Stability conditions derived from spectral theory - Discrete systems with periodic feedback
Stability conditions of linear discrete system with periodic feedback from spectrum location of bounded linear operator acting in Banach space
Davis, J. H.↗
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Working With the Wave Equation in Aeroacoustics: The Pleasures of Generalized Functions
The theme of this paper is the applications of generalized function (GF) theory to the wave equation in aeroacoustics. We start with a tutorial on GFs with particular emphasis on viewing functions as continuous linear functionals. We next define operations on GFs. The operation of interest to us in this paper is generalized differentiation. We give many applications of generalized differentiation, particularly for the wave equation. We discuss the use of GFs in finding Green s function and some subtleties that only GF theory can clarify without ambiguities. We show how the knowledge of the Green s function of an operator L in a given domain D can allow us to solve a whole range of problems with operator L for domains situated within D by the imbedding method. We will show how we can use the imbedding method to find the Kirchhoff formulas for stationary and moving surfaces with ease and elegance without the use of the four-dimensional Green s theorem, which is commonly done. Other subjects covered are why the derivatives in conservation laws should be viewed as generalized derivatives and what are the consequences of doing this. In particular we show how we can imbed a problem in a larger domain for the identical differential equation for which the Green s function is known. The primary purpose of this paper is to convince the readers that GF theory is absolutely essential in aeroacoustics because of its powerful operational properties. Furthermore, learning the subject and using it can be fun.
Farassat, F.↗
Prediction of compressor stall for distorted and undistorted flow by use of a multistage compressor simulation on the digital computer
A simulation technique is presented for the prediction of compressor stall for axial-flow compressors for clean and distorted inlet flow. The simulation is implemented on the digital computer and uses stage stacking and lumped-volume gas dynamics. The resulting nonlinear differential equations are linearized about a steady-state operating point, and a Routh-Hurwitz stability test is performed on the linear system matrix. Parallel compressor theory is utilized to extend the technique to the distorted inlet flow problem. The method is applied to the eight-stage J85-13 compressor.
Daniele, C. J.↗
Prediction of compressor stall for distorted and undistorted flow by use of a multistage compressor simulation on the digital computer
A simulation technique is presented for the prediction of compressor stall for axial-flow compressors for clean and distorted inlet flow. The simulation is implemented on the digital computer and uses stage stacking and lumped-volume gas dynamics. The resulting nonlinear differential equations are linearized about a steady-state operating point, and a Routh-Hurwitz stability test is performed on the linear system matrix. Parallel compressor theory is utilized to extend the technique to the distorted inlet flow problem. The method is applied to the eight-stage J85-13 compressor. Analytical stall prediction for the undistorted stall line shows good agreement with experimental results. The predicted stall line for distorted inlet flow is in agreement with experimental results only for large distortion extents and/or low distortion levels. Results for low distortion extents and high distortion level do not agree with experimental results.
Daniele, C. J.↗
Going through a quantum phase
Phase measurements on a single-mode radiation field are examined from a system-theoretic viewpoint. Quantum estimation theory is used to establish the primacy of the Susskind-Glogower (SG) phase operator; its phase eigenkets generate the probability operator measure (POM) for maximum likelihood phase estimation. A commuting observables description for the SG-POM on a signal x apparatus state space is derived. It is analogous to the signal-band x image-band formulation for optical heterodyne detection. Because heterodyning realizes the annihilation operator POM, this analogy may help realize the SG-POM. The wave function representation associated with the SG POM is then used to prove the duality between the phase measurement and the number operator measurement, from which a number-phase uncertainty principle is obtained, via Fourier theory, without recourse to linearization. Fourier theory is also employed to establish the principle of number-ket causality, leading to a Paley-Wiener condition that must be satisfied by the phase-measurement probability density function (PDF) for a single-mode field in an arbitrary quantum state. Finally, a two-mode phase measurement is shown to afford phase-conjugate quantum communication at zero error probability with finite average photon number. Application of this construct to interferometric precision measurements is briefly discussed.
Shapiro, Jeffrey H.↗
Fluid System Stability Analysis Techniques
Fluid systems, or networks, consist of multiple components that work together to achieve some desired thermofluid state. For any generic application, this fluid state can be a combination of the fluid pressure, flow rate, enthalpy, or species concentration. Fluid system components, such as pumps and valves, are often governed by nonlinear differential equations, resulting in complex component-to-component interactions. System-level fluid network stability occurs when the flow through the system can maintain a steady-state solution in the presence of small perturbations, which depends on these component interactions. System instability, however, can go undetected until issues arise during integrated system testing. This presentation explores a method for system designers to think of the fluid network as an assembly of components, each with their own thermofluid surfaces of partial stability, called nullclines. The intersections of all nullclines yields system-level solutions, called equilibrium points. When designers define operating points, they are tuning system parameters so that these equilibrium points move to the desired location in the thermofluid state plane. However, linearization theory shows us that the dynamic behavior around these points can be unstable. The local stability of these equilibrium points can be assessed analytically with eigen-analysis, or numerically by propagating state-plane samples to construct a phase portrait. Investigating a phase portrait can help designers gain a qualitative understanding of a system’s dynamic performance. This understanding can then help inform requirement definitions, component selection, and operational procedures. This presentation includes an example of the phase portrait technique on a system featuring a centrifugal pump and a back-pressure regulator (BPR). Numerical modeling of this system suggests that equilibrium points on the left-hand side of the pump curve are dynamically unstable.
