Fan Angular Momentum Analysis for Ducted Fan UAVs During Conceptual Design

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January 2012, Nashville, Tennessee AIAA Fan Angular Momentum Analysis for Ducted Fan UAVs During Conceptual Design Brandon Stiltner, and O. John Ohanian III AVID LLC, Blacksburg, Virginia, This paper outlines some of the effects of fan angular momentum on ducted fan UAVs. All aircraft will be affected by their propeller s angular momentum, however, ducted fan vehicles are particularly susceptible due to the relative sizes of the fan and the airframe. During maneuvers, the angular momentum of the fan causes a gyroscopic torque that must be countered by the UAV s control surfaces. If not accounted for during design, the gyroscopic torque can result in a reduction of the vehicle s maneuverability. In this paper, the equations of motion for a ducted fan vehicle are provided to visualize the gyroscopic torque. It is also important to understand how the gyroscopic torque scales with fan radius. Through this knowledge, a constraint equation is formulated that ensures that the UAVs control surfaces can counteract the gyroscopic moments acting on the vehicle. It is also shown that, to ensure the control surfaces are adequate, the control surface effectiveness must scale with the square of the fan s radius if equivalent body angular rates are required. In some cases, however, it may be more beneficial to the vehicle to design a contra-rotating fan system. It is shown how this system eliminates the gyroscopic torque that is present in a conventional single fan UAV. However, contra-rotating fan systems have unique dynamic properties, and modal analysis is used to compare the dynamics of both configurations. I. Introduction Ducted fan aircraft are a relatively new class of aircraft that have emerged with the advent of UAVs (Unmanned Aerial Vehicles). Ducted fan UAVs are a type of VTOL (Vertical Take-Off and Landing) vehicle that behave differently from both fixed wing airplanes and helicopters. To point out some of the unique properties of the ducted fan UAV, first the ducted fan provides more thrust than an an open rotor at a fixed diameter and power input, and the enclosed blades provide improved safety. Ducted fan vehicles can cruise at speeds comparable to fixed-wing vehicles of the same size and can also hover efficiently, an important ability for surveillance missions. Vertical takeoff and landing eliminates the logistics footprint of launching and recovery hardware that many fixed-wing UAVs employ. Typically, ducted fan aircraft are composed of a duct, a fan, a motor, a center-body structure, control surfaces known as vanes, and landing legs. A few ducted fan UAVs are provided in Figure 1. One of the many properties that makes the ducted fan different from other aircraft is its propeller angular momentum. Specifically, the spinning fan imparts a large gyroscopic torque on the vehicle during maneuvers. On a fixed wing aircraft, the gyroscopic torque imparted by the propeller is small relative to the aerodynamic moments acting on the vehicle. On the other hand, a helicopter s rotor has a very large angular momentum. However, the gyroscopic torque is not fully imparted to the body of the helicopter because of the added degrees of freedom in the swash plate. The swash plate allows the body to rotate relative to the rotor, and can be thought of as decoupling the tilt angle between the rotor and vehicle. The ducted fan aircraft is caught somewhere in between these two examples. Unlike the helicopter, the fan is connected to the UAV through a rigid shaft. Thus, the gyroscopic torques acting on the fan are directly imparted to the vehicle. Second, unlike a fixed wing aircraft, the ducted fan UAV does not have large aerodynamic surfaces(wings) to generate significant aerodynamic moments. Therefore, no aerodynamic moments are acting on the vehicle to dampen or help counteract the gyroscopic moment. For the ducted fan Aircraft Design Engineer, AVID LLC, Member AIAA. Senior Aircraft Design Engineer, AVID LLC, Member AIAA. 1of16 Copyright 2012 by AVID LLC. Published by the, Inc., with permission.

2 UAV, the lack of wings also results in low vehicle inertias relative to the fan. Furthermore, the gyro-torque can be large relative to the control moments acting on the vehicle. The gyroscopic torque arises when the vehicle experiences a change in attitude, such as during a maneuver or wind gust. The ducted fan UAV must counteract the gyro-torque with its control surfaces, using control throw that could otherwise have been allocated to the maneuver. In some cases, the gyroscopic torque is large enough that it can restrict the vehicle s maneuverability. To prevent an unnecessary limit on the vehicle s performance, the fan angular momentum must be considered early during the vehicle s design. A review of literature revealed that little to no research has been published on the matter of taking angular momentum into account when designing the control surfaces. However, there are a number of works that discuss vehicle flight controls in the presence of fan/propeller angular momentum 1,2,3. Most of the literature in this respect is focused toward quad-rotor UAVs that are fully controlled by making use of the gyroscopic torque. In other words, these vehicles do not utilize control surfaces, but use differential throttle for each propeller for control. In this paper, the authors provide a means by which the gyroscopic torque can be taken into account earlier. Specifically, the motivation here is to develop methods and knowledge so that the designer can account for the fan s momentum and gyroscopic torque during the conceptual design phase. To start, the equations of motion for a ducted fan UAV are presented in Section II. From the equations of motion, it is seen how the gyroscopic torques imparted by the spinning fan can affect the UAV. It is also necessary to understand how the gyroscopic torque scales with vehicle size. This is presented in Section III. From this, an equation is formulated that states that the control vane effectiveness must increase proportionally to the square of the fan s radius. Similarly, an equation is also derived to serve as a vane sizing constraint during vehicle design. This constraint ensures that the vanes have sufficient control power to counteract the gyro-torque. In other words, the control vanes must be adequately sized and designed so they can provide enough control moment to counteract the gyro-torque. Finally, it is also shown how the vehicle s design parameters can be manipulated to ensure that the constraint is met. Another design approach to negate the gyroscopic torque is a ducted fan UAV with contra-rotating fans. The angular momentum of the counter rotating fans cancel, resulting in no gyroscopic torques imparted to the vehicle by the fans. However, this approach has drawbacks of its own, which are detailed in Section IV. Specifically, an example is provided to show that a contra-rotating fan UAV has faster dynamics than a conventional single fan UAV. This means that the control system must be able to perform faster to ensure that the vehicle is stabilized. Furthermore, comparisons are drawn between the contra-rotating and conventional (single fan) configurations. (a) Aurora GoldenEye 50 4 (b) MASS HeliSpy 5 (c) Honeywell RQ-16 T-Hawk 6 (photo credit U.S. Navy) Figure 1. Examples of ducted fan UAVs. 2of16

