[Abstract] This paper proposes a systematic approach to establish the 6-DOF nonlinear dynamic model for airships. The flight mechanics, aerostatics and complete aerodynamic effects are incorporated into the model. A simulation program is developed from the dynamic model and applied to analyze the control responses of the Skyship-500 airship. A comparison between the simulation and flight test results for different control inputs shows that the dynamic simulation can give a reasonable estimation of the flight behavior.
Nomenclature b = fin semi span
fcc, = airfoil chord and flap chord
00 , dFdH CC = zero-angle axial drag coefficients of the hull and the fins, respectively
MLD CCC ,, = drag, lift and pitching moment coefficients
αα lL CC , = 3-D and 2-D lift curve slopes of the fins
αpC∆ = ( ) α∂∆∂ pC , where pC∆ is the pressure coefficient of the airfoil
ASG FF , = force vectors due to gravity and aerostatics, respectively ( )zFyFNF FFF , = normal forces (in y and z directions) acting on the fins
)(FNHF = normal forces acting on the hull due to the fins ( )zVyVNV FFF , = normal forces (in y and z directions) due to viscosity on the hull
g = acceleration of gravity I ′ = moment of inertia of the displaced air by the hull J = inertia tensor of the airship
kkk ′,, 21 = added-mass factors of ellipsoids
44k = added-mass factor for the added moment of inertia due to the fins
Dk3 = 3-D efficiency factor for the aerodynamics computation of control surface deflection
AMM, = mass matrix and added-mass matrix
zFyFxF MMM ,, = rolling, pitching and yawing moments from NFF
zVyV MM , = pitching and yawing moments from NVF m = total mass of the airship m′ = mass of the displaced air by the hull
FijHij mm , = elements of added-mass matrices of the hull and the fins, respectively SFSFSF mmm 443322 ,, = cross-sectional added mass and moment of inertia due to the fins
20 21 Vq ρ= = dynamic pressure
R = hull cross-sectional radius Vg rr , = position vectors of the C.G. and the C.V. relative to the body-frame origin, respectively
FH SS , = reference areas for the axial drag of the hull and the fins, respectively * PhD Student, Department of Mechanical Engineering, 817 Sherbrooke Street West, Email: [email protected] † Associate Professor, Department of Mechanical Engineering, 817 Sherbrooke Street West, Senior Member AIAA, Email: [email protected]
AIAA Modeling and Simulation Technologies Conference and Exhibit21 - 24 August 2006, Keystone, Colorado
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FAf SS , = flap and fin area, respectively s = spanwise position V = airspeed
BV = airship volume
[ ]T,, wvu=v = translational velocity vector
tn vv , = velocity components perpendicular and parallel to the fin surface, at the center of ¼-chord
dw = downwash due to the fins zyx ,, = coordinates of a point in the body frame
fefs xx , = x-coordinates of the start and end positions of the fins
eF αα , = geometric and effective angles of attack of the fins ε = longitudinal distance from the nose δ = control surface deflection ρ = air density
[ ]T,, rqp=ω = angular velocity vector
[ ]T1 ,, III zyx=η = position vector written in the inertial frame
[ ]T2 ,, ψθφ=η = Euler angles η = efficiency factor for the cross-flow drag due to finite length
dη = correction factor for the flap effectiveness factor
fη = efficiency factor for the fin added-mass due to 3-D effect
AASCGI τττττ ,,,, = forces and moments from inertia, gravity, control, aerostatics and added mass, respectively τ = theoretical flap effectiveness factor
FΦ = angle from oxz plane to the fin surface plane
I. Introduction N the past few years, researchers have become increasingly interested in airships, because these aircraft can provide long endurance, high payload-to-weight ratio and low fuel consumption1-3. The resurgence of airships has
created a need for accurate dynamic models and simulation capabilities to analyze their flight behavior and to design their control systems.
