# Modeling and Analysis of an Air-Breathing Flexible Hypersonic Vehicle

Authors: Xi-bin Zhang; Qun Zong
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

By using light-weighted material in hypersonic vehicle, the vehicle body can be easily deformed. The mutual couplings in aerodynamics, flexible structure, and propulsion system will bring great challenges for vehicle modeling. In this work, engineering estimated method is used to calculate the aerodynamic forces, moments, and flexible modes to get the physics-based model of an air-breathing flexible hypersonic vehicle. The model, which contains flexible effects and viscous effects, can capture the physical characteristics of high-speed flight. To overcome the analytical intractability of the model, a simplified control-oriented model of the hypersonic vehicle is presented with curve fitting approximations. The control-oriented model can not only reduce the complexity of the model, but also retain aero-flexible structure-propulsion interactions of the physics-based model and can be applied for nonlinear control.


---

## Body

## 1. Introduction

Hypersonic vehicle (HSV) which can travel faster than 5 times the speed of sound has wide range of applications in military and civilian areas. The research on such vehicle has received tremendous attention in the fields of aeronautics and astronautics in recent years. Unlike conventional aircraft, the propulsion system of hypersonic vehicle is highly integrated into the airframe, which will cause strong couplings between the airframe and propulsion system. Furthermore, high-speed flight, flexible structure, and low natural frequency can result in the bending of fore-body and aft-body of vehicle which will affect the airflow characteristics through the inlet and exit of the engine and hence the aerodynamic layout on the vehicle surfaces. The high couplings among the aerodynamics, flexible structure, and propulsion system make the modeling and control of such vehicle very challenging [1].From the 1990s, several researchers have paid more attention to the development of the modeling of the hypersonic vehicle. The first attempt at a comprehensive analytical model of a flexible HSV was presented by Chavez et al. Newtonian impact theory was utilized to calculate the forces and moments on the vehicle. The first flexible mode was estimated by a complete NASTRAN analysis, and the couplings between the rigid and flexible dynamic were considered [2]. Clark et al. developed a two-dimensional CFD-based model of a full-scale generic air-breathing hypersonic flight vehicle (CSULA-GHV). The modeling procedure includes configuration design, CFD simulations, and elasticity and aerodynamic interaction [3, 4]. Bolender et al. used a combination of oblique shock, Prandtl-Meyer expansion theory [5], and piston theory [6, 7] to calculate the aerodynamic forces and moments and conducted the first principles model (FPM) with X-43A vehicle geometry. The model is a nonlinear, physics-based model that can capture the couplings among the aerodynamics, flexible structural dynamics, and propulsion system. To overcome the difficulty for controller design, Parker et al. [8], Fiorentini et al. [9], and Sigthorsson et al. [10] presented several control-oriented models by replacing complex aerodynamic force functions in FPM model with curve fitting method, respectively. Frendreis et al. developed a comprehensive aerothermoelastic-propulsion model of a six-degree-of-freedom dynamics of a generic hypersonic vehicle [11, 12]. The three-dimensional model extends the 3D flight dynamics analysis framework to include the effects of flexibility and unsteady aerodynamics. Based on these models, several studies on the robust guidance and nonlinear control systems design have been published in recent years. The controller design methods provide robust stability and performance for the HSV systems under varying flight conditions [13–17].The HSV model should contain the real physical characteristics and can be applied for nonlinear control. The unsteady flexible effects, which are omitted in the existing control-oriented models, are significant in aeroservoelasticity analysis. For conducting a high-fidelity control-oriented model which can capture the inherent couplings of vehicle, we concentrate on the influence of flexible effects on the aerodynamic and propulsion system and present an air-breathing flexible HSV model by analyzing the physical characteristics of vehicle flight. Compressible flow theory and Eckert’s reference temperature method are used to calculate the complex forces and moments which contain the unsteady flexible effects and viscous effects. By replacing the complicated aerodynamic forces with curve fitting approximations, a simplified control-oriented model is obtained with the interactions of aerodynamics, flexible structure, and propulsion system. The stability analysis and pole-zero analysis are used to analyze the dynamic characteristic of the vehicle model. The results show that the control-oriented model reduces the complexity of the physics-based model and is convenient for nonlinear control.


## 2. Hypersonic Vehicle Modeling

The hypersonic vehicle, which typically has a tightly integrated airframe and scramjet propulsion system, is quite different from traditional aircraft. Figure1 shows the side view of X-43A hypersonic vehicle, and the literature [3] presents the detailed parameters of such vehicle. The lower fore-body surface of the vehicle has two functions: one is to produce lift and nose-up pitching moment; another is to act as a compression system to slow the flow velocity and increase the pressure and temperature at the inlet to the engine. The aft-body surface acts as an expansion nozzle to produce lift, thrust, and nose-down pitching moment [2]. The scramjet engine is located underside the vehicle with a translating cowl door, which is used to maintain a shock-on-lip condition for off-design flight conditions [3]. In this vehicle, the airframe and propulsion system cannot separate from each other. Furthermore, the flexible structure bending affects the pressure distribution on the vehicle and the performance of propulsion system. Therefore, the aerodynamic-flexible structure-propulsion interactions should be included in the hypersonic vehicle modeling.Figure 1
The side view of X-43A HSV.


### 2.1. Flexible Structure

We assume that the fuselage structure is a uniform free-free beam, and the displacements are sufficiently small to satisfy Hooke’s Law, and only the transverse displacements of the beam are considered. The assumed modes approach [18] is used to calculate the natural frequencies and mode shapes of the vehicle. This method has the advantage of simplicity and can achieve the desired accuracy. The method is based on Lagrange’s equations:
(1)
d
d
t
(
∂
T
∂
q
˙
i
)
-
∂
T
∂
q
i
+
∂
V
∂
q
i
=
f
i
,
i
=
1,2
,
…
,
where T is the total kinetic energy, V is the potential energy of the system, and f
i is the generalized force.The displacement of the flexible structure is(2)
ω
(
x
,
t
)
=
∑
i
=
1
n
Φ
i
(
x
)
η
i
(
t
)
,
where η
i is the generalized model coordinate and the functions Φ
i
(
x
), i
=
1,2
,
…
,
n, are the assumed modes which must be linearly independent, satisfy all of the geometric boundary conditions of the flexible structure, and possess all the required derivatives. The free-free beam meets the satisfaction, and we can take the mode shapes of the free-free beam to be the assumed modes.Calculating the kinetic energyT and potential energy V and substituting them into (1), then, the Lagrange’s equations can become
(3)
M
η
¨
+
K
η
=
f
,
where f
=
[
f
1
⋯
f
n
] and the mass matrix M
n
×
n and the stiffness matrix K
n
×
n are symmetric.By settingf
=
0, η
¨
=
-
ω
2
η, we can get the unforced harmonic dynamics of the system (
ω
2
I
-
M
-
1
K
)
η
=
0.Then the natural frequencies of the flexible structure are the square roots of the eigenvalues ofM
-
1
K, and the mode shapes are the combinations of the assumed modes Φ
i
(
x
), i
=
1,2
,
…
,
n, with the coefficients of the eigenvectors of M
-
1
K.The primary effects of flexible structure on the aerodynamic and propulsion system are in the form of deflections of the fore-body and aft-body as shown in Figure2, which is a simplification figure of the flexible vehicle model, and only the first flexible mode is considered [3, 7]. Suppose that a bending deflection θ of the beam is sufficiently small; then we can replace the deflection by the deflection angle
(4)
θ
(
x
,
t
)
≈
tan
θ
(
x
,
t
)
=
∂
ω
(
x
,
t
)
∂
x
.
By using the principle of separation of variables, we obtain the deflection caused by the ith flexible mode
(5)
θ
i
(
x
,
t
)
=
∂
∂
x
[
φ
i
(
x
)
η
i
(
t
)
]
=
d
φ
i
(
x
)
d
x
η
i
(
t
)
,
where φ
i
(
x
) is the mode shape and η
i
(
t
) is the generalized coordinate.Figure 2
The flexible vehicle model.Consideringn flexible modes, the fore-body deflection can be written as
(6)
Δ
τ
1
=
∑
i
=
1
n
d
φ
i
(
x
)
d
x
|
1
2
η
i
(
t
)
x
=
0
.
The aft-body deflection is
(7)
Δ
τ
2
=
∑
i
=
1
n
d
φ
i
(
x
)
d
x
|
1
2
η
i
(
t
)
x
=
L
.Bending of the fore-body and aft-body together can change the airflow characteristics of vehicle surfaces; then it will affect the engine inlet condition and alter the aerodynamic performance.


