AIAA 2009-7287 16th AIAA/DLR/DGLR International Space Planes and Hypersonic Systems and Technologies Conf Control-Relevant Modeling, Analysis, and Design for Scramjet-Powered Hypersonic Vehicles Armando A. Rodriguez ∗ Jeffrey J. Dickeson † Srikanth Sridharan Akshay Korad § Jaidev Khatri ¶ ‡ Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Dept. of Electrical Eng., Fulton School of Eng., Arizona State University, Tempe, AZ, 85287, U.S.A. Jose Benavides  Don Soloway ∗∗ Guidance, Navigation, and Control, NASA Ames Research Center, Moffett Field, CA, 94035, U.S.A. Atul Kelkar †† Jerald M. Vogel ‡‡ Dept. of Aerospace Eng., College of Eng., Iowa State University, Ames, IA, 50011, U.S.A. Within this paper, control-relevant vehicle design concepts are examined using a widely used 3 DOF (plus flexibility) nonlinear model for the longitudinal dynamics of a generic carrot-shaped scramjet powered hypersonic vehicle. Trade studies associated with vehicle/engine parameters are examined. The impact of parameters on control-relevant static properties (e.g. level-flight trimmable region, trim controls, AOA, thrust margin) and dynamic properties (e.g. instability and right half plane zero associated with flight path angle) are examined. Specific parameters considered include: inlet height, diffuser area ratio, lower forebody compression ramp inclination angle, engine location, center of gravity, and mass. Vehicle optimizations is also examined. Both static and dynamic considerations are addressed. The gap-metric optimized vehicle is obtained to illustrate how this controlcentric concept can be used to “reduce” scheduling requirements for the final control system. A classic inner-outer loop control architecture and methodology is used to shed light on how specific vehicle/engine design parameter selections impact control system design. In short, the work represents an important first step toward revealing fundamental tradeoffs and systematically treating control-relevant vehicle design. I. INTRODUCTION AND OVERVIEW Motivation. With the historic 2004 scramjet-powered Mach 7 and 10 flights of the X-43A1–4 , hypersonics research has seen a resurgence. This is attributable to the fact that air-breathing hypersonic propulsion is viewed as the next critical step toward achieving (1) reliable, affordable, routine access to space, as well as (2) global reach vehicles. Both of these objectives have commercial as well as military implications. While rocket-based (combined cycle) propulsion systems5 are needed to reach orbital speeds, they are much more expensive to operate because they must carry oxygen. This is particularly costly when traveling at lower altitudes through the troposphere (i.e. below 36,152 ft). Current rocket-based systems also do not exhibit the desired levels of reliability and flexibility (e.g. airplane like takeoff and landing options). For this reason, ∗ Professor, Dept. of Electrical Engineering, Arizona State University, and AIAA Member. This research has been supported, in part, by NASA grant NNX07AC42A. † NASA PhD Fellow, Dept. of Electrical Engineering, Arizona State University, and AIAA Student Member. ‡ MS Student, Dept. of Electrical Engineering, Arizona State University, and AIAA Student Member. § MS Student, Dept. of Electrical Engineering, Arizona State University, and AIAA Student Member. ¶ MS Student, Dept. of Electrical Engineering, Arizona State University, and AIAA Student Member.  Hardware/Controls Engineer, Mission Critical Technologies, Inc. and AIAA Member. ∗∗ Hypersonics Project Associate Principal Investigator, NASA Ames Research Center, and AIAA Member. †† Professor, Dept. of Aerospace Engineering, Iowa State University, and AIAA Member. ‡‡ Emeritus Professor, Dept. of Aerospace Engineering, Iowa State University, and AIAA Member.This research has been supported, in part, by NASA grant NNL08AA38C. 1 of 45 Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. American Institute of Aeronautics and Astronautics much emphasis has been placed on two-stage-to-orbit (TSTO) designs that involve a turbo-ram-scramjet combined cycle first stage and a rocket-scramjet second stage. This paper focuses on control challenges associated with scramjet-powered hypersonic vehicles. Such vehicles are characterized by significant aerothermo-elastic-propulsion interactions and uncertainty1–18 . Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Controls-Relevant Hypersonic Vehicle Modeling. The following significant body of work (20052007)7–9, 19–28 examines aero-thermo-elastic-propulsion modeling and control issues using a first principles nonlinear 3-DOF longitudinal dynamical model which exploits inviscid compressible oblique shock-expansion theory to determine aerodynamic forces and moments, a 1D Rayleigh flow scramjet propulsion model with a variable geometry inlet, and an Euler-Bernoulli beam based flexible model. The vehicle is 100 ft long with weight (density) 6154 lb per foot of depth and has a bending mode at about 21 rad/sec. The controls include: elevator, stoichiometrically normalized fuel equivalency ratio (FER), diffuser area ratio (not considered in this work), and a canard (not considered in this work). A more complete description of the vehicle model can be found in previous works7, 29 . More recent modeling efforts have focused on improved propulsion modeling30, 31 that captures precombustion shocks, dissociation, wall heat transfer, skin friction, fuel-air mixing submodel, and finite-rate chemistry. The computational time associated with the enhanced model is significant, thus making it cumbersome for control-relevant analysis. The simple 1D Rayleigh flow engine model discussed within7, 19, 26, 29 will be used in the current paper. Hypersonic Vehicle Control Issues. Within this paper, we exploit the generic carrot-shaped vehicle 3DOF (plus flexibility) model presented in7, 19, 26, 29 . A myriad of issues exist that make control design for this hypersonic vehicle a potentially challenging problem: • Input/Output Coupling. For this system, velocity control is achieved via the FER input. Flight path angle (FPA) control is achieved with the elevator32 . However, there is significant coupling between F ER and FPA. • Unstable/Nonminimum Phase. Tail controlled vehicles are characterized by a non-minimum phase (right half plane, RHP) zero that is associated with the elevator to FPA map28 . This RHP zero limits the achievable elevator-FPA bandidth (BW) 33–35 . In addition, the rearward situated scramjet and cg (center of gravity), implies an inherent pitch-up vehicle instability. This instability requires a minimum BW for stabilization29 . To address these potentially conflicting specifications, one approach has been to exploit the addition of a canard19, 32, 36–38 . It is understood, of course, that any canard approach would face severe heating, structural, and reliability issues. • Varying Dynamic Characteristics. Within29 , it is shown that the nonlinear model changes significantly as a function of the flight condition. Specifically, it is shown that the vehicle pitch-up instability and non-minimum phase (NMP) zero vary significantly across the vehicle’s trimmable region. In addition, the mass of the vehicle can be varied during a simulation in order to represent fuel consumption. Several methods have been presented in the literature to deal with the nonlinear nature of the model. Papers addressing modeling issues include: nonlinear modeling of longitudinal dynamics28 , heating effects and flexible dynamics9, 24, 39 , FPA dynamics36 , unsteady and viscous effects8, 20 , and high fidelity engine modeling30, 31, 40 . Papers addressing nonlinear control issues include: control via classic inner-outer loop architecture41 , nonlinear robust/adaptive control32 , robust linear output feedback38 , control-oriented modeling19 , linear parameter-varying control of flexible dynamics42 , saturation prevention22, 43, 44, and thermal choking prevention29, 44. • Uncertain Flexible Modes and Coupling to Propulsion. Flexible dynamics have been captured within the model by approximating a free-free Euler-Bernoulli beam using the assumed modes method24 . Three flexible modes are used to approximate the structural dynamics. A damping factor of ζ = 0.02 is assumed. The associated mode frequencies are ω1 = 21.02 rad/sec, ω2 = 50.87 rad/sec, ω3 = 101 rad/sec. These modes must be adequately addressed within the control system design process. While performance can be improved by increasing controller complexity (e.g. higher order notches)42 , one 2 of 45 American Institute of Aeronautics and Astronautics must be wary of, and careful in dealing with, modal/damping uncertainty issues. This is particularly important because structural flexing impacts the bow shock. This, in turn impacts the scramjet’s inlet properties, thrust generated, aft body forces, the associated pitching moments, and hence the vehicle’s attitude. Given the tight altitude-Mach flight regime - within the air-breathing corridor5 - that such vehicle must operate within, the concern is amplified. In short, one must be careful that the control system BW and complexity are properly balanced so that these lightly damped flexible modes are not overly excited. Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 • Control Saturation Constraints. Control saturation is of particular concern for unstable vehicles such as the one under consideration. Two specific saturation nonlinearities are a concern for any control system implementation. – Maximum Elevator/Canard Deflection and Instability. FPA is controlled via the elevator/canard combination36 . Because these dynamics are inherently unstable, elevator saturation can result in instability43 . Classical anti-windup methods may be inadequate to address the associated issues - particularly when the vehicle is open loop unstable. The constraint enforcement method within43, 45 and generalized predictive control46 have been used to address such issues. It should be noted that control surface/actuator rate limits must also be properly addressed by the control system in order to avoid instability. – Thermal Choking/Unity FER: State Dependent Constraint. As heat is added within the combustor, the supersonic air flow is slowed. If enough heat is added, the combustor exit Mach number will approach unity, and the flow is said to be thermally choked47 . If additional heat is added, the upstream conditions can be altered. This can (in principle) lead to engine unstart5 - a highly undesirable condition. The amount of FER that causes thermal choking at a particular flight condition is referred to as the thermal choking FER, or F ERT C . In general, F ERT C depends upon the free-stream Mach, free-stream temperature, pressure, and density (which depend on the altitude), and the flow turn angle (vehicle geometry + AOA + elastic deflection)29, 46 . In addition, since the model does not capture what happens when F ER ≥ 128 , it is natural to restrict FER below unity. Given the above, it follows that the minimum of these two constraints dictates the available FER at a given flight condition. The resulting state dependent FER constraint can be computed (on-line) based on the flight condition, and must be accounted for by the control law. Here, uncertainty is of great concern because of the potential unstart issues - issues not captured within the model. Engineers, of course, would try to “build-in protection” so that this is avoided. As such, engineers are forced to tradeoff operational envelop for enhance unstart protection. Control-Relevant Vehicle Design Issues. Despite the successful integrated approach taken by the X43A team, as well as other prior successful flight control efforts, far too often aerospace vehicle design has not significantly involved the discipline of controls until very late in the vehicle design process or even afterwards. Research programs over the past two decades have suggested that for the anticipated hypersonic vehicles, the traditional “sequential” approach is not likely to work. This is attributable, in part, to complex uncertain nonlinear coupled unstable, non-minimum phase, flexible dynamics together with stringent flight corridor and variable constraints (e.g. specific impulse, fuel use, maximum dynamic pressure, engine temperatures and pressures). For such vehicles, an integrated multidisciplinary “parallel” approach - involving multiple disciplines up front - is essential. This is particularly true when tight flight control specifications must be satisfied in the presence of significant uncertainty. Goals and Contributions of Paper. This paper addresses a myriad of issues that are of concern to both vehicle and control system designers. In short, this paper represents a step toward answering the following critical control-relevant vehicle design questions: 1. How do vehicle/engine design properties impact a vehicle’s static and dynamic properties? 