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Liapunov stability theory applied to class of partial differential equations, and generation of contraction groups related to equivalent norms
Since the Voyager encounters in 1979, it has been known that Jupiter's cloud-top zonal winds violate the barotropic stability criterion. A vortex-tube stretching analysis of the Voyager wind data indicates that the more general Charney-Stern stability criterion is also violated. On the other hand, the zonal winds determined by tracking cloud features in Hubble Space Telescope images taken in 1991 precisely match the zonal winds determined by tracking cloud features in Voyager images, and it is hard to understand how a complicated zonal wind profile like Jupiter's could be unstable and yet not change at all in 12 years. In fact, there are at least two unknown ways to violate the Charney-Stern stability criterion and still have a stable flow. The better known of these is called Fjortoft's theorem, or Arnol'd's 1st theorem for the case of large-amplitude perturbations. Although the Fjortoft-Arnol'd theorem has been extended from the quasi-geostrophic equations to the primitive equations, the basic requirement that the potential vorticity be an increasing function of streamfunction is opposite to the case found in Jupiter, where the Voyager data indicate that the potential vorticity is a decreasing function of streamfunction. But this second case is precisely that which is covered by Arnol'd's 2nd stability theorem. In fact, the Voyager data suggest that Jupiter's zonal winds are neutrally stable with respect to Arnol'd's 2nd stability theorem. Here, we analyze the linear stability problem of a one-parameter family of sinusoidal zonal wind profiles that are close to neutral stability with respect to Arnol'd's 2nd stability theorem. We find numerically that the most unstable mode is always stationary, which may help to explain the slowly moving mode 10 waves observed on Jupiter. We find that violation of Arnol'd's 2nd stability theorem is both necessary and sufficient for instability of sinusoidal profiles. However, there appears to be no simple extension of Arnol'd's 2nd stability theorem to the primitive equations. Nevertheless, the primitive growth rates are small, and the primitive system is still governed by the quasi-geostrophic neutral-stability configuration.
This paper investigates the stability and the stability degree of a flexible cracked rotor supported on different kinds of journal bearings. It is found that no matter what kind of bearings is used, the unstable zones caused by rotor crack locate always within the speed ratio (2/N) (1 - Delta K(sub xi)/4) is less than Omega is less than 2/N when gravity parameter W(sub R) is greater than 1.0, and locate always within the speed ratio (2 Omega(sub alpha)/N) (1 - Delta K(sub xi)/4) is less than Omega is less than 2 Omega(sub alpha)/N when W(sub R) is less than 0.1, where Delta K(sub xi) is the crack stiffness ratio, N = 1, 2, 3, 4, 5, ..., and Omega(sub alpha) = ((1 + 2 alpha)/2 alpha)(exp 1/2). When 0.1 is less than W(sub R) is less than 1.0, there is a region where no unstable zones caused by rotor crack exist. Outside the crack ridge zones, the rotor crack has almost no influence on system's stability and stability degree; while within the crack ridge zones, the stability and stability degree depend both on the crack and system's parameters. In some cases, the system may still be stable even when the crack is very large. For small gravity parameter (W(sub R) is less than 0.1), the mass ratio alpha has large influence on the position of unstable region, but its influence on the stability degree is small. The influence of fixed Sommerfeld number S(sub 0) on the crack stability degree is small although S(sub 0) has large influence on the stability degree of uncracked rotor.
A method of determining the stability of linear systems with many constant time delays is developed. This technique, an extension of the tau-decomposition method, is used to examine not only the stability but also the relative stability of retarded systems with many delays and a class of neutral equations with one delay. Analytical equations are derived for partitioning the delay space of a retarded system with two time delays. The stability of the system in each of the regions defined by the partitioning curves in the parameter plane is determined using the extended tau-decomposition method. In addition, relative stability boundaries are defined using the extended tau-decompositon method in association with parameter plane techniques. Several applications of the extended tau-decomposition method are presented and compared with stability results obtained from other analyses. In all cases the results obtained using the method outlined herein coincide with and extend those of previous investigations. The extended tau-decomposition method applied to systems with time delays requires less computational effort and yields more complete stability analyses than previous techniques.
MATLAB Stability and Control Toolbox (MASCOT) utilizes geometric, aerodynamic, and inertial inputs to calculate air vehicle stability in a variety of critical flight conditions. The code is based on fundamental, non-linear equations of motion and is able to translate results into a qualitative, graphical scale useful to the non-expert. MASCOT was created to provide the conceptual aircraft designer accurate predictions of air vehicle stability and control characteristics. The code takes as input mass property data in the form of an inertia tensor, aerodynamic loading data, and propulsion (i.e. thrust) loading data. Using fundamental nonlinear equations of motion, MASCOT then calculates vehicle trim and static stability data for the desired flight condition(s). Available flight conditions include six horizontal and six landing rotation conditions with varying options for engine out, crosswind, and sideslip, plus three take-off rotation conditions. Results are displayed through a unique graphical interface developed to provide the non-stability and control expert conceptual design engineer a qualitative scale indicating whether the vehicle has acceptable, marginal, or unacceptable static stability characteristics. If desired, the user can also examine the detailed, quantitative results.