### 2.2. Aerodynamic Model

Aerodynamic modeling is an important issue in hypersonic vehicle modeling. The couplings between the flexible dynamics and rigid dynamics always occur through the aerodynamic forces and moments. Oblique shock relations, Prandtl-Meyer theory, and piston theory can be employed to estimate the aerodynamic forces and moments on the vehicle.


#### 2.2.1. Aerodynamic Forces

Shock and expansion waves are common physical phenomena in hypersonic flows. By using the oblique shock and Prandtl-Meyer theory, we can calculate the pressure, Mach number, and temperature behind the shock or expansion wave [3]. Furthermore, the flexible effects affect the wave structure on the vehicle surface and can change the angle of attack and control angles, which in turn change the pressures on vehicle surfaces.For the upper surface, ifα
+
Δ
τ
1
<
τ
1, an oblique shock wave forms on the upper surface; if α
+
Δ
τ
1
>
τ
1, an expansion fan forms on the upper surface; if α
+
Δ
τ
1
=
τ
1, then the upper surface experiences the freestream. By using oblique shock and Prandtl-Meyer theory, we can obtain the force and moment on the upper surface, which can be resolved into x- and z-body axes components:
(8)
F
x
,
up
=
-
p
up
L
tan
τ
1
,
F
z
,
up
=
p
up
L
,
M
up
=
z
up
F
x
,
up
-
x
up
F
z
,
up
.Similar to the upper surface of the vehicle, the forces and moments on the lower fore-body surface, engine nacelle, and elevator surface can be obtained, respectively. The pressure over the aft-body surface is affected by the exhaust gases from the propulsion system. By calculating the lower fore-body surface pressure, the inner pressure in the engine and the position of the shear layer, we can get the force and moment on the aft-body surface [2]. Then the resultant aerodynamic forces and moments on the x- and z
-body axes F
x
r
,
F
z
r
,
M
r are summarized from the individual components.


#### 2.2.2. Unsteady Flexible Effects

When traveling at high speed, the interactions between the fluid and the flexible structure of such vehicle may lead to unsteady aerodynamic effects. Understanding the consequences of unsteady aerodynamic caused by flexible effects is significant to the development of flexible vehicle modeling.Assuming that the flexible effects only occur in thez-direction and the engine is rigid, the unsteady flexible effects here are considered as small perturbations of deformation velocity and angular rate:
(9)
ω
˙
(
x
,
t
)
=
∑
i
=
1
∞
φ
i
(
x
)
η
˙
i
(
t
)
=
Φ
·
η
˙
,
(10)
θ
˙
=
∑
i
=
1
n
d
φ
i
(
x
)
d
x
η
˙
i
(
t
)
=
Φ
˙
·
η
˙
.Piston theory is applied to compute the unsteady aerodynamic effects behind oblique shock waves and expansion fans. The pressure on the surface is given by [7]
(11)
P
P
i
=
(
1
+
γ
-
1
2
w
i
a
i
)
2
γ
/
(
γ
-
1
)
.According to the first-order linear piston theory, the infinitesimal force due to the pressure is(12)
d
F
=
-
P
d
A
n
=
-
(
P
i
+
ρ
i
a
i
[
V
i
·
n
]
)
d
A
n
,
where P
i is the local pressure behind the shock or expansion wave, V
i is the velocity of the surface to the steady flow, ρ
i, a
i are the atmospheric density and sonic speed, respectively, n is the outward pointing surface normal, and d
A is the surface element.For the upper surface, the velocity due to the flexible deformation velocity and angular rate perturbations is(13)
V
up
=
V
1
cos
⁡
τ
1
i
⃑
+
(
V
1
sin
τ
1
+
ω
˙
(
x
,
t
)
)
k
⃑
+
(
Φ
˙
·
η
˙
)
×
r
up
,
where V
1, r
up are the velocity of the flow and the position vector of a point on the upper surface, respectively.Substituting the velocityV
up into (12), the unsteady force and moment on the upper surface can be written as
(14)
Δ
F
up
,
x
=
∫
0
L
ρ
up
a
up
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
+
Φ
·
η
˙
]
sin
τ
1
d
x
,
Δ
F
up
,
z
=
-
∫
0
L
ρ
up
a
up
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
d
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
]
cos
⁡
τ
1
d
x
,
Δ
M
up
=
z
up
Δ
F
up
,
x
-
x
up
Δ
F
up
,
z
.For the lower fore-body surface, the unsteady force and moment are given by(15)
Δ
F
d
f
,
x
=
∫
0
L
1
ρ
d
f
α
d
f
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
]
sin
τ
2
d
x
,
Δ
F
d
f
,
z
=
∫
0
L
1
ρ
d
f
α
d
f
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
d
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
]
cos
⁡
τ
2
d
x
,
Δ
M
d
f
=
z
d
f
Δ
F
d
f
,
x
-
x
d
f
Δ
F
d
f
,
z
.For the lower aft-body surface, the unsteady force and moment are given by(16)
Δ
F
d
a
,
x
=
∫
L
1
+
L
2
L
ρ
d
a
a
d
a
d
d
d
d
d
d
d
×
{
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
dddddd
d
×
sin
(
τ
1
+
τ
3
)
+
(
-
(
Φ
˙
·
η
˙
)
x
+
Φ
·
η
˙
)
d
d
d
d
d
d
d
×
sin
(
τ
1
+
τ
3
)
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
}
d
x
,
Δ
F
d
a
,
z
=
-
∫
L
1
+
L
2
L
ρ
d
a
a
d
a
d
d
d
d
d
d
i
d
×
{
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
dddddddd
×
sin
(
τ
1
+
τ
3
)
+
(
-
(
Φ
˙
·
η
˙
)
x
+
Φ
·
η
˙
)
d
d
d
d
d
i
d
d
×
cos
⁡
(
τ
1
+
τ
3
)
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
}
d
x
,
Δ
M
d
a
=
-
z
d
a
Δ
F
d
a
,
x
-
x
d
a
Δ
F
d
a
,
z
.Summarize the unsteady forces and moments on each surface and denoteF
x
,
fle, F
z
,
fle, M
fle as the unsteady flexible aerodynamic of vehicle; then we can get the total body force and moment:
(17)
F
x
=
F
x
r
+
F
x
,
fle
,
F
z
=
F
z
r
+
F
z
,
fle
,
M
=
M
r
+
M
fle
.


#### 2.2.3. Viscous Effects

The inviscid flow cannot describe the real airflow characteristics. With high Mach flight, the viscous fluid moves relatively to the vehicle which can generate viscous friction, and can result in boundary layer effect and aerodynamic heating. The flow temperature changes and it is difficult to calculate the aerodynamic parameters inside the boundary layer. Similar to the literature [19], Eckert’s reference temperature method is used to compute the viscous skin friction on each surface of the vehicle. The reference temperature which is a function of the Mach number (M) and temperature (T) at the edge of the boundary layer as well as the wall (skin) temperature T
w can be written as
(18)
T
*
=
T
[
1
+
M
2
+
0.58
(
T
w
T
-
1
)
]
.Based on the reference temperature, we can calculate the densityρ
*, the viscosity μ
*, Reynolds number R
e
*, and the skin friction coefficient C
f with the perfect gas law and Sutherland’s formula. Then, the shear stress on the surface can be computed by
(19)
τ
w
=
(
1
2
ρ
∞
V
∞
2
)
C
f
.
Integrating (19), the skin friction drag can be given by
(20)
F
vis
=
∫
τ
w
d
x
.
Then the normal and axial forces are obtained as follows:
(21)
N
vis
=
F
vis
sin
β
,
A
vis
=
F
vis
cos
⁡
β
,
where β is the surface inclination to the body axis.By using the coordinate transformation, the additional lift, drag, and moment of viscous effects can be written as(22)
L
vis
=
N
vis
cos
⁡
α
-
A
vis
sin
α
,
D
vis
=
N
vis
sin
α
-
A
vis
cos
⁡
α
,
M
vis
=
N
vis
x
+
A
vis
z
.We consider the viscous skin friction on the upper surface, lower fore-body surface, lower aft-body surface, surface below the engine nacelle, and control surfaces. By calculating the additional lift, drag, and moment on each surface, we can get the total additional liftL
vis, additional drag D
vis, and additional moment M
vis.Translating the forceF
x, F
z into the lift and drag acting on the vehicle and taking the viscous effects into account, the total lift, drag, and pitching moment are
(23)
L
=
F
x
sin
α
-
F
z
cos
⁡
α
+
L
vis
,
D
=
-
F
x
cos
⁡
α
-
F
z
sin
α
+
D
vis
,
M
=
M
r
+
M
fle
+
M
vis
.The flexible and viscous effects are included in force and moment formulations in (23). If we assume a rigid structure such that η
1
=
η
2
=
η
3
=
η
˙
1
=
η
˙
2
=
η
˙
3
=
0 and viscous effect is excluded, then (23) is steady aerodynamic forces of a rigid body vehicle.


