Modeling and Control Challenges for Airbreathing Hypersonic Vehicles 2 Mar 2007 Dr. David B. Doman...

Modeling and Control Challenges for Airbreathing Hypersonic

Vehicles

2 Mar 2007Dr. David B. Doman

Control Design and Analysis Branch Air Vehicles Directorate

Air Force Research Laboratory

SAE Aerospace Guidance & Control Committee Meeting

Outline

• Background

• Modeling– Aerodynamics– Propulsion– Structural– Thermal– Unsteady Aerodynamics– Control Approaches

• Conclusions

• Highly coupled airframe/propulsion system with aeroelastic interactions.

• Unstable and non-minimum phase • First-principles modeling approach

– Oblique Shock Theory– Prandtl-Meyer Expansion Theory– Quasi-1D flow with heat addition in

combustor– Spillage effects on inlet performance– Aeroelastic modeling – Plume shear layer modeling– Thermal effects on structure – Unsteady aerodynamic effects via nonlinear

piston theory• Advantages

– Flexibility to add/move control surfaces, change sensor locations

– A fundamental understanding of what is being controlled

2-D X-43A Profile Scaled to 100 ft

Scramjet Modeling and Control Overview

Animation

Outline

• Background


• Conclusions

• Oblique Shock and Prandtl-Meyer Expansion Theory used to compute pressures over top, forebody and underbody of engine

• Approximation for plume shear layer location verified against panel solution

• Pressure over rear ramp computed from plume results

• Aerodynamic Forces and Moments computed by integrating pressures over each surface (including deformation due to bending)

• Inlet turning forces included

Aerodynamic Forces & MomentsAerodynamic Forces & Moments

c

g

f

xb

zb

1,U

hi

V

de

q1,L

2

V

e

M, P

M1, P1

Region 1

Region 2

M2, P2

Region 3

M3, P3

Region 5

M5, P5

Region 4

M4, P4

Aerodynamic Flow Regions

Outline

• Background


• Conclusions

Scramjet Modeling• Variable Geometry Diffuser

– Allows pressure at combustor inlet to be adjusted– Assumed to be isentropic

• Constant area combustor with simple heat addition– Heat addition/change in total temperature a function of equivalence ratio

• Fixed area ratio internal nozzle– Assumed to be isentropic

• Very easy for the combustor to become thermally choked!

• Euler-Bernoulli Beam Theory– Beam loading:

• Normal component of surface pressures changes with beam deflection

• Variable point load due to elevon• Heave and rotational effects included

• Deflection is a function of load• Load is a function of deflection• Modal analysis for beam dynamics• On/Off-design conditions modeled

– Shock on lip (no spillage)– Shock forward of lip (spillage)– Shock inside inlet (no spillage)

• Quasi-1D flow with heat addition in combustor associated with fuel flow

• First dynamic model to include nonlinear on/off design effects

• Captures interaction between aero, structure and propulsion system

Aeroelastic Influence on Propulsion System

Outline

• Background


• Conclusions

Refined Structural Model

• Model the effects of mass and temperature on the structural dynamics of a hypersonic aircraft

• Avoid FEM analysis of frequencies/mode shapes• Approximation of fuselage first bending mode

– Assume n = 18 rad/sec and = 0.02 when fully fueled

– Use assumed modes method to estimate mode shapes and frequencies

HSV Mission Analysis

• Mission: – 8000 n.mi. Cruise at Mach

10, 120 kft– Transition from rocket to

scramjet at Mach 5 during ascent

– Descent assumed to be unpowered

LH2: 40% of GTOW

LOX: 31% of GTOW

Assumed Modes Method

• Based on Lagrange’s Equations:

• The transverse displacement along the structure is given by a separation of the time and spatial coordinates

• The shape functions i(x) are the assumed modes.

