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American Institute of Aeronautics and Astronautics

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Stability Analysis for UAVs with a Variable Aspect Ratio Wing

Janisa J. Henry*, Julie E. Blondeau† and Darryll J. Pines‡ University of Maryland, College Park, MD, 20740

Morphing technologies are a major thrust in current unmanned aerial vehicle (UAV) research. A morphing wing is one that can change its geometry to accommodate multiple flight regimes. Wing morphing methods include the following: camber change, aspect ratio change, twist change and wing sweep change. This paper explores the effect of changes in aspect ratio on the stability properties of UAVs. Stability coefficients are calculated in order to verify that variable aspect ratio (VAR) wings maintain stability during symmetric changes in span. Wind tunnel test data is used to validate models of the variation of roll and yaw moment with symmetric changes in aspect ratio. This paper presents a preliminary study of the stick fixed free response of a generic VAR UAV with symmetric changes in span.

Nomenclature b = wing span c = mean chord CL = lift coefficient Cl = roll moment coefficient Cm = pitch moment coefficient Cn = yaw moment coefficient

LC α = lift curve slope

nC β = yaw moment derivative due to sideslip

nrC = yaw moment derivative due to yaw rate

lC β = roll moment derivative due to sideslip

lpC = roll moment derivative due to roll rate pw = port wing sw = starboard wing S = wing surface area Lt, Lv = distance from cg to aerodynamic center of horizontal/vertical tail St, Sv = horizontal/vertical tail area VH = tail volume ratio Vtrim = trim velocity

* Graduate Student, Rotorcraft Center, [email protected], Student Member of AIAA. † Graduate Student, Rotorcraft Center, [email protected] ‡ Professor, Aerospace Engineering Department, [email protected], Senior Member of AIAA.

46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference18 - 21 April 2005, Austin, Texas

AIAA 2005-2044

Copyright © 2005 by Janisa Jernard Henry. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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I. Introduction Morphing technologies comprise a major thrust in current UAV research. A morphing wing is one that can change its geometry to accommodate multiple flight regimes. Wing morphing methods include the following: camber change2,3, variable aspect ratio4, twist change and wing sweep change. The F-14, F-111 and B-1 are examples of aircraft with swept wings1. The focus of this research is the application of VAR wings to UAVs. A VAR telescoping morphing wing is one that can change its wingspan in order to adapt its aerodynamic characteristics to specific mission requirements. For example, wings with large aspect ratios have improved range and fuel efficiency, but lack maneuverability and have relatively low cruise speeds. To the contrary, aircraft with short stubby wings are faster and more maneuverable but show poor aerodynamic efficiency. A variable aspect ratio wing can potentially integrate in a single aircraft the advantages of both designs, making this emerging technology especially attractive for military UAVs. In addition to performance improvements, VAR wings can be used to provide roll control. In a conventional wing configuration, roll control is achieved by the use of ailerons that are deflected such that lift is increased on one wing and decreased on the other, thus creating a roll moment. In the telescoping wing currently under development at the University of Maryland, the use of ailerons becomes problematic due to the overlapping of skins and limited internal wing space. Roll control can be achieved without ailerons when a span difference exists between the two half wings causing a commensurate differential in lift, so generating a rolling moment. This control scheme eliminates the need for additional actuators while minimizing drag since no large span-wise wing discontinuities are present. However, this scheme suffers from adverse yaw due to asymmetric drag distribution in asymmetric configurations The objective of this investigation is to look at experimental data for roll and yaw moment coefficients and to analyze the stability of the symmetric wing configurations as span is varied. This is the first step toward studying how changing span inputs affect the controllability of a generic UAV with a VAR.

II. Wing Tunnel Model Development In this section a description of the wing and its operation are presented. The final wing-body model developed

for wind tunnel testing is also described.

A. VAR Wing Design and Operation The pneumatic telescoping wing under consideration in this paper has numerous advantages over conventional

wing technologies, including weight, compactness, compliance tailoring and minimal moving parts4. Its components are listed below.