2. How do these impact control system design? 3. How should a hypersonic vehicle be designed to permit/facilitate the development of an adequately robust control system? 4. What fundamental tradeoffs exist between vehicle design objectives and vehicle control objectives? 3 of 45 American Institute of Aeronautics and Astronautics More specifically, in this paper we consider how the following parameters impact the static and dynamic properties of a vehicle: • engine inlet height, diffuser area ratio, compression ramp inclination, engine location (distance behind vehicle nose), vehicle cg (center-of-gravity), and vehicle mass Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Vehicle optimization is also considered. It is specifically shown that a gap-optimized vehicle can “reduce” control system scheduling requirements. A classic decentralized inner-outer loop control system architecture is used to illustrate how vehicle/engine parameter selection The gap metric represents a system-theoretic measure that quantifies the “distance” between two dynamical systems and whether or not a common controller can be deployed for the systems under consideration48, 49. Within this paper, the gap metric is used to obtain a “gap-optimized vehicle” which “reduces” how much the vehicle varies throughout the trimmable region is obtained. A nonlinear pull-up maneuver is used to show that a “gap-optimized vehicle” can “reduce” control system scheduling requirements. Future work will examine the utility of pursuing gap-optimized vehicles or optimizing vehicles subject to gap constraints. In short, this paper illustrates fundamental tradeoffs that vehicle and control system designers should jointly consider during the early stages of vehicle conceptualization/design. The paper also sheds light on how specific vehicle/engine parameter selections impact control system design - thus providing a contribution to control-relevant vehicle design. While vehicle designers may want to use a higher fidelity model (e.g. Euler based CFD with boundary layer reconstruction or Navier-Stokes based CFD50 ) to conduct more accurate vehicle trade studies, this paper shows that a (first principles) 3DOF nonlinear engineering model - such as that used in the paper - may be very useful during the early stages of vehicle conceptualization and design. Organization of Paper. The remainder of the paper is organized as follows. • Section II provides an overview of the dynamical model to be used in our studies. • Section III presents engine parameter trade study results as well as a new set of nominal engine parameter values. • Section IV presents vehicle parameter trade study results. • Section V presents vehicle optimization results. • Section VI discusses how control system design is impacted by vehicle/engine design parameter selection. • Section VII summarizes the paper and presents directions for future research. II. DESCRIPTION OF NONLINEAR MODEL In this paper, we consider a first principles nonlinear 3-DOF dynamical model for the longitudinal dynamics of a generic scramjet-powered hypersonic vehicle7–9, 19–28 . The vehicle is 100 ft long with weight (density) 6,154 lb per foot of depth and has a bending mode at about 22 rad/sec. The controls include: elevator, stoichiometrically normalized fuel equivalency ratio (FER), diffuser area ratio (not considered in our work), and a canard. The vehicle may be visualized as shown in Figure 18 . Modeling Approach. The following summarizes the modeling approach that has been used. • Aerodynamics. Pressure distributions are computed using inviscid compressible oblique-shock and Prandtl-Meyer expansion theory10, 16, 28, 47. Air is assumed to be calorically perfect; i.e. constant specific def c heats and specific heat ratio γ = cpv = 1.410, 47 . A standard atmosphere is used. Viscous drag effects (i.e. an analytical skin friction model) are captured using Eckerts temperature reference method8, 10 . This relies on using the incompressible turbulent skin friction coefficient formula for a flat plate at a reference temperature. Of central importance to this method is the so-called wall temperature used. The model assumes a nominal wall temperature of 2500◦ R8 . While our analysis has shown that this assumption is reasonable for conducting preliminary trade studies, the wall temperature 4 of 45 American Institute of Aeronautics and Astronautics 8 Oblique Shock Pu, Mu, Tu 6 4 τ2 Freestream 2 Feet 0 −2 −4 Elevator τu τl CG Inlet Pe, Me, Te Diffuser Nozzle Combustor −6 Shear Layer (Plume) −8 P1, M1, T1 Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 −10 Expansion Fan −20 0 20 Pb, Mb, Tb 40 Feet 60 80 100 Figure 1. Schematic of Hypersonic Scramjet Vehicle used should (in general) depend upon the flight condition being examined. As such, modeling heat transfer to the vehicle via parabolic heat equation partial differential equations (pdes) as well as modeling a suitable thermal protection system is essential for obtaining insight into wall temperature selection9 . This will be addressed more comprehensively in a subsequent publication. Unsteady effects (e.g. due to rotation and flexing) are captured using linear piston theory8, 51 . The idea here is that flow velocities induce pressures just as the pressure exerted by a piston on a fluid induces a velocity. • Propulsion. A single (long) forebody compression ramp provides conditions to the rear-shifted scramjet inlet. The inlet is a variable geometry inlet (variable geometry is not exploited in our work). The model assumes the presence of an (infinitely fast) cowl door which uses AOA to achieve shockon-lip conditions (assuming no forebody flexing). Forebody flexing, however, results in air mass flow spillage28 . At the design cruise condition, the bow shock impinges on the engine inlet (assuming no flexing). At speeds below the design-flight condition and/or larger flow turning angles, the cowl moves forward to capture the shock. At larger speeds and/or smaller flow turning angles, the bow shock is swallowed by the engine. In either case, there is a shock reflected from the cowl or within the inlet (i.e. we have a bow shock reflection). This reflected shock further slows down the flow and steers it into the engine. It should be noted that shock-shock interactions are not modeled. For example, at larger speeds and smaller flow turning angles there is a shock off of the inlet lip. This shock interacts with the bow shock. This interaction is not captured in the model. The model uses liquid hydrogen (LH2) as the fuel. It is assumed that fuel mass flow is negligible compared to the air mass flow. The model also captures linear fuel depletion. Thrust is linearly related to FER for all expected FER values. For large FER values, the thrust levels off. In practice, when FER > 1, the result is decreased thrust. This phenomena28 is not captured in the model. As such, control designs based on this nonlinear model (or derived linear models) should try to maintain FER below unity. The model also captures thermal choking. In what follows, we show how to compute the FER required to induce thermal choking as well as the so-called thermal choking FER margin. The above will lead to a useful FER margin definition - one that is useful for the design of control systems for scramjet-powered hypersonic vehicles. Finally, it should be noted that the model offers the capability for addressing linear fuel depletion. This feature was exploited for the nonlinear simulation presented in this paper. • Structural. A single free-free Euler-Bernoulli beam partial differential equation (infinite dimensional pde) model is used to capture vehicle elasticity. As such, out-of-plane loading, torsion, and Timoshenko 5 of 45 American Institute of Aeronautics and Astronautics effects are neglected. The assumed modes method (based on a global basis) is used to obtain natural frequencies, mode shapes, and finite-dimensional approximants. This results in a model whereby the rigid body dynamics influence the flexible dynamics through generalized forces. This is in contrast to the model described within [28] which uses fore and aft cantilever beams (clamped at the center of gravity) and leads to the rigid body modes being inertially coupled to the flexible modes (i.e. rigid body modes directly excite flexible modes). Within the current model, forebody deflections influence the rigid body dynamics via the bow shock which influences engine inlet conditions, thrust, lift, drag, and moment24 . Aftbody deflections influence the AOA seen by the elevator. As such, flexible modes influence the rigid body dynamics. Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 The nominal vehicle is 100 ft long. The associated beam model is assumed to be made of titanium. It is 100 ft long, 9.6 inches high, and 1 ft wide (deep). This results in the nominal modal frequencies ω1 = 21.02 rad/sec, ω2 = 50.87 rad/sec, ω3 = 101 rad/sec. When the height is reduced to 6 inches, then we obtain the following reduced modal frequencies: ω1 = 10.38 rad/sec, ω2 = 25.13 rad/sec, ω3 = 49.89 rad/sec. Future work will examine vehicle mass-flexibility-control trade studies19 . • Actuator Dynamics. Simple first order actuator models (contained within the original model) were used 20 10 20 , FER - s+10 , canard - s+20 (Note: canard not used in our in each of the control channels: elevator - s+20 study). These dynamics did not prove to be critical in our study. An elevator saturation of ±30◦ was used.22, 43 It should be noted, however, that these limits were never reached in our studies41 . Within this paper, we consider a pull up maneuver that does not result in elevator saturation. Future work will consider more aggressive pull up maneuvers where elevator position and rate saturation become very important given the vehicle’s (open loop) unstable dynamics. A (state dependent) saturation level - associated with FER (e.g. thermal choking and unity FER) - was also directly addressed41 . This (velocity bandwidth limiting) nonlinearity is discussed below. Generally speaking, the vehicle exhibits unstable non-minimum phase dynamics with nonlinear aero-elasticpropulsion coupling and critical (state dependent) FER constraints. The model contains 11 states: 5 rigid body states (speed, pitch, pitch rate, AOA, altitude) and 6 flexible states. Unmodeled Phenomena/Effects. All models possess fundamental limitations. Realizing model limitations is crucial in order to avoid model misuse. Given this, we now provide a (somewhat lengthy) list of phenomena/effects that are not captured within the above nonlinear model. (For reference purposes, flow physics effects and modeling requirements for the X-43A are summarized within [52].) • Dynamics. The above model does not capture longitudinal-lateral coupling and dynamics53 