• i(x) satisfy the geometric boundary conditions and possess all required derivatives

Choice of Assumed Modes• Basis functions are

uniform free-free beam mode shapes – Satisfies geometric

boundary conditions– Expected to be “close” to

real mode shapes• Requires fewer assumed

mode shapes for convergence

– Alternatively assessing viability of Chebyshev Polynomials as basis functions

Assumed Modes MethodMass Effects

Outline

• Background


• Conclusions

Plant CharacterisiticsHeating Effects

• Modulus of Elasticity a function of Temperature

– n= f(E1/2)

– Expect n to vary with T1/2

• 125 deg T predicts 3% change in frequency

• 2% change in first mode freq after 2 hrs

• No effect on aircraft controllability!

From Vosteen, NACA TR 4348

Assumed Modes MethodTemperature Effects

• Uniform Temperature Distribution along beam (1500 deg F)– Vehicle structure

modeled as a Ti “beam”– Only freqs affected by a

uniform T– Spatially varying

temperature will affect mode shapes

• Currently under study…

Aerothermal Modeling

• Sustained Flight at high Mach number requires TPS – Either “passive” or “active”

• Capture effects of heat transfer on the vehicle structural dynamics– Controllability considerations

• Structural Dynamics estimated using the Assumed Modes Method – Frequencies and Mode Shape dependency on vehicle mass and temperature

can be modelled• Modulus of Elasticity a function of temperature

• Thermal Protection System Architecture assumed

Aerothermal ModelingProblem Formulation

• “Three layer” model assumed– PM2000 honeycomb– SiO2 Insulation– Titanium Load Bearing Structure

• Explicit, finite difference method used to calculate unsteady heat transfer during cruise

• 1-D heat transfer only• Discrete Points along Structure

Considered– Boundary Conditions:

• Convective Heating• Radiative Cooling• nth node insulated

Ti Beam

Insulation

PM2000

Honeycomb

1st node

Problem FormulationSolution Method

• Once all the difference equations are obtained, they can be written in the form:

• A is a tri-diagonal matrix

– Convergence requires eigenvalues of A to lie in the unit circle

– Max allowable t given y & material properties

• b is a non-linear forcing term

– Without loss of generality, lump both radiative cooling and convective heating into this term as well

Aerothermal ModelingResults

• Case 1: Temp profile due to constant heating– Point 50 ft behind the nose – dq/dt = 15 BTU/(ft2 s)– Emissivity = 0.6– Radiation to a perfect black

body at T=0 deg R• Steady state reached for

outer TPS • Large T across insulation

layer• Minimal T in structure• Model matches ABACUS

predictions

PM2000

Ti Beam

SiO2

Aerothermal ModelingResults

Outer Layer

Structure


• Temperature Along Structure Determined

Leading Edge

Trailing Edge

Outline

• Background


• Conclusions

Piston Theory

Piston Theory:• Method for Calculating Aerodynamic Loads • Local Pressure from Body’s Motion Is Related to Local

Normal Component of Fluid Velocity• Same Way As Pressure and Velocity Are Related At Face of

Piston Moving in a 1-D Channel• Provides Expression for Time Varying Pressure as a Function

of Local Surface Velocity• Valid for M ≥ 4• Accurate Calculation

Of Pressures

Piston Theory - Basics

Basic Result From Linear Piston Theory

nVaPP

• Vn = Velocity of Surface Normal to Flow• Subscript ‘*’ Refers To Flow Conditions Past Surface, i.e., Behind Shock or

