Document
N94-25113
EFFECT OF AEROELASTIC-PROPULSIVE
INTERACTIONS ON FLIGHT DYNAMICS OF A
HYPERSONIC VEHICLE
David L. Raney and John D. McMlnn NASA Langley Research Center ttamplon, VA 23681 Anthony S. Potolzky Lockheed Engineering Sciences Co.
Christine L. Wooley Unlversily of Cincinnati NASA LaRC Wolkshop on Guidance, Navigation. Controls, and Dynamics for Atmospheric Flight March 19, 1993
Outline
Motivation and Objectives
Model Description
Propulsion Sensitivities
Impact on Flight Dynamics
Concluding Remarks
Pi_KC_D_ PAGE" BLANK t'_OT FJLMtEL_
Motivation and Objectives
com u.o I
J _-" Forebody " Inlet Conditons- Compression Surface
• Assess magnitudes of propulsive force and moment
perturbations
• Examine Impact on longitudinal flight dynamics
MOTIVATION AND OBJECTIVES The desire to achieve orbit-on-demand access to space with rapid turn-around capability and aircraft- like processing operations has given rise to numerous hypersonic aerospace plane design concepts which would take off horizontally from a conventional runway and employ air-breathing scramjet propulsion systems for acceleration to orbital speeds. Most of these air-breathing hypersonic vehicle concepts incorporate an elongated fuselage forebody to act as the aerodynamic compression surface for a scramjet combustor module. This type of airframe-integrated scramjet propulsion system tends to be highly sensitive to inlet conditions and angle-of-attack perturbations. Furthermore, the basic configuration of the fuselage, wkh its elongated and tapered forebody, produces relatively low frequency elastic modes which will cause perturbations in the combustor inlet conditions due to the oscillation of the forebody compressi<m surface. The flexibility of the forebody compression surface, together With sensitivity of _ramjet propulsion systems to inlet conditions, creates the potential for an unprecedented form of aeroelastic-propulsive interaction in which deflections of the vehicle fuselage give rise to propulsion transients, producing force and moment variations that may adversely impact the longitudinal flight dynamics l,nd/or excite the elastic modes. These propulsive force and moment variations may have an appreciable impact on the performance, guidance, and control of a hypersonic aerospace plane. The objectives of this research are (1) to quantify the magnitudes Of propulsive force and moment perturbations resulting from elastic deformation of a representative hypersonic vehicle, and (2) to assess d_e polential impact of these perturbations on the vehicle's longitudinal flight dynamics.
Vehicle Geometry
All-moving wing VEHICLE GEOMETRY A three-view finite element representation of the vehicle concept used in this modeling effort is shown tn the figure. It is a hypersonic lifting body with underslung engine nacelles, very similar in configuration to the proposed X-30 research vehicle. The vehicle length is 150 ft. The wingspan is 60 It, with a wing sweep angle of 70 degrees. Vertical fins project from the upper surface of the aft fu_lage near the wing root. The configuration is equipped with all-moving wing control effectors. At hypersonic speeds, the lower surface of the elongated fuselage forebody acts as a compression wedge for the scramjet combuslor unit, and the lower surface of the aft portion of the vehicle acts as a nozzle. The weight of the vehicle used in this study was 300,000 lb. The configuration was analyzed at two hypersonic flight conditions: Mach 6 at 75,000 feet and Mach 10 at 95,000 feet, representing two points along a typical ascent trajectory. The dynamic pressures at these two flight conditions are 1,840 psf and 2,010 psf, respectively.
Aeroelastic Model
2.9 Hz 3.11Hz 5,5 Hz S.7 Hz 7.7 Hz 11.11 Hz 10.9 Hz • 2 Rigid Body Modes, 7 Elastic Modes, 3rd Order Actuator • General form of model: * =[A]x +[B]u
y:[C]x
I fuselage deflections } Y = I angle of attack X" elastic states / ' rigid body states / actuator states U = {control command } AEROELASTIC MODEL The aeroelastic state-space model used in this study is a longitudinal approximation. The model includes seven symmetric bending modes and two rigid body modes. The rigid body degrees of freedom are pitch and vertical translation (plunge). No translational degree of freedom along the vehicle's longitudlnal axis is included. The general form of this aeroelastic model is shown in equations (I) and (2).
, - [A]x +Is] u (i)
y -- [C] x (2) The state vector, x, contains a total of 21 elements; two for each of the nine second-order dynamic modes, and three states for an actuator model associated with the all-moving wing. The input vector, u, corresponds to the all-moving wing control command. The output vector, y, includes the rigid body angle of attack and pitch rate, as well as displacements, slopes, and accelerations at various locations throughout the fuselage and wing.