#### 2.2.4. Generalized Forces

In this paper, we present the loadings on the upper surface, lower surface, and control surface; the generalized forces for the first three flexible modes can be written as(24)
N
i
=
∫
0
L
φ
i
(
x
)
p
up
d
x
-
∫
0
L
f
φ
i
(
x
)
p
d
f
d
x
-
∫
L
f
+
L
n
L
φ
i
(
x
)
p
d
a
d
x
-
∫
L
f
L
n
φ
i
(
x
)
p
e
n
d
x
+
φ
i
(
x
e
)
F
e
,
dddddddddddddddddddddddddddddddddd
i
=
1,2
,
3
,
where p
up, p
d
f, p
d
a, p
e
n are the pressures on the upper surface, lower fore-body surface, engine nacelle, and lower aft-body surface, respectively. F
e is the force act on the elevator surface; φ
i
(
x
) is the mode shape, which is calculated by the assumed mode method.


### 2.3. Thrust

Similar to the literature [20], the scramjet engine consists of 3 sections: a diffuser, a combustor, and an internal nozzle. We assume that the airflow into the engine is parallel to the direction of the body axis. By avoiding the mass flow impinges at the inlet, a translating cowl door is added to increase the mass flow through the engine module. With quasi-one-dimensional aero/thermoanalysis of the flow in the propulsion system and the momentum theorem, the thrust is given by
(25)
T
=
m
˙
a
(
V
e
-
V
∞
)
+
(
p
e
-
p
∞
)
A
e
b
,
where m
˙
a is the air mass flow through the engine, m
˙
a is dependent on the fore-body deflection Δ
τ
1, angle of attack α, and the freestream, V
e and p
e are the airflow velocity and pressure at the engine exit, respectively, V
∞ is the freestream airflow velocity, and A
e
/
b is the exit area per unit width.


### 2.4. Equations of Motion

Assume that the earth is flat and the fuselage is free-free beam. The equations of motion of the longitudinal dynamics for the hypersonic vehicle based on Lagrange’s equations are presented in [3]. By ignoring the weak inertial coupling terms, the flexible mode shapes are orthogonal to the rigid body rotation, which decouples the flexible modes and the rigid body modes, and the couplings between the rigid and flexible modes are included in the aerodynamic forces and moments. Then the simplified longitudinal dynamics of a flexible vehicle written in the stability axis coordinate system are [8, 10]
(26)
V
˙
=
1
m
(
T
cos
⁡
α
-
D
)
-
g
sin
(
θ
-
α
)
,
α
˙
=
1
m
V
(
-
T
sin
α
-
L
)
+
Q
+
g
V
cos
⁡
(
θ
-
α
)
,
q
˙
=
I
y
y
M
,
θ
˙
=
q
,
h
˙
=
V
sin
(
θ
-
α
)
,
η
¨
i
+
2
ζ
i
ω
i
η
˙
i
+
ω
i
2
η
i
=
N
i
,
i
=
1,2
,
3
,
where the rigid body states are {
V
,
α
,
q
,
θ
,
h
}, in which V, α, q, θ, h are the vehicle speed, angle of attack, pitch rate, pitch acceleration, and height, respectively. The flexible states are {
η
1
,
η
˙
1
,
η
2
,
η
˙
2
,
η
3
,
η
˙
3
}
,
ζ is the structure damping, ω
i is the frequency, and η
i is the generalized coordinate. The control input elevator deflection δ
e, throttle ϕ, and the couplings between the rigid and flexible modes will appear explicitly in the forces and moment L, D, T, M, N
i which are calculated in the above sections.


## 2.1. Flexible Structure

We assume that the fuselage structure is a uniform free-free beam, and the displacements are sufficiently small to satisfy Hooke’s Law, and only the transverse displacements of the beam are considered. The assumed modes approach [18] is used to calculate the natural frequencies and mode shapes of the vehicle. This method has the advantage of simplicity and can achieve the desired accuracy. The method is based on Lagrange’s equations:
(1)
d
d
t
(
∂
T
∂
q
˙
i
)
-
∂
T
∂
q
i
+
∂
V
∂
q
i
=
f
i
,
i
=
1,2
,
…
,
where T is the total kinetic energy, V is the potential energy of the system, and f
i is the generalized force.The displacement of the flexible structure is(2)
ω
(
x
,
t
)
=
∑
i
=
1
n
Φ
i
(
x
)
η
i
(
t
)
,
where η
i is the generalized model coordinate and the functions Φ
i
(
x
), i
=
1,2
,
…
,
n, are the assumed modes which must be linearly independent, satisfy all of the geometric boundary conditions of the flexible structure, and possess all the required derivatives. The free-free beam meets the satisfaction, and we can take the mode shapes of the free-free beam to be the assumed modes.Calculating the kinetic energyT and potential energy V and substituting them into (1), then, the Lagrange’s equations can become
(3)
M
η
¨
+
K
η
=
f
,
where f
=
[
f
1
⋯
f
n
] and the mass matrix M
n
×
n and the stiffness matrix K
n
×
n are symmetric.By settingf
=
0, η
¨
=
-
ω
2
η, we can get the unforced harmonic dynamics of the system (
ω
2
I
-
M
-
1
K
)
η
=
0.Then the natural frequencies of the flexible structure are the square roots of the eigenvalues ofM
-
1
K, and the mode shapes are the combinations of the assumed modes Φ
i
(
x
), i
=
1,2
,
…
,
n, with the coefficients of the eigenvectors of M
-
1
K.The primary effects of flexible structure on the aerodynamic and propulsion system are in the form of deflections of the fore-body and aft-body as shown in Figure2, which is a simplification figure of the flexible vehicle model, and only the first flexible mode is considered [3, 7]. Suppose that a bending deflection θ of the beam is sufficiently small; then we can replace the deflection by the deflection angle
(4)
θ
(
x
,
t
)
≈
tan
θ
(
x
,
t
)
=
∂
ω
(
x
,
t
)
∂
x
.
By using the principle of separation of variables, we obtain the deflection caused by the ith flexible mode
(5)
θ
i
(
x
,
t
)
=
∂
∂
x
[
φ
i
(
x
)
η
i
(
t
)
]
=
d
φ
i
(
x
)
d
x
η
i
(
t
)
,
where φ
i
(
x
) is the mode shape and η
i
(
t
) is the generalized coordinate.Figure 2
The flexible vehicle model.Consideringn flexible modes, the fore-body deflection can be written as
(6)
Δ
τ
1
=
∑
i
=
1
n
d
φ
i
(
x
)
d
x
|
1
2
η
i
(
t
)
x
=
0
.
The aft-body deflection is
(7)
Δ
τ
2
=
∑
i
=
1
n
d
φ
i
(
x
)
d
x
|
1
2
η
i
(
t
)
x
=
L
.Bending of the fore-body and aft-body together can change the airflow characteristics of vehicle surfaces; then it will affect the engine inlet condition and alter the aerodynamic performance.


## 2.2. Aerodynamic Model

Aerodynamic modeling is an important issue in hypersonic vehicle modeling. The couplings between the flexible dynamics and rigid dynamics always occur through the aerodynamic forces and moments. Oblique shock relations, Prandtl-Meyer theory, and piston theory can be employed to estimate the aerodynamic forces and moments on the vehicle.


### 2.2.1. Aerodynamic Forces

Shock and expansion waves are common physical phenomena in hypersonic flows. By using the oblique shock and Prandtl-Meyer theory, we can calculate the pressure, Mach number, and temperature behind the shock or expansion wave [3]. Furthermore, the flexible effects affect the wave structure on the vehicle surface and can change the angle of attack and control angles, which in turn change the pressures on vehicle surfaces.For the upper surface, ifα
+
Δ
τ
1
<
τ
1, an oblique shock wave forms on the upper surface; if α
+
Δ
τ
1
>
τ
1, an expansion fan forms on the upper surface; if α
+
Δ
τ
1
=
τ
1, then the upper surface experiences the freestream. By using oblique shock and Prandtl-Meyer theory, we can obtain the force and moment on the upper surface, which can be resolved into x- and z-body axes components:
(8)
F
x
,
up
=
-
p
up
L
tan
τ
1
,
F
z
,
up
=
p
up
L
,
M
up
=
z
up
F
x
,
up
-
x
up
F
z
,
up
.Similar to the upper surface of the vehicle, the forces and moments on the lower fore-body surface, engine nacelle, and elevator surface can be obtained, respectively. The pressure over the aft-body surface is affected by the exhaust gases from the propulsion system. By calculating the lower fore-body surface pressure, the inner pressure in the engine and the position of the shear layer, we can get the force and moment on the aft-body surface [2]. Then the resultant aerodynamic forces and moments on the x- and z
-body axes F
x
r
,
F
z
r
,
M
r are summarized from the individual components.


