An analytical representation of temperature distributions in gravity gradient rods.
Thermal balance analysis of temperature distributions in cylindrical tube considered to model gravity gradient attitude control rod isolated in space
Florio, F. A.↗
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Results for “Analytic gradients”
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Atmospheric wave-induced instability in the nighttime E-region.
Examination of the perturbed continuity equation when the perturbations are the result of an internal atmospheric gravity wave in the E region. The transient response of the ionization is interpreted as the gradient instability and the values of the vertical and horizontal wave numbers that will induce it are plotted for various heights. Only in the presence of westward directed electric fields, which are believed to occur only at night, will the gravity waves induce the gradient instability. Approximate analytic expressions are obtained for the permitted wave numbers as well as for the instability growth times. In the course of this analysis it is shown that in the D region all irregularities, even those that are field-aligned, will tend to move with the ion velocity.
Beer, T.↗
Active control law synthesis for flexible aircraft
An application of an active control law synthesis procedure is presented, to meet multiple design requirements for a flexible aircraft modeled by a large order state space system. In this synthesis procedure, a linear quadratic Gaussian type cost function is minimized by updating the free parameters of the control law, while satisfying a set of constraints on the design loads, responses and stability margins. Analytical expressions for gradients of the cost function and the constraints, with respect to the control law design variables are used to facilitate rapid numerical convergence. These gradients can also be used for sensitivity study. A stable classical control law as well as an estimator-based full or reduced order control law can be modified, in order to meet individual root-mean-square response limitations as well as minimum singular value restrictions. Both analog and digital control laws can be optimized. Low order, robust control laws were synthesized for flutter suppression of a flexible aircraft.
Mukhopadhyay, V.↗
Metastable State Relaxation in a Gravitational Field
A metastable state relaxation equation for a physical system placed into a gravitational field is constructed for non-critical supersaturated solutions which arc in the immediate neighborhood of the coexistence line. Solutions of this equation are obtained in two different regimes: stationary and dynamic. The sedimentation time which can be defined as the time of the subcritical solute cluster redistribution corresponding to the final steady state in the gravitational field is found. The formation of the concentration gradient is proved analytically and its expression through the model parameters is obtained. The following analysis gives the expression for the sedimentation time which does not depend on the column height. The law of the concentration change with respect to the column height is also found and analyzed.
Izmailov, Alexander F.↗
Quantitation of promethazine and metabolites in urine samples using on-line solid-phase extraction and column-switching
A chromatographic method for the quantitation of promethazine (PMZ) and its three metabolites in urine employing on-line solid-phase extraction and column-switching has been developed. The column-switching system described here uses an extraction column for the purification of PMZ and its metabolites from a urine matrix. The extraneous matrix interference was removed by flushing the extraction column with a gradient elution. The analytes of interest were then eluted onto an analytical column for further chromatographic separation using a mobile phase of greater solvent strength. This method is specific and sensitive with a range of 3.75-1400 ng/ml for PMZ and 2.5-1400 ng/ml for the metabolites promethazine sulfoxide, monodesmethyl promethazine sulfoxide and monodesmethyl promethazine. The lower limits of quantitation (LLOQ) were 3.75 ng/ml with less than 6.2% C.V. for PMZ and 2.50 ng/ml with less than 11.5% C.V. for metabolites based on a signal-to-noise ratio of 10:1 or greater. The accuracy and precision were within +/- 11.8% in bias and not greater than 5.5% C.V. in intra- and inter-assay precision for PMZ and metabolites. Method robustness was investigated using a Plackett-Burman experimental design. The applicability of the analytical method for pharmacokinetic studies in humans is illustrated.
NASA Center JSC↗
Phase-retrieval algorithms for a complicated optical system
Phase-retrieval algorithms have been developed that handle a complicated optical system that requires multiple Fresnellike transforms to propagate from one end of the system to the other including the absorption by apertures in more than one plane and allowance for bad detector pixels. Gradient-search algorithms and generalizations of the iterative-transform phase-retrieval algorithms are derived. Analytic expressions for the gradient of an error metric, with respect to polynomial coefficients and with respect to point-by-point phase descriptions, are given. The entire gradient can be computed with the number of transforms required to propagate a wave front from one end of the optical system to the other and back again, independent of the number of coefficients or phase points. This greatly speeds the computation. The reconstruction of pupil amplitude is also given. A convergence proof of the generalized iterative transform algorithm is given. These improved algorithms permit a more accurate characterization of complicated optical systems from their point spread functions.
Fienup, J. R.↗
Corrections to fringe distortion due to flow density gradients in optical interferometry
An analytical method is formulated to account for distortions of optical interferograms used for studies of flow over airfoils experiencing dynamic stall. It is shown that such distortions are generated primarily due to optical path deflections in the test flow, caused by large density gradients. Such in-flow optical path deflections are neglected in conventional optical techniques for flow studies. The present method employs a ray analysis to determine these in-flow optical path deflections, and accurately predicts the interferogram distortions.