Numerical values for the matrices [A1, [B|, and tC] appearing in equations (1) and (2) were generated using the Interaction of Structures, Aerodynamics, and Controls code, ISAC. Second-order piston theory was used to model the unsteady aerodynamic effects at the two selected hypersonic flight conditions. The shapes and in-vacuo frequencies of the seven elastic modes are shown in the figure. Mode shapes which strongly impact the fuselage geometry are of particular importance, since they are likely to have the greatest influence on the propulsion system. The in-vacuo frequencies of the elastic modes are relatively low and closely spaced.
Propulsion Model
1800 1533 646 47 0 J I I ( Inches )'_-r--] 83 82 81 I -- Nozzle I I-_ Forebody _t Combustor • Undersurface geometry analyzed by SRGULL 2-d forebody and nozzle ° Assumptions: 1-d combustor no unsteady propulsion aerodynamics • General form of rnodeh FA ) FN i = f(81, 82, 83, a) M PROPULSION MODF_I.
The propulsion model was developed using the SROULL code for hypersonic propulsion systems. The SRGULL code uses a two-dimensional inviscid Forcbody and inlet analysis, and a one-dimensional combnstor analysis to address the entire propulsion system flowpath shown in the figure. A variable grid is used to analyze the vehicle nose-to-tail stream tube control volume, determining mass capture, forebody and inlet drag, and combustor and nozzle performance. The nose-to-tail propulsion flowpath consists of the undersurface Of the fuselage forebody, the combustor module, and the undersurface of the fuselage arterbody (serving as the nozzle). Using SRGULL, a database was produced which allows the interpolation of propulsive axial and normal force and pitching moment perturbations resulting from a given structural deflection at a given angle of attack.
To produce the propulsive force and moment database, the SRGULL code was first run at both night conditions (Mach 6 and Mach 10) for the undeflected vehicle geometry over an angle-of-attack range from -I to 3 degrees in one-degree increments. Structural deflections were then generated at selected stations along the fuselage centerline by calculating the RMS elastic responses to a Von Karman spectra turbulence inpl,t. Three stations along the fuselage centerline, designated as 81, 82, and 83 in the figure were chosen to parameterize a set of perturbation geometries. The three stations are located 47 inches, 646 inches, and 1,53::] inches back from the nose of the vehicle. The RM$ deflections were then used to produce a collection of 27 perturbation geometries consisting of the set of all possible combinations of the upward, zero, and downward deflection posilions at each of the three selected fuselage stations, assuming that the combustor section was rigid. Each of the perturbation geometries was then analyzed using SRGULL over the angle-of-attack range from -I to 3 degrees at both Much Numbers to produce a database of axial force, normal force, and pilching moment perturbations as a function o41" fuselage deflections and angle of attack.
The data was then combined into a 4-dimensional interpolation table using angle of attack and the deflections at the tbm¢ fuselage stations as the independent variables. Curve fits to the data were used to increase the number of bmakpoints in the interpolation table. In this way, a database was produced which could be used to estimate the propulsive forces and moments for any deflected geometry by interpolating from the table based on angle of attack and the deflections at the three selected fuselage stations.
Propulsive Force and Moment Data
8.0 4 8.0 s !
I I 1
,£ !
I1_ 4"04 4.0 s !
i_0 2"04 3.08 "2'04 0 2"08 -4,_ .... = .... • .... i .... i._..= ....
1.0 s -2 .I 0 I 2 3 4 -2 -1 0 1 2 3 4 angle of attack, deg angle of attack, deg 1.0 7 8.0 s .tO ° Large variation with alpha = 8"08 _,_ 4.0 e • Symbols: data for undeflected 2.08 geometry 0,0o • Brackets: range of variation due -2._ to perturbed geometries .... ! .... | .... i .... i .... | ....
-4.0 e -2 -I 0 1 2 3 4 angle of attack, deg PROPULSIVE FORCE AND MOMENT DATA The force and moment database produced using SRGULL is plotted against angle of attack in the figure. The solid symbols represent the data for the undeflected vehicle geometry at Mach Numbers of 6 and 10. The brackets about each symbol indicate the range of variation in axial force, normal force, or moment that resulted from the analysis of the 27 perturbation geometries.