1. A telescopic pneumatic spar and its extension/retraction control mechanism 2. Length proximity sensors 3. Ribs fixed at the end of each section of the pneumatic telescopic spar 4. Wing skins that deploy and retract 5. A pressurized air source with associated valving

The Actuator The actuator consists of two pneumatic telescoping spars, developed by Ergo-Help, that are mechanically coupled by a rib at the tip of each moving section as shown in figure 1. The spars cannot act independently of each other as they are connected to the same pressure line. Figure 2 shows the details of the extension/retraction mechanism. An input of pressure at location (1) results in the motion of the middle (2) and tip element (3). Air is exhausted via orifice (4). When the spar is in rest mode, the input and output to the orifices are closed by solenoid valves. In the retraction mode, pressure is fed to location (5) initiating the retraction of the middle element due to the force acting on area (6). The tip element remains fixed relative to the middle element until the latter has completely retracted. Subsequent to the complete retraction of the middle element, the orifice at its tip allows for the tip element retraction due to the pressure input at (8).

Length Sensors A rack and pinion mounted on a limited-number-of-turns potentiometer was chosen as the continuous sensing device. The limited-number-of-turns potentiometer was used to ensure a one-to-one correspondence between voltage and displacement. Each moving rib section was mounted with a rack; the rack on the moving tip element was carefully aligned with the pinion on the lower rib as shown in figure 3. This allowed the displacement of each


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moving section with respect to the fixed lower section to be sensed. The outputs from the two potentiometers were collected and transformed into a sum of displacements by a control program.

Figure 1. Pneumatic Actuators Figure 2. Pneumatic Actuator Functional Design

Skins The skin consists of three telescoping hollow fiberglass shells of 0.045” thickness. The use of a telescopic skin

allows for several rigid sections to support aerodynamic loads while the wing is in any configuration. Each rib was mounted at the tip of each moving section of the spar. The skins were then glued to the corresponding rib using epoxy.

Figure 3. Potentiometer Mounted on Middle Element

B. Wing/Fuselage integration

A simple wing-body model was chosen for the wind tunnel testing. The fuselage is a hollow PVC cylinder, 71” in length, 10.5” diameter. Each telescoping half wing was mounted mid-wing to the fuselage such that the wing zero-lift line was coincident with the fuselage’s x-principal axis.

III. Wind Tunnel Testing Wind tunnel testing was conducted on the VAR wing-body (WB) model in the Glenn L. Martin Wind Tunnel at

the University of Maryland. The major purpose of these tests was to quantify the effect of differential span on the roll and yaw moment coefficients. In addition, data for symmetric configurations with various aspect ratios was collected. This data was used for stability analysis in each symmetric case. Thirty tests were conducted, with more than half of them devoted to asymmetric span configurations. The changing span is the driving factor in the production of aerodynamic roll moment for the VAR wing. When the span configuration is symmetric, the roll and yaw moments generated by each half wing are equal in magnitude and opposite in sense, resulting in zero roll and yaw moments. At any given angle of attack, the total lift and total drag on each half wing differ as the span is varied asymmetrically. The resulting differential lift and differential drag give rise to a roll and yaw moment respectively. It is the roll moment that may be used to control the vehicle during certain maneuvers like coordinated steady turns. In this section, wind tunnel data on roll and yaw moment for asymmetric span is presented and discussed. In the following sections, the data from the symmetric cases is used for a preliminary stability analysis.

PotentiometerPinion


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A. Wind Tunnel Setup and Test Matrix The wing body model was tested at free stream velocities of 20 mph, 25 mph and 30 mph at angles of attack

from –2o to 24o. Ten different combinations of span were tested. The test matrix is displayed below. Table 1: Test Matrix of Angles of Attack for given Airspeed and Wingspan Configuration

Test Speed (mph) AOA (degrees) Left Wing% span Right Wing %span1 20- 30 0 to 24 40 40 2 20- 30 0 to 24 60 60 3 20- 30 0 to 24 80 80 4 20- 30 0 to 24 100 100 5 20- 30 0 to 24 40 60 6 20- 30 0 to 24 40 80 7 20- 30 0 to 24 40 100 8 20- 30 0 to 24 60 80 9 20- 30 0 to 24 60 100 10 20- 30 0 to 24 80 100

Figure 4. Wind Tunnel Tests of Wing Body Model in (a) Retracted and (b) Extended

Configurations

B. Theoretical Aerodynamic Model Finite wing theory is used to predict the roll moment coefficient for the asymmetric span configurations. The lift coefficient slope of a rectangular wing is given as5:

αaCL = (1a)

where

)/(1 0

0

ARaaa

π+= (1b)

The roll moment coefficient generated by differential span is thus given by:

pw swCl Cl Cl= + (2a)

where 2

LC bClc

= (2b)

