Notes on System Theory, Volume VII
System theory - matrices, feedback control system, network synthesis, set theory, stability, shift registers, coding, theorem proving, polynomial roots, channels, and signal flow graphs
FEEDBACK CONTROL SYSTEM↗
SEARCH · Engineering Papers
Results for “stability theory”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Stability of Detached Solidification
Bridgman crystal growth can be conducted in the so-called "detached" solidification regime, where the growing crystal is detached from the crucible wall. A small gap between the growing crystal and the crucible wall, of the order of 100 micrometers or less, can be maintained during the process. A meniscus is formed at the bottom of the melt between the crystal and crucible wall. Under proper conditions, growth can proceed without collapsing the meniscus. The meniscus shape plays a key role in stabilizing the process. Thermal and other process parameters can also affect the geometrical steady-state stability conditions of solidification. The dynamic stability theory of the shaped crystal growth process has been developed by Tatarchenko. It consists of finding a simplified autonomous set of differential equations for the radius, height, and possibly other process parameters. The problem then reduces to analyzing a system of first order linear differential equations for stability. Here we apply a modified version of this theory for a particular case of detached solidification. Approximate analytical formulas as well as accurate numerical values for the capillary stability coefficients are presented. They display an unexpected singularity as a function of pressure differential. A novel approach to study the thermal field effects on the crystal shape stability has been proposed. In essence, it rectifies the unphysical assumption of the model that utilizes a perturbation of the crystal radius along the axis as being instantaneous. It consists of introducing time delay effects into the mathematical description and leads, in general, to stability over a broader parameter range. We believe that this novel treatment can be advantageously implemented in stability analyses of other crystal growth techniques such as Czochralski and float zone methods.
Mazuruk, K.↗
Transition flight experiments on a swept wing with suction
Flight experiments were conducted on a 30 degree swept wing with a perforated leading edge by systematically varying the location and amount of suction over a range of Mach number and Reynolds number. Suction was varied chordwise ahead of the front spar from either the front or rear direction by sealing spanwise perforated strips. Transition from laminar to turbulent flow was due to leading edge turbulence contamination or crossflow disturbance growth and/or Tollmien-Schlichting disturbance growth, depending on the test configuration, flight condition, and suction location. A state-of-the-art linear stability theory which accounts for body and streamline curvature and compressibility was used to study the boundary layer stability as suction location and magnitude varied. N-factor correlations with transition location were made for various suction configurations.
Maddalon, D. V.↗
Transition Flight Experiments on a Swept Wing With Suction
Flight experiments were conducted on a 30 degree swept wing with a perforated leading edge by systematically varying the location and amount of suction over a range of Mach number and Reynolds number. Suction was varied chordwise ahead of the front spar from either the front or rear direction by sealing spanwise perforated strips. Transition from laminar to turbulent flow was due to leading edge turbulence contamination or crossflow disturbance growth and/or Tollmien-Schlichting disturbance growth-depending on the test configuration, flight condition, and suction location. A state-of-the-art linear stability theory which accounts for body and streamline curvature and compressibility was used to study the boundary layer stability as suction location and magnitude varied. N-factor correlations with transition location were made for various suction configurations.
Maddalon, D. V.↗
On the stability of nonlinear, time-varying feedback systems.
Stability theory for nonlinear time-varying feedback control systems
CONTROL STABILITY↗
Linear instability of the supersonic wake behind a flat plate aligned with a uniform stream
A theoretical and numerical study is presented of the growth of linear disturbances in the laminar compressible supersonic wake behind a flat plate aligned with a uniform stream. The approach adopted is that of classical linear stability theory and covers both two- and three-dimensional disturbances. The basic flow is described, and the linear stability problem is formulated that leads to the compressible Rayleigh equation. The stability of short-wave disturbances is then analyzed in the very near wake where the compressible analog of the Goldstein double structure provides an asymptotic description for the unperturbed flow. The behavior of long waves is then considered. The waves are not as long as the plate length but are long compared to the boundary-layer thickness. Numerical solutions of the Rayleigh equation are discussed. Numerical results are then presented for a range of streamwise stations, Mach numbers, and wave propagation angles.
