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Nonlinear feedback solution for minimum-time injection into circular orbit with constant thrust acceleration magnitude
This is the sixth yearly report on the research at Princeton University on the problem of combustion instability in liquid-propellant rocket motors. These investigations have been sponsored by the National Aeronautics and Space Administration under NASA Contract NASr-217 and, in the past, under NASA Grant NsG-99-60. The nonlinear aspects of the problem have been emphasized in both the theoretical and experimental portions of the program. Presently, the theoretical research primarily involves studies of possible mechanisms or forcing functions of the instabilities, the nonlinear gasdynamic oscillations associated with instability in rocket motors, the effect of the exhaust nozzle on the stability, and the effect of damping devices on the stability. In conjunction with the theoretical program, an experimental program of an equally broad range is being conducted. This program includes basic experiments both with rocket motors and "cold" chambers which have been designed to study various combustion processes in order to provide understanding which would be useful in the construction of a theoretical model. It also includes studies of the effects of exhaust nozzles and damping devices upon the instability as well as controlled testing of liquid rocket motors with parametric variations intended to determine the factors which strongly influence the incidence of nonlinear combustion instability.
A simple and rapid analysis method, consisting of a number of modular, 'strength-of-materials-type' models, is presented for predicting the nonlinear response and stiffener separation of postbuckled, flat, composite, shear panels. The analysis determines the maximum principal tensile stress in the skin surface layer under to toe. Failure is said to occur when this stress reaches the mean transverse tensile strength of the layer. The analysis methodology consists of a number of closed-form equations that can easily be used in a 'hand analysis. For expediency, they have been programmed into a preliminary design code called SNAPPS (Speedy Nonlinear Analysis of Postbuckled Panels in Shear), which rapidly predicts postbuckling response of the panel for each value of the applied shear load. SNAPPS response and failure predictions were found to agree well with test results for three panels with widely different geometries, laminates and stiffnesses. Design guidelines are given for increasing the load-carrying capacity of stiffened, composite shear panels.
Combinations of high temperature gradients associated with strong transients in both temperature and rotational speed fields are discussed. Blade definition and results of the analysis are shown. Since the same model should later be used to define some dynamic characteristics and certain areas would require a nonlinear analysis, NASTRAN was chosen as a convenient program to manage the several alternatives.
Active attitude control of a spinning Skylab is analyzed to determine a simple control law that will provide a satisfactory response, considering the dynamics of attached flexible appendages. A simplified model of the complex Skylab vehicle is selected to make it analytically tractable. The vehicle is modeled as a single rigid-core body with two attached flexible massless booms having masses on their tips. The equations of motion describing the attitude dynamics of the model are presented as a linear matrix-differential equation. The states of the vehicle are small perturbations about its steady-state spin. An analysis is performed to determine the domain of stability. Next, attitude dynamics are analyzed; both frequency domain (parameter plane) and time domain (an optimal linear quadratic loss program) techniques are compared. An analysis of the nonlinear effect of control torque saturation of Skylab's control moment gyroscopes is discussed. The results of the analysis compare favorably with a large-scale digital simulation of the Skylab.
This paper discusses the Kalman filter used in the navigation software of the Space Shuttle. The form of the filter will be discussed: the standard Kalman filter versus the square-root version of this filter. Both of these filters have severe nonlinearity problems. Two successful solutions of the nonlinearity problem are presented. A real-time program must have a good method of editing bad data. The data editing scheme is discussed. Those elements of the Kalman filter state vector which are random variables will be discussed.
The effects of wind shear on aircraft flying through thunderstorm gust fronts were investigated. A computer program was developed to solve the two dimensional, nonlinear equations of aircraft motion, including wind shear. The procedure described and documented accounts for spatial and temporal variations of the aircraft within the flow regime. Analysis of flight paths and control inputs necessary to maintain specified trajectories for aircraft having characteristics of DC-8, B-747, augmentor wing STOL, and DHC-6 aircraft was recorded. From the analysis an attempt was made to find criteria for reduction of the hazards associated with landing through thunderstorm gust fronts.
Investigation of the basic requirements for modeling slender member shadowing effects on thermally induced deformation continues. The theoretical formulation is complete and computer programming is underway. The basic requirements for thermal finite elements to model heat transfer in orbiting structures were also studied. The need for planar isothermal elements to model large space structures' antenna meshes was established. Finite Element approaches for modeling space structure cable and membrane components with thermal effects is underway. Three levels of structural modeling and analysis were identified: (1) a linear-elastic small deflection analysis that does not permit cable slackening; (2) a linear-elastic analysis that includes stress stiffening (the ANSYS program), but not large deflections; and (3) full nonlinear large deflection analysis with stress stiffening. Methods of determining cable initial tensions are currently being evaluated.