### 2.2.2. Unsteady Flexible Effects

When traveling at high speed, the interactions between the fluid and the flexible structure of such vehicle may lead to unsteady aerodynamic effects. Understanding the consequences of unsteady aerodynamic caused by flexible effects is significant to the development of flexible vehicle modeling.Assuming that the flexible effects only occur in thez-direction and the engine is rigid, the unsteady flexible effects here are considered as small perturbations of deformation velocity and angular rate:
(9)
ω
˙
(
x
,
t
)
=
∑
i
=
1
∞
φ
i
(
x
)
η
˙
i
(
t
)
=
Φ
·
η
˙
,
(10)
θ
˙
=
∑
i
=
1
n
d
φ
i
(
x
)
d
x
η
˙
i
(
t
)
=
Φ
˙
·
η
˙
.Piston theory is applied to compute the unsteady aerodynamic effects behind oblique shock waves and expansion fans. The pressure on the surface is given by [7]
(11)
P
P
i
=
(
1
+
γ
-
1
2
w
i
a
i
)
2
γ
/
(
γ
-
1
)
.According to the first-order linear piston theory, the infinitesimal force due to the pressure is(12)
d
F
=
-
P
d
A
n
=
-
(
P
i
+
ρ
i
a
i
[
V
i
·
n
]
)
d
A
n
,
where P
i is the local pressure behind the shock or expansion wave, V
i is the velocity of the surface to the steady flow, ρ
i, a
i are the atmospheric density and sonic speed, respectively, n is the outward pointing surface normal, and d
A is the surface element.For the upper surface, the velocity due to the flexible deformation velocity and angular rate perturbations is(13)
V
up
=
V
1
cos
⁡
τ
1
i
⃑
+
(
V
1
sin
τ
1
+
ω
˙
(
x
,
t
)
)
k
⃑
+
(
Φ
˙
·
η
˙
)
×
r
up
,
where V
1, r
up are the velocity of the flow and the position vector of a point on the upper surface, respectively.Substituting the velocityV
up into (12), the unsteady force and moment on the upper surface can be written as
(14)
Δ
F
up
,
x
=
∫
0
L
ρ
up
a
up
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
+
Φ
·
η
˙
]
sin
τ
1
d
x
,
Δ
F
up
,
z
=
-
∫
0
L
ρ
up
a
up
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
d
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
]
cos
⁡
τ
1
d
x
,
Δ
M
up
=
z
up
Δ
F
up
,
x
-
x
up
Δ
F
up
,
z
.For the lower fore-body surface, the unsteady force and moment are given by(15)
Δ
F
d
f
,
x
=
∫
0
L
1
ρ
d
f
α
d
f
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
]
sin
τ
2
d
x
,
Δ
F
d
f
,
z
=
∫
0
L
1
ρ
d
f
α
d
f
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
d
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
]
cos
⁡
τ
2
d
x
,
Δ
M
d
f
=
z
d
f
Δ
F
d
f
,
x
-
x
d
f
Δ
F
d
f
,
z
.For the lower aft-body surface, the unsteady force and moment are given by(16)
Δ
F
d
a
,
x
=
∫
L
1
+
L
2
L
ρ
d
a
a
d
a
d
d
d
d
d
d
d
×
{
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
dddddd
d
×
sin
(
τ
1
+
τ
3
)
+
(
-
(
Φ
˙
·
η
˙
)
x
+
Φ
·
η
˙
)
d
d
d
d
d
d
d
×
sin
(
τ
1
+
τ
3
)
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
}
d
x
,
Δ
F
d
a
,
z
=
-
∫
L
1
+
L
2
L
ρ
d
a
a
d
a
d
d
d
d
d
d
i
d
×
{
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
dddddddd
×
sin
(
τ
1
+
τ
3
)
+
(
-
(
Φ
˙
·
η
˙
)
x
+
Φ
·
η
˙
)
d
d
d
d
d
i
d
d
×
cos
⁡
(
τ
1
+
τ
3
)
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
}
d
x
,
Δ
M
d
a
=
-
z
d
a
Δ
F
d
a
,
x
-
x
d
a
Δ
F
d
a
,
z
.Summarize the unsteady forces and moments on each surface and denoteF
x
,
fle, F
z
,
fle, M
fle as the unsteady flexible aerodynamic of vehicle; then we can get the total body force and moment:
(17)
F
x
=
F
x
r
+
F
x
,
fle
,
F
z
=
F
z
r
+
F
z
,
fle
,
M
=
M
r
+
M
fle
.


### 2.2.3. Viscous Effects

The inviscid flow cannot describe the real airflow characteristics. With high Mach flight, the viscous fluid moves relatively to the vehicle which can generate viscous friction, and can result in boundary layer effect and aerodynamic heating. The flow temperature changes and it is difficult to calculate the aerodynamic parameters inside the boundary layer. Similar to the literature [19], Eckert’s reference temperature method is used to compute the viscous skin friction on each surface of the vehicle. The reference temperature which is a function of the Mach number (M) and temperature (T) at the edge of the boundary layer as well as the wall (skin) temperature T
w can be written as
(18)
T
*
=
T
[
1
+
M
2
+
0.58
(
T
w
T
-
1
)
]
.Based on the reference temperature, we can calculate the densityρ
*, the viscosity μ
*, Reynolds number R
e
*, and the skin friction coefficient C
f with the perfect gas law and Sutherland’s formula. Then, the shear stress on the surface can be computed by
(19)
τ
w
=
(
1
2
ρ
∞
V
∞
2
)
C
f
.
Integrating (19), the skin friction drag can be given by
(20)
F
vis
=
∫
τ
w
d
x
.
Then the normal and axial forces are obtained as follows:
(21)
N
vis
=
F
vis
sin
β
,
A
vis
=
F
vis
cos
⁡
β
,
where β is the surface inclination to the body axis.By using the coordinate transformation, the additional lift, drag, and moment of viscous effects can be written as(22)
L
vis
=
N
vis
cos
⁡
α
-
A
vis
sin
α
,
D
vis
=
N
vis
sin
α
-
A
vis
cos
⁡
α
,
M
vis
=
N
vis
x
+
A
vis
z
.We consider the viscous skin friction on the upper surface, lower fore-body surface, lower aft-body surface, surface below the engine nacelle, and control surfaces. By calculating the additional lift, drag, and moment on each surface, we can get the total additional liftL
vis, additional drag D
vis, and additional moment M
vis.Translating the forceF
x, F
z into the lift and drag acting on the vehicle and taking the viscous effects into account, the total lift, drag, and pitching moment are
(23)
L
=
F
x
sin
α
-
F
z
cos
⁡
α
+
L
vis
,
D
=
-
F
x
cos
⁡
α
-
F
z
sin
α
+
D
vis
,
M
=
M
r
+
M
fle
+
M
vis
.The flexible and viscous effects are included in force and moment formulations in (23). If we assume a rigid structure such that η
1
=
η
2
=
η
3
=
η
˙
1
=
η
˙
2
=
η
˙
3
=
0 and viscous effect is excluded, then (23) is steady aerodynamic forces of a rigid body vehicle.


### 2.2.4. Generalized Forces

In this paper, we present the loadings on the upper surface, lower surface, and control surface; the generalized forces for the first three flexible modes can be written as(24)
N
i
=
∫
0
L
φ
i
(
x
)
p
up
d
x
-
∫
0
L
f
φ
i
(
x
)
p
d
f
d
x
-
∫
L
f
+
L
n
L
φ
i
(
x
)
p
d
a
d
x
-
∫
L
f
L
n
φ
i
(
x
)
p
e
n
d
x
+
φ
i
(
x
e
)
F
e
,
dddddddddddddddddddddddddddddddddd
i
=
1,2
,
3
,
where p
up, p
d
f, p
d
a, p
e
n are the pressures on the upper surface, lower fore-body surface, engine nacelle, and lower aft-body surface, respectively. F
e is the force act on the elevator surface; φ
i
(
x
) is the mode shape, which is calculated by the assumed mode method.


