1 American Institute of Aeronautics and Astronautics
LONGITUDINAL EQUILIBRIUM SOLUTIONS FOR A TOWED AIRCRAFT AND TOW CABLE
Stephen R. Norris* and Dominick Andrisani II†
Purdue University West Lafayette, IN 47907-1282
ABSTRACT
A series of flight tests involving a modified F-106 fighter aircraft towed behind a C-141 cargo aircraft was completed at NASA’s Dryden Flight Research Center in February, 1998, under the title of the Eclipse Project.1 This paper presents a numerical approach used to compute tow cable shapes and altitude offsets in the vertical plane for the towed configuration as a function of trimmed flight condition. Each solution was generated in two distinct steps. First the trim condition of the towed aircraft was found using an existing aerodynamic database. Correction factors for this database were extracted from data measured during untowed steady-state test glides. The second step was numerical integration of the differential equations governing cable shape in the presence of distributed aerodynamic and gravitational loads. The aircraft trim condition from the first part of the solution provided the initial conditions needed for the second part. Computational results for nominal test conditions are compared with flight test measurements of aircraft altitude offset, angle of attack, pitch angle, cable pitch angle, and cable tension. Altitude offsets were measured using highly accurate differential GPS receivers. Families of computed cable shapes are presented as a function of elevator deflection and flight path angle. _______________________ *Systems Engineer, Dynetics Inc., 1000 Explorer Blvd., Huntsville,
AL 35806. Member AIAA. The results presented here are a
synopsis of master’s thesis research2 conducted while a student at
Purdue University and sponsored by a fellowship from the Indiana
Space Grant Consortium. †Associate Professor, Purdue University School of Aeronautics and
A Cable cross-sectional area c Airfoil chord length c Major axis of elliptical cable cross section
xc Cos(x)
C Coefficient d Diameter D Drag E Force per unit length F Force g Gravity G Gravitational force per unit length h Height above mean sea level J Jet engine thrust l Length of tow cable L Lift M Moment n Unit vector normal to cable element N Normal force q Dynamic pressure r Relative position vector s Cable arclength
xs Sin(x)
S Area t Minor axis of elliptical cable cross
section t Unit vector tangent to cable element T Tension V Velocity W Weight
zyx ˆ,ˆ,ˆ Unit vectors used to define right-hand Cartesian coordinate frames
α Angle of attack
eδ Elevator deflection angle, positive down l∆ Change in length of tow cable
ε Cable strain γ Flight path angle
cγ Cable parameter θ Pitch angle σ Cable stress
2 American Institute of Aeronautics and Astronautics
Subscripts
a Attach point of cable A Aerodynamic b Body reference frame c Cable cg Center of gravity D Drag
ld ⋅ Diameter times length for projected cylinder area
e Elliptical f Friction g Glider i Inertial reference frame M Moment N Normal
zyx ,, Vector components corresponding to a Cartesian coordinate frame
∞ Freestream p Pressure r Relative R Reference T Tension Superscripts
i Cable initial (unstressed) state 0 Cable reference state Acronyms
DFRC Dryden Flight Research Center DGPS Differentially-corrected Global
Positioning System GPS Global Positioning System ISGC Indiana Space Grant Consortium KST Kelly Space and Technology NASA National Aeronautics and Space
Administration
INTRODUCTION
The Eclipse Project was conducted by NASA DFRC in conjunction with KST (Kelly Space and Technology, Inc., 294 S. Leland Norton Way, Suite 3A, San Bernardino, CA 92408), in order to demonstrate the feasibility of towing a highly loaded delta wing aircraft behind a cargo-class aircraft. KST holds a patent entitled “Space launch vehicles configured as gliders and towed to altitude by conventional aircraft.” This project was executed to provide proof of that concept. An overview of the Eclipse project, including