and the associated 6DOF effects. • Aerodynamics. Aerodynamic phenomena/effects not captured in the model include the following: boundary layer growth, displacement thickness, viscous interaction, entropy and vorticity effects, laminar versus turbulent flow, flow separation, high temperature and real gas effects (e.g. caloric imperfection, electronic excitation, thermal imperfection, chemical reactions such as 02 dissociation)10 , non-standard atmosphere (e.g. troposphere, stratosphere), unsteady atmospheric effects6 , 3D effects, aerodynamic load limits. • Propulsion. Propulsion phenomena/effects not captured in the model include the following: cowl door dynamics, multiple forebody compression ramps (e.g. three on X-43A54, 55 ), forebody boundary layer transition and turbulent flow to inlet54, 55 , diffuser losses, shock interactions, internal shock effects, diffuser-combustor interactions, fuel injection and mixing, flame holding, engine ignition via pyrophoric silane3 (requires finite-rate chemistry; cannot be predicted via equilibrium methods56 , finite-rate chemistry and the associated thrust-AOA-Mach-FER sensitivity effects31 , internal and external nozzle losses, thermal choking induced phenomena (2D and 3D) and unstart, exhaust plume characteristics, cowl door dynamics, combined cycle issues5 . Within [31], a higher fidelity propulsion model is presented which addresses internal shock effects, diffuser-combustor interaction, finite-rate chemistry and the associated thrust-AOA-Mach-FER sensitivity effects. While the nominal Rayleigh-based model (considered here) exhibits increasing thrustAOA sensitivity with increasing AOA, the more complex model in31 exhibits reduced thrust-AOA sensitivity with increasing AOA - a behavior attributed to finite-chemistry effects. 6 of 45 American Institute of Aeronautics and Astronautics Future work will examine the impact of internal engine losses, high temperature gas effects, and nozzle/plume issues. Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 • Structures. Structural phenomena/effects not captured in the model include the following: out of plane and torsional effects, internal structural layout, unsteady thermo-elastic heating effects, aerodynamic heating due to shock impingement, distinct material properties,57 and aero-servo-elasticity58, 59. – Heating-Flexibility Issues. Finally, it should be noted that Bolender and Doman have addressed a variety of effects in their publications. For example, within [9, 24] the authors address the impact of heating on (longitudinal) structural mode frequencies and mode shapes. Within [9], the authors consider a sustained two hour straight and level cruise at Mach 8, 85 kft. It is assumed that no fuel is consumed (to focus on the impact of heat addition). The paper assumes the presence of a thermal protection system (TPS) consisting of a PM2000 honeycomb outer skin followed by a layer of silicon dioxide (SiO2 ) insulation. The vehicle - modeled by a titanium beam - is assumed to be insulated from the cryogenic fuel. The heat rate is computed via classic heat transfer equations that depend on speed (Mach), altitude (density), and the thermal properties of the TPS materials as well as air - convection and radiation at the air-PM2000 surface, conduction within the three TPS materials. The initial temperature of all three TPS materials was set to 559.67◦R = 100◦ F ). The maximum heat rate (achieved at the flight’s inception) U was approximately 12 fBT t2 sec (1 foot aft of the nose). By the end of the two hour level flight, the average temperature within the titanium increased by 125◦ R and it was observed that the vehicle’s (longitudinal) structural frequencies did not change appreciably (< 2%) [9, page 18]. U When one assumes a constant 15 fBT t2 sec heat rate at the air-PM2000 surface (same initial TPS ◦ ◦ temperature of 559.67 R = 100 F ), then after two hours of level flight the average temperature within the titanium increased by 200◦ R [9, page 19]. In such a case, it can be shown that the vehicle’s (longitudinal) structural frequencies do not change appreciably (< 3%). This high heat rate scenario gives one an idea by how much the flexible mode frequencies can change by. Such information is critical in order to suitably adapt/schedule the flight control system. Comprehensive heating-mass-flexibility-control studies will be examined further in a subsequent publication. • Actuator Dynamics. Future work will examine the impact of actuators that are rate limited; e.g. elevator, fuel pump. It should be emphasized that the above list is only a partial list. If one needs fidelity at high Mach numbers, then many other phenomena become important; e.g. O2 dissociation10 . Longitudinal Dynamics. The equations of motion for the 3DOF flexible vehicle are given as follows:   T cos α − D v̇ = − g sin γ (1) m     v g L + T sin α − +q+ cos γ (2) α̇ = − mv v RE + h M (3) q̇ = Iyy ḣ θ̇ = = v sin γ q η¨i = −2ζωi η˙i − ωi2 ηi + Ni def θ−α  2 RE g0 RE + h γ g = = (4) (5) i = 1, 2, 3 (6) (7) (8) where L denotes lift, T denotes engine thrust, D denotes drag, M is the pitching moment, Ni denotes generalized forces, ζ demotes flexible mode damping factor, ωi denotes flexible mode undamped natural 7 of 45 American Institute of Aeronautics and Astronautics frequencies, m denotes the vehicle’s total mass, Iyy is the pitch axis moment of inertia, g0 is the acceleration due to gravity at sea level, and RE is the radius of the Earth. Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 • States. Vehicle states include: velocity v, FPA γ, altitude h, pitch rate q, pitch angle θ, and the flexible body states η1 , η˙1 , η2 , η˙2 , η3 , η˙3 . These eleven (11) states are summarized in Table 1.  1 2 3 4 5 6 7 8 9 10 11 Symbol v γ α q h η1 η˙1 η2 η˙2 η3 η˙3 Description speed flight path angle angle-of-attack (AOA) pitch rate altitude 1st flex mode 1st flex mode rate 2nd flex mode 2nd flex mode rate 3rd flex mode 3rd flex mode rate Units kft/sec deg deg deg/sec ft - Table 1. States for Hypersonic Vehicle Model • Controls. The vehicle has three (3) control inputs: a rearward situated elevator δe , a forward situated canard δc a , and stoichiometrically normalized fuel equivalence ratio (FER). These control inputs are summarized in Table 2. In this paper, we will only consider elevator and FER; i.e. the canard has been removed.  1 2 3 Symbol F ER δe δc Description stoichiometrically normalized fuel equivalence ratio elevator deflection canard deflection Units deg deg Table 2. Controls for Hypersonic Vehicle Model In the above model, we note that the rigid body motion impacts the flexible dynamics through the generalized forces. As discussed earlier, the flexible dynamics impact the rigid body motion through thrust, lift, drag, and moment. Nominal model parameter values for the vehicle under consideration are given in Table 3. Additional details about the model may be found within the following references7–9, 19–28 . Scramjet Model. The scramjet engine model is that used in28, 60 . It consists of an inlet, an isentropic diffuser, a 1D Rayleigh flow combustor (frictionless duct with heat addition47 ), and an isentropic internal nozzle. A single (long) forebody compression ramp provides conditions to the rear-shifted scramjet inlet. Although the model supports a variable geometry inlet, we will not be exploiting variable geometry in this def 2 paper; i.e. diffuser area ratio Ad = A A1 will be fixed with Ad = 1, see Figure 2). Bow Shock Conditions. A bow shock will occur provided that the flow deflection angle δs is positive; i.e. δs = AOA + forebody flexing angle + τ1l > 0◦ def (9) where τ1l = 6.2◦ is the lower forebody wedge angle (see Figure 1). If δs < 0, a Prandtl-Meyer expansion will occur. Given the above, a bow shock occurs when the following flow turning angle (FTA) condition is satisfied: FTA = AOA + forebody flexing angle > −6.2◦ . def a In (10) this paper, we have removed the canard. Future work will examine the potential utility of a canard as well as its viability. 8 of 45 American Institute of Aeronautics and Astronautics Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Parameter Total Length (L) Forebody Length (L1 ) Aftbody Length (L2 ) Engine Length Engine inlet height hi Upper forebody angle (τ1U ) Elevator position Diffuser exit/inlet area ratio Titanium Thickness First Flex. Mode (ωn1 ) Third Flex. Mode (ωn3 ) Nominal Value 100 ft 47 ft 33 ft 20 ft 3.25 ft 3o (-85,-3.5) ft 1 9.6 in 22.2 rad/s 94.8 rad/s Parameter Lower forebody angle (τ1L ) Tail angle (τ2 ) Mass per unit width Weight per unit width Mean Elasticity Modulus Moment of Inertia Iyy Center of gravity Elevator Area Nozzle exit/inlet area ratio Second Flex. Mode (ωn2 ) Flex. Mode Damping (ζ) Nominal Value 6.2o 14.342o 191.3024 slugs/ft 6,154.1 lbs/ft 8.6482 × 107 psi 86,723 slugs ft2 /ft (-55,0) ft 17 ft2 6.35 48.1 rad/s 0.02 Table 3. Vehicle Nominal Parameter Values Figure 2. Schematic of Scramjet Engine Properties Across Bow Shock. Let (M∞ , T∞ , p∞ ) denote the free-stream Mach, temperature, and def c pressure. Let γ = cvp = 1.4 denote the specific heat ratio for air - assumed constant in the model; i.e. air is calorically perfect.10 The shock wave angle θs = θs (M∞ , δs , γ) can be found as the middle root (weak shock solution) of the following shock angle polynomial28, 47 : sin6 θs + bsin4 θs + csin2 θs + d = 0 where   2 (γ + 1)2 γ−1 +1 2M∞ + c= + sin2 δs 4 2 M∞ 4 M∞ M2 + 2 b = − ∞2 − γsin2 δs M∞ (11) d=− cos2 δs 4 M∞ (12) The above can be addressed by solving the associated cubic in sin2 θs . A direct solution is possible if Emanuel’s 2001 method is used [47, page 143]. After determining the shock wave angle θs , one can determine properties across the bow shock using classic relations from compressible flow [47, page 135]; i.e. Ms , Ts , ps - functions of (M∞ , δs , γ): Ts T∞ ps p∞ Ms2 sin2 (θs − δs ) 2 2 sin2 θs + 1 − γ)((γ − 1)M∞ sin2 θs + 2) (2γM∞ 2 2 sin θ (γ + 1)2 M∞ s   2γ 2 2 = 1+ M∞ sin θs − 1 γ+1 2 M∞ sin2 θs (γ − 1) + 2 = 2 sin2 θ − (γ − 1) 2γM∞ s = 9 of 45 American Institute of Aeronautics and Astronautics (13) (14) (15) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 It should be noted that for large M∞ , the computed temperature Ts across the shock will be larger than it should be because our assumption that air is calorically perfect (i.e. constant specific heats) does not capture other forms of energy absorption; e.g. electronic excitation and chemical reactions10 . Properties Across Prandtl-Meyer Expansion. An expansion fan occurs when there is a flow over a convex corner; i.e. flow turns away from itself. More specifically to the bow, if δs < 0 a Prandtl-Meyer expansion will occur. To determine the properties across the expansion, let (M∞ , T∞ , p∞ ) denote the free-stream (supersonic) Mach, temperature, and pressure, respectively. If we let δ = −δs > 0 denote the expansion ramp angle (in radians), the properties across the expansion fan (Me , Te , pe ) can be calculated as follows28, 47 :    γ+1 γ−1 2 − 1) − tan−1 2 −1 tan−1 (M∞ ν1 = M∞ (16) γ−1 γ+1 ν2 = ν1 + δ (17)    γ+1 γ−1 f (Me ) = tan−1 (M 2 − 1) − tan−1 Me2 − 1 − ν2 = 0 (18) γ−1 γ+1 e Pe P∞ = Te T∞ = γ−1 2 2 M∞ γ−1 + 2 Me2 1+ 1 γ γ−1 (19) γ−1 2 2 M∞ 2 + γ−1 2 Me 1+ 1 (20) ν1 is the angle for which a Mach 1 flow must be expanded to attain the free stream Mach. Translating Cowl Door. The model assumes the presence of an (infinitely fast) translating cowl door which uses AOA to achieve shock-on-lip conditions (assuming no forebody flexing). Forebody flexing, however, results in an oscillatory bow shock and air mass flow spillage28 . A bow shock reflection (off of the cowl or inside the inlet) further slows down the flow and steers it into the engine. Shock-shock interactions are not modeled. • Impact of Having No Cowl Door. Associated with a translating cowl door are potentially very severe