3 A. Related Literature The work presented in this paper is an extension to earlier work presented by Stiltner et al. 7 Specifically, the working presented in Stiltner 7 provides an in depth derivation of the equations of motion for ducted fan UAVs. The desire was to derive and linearize the equations of motion for a ducted fan UAV such that an analytical form was obtained. It was expected that the equations of motion could be simplified to a form that resulted in decoupled longitudinal and lateral motions, similar to fixed wing aircraft. The work also provides a modal analysis of a ducted fan UAV at different trimmed conditions. It was found that the equations of motion do not decouple due to the angular momentum of the spinning fan, and complex aerodynamics. In that work, the significance of the fan s angular momentum was noticed, which led to the research presented in this paper. A large body of research has been published concerning the use of angular momentum for control. Most notably, satellites use flywheels known as Control Moment Gyroscopes (CMGs) for both stabilization and control 8. A few authors have investigated using CMGs for control of VTOL UAVs 3,2. Work has also been published concerning control of quad-rotor (or quad-copter) UAVs 1. These vehicles do not utilize control surfaces, but use variable thrust for each rotor. For these vehicles, the flight controls must account for the gyroscopic torques that occur during maneuvers. Still, other work has been published that concerns controlling traditional ducted fan UAVs. Pflimlin et al. 9 presents a paper concerning hovering flight control for a ducted fan UAV with contra-rotating fans. Additionally, Johnson and Turbe 10 present working on controlling a conventional ducted fan UAV. Because the work presented in this paper is heavily rooted in vehicle and rigid body dynamics, many books were used for reference and understanding of the equations of motion. Two well known works concerning aircraft dynamics, control, and flight mechanics are Etkin s Dynamics of Atmospheric Flight 11 and McRuer et al. 12. Another, relatively newer reference for aircraft dynamics is Stevens and Lewis 13. Furthermore, Schaub and Junkins 8 provides a great reference for rigid body dynamics and kinematics. II. Equations of Motion for Angular Momentum In this section, the Equations of Motion (EOMs) are presented for a ducted fan vehicle. The equations of motion were derived in a previous work, 7 and only the final form is presented here. The equations are then simplified to a form that is used for derivations throughout the remainder of the paper. From these equations, it is shown that the fan imparts a gyroscopic torque on ducted fan UAVs. Before deriving these equations, it is first necessary to introduce the reader to the coordinate frame used in this paper. Figure 2 shows a typical ducted fan coordinate frame. Again note that the EOMs for the ducted fan are not derived here, but are simply presented for reference. A detailed derivation of ducted fan equations of motion can be found in Stiltner 7. In Figure 2, U, V, and W are the velocity components of the vehicle in its body-fixed coordinate frame. Similarly, P, Q, and R are the vehicles angular rate components in the body-fixed frame, and L, M, and N are the components of the aerodynamic moment acting on the vehicle. Although it is not depicted, the fan is assumed to be spinning about the body Z-axis, and let Ω denote the fan s angular rate about the Z-axis. Last, a Euler angle rotation is used to relate the body coordinate frame to the inertial coordinate frame. The Euler angles used are the standard φ, θ, and ψ (roll, pitch, and yaw respectively). The combination of the vehicle s attitude (φ, θ, and ψ), linear velocity (U, V, and W), and angular velocity (P, Q, and R) make up the vehicle s state vector. Euler s equation of motion for a rigid body is expressed in Equation (1). Here, H is the body s angular momentum, Mext are the external moments acting on the body, I is the body s inertia, and ω is the body s angular velocity 8,11,13. d dt H = Mext, H =[I] ω (1) Applying Euler s equation results in the EOMs for a ducted fan vehicle which are provided in Equation(2). [I AC ] ω AC/n = M aero + M ctrl ω AC/n [I AC ]ω AC/n [I f ] ω f/ac ω f/n [I f ]ω f/ac (2) 3of16

