a field of characteristic zero and vertical bar L vertical bar be a base-point free, ample linear system on X. Under suitable hypotheses on (X, vertical bar L vertical bar), we prove that for a very general member Y is an element of vertical bar L vertical bar, the restriction map on divisor class groups Cl(X) -> Cl(Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X subset of P-C(3) of degree >= 4 has Pic(X) congruent to Z.

Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives

A modified form of Green's integral theorem is employed to derive the energy identity in any water wave diffraction problem in a single-layer fluid for free-surface boundary condition with higher-order derivatives. For a two-layer fluid with free-surface boundary condition involving higher-order derivatives, two forms of energy identities involving transmission and reflection coefficients for any wave diffraction problem are also derived here by the same method. Based on this modified Green's theorem, hydrodynamic relations such as the energy-conservation principle and modified Haskind–Hanaoka relation are derived for radiation and diffraction problems in a single as well as two-layer fluid.

Optimal explicit guidance of multistage launch vehicle along three-dimensional trajectory

Details of an efficient optimal closed-loop guidance algorithm for a three-dimensional launch are presented with simulation results. Two types of orbital injections, with either true anomaly or argument of perigee being free at injection, are considered. The resulting steering-angle profile under the assumption of uniform gravity lies in a canted plane which transforms a three-dimensional problem into an equivalent two-dimensional one. Effects of thrust are estimated using a series in a recursive way. Encke's method is used to predict the trajectory during powered flight and then to compute the changes due to actual gravity using two gravity-related vectors. Guidance parameters are evaluated using the linear differential correction method. Optimality of the algorithm is tested against a standard ground-based trajectory optimization package. The performance of the algorithm is tested for accuracy, robustness, and efficiency for a sun-synchronous mission involving guidance for a multistage vehicle that requires large pitch and yaw maneuver. To demonstrate applicability of the algorithm to a range of missions, injection into a geostationary transfer orbit is also considered. The performance of the present algorithm is found to be much better than others.

Parametrisation-free determination of the shape parameters for the pion electromagnetic form factor

Recent data from high-statistics experiments that have measured the modulus of the pion electromagnetic form factor from threshold to relatively high energies are used as input in a suitable mathematical framework of analytic continuation to find stringent constraints on the shape parameters of the form factor at t = 0. The method uses also as input a precise description of the phase of the form factor in the elastic region based on Fermi-Watson theorem and the analysis of the pi pi scattering amplitude with dispersive Roy equations, and some information on the spacelike region coming from recent high precision experiments. Our analysis confirms the inconsistencies of several data on the modulus, especially from low energies, with analyticity and the input phase, noted in our earlier work. Using the data on the modulus from energies above 0.65 GeV, we obtain, with no specific parametrisation, the prediction < r(pi)(2)> is an element of (0.42, 0.44) fm(2) for the charge radius. The same formalism leads also to very narrow allowed ranges for the higher-order shape parameters at t = 0, with a strong correlation among them.

Reduced Grobner bases and Macaulay-Buchberger Basis Theorem over Noetherian rings

In this paper, we extend the characterization of Zx]/(f), where f is an element of Zx] to be a free Z-module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Grobner basis method of computing a k-vector space basis of residue class polynomial rings over a field k (Macaulay-Buchberger Basis Theorem) to rings, i.e. Ax(1), ... , x(n)]/a, where a subset of Ax(1), ... , x(n)] is an ideal. We give some insights into the characterization for two special cases, when A = Z and A = ktheta(1), ... , theta(m)]. As an application of this characterization, we show that the concept of Border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free A-module. (C) 2014 Elsevier B.V. All rights reserved.

Microstructural characterization of ultrafine-grain interstitial-free steel by X-ray diffraction line profile analysis

This paper highlights the microstructural features of commercially available interstitial free (IF) steel specimens deformed by equal channel angular pressing (ECAP) up to four passes following the route A. The microstructure of the samples was studied by different techniques of X-ray diffraction peak profile analysis as a function of strain (epsilon). It was found that the crystallite size is reduced substantially already at epsilon=2.3 and it does not change significantly during further deformation. At the same time, the dislocation density increases gradually up to epsilon=4.6. The dislocation densities estimated from X-ray diffraction study are found to correlate very well with the experimentally obtained yield strength of the samples.

Singularity-free path planning for the Stewart platform manipulator

This paper addresses the problem of singularity-free path planning for the six-degree-of-freedom parallel manipulator known as the Stewart platform manipulator. Unlike serial manipulators, the Stewart platform possesses singular configurations within the workspace where the manipulator is uncontrollable. An algorithm has been developed to construct continuous paths within the workspace of the manipulator by avoiding singularities and ill-conditioning. Given two end-poses of the manipulator, the algorithm finds out safe (well-conditioned) via points and plans a continuous path from the initial pose to the final one. When the two end-poses belong to different branches and no singularity-free path is possible, the algorithm indicates the impossibility of a valid path. A numerical example has also been presented as illustration of the path planning strategy.

