Literal Approximations Research Articles

A Simple, Correct Pedagogical Presentation of Airplane Lateral-Directional Dynamic

This paper offers a clean and correct pedagogical presentation of the theory of airplane lateral-directional dynamics. In this work, the definition of the dynamic (rate) derivatives has been corrected, distinct timescales corresponding to the standard lateral-directional modes have been identified, and a multiple timescale procedure including static residuals has been followed to derive new literal approximations to the modes. The lateral-directional small-per-turbation equations have themselves been written without having to first derive the complete 6-degree of freedom equations of airplane motion. New, physically meaningful results for the Dutch roll and spiral mode parameters are obtained and discussed. This work complements the revised presentation of airplane longitudinal dynamics in a companion paper.

Small-Perturbation Analysis of Airplane Dynamics With Dynamic Stability Derivatives Redefined

The first successful powered and controlled heavier-than-air flight and the subsequent meteoric rise of aviation in the 20th century also brought with it mathematical models to understand the phenomenon of flight. Though successful to a large extent as is obvious from their longevity, these models still suffer from the odd anomalous result. Upon careful examination, the source of this problem can be traced to an inappropriate choice of dynamic stability derivatives that is inconsistent with the aerodynamics. In order to correctly distinguish between the different mechanisms that are responsible for the aerodynamic forces and moments generated due to body angular rates in flight, this paper refines the existing model for flight mechanics by redefining the dynamic stability derivatives with respect to, i) relative rate of rotation between the body and wind axes, and ii) the rate of rotation of the wind axis with respect to the earth axis, sometimes called flow curvature effect. Further, new literal approximations to the longitudinal and lateral-directional modes have been obtained in terms of the redefined dynamic stability parameters. The new approximations, for instance, correctly relate the short-period natural frequency to the longitudinal static stability parameter Cmα.

Compact and Accurate Phugoid Mode Approximation with Residualization

Literal approximations to the phugoid mode are systematically derived by a process of residualization with fast-slow modal decomposition. Approximations with a static residual alone and another including a dynamic residual as well are obtained. The two sets of approximations are shown to be analytically identical to each other, therefore previous attempts by some researchers to obtain "improved" literal approximations by introducing a dynamic residual are seen to be redundant. Our static residualization yields new expressions for the phugoid mode parameters which include an "inertia" term and a "phase damping" term not generally observed in previous approximations. In fact, the "phase damping" term is seen to be the second-most numerically significant damping effect after the drag and is always found to detract from phugoid damping. A third term involving the pitch damping derivative Cmq is relatively insignificant but interestingly is found in many cases to undamp the phugoid motion. Approximations are derived with and without compressibility effects in the form of the derivative CmMa to highlight the contribution of this derivative in obtaining accurate estimates of the phugoid frequency, even at low speeds. Finally, recommended literal approximations to the phugoid frequency and damping are presented which are compact and whose numerical accuracy is established for ten different combinations of airplanes / flight conditions.

Compensation of differential dispersion: application to multiband stellar interferometry

ABSTRACT With the aim of pushing the limiting magnitude of interferometric instruments, the need for wide-band detection channels and for a coordinated operation of different instruments has considerably grown in the field of long-baseline interferometry. For this reason, the Center for High Angular Resolution Astronomy (CHARA), an array of six telescopes, requires a new configuration of longitudinal dispersion compensators to keep the fringe contrast above 95 per cent simultaneously in all spectral bands, while preserving the transmission above 85 per cent. In this paper, we propose a new method for defining the longitudinal dispersion compensators (LDC) suited for multiband observations. A literal approximation of the contrast loss resulting from the dispersion residues enables us to define a general criterion for fringe contrast maximization on several bands simultaneously. The optimization of this criterion leads to a simple solution with only two LDC stages per arm and existing differential delay lines, to the glass choice and a simple linear formula for thickness control of all these media. A refined criterion can also take into account glass transmission. After presenting this criterion, we give the optimal solution (medium, configuration) and its expected performance for the planned observing modes on CHARA.

Comments on "Literal Approximations to Aircraft Dynamics Modes" and "Consistent Approximations to Aircraft Longitudinal Modes"

Since then, the equations ofmotion have not been modi” ed and may be found in any textbookdealingwithaircraftstabilityandcontrol.Nevertheless,theproblemof ” nding literal approximations to aircraft transfer functions, andin particular to aircraft dynamic modes, still continues to attract theattention of the aeronautical community. This is because a literalexpression involving aerodynamicderivatives and inertial data pro-videsphysical insight into the stability of the aircraft motion.How-ever, it is hardly possible to ” nd exact closed-form solutions to theaircraft modes because both the longitudinal and the lateral aircrafteigenvalues come from fourth-order characteristic equations. Thismotivates interest in getting improved literal solutions and explainswhynewexpressionsappearfromtimeto timein theliterature.Thewriter’ s view about this issue is that it is unlikely that at this stagesomething signi” cant can be added about such an old problem. Forthese reasons, in the following I revisit papers by AnanthkrishnanandUnnikrishnan