Expansion Fan or Freestream = density• a = speed of sound

• Isentropic “Simple Wave” Expression for Pressure on Surface of Moving Piston

1

2

n

a

V

2

11

P

P

• Binomial Expansion Gives

a

V1

P

P n

• Use Perfect Gas Law (P=RT) and Speed of Sound (a2=γRT) Yields

Basic Result

• P = Pressure• γ = Ratio of Specific

Heats

Piston Theory - Basics

• Infinitesimal Force Due To Pressure → dF = -P dA n

• F = force (normal/axial)• dA = surface element (unit depth into page)• n = outward pointing normal vector

c

g

f

1,U

hi

V

d e

1,L

2

V

e

h

nVaPP • From Basic Result, Need Vn

nVnVSubscript ‘*’ =• U (upper surface) • Lcd (lower surface cd) • Lgh (lower surface gh)• Lef (lower surface ef)• Infinitesimal Force Becomes

nnVF dAaPd

• Find dA*, n*, and V*

Stability Derivative Approximations for Unsteady Aerodynamic Effects

• Unsteady effects captured in the form of stability deriviatives

• CZα, CXα, CMα, CMQ, CZQ

wzefwzghwzcdwzcfwzwZ dddd

Sq

1d

Sq

1C

FFFFF

VwV

w1

V

wIf

3ef32ef2a12U,1n33fU,222U,111Z LALALAcosLaLcosaLcosa

Sq

VC

Structural mode stability derivatives are also included

Results• Poles of Linearized Model - Mach 8, H=85,000

Without Unsteady Effects

With Unsteady Effects

-1.5081.444-.0004759-5.15e-5 +- j0.0335

-2.4582.314-.00048-4.93e-5 +- j0.0335

More Unstable

Pole-Zero Map

Real Axis

Ima

gin

ary

Ax

is

-2 -1 0 1 2-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-2 -1.5 -1 -0.5 0 0.5 1 1.5-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Pole-Zero Map

Real Axis

Ima

gin

ary

Ax

is

• Continued development of first principles model of scramjet vehicle

• Aero-thermo-servo-elasticity effects captured in multidisciplinary model suitable for control studies

• Unsteady Aero Modeling via Piston Theory:

• Accounts for Fluid-Structure Interaction as Vehicle Vibrates

• Used to Compute Damping and Flex-body stability derivatives

• Steady and Unsteady Aerodynamics in Model

• Significant shifts in pole-zero locations

• Heat transfer and thermal effects on structure modeled

Unsteady Aero Terms – Move Unstable Zero & Pole to Right in S-Plane – Affect Stability and Closed-Loop Bandwidth

-5 -4 -3 -2 -1 0 1 2 3 4 5-20

-15

-10

-5

0

5

10

15

20Pole-Zero Map

Real Axis

Imag

inar

y Ax

is

Unsteady

Steady

Effects of Unsteady Aerodynamics on Airbreathing Hypersonic Vehicle Dynamics

Outline

• Background


• Conclusions

Flexible Aircraft Equations-of-Motion

• Derived using Lagrange’s Equations:

• Kinetic Energy is the sum of rigid & elastic kinetic energies:

• Potential Energy is a superposition of the strain energy of the structure and the altitude potential energy

• Generalized forces:

• “Proportional Damping” on structure (2 n)

• Aerodynamic forces/moments and Thrust

• “Modal forces:”

Flexible Aircraft DynamicsNon-linear Equations-of-Motion

• Non-linear, longitudinal equations-of-motion:

• L, D, M, & Fn are functions of in (x) I

• Variable mass and frequency effects are captured

Plant Characteristics

• Unstable Rigid-body modes: s=-4.47, 4.41

• Very Slow (albeit stable) Altitude mode s=-0.001

• Unstable, very lightly damped phugoid (time to double = 3.6 hrs)

• Non-minimum phase transmission zeros for typical input/output combinations

• Controllable/Observable – Structural modes are the most

Controllable & Observable-10 -8 -6 -4 -2 0 2 4 6 8 10

-100

-80

-60

-40

-20

0

20

40

60

80

100

Pole-Zero Map

Real Axis

Imag

inar

y A

xis

y=[V ]T

u=[e ]T

• Goal: Elimination of Low Frequency Nonminimum Phase Behavior

• Large elevon contributes substantially to total lift

• Increasing forebody AOA requires short-term loss of lift at the elevon to create pitching moment

• Long-term effect-Increased Lift• Short-term effect-Decreased Lift• Canard added with interconnnect

gain to elevator greatly improves control law potential.

