Dynamical modelling of the motorised momentum exchange tether incorporating axial elastic effects

A discretised planar tether model is proposed for the Motorised Momentum Exchange Tether (MMET) in which axial elasticity is accommodated. The model uses a generalised co-ordinate defining angular motion of the tether about its centre of mass, as it travels at constant velocity on a circular orbit in the Earth’s equatorial plane and a generalised coordinate depicting the elastic part of the tether length. The system comprises a symmetrical double payload configuration, with outrigger counter inertia, and it is shown that including axial elasticity permits an enhanced level of modelling accuracy for the tether both in librating and spinning modes. A simulation has been devised in MATLAB and SIMULINK for different data cases. This work will be used later within a spin-up control system and will act as a precursor for an in-depth study into the multi-scale dynamics of MMET tethers and space webs, on more complicated orbits. This, in turn, will be assimilated within new mission architectures

Hybrid fuzzy sliding mode control for motorised space tether spin-up when coupled with axial and torsional oscillation

A specialised hybrid controller is applied to the control of a motorised space tether spin-up space coupled with an axial and a torsional oscillation phenomenon. A seven-degree-of-freedom (7-DOF) dynamic model of a motorised momentum exchange tether is used as the basis for interplanetary payload exchange in the context of control. The tether comprises a symmetrical double payload configuration, with an outrigger counter inertia and massive central facility. It is shown that including axial and torsional elasticity permits an enhanced level of performance prediction accuracy and a useful departure from the usual rigid body representations, particularly for accurate payload positioning at strategic points. A simulation with given initial condition data has been devised in a connecting programme between control code written in MATLAB and dynamics simulation code constructed within MATHEMATICA. It is shown that there is an enhanced level of spin-up control for the 7-DOF motorised momentum exchange tether system using the specialised hybrid controller. hybrid controller

Painlev\'e V and time dependent Jacobi polynomials

In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight $w_0(x)$ defined on an interval by $w_0(x)e^{-tx}.$ It is a well-known fact that under such a deformation the recurrence coefficients denoted as $\alpha_n$ and $\beta_n$ evolve in $t$ according to the Toda equations, giving rise to the time dependent orthogonal polynomials, using Sogo's terminology. The resulting "time-dependent" Jacobi polynomials satisfy a linear second order ode. We will show that the coefficients of this ode are intimately related to a particular Painlev\'e V. In addition, we show that the coefficient of $z^{n-1}$ of the monic orthogonal polynomials associated with the "time-dependent" Jacobi weight, satisfies, up to a translation in $t,$ the Jimbo-Miwa $\sigma$-form of the same $P_{V};$ while a recurrence coefficient $\alpha_n(t),$ is up to a translation in $t$ and a linear fractional transformation $P_{V}(\alpha^2/2,-\beta^2/2, 2n+1+\alpha+\beta,-1/2).$ These results are found from combining a pair of non-linear difference equations and a pair of Toda equations. This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painlev\'e equation

Prandtl number of lattice Bhatnagar-Gross-Krook fluid

The lattice Bhatnagar-Gross-Krook modeled fluid has an unchangeable unit Prandtl number. A simple method is introduced in this letter to formulate a flexible Prandtl number for the modeled fluid. The effectiveness was demonstrated by numerical simulations of the Couette flow.Comment: 4 pages, uuencoded postscript fil

Characterization of the 4-canonical birationality of algebraic threefolds

In this article we present a 3-dimensional analogue of a well-known theorem of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let $X$ be a projective minimal 3-fold of general type with $\mathbb{Q}$-factorial terminal singularities and the geometric genus $p_g(X)\ge 5$. We show that the 4-canonical map $\phi_4$ is {\it not} birational onto its image if and only if $X$ is birationally fibred by a family $\mathscr{C}$ of irreducible curves of geometric genus 2 with $K_X\cdot C_0=1$ where $C_0$ is a general irreducible member in $\mathscr{C}$.Comment: 25 pages, to appear in Mathematische Zeitschrif

Semi-Finite Forms of Bilateral Basic Hypergeometric Series

We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's $_1\psi_1$ summation, Bailey's $_2\psi_2$ transformations, and Bailey's $_6\psi_6$ summation.Comment: 8 pages. accepted by Proc. Amer. Math. So

Calibration of the Pulsed Electroacoustic Technique in the Presence of Trapped Charge

The influence of pulse voltage on the accuracy of charge density distribution in the pulsed electroacoustic technique (PEA) is discussed. It is shown that significant error can be introduced if a low dc voltage and high pulse voltage are used to calibrate charge density. However, our main focus in the present paper is to deal with one of the practical situations where space charge exists in the material prior to any measurements. The conventional calibration method can no longer be used to calibrate charge density due to the interference by the charge on the electrode induced by space charge. A method has been proposed which is based on two measurements. Firstly, the sample containing charge is measured without any applied voltage. The second measurement is carried out with a small external applied voltage. The applied voltage should be small enough so there is no disturbance of the existing charge in the sample. The difference of the two measurements can be used for calibration. An additional advantage of the proposed method avoids the influence of the pulse voltage on calibration and therefore gives a more accurate representation of space charge. The proposed method has been validated