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Kalman-Schmidt filter stability in orbit calculation, dependence on transition matrix, initial covariance matrix, observations, and covariance of noise in measurements
FORTRAN computer subroutines stemming from requirements to process state variable system equations for systems of high order are presented. They find the characteristic equation of a matrix using the method of Danilevsky, the number of roots with positive real parts using the Routh-Horwitz alternate formulation, convert a state variable system description to a Laplace transfer function using the method of Bollinger, and evaluate that transfer function and obtain its frequency response. A sample problem is presented to demonstrate use of the subroutines.
A nonsteady vortex-lattice method is introduced for predicting the dynamic stability derivatives of a delta wing undergoing an oscillatory motion. The analysis is applied to several types of small oscillations in pitch. The angle of attack varied between + or - 1 deg, with the mean held at 0 deg when the flow was assumed to be attached and between + or - 1 deg and the mean held at 15 deg when both leading-edge separation and wake roll-up were included. The computed results for damping in pitch are compared with several other methods and with experiments, and are found to be consistent and in good agreement.
Transonic aeroelastic stability and response analyses are performed for the MBB A-3 supercritical airfoil. Three degrees of freedom are considered: plunge, pitch, and aileron pitch. The control of airfoil stability and response in transonic flow are studied. Stability analyses are performed using a Pade aeroelastic model based on the use of LTRAN2-NLR transonic small disturbance finite difference computer code. Response analyses are performed by coupling the structural equations of motion to the unsteady aerodynamic forces of LTRAN2-NLR. The focus is on transonic time marching transient response solutions using modal identification to determine stability. Frequency and damping of these modes are directly compared in the complex s-plane with Pade model eigenvalues. Transonic stability and response characteristics of 2-D airfoils are discussed and comparisons are made. Application of the Pade aeroelastic model and time marching analyses to flutter suppression using active controls is demonstrated.
Previously cited in issue 21, p. 3626, Accession no. A81-44570
Transonic aeroelastic stability and response analyses are performed for the MBB A-3 supercritical airfoil. Three degrees of freedom are considered: plunge, pitch, and aileron pitch. The objective of this study is to gain insight into the control of airfoil stability and response in transonic flow. Stability analyses are performed using a Padeaeroelastic model based on the use of the LTRAN2-NLR transonic small-disturbance finite-difference computer code. Response analyses are performed by coupling the structural equations of motion to the unsteady aerodynamic forces of LTRAN2-NLR. The focus of the present effort is on transonic time-marching transient response solutions using modal identification to determine stability. Frequency and damping of these modes are directly compared in the complex s-plane with Pademodel eigenvalues. Transonic stability and response characteristics of two-dimensional airfoils are discussed and comparisons are made. Application of the Padeaeroelastic model and time-marching analyses to flutter suppression using active controls is demonstrated.
The stability of two-dimensional infinitesimal disturbances of the steady inviscid incompressible flow produced by infinite rows of finite-cored particles is studied. Numerical results are obtained for the growth rate and oscillation frequencies of disturbances of arbitrary subharmonic wavenumber, and the stability boundaries are calculated. The stabilization of the pairing instability by finite area demonstrated by Saffman and Schatzman (1982) is confirmed, as is Kida's (1982) result that this is not the most unstable disturbance when the area is finite. However, the finite area does not stabilize the inviscid Karman vortex street of finite-area vortices to infinitesimal two-dimensional disturbances of arbitrary wavelength. The finite area is always unstable except for one isolated value of the aspect ratio which depends upon the size of the vortices.
The term vortex breakdown refers to the abrupt and drastic changes of structure that can sometimes occur in swirling flows. It was conjectured that the bubble type of breakdown can be viewed as an axisymmetric wave traveling upstream in a primarily columnar vortex flow. In this scenario the wave's upstream progress is impeded only when it reaches a critical amplitude and it loses stability to some nonaxisymmetric disturbance. The stability of some axisymmetric wavy flows to three dimensional disturbances, viewing the amplitude of the wave as a bifurcation parameter is examined. The stability of a set of related columnar vortex flows, constructed by taking the two dimensional flow at a single axial location and extending it throughout the domain without variation, is investigated. The method used will be to expand the perturbation velocity in a series of divergence free vectors which ensures that the continuity equation for the incompressible fluid is satisfied exactly by the computed velocity field. Projections of the stability equation onto the space of inviscid vector fields eliminated the pressure term from the equation and reduces the differential eigen problem to a generalized matrix eigen problem. Results are presented both for the one dimensional, columnar vortex flows and also for the wavy bubble flow.