Unsteady free convection flow in the stagnation-point region of a rotating sphere

The unsteady free convection boundary-layer flow in the forward stagnation-point region of a sphere, which is rotating with time-dependent angular velocity in an ambient fluid, has been studied. Both constant wall temperature and constant hear flux conditions have been considered. The non-linear coupled parabolic partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. The skin friction and the heat transfer are enhanced by the buoyancy force. The effect of the buoyancy force is found to be more pronounced for smaller Prandtl numbers than for larger Prandtl numbers. For a given buoyancy force, the heat transfer increases with an increase in Prandtl number, but the skin friction decreases.

Unsteady free convection flow in the stagnation-point region of a three-dimensional body

The unsteady free convection flow in the stagnation-point region of a heated three-dimensional body placed in an ambient fluid is studied under boundary layer approximations. We have considered the case where there is an initial steady state that is perturbed by a step-change in the wall temperature. The non-linear coupled partial differential equations governing the free convection flow are solved numerically using a finite difference scheme. The presented results show the temporal development of the momentum and thermal boundary layer characteristics.

Free Energy Gap Dependence of the Electron-Transfer Rate from the Inverted to the Normal Region

numerical study of the free energy gap (FEG) dependence of the electron-transfer rate in polar solvents is presented. This study is based on the generalized multidimensional hybrid model, which not only includes the solvent polarization and the molecular vibration modes, but also the biphasic polar response of the solvent. The free energy gap dependence is found to be sensitive to several factors, including the solvent relaxation rate, the electronic coupling between the surfaces, the frequency of the high-frequency quantum vibrational mode, and the magnitude of the solvent reorganization energy. It is shown that in some cases solvent relaxation can play an important role even in the Marcus normal regime. The minimal hybrid model involves a large number of parameters, giving rise to a diverse non-Marcus FEG behavior which is often determined collectively by these parameters. The model gives the linear free energy gap dependence of the logarithmic rate over a substantial range of FEG, spanning from the normal to the inverted regime. However, even for favorable values of the relevant parameters, a linear free energy gap dependence of the rate could be obtained only over a range of 5000-6000 cm(-1) (compared to the experimentally observed range of 10000 cm(-1) reported by Benniston et al.). The present work suggests several extensions/generalizations of the hybrid model which might be necessary to fully understand the observed free energy gap dependence.

Free energy landscape of a dense hard-sphere system

The topography of the free energy landscape in phase space of a dense hard-sphere system characterized by a discretized free energy functional of the Ramakishnan-Yussouff form is investigated numerically using a specially devised Monte Carlo procedure. We locate a considerable number of glassy local minima of the free energy and analyze the distributions of the free energy at a minimum and an appropriately defined phase-space "distance" between different minima. We find evidence for the existence of pairs of closely related glassy minima("two-level systems"). We also investigate the way the system makes transitions as it moves from the basin of attraction of a minimum to that of another one after a start under nonequilibrium conditions. This allows us to determine the effective height of free energy barriers that separate a glassy minimum from the others. The dependence of the height of free energy barriers on the density is investigated in detail. The general appearance of the free energy landscape resembles that of a putting green: relatively deep minima separated by a fairly flat structure. We discuss the connection of our results with the Vogel-Fulcher law and relate our observations to other work on the glass transition.

Free vibration of a kinked cantilever with attached masses

In this paper free vibration characteristics of a centrally kinked cantilever beam of unit mass carrying masses at the kink (m(k)) and at the tip (m(t)) are analyzed. Frequency factors are presented for the first two modes for different combinations of m(k),m(t) and the kink angle delta. A relationship of the form f(m(k),m(t), delta) = m(k) + m(t)(4 + 10/3 cos delta+ 2/3 cos(2) delta)=const appears to give the same fundamental frequency for a given delta and different combinations of [m(k), m(t)]. Mode shapes as well as bending moments at the support and at the kink are also discussed. The utility of a discrete beam model in understanding the free vibration characteristics is also highlighted.

Extended Kalman filters using explicit and derivative-free local linearizations

We propose three variants of the extended Kalman filter (EKF) especially suited for parameter estimations in mechanical oscillators under Gaussian white noises. These filters are based on three versions of explicit and derivative-free local linearizations (DLL) of the non-linear drift terms in the governing stochastic differential equations (SDE-s). Besides a basic linearization of the non-linear drift functions via one-term replacements, linearizations using replacements through explicit Euler and Newmark expansions are also attempted in order to ensure higher closeness of true solutions with the linearized ones. Thus, unlike the conventional EKF, the proposed filters do not need computing derivatives (tangent matrices) at any stage. The measurements are synthetically generated by corrupting with noise the numerical solutions of the SDE-s through implicit versions of these linearizations. In order to demonstrate the effectiveness and accuracy of the proposed methods vis-à-vis the conventional EKF, numerical illustrations are provided for a few single degree-of-freedom (DOF) oscillators and a three-DOF shear frame with constant parameters.