sample flight test data and computational results, has already been published.1 Figure 1 shows the C-141A that was used as the cargo-class towing aircraft or “ towplane.” The QF-106 target drone shown in Figure 2 was modified for use as a piloted, towed aircraft and renamed the EXD-01. In this paper the towed aircraft will also be referred to as the “glider.” Figure 3 shows the EXD-01 during a towed flight test. Both aircraft were outfitted for the flight tests with instrumentation packages that included 12-channel carrier-phase DGPS receivers. The differentially-corrected GPS data provided highly accurate measurements of position offsets between the two aircraft during flight. In addition, the EXD-01 was outfitted with a load cell and an articulated cable attachment mechanism for measurement of the cable tension vector applied to the glider. The results collected by NASA DFRC during the Eclipse project compose a highly significant data set for towed air vehicle research. The accurate position measurements captured during this series of flight tests were extremely difficult to obtain prior to the recent availability of DGPS systems. This approach produced a very useful and unusually complete set of measurements. Results were collected for the EXD-01 in both “clean” and “dirty” configurations. In the clean configuration, the landing gear and speed brake were retracted, while both were extended in the dirty configuration. A computational approach is presented in this paper for finding the longitudinal equilibrium conditions of the glider and tow cable. Such solutions are important because they provide key flight parameters, including:
• Relative vehicle and cable positions during tow,
• Drag force that must be overcome by the towplane,
• Conditions of minimum drag force for optimum
tow performance, and
• Ability or inability of the glider to trim at a given
climb or descent rate. In addition, these equilibrium solutions can provide initial conditions and benchmark solutions for dynamic analyses of the tow cable and glider combination. The solution strategy was broken into two major parts. First, the two-dimensional tension vector required to trim the glider was determined as a function of elevator deflection. Second, the components of the tension vector at the glider attach point were used as the initial conditions for numerical integration of the differential equations governing cable shape.
3 American Institute of Aeronautics and Astronautics
Propagation of this solution over the entire length of the cable produced relative aircraft positions. The basic approach has been used in other applications,3,4 but the Eclipse flight test measurements provided a unique opportunity to assess its suitability and accuracy in the context of a unique configuration.
Figure 1. The C-141A towplane
Figure 2. The QF-106 in flight
Figure 3. EXD-01 on tow as seen from the C-141A towplane
4 American Institute of Aeronautics and Astronautics
THEORY
Aircraft Trim
In this paper the longitudinal trim problem is considered. The geometry of this problem is presented in Figure 4 as a free-body diagram. Both inertial and body-fixed coordinates are used. All reference frames are standard right-hand Cartesian systems. Three unknowns are needed as a function of elevator deflection: gzx ii
TT α and,, . These unknowns may be found via simultaneous solution of Equations 1, 2, and 3.
giiJcDcLsTF xx θγγ +−−==∑ 0 (1)
giiJsWDsLcTF zz θγγ −++−==∑ 0 (2)
∑ +== TAy MMMb
0 (3) In the force and moment equations lift ( L ), drag ( D ), and pitching moment ( M ) are all functions of elevator deflection angle ( eδ ) and glider angle of attack ( gα ). TM
�
, the moment due to cable tension, is derived from Equation 3 as follows: First, the relative position of the cable attach point, a , with respect to the glider center of gravity is expressed in body-fixed coordinates as shown in Equation 4.
babaacg zzxxr ˆˆ, +=
�
(4) Next, the moment about the glider center of gravity due to cable tension is computed using a vector cross product. The result is given in Equation 5.
( )
( ) bzaxa
xaza
T
acgT
yTxTzc
TxTzsM
TrM
iig
iig ˆ
,
−+
−−=
×=
θ
θ�
�
�
�
(5) NASA DFRC provided a FORTRAN-based simulation of the F-106 for use in this study. The simulation was modified to run the aerodynamic model repeatedly while sweeping angle of attack and elevator deflection. The outputs were used to generate a database of lift, drag, and moment coefficients for the nominal flight test conditions listed in Table 1. This approach produced tables of LC , DC , and MC as a function of α and eδ . These three-dimensional surfaces provided inputs for generation of the trim solution, and are shown in Figure 5. A numerical solution algorithm used these aerodynamic data as inputs to solve for gα ,
ixT , and izT as functions of
elevator deflection angle. The two tension components provided the initial conditions needed to solve for the equilibrium shape of the tow cable.