Papageorgiou, D. T.↗
Research in computational fluid dynamics and analysis of algorithms
Recently, higher-order compact schemes have seen increasing use in the DNS (Direct Numerical Simulations) of the Navier-Stokes equations. Although they do not have the spatial resolution of spectral methods, they offer significant increases in accuracy over conventional second order methods. They can be used on any smooth grid, and do not have an overly restrictive CFL dependence as compared with the O(N(exp -2)) CFL dependence observed in Chebyshev spectral methods on finite domains. In addition, they are generally more robust and less costly than spectral methods. The issue of the relative cost of higher-order schemes (accuracy weighted against physical and numerical cost) is a far more complex issue, depending ultimately on what features of the solution are sought and how accurately they must be resolved. In any event, the further development of the underlying stability theory of these schemes is important. The approach of devising suitable boundary clusters and then testing them with various stability techniques (such as finding the norm) is entirely the wrong approach when dealing with high-order methods. Very seldom are high-order boundary closures stable, making them difficult to isolate. An alternative approach is to begin with a norm which satisfies all the stability criteria for the hyperbolic system, and look for the boundary closure forms which will match the norm exactly. This method was used recently by Strand to isolate stable boundary closure schemes for the explicit central fourth- and sixth-order schemes. The norm used was an energy norm mimicking the norm for the differential equations. Further research should be devoted to BC for high order schemes in order to make sure that the results obtained are reliable. The compact fourth order and sixth order finite difference scheme had been incorporated into a code to simulate flow past circular cylinders. This code will serve as a verification of the full spectral codes. A detailed stability analysis by Carpenter (from the fluid Mechanics Division) and Gottlieb gave analytic conditions for stability as well as asymptotic stability. This had been incorporated in the code in form of stable boundary conditions. Effects of the cylinder rotations had been studied. The results differ from the known theoretical results. We are in the middle of analyzing the results. A detailed analysis of the effects of the heating of the cylinder on the shedding frequency had been studied using the above schemes. It has been found that the shedding frequency decreases when the wire was heated. Experimental work is being carried out to affirm this result.
Gottlieb, David↗
Instabilities in the Wake of Roughness on a Flat Plate in a Quiet Supersonic Tunnel
Roughness-induced transition is an unavoidable reality in practical high-speed vehicles. Typical prediction of transition due to roughness include algebraic correlations and, more recently, semi-empirical methods. In the NASA Langley Research Center Supersonic Low Disturbance Tunnel, a Mach 3.5 quiet tunnel, several transition experiments have been performed in the past decade to better understand the mechanisms by which small roughness causes transition in a supersonic boundary layer. The study started first with isolated roughness elements of different planforms and shapes and progressed to increasingly more complicated geometries before arriving at a pseudorandom roughness, defined by an analytic function. Concurrent computational efforts progressed with these studies as well, starting with the use of linear stability theory and progressing to the use of harmonic linearized Navier-Stokes to predict the growth of boundary layer stabilities.
Amanda Chou↗
Methods for systematic generation of Liapunov functions /part two/
Summary of stability theory relative to generation of Liapunov functions
LIAPUNOV FUNCTION↗
Liapunov's second method.
Liapunov second method and stability theory of autonomous and certain nonautonomous systems
LIAPUNOV FUNCTION↗
Results of Flight Test of an Automatically Stabilized Model C (Swept Back) Four-Wing Tiamat
The results of the first flight test of a swept-back four-wing version of Tiamat (MX-570 model C) which was launched at the NACA Pilotless Aircraft Research Station at W4110PB Island, Va. are presented. In general, the flight behavior was close to that predicted by calculations based an stability theory and oscillating table tests of the autopilot. The flight test thus indicates that the techniques employed to predict automatic stability are valid and practical from an operational viewpoint. The limitations of the method used to predict flight behavior arise from the fact that the calculations assume no coupling among roll, pitch, and yaw, while in actual flight some such coupling does exist.