A number of textbooks, such as Ref. 4, have derived the equations of motion for conventional aircraft, for which certain solid-fluid interaction forces can be neglected, such as buoyancy and those related to the inertia of the surrounding air. However, these forces become important for airships because their flight relies on a light lifting gas rather than aerodynamic lift forces. This distinguishing feature implies that their dynamic models must include a more complete formulation for the interaction forces between the body and the air. A number of airship models have been presented in the literature. Tischler et al. derived the nonlinear equations of motion and developed the simulation program HLASIM for the design of heavy lift airships5; Jex6 applied their model and frequency-domain fitting technique to predict the control responses of the Skyship-500 non-rigid airship. Amann7 followed the aerodynamics prediction method of Jones and DeLaurier8 and developed a dynamic simulation program to analyze the flight characteristics also for the Skyship-500. Cook et al. formulated the linearized equations of motion for the airship stability analysis9. Azinheira et al. investigated how to incorporate the wind effects into the nonlinear equations of motion of airships10. However, difficulties still arise in formulating the dynamic model. For example, in most of the above works, relatively little detail is given on the estimation of the solid-fluid interaction forces; and few works have investigated the influence of fins on the airship’s added mass and moment of inertia.
In this paper, a systematic approach is proposed to assemble the nonlinear equations of motion for the simulation of airship dynamics, with a particular focus on a complete formulation of the interaction forces between the airship and the air. For this purpose, the derivation begins from the equations of motion of a rigid-body vehicle moving in vacuum. Then the relevant solid-fluid interaction forces, both aerostatics and aerodynamics, are incorporated into the equations. The dynamic model is used to perform the flight simulation of the Skyship-500 non-rigid airship and the responses due to control inputs are verified by comparing to available experimental results.
I
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II. Equations of Motion for Airship in Vacuum The modeling begins from the simplest case of a rigid-body airship moving in vacuum. A local body-fixed frame
{ }oxyz and an inertial frame { }III ZYOX are established respectively, shown in Fig. 1. The positive direction of the ZI axis of the inertial frame is vertically downward. The position of the origin of body-fixed frame is described by a vector written in the inertial frame, [ ]T1 ,, III zyx=η ; the airship’s orientation is represented by the Euler (roll, pitch
and yaw) angles, [ ]T2 ,, ψθφ=η . For convenience, the equations of motion for a 6-DOF vehicle are usually derived in the body frame, which can
be located at an arbitrary position without loss of generality. For a body moving in vacuum, these equations have been derived in standard textbooks4,11 using the Newton-Euler approach, and can be summarized in vector form as
CGI τττVM ++=& (1)
where [ ]TTT ,ωvV = and [ ]T,, wvu=v , [ ]T,, rqp=ω denote the translational and angular velocity vectors expressed in the body frame respectively. M is the mass matrix of the body and can be written as
−= ×
××
JrrI
Mg
g
mmm 33 (2)
where m is the total mass, including the hull, gas, gondola, fins, ballonets, etc. J is the inertia tensor and gr is the position vector of the center of gravity (C.G.) from the origin o . Note that J and gr are both expressed in the body frame. The superscript × denotes taking the skew-symmetric matrix form of a vector (corresponding to a cross-product operation). That is, for a vector [ ]T321 ,, sss=s , the corresponding skew-symmetric matrix is
−−
−=×
00
0
12
13
23
ssss
sss
The right hand side of Eq. (1) consists of the external forces and moments. The subscripts I , G and C denote the terms from inertia, gravity and control respectively. The inertial force and moment is calculated as
−−+−
= ×××
×××
Jωωvωrωrωvω
τg
gI m
mm (3)
If the body frame is located at the C.G., 0r =g and the terms related to gr will be zero. The gravitational force and moment is obtained as
= ×
Gg
GG Fr
Fτ , where
−=
φθφθ
θ
coscossincos
sin
mgmg
mg
GF (4)
and g is the acceleration of gravity. If the body frame is located at the C.G., the gravitational moment can be ignored.
The control force and moment Cτ are due to the thruster, the deflection of control surfaces, and the inflation or deflation of ballonets, and are generated by the automatic control system or the pilot’s commands. Some of these forces are generated aerodynamically, such as the control surface deflection, and will be discussed in Section IV.
Hull Tail Fins
Gondola
z
ox
z
OXIZI
y
YI
1η
q
r
p
Figure 1. Body frame and inertial frame.