## 2.2.1. Aerodynamic Forces

Shock and expansion waves are common physical phenomena in hypersonic flows. By using the oblique shock and Prandtl-Meyer theory, we can calculate the pressure, Mach number, and temperature behind the shock or expansion wave [3]. Furthermore, the flexible effects affect the wave structure on the vehicle surface and can change the angle of attack and control angles, which in turn change the pressures on vehicle surfaces.For the upper surface, ifα
+
Δ
τ
1
<
τ
1, an oblique shock wave forms on the upper surface; if α
+
Δ
τ
1
>
τ
1, an expansion fan forms on the upper surface; if α
+
Δ
τ
1
=
τ
1, then the upper surface experiences the freestream. By using oblique shock and Prandtl-Meyer theory, we can obtain the force and moment on the upper surface, which can be resolved into x- and z-body axes components:
(8)
F
x
,
up
=
-
p
up
L
tan
τ
1
,
F
z
,
up
=
p
up
L
,
M
up
=
z
up
F
x
,
up
-
x
up
F
z
,
up
.Similar to the upper surface of the vehicle, the forces and moments on the lower fore-body surface, engine nacelle, and elevator surface can be obtained, respectively. The pressure over the aft-body surface is affected by the exhaust gases from the propulsion system. By calculating the lower fore-body surface pressure, the inner pressure in the engine and the position of the shear layer, we can get the force and moment on the aft-body surface [2]. Then the resultant aerodynamic forces and moments on the x- and z
-body axes F
x
r
,
F
z
r
,
M
r are summarized from the individual components.


## 2.2.2. Unsteady Flexible Effects

When traveling at high speed, the interactions between the fluid and the flexible structure of such vehicle may lead to unsteady aerodynamic effects. Understanding the consequences of unsteady aerodynamic caused by flexible effects is significant to the development of flexible vehicle modeling.Assuming that the flexible effects only occur in thez-direction and the engine is rigid, the unsteady flexible effects here are considered as small perturbations of deformation velocity and angular rate:
(9)
ω
˙
(
x
,
t
)
=
∑
i
=
1
∞
φ
i
(
x
)
η
˙
i
(
t
)
=
Φ
·
η
˙
,
(10)
θ
˙
=
∑
i
=
1
n
d
φ
i
(
x
)
d
x
η
˙
i
(
t
)
=
Φ
˙
·
η
˙
.Piston theory is applied to compute the unsteady aerodynamic effects behind oblique shock waves and expansion fans. The pressure on the surface is given by [7]
(11)
P
P
i
=
(
1
+
γ
-
1
2
w
i
a
i
)
2
γ
/
(
γ
-
1
)
.According to the first-order linear piston theory, the infinitesimal force due to the pressure is(12)
d
F
=
-
P
d
A
n
=
-
(
P
i
+
ρ
i
a
i
[
V
i
·
n
]
)
d
A
n
,
where P
i is the local pressure behind the shock or expansion wave, V
i is the velocity of the surface to the steady flow, ρ
i, a
i are the atmospheric density and sonic speed, respectively, n is the outward pointing surface normal, and d
A is the surface element.For the upper surface, the velocity due to the flexible deformation velocity and angular rate perturbations is(13)
V
up
=
V
1
cos
⁡
τ
1
i
⃑
+
(
V
1
sin
τ
1
+
ω
˙
(
x
,
t
)
)
k
⃑
+
(
Φ
˙
·
η
˙
)
×
r
up
,
where V
1, r
up are the velocity of the flow and the position vector of a point on the upper surface, respectively.Substituting the velocityV
up into (12), the unsteady force and moment on the upper surface can be written as
(14)
Δ
F
up
,
x
=
∫
0
L
ρ
up
a
up
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
+
Φ
·
η
˙
]
sin
τ
1
d
x
,
Δ
F
up
,
z
=
-
∫
0
L
ρ
up
a
up
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
d
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
1
2
-
x
)
]
cos
⁡
τ
1
d
x
,
Δ
M
up
=
z
up
Δ
F
up
,
x
-
x
up
Δ
F
up
,
z
.For the lower fore-body surface, the unsteady force and moment are given by(15)
Δ
F
d
f
,
x
=
∫
0
L
1
ρ
d
f
α
d
f
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
]
sin
τ
2
d
x
,
Δ
F
d
f
,
z
=
∫
0
L
1
ρ
d
f
α
d
f
[
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
d
d
d
d
d
d
d
d
d
d
d
+
Φ
·
η
˙
(
Φ
˙
·
η
˙
)
(
x
tan
τ
2
2
-
x
)
]
cos
⁡
τ
2
d
x
,
Δ
M
d
f
=
z
d
f
Δ
F
d
f
,
x
-
x
d
f
Δ
F
d
f
,
z
.For the lower aft-body surface, the unsteady force and moment are given by(16)
Δ
F
d
a
,
x
=
∫
L
1
+
L
2
L
ρ
d
a
a
d
a
d
d
d
d
d
d
d
×
{
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
dddddd
d
×
sin
(
τ
1
+
τ
3
)
+
(
-
(
Φ
˙
·
η
˙
)
x
+
Φ
·
η
˙
)
d
d
d
d
d
d
d
×
sin
(
τ
1
+
τ
3
)
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
}
d
x
,
Δ
F
d
a
,
z
=
-
∫
L
1
+
L
2
L
ρ
d
a
a
d
a
d
d
d
d
d
d
i
d
×
{
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
dddddddd
×
sin
(
τ
1
+
τ
3
)
+
(
-
(
Φ
˙
·
η
˙
)
x
+
Φ
·
η
˙
)
d
d
d
d
d
i
d
d
×
cos
⁡
(
τ
1
+
τ
3
)
Φ
˙
·
η
˙
[
(
L
-
x
)
tan
(
τ
1
+
τ
3
)
-
L
tan
τ
1
]
}
d
x
,
Δ
M
d
a
=
-
z
d
a
Δ
F
d
a
,
x
-
x
d
a
Δ
F
d
a
,
z
.Summarize the unsteady forces and moments on each surface and denoteF
x
,
fle, F
z
,
fle, M
fle as the unsteady flexible aerodynamic of vehicle; then we can get the total body force and moment:
(17)
F
x
=
F
x
r
+
F
x
,
fle
,
F
z
=
F
z
r
+
F
z
,
fle
,
M
=
M
r
+
M
fle
.


## 2.2.3. Viscous Effects

The inviscid flow cannot describe the real airflow characteristics. With high Mach flight, the viscous fluid moves relatively to the vehicle which can generate viscous friction, and can result in boundary layer effect and aerodynamic heating. The flow temperature changes and it is difficult to calculate the aerodynamic parameters inside the boundary layer. Similar to the literature [19], Eckert’s reference temperature method is used to compute the viscous skin friction on each surface of the vehicle. The reference temperature which is a function of the Mach number (M) and temperature (T) at the edge of the boundary layer as well as the wall (skin) temperature T
w can be written as
(18)
T
*
=
T
[
1
+
M
2
+
0.58
(
T
w
T
-
1
)
]
.Based on the reference temperature, we can calculate the densityρ
*, the viscosity μ
*, Reynolds number R
e
*, and the skin friction coefficient C
f with the perfect gas law and Sutherland’s formula. Then, the shear stress on the surface can be computed by
(19)
τ
w
=
(
1
2
ρ
∞
V
∞
2
)
C
f
.
Integrating (19), the skin friction drag can be given by
(20)
F
vis
=
∫
τ
w
d
x
.
Then the normal and axial forces are obtained as follows:
(21)
N
vis
=
F
vis
sin
β
,
A
vis
=
F
vis
cos
⁡
β
,
where β is the surface inclination to the body axis.By using the coordinate transformation, the additional lift, drag, and moment of viscous effects can be written as(22)
L
vis
=
N
vis
cos
⁡
α
-
A
vis
sin
α
,
D
vis
=
N
vis
sin
α
-
A
vis
cos
⁡
α
,
M
vis
=
N
vis
x
+
A
vis
z
.We consider the viscous skin friction on the upper surface, lower fore-body surface, lower aft-body surface, surface below the engine nacelle, and control surfaces. By calculating the additional lift, drag, and moment on each surface, we can get the total additional liftL
vis, additional drag D
vis, and additional moment M
vis.Translating the forceF
x, F
z into the lift and drag acting on the vehicle and taking the viscous effects into account, the total lift, drag, and pitching moment are
(23)
L
=
F
x
sin
α
-
F
z
cos
⁡
α
+
L
vis
,
D
=
-
F
x
cos
⁡
α
-
F
z
sin
α
+
D
vis
,
M
=
M
r
+
M
fle
+
M
vis
.The flexible and viscous effects are included in force and moment formulations in (23). If we assume a rigid structure such that η
1
=
η
2
=
η
3
=
η
˙
1
=
η
˙
2
=
η
˙
3
=
0 and viscous effect is excluded, then (23) is steady aerodynamic forces of a rigid body vehicle.