• Moves center of rotation from off-vehicle to near CG.

Configuration Changes for NMP Mitigation

Control of Unstable/NMP VehicleOutput Modification

• New controlled variable: flight path angle measured ahead of cg– p := azp/(U0 s)

• Accelerometer placed ahead of Instantaneous Center-of-Rotation

• Real Zeros of N(s)

bifurcate to complex conjugate pair in LHP

– Improved Tracking Performance since NMP behavior is eliminated

• Control Performance Limitations– Feedback control required (O.L unstable)– Low frequency non-minimum phase

behavior prevalent– Limited speed of response

• Dynamic Inversion with NMP coupling strategically ignored

• LQR & Classical Loop Shaping• Modified Dynamic Inversion

– Place poles at mirror image of NMP zeros– Result: Decoupled bank of integrators & all-

pass filters

Modified Dynamic Inversion Results

Control Approaches

Outline

• Background


• Conclusions

Conclusions• First principles based model that captures coupling of aerodynamic,

structural, and propulsion systems– Mass flow spillage effects on thrust added

• Control of flexible hypersonic aircraft is a difficult problem, extreme coupling, nonlinearities, NMP behavior, unstable

• Low-frequency flexible mode < actuator bandwidth (20 rad/sec)– Possible interactions between structural dynamics and control

system• Non-minimum phase transmission zero due to location of center-of-

rotation limits controller bandwidth • Unsteady aero effects captured using piston theory in conjunction

with quasi-steady gas dynamics. • Unsteady aero appears to significantly affect pole-zero locations

Conclusions

• Model validation; comparisons to X-43 flight test data show aero-results are reasonable.

• Thermal and mass effects included in structural model– Frequencies Increase and Mode Shapes change with fuel

burn– Structural frequencies change due to aerodynamic heating

(lowered by 2-3%)• Future work:

– Improve scramjet model by integrating cycle code– Propulsion mode transition and low speed modeling– Viscous effects

Backup Slides

Assumed Modes MethodKinetic Energy

• The kinetic energy is

• Which can be written in vector-matrix form

• Where:

Assumed Modes MethodPotential Energy

• Similarly, the potential energy is:

• In vector-matrix form becomes:

• Where

Problem FormulationSurface Node

• At the surface, the energy balance is:

Which, after discretizing, becomes

Where:

pth Time step

Node index

Problem FormulationInterior Point Nodes

• For the interior points, the energy balance gives:

When discretized, this becomes:

Problem FormulationMaterial Intefaces

• Again compute the energy balance at the ith node:

Applying the following:

Gives:

• Euler-Bernoulli Beam Theory– Beam loading:

• Normal component of surface pressures changes with beam deflection

• Variable point load due to elevon• Heave and rotational effects included

• Deflection is a function of load• Load is a function of deflection• Modal analysis for beam dynamics• On/Off-design conditions modeled

– Shock on lip (no spillage)– Shock forward of lip (spillage)– Shock inside inlet (no spillage)

• Quasi-1D flow with heat addition in combustor associated with fuel flow

• First dynamic model to include nonlinear on/off design effects

• Captures interaction between aero, structure and propulsion system

Aeroelastic Influence on Propulsion System

• Dynamic model derived using Lagrangian method– Captured both heave and

rotational effects on structure– Captured on/off design effects

propulsion (use gas dynamics versus Newtonian flow)

– Include effects of uncommanded control rotation due to flexible structure

• Linearized Model reveals – Short Period – Phugoid modes– Height mode– Fore/aft first bending modes– Open-loop unstable– Non-minimum phase– Off-vehicle center of rotation