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III. Interaction Forces and Moments Between Airship and Air: Aerostatics We can now start to incorporate the interaction forces and moments between the vehicle and the air, which
include two components: aerostatics (also called buoyancy) and aerodynamics. The former is due to the static air pressure and is independent of the motion of the body while the latter is related to its motion.
If the position vector of the center of volume (C.V.) relative to the origin o is Vr , then the aerostatic force and moment expressed in the body frame is
= ×
ASV
ASAS Fr
Fτ , where
−−=
φθρφθρ
θρ
coscossincos
sin
B
B
B
AS
gVgV
gVF (5)
BV is the volume of the body, while ρ is the air density. If the body frame is established at the C.V., 0=Vr and the
aerostatic moment is zero. To incorporate the aerostatics into the equations of motion, ASτ is added to the right hand side of Eq. (1).
IV. Interaction Forces and Moments Between Airship and Air: Aerodynamics The aerodynamic forces are categorized into various terms based on different physical effects. Estimation
methods for each of these terms are now investigated.
A. Added-mass force and moment This subsection first reviews the complete expression for the added-mass force and moment, followed by
estimation methods for the added-mass matrix of airships.
1. Added-mass force and moment computation from the added-mass matrix The added-mass force and moment can be considered as the pressure-induced fluid-structure interaction terms
based on the potential flow assumption11. For a body completely submerged in an unbounded fluid, the added-mass terms can be derived by an energy approach in terms of Kirchhoff’s equations11-12 or alternatively by using Bernoulli’s equation to find the pressure distribution over the body13. Here we will not repeat the detailed derivation but just review the results. That is, if the 66× symmetric added-mass matrix is written as
=
2221
1211
MMMM
M A (6)
where 11M , 12M , 21M , 22M are 33× matrices, whose elements can be estimated using methods discussed in Section IV.A.2. The corresponding added-mass force and moment can be obtained in vector form as
( )
( ) ( )
++++
−
−=
××
×
ωMvMωωMvMvωMvMω
ωv
MMMM
τ22211211
1211
2221
1211
&
&A (7)
Thus, Aτ includes two terms: one related to the time rates of change of the linear and angular velocities, the other related to the coupling of the linear and angular velocities. To incorporate these terms into the dynamic model, the first term is written on the left hand side of Eq. (1) so that the mass matrix M is replaced by AMM + ; while the second term is added to the right hand side of Eq. (1).
2. Estimation for the added-mass matrix
Most prior works only include the added mass of the hull while neglecting that of the fins. However, the total added mass matrix of the airship should include the contributions of both the hull and the fins, i.e.,
AFAHA MMM += (8)
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In practice, a simple approach to obtain the added mass and moment of inertia of the hull, AHM , is to approximate the hull as an ellipsoid of revolution. If the body frame is located at the C.V., with the x-axis along the centerline and positively toward the nose and the z-axis positively downward, then all the off-diagonal terms in the added-mass matrix of the hull are zero, and we have11
where m′ is the mass of air displaced by the hull and I ′ is the moment of inertia of the displaced air. 1k , 2k and k′ are added-mass factors plotted in Fig. 2 as functions of the fineness ratio DL , where L is the length of the hull and D is its maximum diameter.
The added mass and moment of inertia due to the fins can be computed by integrating the 2-D added mass of the cross section over the fin region. For example, for the cross section with cruciform fins shown in Fig. 3, the 2-D transverse added mass in y and z directions can be computed using potential flow theory as14
( )
−+== 2
2222
3322 bRbRmm SS ππρ (10)
where R is the body cross-sectional radius and b is the fin semi span. The first term in Eq. (10) represents the contribution of the hull while the second term is due to the fins. Thus the fin contribution is
( )2
222
3322 bRbmm SFSF −
== ρπ (11)
Similarly, the added moment of inertia about the x-axis due to the fins is given as14
πρ 4
44442 bkmSF = (12)
where the factor 44k is plotted in Fig. 4 as a function of bR . The 2-D added mass of cross sections with other fin arrangements can be also found in Ref. 14.
