Chaotic Motion in a Spinning Spacecraft with Circumferential Nutational Damper

The dynamics of a simplified model of a spinning spacecraft with a circumferential nutational damper is investigated using numerical simulations for nonlinear phenomena. A realistic spacecraft parameter configuration is investigated and is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. Such a torque, in practice, may arise in the platform of a dual-spin spacecraft under malfunction of the control system or from an unbalanced rotor or from vibrations in appendages. The equations of motion of the model are derived with Lagrange's equations using a generalisation of the kinetic energy equation and a linear stability analysis is given. Numerical simulations for satellite parameters are performed and the system is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. The motion is studied by means of time history, phase space, frequency spectrum, Poincaré map, Lyapunov characteristic exponents and Correlation Dimension. For sufficiently large values of torque amplitude, the behaviour of the system was found to have much in common with a two well potential problem such as a Duffing oscillator. Evidence is also presented, indicating that the onset of chaotic motion was characterised by period doubling as well as intermittency.     

meso-Aryl-3-alkyl-2-oxachlorins

The formal replacement of a pyrrole moiety of meso-tetraarylporphyrin 1 by an oxazole moiety is described. The key step is the conversion of porpholactones 4 (prepared by a known two-step oxidation procedure from 1) by addition of alkyl Grignard reagent to form meso-tetraaryl-3-alkyl-2-oxachlorins 9 (alkyloxazolochlorins; alkyl = Me, Et, iPr). Hemiacetal 9 can be converted to an acetal, reduced to an ether, or converted to bis-alkyloxazolochlorins 11. The optical properties (UV-visible and fluorescence spectroscopy) are described. The chlorin-like optical properties of the alkyloxazolochlorins are compared to regular chlorins, such as 2,3-dihydroxychlorins and nonalkylated oxazolochlorins made by reduction from porpholactone 4. The conformations of the mono- and bis-alkylated 2-oxachlorins, as determined by single crystal X-ray diffractometry, are essentially planar, thus proving that their optical properties are largely due to their intrinsic electronic properties and not affected by conformational effects. The mono- and bis-3-alkyl-2-oxachlorins are a class of readily prepared and oxidatively stable chlorins.     

High yield synthesis,molecular structure and n.m.r. studies of the seven-coordinate complex [WI2(CO)3(PEt3)2]

The reaction of [WI2(CO)3(NCMe)2] with two equivalents of PEt3 in Et2O/CH2Cl2 gives the MeCN-displaced product [WI2(CO)3(PEt3)2] (1) in high yield. The crystallographically determined solid state structure of (1) shows the complex to be capped octahedral with a carbonyl ligand capping an octahedral face containing the two octahedral carbonyls and one PEt3 group. The other face contains the two iodo ligands trans to carbonyl groups, and the second PEt3 ligand trans to the first PEt3 group. The low temperature (-30°C) 13C-{1H}-n.m.r. spectrum of (1) correlates with the solid state structure, and the fluxional properties of (1) have been investigated by VT 31P-{1H}-n.m.r.spectroscopy. This revised version was published online in June 2006 with corrections to the Cover Date.     

Molecular structure and iodine oxidative-thermal decarbonylation of the seven-coordinate complex [MoI2(CO)3(NCMe)2] to give the crystallographically characterised complex mer-[MoI3(NCMe)3]

The X-ray crystal structure of the seven-coordinate complex [MoI2(CO)3(NCMe)2] (1) has been determined and shows discrete molecules. The geometry of the metal coordination sphere conforms to a distorted capped octahedron with a carbonyl group in the unique capping position. The capped face contains two carbonyl ligands and an iodo group, the uncapped face two MeCN ligands and an iodo group. The low temperature 13C n.m.r. spectrum (–70 °C, CD2Cl2) shows a single band at = 222.8 ppm, which suggests that the complex is rearranging more rapidly than the n.m.r. timescale at this temperature. Refluxing a 2:1 mixture of [MoI2 (CO)3(NCMe)2] and I2 in MeCN for 2h gives the crystallographically characterised molybdenum(III) complex mer-[MoI3(NCMe)3] (2) in quantitative yield.     

