Angle-of-attack convergence and windward- meridian rotation rate of rolling re-entry vehicles

Platus, D. H.

doi:10.2514/3.5535

Cited by 18 publications

(6 citation statements)

References 2 publications

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“…The undamped, small-angle equations of missile angular motion in classical Euler coordinates, for only a linear static moment and constant roll rate, are 1 …”

Section: Discussionmentioning

confidence: 99%

“…If the net trim remained sufficiently small, then the epicyclic motion would persist until it damped out from inherent pitch and normal force damping. 1 Because successive points of minimum !</> I and presumed shape change occur on essentially opposite meridians (from the point of view of an observer fixed in space), the lift nonaveraging dispersion associated with equal increments of shape change should be minimal. 5 ' 6 It is emphasized that these conclusions are based on the assumption that very small "incipient" ablation or erosion asymmetries exist which do not influence the motion.…”

Section: Application To Asymmetry Formationmentioning

confidence: 99%

“…1 Experimental investigations have shown this flight instability to be produced by motion of the liquid fill and to have a maximum effect for a liquid kinematic viscosity on the order of 100 K cs. 2 Other experimental studies 3 ' 4 indicate that this instability appears to be fundamentally different from the Stewartson type usually associated with projectiles having low viscosity liquid fills.…”

Section: Introductionmentioning

confidence: 98%

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Angular motion influence on re-entry vehicle ablation or erosion asymmetry formation

Platus

1985

Journal of Guidance, Control, and Dynamics

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mode, so that its impulse response is the sum of a ramp function and damped sinusoids (see Ref. 11 for details). Consequently, Wilson's method is not directly applicable. Figure 1 compares the impulse response of the original system with that of the optimal fourth-order model obtained for several values of a. It appears, as expected, that the impulse response is approximated over an interval of duration proportional to 1/ce. ConclusionWe have shown how methods now existing for optimal model reduction of asymptotically stable sytems can be extended to nonasymptotically-stable systems. This extension can be very simply implemented in software now existing, making it useful and convenient for the optimal model reduction of marginally stable and unstable systems. Nomenclaturea = E/2W;(B 2 max +B 2 min )/2 b = (E 2 -4K 2 &) " /2tf; (0 max -e 2 min )/2 E = constant determined by initial conditions, Eq. (6) / = pitch or yaw moment of inertia I x = roll moment of inertia K = constant determined by initial conditions, Eq. (5) p = roll rate p r = roll rate parameter, pp/2 t =time B = pitch angle (Euler angle) 0 max = maximum value of B during epicyclic oscillation 0 min = minimum value of B during epicyclic oscillation B = pitch rate At =I X /I T = phase angle, Eq. (8) 0 = roll angle relative to wind (Euler angle) 0 = windward-meridian rotation rate max = maximum value of 0 during epicyclic oscillation 0min = minimum value of 0 during epicyclic oscillation j/ = precession rate W = undamped natural pitch frequency

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“…The undamped, small-angle equations of missile angular motion in classical Euler coordinates, for only a linear static moment and constant roll rate, are 1 …”

Section: Discussionmentioning

confidence: 99%

Section: Application To Asymmetry Formationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Angular motion influence on re-entry vehicle ablation or erosion asymmetry formation

Platus

1985

Journal of Guidance, Control, and Dynamics

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“…Figure 3 shows a comparison of angle-of-attack histories for the two extreme cases of exoatmospheric motion that result in nearly circular motion in both the positive and negative precession modes. 7 Also shown, for comparison, is the angle-of-attack behavior for a negative precession case in which unbalanced fin forces are nonexistent. The theoretical approximation, Eq.…”