Seacord, Charles L., Jr.↗
Elastic and Plastic Stability of Orthotropic Cylinders
By utilizing linear stability theory, solutions for elastic buckling of short and moderate length orthotropic cylinders under axial compression are presented and correlated with experimental results on circumferentially stiffened cylinders. The plastic buckling of short and moderate length isotropic and orthotropic cylinders is also investigated and the theoretical results correlated with available experimental data. A discussion of the effects of finite deflection and initial imperfection is presented in order to explain the correlation obtained between the theory and the experimental data.
ELASTIC BUCKLING↗
Stability of streamwise vortices
A brief overview of some theoretical and computational studies of the stability of streamwise vortices is given. The local induction model and classical hydrodynamic vortex stability theories are discussed in some detail. The importance of the three-dimensionality of the mean velocity profile to the results of stability calculations is discussed briefly. The mean velocity profile is provided by employing the similarity solution of Donaldson and Sullivan. The global method of Bridges and Morris was chosen for the spatial stability calculations for the nonlinear eigenvalue problem. In order to test the numerical method, a second order accurate central difference scheme was used to obtain the coefficient matrices. It was shown that a second order finite difference method lacks the required accuracy for global eigenvalue calculations. Finally the problem was formulated using spectral methods and a truncated Chebyshev series.
Khorrami, M. K.↗
An introduction to the physical aspects of helicopter stability
In order to provide engineers interested in rotating-wing aircraft, but with no specialized training in stability theory, some understanding of the factors that influence the flying qualities of the helicopter, an explanation is made of both the static stability and the stick-fixed oscillation in hovering and forward flight in terms of fundamental physical quantities. Three significant stability factors -- static stability with angle of attack, static stability with speed, and damping due to a pitching or rolling velocity -- are explained in detail.
Gessow, Alfred↗
Stability of an oscillating boundary layer
Levchenko and Solov'ev (1972, 1974) have developed a stability theory for space periodic flows, assuming that the Floquet theory is applicable to partial differential equations. In the present paper, this approach is extended to unsteady periodic flows. A complete unsteady formulation of the stability problem is obtained, and the stability characteristics over an oscillating period are determined from the solution of the problem. Calculations carried out for an oscillating incompressible boundary layer on a plate showed that the boundary layer flow may be regarded as a locally parallel flow.
Levchenko, V. Y.↗
Transition in hypersonic boundary layers
Linear stability theory for hypersonic boundary layers is presented. The theory is used to study the effects of real gas (under the assumption of local chemical equilibrium) and small nose bluntness on hypersonic boundary layer stability. It is found that chemical reactions have a stabilizing effect on the first mode instability and a destabilizing effect on the second mode instability in hypersonic boundary layers. There is also a tendency for the second-mode instability to shift to lower frequencies. The effect of small nose bluntness is found to be stabilizing.
Malik, Mujeeb R.↗
The modified equation approach to the stability and accuracy analysis of finite-difference methods
The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by certain algebraic manipulations. The connection between 'heuristic' stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.
Warming, R. F.↗
An Experimental Study of Pressure Oscillation in a Capillary Pumped Loop with Multiple Evaporators and Condensers
The heat transport capability of a capillary pumped loop (CPL) is limited by the pressure drop that its evaporator wick can sustain. The pressure drop in a CPL is not constant even under seemingly steady operation, but rather exhibits an oscillatory behavior. A hydrodynamic theory based on a mass-spring-dashpot model was previously developed to predict the pressure oscillation in a CPL with a single evaporator and a single condenser. The theory states that the pressure oscillation is a function of physical dimensions of the CPL components and operating conditions. Experimental data agreed very well with theoretical predictions. The hydrodynamic stability theory has recently been extended to predict the pressure oscillations in CPLs with multiple evaporators and multiple condensers. Concurrently, an experimental study was conducted to verify the theory and to investigate the effects of various parameters on the pressure oscillation. Four evaporators with different wick properties were tested using a test loop containing two condenser plates. The test loop allowed the four evaporators to be tested in a single-pump, two-pump or four-pump configuration, and the two condenser plates to be plumbed either in parallel or in series. Test conditions included varying the power input, the reservoir set point temperature, the condenser sink temperature, and the flow resistance between the reservoir and the loop. Experimental results agreed well with theoretical predictions.
Ku, Jen-Tung↗