Table 1. Nominal flight test conditions
Variable Value h 10,000 ft V 371.45 ft/s q 121.11 lbf / ft
2 ρ 0.001756 slugs / ft3 γ 0.0o
Aerodynamic Offsets
During the course of this research, several factors indicated the need to modify the aerodynamic database of Figure 5. During flight tests, the engine on the EXD-01 was run at an idle throttle setting. This allowed the EXD-01 to disconnect and fly away from any dangerous tow situation that might develop. The aerodynamic database did not account for this flight-idle thrust, which was reported to be approximately 800 lbf at sea level.5 In addition, engineers at NASA DFRC suspected some inaccuracies in the aerodynamic data used by the FORTRAN simulation, in part because predicted and measured cable tensions initially showed significant disagreement. In response to these issues, constant offsets were computed for use as correction terms for the baseline aerodynamic data. These offsets were extracted from flight test data obtained by the instrumented EXD-01 while flying in an untowed configuration. After climbing past the nominal test altitude of 10,000 ft, the pilot trimmed the aircraft for a steady-state descent at the nominal test airspeed. Table 2 shows key parameters measured as the untowed EXD-01 passed through the target altitude. The two test glides were conducted consecutively, so there was minimal weight change due to fuel burn between the two events.
Table 2. Key parameters measured dur ing untowed test glides at nominal flight conditions
Variable Value for Clean
Aircraft Value for Dirty
Aircraft V 371 ft/s 371 ft/s h 10,000 ft 10,000 ft
W 28,577 lbf 28,577 lbf
eδ -4.0o -3.3o α 10.1o 10.5o γ -4.5o -12.5o
5 American Institute of Aeronautics and Astronautics
iz
WD D
bz
cθ
ix
x
T
bx
x
gθ
γ
α
L D
cgJ
D
a
γ
γV
Figure 4. Geometry of the longitudinal equilibr ium problem
−20−10
0
−100
1020
30−1
0
1
2
δe , degα, deg
CL
−20−10
0
−100
1020
300
0.5
1
δe , degα, deg
CD
−20−10
0
−100
1020
30
−0.2
−0.1
0
0.1
0.2
α, degδ
e , deg
CM
Figure 5. Aerodynamic database extracted from the F-106 FORTRAN simulation
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The data from these two test points were used as inputs to an algorithm that computed offsets in DC and
MC . Equations 1, 2, and 3 were again solved numerically. However, this time eδ and γ were fixed at the values measured during the untowed test glides, while α , M , and D were treated as unknowns. The tension vector was set to zero since the cable was not attached for these trimmed descents. Once the trim solution was obtained for the three unknown variables, constant offsets were computed for DC and MC using Equations 6 and 7. In these equations the subscript “experimental” indicates a value measured during flight test, “ table” indicates a value extracted from the aerodynamic database, and “computed” indicates a value resulting from the numeric solution of Equations 1, 2, and 3.
sq
DC
DDD
DDD
D∆=∆
−=∆
∆+=
tablecomputed
tablecomputed
(6)
In this case, ( )computedalexperiment,table , αδ efD = .
scq
MC
MMM
MMM
M∆=∆
−=∆
∆+=
tablecomputed
tablecomputed
(7)
Here ( )computedalexperiment,table , αδ efM = . For this approach the presence of thrust at flight idle was treated as a simple offset in the drag model. Thus the value for DC∆ from Equation 6 was used to compensate for both flight idle thrust and for any drag error inherent in the data tables of Figure 5. The resulting offsets are listed in Table 3. These correction terms were then added to the drag and moment coefficients in the aerodynamic data tables.