4 U X C G L,P M,Q N,R W Z V Y (a) Coordinate frame fixed to vehicle. (b) Linear and rotational coordinate performance parameters. Figure 2. Vehicle body-fixed coordinate frame. A description of the notation used above is as follows. The term I is used to denote an inertia matrix, ω denotes angular velocity, the dot notation is used to designate a time derivative, and M is used to denote an external moment acting on the aircraft. The subscript notation AC denotes the Air Craft reference frame, the subscriptf denotes the fan s reference frame, and n denotes a ground fixed inertial reference frame. Finally, the notation a/b is used to denote frame a relative to frame b. Therefore, the term ω AC/n is read as the angular velocity of the aircraft relative to the inertial frame n. For analysis during conceptual design, not all of the terms in Equation (2) are necessary. Therefore, some assumptions are made to reduce the size of the equations of motion so they are easier to work with. From left to right in Equation (2), the third term from the equal sign can be neglected which is the cross product term (the Coriolis term). This term is written explicitly in Equation (3) for reference. Tosimplifythisterm, Equation(3)iswrittensuchthattheinertiamatrixoftheaircraftI AC issymmetric. While this is not necessarily true, the off-diagonal terms of the aircraft s inertia are usually much smaller than the diagonal terms. QR (Izz Iyy) I xx ω AC/n [I AC ]ω AC/n = PR (Ixx Izz) I yy (3) PQ (Iyy Ixx) I zz Equation (3) is ignored for several of reasons. The first two terms can be neglected by assuming that the vehicle is not rotating about its z-axis during a maneuver (i.e. R = 0). The third term of Equation (3) is neglected by assuming that the vehicle is only rotating about one axis at a time during a maneuver. The vehicle has the ability to rotate about each of these axes simultaneously. However, for this analysis the vehicle s motion is restricted to simplify the equations of motion. For example, assume a transition maneuver from hover to forward flight. In this case, the vehicle should only pitch forward, while the roll and yaw attitudes should remain constant. This assumption is expressed mathematically by the following inequalities. P>0, and Q,R= 0 (4) Q>0, and P,R= 0 (5) The next term that is neglected is the fourth term on the right, which represents the fan s angular acceleration ( ω f/ac in Equation (2)). This term is the reaction torque caused by accelerating or decelerating the mass of the fan. This term is ignored by assuming that the fan accelerates slower than the duration of the 4of16

5 maneuver. In other words, assume that the fan speed is constant during a maneuver, which is approximately one second. This assumption is written as follows. [I f ] ω f/ac =0, ω f/ac = const. (6) Rewriting Equation (2) with the assumptions above results in Equation (7). From right to left, the first term on the right of the equal sign is the aerodynamic moments acting on the aircraft. The second term is the control moments acting on the aircraft, denoted as aileron, elevator, and rudder. Finally, the last term on the right are the gyroscopic torques imparted on the aircraft by the spinning fan. As shown, the gyroscopic terms are non-zero only if the vehicle is rotating about its x or y body axes. Hence, when P or Q is non-zero. Additionally, it is assumed that the inertia matrix of the fan is a symmetric matrix, and the term I zzf is the component of the fan s inertia about its spin axis (body z-axis). L M ail QΩI [I AC ] ω AC/n = M aero + M zzf ctrl ω f/n [I f ]ω f/ac = M + M ele P ΩI zzf (7) N 0 An interesting result from Equation (7) is that a pitch rate Q causes a gyroscopic roll moment (moment about x-axis). Similarly, a roll rate P causes a gyroscopic pitch moment (moment about y-axis). In the following sections, Equation (7) will be the baseline equation for which all other equations are derived. The next section presents the motivation for the studies performed during this research, how the gyroscopic torque impacts the ducted fan vehicle. M rud III. Impacts on Conceptual Design This section presents a three ways that the gyroscopic torque can be accounted for during conceptual design. The first method is understanding the how the fan s inertia and angular velocity scale with radius. Because the gyroscopic torque is linearly dependent on the fan s inertia, keeping the inertia low is critical to mitigating the impact of the gyroscopic torque. The second method to account for the gyroscopic torque is to ensure that the vanes have adequate control power to counteract the gyro-torque. In this section, a new control vane constraint equation is developed that ensures that the vanes are able to counteract the gyro-torque during maneuvers. Last, a third method to consider during design is a contra-rotating fan concept. Here, it is shown that the angular momentum of contra-rotating fans will negate, thus resulting in zero gyro-torques acting on the aircraft. A. Fan Inertia and Angular Momentum Scaling The angular momentum present in a ducted fan vehicle is dependent on several design parameters. The angular momentum is directly proportional to the fan angular speed and moment of inertia, which are dictated by other design parameters. The angular speed and inertia are strongly coupled to the fan radius and several other design features that affect vehicle performance. The tip speed of the fan (usually stated in terms of dimensionless Mach number) combined with the fan radius will dictate the fans angular speed, or revolutions per minute (RPM). This is shown below in Equation (8). Ω= V tip (8) r Furthermore, the thrust achievable at a given RPM is based on the fan s airfoil cross-section (and corresponding lift coefficient), blade planform (chord length versus radius), and twist distribution. These geometric properties, in conjunction with material densities and a chosen fabrication technique, will result in a fan moment of inertia value. Typically, fans that produce more thrust at a particular tip speed and radius will require more blade area, resulting in higher fan inertia. An alternative approach to increasing thrust is to increase the fan radius, which also directly impacts fan inertia and angular momentum. The fan radius will also affect the power required to turn the fan, and is therefore a key parameter in vehicle design that deserves close attention. A geometric study was performed to identify how fan inertia scales with fan radius for two conceptual fabrication techniques. The first fabrication approach is a solid fan with uniform mass density, approximating 5of16