Nonlinear Longitudinal Equations of Motion

Nonlinear Model and Linearization Results

• Control Performance Limitations– Feedback control required (O.L unstable)– Low frequency non-minimum phase

behavior prevalent– Limited speed of response

• Dynamic Inversion with NMP coupling strategically ignored

• LQR & Classical Loop Shaping• Modified Dynamic Inversion

– Place poles at mirror image of NMP zeros– Result: Decoupled bank of integrators &

all-pass filters

Modified Dynamic Inversion Results

Control Approaches

Work-in-Progress– Include thermal effects on structure– Include unsteady aerodynamic

effects via nonlinear piston theory– Include higher order structural

modes– Lateral directional dynamics– Advanced control methodologies as

required

Current Status

Mode Shapes

ColdHot

“Aerothermoelasticity”

Unsteady Aero. Piston Theory

1

2

2

11

),(

a

v

p

txp n

Thermal Modeling and Implications for HSV Control

• Problem: Capture heat transfer effects on structural dynamics

– Structural Frequencies depend upon:• Vehicle mass

– Frequencies increase with decreasing mass

• Temperature of the underlying structure

– ~10% reduction in frequencies for a “hot” structure

– Temperature effects calculated using unsteady, heat transfer code

• Explicit Finite Difference Method– Ti Multiwall TPS Architecture w/

Aerodynamic heating function of flight condition

• Integrated with vehicle model– On-going research effort to quantify

heat transfer effects on vehicle controllability & required robustness of controllers

Aerodynamic Heating at cruise

Vehicle Configuration

c

g h

f

xb

zb

47’=Lf 20’=Ln 33’=La

L=100’

1,U=3

3.5=hi

V d

e

q1,L=6

14.4=2

V

s

e

Xcs

Zcs

'55x f '45x a

Outline

• Piston Theory

• Forces

• Flow Analysis

• Stability Derivatives

• Engine

• Simulation

• Conclusions

Assumed Modes MethodEigenproblem

• From Lagrange’s Equation:

• Assuming simple harmonic motion for the free-vibration of the structure, set

• The unknown is found from the eigenproblem

• The eigenvectors ui determines the mode shape of the structure from

Aerothermal ModelingTPS Model

• TPS consists of PM2000 Honeycomb Outer wall and SiO2 Insulation

• Structure (aircraft) idealized as a Titanium beam with varying mass and stiffness

Plant CharacteristicsA Word On Zeros

• Origin of NMP zero– Due to loss of lift on aircraft when elevator deflected to

pitch aircraft up• True for any tail controlled aircraft

– Position of zero a function of instantaneous center-of-rotation (for a rigid aircraft):

Field, E., Armor, J., Rossitto, K., and Mitchell, D., “Effects of Instantaneous Center of Rotation Location on Flying Qualities,” Proceedings of the AIAA Atmospheric Flight Mechanics Conference [CD-ROM], Aug. 5-8 2002, AIAA 2002-4799.


• Modification of ICR will move the transmission zeros – Ideally, would like to apply a couple to the aircraft in

order to minimize the reduction in lift that is experienced when deflecting the elevator

• How do we accomplish this?– Augment HSV model with a canard

1. Choose the mean aero. chord2. Interconnect with elevator using a gain, k3. Deflect canard in opposite direction of elevator

• Redefine the ICR:– We can pick k, M c, and Z c to place the ICR where we

want and thus move the offending zero to a “better” location

– Optimal k exists that moves ICR to origin & zero to 1


• Effect of flexibility is to move zero back to the j axis– Incomplete cancellation of