Table 3. Aerodynamic offsets
Configuration DC∆ MC∆
clean -0.0274 0.0040 dirty -0.0242 0.0134
Cable Differential Equations
Solution for the equilibrium cable shape was restricted to displacements in the longitudinal plane. Equilibrium cable shape is governed by a set of five
coupled first-order differential equations. The presentation of these equations is provided here in abbreviated form, but closely follows the development by Leonard and Nath, whose paper should be referenced for additional details.4 The appropriate differential equations were numerically integrated with respect to unstressed cable length to obtain equilibrium cable shapes that correspond to a particular trim solution for the glider. In this discussion of cable equations, the superscript i is used to specify an “ initial,” unstressed condition, while the superscript 0 refers to a pre-stressed “ reference” condition. The distinction between these two states is important for some dynamic analyses. However, for the static solutions provided here both initial and reference cable conditions are defined to be the unstressed cable state. The distinction in notation is maintained to preserve the generality of the development.
Cable Physical Properties Several key physical properties of the tow cable were determined experimentally through cyclical loading of sample tow cable material by researchers at NASA DFRC.1 The test results showed that cable tension could be described as quadratic with respect to changes in cable length, and linear with respect to the rate of change of the cable length. The form of this expression is shown in Equation 8.
( ) ( ) ( )LcLbLaT�
′+∆′+∆′= 2 (8) The experimentally determined numerical values for the tension coefficients are listed in Table 4. The rate-dependent term provides damping for tow cable dynamics, but does not affect equilibrium solutions.
Table 4. Cable tension coefficients
Coefficient Value Units
a ′ 1833 lbf / ft b′ 47.2 lbf / ft
2 c ′ 150 lbf / (ft/s)
Cable strain may be defined according to Equation 9.
∆=
iL
Lε
(9)
An expression for cable stress is obtained by dividing Equation 8 by iA , and then using the expression for
7 American Institute of Aeronautics and Astronautics
strain from Equation 9 in place of L∆ . These substitutions yield Equation 10.
( ) ( )ii
iii
LA
bL
A
a
A
T εε′
+′
=2
(10)
Terms in Equation 10 may be combined to yield an expression for cable stress, ii AT /=σ , as a quadratic function of the change in cable length, L∆ . Here the superscript i indicates that the stress term is computed using the cross-sectional area of the cable in its initial, unstressed condition.
εεσ bai += 2 (11) The coefficients a and b from Equation 11 may be computed using the empirically derived tension coefficients from Table 4 in combination with the cable parameters listed in Table 5. Cable strain may in turn be described as a function of cable stress by solving Equation 11 for ε .
( )a
abb i
2
42 σε
++−=
(12)
Equations Governing Cable Length
The differential equation governing cable arclength is a function of the cable parameter cγ .4
cds
ds γ210
+=
(13)
Here cγ is a function of cable strain ε .
01
121
εεγ
++=+ c
(14)
The reference condition is unstressed, so 00 =ε . Cable strain, ε , is available from Equation 12.
Equations Governing Cable Tension Cable tension varies with cable arclength as a function of the distributed loads acting on the cable. Figure 6 shows a free body diagram for a differential cable element of length ds . Summing the forces in the
ix direction and setting that sum to zero yields Equation 15.
( ) ( )( ) 0
cos
cos=
+−
++dsET
ddTT
xx
xx
φφφ
(15)
Expanding this expression via trigonometric identity and taking the limit as 0→ds and 0→xdφ produces Equation 16.
( ) dsEdT xx −=φcos (16) The left-hand side of Equation 16 may be further simplified by recognizing that in the limit as 0→ds , there is no cable length available for bending, so
( )xx dTdT φcos→ . Performing this substitution and dividing by the differential cable arclength in the cable reference state results in Equation 17.
−=
00 ds
dsE
ds
dTx
x
(17)
The same reasoning may be used to obtain the analogous equation for the iz -direction, as shown in Equation 18.
−=
00 ds
dsE
ds
dTz
z
(18)
dTT ++xdφφ +
dTT + ds
T
zE T +
xE
T +ix
T + iz
+
xφ
T + Figure 6. Geometry of a differential cable element
Equations Governing Cable Displacement
Two more differential equations are still needed in order to determine the shape of the cable at equilibrium. For the ix -direction, the next step is to take the derivative of position with respect to reference state arclength to get Equation 19.