Comments on "Literal Approximations to Aircraft Dynamics Modes" and "Consistent Approximations to Aircraft Longitudinal Modes"

Comments on "Literal Approximations to Aircraft Dynamics Modes" and "Consistent Approximations to Aircraft Longitudinal Modes"

Rigid-Body Trajectory Reconstruction

References 1Lanchester, F. W., Aerodonetics, Archibald Constable, London, 1908. 2Etkin, B., and Reid, L. D., Dynamics of Flight: Stability and Control, Wiley, New York, 1996. 3Nelson, R. C., Flight Stability and Automatic Control, McGraw–Hill, New York, 1990. 4Roskam, J., Airplane Flight Dynamics and Automatic Flight Controls, Vol. 1, Roskam Aviation and Engineering Corp., Lawrence, KS, 1979. 5Pradeep, S., “A Century of Phugoid Approximations,” Aircraft Design, Vol. 1, 1998, pp. 89–104. 6Ananthkrishnan, N., and Unnikrishnan, S., “Literal Approximations to Aircraft Dynamic Modes,” Journal of Guidance, Control, and Dynamics, Vol. 24, No. 6, 2001, pp. 1196–1203. 7Phillips, W. F., “Phugoid Approximation for Conventional Airplanes,” Journal of Aircraft, Vol. 37, No. 1, 2000, pp. 30–36.

Literal Approximations to Aircraft Dynamic Modes

Derivation of literal approximations to the aircraft dynamicmodes remains one of the central problems in the theory of aircraft  ight dynamics, and one that has never been satisfactorily resolved. Although approximations to the fast modes (short period, roll) are generally good, the slow modes (phugoid, dutch roll, spiral) seem to have several competing approximations, none of which are uniformly satisfactory. This paper examines the fundamental assumptions behind the traditional approach to the derivation of literal approximations.A major  aw in all previous derivations resulting in incorrect slow-mode approximations is uncovered. In following a formal procedure, improved literal approximations are derived for the slow modes. In this process, a new condition for onset of directional departure is obtained, which is related to the dutch-roll frequency. Predictions from the literal approximationsderived in this paper are compared with actual numerical values for an example aircraft to illustrate conditions under which the approximationswork well, and to point out the fundamental limitations of the literal approximationsderived by the traditional approach.

Literal approximations to aircraft dynamic modes

Literal approximations to aircraft dynamic modes

Unified literal approximations for longitudinal dynamics of flexible flight vehicles

The literal approximation method has been recently used to obtain simple literal (analytical) approximations for key longitudinal dynamic characteristics of a given large, flexible aircraft. These approximations constitute a useful additional tool to existing design and simulation procedures for the particular flexible flight vehicle for which they were derived. A desirable feature of this tool is generality, i.e., the suitability of one set of unified literal approximations to adequately describe the dynamics of significantly different flexible flight vehicles. This problem is addressed by incorporating symbolic manipulation software to facilitate both the ease of use and the derivation accuracy of the otherwise algebraically intensive literal approximation method used. The resulting automated literal method is used to independently derive literal approximations for the longitudinal dynamics of three significantly different flexible flight vehicle configurations, namely, a large aircraft, a large supersonic missile, and a hypersonic vehicle. The unified set of literal approximations is then synthesized from these three sets of independently derived literal approximations, yielding a new, unified formulation of the flexibility effects on the poles and zeros of a general flexible flight vehicle. The unified approximations lead to marked improvement in modeling accuracy over the traditional approximations (derived using the decoupled rigid body and elastic modes). These improvements are demonstrated using detailed tables and an extensive set of frequency response plots.

Improved Literal Approximation for Lateral-Directional Dynamics of Rigid Aircraft

Improved Literal Approximation for Lateral-Directional Dynamics of Rigid Aircraft

Literal singular-value-based flight control system design techniques

Recent control system research has resulted in new methods for assessing stability and performance robustness with respect to plant and controller uncertainty. The emphasis has been on using frequency domain matrix singular values for multi-input/multi-output systems. Problems originating from aircraft flight control systems have been an important motivation for these developments, and applications are currently being carried out at government research centers and in industry. These new robustness methods are, however, only slowly becoming part of the working flight control designer's practical tools. Before becoming routine, means must be provided for better physical insight. This paper presents a distinctly different interpretation based on literal (symbolic) formulations of singular values. This approach is an outgrowth of and is related to classical multivariable control systems methods based on coupling numerators. A lateral-directional flight control system design for an advanced fighter is used as an example. Robustness of stability with respect to input uncertainties is first assessed using standard numerical techniques. Literal approximations are then given for openand closed-loop singular values. The literal approximations are consistent with the numerical results. More importantly, they provide a means for decomposing and diagnosing robustness problems by providing explicit connections to critical aircraft and controller parameters. Both unstructured and structured uncertainties are treated, using respectively literal approximations of regular and structured singular values.