6 an injection-molded plastic part or a fan machined from wood or metal. The second fabrication approach is a thin shell representation, which approximates a composite skin structure, which could be either hollow or contain a lightweight foam core. The inertia of a 7-bladed fan geometry was evaluated in the ducted fan conceptual design code, AVID OAV (Organic Air Vehicle) 14. Scaling was applied equally in every direction. The geometry used in this scaling study is shown in Figure 3, along with the non-dimensional inertia scaling trends. The results are presented relative to any chosen fan radius, r 0, with rotational inertia I zz0. (a) Example geometry of duct and fan to evaluate inertia scaling. (b) Fan inertia scaling with radius for a seven bladed fan with mass properties calculated as both a solid and a shell. Figure 3. Fan inertia scaling versus fan radius. The plot in Figure 3 shows that for a solid fan with uniform density, the inertia scales with radius to the 5th power. For a thin shell composite fan the inertia scales with radius to the 4th power (proportional to the surface area rather than the volume). Each of the polynomial curve fits could be decreased by one order with marginal sacrifice in accuracy (solid fan at 4 th order R 2 = , thin shell fan at 3 rd order R 2 = ). The key observation is the difference in the order of magnitude in fan inertia between the two fabrication approaches, with the composite shell fabrication resulting in a lower inertia design as expected. These scaling trends in fan inertia will be essential for understanding the angular momentum and resulting constraints on the vehicle design. In general, if the tip speed is held constant as the ducted fan is scaled up, the RPM will be inversely proportional to fan radius as shown in Equation (8). This implies that the angular momentum of the ducted fan will scale with radius to the 4th power for solid fans, and radius to the 3rd power for thin shell composite fans. Figure 4 shows the angular momentum for a thin composite shell fan. This plot was generated for a constant fan tip speed with radius (i.e. V tip = const). As can be seen, the fan s momentum grows with radius to the 3rd power. B. Control Vane Constraint Equation In this section, a design equation is derived that is used as a constraint equation for sizing a ducted fan s control vanes. The resulting equation is a necessary condition, but not a sufficient condition for sizing the ducted fan s control vanes. Other equations and constraints will be taken into account to adequately size the vanes, such as control power and the vane trim deflections over the vehicle s flight speed envelope. To derive the constraint equation based on angular momentum, start with Equation (7) which is restated again below for reference. 6of16

7 $2&"!"#$%&"'($)'*+,")-.,'/0/ 1 '' $,&" $&&" 3&" -&" 2&",&" :";<=>5?7="@ABA&C" D<8!%"@456789:";<=>5?7="@ABA&CC"!"#"$%&''() * "+"&%*,-'), "."&%(/,')" 01"#"&%''''-" &" &" $"," *" 2" (" -"!"#$%&"'($)'!$23.4'!0! 1 ' Figure 4. Fan angular momentum scaling with fan radius for a thin composite shell fan construction. L [I AC ] ω AC/n = M + N M ail M ele M rud QΩI zzf P ΩI zzf 0 (9) Next, assume that the vehicle has a constant angular rotation (no angular acceleration), and assume that the aerodynamic moments will be assessed separately. The resulting derivation provides a simple equation that serves as a necessary constraint for the control vane sizing. Applying these assumptions to the equation results in the following. M ail M ele M rud QΩI zzf > P ΩI zzf (10) 0 Equation (10) has been reduced to state that the available control moments must be greater than the gyroscopic torque caused by the spinning fan. Next, the derivation of the control vane moment is provided. For this, only the equations are derived for an arbitrary control vane, as the equations are the same for both elevator and aileron. In either case, the moment that a control vane can generate about the aircraft s center of gravity (CG) is the product of the vane s lift and moment arm. The moment arm is roughly the distance between the control vane and the aircraft s CG. In equation form, this is written as M vane = 0.5C L ρve 2 S l (11) In Equation (11), the term inside the parenthesis is the control vane s lift, while l designates the vane s moment arm. Here, C L is the vane s lift coefficient, ρ is the air density, V e is the air speed at the duct s exit, and S is the surface area of the vane. Furthermore, for a ducted fan UAV in hover, the exit velocity is a function of the duct s thrust and exit area. V 2 e = T (Ω) ρa e (12) 2(2r) T = C T ρη 2 d 4 = C T ρ Ω 4 2π In Equation (12), T is the vehicle s thrust which is a function of the fan speed Ω. The relationship between thrust and fan speed is shown in Equation (13). In the equation, C T is the static thrust coefficient (13) 7of16