The added mass and moment of inertia due to the fins is obtained by integrating Eqs. (11) and (12) over the fin region, i.e.,
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where fsx and fex are the x-coordinates of the start and end positions of the fins. An efficiency factor fη (shown in Fig. 5) is included to account for 3-D effects; it is calculated from the potential flow theory for the added mass of a thin plate15-16 and determined based on the aspect ratio. Then the added-mass matrix of the fins is obtained as
=
6626
5535
44
3533
2622
00000000000000000
0000000000
FF
FF
F
FF
FF
AF
mmmm
mmm
mm
M (14)
3. Added-mass matrix transformation between different frames In the estimation approach for the hull, the body frame is located at the C.V. and oriented along the hull’s
principal axes. In order to obtain alternative formulations of the equations of motion, with a body frame located at a different position or with different orientation, it is relevant to be able to transform the added-mass matrix between different body frames.
Consider two body frames { }oxyz and { }zyxo ′′′′ , with the position vector of o′ in the frame { }oxyz being r while the rotation matrix from frame { }zyxo ′′′′ to { }oxyz is denoted as R . If the added-mass matrix written in the frame { }oxyz is AM , then it can be written in { }zyxo ′′′′ using the following equation
UMUM AAT=′ (15)
where ( )
=
×
××
33
T33
I0rRI
R00R
U . Eq. (15) can be proven from the invariance of the kinetic energy of the fluid.
B. Viscous effect on the hull, normal to the centerline Wind tunnel tests on the aerodynamics of bodies of revolution at angles of attack have shown that a prediction
based on potential flow assumption can cause considerable error because of the effects of viscosity, especially in the rear part of the body. Ref. 17 and 18 give a semi-experimental estimation approach for the aerodynamics of bodies of revolution; in this approach, the force normal to the centerline due to viscous effects can be computed as
( ) ∫∫ ⋅+−⋅−=L
dc
L
NV RdCqdddSkkqF
00
2sin2sin 20120 εε
εαηεε
α (16)
where 0q is the dynamic pressure, α is the angle of attack defined as the angle between the centerline and the velocity vector (shown in Fig.6) as
+= −
uwv 22
1tanα (17)
dcC is the cross-flow drag coefficient of an infinite-length circular cylinder. η is an efficiency factor accounting for the finite length of the body and determined from the fineness ratio of the body, shown in Fig. 7. R and S are the local cross-sectional radius and area. ε denotes the longitudinal position from the nose and 0ε , the location at which the flow ceases to be potential, is empirically determined as18
u
v
w
x
y
z
α
NVFzVF
yVFo
Figure 6. Normal force decomposition.
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10 527.0378.0 εε += L
where 1ε denotes the position at which εddS has a maximum negative value. The first term in Eq. (16) effectively removes the inviscid flow contribution upstream of 0ε while the second term replaces it with a viscous flow contribution. NVF can be further decomposed into
22 wv
vFF NVyV+
−= , 22 wv
wFF NVzV+
−= (18)
The corresponding moment about the origin of body frame is computed as
( ) ( ) ( )∫∫ −⋅+−−⋅−=L
mdc
L
mV dRCqdddSkkqM
00
2sin2sin 20120 εε
εεεαηεεεε
α (19)
where mε is the position of the origin of body frame from the nose. The pitching and yawing moments are obtained from VM as
22 wv
wMM VyV+
= , 22 wv
vMM VzV+
−= (20)
Eqs. (18) and (20) are applied to compute the normal forces due to viscosity on the hull and the corresponding moments. These forces and moment are then added to the right hand side of Eq. (1).
C. Force acting on the fins, normal to the centerline We now turn our attention to the force produced by the fins, normal to the airship centerline. This is obtained by
estimating the force distribution and integrating this over the fin area. If we consider a point P on the fin planform (shown in Fig. 8), with longitudinal position x and spanwise
position s , the normal force per unit area at this point is predicted as
( ) ( )sxsxCqdxdsdF
epNF ,,0 αα∆= (21)
where αα ∂∆∂≡∆ pp CC , and pC∆ is the pressure coefficient of the airfoil; αpC∆ is determined by the local chordwise position and can be obtained from experiments or from CFD results of the pressure distribution of the airfoil. eα is the effective angle of attack computed as
Fl
Le s
RCC
ααα
α
+= 2
2
1 (22)
where α
α
l
L
CC
is a correction factor for 3-D effects and can be obtained from finite wing theories, such as those in Ref.