Simulation Layout
Aeroelastic Model Propulsion Model Deflection - Deflectl°ns vJ _- & Normal Force Contr°l I 7 Elastic M°dea I Fuselage -"[ _ ]- A Axial F°rce---'lZ" _'[ /
--- y = [C] x _ = [A] x + [B] u I I I I I II fromSRGULL FA Moment
(X o = 1.5 dog • Removed rigid body modes from aeroelastlc model • Excited elastic modes with control doublet • Examined magnitudes of elastically-induced propulsive perturbations SIMULATION LAYOUT In order to ascertain the approximate magnitudes of elastic deflections and resulting propulsive perturbations which can be expected in respon_ to a typical control input, a simulation was constructed incorporating the aeroclastic and propulsion models in the general structure shown in the figure. In this simulation, the aeroelastic model is driven by control surface deflections to yield an output vector, y, consisting of angle-of-attack perturbations and elastic deflections which are then fed into the propulsive interpolation database to produce time histories of the resulting force and moment perturbations. The rigid body dynamics of the aeroelastic model are unstable at both flight conditions and were removed so that these time histories could be produced in the absence of the divergent rigid body motion. The time histories were generated using the propulsive force and moment interpolation database at an angle of attack of 1.5 degrees. This angle of attack did not change during the lime histories, since the angle-of-attack perturbations did not occur in the absence of the rigid body modes. Therefore, the propulsive perturbations produced by the control doublet are entirely the result of elastic fuselage deformations and not of angle-of-attack variations. Also, the perturbations do not include the aerodynamic lift, drag. c_r moment acting on the control surface itself. The responses do not represent worst case perturbatio,s, but rather are intended to provide insight into the magnitude of the propulsion system sensitivity to elastic deformation of the vehicle.
Perturbation Time Histories
2 Math 10 w Q • t ; , ,t ,,, , .
o o /'t,,;A ,, ^;,, ,,=,, ;;,,..
._= ¢,o -2
V
-3 • , , , 1 2 3 4 Time, sec .O 8000 "_" 4000 L_ . , ;I ' ;, " .
X,A:- ",) ,: _l :, ,q ,-, .
o max - 2% from nominal it II U rd _,' |1 ,'J v N v:,, F, :v ,_;- ,,.,-- ( about 0.02 g ) (ll H : I _ r# . r! , , , , ' , =, " , ,, _ w I i , t , . i ,, iw E '- -4000 '., .. '1 ,, ii _ ,i ;,, ,, Z q •_ -8000 2 3 ' 4 Time, sec PERTURBATION TIME HISTORIES Time histories of the elastic deflections at the first station along the fuselage centerline designated as _51 (47 from the nose of the vehicle) are shown in the figure for the Mach 6 and Mach 10 flight conditions.
The doublet was initiated 0.5 seconds into the run. The largest fuselage deflections reach about 2.4 inches at this fuselage station. These deflections appear to represent relatively minor distortions of the aerodynamic compression surface of the integrated airframe-propulsion system. The deflections did not produce appreciable accelerations at the pilot station.
Time histories of the propulsive force and moment perturbations resulting from the elastic deflections are also shown. The largest normal force perturbations range from 6,390 Ibs for the Mach 6 case to 7,580 Ibs for Mach I 0 case.
Perturbation Time Histories
..... Mach
[ ..Oh ol
JD m o _h . i I ,J ,i ,i • % L.
max - 4% from nominal _ , , ,'_.,., O U.
( about 0.01 g )
" ,',V, 1
.,._v_ _., .,.. -
,m -1000 x
=j
,< i -2000
' _ ' _ 4
Time, sec 8 xlO s 6 xlOS 4 x10 s I_ 2 x10s • ". , . .'t ....
, • , : , ; ; ,. ,, ,-, max > 10% from nominal @ E o trim moment 0 -2 x10s -4 xlO s -6 xl0 s 0 1 2 3 4 Time, sec PERTURBATION TIME HISTORIES (continued) The largest axial force perturbations range from 1.770 Ibs to 1,410 Ibs, and the pitching moment perturbations range from 6.32 x 105 ft-lbs to 3.60 x 105 ft-lbs in the Mach 6 and Mach 10 ca_s, respectively. The maximum normal force perturbalions represent approximately 2 percent variations from nominal and would produce vertical acceleration transients of about 0.02 g's for a vehicle weighing 300.000 lb. 1"he maximum axial force perturbations represent approximately 4 percent variations from nominal and would produce longitudinal acceleration transients of less than 0.01 g's for a vehicle weighing 300,000 lb. The maximum moment perturbations, however, represent greater than l0 percent variations from the nominal trim moment and may require substantial control deflections to maintain stable trimmed flighl. It is important Io remember that these force and moment perturbations am due solely to propulsion system sensitivity to elastic deflections. They do not include the effect of angle-of-attack perturbations on the propulsion model. The large pitching moment variation is due mainly to the aerodynamic contribution of the forebody. Maintaining trim in presence of the large pitching moment perturbations may require excexsive control activity in hypersonic flight, which could translate into a substantial drag increment when integrated over the duration of a mission, implying reduced fuel efficiency and decrea_d payload capacity.