## 2.2.4. Generalized Forces

In this paper, we present the loadings on the upper surface, lower surface, and control surface; the generalized forces for the first three flexible modes can be written as(24)
N
i
=
∫
0
L
φ
i
(
x
)
p
up
d
x
-
∫
0
L
f
φ
i
(
x
)
p
d
f
d
x
-
∫
L
f
+
L
n
L
φ
i
(
x
)
p
d
a
d
x
-
∫
L
f
L
n
φ
i
(
x
)
p
e
n
d
x
+
φ
i
(
x
e
)
F
e
,
dddddddddddddddddddddddddddddddddd
i
=
1,2
,
3
,
where p
up, p
d
f, p
d
a, p
e
n are the pressures on the upper surface, lower fore-body surface, engine nacelle, and lower aft-body surface, respectively. F
e is the force act on the elevator surface; φ
i
(
x
) is the mode shape, which is calculated by the assumed mode method.


## 2.3. Thrust

Similar to the literature [20], the scramjet engine consists of 3 sections: a diffuser, a combustor, and an internal nozzle. We assume that the airflow into the engine is parallel to the direction of the body axis. By avoiding the mass flow impinges at the inlet, a translating cowl door is added to increase the mass flow through the engine module. With quasi-one-dimensional aero/thermoanalysis of the flow in the propulsion system and the momentum theorem, the thrust is given by
(25)
T
=
m
˙
a
(
V
e
-
V
∞
)
+
(
p
e
-
p
∞
)
A
e
b
,
where m
˙
a is the air mass flow through the engine, m
˙
a is dependent on the fore-body deflection Δ
τ
1, angle of attack α, and the freestream, V
e and p
e are the airflow velocity and pressure at the engine exit, respectively, V
∞ is the freestream airflow velocity, and A
e
/
b is the exit area per unit width.


## 2.4. Equations of Motion

Assume that the earth is flat and the fuselage is free-free beam. The equations of motion of the longitudinal dynamics for the hypersonic vehicle based on Lagrange’s equations are presented in [3]. By ignoring the weak inertial coupling terms, the flexible mode shapes are orthogonal to the rigid body rotation, which decouples the flexible modes and the rigid body modes, and the couplings between the rigid and flexible modes are included in the aerodynamic forces and moments. Then the simplified longitudinal dynamics of a flexible vehicle written in the stability axis coordinate system are [8, 10]
(26)
V
˙
=
1
m
(
T
cos
⁡
α
-
D
)
-
g
sin
(
θ
-
α
)
,
α
˙
=
1
m
V
(
-
T
sin
α
-
L
)
+
Q
+
g
V
cos
⁡
(
θ
-
α
)
,
q
˙
=
I
y
y
M
,
θ
˙
=
q
,
h
˙
=
V
sin
(
θ
-
α
)
,
η
¨
i
+
2
ζ
i
ω
i
η
˙
i
+
ω
i
2
η
i
=
N
i
,
i
=
1,2
,
3
,
where the rigid body states are {
V
,
α
,
q
,
θ
,
h
}, in which V, α, q, θ, h are the vehicle speed, angle of attack, pitch rate, pitch acceleration, and height, respectively. The flexible states are {
η
1
,
η
˙
1
,
η
2
,
η
˙
2
,
η
3
,
η
˙
3
}
,
ζ is the structure damping, ω
i is the frequency, and η
i is the generalized coordinate. The control input elevator deflection δ
e, throttle ϕ, and the couplings between the rigid and flexible modes will appear explicitly in the forces and moment L, D, T, M, N
i which are calculated in the above sections.


## 3. Control-Oriented Modeling

### 3.1. Flexible Modes

The natural frequencies of the flexible structure are the square roots of the eigenvalues ofM
-
1
K; by using the assumed modes method in Section 2.1, we obtain the first three natural frequencies:
(27)
ω
1
=
19.7437
rad
/
s
,
ω
2
=
47.7851
rad
/
s
,
ω
3
=
94.8203
rad
/
s
.The first three mode shapes are shown in Figure3; the deflections of the fore-body and aft-body are presented by
(28)
Δ
τ
1
=
Φ
1
(
η
1
η
2
η
3
)
T
,
Δ
τ
2
=
Φ
2
(
η
1
η
2
η
3
)
T
,
where Φ
1
=
(
φ
1
′
(
0
)
,
φ
2
′
(
0
)
,
φ
3
′
(
0
)
) and Φ
2
=
(
φ
1
′
(
L
)
,
φ
2
′
(
L
)
,
φ
3
′
(
L
)
).Figure 3
The mode shapes of the flexible structure.We assume that the generalized coordinates are constant; thenΔ
τ
1
=
-
2.3835
deg
⁡, Δ
τ
2
=
0.7047
deg
⁡. It can be seen that the fore-body deflection is changed significantly compared to the aft-body deflection.


### 3.2. Flexible and Viscous Effects

For analyzing the contributions of flexible and viscous effects to the aerodynamic, we observe the trend of forces with angle of attack and other variables frozen at certain conditions. From Figures4 and 5, we can see that the viscous drag is large compared to the lift. This is because the viscous forces act parallel to the surface and there is an increase in the drag acting on the vehicle. On the other hand, the flexible effects are relatively small compared to the viscous effects.Figure 4
The trend of lift with AOA.Figure 5
The trend of drag with AOA.We can conclude that the viscous effects are significant contributor to the drag and should be included in the aerodynamic modeling. The flexible effects do not make significant contribution to the forces but cannot be ignored here because the lowest frequency will change with the aerodynamic heating and mass decreasing [17] and will be close to the rigid modes, which can bring the coupling between the rigid and flexible dynamic [21].Figure6 shows the trend of thrust with the fore-body deflection. The thrust decreases as the fore-body deflection increases; this is because the bending of vehicle body can result in not enough airflow flown into the engine, which will then affect the propulsion system.Figure 6
Thrust versusΔ
τ
1.