(a) (b)


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C. Aerodynamic Results and Comparison

The figure below shows how the roll moment varies linearly with percentage span increase from a symmetric reference condition. At the symmetric reference conditions, the total span is

0b . These conditions represent three (40-40, 60-60 and 80-80) of the symmetric cases identified in the test matrix. The 100-100 case was not included as only span increases from a symmetric reference condition were considered. From this reference condition, the span of the starboard half wing was varied while the port wing was held fixed. As the total span,

0b of the symmetric reference condition increases, the available roll moment increases as well. Recent work demonstrates that prior to stall, at any given angle of attack, a VAR morphing wing generates more roll moment than conventional ailerons9. Experimental data show good agreement with theory. The solid line gives the theoretical prediction. The plots shown below are at angle of attack of 6°. The trends illustrated here are representative of coefficients at other pre-stall angles of attack.

Figure 6. Roll Coefficient due to Asymmetric Span

for 6 deg Angle of Attack Figure 7. Experimental Yaw Coefficient due to

Asymmetric Span for 6 deg Angle of Attack

The VAR wing suffers from large amounts of adverse yaw. This is the result of the asymmetric drag produced

as asymmetric lift is generated. This drag is primarily due to profile drag. Adverse yaw can hinder the use of asymmetric span as a means of rolling to initiate a coordinated turn as the sense of the adverse yaw is opposite the rolling moment. Figure 7 above shows the experimental data for the magnitude of the adverse yaw corresponding to the same symmetric reference conditions shown in figure 6 above. The experimental yaw coefficient is always significantly less than the rolling moment coefficient.

IV. Correction of Wind Tunnel Data for Static Stability The static longitudinal and directional instability inherent in the WB model necessitated the addition of

horizontal and vertical tail data to the WB wind tunnel data prior to dynamic stability analyses. Static longitudinal criteria for stability are satisfied if the zero-lift pitching moment coefficient, 0 0MC > and slope of moment curve

0MC α < . Total pitching moment about the cg is given by6:

( ) ( ), , 01wb wb

tM cg M ac a ac H H t t

aC C a h h V V a i

a aεα ε ∂ � �= + − − − + + � �∂ �

(3)


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This equation clearly demonstrates the strong dependence of 0MC and MC α on the tail incidence angle, ti and

the tail volume ratio VH . The downwash angle at zero-lift, 0ε and εα

∂∂

are difficult to predict theoretically and

were assumed to be negligible. The plots below are representative wing-body (WB) and wing-body-tail (WBT) pitching coefficient curves.

-0.010

0.010.020.030.040.05

-10 0 10 20 30

angle of attack, deg

CM

40-40

-3-2-10123

-10 0 10 20 30


CM

40-40Vtrim=120mph

Figure 8. Pitching coefficient curve of WB Figure 9. Pitching coefficient curve of WBT

Directional stability, most of which is provided by the vertical tail, requires that 0nC β > . An estimate of this

parameter is given by7:

( )1n t v vC V aβ βη ε= − (4)

For preliminary estimates, the sidewash factor

βε was taken to be zero. The dynamic ratio tη was taken as 1 since no there was no flow interference due to a propellor slipstream. Data on the horizontal and vertical tail are given below.

• Horizontal tail: NACA 0012, c=12”, b= 4.2 ft, tl = 38”, ti =10 deg,

• Vertical tail: NACA 0012, c=12”, b=1.68 ft, tl = 38”

V. Stability Analysis of Free Response of Symmetric Model Symmetric changes in span can be used to optimize UAV performance for a specific mission segment. It is

important to ensure that the vehicle remains stable in all symmetric configurations. This section discusses the changes in stability for symmetric cases as span is varied. The linearized, decoupled equations that describe an aircraft’s lateral-directional motion are shown below. These equations describe the motion of the aircraft out of the plane of symmetry, specifically, the translational velocities along the y body axis and roll and yaw motions. In state space form, the equations are given as6:

x Ax Bu= +� (5)