8 of the fan and η is the speed of the fan in revolutions per second. Furthermore, the vane area S can be represented as a function of radius, as can the exit area A e. These are provided in the following relationships. S =2rcn A e = πr 2 (14) In Equation (14), it is assumed that the vane spans the diameter of the duct, hence the factor of 2 in the vane area equation. Also, c is the vane s chord length. In some cases, there are more than one blade per vane set. Thus, n designates multiple control vanes in the vane area equation. Substituting Equation (8), Equation (12), Equation (13), and Equation (14) into Equation (11) and simplifying results in Equation (15). M vane = 4 π C LC T ρv 2 tiprncl (15) Equation (15) shows that the moment produced by the control vane is a function of both vane parameters as well as fan parameters. The fan parameters being V tip, C T, and r. Last, substituting Equation (15)into Equation (10) results in the design constraint equation for the control vane effectiveness. Here, the design equation is written for the aileron vane, and a similar equation can be written for the elevator. 4 π C LC T ρv tip r 2 ncl > QI zzf (16) To use Equation (16) as a constraint, the constraining condition must first be determined. The constraining condition occurs when certain parameters in Equation (16) are at a minimum or maximum. One constraint occurs when the control vane lift is at a minimum (i.e. when the dynamic pressure over the vanes is at a minimum). In therms of Equation (16), this occurs when V tip and ρ (air density) are at a minimum. Realistically, this condition occurs at the maximum design altitude and minimum throttle. On the other hand, the constraining condition for the gyroscopic torque is when the vehicle is required to rotate at its maximum rate. Therefore, when pitch rate Q is at a maximum. From this, the vane sizing constraint is rewritten as Equation (17) below. Here, the asterisk is used to denote maximum values. Furthermore, the equation has been rearranged so that the parameters affecting the fan are on the right of the equation, while the control vane parameters are on the left. 4CL ρmin ncl I zzf πq > C T Vtip min (17) r2 An example of how Equation (17) is used during conceptual design is provided. Assume that a ducted fan vehicle has been conceptualized and is in the early stages of aircraft design. At this stage, gross geometric values for the vehicle are known such as duct height, fan diameter, control vane size, etc. Furthermore, assume that a fan geometry has been designed and the fan s Thrust vs RPM curve is also known. So, in terms of Equation (17), Vtip, C L, C T, c, l, r, and I zzf are all known. The maximum pitch rate Q is typically provided as a design requirement. Similarly, the maximum altitude is also a design requirement and therefore ρ min is also known. If the constraint in Equation (17) is not met, then the current design is inadequate. In this event, the parameters in Equation (17) must be varied until the constraint is met. First, the vane parameters may be varied until this constraint is met. For example, the designer can increase the effectiveness of the control vanes by increasing the vane s lift coefficient, chord length, or moment arm. Similarly, the fan parameters can also be varied by reducing the fan s inertia or increasing the fan s thrust coefficient C T. Another way to look at the constraint in Equation (17) is by investigating how the constraint scales with radius. Here, a term known as disk loading is introduced, which is a well known ducted fan design parameter. Equation (18) demonstrates how disk loading relates to thrust and fan radius. DL = T A disk = T πr 2 (18) Disk loading is a quantity that is invariant of size and translates to designs of any scale. Substituting Equation (18) into Equation (12) and re-deriving the constraint equation results in Equation (19). In the equation, all of the radius dependent terms have been moved to the right of the equation. DLC L ncl QV tip > I zz f r 2 O(r 2 ) (19) 8of16

9 The relationship in Equation (19) shows that the effects of the angular momentum and gyroscopic torque grow to the 2 nd power of fan radius for thin shell composite fans. For a solid fan, the term on the right side of Equation (19) grows to the 3 rd power with radius. Potential solutions for satisfying this constraint include increasing the disk loading, vane lift coefficient, vane area, and vane moment arm, or decreasing the fan speed or desired body angular rate. Last, it should be noted that Equation (17) and Equation (19) are not exclusive equations, but are the same equation represented in two different forms. Equation (19)is provided to show how the vane parameters must scale as vehicle size increases to ensure that the gyroscopic torque can be counteracted. The previous examples show how Equation (17) or Equation (19) can be used as a constraint during vehicle design. More specifically, Equation (17) will be used in combination with other constraints that are placed on the control vanes. For instance, the vane s are often sized such that their required trim deflections are minimized. Therefore, an optimization problem can be formed for a specific vehicle design to find the minimum vane size that meets the constraint in Equation (17) and also result in minimal trim deflections. From Equation (19), it is shown that the vane effectiveness must grow with the square of the radius. For large ducted fan UAV designs, it may become counter productive to ensure that this constraint is met. In this event, it is more beneficial to approach the vehicle design from a different view point, such as contra-rotating fans. C. Contra-Rotating Fans In the previous two subsections, it is shown how the fan s inertia and angular momentum scale with various powers of the fan s radius. Furthermore, a constraint equation is presented which ensures that the vanes will be able to counter the gyroscopic torque. In this section, an alternative method to account for the gyroscopic torque is presented. Specifically, a ducted fan design with contra-rotating fan s negates the gyroscopic torque. To show this, the angular momentum equations for a contra-rotating fan vehicle are presented. From this, it is easily shown that the angular momentum of contra-rotating fans cancel. Thus resulting in zero gyroscopic moments imparted to the aircraft. Recall from Equation (1) that the angular momentum of a rigid body is the product of the body s inertia and angular velocity. Therefore, the total angular momentum for a ducted fan with contra-rotating fan s is written as that shown in Equation (20). H total = H AC + H f1 + H f2 (20) Writing the angular momentum for a fan explicity results in Equation (21). H f1 =[I f1 ]ω f1 (21) Now, by assuming that the both fans have equal inertias, and the fan s rotation speeds are equal magnitudes but opposite directions, Equation (20) is rewritten as the following. H total = H f1 + H f2 = H AC +[I f1 ]ω f1 [I f1 ]ω f1 = H AC (22) Equation (22) holds as long as both fan s rotate at the same speed. However, there could be some contra-rotating systems where the fans are driven by two separate motors that have the capability to rotate at different speeds. In those cases, the difference in angular momentum between the two fans would cause a gyroscopic torque during roll and pitch maneuvers. However, the gyro-torque is small because its magnitude is proportional to the difference in fan rotational speed. In general, the contra-rotating design configuration results in no, or minimal gyroscopic torques. While the contra-rotating system appears to have an advantage for maneuverability, care should be taken before proceeding with this configuration. First, a dual fan and dual motor configuration will likely result in a heavier vehicle than a single fan configuration. This is because extra structure is necessary to support the second motor and fan. Furthermore, additional electronics may be required to control the speed of the second motor, resulting in additional weight and volume impacts. Lastly, a dual fan and dual motor configuration may negatively impact the vehicle s CG location. Typical ducted fan vehicle CGs are located as high as possible to maximize vane control power. A coaxial, contra-rotating system may result in CG locations that are lower than desired. Many variables exist in the design of a ducted fan UAV, and therefore it is difficult to provide a simple criterion for when contra-rotating systems are preferred. Certain constraints may result in concepts where 9of16