lift– Exact details being worked

• Small perturbations in Z c can move zeros significantly

Surface Velocities

• Consider Small Perturbations From Steady Flight Condition at M in u,w, and q

• Velocity of Point on Upper Surface Due To Perturbations

cfU,11U,11cf kwsinViucosV rωV

k,i = Unit Vectors in X and Z directions

= Angular Rate Vectorjqω

cfr = Position Vector of Point on Upper Surface

c

g

f

1,U

hi

d e

1,L

2e

h

fa

fU,1cf

xxx

kxxtanix

r

kcosisin U,1U,1cf n

dxsec)1(tan1dx)1(dxtandx)1(dzdx)1(dA U,1U,122

U,12222

cf

• Integrate Differential Force

Incremental Forces

kZiXkcosisinsecxPFaaa cfcfU,1U,1U,1a1cf

kZiXkcosisinsecxPFfff cfcfU,1U,1U,1f1cf

kZiXkcosisinsecLPF cdcdL,1L,1L,1f2cd

kZiXkLPF ghghn3gh

Upper Surface

Lower Surfaces

• Control Surface (Elevator)• Modeled As Flat Plate Hinged at Midpt.• Le = Length of Elevator e = Elevator Deflection (+ trailing edge down)• xcs, zcs = x & z Position of Elevator Relative to CG

kZiXseckcosisincosLP

kZiXseckcosisincosLP

eUeUeeeee5eU

eLeLeeeee4eL

F

F

Afterbody• Flow On Afterbody Bounded By Vehicle Surface Shear Layer• Pef = Pressure on Afterbody• Pe = Pressure at Engine Exit• P = Freestream Pressure

c

g

f

1,U

hi

V

de

1,L

2

V

e

h

e

s

1PP

cosL

s1

P)s(P

e

2U,1

a

eef

*F. Chavez and D. Schmidt, “Analytical Aeropropulsive /Aeroelastic Hypersonic Vehicle Model with Dynamic Analysis,” JGCD, Vol. 17, No. 6, 1994.

ds)s(PF1s

0

efef

kZiXkcosisinP

Pln

PPcos

PPLefef2U,12U,1

e

e2U,1

eaef

F

Afterbody, cont.

• For Stability Derivatives Need Force on Rear Ramp Due To Perturbations

efefefefefefefef dAaPd nnVF

aaa

efU,1a2U,1ef

xLxx

krixktanLxxtanixz

r

kcosisin 2U,12U,1ef n

dxsecdA 2U,1ef

efef2U,12U,1efefefefef dAcosqxwsinqruaPdz

nF

Unsteady Portion Used To Compute Contributions to Stab. Der.

Flow Analysis, cont.

Let Bowshock angle be denoted by

bowshockbowshock

Shock forward of pt. g Shock on lip or inside engine inlet

Expansion Fan

Flow Behind ObliqueShock from cd Used As I.C.

Wedge Angle = Freestream I.C

L,1

α < 0 Expansion Fan

Wedge Angle = α

α > 0 Shock

Rigid Body Moments

- +

Sign Convention U,12

2f1

cf tan12

xPM

f

2

xP

2

xxtanxPM

2a

1a

fU,12

a1cfa

2

LxLP

2

tanLtanLPM f

ff2L,1f

L,1f2cd

2L,1Fefnffef2L,1FefL,1fefef cosxLLxZsinxtanLXM

+

-

0sinxtanL 2U,1FefL,1f

0sinxtanL 2U,1FefL,1f

2

LLxLPM n

ffn3gh

csee5csee5eU xcosLPzsinLPM

csee4csee4eL xcosLPzsinLPM

Outline

• Piston Theory• Forces• Flow Analysis• Stability

Derivatives• Engine• Simulation• Conclusions

Scramjet Engine

A1 A2 A3 Me, PeMein , Pein

T0

Diffuser Combustor Internal Nozzle

1

2D

A

AA 23 AA

3

eN

A

AA

• Working Fluid – perfect gas, constant specific heats

• Two Controls – Diffuser Area Ratio, Ad and Temperature Addition in Combustor, To