heating issues. For our vehicle, the translating cowl door can extend a great deal. For example, at Mach 5.5, 70kft, the trim FTA is 1.8◦ and the cowl door extends 14.1 ft. Of particular concern, due to practical cowl door heating/structural issues, is what happens when the cowl door is over extended through the bow shock. This occurs, for example, when structural flexing results in a smaller FTA (and hence a smaller bow shock angle) than assumed by the rigid-body shock-on-lip cowl door extension calculation. This is certainly a major concern. It leads one to ask the question: What happens to the vehicle properties if no cowl door is present? When the FTA is large or when the vehicle Mach is low, the shock angle increases and more air mass spillage would occur. Our analysis shows that the impact of neglecting the cowl door on the vehicle’s static properties is significant while the impact on the vehicle’s dynamic properties is negligible. This will receive further examination in a subsequent publication. Inlet Properties. The bow reflection turns the flow parallel into the scramjet engine28 . The oblique shock relations are implemented again, using Ms as the free-stream input, δ1 = τ1l as the flow deflection angle to obtain the shock angle θ1 = θ1 (Ms , δ1 , γ) and the inlet (or diffuser entrance) properties: M1 , T1 , p1 functions of (Ms , θ1 , γ). Diffuser Exit-Combustor Entrance Properties. The diffuser is assumed to be isentropic. The combustor entrance properties are therefore found using the formulae in28 , [47, pp. 103-104] - M2 = M2 (M1 , Ad , γ), 10 of 45 American Institute of Aeronautics and Astronautics T2 = T2 (M1 , M2 , γ), p2 = p2 (M1 , M2 , γ):  1+ γ−1 2 2 M2 2 M2 γ+1  γ−1  = A2d  T2 = T1  p2 = p1 1+ 1+ 1+  γ+1 γ−1 2 γ−1 2 M1 2 M1 1 2 2 (γ − 1)M1 1 2 2 (γ − 1)M2 1 + 12 (γ − 1)M12 1 + 12 (γ − 1)M22 (21) (22) γ  γ−1 (23) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 def 2 where Ad = A A1 is the diffuser area ratio. Also, one can determine the total temperature Tt2 = Tt2 (T2 , M2 , γ) at the combustor entrance can be found using [47, page 80]:   γ−1 2 M2 T2 . (24) Tt2 = 1 + 2   2 Since Ad = 1 in the model, it follows that M2 = M1 , T2 = T1 , p2 = p1 , and Tt2 = 1 + γ−1 2 M1 T1 = Tt1 . FER. The model uses liquid hydrogen (LH2) as the fuel. If f denotes fuel-to-air ratio and fst denotes stoichiometric fuel-to-air ratio, then the stoichiometrically normalized fuel equivalency ratio is given by def F ER = ffst ,5 .28 FER is the engine control. While FER is primarily associated with the vehicle velocity, its impact on FPA is significant (since engine is situated below vehicle cg). This coupling will receive further examination in what follows. Combustor Exit Properties. In this model, we have a constant area combustor where the combustion process is captured via heat addition. To determine the combustor exit properties, one first determines the change in total temperature across the combustor28 :    fst F ER Hf ηc − Tt2 ΔTc = ΔTc (Tt2 , F ER, Hf , ηc , cp , fst ) = (25) 1 + fst F ER cp where Hf = 51, 500 BTU/lbm is the heat of reaction for liquid hydrogen (LH2), ηc = 0.9 is the combustion efficiency, cp = 0.24 BTU/lbm◦ R is the specific heat of air at constant pressure, and fst = 0.0291 is the stoichiometric fuel-to-air ratio for LH25 . Given the above, the Mach M3 , temperature T3 , and pressure p3 at the combustor exit are determined by the following classic 1D Rayleigh flow relationships28 , [47, pp. 103-104]:       M22 1 + 12 (γ − 1)M22 M32 1 + 12 (γ − 1)M32 M22 ΔTc = + (26) (γM32 + 1)2 (γM22 + 1)2 (γM22 + 1)2 T2  2  2 1 + γM22 M3 (27) T3 = T2 1 + γM32 M2   1 + γM22 p3 = p2 . (28) 1 + γM32 c Given the above, one can then try to solve equation (26) for M3 = M3 M2 , ΔT T2 , , γ . This will have a solution provided that M2 is not too small, ΔTc is not too large (i.e. F ER is not too large or T2 is not too small. See discussion below. Thermal Choking FER (M3 = 1). Once the change in total temperature ΔTc = ΔTc (Tt2 , F ER, Hf , ηc , cp , fst ) across the combustor has been computed, it can be substituted into equation (26) and one can “try” to solve for M3 . Since the left hand side of equation (26) lies between 0 (for M3 = 0) and 0.2083 (for M3 = 1), it follows that if the right hand side of equation (26) is above 0.2083 then no solution for M3 exists. Since the first term on the right hand side of equation (26) also lies between 0 and 0.2083, it follows that this occurs when ΔTc is too large; i.e. too much heat is added into the combustor or too high an FER. In short, a solution M3 will exist provided that FER is not too large, T2 is not too small (i.e. altitude not too high), and the combustor entrance Mach M2 is not too small (i.e. FTA not too large). When M3 = 1, a condition referred to as thermal choking 5, 47 is said to exist. The FER that produces this we call the thermal choking FER - denoted F ERT C . In general, F ERT C will be a function of the following: M∞ , T∞ , and FTA. 11 of 45 American Institute of Aeronautics and Astronautics Physically, the addition of heat to a supersonic flow causes it to slow down. If the thermal choking FER (F ERT C ) is applied, then we will have M3 = 1 (i.e. sonic combustor exit). When thermal choking occurs, it is not possible to increase the air mass flow through the engine. Propulsion engineers want to operate near thermal choking for engine efficiency reasons5. However, if additional heat is added, the upstream conditions can be altered and it is possible that this may lead to engine unstart. This is highly undesirable. For this reason, operating near thermal choking has been described by some propulsion engineers as “operating near the edge of a cliff.” In general, thermal choking will occur if FER is too high, M∞ is too low, altitude is too high (T∞ too low), FTA is too high. See discussion below. Internal Nozzle. The exit properties Me = Me (M3 , An , γ), Te = Te (M3 , Me , γ), pe = pe (M3 , Me , γ) of the scramjet’s isentropic internal nozzle are founds as follows: Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287  1+ γ−1 2 2 Me 2 Me γ+1  γ−1 = A2n  1+  Te = T3  pe def Ae A3 where An = = p3 1+ 1+  γ+1 γ−1 2 γ−1 2 M3 M32 1 2 2 (γ − 1)M3 1 2 2 (γ − 1)Me 1 + 12 (γ − 1)M32 1 + 12 (γ − 1)Me2 (29) (30) γ  γ−1 (31) is the internal nozzle area ratio (see Figure 2). An = 6.35 is used in the model. Thrust due to Internal Nozzle. The purpose of the expanding internal nozzle is to recover most of the potential energy associated with the compressed (high pressure) supersonic flow. The thrust produced by the scramjet’s internal nozzle is given by47 Thrustinternal = ṁa (ve − v∞ ) + (pe − p∞ )Ae (32) where ṁa is the air mass flow through the engine, ve is the exit flow velocity, v∞ is the free-stream flow velocity. pe is the pressure at the engine √ exit plane, A1√is the engine inlet area, Ae is the engine exit area, ve = Me sose , v∞ = M∞ sos∞ , sose = γRTe , sos∞ = γRT∞ , and R is the gas constant for air. Because we assume that the internal nozzle to be symmetric, this internal thrust is always directed along the vehicle’s body axis. The mass air flow into the inlet is given as follows:   ⎧  sin(τ1l −α) γ ⎪ p L M + h cos(α) Oblique bow shock (swallowed by engine) ⎪ ∞ ∞ 1 i ⎪ ⎨   RT∞  tan(τ1l ) sin(θs )cos(τ1l ) Oblique bow shock - shock on lip p∞ M∞ RTγ∞ hi sin(θ (33) ṁa = s −α−τ1l ) ⎪  ⎪ ⎪ γ ⎩ p∞ M ∞ Lower forebody expansion fan RT∞ hi cos(τ1l ) External Nozzle. The purpose of the expanding external nozzle is recover the rest of the potential energy associated with the compressed supersonic flow. A nozzle that is too short would not be long enough to recover the stored potential energy. In such a case, the nozzle’s exit pressure would be larger than the free stream pressure and we say that it is under-expanded [47, Page-209]. The result is reduced thrust. A nozzle that is too long would result in the nozzle’s exit pressure being smaller than the free stream pressure and we say that it is over-expanded [47, Page-209]. The result, again, is reduced thrust. When the nozzle length is “properly selected,” the exit pressure is equal to the free stream pressure and maximum thrust is produced. Within [61, page 5],62 the authors say that the optimum nozzle length is about 7 throat heights. This includes the internal as well as the external nozzle. For our vehicle, the internal nozzle has no assigned length. This becomes an issue when internal losses are addressed. For the Bolender, et. al. model, the external nozzle length is 10.15 throat heights (with throat height hi = 3.25 ft). For the new engine design presented later on in this paper, the external nozzle length is 7.33 throat heights (with throat height hi = 4.5 ft). The external nozzle contributes a force on the upper aft body. This force can be resolved into 2 components - the component along the fuselage water line is said to contribute to the total thrust. This component is given by the expression: ⎤ ⎡ pe   pe ⎣ ln p∞ ⎦ tan(τ2 + τ1U ). (34) Thrustexternal = p∞ La pe p∞ p∞ − 1 12 of 45 American Institute of Aeronautics and Astronautics Plume Assumption. The engine’s exhaust is bounded above by the aft body/nozzle and below by the shear layer between the gas and the free stream atmosphere. The two boundaries define the shape of the external nozzle. Within [60, page 1315],7 , a critical assumption is made regarding the shape of the external nozzle-and-plume in order to facilitate (i.e. speed up) the calculation of the aft body pressure distribution. In short, the so-called “plume assumption” implies that the external nozzle-and-plume shape does not change with respect to the vehicle’s body axes. This implies that the plume shape is independent of the flight condition. Our (limited) studies to date show that this assumption is suitable for preliminary trade studies but a higher fidelity aft body pressure distribution calculation is needed to understand how properties change over the trimmable region. In short, our fairly limited studies suggest that the plume assumption impacts static properties significantly while dynamic properties are only mildly impacted. The impact of the plume assumption will be examined further in a subsequent publication. Trimmable Region and Vehicle Properties. Within this paper (and all our work to date), trim refers to a non-accelerating state; i.e. no translational or rotational acceleration. Moreover, all trim analysis has focused on level flight. Figure 3 shows the level-flight trimmable region for the nominal vehicle being considered7, 19, 26, 29, 41 (using the original nominal engine parameters). We are interested in how the static and dynamic properties of the vehicle vary across this region. Static properties of interest include: trim controls (FER and elevator), internal engine variables (e.g. temperature and pressure), thrust, thrust margin, AOA, L/D. Dynamic properties of interest include: vehicle instability and RHP transmission zero associated with FPA. Understanding how these properties vary over the trimmable region is critical for designing a robust nonlinear (gain-schedulted/adaptive) control system that will enable flexible operation. For example, consider a TSTO flight. The mated vehicles might fly up along q = 2000 psf to a desired altitude, then conduct a pull-up maneuver to reach a suitable staging altitude. 120 FER = 1 115 500 psf 110 Altitude (kft) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Total Thrust. The total thrust is obtained by adding the thrust due to the internal and external nozzles. 105 Thermal Choking 100 95 90 2000 psf 85 80 100 psf increments 75 70 4 5 6 7 8 9 10 11 12 13 Mach Figure 3. Visualization of Trimmable Region: Level-Flight, Unsteady-Viscous Flow, Flexible Vehicle, 2 Controls III. Engine Parameter Studies This section examines the impact of varying the engine inlet height hi and the diffuser area ratio Ad . Three basic engine designs were considered: (1) current (nominal, slow or small), (2) new (intermediate speed or size), and (3) aggressive (fast or large). In what follows, he denotes the internal nozzle exit height and An is the internal nozzle area ratio. 