Simulation Layout
Aeroelastic Model Linearized Propulsion Model Pitch, Plunge, 7 Elastic Modes Control Input -- Deflections -_ FA j- .-I_ Fuselage _J__ IZ _,
y--[cJx
Propulsive Perturbations • Examine Impact of propulsive perturbations on aeroelastic model.
• Equates to augmenting the stability matrix: i=[A]x +[B]u + [B '][P] [C] x , or i--[A+A']x+[B]u where A ' = [B '][P] [C] • Propulsive force and moment perturbations treated as "virtual inputs" applied at cg.
SIMULATION LA YOUT The simulation was used to assess the impact of Ihe propulsive perturbations on the dynamics of the combined aeroclastic-propulsive system. This was accomplished by feeding the propulsive force and moment perturbations back into the aeroelastic model as indicated by the dashed line in the figure. As shown in the figure, this simply equates to augmenting the stability matrix with the effect of the linearizcd propulsion sensitivitie,s. The model contains a further approximation in that the propulsive force and moment perturbations are applied at the cg, rather than being distributed over the aft nozzle area and cowl structure of the vehicle. Application of forces and moments to the aeroelastic model at the cg produces acceptable results regarding the impact of the propulsive perturbations on the vehicle's rigid body dynamics (pitch and plunge), but should not he used to assess the impact of propulsive perlurbations on the elastic modes. In order to achieve the latter, it would be necessary to apply the propulsive perturbation .
loads to the structure using an appropriate load distribution function.
Impact on Rigid Body Modes
Mach 6 Mach 10
1.5 1.5 0.5
J
e J. _.:._ _= A A|A A _s pitch ptunge pitch pitch -0.5 -0.5 -1 -1 +B -1.5 I -1.5 0 1 2 3 -3 -2 -1 -3 -2 -1 0 1 2 3 Real Real • Illustrates Impact on rigid body modes due to augmenting stability matrix with propulsion system sensitivities.
• Propulsion Iinearization conditions: (X0 = 1.5 deg, (_0 = 0 • Final position of eigenvalues was strongly dependent upon linearizatlon conditions of propulsion model.
IMPACT ON RIGID BODY MODES The figure illustrates the effect of the propulsive perturbations on the rigid body dynamics for the Mach 6 and Mach l0 flight conditions. The roots labeled "A" in the figure represent the statically unstable pitch and plunge modes when the propulsive perturbations arc not being fed into the aeroelastic model.
When the propulsive perturbations are fed into the aeroelastic model, two of the poles associated with the rigid body modes are observed to couple, producing a new oscillatory mode. The frequency of the unstable pole associated with the pitch mode is also observed to vary. The final position of the rigid body roots is indicated by the points labeled "B" in the figure. The Mach 10 case exhibits a slightly greater variation in the frequency of the unstable root of the pitch mode.
Eigenvalue Dispersion Caused by Varying
Linearization Conditions
Mach 6 Mach 10
t
f
4,
I
+ t .-.-iit_l..- _ + .40 m .....
1,41l-4-4g-4--_1---.-41--., lIB- - 4141i- -41F,--4- _NI-- + I" -1 -1
t
t
J -2 -2 ' .4 -2 0 2 4 -4 -2 0 2 4 Real Real • Varied (X0 and _0 used to Iinearize propulsion sensitivities.
• Illustrates range of possible dynamic characteristics due to nonlinear propulsion database.
• Can be viewed as uncertainty associated with rigid body flight dynamics.
EIGENVALUE DISPERSIONS II was found that the final position of the poles of the augmented stability matrix varied depending on the angle of allack and nominal fuselage deflections about which the propulsion model was linearized.