A large array of models were applied in a unified manner to solve problems in space flight physiology. Mathematical simulation was used as an alternative way of looking at physiological systems and maximizing the yield from previous space flight experiments. A medical data analysis system was created which consist of an automated data base, a computerized biostatistical and data analysis system, and a set of simulation models of physiological systems. Five basic models were employed: (1) a pulsatile cardiovascular model; (2) a respiratory model; (3) a thermoregulatory model; (4) a circulatory, fluid, and electrolyte balance model; and (5) an erythropoiesis regulatory model. Algorithms were provided to perform routine statistical tests, multivariate analysis, nonlinear regression analysis, and autocorrelation analysis. Special purpose programs were prepared for rank correlation, factor analysis, and the integration of the metabolic balance data.
Delamination growth caused by local buckling of a delaminated group of plies was investigated. Delamination growth was assumed to be governed by the strain energy release rates, G(1), G(2) and G(3). The strain energy release rates were calculated using a geometrically nonlinear three-dimensional finite element analysis. The program is described and several checks of the analysis are discussed. Based on a limited parametric study, the following conclusions were reached: (1) the problem is definitely mixed mode (in some cases G(1) is larger than G(2), for other cases the opposite is true); (2) in general, there is a large gradient in the strain energy release rates along the delamination front; (3) the locations of maximum G(1) and G(2) depend on the delamination shape and the applied strain; (4) the mode 3 component was negligible for all cases considered; and (5) the analysis predicted that parts of the delamination would overlap. The results presented did not impose contact constraints to prevent overlapping. Further work is needed to determine the effects of allowing the overlapping.
Delamination growth caused by local buckling of a delaminated group of plies was investigated. Delamination growth was assumed to be governed by the strain energy release rates, G(1), G(2) and G(3). The strain energy release rates were calculated using a geometrically nonlinear three-dimensional finite element analysis. The program is described and several checks of the analysis are discussed. Based on a limited parametric study, the following conclusions were reached: (1) the problem is definitely mixed mode (in some cases G(1) is larger than G(2), for other cases the opposite is true); (2) in general, there is a large gradient in the strain energy release rates along the delamination front; (3) the locations of maximum G(1) and G(2) depend on the delamination shape and the applied strain; (4) the mode 3 component was negligible for all cases considered; and (5) the analysis predicted that parts of the delamination would overlap. The results presented did not impose contact constraints to prevent overlapping. Further work is needed to determine the effects of allowing the overlapping.
This report attempts to unify in a single document the results of a series of studies on fighter aircraft agility funded by the NASA Ames Research Center, Dryden Flight Research Facility and conducted at the University of Kansas Flight Research Laboratory during the period January 1989 through December 1993. New metrics proposed by pilots and the research community to assess fighter aircraft agility are collected and analyzed. The report develops a framework for understanding the context into which the various proposed fighter agility metrics fit in terms of application and testing. Since new metrics continue to be proposed, this report does not claim to contain every proposed fighter agility metric. Flight test procedures, test constraints, and related criteria are developed. Instrumentation required to quantify agility via flight test is considered, as is the sensitivity of the candidate metrics to deviations from nominal pilot command inputs, which is studied in detail. Instead of supplying specific, detailed conclusions about the relevance or utility of one candidate metric versus another, the authors have attempted to provide sufficient data and analyses for readers to formulate their own conclusions. Readers are therefore ultimately responsible for judging exactly which metrics are 'best' for their particular needs. Additionally, it is not the intent of the authors to suggest combat tactics or other actual operational uses of the results and data in this report. This has been left up to the user community. Twenty of the candidate agility metrics were selected for evaluation with high fidelity, nonlinear, non real-time flight simulation computer programs of the F-5A Freedom Fighter, F-16A Fighting Falcon, F-18A Hornet, and X-29A. The information and data presented on the 20 candidate metrics which were evaluated will assist interested readers in conducting their own extensive investigations. The report provides a definition and analysis of each metric; details of how to test and measure the metric, including any special data reduction requirements; typical values for the metric obtained using one or more aircraft types; and a sensitivity analysis if applicable. The report is organized as follows. The first chapter in the report presents a historical review of air combat trends which demonstrate the need for agility metrics in assessing the combat performance of fighter aircraft in a modern, all-aspect missile environment. The second chapter presents a framework for classifying each candidate metric according to time scale (transient, functional, instantaneous), further subdivided by axis (pitch, lateral, axial). The report is then broadly divided into two parts, with the transient agility metrics (pitch lateral, axial) covered in chapters three, four, and five, and the functional agility metrics covered in chapter six. Conclusions, recommendations, and an extensive reference list and biography are also included. Five appendices contain a comprehensive list of the definitions of all the candidate metrics; a description of the aircraft models and flight simulation programs used for testing the metrics; several relations and concepts which are fundamental to the study of lateral agility; an in-depth analysis of the axial agility metrics; and a derivation of the relations for the instantaneous agility and their approximations.