Engine

• Engine Inlet Conditions – Flow Turned Parallel to Surface cd

nd

e

L,1222z

nd

e

L,1222x AA

bA

sinPMFAAb

A

cos1PMFinletinlet

• Force Acts At: L,1fffinlet tanLLx r

• Turning Force Moment inletinlet zffxL,1finlet FLxFtanLM

• Engine Thrust

ND

e

2e

eeo AAb

A

ppbAppVVmT

T2

htanLM i

L,1fengine

• Engine Moment

Engine Thrust and Moment

inletaf z0

Z0

ZeUeLefghcdcfcftotal Fqq

q

C

V2

cSqCSqZZZZZZZZ

inletaf x0

XeUeLefghcdcfcftotal F

CSqXXXXXXXX

engineinlet0M

0M

eUeLefghcdcfcftotal MMqqq

C

V2

cSqCcSqMMMMMMMM

af

• Total Forces and Moments

Mass Flow/Capture Area

c

g h

f

xb

zb

1,U

hi

V

d e

q1,L

2

V

e

Shock

A1

A0

As

0o ART

MPm

cd

cd

cdiL,1fff0 sin

cos

cothtanLL2LA

Outline

• Piston Theory

• Forces

• Flow Analysis

• Stability Derivatives

• Engine

• Simulation

• Conclusions

Stability Derivatives – Rear Ramp

• Stab. Der. Functions of density and speed of sound (Not Constant on Rear Ramp)

• Let Temperature on Rear Ramp Be Similar To Pressure, i.e.,

1T

T

cosL

s1

T)s(T

e

2U,1

a

eef

• Thenef

efefefef RT

PRTa No Closed Form Solution

• Instead, Let eaa

a

eef

eaaa

eef

xLxL

axLxL

aaa


• Two hour cruise @ 85000 ft, 2000 psf (~=Mach 8)

• Compression due to oblique shock over a 3 deg ramp– Mass changes neglected

• Convective heating due to hypersonic flow– Eckert’s reference Temp.

Method– Turbulent boundary layer

assumed• Radiative cooling occurs (same

as before)• Net Heating determined at

discrete points along structure

Convective heating:

Recovery Factor:

Reference Temperature:

The local Heat Transfer Coefficient is

found from:

HSV Geometry

• Capture area function of Mach and – Mass flow spillage effects included in thrust computation

• Motivation for using oblique shock and expansion theory• Elevator modeled as a flat plate hinged at mid-chord

– “all-moving” elevator• Inlet turning force included in total aero force calculation

Reflected Shock

Aircraft has “Unit depthinto paper”

Spill Flow

Piston Theory Conclusions

• Hypersonic Vehicles:• Highly Flexible – Long and Slender• Integrated Airframe/Propulsion • Airframe Provides External Compression/Expansion

• Piston Theory:• Used to Calculate Aerodynamic Loads • Local Pressure from Body’s Motion Is Related to Local Normal Component

of Fluid Velocity• Same Way As Pressure and Velocity Are Related At Face of Piston Moving

in a 1-D Channel• Provides Expression for Time Varying Pressure as a Function of Local

Surface Velocity• Unsteady Effects Appear Significant

Flow Analysis

Determine Properties of Flow

Upper Surface:

U,1

U,1

U,1

Freestream

Expansion Fan

Compression (Shock)

Lower Surface cd:

Engine Constraint

L,1

L,1 Compression (Shock)

Lower Surface gh: Find angle at which shock exactly impinges on point g of engine nacelle

f

L,1fi1

bowshock L

tanLhtan

Rigid Body Forces and Moments

c

g h

f

xb

zb

1,U

hi

V d

e

q1,L

2

V

e

FUf

FUa

FeU

FeL

Fa

Flgh

Flcd

mg

Control Approaches

• Approximate Feedback Linearization– Team Lead: Prof Andrea

Serrani

– Controls Oriented Model Developed from AFRL Model

• Analytically intractable otherwise

– Neglect elevator contribution to lift in the inversion (a la Sastry)

• Otherwise zero dynamics are unstable

• Ongoing research by CCCS Team to apply robust non-linear and adaptive control methods to the HSV model

Modeling and Control Challenges for Airbreathing Hypersonic Vehicles 2 Mar 2007 Dr. David B. Doman...

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