Period-doubling behavior in frontal polymerization of multifunctional acrylates

Front dynamics in the frontal polymerization of two multifunctional acrylate monomers, 1,6-hexanediol diacrylate (HDDA) and trimethylolpropane ethoxylate triacrylate (TMPTA), with Lupersol 231 [1,1-di(t-butylperoxy)-3,3,5-trimethylcyclohexane] as the initiator, are studied. In most frontal polymerization systems, the dynamics are associated with a planar front propagating through the sample. However, in some cases, front behavior can be altered: the front becomes nonplanar characterized by complex patterns like spin modes and pulsations. To determine how these periodic and aperiodic modes arise, reactant solutions consisting of HDDA diluted with diethyl phthalate (DEP) and TMPTA diluted with dimethyl sulfoxide (DMSO) were used in the study. In the study we reveal frontal behavior characteristic of period-doubling behavior, a doubling of spin heads that degenerate into an apparently chaotic mode. Also, a pulsating symmetric mode has been observed. These observations have a striking similarity to observations made in studies of self-propagating high-temperature synthesis (SHS) in which the addition of an inert diluent afforded a rich variety of dynamical behavior. The degree of cross-linking has also been found to be a bifurcation parameter. The energy of activation of multifunctional acrylate polymerization is a strong function of the degree of polymerization. By adding a monoacrylate (benzyl acrylate: BzAc), such that the front temperature was invariant, we observed a period-doubling bifurcation sequence through changes in the energy of activation, which has not been previously reported. (c) 1999 American Institute of Physics.     

Solid-state conformations of medium-sized-ring phosphorus heterocycles: Preparation and x-ray crystal structure of a benzo-fused 1,3,2-dioxaphosphepane in a twist conformation

The crystal and molecular structure of the benzo-fused seven-membered-ring phosphorus heterocycle, 2-phenyl-2-oxo-5,6-benzo-1,3,2-dioxaphosphepane, has been determined by single-crystal X-ray techniques. The compound crystallizes in the monoclinic space group P 2₁/c with four molecules per unit cell of dimensions a = 13.312(1) Å, b = 7.6132(8) Å, c = 12.119(2) Å, and β = 95.990(9)°. Full-matrix leastsquares refinement led to R = 4.4% and R_w = 5.9%. The conformation adopted by the 1,3,2-dioxaphosphepane ring is a twist with approximate C₂ symmetry.     

High Yield Synthesis,Molecular Structure,and Reactions of the 16‐Electron Norbornadiene Complex, [WI2(CO)2(nbd)]

Reaction of equimolar amounts of [WI₂(CO)₃(NCMe)₂] and norbornadiene (nbd) in toluene at 95 °C for 3h gave the 16‐electron crystallographically characterised complex, [WI₂(CO)₂(nbd)] (1) in 96 % yield. The structure of 1 has a distorted octahedral geometry, with the two cis‐ iodo ligands opposite to the two alkene groups in the equatorial plane, with the carbonyl groups in the axial sites. Treatment of 1 with two equivalents of PhC₂Ph in CH₂Cl₂ at room temperature afforded the bis(alkyne) complex [WI₂(CO)₂(η²—PhC₂Ph)₂] (2) . Equimolar quantities of 1 and 4, 4′‐bipyridine react in CH₂Cl₂ at room temperature to yield the seven‐coordinate complex, [WI₂(CO)₂(4, 4′‐bipyridine)(nbd)] (3) .     

Nonlinear dynamics of a satellite with deployable solar panel arrays

The multibody dynamics of a satellite in circular orbit, modeled as a central body with two hinge-connected deployable solar panel arrays, is investigated. Typically, the solar panel arrays are deployed in orbit using preloaded torsional springs at the hinges in a near symmetrical accordion manner, to minimize the shock loads at the hinges. There are five degrees of freedom of the interconnected rigid bodies, composed of coupled attitude motions (pitch, yaw and roll) of the central body plus relative rotations of the solar panel arrays. The dynamical equations of motion of the satellite system are derived using Kane's equations. These are then used to investigate the dynamic behavior of the system during solar panel deployment via the 7-8th-order Runge-Kutta integration algorithms and results are compared with approximate analytical solutions. Chaotic attitude motions of the completely deployed satellite in circular orbit under the influence of the gravity-gradient torques are subsequently investigated analytically using Melnikov's method and confirmed via numerical integration. The Hamiltonian equations in terms of Deprit's variables are used to facilitate the analysis.     

Control of Chaotic Motion in a Spinning Spacecraft with a Circumferential Nutational Damper

Control of chaotic vibrations in a simplified model of a spinning spacecraft with a circumferential nutational damper is achieved using two techniques. The control methods are implemented on a realistic spacecraft parameter configuration which has been found to exhibit chaotic instability when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. Such a torque, in practice, may arise in the platform of a dual-spin spacecraft under malfunction of the control system or from an unbalanced rotor or from vibrations in appendages. Chaotic instabilities arising from these torques could introduce uncertainties and irregularities into a spacecraft's attitude and consequently could have disastrous affects on its operation. The two control methods, recursive proportional feedback (RPF) and continuous delayed feedback, are recently developed techniques for control of chaotic motion in dynamical systems. Each technique is outlined and the effectiveness of the two strategies in controlling chaotic motion exhibited by the present system is compared and contrasted. Numerical simulations are performed and the results are studied by means of time history, phase space, Poincaré map, Lyapunov characteristic exponents and bifurcation diagrams.