Cho, Y. C.↗
Transonic laminar boundary layers with surface curvature.
The effect of surface curvature (both longitudinal and transverse) and the associated pressure gradient across the flow is investigated analytically for a laminar boundary layer subjected to pressure gradients along the flow. Property variation which results from heat transfer and compressibility is taken into account. Numerical solutions of the boundary layer equations are obtained for locally similar sonic flow through the throat of a nozzle for a range of flow conditions and for various shaped nozzle surfaces with different amounts of wall cooling. A few solutions were also obtained for the analogous flow around the shoulder of a flat-faced body in a supersonic flow. The effect of various parameters that arise in the equations upon application of the Levy-Mangler transformation are investigated and discussed with respect to their influence on the velocity and total enthalpy profiles and the corresponding profile slopes at the surface to which the shear stress and heat transfer are related. An important finding is that at throat Reynolds numbers less than 100,000 the heat transfer parameter at a nozzle throat decreases as the throat radius of curvature decreases.
Back, L. H.↗
Gradient-Based Optimization of the Common Research Model Wing Subject to CFD-Based Gust and Flutter Constraints
The linearized frequency-domain method was recently implemented in the stabilized finite element solver in NASA’s FUN3D code. Previous work by the authors used this method for enforcing flutter constraints during gradient-based optimizations. More recently, the solver was expanded to account for continuous (also known as stochastic) gust responses. This paper expands on recent Common Research Model wing optimization work, which demonstrated gradient-based optimization with flutter and stochastic gust constraints, among others. While that work utilized FUN3D for static aeroelastic solutions but relied on doublet lattice aerodynamics for gust and flutter responses, the present work replaces these unsteady aerodynamic analyses with those of FUN3D’s linearized frequency-domain solver. With analytic derivatives available, gradient-based optimization is performed through the use of the OpenMDAO/MPhys libraries with over 700 shape, structural, and aerodynamic design variables and over 10 nonlinear constraints. Comparisons of analysis results and optimized designs are made between doublet lattice and linearized frequency-domain solutions.
aeroelasticity↗
Parallel Aircraft Trajectory Optimization with Analytic Derivatives
Trajectory optimization is an integral component for the design of aerospace vehicles, but emerging aircraft technologies have introduced new demands on trajectory analysis that current tools are not well suited to address. Designing aircraft with technologies such as hybrid electric propulsion and morphing wings requires consideration of the operational behavior as well as the physical design characteristics of the aircraft. The addition of operational variables can dramatically increase the number of design variables which motivates the use of gradient based optimization with analytic derivatives to solve the larger optimization problems. In this work we develop an aircraft trajectory analysis tool using a Legendre-Gauss-Lobatto based collocation scheme, providing analytic derivatives via the OpenMDAO multidisciplinary optimization framework. This collocation method uses an implicit time integration scheme that provides a high degree of sparsity and thus several potential options for parallelization. The performance of the new implementation was investigated via a series of single and multi-trajectory optimizations using a combination of parallel computing and constraint aggregation. The computational performance results show that in order to take full advantage of the sparsity in the problem it is vital to parallelize both the non-linear analysis evaluations and the derivative computations themselves. The constraint aggregation results showed a significant numerical challenge due to difficulty in achieving tight convergence tolerances. Overall, the results demonstrate the value of applying analytic derivatives to trajectory optimization problems and lay the foundation for future application of this collocation based method to the design of aircraft with where operational scheduling of technologies is key to achieving good performance.
aircraft↗
Analytical model for the density distribution in the Io plasma torus
An analytical model is developed for the diffusive equilibrium plasma density distribution in the Io plasma torus. The model has been employed successfully to follow the ray path of plasma waves in the multi-ion Jovian magnetosphere; it would also be valuable for other studies of the Io torus that require a smooth and continuous description of the plasma density and its gradients. Validity of the analytical treatment requires that the temperature of thermal electrons be much lower than the ion temperature and that superthermal electrons be much less abundant than the thermal electrons; these two conditions are satisfied in the warm outer region of the Io torus from L = 6 to L = 10. The analytical solutions agree well with exact numerical calculations for the most dense portion of the Io torus within 30 deg of the equator.
Mei, YI↗
Constrained Burn Optimization for the International Space Station
In long-term trajectory planning for the International Space Station (ISS), translational burns are currently targeted sequentially to meet the immediate trajectory constraints, rather than simultaneously to meet all constraints, do not employ gradient-based search techniques, and are not optimized for a minimum total deltav (v) solution. An analytic formulation of the constraint gradients is developed and used in an optimization solver to overcome these obstacles. Two trajectory examples are explored, highlighting the advantage of the proposed method over the current approach, as well as the potential v and propellant savings in the event of propellant shortages.