10 a contra-rotating system is advantageous. For example, a double duct design with side-by-side ducts is one such example where a contra-rotating system is easily implemented and preferred. IV. Impacts of the Gyroscopic Torque on Ducted Fan UAVs The previous section provides some equations and methodologies that take the fan s inertia and angular momentum into account during conceptual design. In this section, those methodologies are illustrated on a proven ducted fan design. First, an example is provided that shows how the vehicle s maneuverability can be limited if the gyroscopic torque is not accounted for during design using Equation (17). Second, the dynamics of a ducted fan are investigated for both a conventional and contra-rotating ducted fan. Both of these examples are presented for a ducted fan vehicle for which extensive wind tunnel and mass property data exists. The vehicle is an axisymmetric, aerodynamically clean ducted fan and provides a good baseline for investigating the fan angular momentum effects. A picture of the ducted fan is shown in Figure 5. Figure 5. Axisymmetric ducted fan vehicle baseline. A. Impact on Maneuver Performance From Equation (7) it is seen that the fan imparts gyroscopic torques about the vehicle s x and y axes. Specifically, a pitch rate Q causes a torque about the roll-axis, and a roll rate P causes a torque about the vehicles pitch-axis. Also, for a fixed pitch rate, the resulting torque about the roll-axis is proportional to the angular momentum of the fan (ΩI zzf ). In the following example, the aerodynamic moments in Equation (7) are omitted, leaving only the gyro torque and the control moments. The aerodynamic moment is addressed separately by other constraints. This simplification is used here to show how the gyro torque affects the vehicles maneuverability. During a maneuver, the gyro torque must be countered by the vehicle s control vanes. A good maneuver example is a ducted fan vehicle transitioning from hover to forward flight. To perform this maneuver, the ducted fan must pitch forward so that a component of its thrust is in the forward direction. By pitching forward, i.e. Q > 0, the vehicle must use its aileron vanes to counter the gyroscopic roll moment of magnitude QΩI zzf. If this value is too large, the aileron vanes will not be able to counter the gyroscopic torque generated during the maneuver. Thus making the maneuver infeasible. Recall this condition from Equation (17) in the previous section. If the gyro-torque is too large, the vehicle must pitch forward at a slower rate to safely perform the transition from hover. Thus, the pitch rate is limited by the spinning fan. Tobettervisualizethisexample, thegyroscopicrollmoment(qωi zzf )isplottedfortheductedfanvehicle shown in Figure 5. Figure 6 provides a plot of the gyroscopic torque plotted against fan speed (RPM) at varying pitch rates (Q) in degrees per second. As can be seen, the gyroscopic torque is linearly proportional to both the pitch rate and fan speed. Also, in this analysis the fan s inertia (I zzf ) is held constant. In Figure 6, the upper right corner represents the maximum gyroscopic torque that the fan will exert on the 10 of 16

11 vehicle, which occurs at a pitch rate of 360 degrees per second and a fan speed of RPM. Figure 6. Gyroscopic torque versus fan speed with aileron moment at varying C L. Also in the plot are the red curves, which are the aileron vane s control moment at varying lift coefficients C L. Recall Equation (15), which provides the moment that a control vane can generate about the vehicle mass center (CG). This equation is restated below for reference. M vane = T 0.5C L A e S l (23) Essentially, this plot visualizes the constraint provided in Equation (17). The circle on the upper left corner of the surface plot is the design point mentioned in Section III. This shows that if the maximum pitch rate is 360 degrees per second, this vehicle design does not meet the constraint. Another point that can be madefromthisplotisthatthedesignpointiscorrect. Ascanbeseen, thegyro-torqueislinearlyproportional with fan speed, while the vane moment is quadratic with fan speed. This is shown in Equation (13). Because the vane moment is quadratic with fan speed, meeting the constraint in Equation (17) guarantees that the vane can provided enough moment to counter the maximum gyro torque (upper right corner of the surface plot in Figure 6). At first glance, this flight condition seems unlikely. However, this is the limiting condition for the control vane power concerning the gyroscopic torque. If the vanes were designed using Equation (17) as a constraint, the curve for C L =1.2 would be larger than gyroscopic torque at all conditions in Figure 6. As mentioned in Section III, to ensure that the constraint is met, the vane s moment arm or surface area could be increased. Or, in more extreme cases, the ducted fan UAVs fan could be redesigned to be lighter, or provide more thrust at the same radius. While the vehicle may never be required it to perform a 360 degree per second pitch maneuver, one must still consider the possibility of disturbances. Assume that a disturbance caused a 360 degree per second pitch rate. In that event, the vane design depicted in Figure 6 would not be able to fully counteract the gyroscopic torque. Also, recall that the aerodynamic moments were omitted in this analysis. Including the aerodynamic moment has the virtual affect of reducing the aileron vane s available allocation. For example, some of the vane deflections are already used to counter the aerodynamic moments acting on the vehicle. Put another way, the aileron vane must be able to provide enough moment to at least counter the aerodynamic and gyroscopic torques acting on the vehicle. 11 of 16