### 3.3. Control-Oriented Model

The model in Section2 captures interactions among the aerodynamics, structural dynamics, and propulsion of the hypersonic vehicle and the strong couplings between the rigid and flexible modes result in intractable systems for nonlinear control. By replacing complex force and moment functions with curve fitting approximations, neglecting certain weak couplings, a simplified control-oriented model can be obtained. The model should be simple enough to facilitate controller design and capture the inherent features which are included in physics-based model. By analyzing the couplings from the aerodynamic data of the model, we choose the variables and the functions of forces and moments which are
(29)
L
≈
q
-
S
C
L
(
M
a
,
α
,
δ
e
,
η
,
η
˙
)
,
D
≈
q
-
S
C
D
(
M
a
,
α
,
δ
e
,
η
,
η
˙
)
,
T
≈
q
-
S
C
T
(
M
a
,
α
,
ϕ
,
δ
e
,
η
,
η
˙
)
,
M
≈
z
T
T
+
q
-
c
-
S
C
M
(
M
a
,
α
,
δ
e
,
η
,
η
˙
)
,
N
i
≈
q
-
S
C
N
i
(
M
a
,
α
,
δ
e
,
η
,
η
˙
)
,
i
=
1,2
,
3
,
where η
=
(
η
1
,
η
2
,
η
3
) and η
˙
=
(
η
˙
1
,
η
˙
2
,
η
˙
3
) reflect the unsteady effects on the aerodynamic forces and moments.Based on the aerodynamic data from the physics-based model, we can get the forces and moments coefficients by using the curve fitting method:(30)
C
L
=
C
L
M
a
M
a
+
C
L
α
α
+
C
L
δ
e
δ
e
+
C
L
η
1
η
1
+
C
L
η
3
η
3
+
C
L
η
˙
2
η
˙
2
+
C
L
0
,
C
D
=
C
D
M
a
M
a
+
C
D
α
α
+
C
D
δ
e
δ
e
+
C
D
η
1
η
1
+
C
D
η
3
η
3
+
C
D
M
a
α
M
a
·
α
+
C
D
M
a
δ
c
M
a
·
δ
e
+
C
D
α
δ
e
α
·
δ
e
+
C
D
α
2
α
2
+
C
D
δ
e
2
δ
e
2
+
C
D
o
,
C
T
=
C
T
M
a
M
a
+
C
T
α
α
+
C
T
ϕ
ϕ
+
C
T
α
3
ϕ
α
3
ϕ
+
C
T
α
2
ϕ
α
2
ϕ
+
C
T
α
ϕ
α
·
ϕ
+
C
T
M
a
ϕ
M
a
·
ϕ
+
C
T
η
1
η
1
+
C
T
η
2
η
2
+
C
T
η
3
η
3
+
C
T
o
,
C
M
=
C
M
M
a
M
a
+
C
M
α
α
+
C
M
δ
e
δ
e
+
C
M
η
1
η
1
+
C
M
η
˙
2
η
˙
2
+
C
M
η
˙
3
η
˙
3
+
C
M
0
,
C
N
i
=
C
N
i
M
a
M
a
+
C
N
i
α
α
+
C
N
i
δ
e
δ
e
+
C
N
i
η
1
η
1
+
C
N
i
η
2
η
2
+
C
N
i
η
3
η
3
+
C
N
i
0
,
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
i
=
,
1,2
,
3
.The forces and moments coefficients are all the functions of rigid states, flexible modes, and control inputs, where the lift and general force functions contain linear terms with respect to each variable. Throttle term is omitted in the function of the lift, drag, and general force because of little influence on the fitting accuracy. Adding the cross-terms and higher-order terms can increase the fit accuracy in the drag and thrust functions. The forces and moment contain steady states and unsteady flexible effects.The fit functions (30) reduce the complexity of the aerodynamic model in Section 2, and the containing of the unsteady flexible effects η
˙ in polynomials reflects the couplings between the flexible structure and aeropropulsion system. The ranges of variables in the fit functions are shown in Table 1, and Tables 2, 3, 4, 5, and 6 show the values of the fit coefficients.Table 1
The range of the variables.
Variables
Range
Variables
Range
V (ft/s)
7500~11000
η
2
−0.03~0.04
α (deg)
−5~5
η
3
−0.003~0.004
δ
e (deg)
−10~10
η
˙
1
−3~4
ϕ
0.05~1.5
η
˙
2
−0.3~0.4
η
1
−0.3~0.4
η
˙
3
−0.03~0.04Table 2
The coefficients of fits for the lift.
Terms
Values
Terms
Values
C
L
Ma
−0.00578253
C
L
η
3
−0.0954033
C
L
α
5.84425
C
L
η
˙
2
−0.00413381
C
L
δ
e
0.755975
C
L
0
0.0621356
C
L
η
1
−0.0522466Table 3
The coefficients of fits for the drag.
Terms
Values
Terms
Values
C
D
Ma
0.0201306
C
D
Ma
δ
e
−0.00383177
C
D
α
0.203794
C
D
α
δ
e
1.98468
C
D
δ
e
0.0325529
C
D
α
2
7.72588
C
D
η
1
0.0025028
C
D
δ
e
2
0.945984
C
D
η
3
0.00496105
C
D
0
−0.109122
C
D
Ma
α
−0.0241477Table 4
The coefficients of fits for the thrust.
Terms
Values
Terms
Values
C
T
Ma
0.0059
C
T
Ma
ϕ
−126.8976
C
T
α
−0.4255
C
T
η
1
132.3165
C
T
ϕ
−0.7297
C
T
η
2
−9.3163
C
T
α
3
ϕ
0.0081
C
T
η
3
0.1239
C
T
α
2
ϕ
0.01
C
T
0
−0.0429
C
T
α
ϕ
0.0157Table 5
The coefficients of fits for the pitching moment.
Terms
Values
Terms
Values
C
M
Ma
0.52405
C
M
η
˙
2
−0.116657
C
M
α
31.7904
C
M
η
˙
3
−0.0868456
C
M
δ
e
−25.1464
C
M
0
−0.887385
C
M
η
1
−0.325935Table 6
The coefficients of fits for the generalized forces.
Terms
Values
Terms
Values
C
N
1
Ma
−0.0003
C
N
2
η
1
0.0009
C
N
1
α
−0.4515
C
N
2
η
2
−0.0012
C
N
1
δ
e
−0.1517
C
N
2
η
3
0.0021
C
N
1
η
1
0.0042
C
N
2
0
0.0015
C
N
1
η
2
0.001
C
N
3
Ma
0.0005
C
N
1
η
3
0.0087
C
N
3
α
−0.1365
C
N
1
0
−0.0027
C
N
3
δ
e
−0.0527
C
N
2
Ma
−0.0006
C
N
3
η
1
0.0009
C
N
2
α
−0.1236
C
N
3
η
2
0.0003
C
N
2
δ
e
−0.0869
C
N
3
η
3
0.002
C
N
2
δ
c
−0.0926
C
N
3
0
−0.0044
C
N
2
δ
e
−0.0864To validate the goodness of the fit functions, we examine the three-dimensional plots of the aerodynamic forces and moments with variables under certain flight conditions. Figures7 and 8 show the trend of lift and pitching moment with α, δ
e, and M
a; we can see that the trend of the fit curve and the interpolation surface of the experimental data is approximated. The P value is 0, and the coefficient of determination is above 0.9 in the fitting, which shows that the fit functions are very significant. The fitting errors are 0.0672, 0.009, 0.232, 0.0057, 0.0066, 0.0029, and 0.0021, respectively. Then we can conclude that the goodness of the fit is high and the fittings meet the accuracy requirement.Figure 7
Lift versusα and M
a.Figure 8
Pitching moment versusα and δ
e.


## 3.1. Flexible Modes

The natural frequencies of the flexible structure are the square roots of the eigenvalues ofM
-
1
K; by using the assumed modes method in Section 2.1, we obtain the first three natural frequencies:
(27)
ω
1
=
19.7437
rad
/
s
,
ω
2
=
47.7851
rad
/
s
,
ω
3
=
94.8203
rad
/
s
.The first three mode shapes are shown in Figure3; the deflections of the fore-body and aft-body are presented by
(28)
Δ
τ
1
=
Φ
1
(
η
1
η
2
η
3
)
T
,
Δ
τ
2
=
Φ
2
(
η
1
η
2
η
3
)
T
,
where Φ
1
=
(
φ
1
′
(
0
)
,
φ
2
′
(
0
)
,
φ
3
′
(
0
)
) and Φ
2
=
(
φ
1
′
(
L
)
,
φ
2
′
(
L
)
,
φ
3
′
(
L
)
).Figure 3
The mode shapes of the flexible structure.We assume that the generalized coordinates are constant; thenΔ
τ
1
=
-
2.3835
deg
⁡, Δ
τ
2
=
0.7047
deg
⁡. It can be seen that the fore-body deflection is changed significantly compared to the aft-body deflection.


## 3.2. Flexible and Viscous Effects

For analyzing the contributions of flexible and viscous effects to the aerodynamic, we observe the trend of forces with angle of attack and other variables frozen at certain conditions. From Figures4 and 5, we can see that the viscous drag is large compared to the lift. This is because the viscous forces act parallel to the surface and there is an increase in the drag acting on the vehicle. On the other hand, the flexible effects are relatively small compared to the viscous effects.Figure 4
The trend of lift with AOA.Figure 5
The trend of drag with AOA.We can conclude that the viscous effects are significant contributor to the drag and should be included in the aerodynamic modeling. The flexible effects do not make significant contribution to the forces but cannot be ignored here because the lowest frequency will change with the aerodynamic heating and mass decreasing [17] and will be close to the rigid modes, which can bring the coupling between the rigid and flexible dynamic [21].Figure6 shows the trend of thrust with the fore-body deflection. The thrust decreases as the fore-body deflection increases; this is because the bending of vehicle body can result in not enough airflow flown into the engine, which will then affect the propulsion system.Figure 6
Thrust versusΔ
τ
1.