Brown, Aaron J.↗
Solving the optimal attention allocation problem in manual control
Within the context of the optimal control model of human response, analytic expressions for the gradients of closed-loop performance metrics with respect to human operator attention allocation are derived. These derivatives serve as the basis for a gradient algorithm that determines the optimal attention that a human should allocate among several display indicators in a steady-state manual control task. Application of the human modeling techniques are made to study the hover control task for a CH-46 VTOL flight tested by NASA.
Kleinman, D. L.↗
Digital robust control law synthesis using constrained optimization
Development of digital robust control laws for active control of high performance flexible aircraft and large space structures is a research area of significant practical importance. The flexible system is typically modeled by a large order state space system of equations in order to accurately represent the dynamics. The active control law must satisy multiple conflicting design requirements and maintain certain stability margins, yet should be simple enough to be implementable on an onboard digital computer. Described here is an application of a generic digital control law synthesis procedure for such a system, using optimal control theory and constrained optimization technique. A linear quadratic Gaussian type cost function is minimized by updating the free parameters of the digital control law, while trying to satisfy a set of constraints on the design loads, responses and stability margins. Analytical expressions for the gradients of the cost function and the constraints with respect to the control law design variables are used to facilitate rapid numerical convergence. These gradients can be used for sensitivity study and may be integrated into a simultaneous structure and control optimization scheme.
Mukhopadhyay, Vivekananda↗
Digital robust active control law synthesis for large order flexible structure using parameter optimization
A generic procedure for the parameter optimization of a digital control law for a large-order flexible flight vehicle or large space structure modeled as a sampled data system is presented. A linear quadratic Guassian type cost function was minimized, while satisfying a set of constraints on the steady-state rms values of selected design responses, using a constrained optimization technique to meet multiple design requirements. Analytical expressions for the gradients of the cost function and the design constraints on mean square responses with respect to the control law design variables are presented.
Mukhopadhyay, V.↗
Designing reduced-order linear multivariable controllers using experimentally derived plant data
An iterative numerical algorithm for simultaneously improving multiple performance and stability robustness criteria for multivariable feedback systems is developed. The unsatisfied design criteria are improved by updating the free parameters of an initial, stabilizing controller's state-space matrices. Analytical expressions for the gradients of the design criteria are employed to determine a parameter correction that improves all of the feasible, unsatisfied design criteria at each iteration. A controller design is performed using the algorithm with experimentally derived data from a large space structure test facility. Experimental results of the controller's performance at the facility are presented.
Frazier, W. G.↗
Gain weighted eigenspace assignment
This report presents the development of the gain weighted eigenspace assignment methodology. This provides a designer with a systematic methodology for trading off eigenvector placement versus gain magnitudes, while still maintaining desired closed-loop eigenvalue locations. This is accomplished by forming a cost function composed of a scalar measure of error between desired and achievable eigenvectors and a scalar measure of gain magnitude, determining analytical expressions for the gradients, and solving for the optimal solution by numerical iteration. For this development the scalar measure of gain magnitude is chosen to be a weighted sum of the squares of all the individual elements of the feedback gain matrix. An example is presented to demonstrate the method. In this example, solutions yielding achievable eigenvectors close to the desired eigenvectors are obtained with significant reductions in gain magnitude compared to a solution obtained using a previously developed eigenspace (eigenstructure) assignment method.
Davidson, John B.↗
Multidisciplinary Optimization of a Turboelectric Tiltwing Urban Air Mobility Aircraft
Urban air taxis, also known as urban air mobility (UAM) vehicles, are anticipated to be an area of significant market growth in the near future. These vehicles are typically vertical take-off and landing (VTOL) designs which are capable of carrying 1 to 30 passengers in an intra-urban environment with flights of less than 50 nautical miles. Development of UAM vehicles and their integration into the airspace will be enabled by advancements in a number of areas including electrified propulsion systems, structures, acoustics, automation, and controls. However, the strong multidisciplinary interactions for these unique vehicles presents a significant new design challenge. This work describes the development of a multidisciplinary analysis and optimization environment which can be used to support the conceptual design of these UAM vehicles, using efficient gradient based optimization with analytic derivatives. The tools included in this multidisciplinary analysis model the aircraft trajectory, vehicle aerodynamics, structures, and electrified propulsion system. The multidisciplinary environment created in this research is unique in that all the physics tools are tightly integrated together, with the trajectory model directly calling the aerodynamics, structures, and propulsion models. This multidisciplinary analysis environment is then demonstrated in the design optimization of a turboelectric tiltwing UAM vehicle concept.
Hendricks, Eric S.↗