Aerogels demonstrate extraordinarily low thermal conductivity and low density, presenting a promising form of lightweight insulation for aerospace applications. The primary challenge is stabilization of the highly porous structure at temperatures to 1200 °C. Upon collapse of the pore structure, the favorable thermal properties and low density are diminished. Yttria-stabilized zirconia (YSZ) aerogels are synthesized to investigate the effect of composition on morphology, phase behavior, and thermal stability (specific surface area, pore size distribution). This work demonstrates the improvement in thermal stability of zirconia aerogels with increased yttria content. To develop insight on the influence of structure on thermal stability, the effects of water content and solids loading are studied and in elucidation of the mechanism by which yttria stabilizes the pore structure. With an improved understanding of aerogel structure and performance, metal oxide aerogels can be developed to withstand the extreme environments of space exploration.
A method is presented for obtaining the value of the phase stability from time domain frequency stability measurements. Definition of frequency and phase stability are presented. The various types of noise sources in an oscillator and how their location in the oscillator circuitry determines the resultant phase and frequency noise spectrum are described. With this knowledge, the type of noise spectrum from time domain frequency stability measurements is determined. Using certain conversions, the total phase noise spectrum is obtained, which is integrated to obtain the phase stability. Examples of the conversion process are also presented.
A comparison is made between the stability criteria of Hill and that of Laplace to determine the stability of outer planetary orbits encircling binary stars. The restricted, analytically determined results of Hill's method by Szebehely and coworkers and the general, numerically integrated results of Laplace's method by Graziani and Black (1981) are compared for varying values of the mass parameter mu. For mu = 0 to 0.15, the closest orbit (lower limit of radius) an outer planet in a binary system can have and still remain stable is determined by Hill's stability criterion. For mu greater than 0.15, the critical radius is determined by Laplace's stability criterion. It appears that the Graziani-Black stability criterion describes the critical orbit within a few percent for all values of mu.
Part I of this report presents a brief review of methods available to the helicopter designer for obtaining desired stability characteristics by modifications to the airframe design. The discussion is based on modifications made during the establishment of flying-qualities criteria and includes sample results of theoretical studies of additional methods. The conclusion is reached that it is now feasible to utilize combinations of methods whereby stability-parameter values are realized which in turn provide the desired stability characteristics. Part II reviews some of the methods of predicting rotor stability derivatives. The procedures by which these rotor derivatives are employed to estimate helicopter stability characteristics have been summarized.
Maintaining high surface area and porosity at high temperatures is important when considering aerogels for use in thermal management systems. The mesoporous structure of aerogels results in extremely low thermal conductivity, making them lightweight, high performance insulating materials. Maintaining these properties requires innovative routes to suppress sintering and pore collapse as use temperatures rise. The current work aims to improve the pore structure stability of yttria-stabilized zirconia aerogels by the addition of a SiO 2 coating. The functionalization of surface hydroxyl groups by the addition of SiO 2 is hypothesized to mitigate condensation reactions, which are a driving force for shrinkage and pore structure collapse. Zirconia aerogels with 0, 10 and 30 mol% yttria (YO 1.5 ) additions were coated in a tetraethyl orthosilicate (TEOS) solution and exposed to temperatures up to 1200°C. Crystal structure, pore structure, and aerogel morphology were investigated to understand changes in aerogel thermal stability. The SiO 2 coating exhibited a greater influence on pore stability over yttria concentration, with specific surface area of the coated aerogels being twice that of the uncoated aerogels up to 1000°C. However, the presence of the SiO 2 coating promoted rapid sintering and densification at 1200°C, establishing an upper use temperature for SiO 2 and a need to develop other coating chemistries.
A summary of methods for making dynamic lateral stability and response calculations and for estimating the aerodynamic stability derivatives required for use in these calculations is presented. The processes of performing calculations of the time histories of lateral motions, of the period and damping of these motions, and of the lateral stability boundaries are presented as a series of simple straightforward steps. Existing methods for estimating the stability derivatives are summarized and, in some cases, simple new empirical formulas are presented. Reference is also made to reports presenting experimental data that should be useful in making estimates of the derivatives. Detailed estimating methods are presented for low-subsonic-speed conditions but only a brief discussion and a list of references are given for transonic- and supersonic-speed conditions.