13 of 45 American Institute of Aeronautics and Astronautics 1. Current (Nominal, Slow or Small) Engine Design. The current (nominal, slow or small) engine design parameters are as follows7 : • hi = 3.25 he = 5 Ad = 1 An = 6.35. These parameters are not geometrically compatible with the vehicle shown in Figures 1 and 2; i.e. it would be impossible for the vehicle to have the pictorially implied flat base; i.e. internal nozzle exit height he equal to inlet height hi . Given the above, we set out to examine engines with he = hi . This implies that An = 1 Ad . Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 2. New (Intermediate Speed or Size) Engine Design. The new (intermediate speed or size) engine design parameters were selected as follows: • he = hi = 4.5 Ad = 0.15 1 Ad An = = 6.67. It should be noted that the value Ad = 0.1 was used within30, 31 . This new engine design will be used later in the paper for analysis and control system design purposes. 3. Aggressive (Fast or Large) Engine Design. An aggressive (fast or large) engine design was also considered: • he = hi = 6 Ad = 0.125 An = 1 Ad = 8. Constraints for Engine Parameter Trade Studies (Mach 8, 85 kft, Level Flight). The above engines were obtained by conducting parametric trade studies at Mach 8, 85 kft, level flight. The following constraints were assumed in our studies: • Flat base (internal nozzle exhaust height he equal to inlet height hi ); i.e. he = hi and An = A−1 d ; • Inlet height hi was varied between ±50% of nominal 3.25 ft; • Engine mass mengine was varied between ±50% of nominal 10 klbs; • Diffuser area ratio Ad was varied between 0.1 and 0.35. III..1. Impact of Engine Parameters on Static Properties (Mach 8, 85 kft, Level Flight) Figure 4 shows the impact of varying (hi , Ad ) on FER, combustor temperature (assuming calorically perfect air), thrust, thrust margin at Mach 8, 85 kft, level flight. Trim FER. From Figure 4 (upper left), one observes that the: • trim FER decreases with decreasing Ad for a fixed hi ; • trim FER decreases with increasing hi when hi < 7. These suggests choosing Ad small (i.e. significant diffuser compression) and hi large (i.e. large air mass flow) in order to achieve a small trim FER. The above, however, does not tell the full story since fuel consumption (trim fuel rate) - shown in Figure 4 (upper right) - increases with increasing hi , and the thrust margin decreases for Ad < 0.125. Trim Combustor Temperature. From Figure 4 (lower left), one also observes that: • Trim combustor temperature is a concave up function of (hi , Ad ) - minimized at hi ≈ 5.5, Ad ≈ 0.125. • Trim combustor temperature exhibits a steep gradient for Ad > 0.2 14 of 45 American Institute of Aeronautics and Astronautics FER at Mach 8, 85kft Fuel consumption (slugs/s) at Mach 8, 85kft 0.0 0.05 00 4500 0.35 0.05 7000 0 00 5 00 40 4000 00 30 4000 3000 00 20 2000 2000 0.15 0.2 0.25 0.3 Engine Diffuser Area Ratio 2 0.1 5000 00 30 3000 3 2000 6000 6000 20 6 50 00 0 600 5 5000 04 4000 0.35 00 7000 00 0 0 700 55 60 0 550 00 0 70 5500 600 2 0.1 0.15 0.2 0.25 0.3 Engine Diffuser Area Ratio 5000 7 40 5500 5000 4500 00 5000 0.3 00 00 0.2 0.25 Engine Inlet Height (ft) Thrust Margin at Mach 8, 85kft 500 40 20 6000 0 00 00 7000 00 50 0.15 30 70 5500 6000 00 0 4 600 45 50 0 0.1 5 3 6500 6000 0.15 0.1 4 8 64 0.2 5 0.1 7000 0.35 0.25 0.1 5 Combustor Temperature (R) at Mach 8, 85kft 55 5000 00 0 50 3 0.1 2 0.1 0.2 0.25 0.3 Engine Diffuser Area Ratio 5 0.2 0.15 00 0 .7 0.6 0.6 6 0.1 0.15 8 000 7 0.2 5 0.3 0.2 0.2 0.1 0.1 30 0.5 7 00 0.4 Engine Inlet Height (ft) 3 0.3 0. 7 0. 4 0.5 0.3 4 5 0.3 0.2 5 0.2 0.1 6000 0.7 0.5 0.4 0 500 Engine Inlet Height (ft) 0.5 0.4 Engine Inlet Height (ft) 5 2 0.1 0.35 1000 Figure 4. Trim FER, Combustor Temperature, Thrust, Thrust Margin: Dependence on hi , Ad (Mach 8, 85 kft) Since air is assumed to be calorically perfect, it follows that high temperature effects63 are not captured within the model. As such, the combustor temperatures in Figure 4 (lower left) may be excessively large. Future work will consider high temperature gas effects within the combustor. This is important because material temperature limits within the combustor are stated as 4500◦R within64 . Trim Thrust Margin. From Figure 4 (lower right), we also observe that • Trim thrust margin is a concave down function of (hi , Ad ) - maximized at hi ≈ 6, Ad ≈ 0.125. Trim Elevator and AOA. Figure 5 shows how trim elevator and AOA depend on (hi , Ad ). From Figure 5, one observes that the: 6 6.5 6 6.5 7 7 6. 7 4 5 6 5 5. 6 5 5.5 5 6. 3 5 5. 6 5 5 6 6.5 7 2 0.05 0.1 6 8 5.5 5 0.15 0.2 0.25 Engine Diffuser Area Ratio 0.3 0.35 Engine Inlet Height (ft) 76 7 7 6.5 7 7 6 5 4 3 Angle of Attack (deg) at Mach 8, 85kft 7.5 7.5 7.5 7.5 7 7 7 7 7 6.5 6.5 6.5 6.5 6 6 6 6 6 5.5 5.5 5.5 5.5 5 5 5 5 4.5 4.5 4 4 4 4 4.54 4 44.5 4 3.5 3.5 3.5 3.5 3.5 3 3 3 3 3 3 2.5 2.5 2.5 2.5 2.5 2 2 2 2 2 1.5 1.5 1.5 1.5 6 Elevator Deflection (deg) at Mach 8, 85kft 8 Engine Inlet Height (ft) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 0.6 0.6 3 0. 0.6 6 0 .5 0.4 0.3 7 8 50 0.7 Engine Diffuser Area Ratio 8 2 0.05 5 0.1 0.15 0.2 0.25 Engine Diffuser Area Ratio 0.3 0.35 Figure 5. Trim Elevator Deflection and Trim AOA: Dependence on (hi , Ad ) - Mach 8, 85 kft, Level Flight • Trim elevator increases with increasing hi for a fixed Ad ; • Trim elevator increases with decreasing Ad for a fixed hi ; 15 of 45 American Institute of Aeronautics and Astronautics 7 6 5 4 3 2 • Trim AOA increases with increasing hi for fixed Ad . Trim AOA decreases with increasing Ad for fixed hi . (For hi sufficiently large, trim AOA becomes nearly independent of Ad .) III..2. Impact of Engine Parameters on Dynamic Properties (Mach 8, 85 kft, Level Flight) Right Half Plane Zero at Mach 8, 85kft 8 Engine Inlet Height (ft) Right Half Plane Pole at Mach 8, 85kft 8 2.4 2.3 2.3 3.2 .4 2 2.5 2.4 7 2.4 5 2. .6 2 2.5 2.5 2.6 2.7 2.6 3 2.7 6 2.6 2.7 2.8 2.7 2.8 2.8 2.9 2.8 2.9 5 2.8 2.9 3 2.9 3 3 4 3 3.1 2.6 3.1 3.1 3.2 3 3.2 3.2 2.4 3.2 3.3 2 0.1 0.15 0.2 0.25 0.3 0.35 Engine Diffuser Area Ratio Engine Inlet Height (ft) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 The following figure shows the impact of hi and Ad on the vehicle instability and RHP transmission zero associated with FPA. 7 8 5.5 5.5 7.5 6 6 6 6 66 5 5.5 6.5 6.5 6.5 6.5 7 7 7 7 7 6.5 4 7.5 7.5 7.5 6 3 8 8 2 0.1 0.15 8 0.2 0.25 0.3 Engine Diffuser Area Ratio 0.35 5.5 Figure 6. Right Half Plane Pole and Zero: Dependence on (hi , Ad ) - Mach 8, 85 kft, Level Flight From Figure 6, one observes that the: • RHP pole increases with increasing Ad (for a fixed hi ) and decreasing hi (for a fixed Ad ); • RHP zero is constant with respect to Ad (for a fixed hi ); it decreases with increasing hi (for a fixed Ad ). III..3. Comparison of Engine Designs (Mach 8, 85 kft, Level Flight) In the previous sections, we considered the impact of increasing the engine height hi and diffuser area ratio Ad . We consider hi ≤ 6 (bound chosen due to combustor temperature effects) and Ad ≥ 0.125 (bound chosen due to thrust margin effects). Within this range, we observe the following trade-offs: • Increasing hi (fixed Ad ) – PROS: Trim FER reduces, trim combustor temperature decreases (till hi ≈ 5.5 at Ad = 0.125), trim thrust margin increases, trim lift-to-drag increases (for hi > 4.0 at Ad = 0.125, not shown),trim drag decreases (for hi > 4.0 at Ad = 0.125, not shown), RHP pole reduces; – CONS: Trim fuel rate increases, trim elevator increases, trim AOA increases, RHP zero decreases, trim lift-to-drag decreases (for hi < 4.0 at Ad = 0.125, not shown), trim drag increases (for hi < 4.0 at Ad = 0.125, not shown); • Decreasing Ad (fixed hi ) – PROS: Trim FER decreases, trim fuel rate decreases, trim combustor temperature decreases, trim thrust margin increases, RHP pole decreases (marginally); – CONS: Trim elevator increases, trim AOA increases (marginally), trim lift-to-drag decreases (not shown), trim drag increases (not shown). Table 4 shows a comparison of the three engine designs described above. The first is the nominal engine design presented in7–9, 19, 26, 28, 36, 38 As stated earlier, this configuration is geometrically unfeasible with respect to the implied flat base vehicle diagram shown in Figures 1 and 2. As can be seen from the table, it is generally “slow” with a small maximum acceleration capability. The second engine design will be used throughout the remainder of this paper. It satisfies each of the constraints listed at the beginning of Section III. The third configuration is a faster configuration that also obeys the constraints. 16 of 45 American Institute of Aeronautics and Astronautics Engine Nominal New Fast hi 3.25 ft 4.5 ft 6 ft Ad 1 0.15 0.125 Engine Nominal New Fast Trim FER 0.47 0.35 0.3 An 6.35 6.67 8 he 5 ft 4.5 ft 6 ft Trim Temp. 4500◦R 4750◦R 4400◦R Trim L2D 2.17 4.32 4.53 Trim Thrust 1250 lbf 1380 lbf 1445 lbf Trim Fuel Rate 0.051 slugs/s 0.071 slugs/s 0.094 slugs/s Trim Elev. 9.7◦ 6.25◦ 6.9◦ RHP Pole 3.1 2.9 2.6 Trim AOA 1◦ 3.1◦ 4.6◦ RHP Zero 8.5 6.9 6.1 Max Thrust 2834 lbf 4647 lbf 6582 lbf Z/P Ratio 2.7 2.4 2.3 Max Acceleration 11.1 ft/s2 18.2 ft/s2 28.8 ft/s2 Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Table 4. Comparison of 3 Engine Designs (Mach 8, 85 kft, Level Flight) Table 4 shows that with respect to the nominal (slow or small) engine, the new (intermediately fast and sized) engine has the following associated PROS and CONS at Mach 8, 85 kft, level flight: • PROS: smaller trim elevator, smaller trim FER, larger maximum thrust, larger thrust margin, larger maximum acceleration, smaller RHP pole; • CONS: larger engine, larger mass, larger trim thrust, larger trim combustor temperature, larger trim AOA, smaller RHP zero, smaller RHP zero-pole ratio. Future work will examine the above tradeoffs more precisely. IV. Vehicle Parameter Studies This section examines the impact of the following vehicle parameters: • Engine location with respect to vehicle nose, lower forebody compression ramp inclination, center of gravity (CG) location with respect to vehicle nose, and vehicle mass across the vehicle’s level-flight trimmable region. IV.A. Engine Location with respect to Vehicle Nose In this section, we examine the impact of moving the engine rearward with respect to the vehicle nose. The following assumptions will be made: • The new engine parameters are being used; i.e. he = hi = 4.5 ft, Ad = 0.15, An = 1 Ad = 6.67. • The engine location is varied between 40 and 60 ft from the vehicle nose. • The vehicle’s center of gravity (cg) moves with the engine location (assumed fixed near engine combustor). IV.A.1. Impact of Engine Location on Static Properties (Level Flight) Figure 7 shows how the trimmable region changes with the engine location. Specifically, it shows that the trimmable region shrinks as the engine is shifted rearward. Trim AOA. From Figure 8, we observe the following: • Trim AOA decreases as the engine is moved rearward for a fixed Mach and altitude. • For a fixed altitude, trim AOA dependence on engine location is a bit complex. When the engine is closer to the nose, trim AOA increases with Mach. This is because a forward situated engine results in a forward CG shift and hence a more stable aircraft. When the engine is moved rearward, trim AOA becomes insensitive to Mach variations. This dependence requires further examination. • For a fixed engine location and Mach, trim AOA increases with increasing altitude. 