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The discussion of the structural methods for obtaining lateral stability discloses the remarkable influence of the constant fuselage and wing proportions to the yawing moments. For the effectiveness of modifications in vertical tail surfaces and tail length, these quotas - little observed heretofore, in this connection - are decisive. This also applies to the amount of dihedral of the wing with regard to the roll stability of the complete wing already existing without angle of the dihedral.
The flapping equation for a helicopter in forward flight has coefficients which are periodic in time, and this effect complicates the calculation of stability. This paper presents a constant coefficient approximation which will allow the use of all the well known methods for analyzing constant coefficient equations. The flapping equation is first transformed into the nonrotating coordinate frame, where some of the periodic coefficients are transformed into constant terms. The constant coefficient approximation is then made by using time averaged coefficients in the nonrotating frame. Stability calculations based on the approximation are compared to results from a theory which correctly includes all of the periodicity. The comparison indicates that the approximation is reasonably accurate at advance ratios up to 0.5.
The flapping equation for a helicopter in forward flight are reported which have coefficients that are periodic in time, and this effect complicates the calculation of stability. A constant coefficient approximation which will allow the use of all the well known methods for analyzing constant coefficient equations are presented. The flapping equation is first transformed into the nonrotating coordinate frame, where some of the periodic coefficients are transformed into constant terms. The constant coefficient approximation is then made by using time averaged coefficients in the nonrotating frame. Stability calculations based on the approximation are compared to results from a theory which correctly includes all of the periodicity. The comparison indicates that the approximation is reasonably accurate at advance ratios up to 0.5.
The problem of calculating the stability of steady state solutions of differential equations is addressed. Leading eigenvalues of large matrices that arise from discretization are calculated, and an efficient multigrid method for solving these problems is presented. The resulting grid functions are used as initial approximations for appropriate eigenvalue problems. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a nonstandard way in which the right-hand side of the coarse grid equations involves unknown parameters to be solved on the coarse grid. This leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem are presented which demonstrate the effectiveness of the method.
The problem of calculating the stability of steady state solutions of differential equations is treated. Leading eigenvalues (i.e., having maximal real part) of large matrices that arise from discretization are to be calculated. An efficient multigrid method for solving these problems is presented. The method begins by obtaining an initial approximation for the dominant subspace on a coarse level using a damped Jacobi relaxation. This proceeds until enough accuracy for the dominant subspace has been obtained. The resulting grid functions are then used as an initial approximation for appropriate eigenvalue problems. These problems are being solved first on coarse levels, followed by refinement until a desired accuracy for the eigenvalues has been achieved. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a non-standard way in which the right hand side of the coarse grid equations involves unknown parameters to be solved for on the coarse grid. This in particular leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem demonstrate the effectiveness of the method proposed. Using an FMG algorithm a solution to the level of discretization errors is obtained in just a few work units (less than 10), where a work unit is the work involved in one Jacobi relization on the finest level.
Graphical test helps assure numerical stability of calculations of transient temperature or diffusion in composite medium. Rectangular grid forms basis of two-dimensional finite-difference model for heat conduction or other diffusion like phenomena. Model enables calculation of transient heat transfer among up to four different materials that meet at grid point.
Subsonic span loads and the resulting stability derivatives have been calculated using the discrete-horseshoe-vortex method for a systematic series of horizontal tails in combination with a vertical tail of aspect ratio 1.0 in order to provide information on the effect of varying the chord of the horizontal tail for isolated tail assemblies performing sideslip and steady-roll motions. In addition, the effects of horizontal-tail dihedral angle for the sideslip case were obtained. Each tail surface considered had a taper ratio of 0.5 and an unswept quarter-chord line. The investigation covered variations in horizontal-tail chord, horizontal-tail span, and vertical location of the horizontal tail. The span loads and the resulting total stability derivatives as well as the vertical- and horizontal-tail contributions to these tail-assembly derivatives are presented in the figures for the purpose of showing the influence of the geometric variables. The results of this investigation showed trends that were in agreement with the results of previous investigations for variations in horizontal-tail span and vertical location of the horizontal tail. Variations in horizontal-tail chord expressed herein in terms of the root-chord ratio, that is, the ratio of horizontal-tail root chord to vertical-tail root chord, were found to have a pronounced influence on most of the span loads and the resulting stability derivatives. For most of the cases considered, the rate of change of the span load coefficients and the stability derivatives with the root-chord ratio was found to be a maximum for small values of root-chord ratio and to decrease as root-chord ratio increased.
Machine learning can be used to screen millions of compounds and unearth relationships between electronic structure, chemistry, thermodynamic stability, formability and band gap. We know how to calculate thermodynamic stability and calculate approximate band gap, but it is very complicated to estimate synthesizability.