00 ds
ds
ds
dx
ds
dx =
(19)
8 American Institute of Aeronautics and Astronautics
In the limit as 00 →ds , ( )xdsdx φcos/ → , or equivalently, TTdsdx x // → . These substitutions may be used to obtain Equation 20. The same approach yields Equation 21 when applied to the iz -direction.
00 ds
ds
T
T
ds
dx x=
(20)
00 ds
ds
T
T
ds
dz z=
(21)
Cable Equation Summary
The five coupled differential equations needed to produce in-plane equilibrium solutions for the tow cable have now been presented. Equation 13 defines the derivative of stressed cable arclength with respect to unstressed cable arclength. Equations 17 and 18 provide the derivatives of tension with respect to unstressed cable arclength. Equations 20 and 21 give the derivatives of position, again with respect to unstressed cable arclength. Further details may be found in the paper by Leonard and Nath.4 The solution for equilibrium cable shape and tension was generated by numerical integration of all five of these equations over the unstressed cable length, which was known from preflight measurements. Initial conditions for the integration of these differential equations were obtained from the solution to the aircraft trim problem which was described previously.
Distributed Cable Forces
It is clear from both physical intuition and from Equations 17 and 18 that distributed forces must be considered in order to estimate equilibrium cable shapes. For the towed aircraft problem, there are two primary sources of these distributed forces: gravity and aerodynamic drag. Figure 7 shows the coordinate system used to define the orientation of the distributed force vectors acting on a differential cable element. A local angle of attack and pitch attitude angle are associated with each differential cable element. Table 5 lists physical attributes of the tow cable used during flight test.
n
cαcθ
ix
p
t
izcα
rV�
Figure 7. Cable coordinate system
Gravitational Loads
An expression for the distributed gravitational load acting on a differential cable element may be developed by recognizing that the element’s weight is the same before and after stretching, as expressed in Equation 22.
iiicc dsgAdsgA ρρ =
= weightdunstretche weight Stretched
(22) By rearranging Equation 22 and then substituting from Equation 13, Equation 14, and dsgAG cz ρ= , we arrive at an expression for the gravitational force per unit length of stretched cable.
( )ερ
+=
1
gAG
iic
z
(23)
Table 5. Cable parameters
Parameter Value iA 0.0031 ft2 (assumed circular)
d 0.75 in icl 1050 ft
cW 205 lbf icρ 1.98 slugs/ft3
Aerodynamic Loads
Aerodynamic forces were assumed to act along the entire length of the cable. Downwash from the towing aircraft was shown to have only minor effects on computational results2, and was ignored for the analysis presented in this paper. Estimations of aerodynamic loading on cable elements have been made by numerous researchers by treating each element as a cylinder inclined at some
9 American Institute of Aeronautics and Astronautics
angle of attack to the relative freestream flow.6,7,8 For subcritical (i.e. non-turbulent) flow, the “crossflow” principle is known to be a good assumption.7 This approach assumes that the normal aerodynamic force exerted on an inclined cylinder, and thus on a differential cable element, is dependent solely on the component of the freestream velocity perpendicular to the cylinder axis. However, the tow cable used in this experiment had a rough surface, and the normal Reynolds numbers were relatively high for large cable angles of attack. For these reasons the flow around the cable was assumed to be turbulent. Bootle has presented a method for estimating the aerodynamic forces acting on a cylinder at supercritical Reynolds numbers (i.e. after transition to turbulent flow).9 Bootle’s approach takes a horizontal cross-section of the cylinder parallel to the freestream flow and treats it as an elliptical section. The semi-empirical relationship of Equation 24 is then employed to compute the elliptical section drag coefficient.
( )2/120 ctCCturbulente fD = (24)
The ratio ( )ct / is the length of the elliptic section minor axis divided by the major axis, and is equal to
( )cαsin . eDC represents only the streamwise
component of a larger normal pressure force. eS , the reference area used to non-dimensionalize the streamwise pressure drag term, is a function of the local angle of attack. The elliptical section drag coefficient is related to the normal pressure drag coefficient by Equation 25.