17 of 45 American Institute of Aeronautics and Astronautics Envelope Variations with Engine Location 120 50 ft 55 ft 60 ft 50 0 115 110 00 21 0 100 21 00 50 Altitude (kft) 105 95 90 00 85 21 75 70 4 6 8 10 Mach 12 14 16 Figure 7. Impact of Engine Location on (Level Flight) Trimmable Region AOA vs. Engine Position, h=100 kft Mach 8 Mach 9 Mach 10 Mach 11 6 AOA (deg) AOA vs. Engine Position, M=8 4 6 2 0 40 85 kft 90 kft 95 kft 100 kft 5 AOA (deg) 8 4 3 2 45 50 Engine Position (ft) 55 1 40 60 45 50 Engine Position (ft) 55 60 Figure 8. Impact of Engine Location on Trim AOA A smaller AOA is typically desirable when a designer wishes to increase the vehicle’s stall margin. (Generally speaking, an AOA that is too large results in flow separation and loss of lift.) Elevator vs. Engine Position, h=100 kft Elevator vs. Engine Position, M=8 Mach 8 Mach 9 Mach 10 Mach 11 15 10 5 0 40 45 50 Engine Position (ft) 55 20 Elevator deflection (deg) 20 Elevator deflection (deg) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 80 85 kft 90 kft 95 kft 100 kft 15 10 60 5 0 40 45 50 Engine Position (ft) Figure 9. Impact of Engine Location on Trim Elevator 18 of 45 American Institute of Aeronautics and Astronautics 55 60 Trim Elevator. From Figure 9, we observe the following: • The trim elevator deflection increases as the engine is moved rearward for a fixed Mach and altitude. • For a fixed altitude, the dependence of trim elevator on engine location is nearly linear with the slope decreasing with increasing Mach. When the engine is closer to the nose (more stable vehicle), trim elevator increases with increasing Mach. When the engine is closer to the rear (more unstable vehicle), trim elevator increases with decreasing Mach. From this, it follows that flight at a fixed altitude and a low (high) Mach requires less (more) elevator for a forward situated engine (more stable vehicle), and more (less) elevator for a rearward situated engine (more unstable vehicle). • For a fixed Mach, the dependence of trim elevator on engine location is almost linear with the slope increasing slightly with increasing altitude. When the engine location is also fixed, the trim elevator increases with increasing altitude. Trim FER. Figure 10 illustrates how trim FER depends on engine location. From the figure, the following FER vs. Engine Position, h=100 kft 0.9 85 kft 90 kft 95 kft 100 kft 0.65 0.6 0.55 0.7 FER FER FER vs. Engine Position, M=8 Mach 8 Mach 9 Mach 10 Mach 11 0.8 0.6 0.5 0.45 0.4 0.5 0.35 0.4 40 45 50 Engine Position (ft) 55 60 40 45 50 Engine Position (ft) 55 60 Figure 10. Impact of Engine Location on Trim FER is observed: • Trim FER is a concave up function with respect to engine location for a fixed altitude and Mach - with trim FER being minimized near 45 ft for most flight conditions. • For a fixed engine location and altitude (or Mach), trim FER increases with increasing Mach (or altitude). IV.A.2. Impact of Engine Location on Dynamic Properties (Level Flight) RHP Pole vs. Engine Position, h=100 kft RHP Pole vs. Engine Position, M= 8 Mach 8 Mach 9 Mach 10 Mach 11 4 3 2 1 40 6 85 kft 90 kft 95 kft 100 kft 5 RHP Pole 5 RHP Pole Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 • Trim elevator deflection increases with increasing altitude. 4 3 2 45 50 Engine Position (ft) 55 60 1 40 45 50 Engine Position (ft) Figure 11. Impact of Engine Location on Right Half Plane Pole RHP Pole. Figure 11 shows that the 19 of 45 American Institute of Aeronautics and Astronautics 55 60 • instability increases (roughly linearly) as the engine is moved rearward; • instability increases with increasing Mach and decreasing altitude. Motivation for Increased Instability. From the above, a designer may wish to increase the vehicle instability in order to make the vehicle more maneuverable in terms of following aggressive flight path angle or vertical acceleration commands. It may also be desirable in order to facilitate the attenuation of high frequency wind disturbances. (The link between instability and maneuverability was understood by the Wright Brothers early on in their work [65, page 39]. This could be important for a missile going after agile targets. Such might be the case for military applications. In such a case, one should note that a larger instability requires a larger minimum control system bandwidth for vehicle stabilization.41 This, however, may conflict with higher frequency non-minimum phase, structural, aero-elastic, and actuator dynamics. In the same spirit, a larger bandwidth at the elevator would typically require a faster control surface actuator. Such considerations must be rigorously addressed at some point in the design process - the sooner, the better. RHP Zero vs. Engine Position, h=100 kft RHP Zero vs. Engine Position, M= 8 Mach 8 Mach 9 Mach 10 Mach 11 6 5.5 7.5 85 kft 90 kft 95 kft 100 kft 7 RHP Zero 6.5 RHP Zero Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Moving the engine rearward, moves the center of gravity (cg) rearward with respect to the aerodynamic center (ac) - thus making the airplane more unstable. 6.5 6 5.5 5 5 4.5 40 45 50 Engine Position (ft) 55 60 4.5 40 45 50 Engine Position (ft) 55 60 Figure 12. Impact of Engine Location on Right Half Plane Zero RHP Zero. From Figure 12, one observes that the: • RHP zero varies little with engine position for a fixed altitude and Mach • RHP zero increases with increasing Mach and decreasing altitude Increasing RHP Zero: Moving Engine Rearward. From the above, it follows that one might move the engine rearward (making the vehicle more unstable) in order to maximize the right half plane zero. By so doing, a vehicle designer can (in principle) increase the maximum achievable flight path angle bandwidth.41 One must, of course, note that flexible modes, the associated uncertainty, and the control system simplicity can also limit the achievable bandwidth. This will be the case when the flexible modes lie within a decade of the right half plane zero. Additional pros associated with moving the engine rearward include: less fuel usage - minimized near 55 ft (not shown). Associated cons include the following: trim L/D drops (monotonically for 40 to 60 ft interval), trim FER increases (FER/thrust/acceleration margin decreases). IV.B. Lower Forebody Inclination In this section, we examine the impact of varying the lower forebody inclination angle. The following is assumed: • New engine parameters; i.e. he = hi = 4.5 ft, Ad = 0.15, An = 1 Ad = 6.67. • Lower forebody inclination varied from 4.2◦ to 8.2◦ (nominal value = 6.2◦ ) • All lengths (forebody, aftbody, engine length), upper forebody angle kept constant 20 of 45 American Institute of Aeronautics and Astronautics • Tail angle and total vehicle height change as a result • CG assumed to be fixed • Heating effects due to slender nose are not considered IV.B.1. Impact of Lower Forebody Inclination on Static Properties (Level Flight) Trimmable Region. Figure 13 shows how the trimmable region changes with lower forebody inclination angle. From Figure 13, one observe that the: Envelope Variations with Lower Forebody Inclination 50 0 120 115 00 21 0 100 00 50 Altitude (kft) 105 21 95 90 21 00 85 80 75 70 4 6 8 10 Mach 12 14 16 Figure 13. Impact of Lower Forebody Inclination on (Level Flight) Trimmable Region • Trimmable region shrinks with increasing lower forebody inclination • Pinch point moves toward a higher Mach and lower altitude with increasing lower forebody inclination Trim AOA. Figure 14 shows how AOA varies with lower forebody inclination angle. From Figure 14, one observes that the: AOA vs. Lower forebody inclination, M=8 AOA vs. Lower forebody inclination, h=100 kft 7 Mach 8 Mach 9 Mach 10 Mach 11 6 5.5 5 5 4 3 4.5 4 4 85 kft 90 kft 95 kft 100 kft 6 AOA (deg) 6.5 AOA (deg) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 110 4.2 deg 5.2 deg 6.2 deg 7.2 deg 8.2 deg 5 6 7 8 Lower forebody inclination (deg) 9 2 4 5 6 7 8 Lower forebody inclination (deg) Figure 14. Impact of Lower Forebody Inclination on Trim AOA • Trim AOA decreases linearly with increasing lower forebody inclination for a fixed altitude 21 of 45 American Institute of Aeronautics and Astronautics 9 • Trim AOA decreases with Mach at lower Mach numbers for a fixed altitude; • Trim AOA increases with increasing altitude for a fixed Mach Trim Elevator. Figure 15 shows how elevator varies with lower forebody inclination angle. From Figure 15, one observes that the: Elevator AOA vs. Lower forebody inclination, M=8 Elevator AOA vs. Lower forebody inclination, h=100 kft 13 12.5 12 Elevator AOA (deg) Elevator AOA (deg) Mach 8 Mach 9 Mach 10 Mach 11 13.5 85 kft 90 kft 95 kft 100 kft 12 10 8 11.5 11 4 5 6 7 8 Lower forebody inclination (deg) 6 4 9 5 6 7 8 Lower forebody inclination (deg) 9 Figure 15. Impact of Lower Forebody Inclination on Trim Elevator • Trim elevator deflection increases linearly with increasing forebody inclination • Trim elevator deflection increases with increasing Mach, increasing altitude Trim FER. Figure 16 shows how FER varies with lower forebody inclination angle. FER vs. Lower forebody inclination, h=100 kft 0.75 FER vs. Lower forebody inclination, M=8 0.5 Mach 8 Mach 9 Mach 10 Mach 11 0.7 0.6 85 kft 90 kft 95 kft 100 kft 0.45 FER 0.65 FER Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 14 14 0.4 0.55 0.35 0.5 0.45 4 5 6 7 8 Lower forebody inclination (deg) 9 4 5 6 7 8 Lower forebody inclination (deg) 9 Figure 16. Impact of Lower Forebody Inclination on Trim FER From Figure 16, one observes that the: • Trim FER increases almost linearly with increasing lower forebody inclination IV.B.2. Impact of Lower Forebody Inclination on Dynamic Properties (Level Flight) RHP Pole. Figure 17 shows how the RHP pole varies with lower forebody inclination. From Figure 17, one observes that the: • RHP pole increasing with increasing lower forebody inclination • RHP pole increases with Mach and decreasing altitude RHP Zero. Figure 18 shows how the RHP pole varies with lower forebody inclination. From Figure 18, one observes that the: • RHP zero increases with increasing lower forebody inclination • RHP zero increases with increasing Mach and decreasing altitude 22 of 45 American Institute of Aeronautics and Astronautics RHP Pole vs. Lower forebody inclination, h=100 kft Mach 8 Mach 9 Mach 10 Mach 11 2.6 RHP Pole RHP Pole vs. Lower forebody inclination, M= 8 2.4 3.5 RHP Pole 2.8 85 kft 90 kft 95 kft 100 kft 3 2.5 2.2 2 4 5 6 7 8 Lower forebody inclination (deg) 2 4 9 5 6 7 8 Lower forebody inclination (deg) 9 RHP Zero vs. Lower forebody inclination, M= 8 RHP Zero vs. Lower forebody inclination, h=100 kft Mach 8 Mach 9 Mach 10 Mach 11 6 5.5 7.5 85 kft 90 kft 95 kft 100 kft 7 RHP Zero 6.5 RHP Zero Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Figure 17. Impact of Lower Forebody Inclination on Right Half Plane Pole 6.5 6 5.5 5 5 4.5 4 5 6 7 8 Lower forebody inclination (deg) 9 4.5 4 5 6 7 8 Lower forebody inclination (deg) 9 Figure 18. Impact of Lower Forebody Inclination on Right Half Plane Zero IV.C. Center of Gravity This section examines the impact of varying the center of gravity (cg) location. The following assumptions are made: • New engine parameters; i.e. he = hi = 4.5 ft, Ad = 0.15, An = 1 Ad = 6.67. • CG varied from 45ft to 65ft artificially (no internal changes are made to achieve this shift) • For dynamic properties, we only consider CG locations for which vehicle is open loop unstable IV.C.1. Impact of Center of Gravity on Static Properties (Level Flight) Trimmable Region. Figure 19 shows how the trimmable region changes with a shifting vehicle CG. From Figure 19, one observes that the • trimmable region decreases slightly as the vehicle CG is moved rearward. Trim AOA. Figure 20 shows how trim AOA depends on the vehicle’s CG location. Figure 20 shows that the • Trim AOA decreases as the vehicle CG moves rearward. Trim Elevator. Figure 21 shows how trim elevator depends on cg location. • Trim elevator deflection increases as the vehicle CG moves rearward. Trim FER. Figure 22 shows how trim FER depends on vehicle CG location. Figure 22 shows that the • Trim FER increases as the vehicle CG moves rearward. 