Fluid System↗
Fluid System Stability Analysis Techniques
Fluid systems, or networks, consist of multiple components that work together to achieve some desired thermofluid state. For any generic application, this fluid state can be a combination of the fluid pressure, flow rate, enthalpy, or species concentration. Fluid system components, such as pumps and valves, are often governed by nonlinear differential equations, resulting in complex component-to-component interactions. System-level fluid network stability occurs when the flow through the system can maintain a steady-state solution in the presence of small perturbations, which depends on these component interactions. System instability, however, can go undetected until issues arise during integrated system testing. This presentation explores a method for system designers to think of the fluid network as an assembly of components, each with their own thermofluid surfaces of partial stability, called nullclines. The intersections of all nullclines yields system-level solutions, called equilibrium points. When designers define operating points, they are tuning system parameters so that these equilibrium points move to the desired location in the thermofluid state plane. However, linearization theory shows us that the dynamic behavior around these points can be unstable. The local stability of these equilibrium points can be assessed analytically with eigen-analysis, or numerically by propagating state-plane samples to construct a phase portrait. Investigating a phase portrait can help designers gain a qualitative understanding of a system’s dynamic performance. This understanding can then help inform requirement definitions, component selection, and operational procedures. This presentation includes an example of the phase portrait technique on a system featuring a centrifugal pump and a back-pressure regulator (BPR). Numerical modeling of this system suggests that equilibrium points on the left-hand side of the pump curve are dynamically unstable.
Fluid System↗
SPAR thermal analysis processors reference manual, system level 16. Volume 1: Program executive. Volume 2: Theory. Volume 3: Demonstration problems. Volume 4: Experimental thermal element capability. Volume 5: Programmer reference
User instructions are given for performing linear and nonlinear steady state and transient thermal analyses with SPAR thermal analysis processors TGEO, SSTA, and TRTA. It is assumed that the user is familiar with basic SPAR operations and basic heat transfer theory.
Marlowe, M. B.↗
Optimal digital control of a Stirling cycle cooler
This short paper describes work in progress on the conceptual design of a control system for a cryogenic cooler intended for use aboard spacecraft. The cooler will produce 5 watts of cooling at 65 K and will be used to support experiments associated with the following: earth observation; atmospheric measurements; infrared, x-ray, and gamma-ray astronomy; and magnetic field characterization. The cooler has been designed and constructed for NASA/GSFC by Philips Laboratories and is described in detail. The cooler has a number of unique design features intended to enhance long life and maintenance free operation in space including use of the high efficiency Stirling thermodynamic refrigeration cycle, linear magnetic motors, clearance-seals, and magnetic bearings. The proposed control system design is based on optimal control theory and is targeted for custom integrated circuit implementation. The resulting control system will meet the following mission requirements: efficiency, reliability, optimal thermodynamic, electrical, and mechanical performance; freedom from operator intervention; light weight; and small size.
Feeley, J.↗
Optimal discrete-time LQR problems for parabolic systems with unbounded input: Approximation and convergence
An abstract approximation and convergence theory for the closed-loop solution of discrete-time linear-quadratic regulator problems for parabolic systems with unbounded input is developed. Under relatively mild stabilizability and detectability assumptions, functional analytic, operator techniques are used to demonstrate the norm convergence of Galerkin-based approximations to the optimal feedback control gains. The application of the general theory to a class of abstract boundary control systems is considered. Two examples, one involving the Neumann boundary control of a one-dimensional heat equation, and the other, the vibration control of a cantilevered viscoelastic beam via shear input at the free end, are discussed.
Rosen, I. G.↗
A Simplified Theory of Coupled Oscillator Array Phase Control
Linear and planar arrays of coupled oscillators have been proposed as means of achieving high power rf sources through coherent spatial power combining. In such - applications, a uniform phase distribution over the aperture is desired. However, it has been shown that by detuning some of the oscillators away from the oscillation frequency of the ensemble of oscillators, one may achieve other useful aperture phase distributions. Notable among these are linear phase distributions resulting in steering of the output rf beam away from the broadside direction. The theory describing the operation of such arrays of coupled oscillators is quite complicated since the phenomena involved are inherently nonlinear. This has made it difficult to develop an intuitive understanding of the impact of oscillator tuning on phase control and has thus impeded practical application. In this work a simpl!fied theory is developed which facilitates intuitive understanding by establishing an analog of the phase control problem in terms of electrostatics.