( )[ ] ( )
( )[ ] ( )
pe
pe
pe
ND
cldNcldD
cldNeD
cppe
CC
SqCSqC
SqCSqC
ND
=
=
=
=
⋅∞⋅∞
⋅∞∞
αα
α
α
sinsin
sin
sin,
(25)
Equation 25 shows that the elliptical section drag force coefficient may be used in place of the normal pressure drag coefficient if the correct reference areas are used. Equation 26 expresses the coefficient of normal pressure force per unit length of cylinder as a function of the elliptical section drag coefficient.
( )( ) dqClN
dqClN
SqCN
pe
p
p
Dp
Np
dlNp
∞
∞
∞
=
=
=
,/
/ (26)
The direction of action for the normal aerodynamic force imposed on a given differential cable element is determined by using a vector cross product. Since the relative velocity vector and the unit tangent vector are constrained to lie within the longitudinal plane, their cross product can be used to define a unit vector perpendicular to the longitudinal plane. This operation is detailed in Equation 27, and the corresponding geometry is presented in Figures 7 and 8.
( )
( )cr
r
crr
tV
tVp
ptVtV
α
α
sinˆ
ˆˆ
ˆsinˆˆ
�
�
��
×=
=×
(27) One more vector cross product is required to locate the unit normal vector, as shown in Equation 28.
ptn ˆˆˆ ×= (28) The n unit vector defines the direction of action for the normal aerodynamic force acting on a differential cable element. Bootle computes skin friction drag by considering cylindrical cross sections taken perpendicular to the lengthwise centerline. Equation 29 shows the simple expression for frictional force per unit length that results from this approach. fC was set to 0.008 based on Bootle’s suggestion for stranded cables with a Reynolds number greater than 100,000.9
tdqClF
tldqCF
ff
ff
ˆ/
ˆ
π
π
∞
∞
=
=�
�
(29)
Bootle’s model was selected over other approaches due to the physical reasoning concerning turbulent flow around the tow cable. However, other methods were also tested and found to give similar results for this tow configuration and flight condition.2
10 American Institute of Aeronautics and Astronautics
d pN
peD ,
l
cα
( )cl αsin
rV
cα
Figure 8. Geometry for cable aerodynamic forces
RESULTS
Comparisons with Flight Test Results
Figures 9 and 10 shows comparisons between computed and experimental results for five key parameters:
• z∆ , altitude offset between glider and towplane • gα , glider angle of attack
• gθ , glider pitch attitude angle
• cθ , cable pitch attitude angle
• T , cable tension at the glider attach point The flight numbers referenced in the legend of Figures 9 and 10 designate the particular test flights during which the plotted data points were collected. The computed altitude offset, z∆ , is directly dependent on the entire algorithm used to solve for cable shape. The accuracy of this parameter gives an indication of the overall quality of the computational approach. The difference in iz -position at the two ends of the cable defines the altitude offset. Negative values indicate that the glider is below the towplane. The computational results for z∆ are presented with and without the effects of distributed loads on the cable. The results for the case without distributed loads correspond to straight-line solutions for the cable shape. As expected, the inclusion of the distributed loads improves the agreement between experimental and computational results. This effect is most evident in the results for the dirty configuration in Figure 10. These computational results assume steady, level flight, and thus a flight path angle of zero. Replacing this assumption with the measured flight path angle for each test point improves the agreement between computational and experimental results, but complicates the presentation of the data.2
The results for cable tension show a strong dependence on elevator deflection angle. A minimum value for cable tension is evident in for both the clean and dirty cases. As expected, the magnitude of the tension vector is significantly larger for the dirty configuration than for the clean configuration. Since the towplane must overcome tension forces acting to oppose its forward motion, the presence of a tension minimum indicates that the operational efficiency of the towed configuration is a function of the glider’s elevator deflection angle.