## 3.3. Control-Oriented Model

The model in Section2 captures interactions among the aerodynamics, structural dynamics, and propulsion of the hypersonic vehicle and the strong couplings between the rigid and flexible modes result in intractable systems for nonlinear control. By replacing complex force and moment functions with curve fitting approximations, neglecting certain weak couplings, a simplified control-oriented model can be obtained. The model should be simple enough to facilitate controller design and capture the inherent features which are included in physics-based model. By analyzing the couplings from the aerodynamic data of the model, we choose the variables and the functions of forces and moments which are
(29)
L
≈
q
-
S
C
L
(
M
a
,
α
,
δ
e
,
η
,
η
˙
)
,
D
≈
q
-
S
C
D
(
M
a
,
α
,
δ
e
,
η
,
η
˙
)
,
T
≈
q
-
S
C
T
(
M
a
,
α
,
ϕ
,
δ
e
,
η
,
η
˙
)
,
M
≈
z
T
T
+
q
-
c
-
S
C
M
(
M
a
,
α
,
δ
e
,
η
,
η
˙
)
,
N
i
≈
q
-
S
C
N
i
(
M
a
,
α
,
δ
e
,
η
,
η
˙
)
,
i
=
1,2
,
3
,
where η
=
(
η
1
,
η
2
,
η
3
) and η
˙
=
(
η
˙
1
,
η
˙
2
,
η
˙
3
) reflect the unsteady effects on the aerodynamic forces and moments.Based on the aerodynamic data from the physics-based model, we can get the forces and moments coefficients by using the curve fitting method:(30)
C
L
=
C
L
M
a
M
a
+
C
L
α
α
+
C
L
δ
e
δ
e
+
C
L
η
1
η
1
+
C
L
η
3
η
3
+
C
L
η
˙
2
η
˙
2
+
C
L
0
,
C
D
=
C
D
M
a
M
a
+
C
D
α
α
+
C
D
δ
e
δ
e
+
C
D
η
1
η
1
+
C
D
η
3
η
3
+
C
D
M
a
α
M
a
·
α
+
C
D
M
a
δ
c
M
a
·
δ
e
+
C
D
α
δ
e
α
·
δ
e
+
C
D
α
2
α
2
+
C
D
δ
e
2
δ
e
2
+
C
D
o
,
C
T
=
C
T
M
a
M
a
+
C
T
α
α
+
C
T
ϕ
ϕ
+
C
T
α
3
ϕ
α
3
ϕ
+
C
T
α
2
ϕ
α
2
ϕ
+
C
T
α
ϕ
α
·
ϕ
+
C
T
M
a
ϕ
M
a
·
ϕ
+
C
T
η
1
η
1
+
C
T
η
2
η
2
+
C
T
η
3
η
3
+
C
T
o
,
C
M
=
C
M
M
a
M
a
+
C
M
α
α
+
C
M
δ
e
δ
e
+
C
M
η
1
η
1
+
C
M
η
˙
2
η
˙
2
+
C
M
η
˙
3
η
˙
3
+
C
M
0
,
C
N
i
=
C
N
i
M
a
M
a
+
C
N
i
α
α
+
C
N
i
δ
e
δ
e
+
C
N
i
η
1
η
1
+
C
N
i
η
2
η
2
+
C
N
i
η
3
η
3
+
C
N
i
0
,
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
i
=
,
1,2
,
3
.The forces and moments coefficients are all the functions of rigid states, flexible modes, and control inputs, where the lift and general force functions contain linear terms with respect to each variable. Throttle term is omitted in the function of the lift, drag, and general force because of little influence on the fitting accuracy. Adding the cross-terms and higher-order terms can increase the fit accuracy in the drag and thrust functions. The forces and moment contain steady states and unsteady flexible effects.The fit functions (30) reduce the complexity of the aerodynamic model in Section 2, and the containing of the unsteady flexible effects η
˙ in polynomials reflects the couplings between the flexible structure and aeropropulsion system. The ranges of variables in the fit functions are shown in Table 1, and Tables 2, 3, 4, 5, and 6 show the values of the fit coefficients.Table 1
The range of the variables.
Variables
Range
Variables
Range
V (ft/s)
7500~11000
η
2
−0.03~0.04
α (deg)
−5~5
η
3
−0.003~0.004
δ
e (deg)
−10~10
η
˙
1
−3~4
ϕ
0.05~1.5
η
˙
2
−0.3~0.4
η
1
−0.3~0.4
η
˙
3
−0.03~0.04Table 2
The coefficients of fits for the lift.
Terms
Values
Terms
Values
C
L
Ma
−0.00578253
C
L
η
3
−0.0954033
C
L
α
5.84425
C
L
η
˙
2
−0.00413381
C
L
δ
e
0.755975
C
L
0
0.0621356
C
L
η
1
−0.0522466Table 3
The coefficients of fits for the drag.
Terms
Values
Terms
Values
C
D
Ma
0.0201306
C
D
Ma
δ
e
−0.00383177
C
D
α
0.203794
C
D
α
δ
e
1.98468
C
D
δ
e
0.0325529
C
D
α
2
7.72588
C
D
η
1
0.0025028
C
D
δ
e
2
0.945984
C
D
η
3
0.00496105
C
D
0
−0.109122
C
D
Ma
α
−0.0241477Table 4
The coefficients of fits for the thrust.
Terms
Values
Terms
Values
C
T
Ma
0.0059
C
T
Ma
ϕ
−126.8976
C
T
α
−0.4255
C
T
η
1
132.3165
C
T
ϕ
−0.7297
C
T
η
2
−9.3163
C
T
α
3
ϕ
0.0081
C
T
η
3
0.1239
C
T
α
2
ϕ
0.01
C
T
0
−0.0429
C
T
α
ϕ
0.0157Table 5
The coefficients of fits for the pitching moment.
Terms
Values
Terms
Values
C
M
Ma
0.52405
C
M
η
˙
2
−0.116657
C
M
α
31.7904
C
M
η
˙
3
−0.0868456
C
M
δ
e
−25.1464
C
M
0
−0.887385
C
M
η
1
−0.325935Table 6
The coefficients of fits for the generalized forces.
Terms
Values
Terms
Values
C
N
1
Ma
−0.0003
C
N
2
η
1
0.0009
C
N
1
α
−0.4515
C
N
2
η
2
−0.0012
C
N
1
δ
e
−0.1517
C
N
2
η
3
0.0021
C
N
1
η
1
0.0042
C
N
2
0
0.0015
C
N
1
η
2
0.001
C
N
3
Ma
0.0005
C
N
1
η
3
0.0087
C
N
3
α
−0.1365
C
N
1
0
−0.0027
C
N
3
δ
e
−0.0527
C
N
2
Ma
−0.0006
C
N
3
η
1
0.0009
C
N
2
α
−0.1236
C
N
3
η
2
0.0003
C
N
2
δ
e
−0.0869
C
N
3
η
3
0.002
C
N
2
δ
c
−0.0926
C
N
3
0
−0.0044
C
N
2
δ
e
−0.0864To validate the goodness of the fit functions, we examine the three-dimensional plots of the aerodynamic forces and moments with variables under certain flight conditions. Figures7 and 8 show the trend of lift and pitching moment with α, δ
e, and M
a; we can see that the trend of the fit curve and the interpolation surface of the experimental data is approximated. The P value is 0, and the coefficient of determination is above 0.9 in the fitting, which shows that the fit functions are very significant. The fitting errors are 0.0672, 0.009, 0.232, 0.0057, 0.0066, 0.0029, and 0.0021, respectively. Then we can conclude that the goodness of the fit is high and the fittings meet the accuracy requirement.Figure 7
Lift versusα and M
a.Figure 8
Pitching moment versusα and δ
e.


## 4. Dynamic Analysis

For the analysis of the dynamic stability, the vehicle is trimmed in steady level flight at an altitude of 85,000 ft and Mach 8; then the vehicle model is linearized at the trim condition in Table7. The pole-zero map is shown in Figure 9, where the input is the elevator deflection δ
e and the output is the velocity V. We can see that the vehicle has an unstable short period mode, an unstable long period mode, and three stable flexible modes. The flexible modes are pairs of complex conjugate poles and have negative real parts. The RHP zero in Figure 9 shows that the vehicle model is unstable and the nonminimum phase phenomenon exists.Table 7
Trim at Mach 8,85000 ft height for the model.
States
Value
States
Value
V (Ma)
8
η
3 (ft·slug)
−0.0019934
h (ft)
85000
δ
e (°)
7.9968
α (°)
0.5758
ϕ
0.51914
η
1 (ft·slug)
−0.15266
η
2 (ft·slug)
−0.013716Figure 9
The pole-zero map.


## 5. Conclusions

Based on the longitudinal geometry of an  air-breathing hypersonic vehicle, this work uses the engineering estimated method to calculate the aerodynamics steady forces and moments, viscous effects, and unsteady flexible effects. A physics-based vehicle model which contains aeroflexible structure-propulsion interactions is developed. With the curve fitting approximation method, a simplified control-oriented model is presented, where the forces, moments, and thrust are the polynomial functions of the states and control inputs. The simulations show that the flexible and viscous effects can change the aerodynamic layout on the vehicle and should not be omitted in vehicle modeling. The control-oriented model retains the couplings between the rigid and flexible dynamics and can be applied for controller design.



---

*Source: 264247-2014-04-28.xml*