19. The factor
+ 2
2
1sR accounts for the influence of the hull on the fins20. Fα is the
geometric angle of attack computed from the local velocity at the center of the ¼-chord. Fig.9 shows a fin located in the plane inclined at an angle FΦ from the oxz plane. The velocity component in x direction at its center of the ¼-chord is Fu and the transverse velocity is decomposed into nv , perpendicular to the fin surface, and
tv , parallel to the surface. Then the geometric angle of attack is computed as
PAB
Rs
Exposed finarea
b
x
Figure 8. Fin planform.
Fineness ratio0 5 10 15 20
0.5
0.6
0.7
0.8
η
Figure 7. Efficiency factor η.
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F
nF u
v1tan −=α (23)
If Fα is beyond the angle stallα at which the stall occurs, we use stallα to calculate the effective angle of attack in Eq. (22). The value of stallα can be estimated by the methods in Ref. 17.
The total normal force on the fin is obtained by integrating the force distribution, from Eq. (21), over the exposed fin area, and then decomposed into yFF and zFF , i.e.,
( )∫ ∫
+∆= fe
fs
x
x
b
R pl
LNF dxds
sRsxC
CC
qF 2
2
0 1,αα
α α , FNFyF FF Φ= cos , FNFzF FF Φ−= sin (24)
The corresponding rolling, pitching and yawing moments on the fin are
( )∫ ∫
+∆−= fe
fs
x
x
b
R pFl
LxF dxds
sRxCs
CC
qM 2
2
0 1αα
α α , ( )∫ ∫
+∆Φ= fe
fs
x
x
b
R pFl
LFyF dxds
sRsxCx
CC
qM 2
2
0 1,sin αα
α α ,
( )∫ ∫
+∆Φ= fe
fs
x
x
b
R pFl
LFzF dxds
sRsxCx
CC
qM 2
2
0 1,cos αα
α α (25)
Eqs. (24) and (25) can be applied to compute the normal forces and corresponding moments on each fin, which are then added to the right hand side of Eq. (1).
D. Force acting on the hull due to the fins, normal to the centerline Based on results from wind-tunnel tests on the aerodynamics of airships, such as Akron21,22, it has been found
that the presence of the fins can lead to extra normal force on the hull, due to the fin-induced downwash over the airflow near the hull. The extra normal force per unit length on the hull can be obtained as23
dx
dwVR
dxdF dFNH 2)( πρ= (26)
where V is the air speed, dw is the local fin-induced downwash. The variation of the downwash along the centerline can be computed from the force distribution on the fins24, as given by Eq. (21), and we have
( ) ( )∫ ∫
+∆
+−= fe
fs
x
x F
b
R FpFl
LFd dxds
sRxC
xxddCC
Vxw 2
2
18
1)( αα
α
πα (27)
where ( ) 22 sxxd F +−= . Thus, Eq. (26) can be applied to calculate the fin-induced normal force on the hull and the corresponding pitching and yawing moments, which are then added to the right hand side of Eq. (1).
E. Axial drag The axial force is composed of two components, the contributions from the hull and the fins, respectively. At
low angles of attack, these forces can be obtained as
α200 cosHdHxH SCqF −= , α2
00 cosFdFxF SCqF −= (28)
y
z
FΦNFF
nvtv
FΦyFF
zFF
Figure 9. Normal force on a fin.
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where 0dHC and 0dFC are the zero-angle axial drag coefficients of the hull and the fins respectively and HS and FS are the corresponding reference areas. For example, these drag coefficients can be obtained in Ref. 19. To incorporate the axial drag, the forces from Eq. (28) are added to the right hand side of Eq. (1).
F. Force and moment due to control surface deflection The force and moment due to the control surface deflections also needs to be estimated. An estimation method
for the effects of flap deflection on the aerodynamics of a 2-D airfoil section is given in Ref. 19, and this is now extended to 3-D fins. That is, the lift coefficient from the deflection of control surface can be computed as
δητα ⋅⋅⋅⋅=∆ DdLL kCC 3 (29)
where δ is the deflection angle of the control surface. αLC is the 3-D lift curve slope. The theoretical effectiveness factor τ is derived from potential flow theory and plotted in Fig. 10 as a function of the ratio between the flap chord and airfoil chord, cc f . dη is a correction factor based on experiments and given as a function of δ for plain flap, shown in Fig. 11. Dk3 is an efficiency factor accounting for 3-D effects, which is a function of τ and aspect ratio17; for example, Fig. 12 shows Dk3 for the case where 5.0=τ .