where px

r

β

φ

∆ ∆ = ∆ ∆

,

0

0 0 0 0

cos1

00

0 1 0 0

p r

p r

p r

YY gYu u u u

A L L LN N N

β

β

β

θ − −

=

, 0

0

0 0

r

b r

b r

Yu

L LBN N

δ

δ δ

δ δ

=

and b

r

uδδ

∆ = ∆


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An examination of the eigenvalues of the A matrix for each of four symmetric span configurations will yield information on the stability of the lateral modes. The elements in the A matrix are functions of the UAV’s inertia characteristics and its dimensionless stability derivatives. Calculation of these entries requires a determination of the changing xI and zI principal moments of inertia with changing wingspan. This was accomplished by subjecting the wing to a torsional pendulum test (see figure 8). In order to achieve purely rotational motion in this experiment, it was essential that the axis of the torsional rod pass through the center of gravity location as it migrated with the wing’s changing span. Use of a digital inclinometer ensured careful tracking of the migrating center of gravity location. Inertial products vanish because of vehicle symmetry. The inertias were then determined from the following formula.

2 iIT πκ

= (6)

where iI is the inertia corresponding to the axis of rotation and κ is the torsion constant of the torsional rod.

Figure 8: Torsional Pendulum Test for (a) Ixx and (b) Izz

The table below presents the results of this experiment for four symmetric span configurations. Table 2. Determination of wing principal moments of inertia

Number of Average Configuration Ix T1, s T2, s T3, s oscillations Period, s Ix, slug ft2

40-40 18.29 18.3 18.31 10 1.8300 0.94860-60 23.19 23.23 23.24 2.3220 1.55880-80 29.09 29.14 29.15 2.9127 2.481100-100 33.85 33.87 33.91 3.3877 3.375

Number of Average Configuration Iz T1, s T2, s T3, s oscillations Period, s Iz, slug ft2

40-40 19.01 18.65 18.91 10 1.8857 0.94960-60 23.72 23.94 23.99 2.3883 1.59080-80 29.34 29.76 29.49 2.9530 2.491100-100 33.96 33.81 33.84 3.3870 3.312

Stability derivatives are always evaluated at trim. Hence, in order to determine these derivatives, it was necessary to find a trim solution for each span configuration at which these derivatives are calculated. Aerodynamic data from wind tunnel tests on the WB were used in conjunction with theoretical data from the horizontal and vertical tail were used to find trim solutions for each of the four configurations under consideration. At trim, the resultant moments and forces vanish. As the wingspan changes, new equilibrium values of angle of attack and/or airspeed must be

(a) (b)

Torsion Rod

VAR Wing

Torsion Rod

VAR Wing


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sought. The absence of an elevator on the model precluded changing the trim angle of attack eα for each different configuration. Instead, the trim angle of attack was fixed from the pitching moment coefficient curves (see figure 9.) Since trim angle of attack was fixed, trim airspeed had to be changed as the span changed. Lift force data was extrapolated to appropriate speeds until the lift available at a particular speed balanced the weight. This approach is appropriate because changes in span are expected to correspond to changes in airspeed at steady cruise conditions. Higher airspeeds were required to trim configurations with shorter spans because of their correspondingly smaller lift curve slopes (see figure 9).

-3-2-10123

-10 0 10 20 30


CM

40-40Vtrim=120mph

-3-2-10123

-10 0 10 20 30

angle of attack

CM

60-60Vtrim=100mph

-2-1.5

-1-0.5

00.5

11.5

2

-10 0 10 20 30

angle of attack

CM

80-80Vtrim=90mph

-1.5-1

-0.50

0.51

1.5

-10 0 10 20 30


CM

100-100Vtrim=80mph

Figure 9: Pitching Moment Coefficient Curves for Four Symmetric Span Configurations Dimensionless stability derivatives were evaluated using the Advanced Aircraft Analysis Program12, a software package by the Design, Analysis and Research Corporation, (DARcorp). It is instructive to construct a root locus of the modes as a function of span (see figure 10). The spiral root remains unstable for each span configuration shown, becoming slightly more damped with a smaller time constant, τ ,as span increases. The spiral divergence displayed by this model is typical of most aircraft. At this stage it is not known if the spiral mode ever becomes stable. Further investigation needs to be done to determine if changing span inputs can be used to control the spiral divergence displayed here. The other two roots display similar trends in stability. Increasing span has the effect of increasing the damping of both the dutch roll and roll modes and decreasing τ . The natural frequency, nω of the dutch roll mode increases with increasing span. The effects of span on the modes derive primarily through the

nrC , nC β and lpC and lC β terms in the Α matrix14. At subsonic speeds, these derivatives are heavily influenced by

aspect ratio.13 Increasing nrC and nC β damp the dutch roll and increase its natural frequency. The roll subsidence

approaches steady state more quickly as the roll damping parameter lpC is increased. Spiral roots migrate to the left

half plane as lC β becomes more negative.