Pogorzelski, R. J.↗
Phase-locked loops and their application
A collection of papers is presented on the characteristics and capabilities of phase-locked loops (PLLs), along with some applications of interest. The discussion covers basic theory (linear and nonlinear); acquisition; threshold; stability; frequency demodulation and detection; tracking; cycle slipping and loss of lock; phase-locked oscillators; operation and performance in the presence of noise; AGC, AFC, and APC circuits and systems; digital PLL; and applications and miscellaneous. With the rapid development of IC technology, PLLs are expected to be used widely in consumer electronics.
Lindsey, W. C.↗
Effects of supply conditions on film thickness in lubricated Hertzian contacts
A generalization of the hydrodynamic expression for Hertzian contacts is described and various methods for calculating the thickness of the oil film winter steady-state, isothermal conditions are given. This is important for engineering applications such as gears and bearings because these results are closer to real operating conditions. Theories of lubrication are discussed, and the mathematics involved are presented using approximately 30 equations and 13 figures. For lubricated, linear, elliptical or point Hertzian contacts it is demonstrated how to calculate the thickness of the oil film at the center of the contact for steady-state isothermal conditions.
Dalmaz, G.↗
Application of a First-Principles Anomalous Transport Model for Electrons to Multiple Hall Thrusters and Operating Conditions
We have developed a physics-based model based on a pseudo-particle description of the electron cyclotron drift instability. A key improvement of the model with respect to previous work is that linear theory is not applied in the event of wave saturation and deviations of electrons or ions from a Maxwellian distribution function. In the acceleration region, the anomalous collision frequency is computed as the minimum value necessary to prevent the electron drift velocity from exceeding the thermal velocity. A functional based on the electron equilibration time is defined to control the transition from high to low resistivity regions. The model was previously applied to a single Hall thruster at its nominal operating condition, showing promising results that captured accurately the location of the thruster’s acceleration region. In this paper, we extend the use of this firstprinciples models to two additional thrusters, also considering multiple operating conditions for each of them. Numerical results are compared to experimental measurements obtained with non-invasive laser induced fluorescence. In general, the agreement between experiments and simulations is good. The model is able to predict the location of the acceleration region for all cases. We observe however that fine details, such as changes in the plasma potential gradient within the acceleration regions, are not captured. The model is also insensitive to changes in the magnetic field strength while experiments show that small shifts in location (of less than 5% of the acceleration channel length) occur. We plan to address the weaknesses of our method with the help of physical insight gained from kinetic simulations of the acceleration region.
Chaplin, Vernon H.↗
Glass fining experiments in zero gravity
Ground based experiments were conducted to demonstrate that thermal migration actually operated in glass melts. Thermal migration consistent with the theory was found in one experiment on a borax melt, i.e., there was an approximately linear relation between the bubble diameter and bubble velocity for a given temperature and temperature gradient. It also appeared that nearby bubbles were attracted to one another, which could greatly aid fining. Interpretation of these results was not possible because of complications arising from gravity, i.e., floating of the bubbles, circulation currents due to buoyancy-driven natural connection, and flow of the melt out from the cell.
Smith, H. D.↗
Algorithm For Control Of Large Antenna
Alternative position-error feedback modes provided. Modern control theory basis for computer algorithm used to control two-axis positioning of large antenna. Algorithm - incorporated into software of real-time control computer - enables rapid intertarget positioning as well as precise tracking (using one of two optional position-feedback modes) without need of human operator intervention. Control system for one axis of two-axis azimuth/elevation control system embodied mostly in software based on advanced control theory. System has linear properties of classical linear feedback controller. Performance described by bandwidth and linear error coefficients.
Hill, Robert E.↗
Experimental verification of the photodiode theory of SIS mixers
The authors describe the characterization and interpretation of the performance of SIS receivers within the framework of the photodiode theory of mixing. The quantum efficiency plays a dominant role in the theory, and a simple method of accurately measuring this parameter is presented. It is demonstrated that the quantum efficiency measurements can be conveniently made on a standard radio astronomy receiver and combined with the usual hot and cold load characterization to improve the understanding of the receiver's performance. The measurements verify that the photodiode theory of mixing accurately describes the receiver noise even at local-oscillator power levels well above the linear response range. The results for receivers operating at 100 and 240 GHz verify the utility of this approach. These methods should also prove useful in evaluating submillimeter receivers.
Woody, David P.↗
Studies in Thermocapillary Convection of the Marangoni-Benard Type
The effects of imposed nonlinear oscillatory shear upon the onset of Marangoni-Bernard convection, as predicted by linear theory, in a layer of liquid with a deformable free surface were reported upon by Or and Kelly for small amplitude oscillations. Depending on the operating conditions, either stabilization or destabilization might occur. The aim of the current paper is to report the results for finite amplitude imposed oscillations so that the actual amount of stabilization or destabilization can be determined for prescribed operating conditions.
Kelly, Robert E.↗