The increment in the drag coefficient, DC∆ , due to the flap deflection is given approximately for plain flaps as
δ238.1
sin7.1
=∆
FA
ffD S
Sc
cC (30)
where fS and FAS are the flap and fin area respectively. The influence of flap deflection on the pitching moment coefficient can be estimated from thin airfoil theory, i.e, the ratio of 41MC∆ to LC∆ is written as19
( )ff
ff
L
M
CC
θθπθθ
sin82sinsin241
+−
−−=
∆
∆ (31)
where 41MC∆ is the pitch moment coefficient about the ¼-chord and ( )12cos 1 −= − cc ffθ . The control force due to the rudder and elevator deflection can be estimated using Eqs. (29)-(31), and these
forces and moments are then added to the right hand side of Eq. (1).
V. Numerical Simulation The numerical simulation results presented in this section include two parts. First, the added-mass terms and
steady-state aerodynamic force estimates are shown and compared to CFD calculation or wind tunnel test results. Next, the dynamic simulation results for the Skyship-500 airship are shown and the responses due to the control surface deflection are analyzed.
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A. Validation of the aerodynamics model
1. Added mass and moment of inertia of the Lotte airship CFD packages have been developed to compute the aerodynamics of 3-D bodies in potential flow and these have
been applied to the added mass calculation for airships. For example, in Ref. 25, a CFD package developed at the University of Stuttgart was applied to compute the added mass and moment of inertia for the Lotte airship, shown in Fig.13. This CFD method utilizes a distribution of source density on the body surface and solves the distribution necessary to meet the boundary conditions. These results can be used to evaluate the estimation method given in Section IV.A. The CFD results for the added mass and moment of inertia are listed in Tab. 1 and compared to our estimation results. We can see that the estimation method presented here can lead to a reasonable approximation for the added mass of the bare hull and the hull-fin combination. As well, we note that the fins can have a considerable effect on the added moment of inertia.
2. Steady-state aerodynamics of the Akron airship model
A great deal of wind tunnel experimental results on the steady-state aerodynamics can be found in the literature for older rigid airships. In this paper, the experimental results for the Akron airship model 21 (shown in Fig. 14) are used to test our aerodynamics computation. In these experiments, the aerodynamic coefficients were measured in a wind tunnel at various angles of attack. The measured normal force and pitching moment (about the C.V.) coefficients are compared to our estimation results in Figs. 15 and 16, for both bare hull and hull-fin combination. We can see that the estimates from Section IV can match the experimental results.
B. Dynamic simulation for the Skyship-500 airship A dynamic simulation program has been developed in the MATLAB environment to implement the non-linear
dynamic model discussed in Section II to IV. In the numerical simulation, the Skyship-500 non-rigid airship, shown in Fig. 17, is used as an example, because the dimensional and inertial parameters, and flight test data are available for this airship6,26. This subsection presents the simulation results for the responses due to elevator or rudder deflection and compares them to flight test results.
1. Responses due to elevator deflection
The dimensions, mass and moment of inertia used in the simulation are obtained from Refs. 6 and 26. The body frame is located at the C.V. of the hull. In order to predict the responses from the dynamic simulation program, some control force and moment are first applied in the simulation so that the airship begins its flight in a trim condition at a constant airspeed of 25 knots. Then the elevator deflection is applied.
In this example, the elevator deflection input (positive trailing edge downward) is shown in Fig. 18-a, which is obtained by discretizing the
x (m)
y (m
)
0 1 2 3 4 5
-2
-1
0
1
2
0 2 4 6 8 10 12
0
0.05
0.1
0.15
0.2Bare hull (estimated)Bare hull (experiment)Hull + fins (estimated)Hull + fins (experiment)
Angle of attack (deg)
Nor
mal
forc
e co
effic
ient
Angle of attack (deg)
Pitc
hing
mom
ent c
oeffi
cien
t
0 2 4 6 8 10 120
0.05
0.1
0.15
0.2
0.25 Bare hull (estimated)Bare hull (experiment)Hull + fins (estimated)Hull + fins (experiment)
Figure 14. Profile of Akron model. Figure 15. Normal force result. Figure 16. Pitch moment result.