12 B. Impact on Vehicle Dynamics Fan angular momentum also affects the vehicle s dynamics. In the following section, the vehicle s modes of motion are analyzed for a ducted fan vehicle in hover. Again, the vehicle shown in Figure 5 is used for this analysis. To make comparisons, the vehicle s dynamics are presented for two different configurations: a single fan and contra-rotating fan configuration. For the single fan configuration, the equations of motion are the same as those given in Equation (7). However, with the contra-rotating configuration, the gyroscopic moment is not present because the angular momentum of the two fans cancel. This was shown Section III, as the angular momentum of contra-rotating fans counteract one another. The result is that the third term of Equation (7) is now zero. It should also be noted that the aerodynamic data available for the ducted fan is for the single fan configuration only. Therefore, this analysis assumes that the addition of contra-rotating fans to the vehicle will not significantly change the vehicles aerodynamic properties. A modal analysis requires knowledge of a vehicle s linearized dynamics at a particular trimmed equilibrium, which is hover for this example. The linearized vehicle dynamics have the form shown in Equation (24), where A is known as the state matrix, and x is the vehicle s state vector. The state vector is defined in Section II in this paper. A modal analysis is performed by computing the eigenvalues and eigenvectors of the state matrix. A detailed modal analysis of a ducted fan vehicle is provided in Stiltner 7. x = Ax (24) Ducted fan vehicles are typically unstable. To better understand the gross instability of the two configurations, the time to double for the two configurations was calculated. Time to double is the time for the largest unstable mode to double in amplitude. Therefore, this value represents how quickly the vehicle will diverge from equilibrium if uncontrolled, and is calculated by Equation (25). Here, ln is the natural logarithm, and the denominator is the real part (RE) of the largest unstable eigenvalue λ of the state matrix. t 2 = ln(2) RE(λ ) Figure 7 provides a comparison of the time to double for the two configurations mentioned above (single fan and contra-rotating fans). In the figure, time to double is plotted with respect to forward flight speed. At each flight speed, the aerodynamic model of the vehicle is trimmed for equilibrium, and a new state matrix and time to double is computed. This plot provides an interesting result concerning the two configurations. The figure shows that the conventional ducted fan vehicle is slightly less unstable than the dual fan configuration. This result is somewhat expected by drawing comparisons to a spinning top. A spinning top tends to resist changingtheplaneinwhichitsmassisrotating. Toinvestigatethisaffectfurther, welooktoamorethorough modal analysis. The easiest way to investigate the vehicle s modes is by computing what is known as a Sensitivity matrix. The eigenvalues and eigenvectors provide all the necessary information to evaluate a vehicle s dynamics in a particular equilibrium (in this case, hover). However, the eigenvalues and eigenvectors can be difficult to decipher. The Sensitivity matrix can be thought of as a manipulation of the eigenvectors that allows the engineer to visualize which states are dominant in the response of each mode. A detailed definition of the sensitivity matrix can be found in Stiltner 7. Table 1 provides the sensitivity matrix for the conventional ducted fan aircraft (i.e. single fan configuration). Table 2 provides the sensitivity matrix for the ducted fan aircraft with contra-rotating fans. In the sensitivity matrix, each column represents a particular mode. The first row lists the eigenvalue corresponding to that mode in bold font. Each row of the matrix designates one of the vehicle s nine states. The states are depicted in Figure 2, and are composed of velocity, attitude, and angular rates. Also note that values less than E-6 have been truncated to 0.0 in the sensitivity matrices. The sensitivity matrix is interpreted in the following way. Each mode can be excited individually, and the resulting motion will affect some, or all of the vehicle s states. The sensitivity matrix allows the engineer to determine which states are affected if a mode is excited. Because the values in the Sensitivity matrix have been normalized, any value less than 0.1 can be ignored in the response of that particular mode. In other words, that state is affected very little by the excitation of the mode. In Table 1 and Table 2, some of the values in the matrix are highlighted in blue font. These represent the states that are affected if that (25) 12 of 16

13 Conventional Single Fan Contra Rotating Fans Time to Double (s) Forward Flight Speed (ft/s) Figure 7. Time-to-double for conventional and dual-fan vehicle configurations versus forward flight speed. particular mode is excited. Furthermore, the value shows the relative magnitude of which that state will be affected. Mode 2 of the sensitivity matrix in Table 1 represents the gyroscopic mode. As can be seen, this mode only affects the states P and Q, which are the roll and pitch rates. Because this mode is represented by a complex conjugate pair of eigenvalues, the resulting motion of this mode is oscillatory. Furthermore, because the real part of this mode is positive (i.e. unstable), its time to double is about 3.3 seconds. If excited, this mode causes the duct to begin wobbling with increasing tilt angles from hover, similar to a spinning top when it looses energy. For a top, this motion is known as precession. Comparing the sensitivity matrices for the two ducted fan configurations, one finds that the gyroscopic mode is no longer present in the contra-rotating dynamics. Instead, the contra-rotating configuration now has two highly unstable modes that contain coupled longitudinal and lateral states. These are Modes 2, 3, and 4 in Table 2. With the removal of the angular momentum coupling of roll and pitch dynamics, on might expect that the dynamics for the contra-rotating system would decouple into pure longitudinal and lateral modes. However, there are aerodynamic moments that still couple the roll and pitch dynamics, although not as strongly. Specifically, there is a small roll moment that is present when the vehicle is near, but not at hover. This effect was also presented in Stiltner et al. 7. In the ideal case with no aerodynamic coupling, the elimination of angular momentum would decouple the dynamics, allowing for a simpler control law design. However, the instabilities would be faster, requiring a control system with faster response times. Another interesting result is that Modes 1, 5, and 6 of both sensitivity matrices are essentially the same. Mode 1 is a neutrally stable mode that only affects the yaw attitude angle ψ. Similarly, Mode 6 is another neutrally stable mode that only affects the yaw rate R. Finally, Mode 5 is a stable mode that only affects the vehicle vertical rate W. This can be thought of as a vertical climb damping mode. The modal analysis and comparison of the two configurations shows that the contra-rotating system has faster unstable dynamics than a conventional system, while both have highly coupled dynamics. As such, the contra rotating system will have more stringent requirements on the speed of the vehicle s control system 13 of 16