Lightning Optical Corporation, under an SBIR (Small Business Innovative Research) agreement with Langley Research Center, manufactures oxide and fluoride laser gain crystals, as well as various nonlinear materials. The ultimate result of this research program is the commercial availability in the marketplace of a reliable source of high-quality, damage resistant laser material, primarily for diode-pumping applications.
The Ko displacement theory, formulated for weak nonuniform (slowly changing cross sections) cantilever beams, was applied to the deformed shape analysis of the doubly-tapered wings of the Ikhana unmanned aircraft. The two-line strain-sensing system (along the wingspan) was used for sensing the bending strains needed for the wing-deformed shapes (deflections and cross-sectional twist) analysis. The deflection equation for each strain-sensing line was expressed in terms of the bending strains evaluated at multiple numbers of strain-sensing stations equally spaced along the strain-sensing line. For the preflight shape analysis of the Ikhana wing, the strain data needed for input to the displacement equations for the shape analysis were obtained from the nodal-stress output of the finite-element analysis. The wing deflections and cross-sectional twist angles calculated from the displacement equations were then compared with those computed from the finite-element computer program. The Ko displacement theory formulated for weak nonlinear cantilever beams was found to be highly accurate in the deformed shape predictions of the doubly-tapered Ikhana wing.
The LINFLUX-AE computer code predicts flutter and forced responses of blades and vanes in turbomachines under subsonic, transonic, and supersonic flow conditions. The code solves the Euler equations of unsteady flow in a blade passage under the assumption that the blades vibrate harmonically at small amplitudes. The steady-state nonlinear Euler equations are solved by a separate program, then equations for unsteady flow components are obtained through linearization around the steady-state solution. A structural-dynamics analysis (see figure) is performed to determine the frequencies and mode shapes of blade vibrations, a preprocessor interpolates mode shapes from the structural-dynamics mesh onto the LINFLUX computational-fluid-dynamics mesh, and an interface code is used to convert the steady-state flow solution to a form required by LINFLUX. Then LINFLUX solves the linearized equations in the frequency domain to calculate the unsteady aerodynamic pressure distribution for a given vibration mode, frequency, and interblade phase angle. A post-processor uses the unsteady pressures to calculate generalized aerodynamic forces, response amplitudes, and eigenvalues (which determine the flutter frequency and damping). In comparison with the TURBO-AE aeroelastic-analysis code, which solves the equations in the time domain, LINFLUX-AE is 6 to 7 times faster.
A general method for parallel and vector numerical solutions of stochastic dynamic programming problems is described for optimal control of general nonlinear, continuous time, multibody dynamical systems, perturbed by Poisson as well as Gaussian random white noise. Possible applications include lumped flight dynamics models for uncertain environments, such as large scale and background random atmospheric fluctuations. The numerical formulation is highly suitable for a vector multiprocessor or vectorizing supercomputer, and results exhibit high processor efficiency and numerical stability. Advanced computing techniques, data structures, and hardware help alleviate Bellman's curse of dimensionality in dynamic programming computations.
Development was extended of a simplified inelastic analysis computer program (ANSYMP) for predicting the stress-strain history at the critical location of a thermomechanically cycled structure from an elastic solution. The program uses an iterative and incremental procedure to estimate the plastic strains from the material stress-strain properties and a plasticity hardening model. Creep effects can be calculated on the basis of stress relaxation at constant strain, creep at constant stress, or a combination of stress relaxation and creep accumulation. The simplified method was exercised on a number of problems involving uniaxial and multiaxial loading, isothermal and nonisothermal conditions, dwell times at various points in the cycles, different materials, and kinematic hardening. Good agreement was found between these analytical results and nonlinear finite-element solutions for these problems. The simplified analysis program used less than 1 percent of the CPU time required for a nonlinear finite-element analysis.
Techniques used in simulating an electrical circuit with nonlinear elements for use in computer-aided circuit analysis programs are described. Elements of the circuit include capacitors, resistors, inductors, transistors, diodes, and voltage and current sources (constant or time varying). Simulation features are discussed for dc, ac, and/or transient circuit analysis. Calculations are based on the model approach of formulating the circuit equations. A particular solution of transient analysis for nonlinear storage elements is described.