Figure 13. Profile of Lotte. Table 1: Added-mass result.
0 10 20 30 40 50
-10
0
10
x (m)
y (m
)
Figure 17. Profile of Skyship-500.
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elevator input in the flight test, shown in Fig. 19-a. The time history of the pitch rate from the simulation is shown in Fig. 18-b. We can see that the simulated response is very close to the flight test data in Fig. 19-b.
2. Responses due to rudder deflection
The simulation setup for the responses due to rudder deflection is similar to the elevator case, that is, the control force and moment include the control input necessary for the trim condition at 25 knots, and the force and moment due to the rudder deflection. A proportional controller is applied to the thrust input so that the airship maintains a constant airspeed. The step-like rudder input (positive trailing edge left) and the response results of yaw rate are plotted in Fig. 20 and compared to the flight test data in Fig. 21. We can see that the predicted yaw rate is higher than the test data by about 20%, but generally speaking, the dynamic simulation program provides a reasonable match to the major trends in the flight test.
3. Yaw rate in steady turning
A steady rudder deflection leads to a steady turning flight for the airship. In this example, the yaw rates in steady turning are computed for various rudder deflection angles (from –30 to 30 deg) at 25 knots. The simulated results are plotted and compared to the flight test results in Fig. 22. We can see that the turning rate is a nonlinear function of the rudder deflection and this function can be well estimated by the dynamic simulation program.
Time (s) Time (s)
Rudd
er de
flectio
n (de
g)
r (deg
/s)
0 20 40 60 80-5
-2.5
0
2.5
5
0 20 40 60 80
-10
-5
0
5
(a) Rudder deflection. (b) Time history of yaw rate.
Figure 20. Rudder input and response results at 25 knots (in the simulation).
(a) Rudder deflection. (b) Time history of yaw rate.
Figure 21. Rudder input and response results at 25 knots (in the flight test6).
Time (s) Time (s)
Eleva
tor de
flecti
on (d
eg)
q (de
g/s)
0 10 20 30 40 50
-3.75-2.5
-1.250
1.252.5
3.75
0 10 20 30 40 50
-15-10-505
1015
(a) Elevator deflection. (b) Time history of pitch rate. Figure 18. Elevator input and response results at 25 knots (in the simulation).
(a) Elevator deflection. (b) Time history of pitch rate.
Figure 19. Elevator input and response results at 25 knots (in the flight test6).
American Institute of Aeronautics and Astronautics
12
VI Conclusions A systematic procedure for deriving the 6-DOF nonlinear dynamic
models for airships is presented. A complete formulation of the solid-fluid interaction forces is incorporated. Estimation methods are given for various aerodynamic effects, and then verified by comparing to CFD or wind tunnel experimental results. We can see that the fins have a significant impact on the added moment of inertia.
A simulation program is developed from the dynamic model, which can reasonably predict the transient response and flight behavior for the Skyship-500 airship, based on a comparison to the flight test data. The dynamic model and simulation program are effective tools in estimating the dynamic characteristics and designing the control systems for airships.
References 1Elfes A., Bueno S. S., Bergerman M. and Ramos J. G., A Semi-autonomous Robotics Airship for Environmental Monitoring
Missions, IEEE International Conference on Robotics and Automation, Leuven, Belgium, May 1998, pp. 3449-3455. 2Wilson J. R., A New Era for Airships, Aerospace America, Vol. 42, No. 5, May 2004, pp. 27-31. 3Hygounenc E., Jung I. Soueres P. and Lacroix S., The Autonomous Blimp Project of LAAS-CNRS: Achievements in Flight
Control and Terrain Mapping, The International Journal of Robotics Research, Vol. 23, No. 4-5, April-May 2004, pp. 473-511. 4Etkin B., Dynamics of Flight: Stability and Control, 3rd Edition, Wiley, New York, 1996, Chapter 4. 5Tischler M. B., Ringland R. R. and Jex H. R., Heavy Airship Dynamics, Journal of Aircraft, Vol. 20, No. 5, May 1983, pp.