14 Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode ±14.97i -2.13±0.83i 1.37±1.0i U 0 1.9E V 0 1.9E W φ 0 1.9E θ 0 1.9E ψ P E Q E R Table 1. Sensitivity matrix for the ducted fan in Figure 5 with the single fan configuration. Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode ±0.72i 2.35±2.92i 1.09±3.64i U V W φ θ ψ P Q R Table 2. Sensitivity matrix for the ducted fan in Figure 5 with the contra-rotating fans. 14 of 16

15 than a conventional ducted fan. For example, the shortest time to double for the contra-rotating system is 0.1 seconds, and therefore the control system must be able to run much faster. In comparison, the shortest time to double for the conventional ducted fan is about 0.2 seconds, twice as long as the contra-rotating system. On the other hand, the control system of the conventional ducted fan UAV must account for the strong gyroscopic coupling due to the fan s angular momentum. V. Conclusions and Discussion The effects of fan angular momentum and the resulting gyroscopic torques during vehicle maneuvers have been explored for ducted fan UAVs. A design constraint equation for control vane sizing has been developed for conventional single ducted fan UAVs to ensure the ability to adequately counter gyroscopic induced torques. The inertia of the fan was shown to scale with fan radius to the 4 th and 5 th powers for thin shell composite construction and uniform density solid construction, respectively. The overall effects of angular momentum were shown to scale with the 2nd and 3rd power of fan radius, again depending on fan construction technique. These gyroscopic effects can be addressed through varying the vehicle design s disk loading, vane lift coefficient, vane area, vane moment arm, fan RPM, and vehicle angular rate goals. Use of contra-rotating fans largely eliminates the effects of angular momentum and gyroscopic torque in a ducted fan vehicle. The vehicle dynamics of a conventional single fan design and a contra-rotating ducted fan design were analyzed using linearized models derived from wind tunnel data. Modal analysis showed that the gyroscopic precession mode present in the conventional single fan design was eliminated in the contra-rotating design. However, the roll and pitch dynamics of the contra-rotating design were not completely decoupled because there are aerodynamic moments that couple the lateral and longitudinal vehicle states. In addition, the unstable modes in the contra-rotating ducted fan grew on average twice as fast as the unstable modes of the conventional design. While the fan angular momentum in a single fan design may limit the body angular rates, this analysis shows its presence provides some benefit in reducing vehicle instability. Choosing a contra-rotating design to eliminate the gyroscopic coupling may enable a simpler control law design, but the requirements on the speed of the sensors and control system may be significantly higher. VI. Acknowledgements The authors would like to acknowledge Dr. M. Christopher Cotting for his initial support of this research, as well as his guidance and input concerning the work performed. We would also like to acknowledge the rest of our coworkers at AVID who provided input and ideas toward the completion of this research. References 1 Pinder, S., Control Strategy for a Four-Rotor VTOL UAV, No in 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January Shin, J.-Y., Lim, K., and Moerder, D., Attitude Control for an Aero-Vehicle Using Vector Thrusting and Variable Speed Control Moment Gyros, No in AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California, August Lim, K. and Moerder, D., CMG-Augmented Control of a Hovering VTOL Platform, No in AIAA Guidance, Navigation, and Control Conference and Exhibit, Hilton Head, South Carolina, August anon, Hovering UAVs Provide Better Access in Difficult Situations, docs/auroraflight pdf, anon, Autonomous UAV Mission Systems, 6 White, A., Upgrades for gmav in light of Iraq ops, upgrades-for-gmav-in-light-of-iraq-ops/6185/,january Stiltner, B., Cotting, M. C., and Ohanian, J., Derivation and Analysis of the Equations of Motion for a Ducted Fan UAV, AIAA Atmospheric Flight Mechanics Conference, Portland, Oregon, August Schaub, H. and Junkins, J. L., Analytical Mechanics of Space Systems,AmericanInstituteofAeronauticsandAstronautics, Inc., Pflimlin, J. M., Soueres, P., and Hamel, T., Hovering Flight Stabilization in Wind Gusts for Ducted Fan UAV, 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, December 2004, pp Johnson, E. N. and Turbe, M. A., Modeling, Control, and Flight Testing of a Small Ducted-Fan Aircraft, Journal of Guidance, Control, and Dynamics,Vol.29,No.4,July-August2006,pp Etkin, B., Dynamics of Atmospheric Flight,Dover,Mineloa,NewYork, McRuer, D., Ashkenas, I., and Graham, D., Aircraft Dynamics and Automatic Control, PrincetonUniversityPress, Princeton, New Jersey, of 16