A frequency analysis of the ν9 band of CD3CD3: experiment and ab initio calculations

The infrared gaseous spectrum of CD₃CD₃ has been measured in the range of 530–670cm^?1 to investigate vibration—torsion effects in the ν₉ band. Three separate spectra all taken under different experimental conditions were recorded. The lines with (ΔK = ?1) and with high values of K show torsional splittings that are substantially larger than expected from the observed barrier height. These splittings are caused primarily by Coriolis-type interactions between the torsional stack of ν₉ = 1 and the corresponding stack for the ground vibrational state. Because of a near-degeneracy that exists between the states (ν₉ = 0, ν₄ = 3) and (ν₉ = 1, ν₄ = 0), three subbands (K, σ) = (15,1), (16,2), (17,3) are resonantly perturbed. For these cases, perturbation-allowed 3ν₄ torsional transitions have been identified. Here σ= 0, 1, 2 or 3 labels the torsional sublevels. Measurements from the ν₉ and 3ν₄ bands, frequencies from the far-infrared torsional spectra in the ground vibrational state, and lower state combination differences from the ν₉ + ν₄ ? ν₄ band were fitted to within experimental uncertainty using an effective Hamiltonian which considered three torsional stacks; one for the ground vibrational state and two for ν₉ = 1. In all, 22 parameters were determined using a total of 2001 lines. Of these, three parameters were the interstack couplings, eight are from the ground vibrational state and 11 are from the excited vibrational state. Two barrier-dependent torsion—rotation parameters, which were essential for obtaining a satisfactory fit, were calculated by ab initio methods.     

A Generalization of Algebraic Stability for Runge--Kutta Methods

The theory of algebraic stability does not give detailed informationabout the stability properties of explicit methods. This theoryis generalized by considering a monotonic and bounded test problem.The results give additional information about the stabilityproperties of explicit and implicit methods and show how particularexplicit methods may be selected.     

Stability of Runge-Kutta Methods for Trajectory Problems

A solution of a system of m autonomous differential equationsdefines a trajectory in m-dimensional space and, in particular,may give a closed orbital path. Typical trajectories are describedby a model nonlinear problem introduced in this article. Forthis problem, a trajectory lies on a surface characterized bya real symmetric matrix. It is shown that some Runge-Kutta methodspossess a property which ensures that, for this model problem,the numerical solution lies on the same surface as the trajectory.When m = 2, the numerical solution lies on the trajectory. Thisproperty is related to algebraic stability. A weaker propertysuffices for normalized differential systems.     

CHARACTERIZATION OF THE BEHAVIOUR OF A SIMPLE AEROSERVOELASTIC SYSTEM WITH CONTROL NONLINEARITIES

The characterization of the behaviour of nonlinear aeroelastic systems has become a very important research topic in the Aerospace Industry. However, most work carried to-date has concentrated upon systems containing structural or aerodynamic nonlinearities. The purpose of this paper is to study the stability of a simple aeroservoelastic system with nonlinearities in the control system and power control unit. The work considers both structural and control law nonlinearities and assesses the stability of the system response using bifurcation diagrams. It is shown that simple feedback systems designed to increase the stability of the linearized system also stabilize the nonlinear system, although their effects can be less pronounced. Additionally, a nonlinear control law designed to limit the control surface pitch response was found to increase the flutter speed considerably by forcing the system to undergo limit cycle oscillations instead of fluttering. Finally, friction was found to affect the damping of the system but not its stability, as long as the amplitude of the frictional force is low enough not to cause stoppages in the motion.     

On the Existence of Solutions for Algebraically Stable Runge--Kutta Methods

An implicit Runge—Kutta method, applied to an initial-valueproblem, gives systems of algebraic equations. It is shown that,under natural assumptions concerning the differential system,these equations have unique solutions if the method satisfiesa condition related to algebraic stability. In particular, thiscondition is satisfied if the method is irreducible and (k,l)-algebraically stable for some l 0.