425-433. 6Jex H. R. and Gelhausen P., Pre- and Post-Flight-Test Models Versus Measured Skyship-500 Control Responses, 7th AIAA
Lighter-Than-Air Technology Conference, Monterey, CA, August 17-19, 1987, pp. 87-97. 7Amann J. H., A Comparison of a Nonlinear Flight Dynamic Simulation of an Airship with Flight Test Results, 7th AIAA
Lighter-Than-Air Technology Conference, Monterey, CA, August 17-19, 1987, pp. 78-86. 8Jones S. P. and DeLaurier J. D., Aerodynamic Estimation Techniques for Aerostats and Airships, Journal of Aircraft, Vol.
20, No. 2, 1983, pp. 120-126. 9Cook M. V., Lipscombe J. M. and Goineau F., Analysis of the Stability Modes of the Non-Rigid Airship, The Aeronautical
Journal, Vol. 104, No. 1036, June 2000, pp. 279-290. 10Azinheira J. R., de Paiva E. C and Bueno S. S., Influence of Wind Speed on Airship Dynamics, Journal of Guidance,
Control and Dynamics, Vol. 25, No. 6, 2002, pp. 1116-1124. 11Fossen, T. I., Guidance and Control of Ocean Vehicles, John Wiley & Sons, New York, 1998, Chapter 2. 12Lamb H., Hydrodynamics, 6th Ed., Dover, New York, 1945, Chapter 6. 13Newman J. N., Marine Hydrodynamics, MIT press, Cambridge, MA, 1977, Chapter 4. 14Nielsen J. N., Missile Aerodynamics, AIAA, 1988, Chapter 10. 15Blevins R.D., Formulas for Natural Frequency and Mode Shape, Robert. E. Kieger Publishing Company, New York, 1979,
Chapter 14. 16Meyerhoff W. K., Added Masses of Thin Rectangular Plates Calculated from Potential Theory, Journal of Ship Research,
Vol. 14, No. 2, 1970, pp. 100-111. 17Fink, USAF Stability and Control DATCOM, 1978, Section 4.1, 4.2 and 6.1.4. 18Hopkins E. J., A Semi-empirical Method for Calculating the Pitching Moment of Bodies of Revolution at Low Mach
Numbers, NACA RM A51C14, 1951. 19McCormick B. W., Aerodynamics, Aeronautics and Flight Mechanics, John Wiley & Sons, 1995, Chapter 3,4. 20Pitts W. C., Nielsen J. N., and Kaattari G. E., Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic,
Transonic and Supersonic Speeds, NACA TR 1307, 1957. 21Freeman H. B., Force Measurements on a 1/40-Scale Model of the U.S. Airship Akron, NACA TR 432, 1932. 22Freeman H. B., Pressure Distribution Measurements on the Hull and Fins of a 1/40-Scale Model of the U.S. Airship Akron,
NACA TR 443, 1933. 23Lawrence H. R. and Flax A. H., Wing-Body Interference at Subsonic and Supersonic Speeds – Survey and New
Developments, Journal of the Aeronautical Sciences, Vol. 21, No. 5, 1954, pp. 289-324. 24von Karman Th., Burgers J. M., General Aerodynamic Theory – Perfect Fluids, Aerodynamic Theory, Durand W. F.,
Editor, Vol. II, Div. E, Durand Reprinting Committee, 1943, pp. 153. 25Lutz T., Fund P., Jakobi A. and Wagner S., Summary of Aerodynamic Studies on the Lotte Airship, Proceeding of the 4th
International Airship Convention and Exhibition, July 28-31, Cambridge, England, 2002. 26Jex H. R. and Gelhausen P. Control Response Measurements of the Skyship-500 Airship, 6th AIAA Lighter-Than-Air
Technology Conference, Norfolk, VA, June 26-28, 1985, pp. 130-141.
-30-1501530-10
-5
0
5
10
simulation resultsflight test results26
Rudder deflection (deg)
Yaw
rate
(deg
/s)
Figure 22. Yaw rate in steady turning(at 25 knots).
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