23 of 45 American Institute of Aeronautics and Astronautics Envelope Variations with CG Location 50 0 120 −65 ft −62 ft −60 ft −58 ft −55 ft 110 00 100 50 0 Altitude (kft) 21 00 21 90 00 21 70 4 6 8 10 Mach 12 14 16 Figure 19. Impact of Vehicle CG Location on (Level Flight) Trimmable Region AOA vs. CG Location, h=100 kft Mach 8 Mach 9 Mach 10 Mach 11 6 AOA (deg) AOA vs. CG Location, M=8 5 7 4 3 −45 85 kft 90 kft 95 kft 100 kft 6 AOA (deg) 7 5 4 3 −50 −55 CG Location (ft) −60 −65 2 −45 −50 −55 CG Location (ft) −60 −65 Figure 20. Impact of Vehicle CG Location on Trim AOA Elevator vs. CG Location, h=100 kft Elevator vs. CG Location, M=8 Mach 8 Mach 9 Mach 10 Mach 11 10 5 0 −45 −50 −55 CG Location (ft) −60 −65 15 Elevator deflection (deg) 15 Elevator deflection (deg) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 80 85 kft 90 kft 95 kft 100 kft 10 5 0 −45 −50 −55 CG Location (ft) −60 −65 Figure 21. Impact of Vehicle CG Location on Trim Elevator IV.C.2. Impact of Center of Gravity on Dynamic Properties (Level Flight) RHP Pole. Figure 23 shows how the vehicle instability depends on the CG location. Figure 23 shows that the vehicle 24 of 45 American Institute of Aeronautics and Astronautics FER vs. CG Location, h=100 kft FER vs. CG Location, M=8 0.75 Mach 8 Mach 9 Mach 10 Mach 11 0.7 0.6 85 kft 90 kft 95 kft 100 kft 0.5 FER 0.65 FER 0.55 0.45 0.4 0.55 0.35 0.5 0.45 −45 −50 −55 CG Location (ft) −60 −65 −45 −50 −55 CG Location (ft) −60 −65 RHP Pole vs. CG Location, h=100 kft RHP Pole vs. CG Location, M= 8 Mach 8 Mach 9 Mach 10 Mach 11 3 2.5 4.5 85 kft 90 kft 95 kft 100 kft 4 RHP Pole RHP Pole 3.5 3.5 3 2.5 2 0 −2 −4 −6 CG Location ft −8 2 0 −10 −2 −4 −6 CG Location ft −8 −10 Figure 23. Impact of Vehicle CG Location on Right Half Plane Pole • RHP pole increases linearly as the vehicle CG moves rearward. RHP Zero. Figure 24 shows how the RHP zero depends on the CG location. Figure 24 shows that the vehicle RHP Zero vs. CG Location, h=100 kft RHP Zero vs. CG Location, M= 8 Mach 8 Mach 9 Mach 10 Mach 11 5.5 5 7 85 kft 90 kft 95 kft 100 kft 6.5 RHP Zero 6 RHP Zero Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Figure 22. Impact of Vehicle CG Location on Trim FER 6 5.5 5 4.5 0 −2 −4 −6 CG Location ft −8 −10 4.5 0 −2 −4 −6 CG Location ft −8 −10 Figure 24. Impact of Vehicle CG Location on Right Half Plane Zero • RHP zero decreases linearly as the vehicle CG moves rearward. IV.D. Vehicle Mass This section examines the impact of varying the vehicle’s (total) mass. The following assumptions are made: • New engine parameters; i.e. he = hi = 4.5 ft, Ad = 0.15, An = 1 Ad = 6.67. 25 of 45 American Institute of Aeronautics and Astronautics • Mass of vehicle modified without changing any material or subsystem properties. IV.D.1. Impact of Vehicle Mass on Static Properties Trimmable Region. Figure 25 shows how the trimmable region depends on vehicle mass. Envelope Variations with Mass Ratio 140 0.5 0.8 1 1.2 1.5 130 0 100 00 50 Altitude (kft) 00 21 110 21 00 90 21 80 70 4 6 8 10 Mach 12 14 16 Figure 25. Impact of Vehicle Mass on (Level Flight) Trimmable Region From Figure 25, one observes that the: • Trimmable region shrinks as the vehicle mass is increased. Trim AOA. Figure 26 shows how trim AOA depends on vehicle mass. The figure shows that the: AOA vs. Mass Ratio, M=8 AOA vs. Mass Ratio, h=100 kft 7 Mach 8 Mach 9 Mach 10 Mach 11 8 6 85 kft 90 kft 95 kft 100 kft 6 AOA (deg) 10 AOA (deg) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 50 0 120 5 4 3 4 2 2 0.5 1 Mass Ratio 1.5 2 1 0.5 1 Mass Ratio 1.5 2 Figure 26. Impact of Vehicle Mass on Trim AOA • Trim AOA increases as the vehicle mass is increased. Trim Elevator. Figure 27 shows how trim elevator depends on vehicle mass. The figure shows that the: • Trim elevator deflection increases as the vehicle mass is increased. Trim FER. Figure 28 shows how trim FER depends on vehicle mass. The figure shows that the: • Trim FER increases as the vehicle mass is increased. 26 of 45 American Institute of Aeronautics and Astronautics Elevator vs. Mass Ratio, h=100 kft Elevator vs. Mass Ratio, M=8 Mach 8 Mach 9 Mach 10 Mach 11 11 10 9 8 7 6 0.5 1 Mass Ratio 1.5 12 Elevator deflection (deg) Elevator deflection (deg) 12 85 kft 90 kft 95 kft 100 kft 10 8 6 4 0.5 2 1 Mass Ratio 1.5 2 FER vs. Mass Ratio, h=100 kft 1 0.6 0.4 0.2 0.5 85 kft 90 kft 95 kft 100 kft 0.45 FER FER FER vs. Mass Ratio, M=8 0.5 Mach 8 Mach 9 Mach 10 Mach 11 0.8 0.4 0.35 1 Mass Ratio 1.5 2 0.5 1 Mass Ratio 1.5 2 Figure 28. Trim FER vs Vehicle Mass IV.D.2. Impact of Vehicle Mass on Dynamic Properties (Level Flight) RHP Pole. Figure 29 shows how the vehicle instability depends on vehicle mass. The figure shows that the: RHP Pole vs. Mass Ratio, h=100 kft RHP Pole vs. Mass Ratio, M= 8 Mach 8 Mach 9 Mach 10 Mach 11 3.5 3 2.5 4.5 3.5 3 2.5 2 1.5 0.5 85 kft 90 kft 95 kft 100 kft 4 RHP Pole 4 RHP Pole Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Figure 27. Impact of Vehicle Mass on Trim Elevator 2 1 Mass Ratio 1.5 2 1.5 0.5 1 Mass Ratio 1.5 Figure 29. Impact of Vehicle Mass on Right Half Plane Pole • RHP pole decreases as the vehicle mass is increased. RHP Zero. Figure 30 shows how the RHP zero depends on vehicle mass. The figure shows that the: • RHP zero decreases as the vehicle mass is increased. 27 of 45 American Institute of Aeronautics and Astronautics 2 RHP Zero vs. Mass Ratio, h=100 kft Mach 8 Mach 9 Mach 10 Mach 11 7 RHP Zero RHP Zero vs. Mass Ratio, M= 8 6 10 85 kft 90 kft 95 kft 100 kft 9 RHP Zero 8 8 7 6 5 5 4 0.5 1 1.5 Mass Ratio 2 4 0.5 1 Mass Ratio 1.5 2 Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Figure 30. Impact of Vehicle Mass on Right Half Plane Zero IV.E. Summary of Level Flight Trim Trade Studies The following tables summarize each of the conducted level-flight trim trade studies. Rearward Engine Shift. Table 5 summarizes trends for a rearward shifting engine. The engine is shifted rearward (with the CG) with the vehicle height kept constant; i.e. the lower forebody inclination angle is decreasing, thus making the vehicle sharper. As the engine is shifted rearward, we observe specific monotonic trends and tradeoffs that result in the following PROS and CONS: • PROS: trim AOA decreases, trim lift remains nearly constant; • CONS: trim elevator increases (CG rearward), trim drag increases, trim L/D decreases, RHP pole increases. We also observe the following more complex (non-monotonic) behavior: • trim fuel rate decreases (min near 55 ft) and then increases; • trim FER decreases (min near 45 ft - thrust/acceleration margin increases) and then increases (thrust/acceleration margin decreases). Property Pro Con Trim Trim Trim Trim Trim Trim Trim RHP RHP RHP Almost Constant Almost Constant Increases monotonically Decreases monotonically Lift Drag L/D AOA Elevator FER Fuel Rate Pole Zero Z/P Ratio Decreases monotonically Decreases till 45 ft Decreases till 55 ft Increases after 45 ft Increases monotonically Increases after 45 ft Increases after 55 ft Increases monotonically Decreases till 45 ft Decreases monotonically Table 5. Trends for Rearward Engine Shift Lower Forebody Angle Increase. Table 6 summarizes trends for an increasing lower forebody angle. Here, the horizontal engine location is fixed and the engine is moved downward - thus increasing the height of the vehicle. As the lower forebody angle is increased, we observe specific monotonic trends and tradeoffs that result in the following PROS and CONS: 28 of 45 American Institute of Aeronautics and Astronautics • PROS: trim AOA decreases, RHP zero increases, RHP zero increases, RHP zero-pole ratio almost constant; • CONS: trim elevator increases, trim drag increases, trim L/D decreases, trim FER increases, trim fuel rate increases, RHP pole increases. Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Non-monotonic patterns are not observed in this case. Property Pro Con Trim Trim Trim Trim Trim Trim Trim RHP RHP RHP Almost Constant Almost Constant Lift Drag L/D AOA Elevator FER Fuel Rate Pole Zero Z/P Ratio Decreases monotonically Decreases monotonically Increases monotonically Increases monotonically Decreases monotonically Increases monotonically Increases monotonically Almost constant Almost constant Table 6. Trends for Increasing Lower Forebody Inclination Rearward CG Shift. Table 7 summarizes trends for a rearward shifting CG. As the CG is shifted rearward, we observe specific monotonic trends and tradeoffs that result in the following PROS and CONS: • PROS: trim AOA decreases, trim fuel rate almost constant; • CONS: trim elevator increases, trim drag increases, trim L/D decreases, trim FER increases, RHP pole increases, RHP zero decreases, RHP zero-pole ration decreases. Non-monotonic patterns are not observed in this case. Property Pro Con Trim Trim Trim Trim Trim Trim Trim RHP RHP RHP Almost Constant Almost Constant Increases monotonically Decreases monotonically Lift Drag L/D AOA Elevator FER Fuel Rate Pole Zero Z/P Ratio Decreases monotonically Increases monotonically Increases monotonically Almost constant Increases monotonically Decreases monotonically Decreases monotonically Table 7. Trends for Rearward CG Shift Vehicle Mass Increase. Table 8 summarizes trends for increasing mass. As the vehicle mass is increased, we observe specific monotonic trends and tradeoffs that result in the following PROS and CONS: • PROS: RHP pole decreases, RHP zero-pole ratio almost constant; • CONS: trim elevator and trim AOA increase, trim FER and fuel rate increase, RHP zero decreases. We also observe the following more complex (non-monotonic) behavior: 29 of 45 American Institute of Aeronautics and Astronautics Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 • trim L/D is a concave down function of mass for a fixed Mach and altitude; mass at which peak occurs increases with Mach (altitude fixed); decreases with altitude (Mach fixed). Property Pro Trim Trim Trim Trim Trim Trim Trim RHP RHP RHP Increases monotonically Lift Drag L/D (peak) AOA Elevator FER Fuel Rate Pole Zero Z/P Ratio Con Increases with Mach (altitude fixed) Increases monotonically Decreases with altitude (Mach fixed) Increases monotonically Increases monotonically Increases monotonically Increases monotonically Decreases monotonically Decreases monotonically Almost constant Almost constant Table 8. Trends for Increasing Mass V. Vehicle Optimization Various schemes have been considered for the optimization of space vehicles. Within66 , a conceptual design process - that takes factors like cost analysis into consideration - is described. The non-hierarchical nature of the design process is illustrated and several optimization methods - parameter based, gradient based, and stochastic methods - are examined. Within67 , a probabilistic approach to vehicle design is taken in order to account for uncertainties. For air-breathing hypersonic aircraft, the coupling between the airframe and the engine introduces additional constraints into the design process. The importance of a multidisciplinary design optimization approach is illustrated within68 . A collaborative approach69 for launch vehicle design is considered within70 . The design is decomposed into several subsystem design problems. A system level optimizer coordinates the integration of subsystem designs while taking into account intersubsystem coupling and constraints. Such a subsystem-based algorithm is well suited to exploit parallel computing architectures. In this section, we address optimizing the generic carrot shaped hypersonic vehicle under consideration. The flat base assumption is made; i.e. he = hi , An = A1d . The goal is twofold: (1) understand what configurations result when very simple optimizations are considered, (2) understand how the gap matric can be used to design a control-friendly vehicle. Admissible Parameter Values. Optimizations are conducted over the following control surface and engine parameter values: • Elevator area Selev : [8.5, 25.5] ft2 (nominal: 17 ft2 ) • Horizontal elevator position xelev : [70, 90] ft from nose (nominal: 85 ft) • Inlet height hi : [2.25, 6.75] ft (nominal: 4.5 ft) • Lower forebody inclination τ1L : [4.2◦ , 8.2◦ ] (nominal: 6.2◦ ) Overview of Trim Optimization Algorithm. A nonlinear simplex optimization method was used. No derivatives are calculated because trimming the vehicle results in each iteration being expensive.71, 72 An overview of the vehicle optimization algorithm is as follows: 1. Choose cost function (e.g. trim AOA, trim fuel, trim elevator, trim FER, trim L/D); 2. Select values for parameters being optimized over; e.g. Selev , xelev , hi , τ1L ; 3. Trim the vehicle at selected flight condition; i.e. Mach 8, 85 kft, level flight; 30 of 45 American Institute of Aeronautics and Astronautics 4. Evaluate cost function using obtained trim values; 5. Repeat 2-4 until optimal configuration for chosen cost function is obtained. Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 ASU High performance Computing Cluster. All optimizations presented in this paper were performed using the Arizona State University (ASU) High Performance Computing (HPC) cluster. This cluster is part of ASU’s Ira A. Fulton School of Engineering High Performance Computing Initiative. The HPC is composed of 220 dual quadcore Intel Xeon EM64T (Extended Memory 64 Technology) nodes - each with 16 gigabytes of RAM. The cluster also has a partition for running large numbers of serial jobs comprised of 185 nodes with dual Xeon MP 64bit processors. Static Optimization Using ASU HPC: Computational Issues. While individual optimizations proceed in a serial fashion, the search space is partitioned into subspaces. Due to the nonlinear nature of the problem, a multistart algorithm was used (with at most 44 = 256 initial conditions per optimization problem); i.e. several initial starting points are considered in each subspace. Optimization from an initial guess for a single operating point using the nonlinear (3DOF + flexibility) model takes approximately 15 minutes (on a 2.5GHz CPU with 1GB of memory) to converge to a local minimum (for the static objectives). The results from all the initial conditions are examined and the optimum is chosen. On average, 256 initial guesses are chosen for each static objective minimization. The q̄ = 2000 psf fuel consumption objective optimization takes approximately an hour to converge to a local minimum (on a 2.5GHz CPU with 1GB of memory) from an initial guess (in each iteration the plant must be trimmed at 7 points along the q2000 trajectory). Similar to the static optimization, a multistart algorithm was used (with at most 44 = 256 initial conditions). Trim Optimization Results (Mach 8, 85kft, Level Flight). Table 9 shows the results obtained by optimizing the following trim vehicle variables at Mach 8, 85kt (level flight): trim AOA, trim fuel, trim elevator deflection, trim FER, and trim L/D. Table 10 shows the parameters that result from the respective optimizations. Table 11 contains corresponding dynamical characteristics. Objective Trim L/D Trim FER Trim Fuel Rate( slugs sec ) Trim Elevator (deg) Trim AOA (deg) (1) (2) (3) (4) (5) (6) (7) 6.63 5.95 5.95 5.89 4.38 4.30 3.28 0.29 0.27 0.27 0.27 0.47 0.35 0.49 0.116 0.102 0.102 0.102 0.0361 0.0708 0.0616 2.19 2.65 1.11 3.11 3.93 6.44 10.14 6.39 6.64 6.64 6.64 2.19 2.88 0.00271 Maximize Trim L/D Minimize Trim FER Minimize Trim Elevator Minimize Fuel Consumed (const. q̄) Minimize Trim Fuel Rate Nominal (with New Engine) Minimize Trim AOA Table 9. Optimized Static Properties - Resulting Trim Values (Mach 8, 85 kft, Level Flight) Objective (1) (2) (3) (4) (5) (6) (7) Maximize Trim L/D Minimize Trim FER Minimize Trim Elevator Minimize Fuel Consumed (const. q̄) Minimize Trim Fuel Rate Nominal (with New Engine) Minimize Trim AOA Selev (ft2 ) xelev (ft) hi (ft) τ1L (deg) 25.5 19.55 25.5 18.19 16.15 17 19.89 90 90 90 90 81.76 85 73 6.75 6.75 6.75 6.75 2.25 4.5 3.15 5.24 4.2 4.2 4.2 4.2 6.2 8.2 Table 10. Optimized Static Properties - Resulting Vehicle Configurations/Parameters (Mach 8, 85 kft, Level Flight) 31 of 45 American Institute of Aeronautics and Astronautics Objective (1) (2) (3) (4) (5) (6) (7) Maximize Trim L/D Minimize Trim FER Minimize Trim Elevator Minimize Fuel Consumed (const. q) Minimize Trim Fuel Rate Nominal (with New Engine) Minimize Trim AOA RHP Pole RHP Zero RHP Zero-Pole ratio 1.60 (-45.18%) 1.95 (-33.43%) 1.48 (-49.56%) 2.05 (-30.07%) 3.14 (+7.23%) 2.93 (0.00%) 3.55 (+21.23%) 5.83 (-16.02%) 5.62 (-19.06%) 5.80 (-16.46%) 5.61 (-19.15%) 7.32 (+5.44%) 6.94 (0.00%) 6.34 (-8.59%) 3.6325 (+53.18%) 2.8833 (+21.59%) 3.928 (+65.64%) 2.7416 (15.61%) 2.3319 (-1.66%) 2.37 (0.00%) 1.788 (-24.60%) Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Table 11. Optimized Static Properties - Resulting Dynamic Characteristics (Mach 8, 85 kft, Level Flight) Within Table 10, over bars imply that the maximum admissible parameter was achieved; underlines imply that the minimum admissible parameter bound was achieved. From Tables 9-10, one observes the following: 1. Maximizing Trim L/D. Maximizing trim L/D results in a small FER - marginally above that obtained when trim FER is directly minimized. As when trim FER is minimized, maximizing L/D also results in a large trim AOA. In short, maximizing trim L/D yields results comparable to minimizing trim FER. Maximizing trim L/D, however, results in a larger elevator surface area and lower forebody inclination vis-a-vis when trim FER is minimized. For this case, Selev , xelev , and hi , were maximized to yield a maximally large elevator, a maximally effective elevator, a maximally large engine. 2. Minimizing Trim FER. Minimizing trim FER produces results that are comparable to those obtained when L/D is maximized - with L/D reduced 10.3% from the maximum L/D, FER reduced by 12.1%, and AOA increased by only 3.9%. In contrast to maximizing L/D, minimizing FER results in a much smaller elevator surface area (23.3%) and lower forebody angle (19.9%). For this case, xelev , and hi were maximized while τ1L was minimized to yield a maximally effective elevator, a maximally large engine, and a very aerodynamic lower forebody. 3. Minimizing Trim Elevator. Minimizing trim elevator produces results (L/D, Fuel, FER, AOA) that are comparable to minimizing trim FER with less than half the elevator. For this case, Selev , xelev , and hi were maximized while τ1L was minimized to yield a maximally large elevator and effective elevator, a maximally large engine, and a very aerodynamic lower forebody. 4. Minimizing Fuel Consumed (const. q̄). Minimizing the fuel consumed along a constant dynamic pressure trajectory produces results that are comparable to minimizing trim FER with xelev and hi maximized and τ1L minimized to yield a maximally effective elevator, a maximally large engine, and a very aerodynamic lower forebody. This optimization is more involved than the others being considered. Hence some additional explanation is required. The objective here is to minimize the total fuel consumed while flying along the q̄ = 2000 psf altitudeMach profile from Mach 5.52 at 70kft to Mach 11.08 at a 100kft. To do so, we approximate the total fuel consumed. For simplicity, vehicle mass changes are ignored. The method used to approximate fuel consumption is described below. (a) Trim at Selected Points. Select points along the stated q̄ = 2000 psf profile spaced 5 kft. This yields the altitudes: [70, 75, 80, 85, 90, 95, 100] kft and the corresponding Machs: [5.52, 6.21, 6.98, 7.85, 8.81, 9.88, 11.08]. This divides the profile into 6 legs. Trim the vehicle at each point. (b) Approximate Maximum Accelerations. Approximate the maximum horizontal acceleration possible at the left end point of each leg. Do so using the FER margin at each left end point. The FER margin is used to determine the maximum FER. This is used to compute the associated thrust T h(F ERmax )i . The trim drag Di and trim to approximate the maximum   AOA αi are then used T h(F ERmax )i cos αi −Di where m is the total mass of the acceleration at the left end point: ai ≈ m vehicle. 32 of 45 American Institute of Aeronautics and Astronautics (c) Determine Average Accelerations for Each Leg. Assume that the average acceleration in leg i i+1 is the mean of the two end-point accelerations; i.e. ai = ai +a where ai is the approximate 2 maximum acceleration at the left end point of the ith leg and ai+1 is that at the right end point. (d) Approximate Time to Fly Each Leg. Approximate the time to fly leg i as Ti ≈ vi+1 −vi . ai (e) Approximate the average fuel rate for leg i (Ḟi ) as the mean of the fuel rates at the end points ˙ ˙ (where we have maximum acceleration); i.e. Ḟi = Fi +2Fi+1 (f) Approximate Fuel Consumed During Each Leg. Approximate the fuel consumed during leg i as Fi ≈ Ḟi × Ti (g) Approximate Total Fuel Consumed Over Flight Profile. Approximate the total fuel consumed (objective to be minimized) as Downloaded by WESTERN MICHIGAN UNIV on October 13, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2009-7287 Total fuel consumed ≈ 6  Fi (35) i=1 (h) Inadmissible Designs. We also want to maintain a reasonable FPA. The average FPA for the ith leg is approximated as γ≈ hi+1 − hi vi × Ti (36) where (vi , hi ) are the velocity and altitude at the left end point of the ith leg and hi+1 is the altitude at the right end point. Designs for which the average FPA in any leg exceeded 3◦ were deemed inadmissible and discarded. 5. Minimizing Trim Fuel Rate. Minimizing trim fuel rate produces results that lie between those obtained when trim AOA is minimized and elevator is minimized (or trim FER is minimized or trim L/D is maximized). Surprisingly, for this case both hi and τ1L are minimized to yield a maximally small engine with a very aerodynamic lower forebody. This optimization results in low trim air mass flow to the engine (and low trim AOA) so that even with the resulting larger trim FER and increased trim thrust, the trim fuel rate is small. 6. Nominal Vehicle. Observe that the nominal values Selev = 17, hi = 4.5, τ1L = 6.2 are midpoints while the nominal value xelev = 85 is near the right end point. Note that this case can be viewed as a tradeoff between minimizing trim fuel rate and trim AOA when looking at trim L/D and elevator, elevator area, and lower forebody angle. Otherwise, trim FER, fuel rate, AOA, elevator rearward distance, and engine inlet height are larger. 7. Minimizing Trim AOA. Minimizing trim AOA uses excessive trim elevator for lift. This results in excessive drag and results in a small trim L/D. It also produces a large trim FER and small trim fuel rate. The resulting low trim fuel rate is due to the reduced trim air mass flow to the engine (due to the smaller trim AOA). From a vehicle configuration perspective, this case results in the largest lower forebody inclination (least aerodynamic shape) - achieving the maximum admissible value for τ1L . Motivation for Gap Metric Based Optimization. The gap metric represents a system-theoretic measure that quantifies the “distance” between two dynamical systems and whether or not a common controller can be deployed for the systems under consideration48 . Within49 , the gap between two LTI dynamical systems (P1 , P2 ) is defined as follows: def g(P1 , P2 ) = max{ inf Q∈H∞   D1  N1 −  D2  N2 Q ∞ , inf Q∈H∞   D2  N2 −  D1  N1 Q ∞ } (37) where P1 = N1 D1−1 , P2 = N2 D2−1 , and (Ni , Di ) denotes a normalized right coprime factorization for Pi (i = 1, 2) in the sense of 73 . The gap metric (and the ν gap74 ) has often been considered from a robustness perspective in the stabilization of feedback systems75 . Within76 ,...
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