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Figure 10: Root Locus Plot Showing Variation of Lateral Roots with Change in Symmetric Span

VI. Conclusions It has been demonstrated that the generic VAR UAV under consideration is dynamically stable in the dutch roll and roll modes for the symmetric span configurations. Increasing the total span decreases the time constant and increases the damping of dutch roll mode, making it more stable. Roll modes also become more stable with increase in total span. Additional work needs to be performed in order to determine if the spiral mode can be controlled by differential span inputs. In addition, wind tunnel test data was used to verify that asymmetric span configurations generate roll moments that can be used to control roll maneuvers.

VII. Future Work The linearized decoupled six-degree of freedom (6 DOF) equations of motion are suitable for the analysis of most flight control problems. Two assumptions enable the decoupling of these equations into a lateral and longitudinal set: (1) a symmetric reference trim condition and (2) constant propulsive forces. These assumptions imply that the side velocity (v0), angular rates (p0, q0 and r0) as well as the roll angle (Φ0) and yaw angle (Ψ0) are zero.8 However, these assumptions are no longer valid when the aircraft morphs into a geometrically asymmetric configuration in order to execute some maneuver. Under these conditions, certain trim variables take on non-zero values. The new trim condition may involve finite values of sideslip, roll and yaw angle, thus coupling the lateral and longitudinal sets. Hence, it becomes necessary to revisit the nonlinear, coupled 6 DOF motions and attempt to linearize about the new geometrically asymmetric steady state condition11. Analysis of these coupled linearized equations would enable the study of the modes of the UAV in asymmetric configurations. Controllability studies for various maneuvers such as coordinated turns can then be conducted.


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Acknowledgments The authors would like to thank the Glenn L. Martin Wind Tunnel staff for their help in the testing of the UAV

with Variable Aspect Ratio Wing. References 1Bowman, J., Sanders, B., and Weisshaar, T., “Evaluating the Impact of Morphing Technologies on Aircraft Performance,”

AIAA Paper 2002-1631, April 2002 2Sanders, B., Eastep, F. E., and Foster, E., “Aerodynamic and Aeroelastic Characteristics of Wings with Conformal Control

faces for Morphing Aircraft,” Journal of Aircraft, Vol. 40, No.1, 2003, pp. 94-99. 3Amprikidis, M., and Cooper J. E., “Development of Smart Spars for Active Aeroelastic Structures,” AIAA Paper 2003-1799, April 2003 4Blondeau, J., Richeson, J., and Pines, D. J., “Design, Development and Testing of a Morphing Aspect Ratio Wing Using an Inflatable Telescopic Spar,” AIAA Paper 2003-1718, April 2003 5Anderson, John D., Fundamentals of Aerodynamics, 2nd ed., McGraw Hill, New York, 1984, p.343 6Nelson, Robert C., Flight Stability and Automatic Control, 1st ed McGraw Hill, New York, 1989 p 93. 7McCormick, Barnes, Aerodynamics, Aeronautics and Flight Mechanics, 2nd ed, Wiley, New York, 1995, p. 514

8Stevens, Brian, L. and Lewis, Frank L, Aircraft Control and Simulation, Wiley, New York, 1992, p. 91. 9Bae, Jae-Sung, Seigler, Michael T., Inman, Daniel J. and Lee, In “Aerodynamic and Aeroelastic Considerations of a

Variable-Span Morphing Wing,” in Proceedings of the 45th AIAA/ASME/AHS/ASC Structures, Structural Dynamics and Materials Conference 2004

10Garcia, Helen M., Abdulrahim, and Lind, Rick, “Roll Control for a Micro Air Vehicle Using Active Wing Morphing”, AIAA Paper 2003-5347, August 2003

11Saraf, Aditya, “Robust Flight Control for Coordinated Turns,” Master’s thesis, Aeronautics and Astronautics Dept., University of Cincinnati, Cincinnati, OH, 2003.

12AAA, Advanced Aircraft Analysis, Software Package, Ver. 2.5 , Lawrence, KS 13USAF Stability and Control DATCOM, Vols. 1-4, 1960 Wright Patterson Airforce Base, OH 14Blakelock, John H., Automatic Control of Aircraft and Missiles, 2nd ed., Wiley Interscience, New York, 1991, p.142

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