This variation is a direct result of nonlinearities in the propulsive force and moment database. The nonlinearities introduce uncertainty into the system regarding the position of the rigid body poles, because lbe pole locations vary as the slmcture deforms and as angle of attack varies. In order to chart the pole variation resulting from nonlinearities in the propulsive force and moment database the propulsion linearizafion conditions were varied and corresponding eigenvalues of the augmented stability matrix were plolled. The angle of a(lack was varied over the range of -I to 3 degrees, and nominal fuselage deflections were simullaneously varied according to the deflection time history shown in the previous figures. The resulting eigenvalue dispersions are shown in this figure. The variation of ]inearization conditions caused a wide range of dynamic characteristics to be observed at both flight conditions. This variation in dynamic characteristics due to the propulsion nonlinearities may be viewed as uncertainty associaled with the rigid body modes.
Future Research
Robust Control Law ........ IK,s, t........
Aeroelastic Model Nonlinear Propulsion Pilch, Plunge, Model L- . -Ib, 7 Elastic Modes Control
Fuselage _=.J _ _-/% FA--_
Input
: / I s_i"
Deflecti°n7 J ba_a_r°=m' I-A FN- _
-e---A_---.--_ I l i i 7-i J-A M _.-
Propulsive Perturbations • Improve integration of aeroelastic/propulsion models.
• Investigate control solutions for robust performance
in the presence of ASPE interactions, FUTURE WORK Additional research is needed to refine the integration of the aeroelastic and propulsive models. Future work will also involve the formulation of uncertainty bounds on the various elements of the stability matrix resulti.g from the feedback of the propulsive perturbations into the aeroelastic model. These uncerlainty bounds could then be used to synthesize a robust rigid body controller.
Concluding Remarks
• Quantified propulsion sensitivities to angle-of-
attack and fuselage elasticity.
• Elastically-induced propulsive perturbations did
not cause excessive accelerations.
• Propulsive moment perturbations may require
excessive control deflection to maintain stable
trimmed flight.
• Propulsion sensitivities significantly alter rigid
body flight dynamics.
• Nonlinearities in propulsion sensitivities may be
viewed as uncertainty in rigid body dynamics.
CONCLUDING REMARKS A study has been conducted to investigate the impact of aeroelastic-propulsive interactions on the longitudinal flight dynamics of an air-breathing hypersonic vehicle. A model was developed based on a finite element representation of a hypersonic configuration at two flight conditions of Mach 6 and Mach I0. The model included rigid body pitch and plunge modes and seven elastic modes, as well as propulsion system sensitivities to angle-of-attack variations and structural deflections. The model was incorporated into a simulation to produce time histories of propulsion force and moment perlurbations in response to elastic deflections. The force and moment perturbations were then fed back into the aeroelastic model to allow their impact on the dynamics of the combined aeroelastic-propulsive system to be assessed.
The propulsion model exhibited a pronounced sensitivity to angle-of-attack variations and elastic fuselage deflections. Significant nonlinearities were observed in the propulsion system sensitivities.
Elastic responses to a representative control input appeared acceptable. The normal and axial force perturbations induced by the elastic deflections were appreciable, but did not produce excessive vertical or longitudinal acceleration transients for the subject configuration. Moment perturbations induced by the elastic deflections, however, appeared quite large and might require significant control activity to maintain stable trimmed flight. A high level of control activity at hypersonic speeds could compromise fuel efficiency, thereby reducing payload capacity or range.
A significant impact on the rigid body flight dynamics was observed when the propulsive force and moment perturbations were fed back into the aeroelastic model. At both flight conditions, the propulsive perlurbations caused a coupling of two poles associated with the rigid body flight dynamics. It was also found that the eigenvalues of the rigid body modes were highly sensitive to the angle of attack and nominal fuselage deflections chosen as the linearization condition of the model. This sensitivity is a direct result of nonlinearities in the propulsive force and moment database, and can be thought of as uncertainly associated with the vehicle's rigid body stability coefficients. Considerable variation in the rigid body modes was observed, emphasizing robustness as a critical factor in the design of flight control laws for air-breathing hypersonic vehicles.