Families of Cable Shapes as a Function of Flight Path Angle
Families of cables shapes were generated for the clean configuration by repeatedly executing the overall solution algorithm while sweeping the input value of γ , the flight path angle. Figure 11 shows the results from this parametric study when performed with a flight path angle of zero. These results correspond directly to the experimental data presented previously in Figure 9. The large aircraft symbol represents the towplane, while the small aircraft symbol represents the glider. Each line represents the side view of a cable shape generated for a different elevator deflection. The curvature of the cable shape becomes more pronounced as the altitude offset between the two aircraft increases. Additional computations were made to investigate cable shapes for descending flight conditions. Figures 12, 13, and 14 show additional families of equilibrium cable shapes for flight path angles of –5o, -6o, and –7o, respectively. Other flight parameters were kept at the nominal values listed in Table 1. These families of cable shapes show several interesting features. Cable curvature generally becomes more pronounced as the flight path angle becomes more negative. For some flight conditions, cable shapes appear that curve aft (downstream) of the attach point at the towed aircraft. In such cases the specified trim condition requires a downstream component of the cable tension vector to restrain the towed aircraft and prevent its acceleration to higher speed. Figure 14 shows a relatively extreme example of this, where one of the equilibrium conditions places the glider forward of the towplane. For a given flight path angle, the sensitivity of the cable shape to elevator deflection is a strong function of the trim condition. The presence of several regions of high sensitivity suggest that vehicle dynamics may be undesirable during transitions from a “high tow” to a “ low tow” position during descending, towed flight.
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−8 −6 −4 −2 0−800
−600
−400
−200
0
200∆
z, ft
Altitude offset vs. elevator angle
−8 −6 −4 −2 09
9.5
10
10.5
11
11.5
12
12.5
13
α g , de
g
Glider angle of attack vs. elevator angle
−8 −6 −4 −2 09
9.5
10
10.5
11
11.5
12
12.5
13
θ g , de
g
Glider pitch angle vs. elevator angle
−8 −6 −4 −2 0−10
0
10
20
30
40
δe , deg
θ c , de
g
Cable pitch angle at glider vs. elevator angle
−8 −6 −4 −2 02000
2500
3000
3500
4000
4500
δe , deg
T, l
bf
Cable tension at glider vs. elevator angle
Flight 7
Flight 8
distributed loads on
distributed loads off
Figure 9. Compar ison of computational and flight test results for the clean configuration
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−15 −10 −5 0 5−600
−500
−400
−300
−200
−100
0
100∆
z, ft
Altitude offset vs. elevator angle
−15 −10 −5 0 57
8
9
10
11
12
13
14
15
α g , de
g
Glider angle of attack vs. elevator angle
−15 −10 −5 0 57
8
9
10
11
12
13
14
15
θ g , de
g
Glider pitch angle vs. elevator angle
−15 −10 −5 0 5−5
0
5
10
15
20
25
30
35
δe , deg
θ c , de
g
Cable pitch angle at glider vs. elevator angle
−15 −10 −5 0 56000
6500
7000
7500
8000
8500
9000
δe , deg
T, l
bf
Cable tension at glider vs. elevator angle
Flight 7
Flight 8
distributed loads on
distributed loads off
Figure 10. Compar ison of computational and flight test results for the dir ty configuration
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0 200 400 600 800 1000 1200
−200
0
200
400
600
Cable x vs. −z for a range of δe values, rho = 0.0017556 slugs/ft3, V
t = 371.45 ft/s, γ = 0 deg
x, ft
−z,
ftδ
e = +8°
δe = −24°
δe = 8.0
δe = 6.0
δe = 4.0
δe = 2.0
δe = 0.0
δe = −2.0
δe = −4.0
δe = −6.0
δe = −8.0
δe = −10.0
δe = −12.0
δe = −14.0