A summary of methods for making dynamic lateral stability and response calculations and for estimating the aerodynamic stability derivatives required for use in these calculations is presented. The processes of performing calculations of the time histories of lateral motions, of the period and damping of these motions, and of the lateral stability boundaries are presented as a series of simple straightforward steps. Existing methods for estimating the stability derivatives are summarized and, in some cases, simple new empirical formulas are presented. Detailed estimation methods are presented for low-subsonic-speed conditions but only a brief discussion and a list of references are given for transonic and supersonic speed conditions.
A theoretical investigation has been made to determine the effect of nonlinear stability derivatives on the lateral stability of an airplane. Motions were calculated on the assumption that the directional-stability and the damping-in-yawing derivatives are functions of the angle of sideslip. The application of the Laplace transform to the calculation of an airplane motion when certain types of nonlinear derivatives are present is described in detail. The types of nonlinearities assumed correspond to the condition in which the values of the directional-stability and damping-in-yawing derivatives are zero for small angle of sideslip.
This viewgraph presentation discusses a major potential source of noise for the Laser Interferometer Space Antenna (LISA) that is the laser frequency noise and the proposed mechanism to suppress the unstabilized frequency fluctuations. These fluctuations must be suppresed by about 12 orders of magnitude to achieve a stability that is sufficient for the detection of gravitational waves. This presentation reviews present a modification to the traditional cavity locking technique that allows the laser to be locked to a cavity resonance with an adjustable frequency offset. This presentation also discusses measurements of the system stability, demonstrating that the pre-stabilization level satisfies LISA requirements and a demonstration of a phase-lock loop which utilizes the tunable sideband locking technique as a pre-stabilization stage.
The dynamics and stability of a spin stabilized spacecraft with a hinged appendage system are treated analytically and numerically. The hinged system consists of a central hub with masses attached to massless booms of fixed length whose orientation relative to the main part can change. The equations of motion for the hinged system, with viscous damping at both hinge points, are linearized about the nominal equilibrium position where the booms are orthogonal to the nominal spin axis. Analytic stability criteria indicate that hinge damping must be present and that for limiting cases, where the spin axis is an axis of symmetry, certain conditions relating the hinge point offset coordinates to the moment of inertia ratio and end masses must be satisfied. The hinge damping is always required for the nominal deployment of hinge members.
A stability-bleed system was installed in a YF-12 flight inlet that was subjected to internal and external airflow disturbances in the NASA Lewis 10 by 10 foot supersonic wind tunnel. The purpose of the system is to allow higher inlet performance while maintaining a substantial tolerance (without unstart) to internal and external disturbances. At Mach numbers of 2.47 and 2.76, the inlet tolerance to decreases in diffuser-exit corrected airflow was increased by approximately 10 percent of the operating-point airflow. The stability-bleed system complemented the terminal-shock-control system of the inlet and did not show interaction problems. For disturbances which caused a combined decrease in Mach number and increase in angle of attack, the system with valves operative kept the inlet started 4 to 28 times longer than with the valves inoperative. Hence, the stability system provides additional time for the inlet control system to react and prevent unstart. This was observed for initial Mach numbers of 2.55 and 2.68. For slow increase in angle of attack at Mach 2.47 and 2.76, the system kept the inlet started beyond the steady-state unstart angle. However, the maximum transient angles of attack without unstart could not be determined because wind-tunnel mechanical-stop limits for angle of attack were reached.
The parabolized stability equation (PSE) approach is employed to study linear and nonlinear compressible stability with an eye to providing a capability for boundary-layer transition prediction in both 'quiet' and 'disturbed' environments. The governing compressible stability equations are solved by a rational parabolizing approximation in the streamwise direction. Nonparallel flow effects are studied for both the first- and second-mode disturbances. For oblique waves of the first-mode type, the departure from the parallel results is more pronounced as compared to that for the two-dimensional waves. Results for the Mach 4.5 case show that flow nonparallelism has more influence on the first mode than on the second. The disturbance growth rate is shown to be a strong function of the wall-normal distance due to either flow nonparallelism or nonlinear interactions. The subharmonic and fundamental types of breakdown are found to be similar to the ones in incompressible boundary layers.
This report presents the results of an investigation conducted in the Langley free-flight tunnel to determine the effects of large artificial variations of several rotary lateral-stability derivatives on the dynamic lateral stability and control characteristics of a 45 degree sweptback-wing airplane model. Calculations of the period and damping of the lateral motions and of the response to roll and yaw disturbances were made for correlation with the experimental results. The calculated results were in qualitative agreement with the experimental results in predicting the general trends in flight characteristics produced by large changes in the stability derivatives, but in some cases the theory with the assumption of zero lag was not in good quantitative agreement with the experimental results.