Section: Re-entry Vehicle Angle-of-attack Convergence Envelopementioning

confidence: 98%

Dynamic instability of finned missiles caused by unbalanced fin forces

Platus

1971

AIAA Journal

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A dynamic instability is described for rolling, finned missiles with canted fins. The instability occurs as an undamping of the angle of attack and is caused by differential lift and drag from unequal effectiveness of the windward and leeward fins. The instability is similar to a Magnus instability in that a yawing moment is produced that unequally damps the positive and negative precession motions. The equivalence of the Magnus coefficient and the differential fin forces is derived, and the instability is demonstrated with computer simulations of the equations of motion for the angle-of-attack convergence of a re-entry vehicle. The computer results are compared with a closed-form solution for the angle-of-attack convergence envelope. NomenclatureA -slope of fin lift effectiveness vs angle of attack B -slope of fin drag effectiveness vs angle of attack Cio = aerodynamic fin-induced roll moment coefficient Ci s = aerodynamic fin lift derivative C mq = aerodynamic pitch damping derivative Cnpa -aerodynamic Magnus moment derivative C Na == aerodynamic normal force derivative C' Na = 2C Na I/md* d -aerodynamic reference diameter D = fin drag force / = pitch or yaw moment of inertia I x -roll moment of inertia L = fin lift force m = vehicle mass M{ -aerodynamic yaw moment Mq -aerodynamic pitch moment M% = aerodynamic roll moment p = roll rate p r = reduced roll rate, pp/2 q = dynamic pressure S = aerodynamic reference area u -vehicle velocity Xf -distance of fins aft of center of mass x s t = static margin (distance of center of pressure aft of center of mass) x ,y,z = body-fixed coordinates X,Y,Z = inertial coordinates 5 == effective fin cant angle 5 0 = fixed fin cant angle relative to body axis of symmetry f = yaw axis t\ -pitch axis 0 -total angle of attack (Euler angle) K = roll moment coefficient, Ci 0 qSd/I x \ = fin drag coefficient, C.o 5

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“…8 The low-frequency damping rate \ L is composed of the usual aerodynamic and so-called density damping, cb 0 /2co 0 , and additionally, a small term due to steady roll acceleration /?, which has been determined previously by Flatus. 10 Separating Eq. (7) into real and imaginary parts, the individual equations for ot L and $ L are…”

Section: Simple Numerical Model For Calculation Of Entry Vehicle Trimmentioning

confidence: 99%

Simple numerical model for calculation of entry vehicle trim response

Burton

1978

Journal of Spacecraft and Rockets

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T HE influence of mass and configurational asymmetries on the dynamics of slender, spinning entry vehicles is an important aspect of vehicle design and performance assessment. High-altitude roll resonance and lower-altitude roll/trim effects influence vehicle loading and attendant impact miss. Descriptions of aerodynamic trim effects on reentry vehicle (RV) angle of attack for vehicles with linear aerodynamics are well known. 1 ' 4 Influence of nonlinear aerodynamics on resonant and nonresonant response have been considered by Murphy 5 and by Nayfeh and Saric. 6 ' 7 Although these solutions are useful in characterizing angle-ofattack response, it is still generally necessary to numerically integrate the six-degree-of-freedom equations of motion in order to quantify the resulting trim-induced dispersion for a given re-entry system. This is often done using a Monte Carlo approach, necessitating significant amounts of computer time.Presented in this Note is a simple trim response model which is numerically efficient and which can be incorporated into standard point-mass trajectory simulators for calculation of trim-induced dispersion and angle-of-attack/load behavior. This model is based on the fact that the coupling of trim and body fixed low-frequency (in the sense of Nelson 8 ) oscillatory motion component dominates the trim-induced response. The high-frequency component/trim coupling is not significant, except in cases where sudden trim changes are encountered, e.g., due to rapid configurational change. By using standard asymptotic solutions to linear differential equations with time varying coefficients, 9 the two secondorder equations in angle-of-attack a and sideslip 0 can be reduced to two first-order equations in which the highfrequency contribution is eliminated. This allows for a much coarser integration increment in calculating the trim-induced response.In the present model, time variations in roll rate, asymmetries, and aerodynamic coefficients are taken into account. Longitudinal principal axis misalignments in pitch and yaw are also included, which was not done in the previously cited work. l ' 7 In addition to the reduced equations of motion, results are obtained for the effect of roll acceleration and socalled density damping on the steady-state trim angle relations. The simplified model is shown to provide adequate accuracy in calculating the combined low-frequency/trim contribution to angle-of-attack.The angle-of-attack behavior is formulated in body fixed coordinates as used by Nelson. 8 For small angles of attack, linear aerodynamics, and with inclusion of products of inertia and aerodynamic asymmetries for an otherwise symmetric vehicle, the equations of rotational motion may be written as follows:. j3= -r+pa-(<7oo5C La /mK)]3(1)Here, a and 0 are angles of attack and sideslip, respectively; /?, q, and r are roll, pitch, and yaw rates, respectively; A and B are roll and pitch moments of inertia (pitch and yaw moments of inertia are assumed equal); vehicle mass and aerodynamic reference area and ...

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Angle-of-attack convergence and windward- meridian rotation rate of rolling re-entry vehicles

Cited by 18 publications

References 2 publications

Angular motion influence on re-entry vehicle ablation or erosion asymmetry formation

Angular motion influence on re-entry vehicle ablation or erosion asymmetry formation

Dynamic instability of finned missiles caused by unbalanced fin forces

Simple numerical model for calculation of entry vehicle trim response

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