δe = −16.0
δe = −18.0
δe = −20.0
δe = −22.0
δe = −24.0
Figure 11. Cable shape as a function of elevator deflection angle for 0o flight path angle
−200 0 200 400 600 800 1000 1200 1400 1600
−400
−200
0
200
400
600
800
Cable x vs. −z for a range of δe values, rho = 0.0017556 slugs/ft3, V
t = 371.45 ft/s, γ = −5 deg
x, ft
−z,
ft
δe = +8°
δe = −24°
δe = 8.0
δe = 6.0
δe = 4.0
δe = 2.0
δe = 0.0
δe = −2.0
δe = −4.0
δe = −6.0
δe = −8.0
δe = −10.0
δe = −12.0
δe = −14.0
δe = −16.0
δe = −18.0
δe = −20.0
δe = −22.0
δe = −24.0
Figure 12. Cable shape as a function of elevator deflection angle for -5o flight path angle
14 American Institute of Aeronautics and Astronautics
−400 −200 0 200 400 600 800 1000 1200 1400
−400
−200
0
200
400
600
800
Cable x vs. −z for a range of δe values, rho = 0.0017556 slugs/ft3, V
t = 371.45 ft/s, γ = −6 deg
x, ft
−z,
ft
δe = +8°
δe = −24°
δe = 8.0
δe = 6.0
δe = 4.0
δe = 2.0
δe = 0.0
δe = −2.0
δe = −4.0
δe = −6.0
δe = −8.0
δe = −10.0
δe = −12.0
δe = −14.0
δe = −16.0
δe = −18.0
δe = −20.0
δe = −22.0
δe = −24.0
Figure 13. Cable shape as a function of elevator deflection angle for -6o flight path angle
−600 −400 −200 0 200 400 600 800 1000 1200 1400
−600
−400
−200
0
200
400
600
800
Cable x vs. −z for a range of δe values, rho = 0.0017556 slugs/ft3, V
t = 371.45 ft/s, γ = −7 deg
x, ft
−z,
ft
δe = +8°
δe = −24°
δe = 8.0
δe = 6.0
δe = 4.0
δe = 2.0
δe = 0.0
δe = −2.0
δe = −4.0
δe = −6.0
δe = −8.0
δe = −10.0
δe = −12.0
δe = −14.0
δe = −16.0
δe = −18.0
δe = −20.0
δe = −22.0
δe = −24.0
Figure 14. Cable shape as a function of elevator deflection angle for -7o fight path angle
15 American Institute of Aeronautics and Astronautics
CONCLUSIONS
The equilibrium conditions computed for an aircraft and tow cable combination showed excellent agreement with parameters measured during flight tests. Altitude offset, glider angle of attack, glider pitch angle, cable pitch angle, and cable tension at the glider attach point all showed good agreement with measured data when examined as a function of elevator deflection angle. The computational approach presented in this paper effectively generated longitudinal equilibrium solutions for the towed aircraft and tow cable combination at the flight condition of interest. Families of cable shapes were generated as a function of flight path angle using the same computational techniques. The results revealed that equilibrium cable shape has a strong dependency on the glider’s flight path angle and elevator deflection angle. At some descending flight path angles and ranges of elevator deflection, the equilibrium cable shape is highly sensitive to changes in elevator deflection. Some trim conditions actually place the glider ahead of the towplane.
Solutions for a Towed Aircraft, Master’s Thesis, Purdue University, August, 1998.
3Gilbert, N.E., The Shape of a Cable Used for Towing a Pitot-Static Probe from a Helicopter, Aerodynamics Note 397, Aeronautical Research Labs, Melbourne, Australia, June 1980. NTIS index AD-A095017.
4Leonard, John W., and John H. Nath, Comparison of Finite Element and Lumped Parameter Methods for Oceanic Cables, Engineering Structures, 3:153-167, July, 1981.
5Boutwell, R. F., Performance Data Substantiation Report for F-106B Flight Manual T. O. 1F-106B-1(Appendix I), dated 22 April 1959, and F106B Standard Aircraft Characteristics Chart, dated 1 May 1959, Technical Report ZA-8-513, Convair, 1 August 1959. NASA index N96-72238.
6DeLaurier, James D., A First Order Theory for Predicting the Stability of Cable Towed and Tethered Bodies where the Cable has a General Curvature and Tension Variation, Technical Note 68, von Karmaan Institute for Fluid Dynamics, Rhode-Saint-Genese, Belgium, December, 1970.