Launch-Vehicle Trajectory Solutions
with Dynamic-Pressure Constraints
Via Finite Elements and Shooting
Robert R. Bless*, LESC
Hans Seywald*, AMA
Dewey H. Hodges, Georgia Tech
*Guidance Group, Spacecraft Controls Branch
NASA LaRC Workshop on Guidance, Navigation,
Controls, and Dynamics for Atmospheric Flight
March 18 - 19
Outline
• Optimal Control Problem Definition
• State Constraints
• Methods of Solution
- Multiple Shooting
- Finite Elements
• Launch-Vehicle Model
• Results
• Summary
Problem Definition
minimize
J= ¢[x(tf),tf] + L(x,u,t)dt
_00 t l
subject to state equations 5c = f(x, u)
boundary conditions ¢[x(t/),t.f] = 0
control constraints
c(x, u, t) <_o
and state constraints
s(x, t) <_o
Result is a nonlinear multi-point boundary-value
problem
State Constraints
Consider active state constraint for tl _< t _< t2
s(=) =o
is equivalent to
s[=(t,)] = o
dO-1)S[x(tl)]
=0
dt(q-])
and
dqS
for tl _< t < t2
= s(q)(x, u) = 0
dtq
Multiple Shooting Method
• Initial guesses chosen for states and costates
• Differential equations integrated forward
• Guesses updated via zero-finding method
• Process repeats until all boundary conditions
are satisfied
Finite Element Method
continuous-time necessary
• Discretization of
conditions
• Set of nonlinear algebraic equations gener-
ated
• Initial guesses required for each element
along trajectory
• Nonlinear equations can be solved by Newton-
Raphson method
Comparison of Methods
• Shooting
- Sensitive to initial guesses
- Slow iteration process due to integration
- Numerically exact answer is found
• Finite Element
- Initial guesses more easily obtained
- Fast iteration process (sparse Jacobian)
- Second-order accuracy
Finite element solutions can provide guesses for
shooting
Launch-Vehicle Model
• NLS two-stage rocket (point-mass model)
and
- States are mass, altitude, velocity,
flight-path angle
- Scalar control is angle-of-attack
- Fixed staging time; change in thrust and
mass
Exponential atmosphere
coeffi-
Piecewise constant aerodynamic
cients
Launch-Vehicle Model (continued)
• Mission
- Maximize final mass
- Perigee injection of 80 x 150 NM orbit
• Constraint on maximum dynamic pressure
• Engine out on pad
Results
60_ "_' 50- EL (3) 40- t,- : ', _ q unconstrained; Shooting ,: /. -... ', ,, q unconstrained; FE(8:8) (/) I- ',, ', O) ... - ,, ....... max q = 45kPa; Shooting (3) 30" L_ / \",,, ............. max q = 35kPa; Shooting EL i/ _\ -,,_ ..... max q = 25kPa; Shooting O t_ \ \",_ • max q = 25kPa; FE(4:4:4:8) E 20-
f ',..\\\
_i V,..
¢-
E3 lo-
100 200 300 400 500
Time (secs)
Results
1.2-- -% _ -_ 1.0" 0.8-
f
_N 0.6- /t,,': -'_ _q unconstrained; Shooting :_ 0.4.
_y • q unconstrained; FE(8:8) .,._ ...... max q ,, 45kPa; Shooting
Z
....... max q - 35kPa; Shooting 0.2- SJ ..... -max q - 25kP,; Shooting f • max q ,, 25kPa; FE(4:4:4:8) 0.0 0.2 0.4 0.6 0.8 1.0 0.0
Normalized Velocity
Results
-- q unconstrained; Shooting • q unconstrained; FE(8:8) ....... max q = 45kPa; Shooting \. ............. max q = 35kPa; Shooting ..... max q = 25kPa; Shooting • max q = 25kPa; FE(4:4:4:8) I 1O0 200 300 400 500 Time (secs)
Results
0.41 0.3" 0.2' • - tI _ 0.1- "_ 0.0-' '_= -0.1 - i.6 i ._ -- q unconstrained; Shooting
9 -
q ,, unconstrained, FE(8:8) ....... max q = 45kPa; Shooting ............ max q = 35kPa; Shooting c- -0.3" < ..... max q = 25kPa; Shooting • max q = 25kPa; FE(4:4:4:8) -0.4" -0.5 I ' I ' I ' I 1O0 200 300 4O0
Time (secs)
Results
£
<: lo-
(1) Reduction of Final Mass -1500 ¢/) .C: 8- O ..J (/) -1000 t-- 6- ¢t) ¢1 (2) Length of Constrained Ar_/ '4" 4" r" 2.
t- -0 55 50 45 40 35 30 6O
Dynamic Pressure Limit (kPa)
Summary
The finite element and shooting methods are
(1)
complementary algorithms
The work has produced a family of dynamic-
- (2)
pressure constrained solutions
An uncommon first-order touch-point solution
(3)
has occurred (not observed in literature yet)
This work will be reported on at the AIAA GNC
(4)
Conference in August
(a) Derivation of finite element method
(b) Discuss touch-point behavior
48O