Scholarly article on topic 'Aeroservoelastic modeling and analysis of a canard-configured air-breathing hypersonic vehicles'

Aeroservoelastic modeling and analysis of a canard-configured air-breathing hypersonic vehicles Academic research paper on "Mechanical engineering"

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{Aeroservoelasticity / Canard / "Flight dynamics" / "Hypersonic vehicles" / Pivot / Stability}

Abstract of research paper on Mechanical engineering, author of scientific article — Zeng Kaichun, Xiang Jinwu, Li Daochun

Abstract Air-breathing hypersonic vehicles (HSVs) are typically characterized by interactions of elasticity, propulsion and rigid-body flight dynamics, which may result in intractable aeroservoelastic problem. When canard is added, this problem would be even intensified by the introduction of low-frequency canard pivot mode. This paper concerns how the aeroservoelastic stability of a canard-configured HSV is affected by the pivot stiffnesses of all-moveable horizontal tail (HT) and canard. A wing/pivot system model is developed by considering the pivot torsional flexibility, fuselage vibration, and control input. The governing equations of the aeroservoelastic system are established by combining the equations of rigid-body motion, elastic fuselage model, wing/pivot system models and actuator dynamics. An unsteady aerodynamic model is developed by steady Shock-Expansion theory with an unsteady correction using local piston theory. A baseline controller is given to provide approximate inflight characteristics of rigid-body modes. The vehicle is trimmed for equilibrium state, around which the linearized equations are derived for stability analysis. A comparative study of damping ratios, closed-loop poles and responses are conducted with varying controller gains and pivot stiffnesses. Available bandwidth for control design is discussed and feasible region for pivot stiffnesses of HT and canard is given.

Academic research paper on topic "Aeroservoelastic modeling and analysis of a canard-configured air-breathing hypersonic vehicles"

Chinese Journal of Aeronautics, 2013,26(4): 831-840

JOURNAL OF

AERONAUTICS

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cn www.sciencedirect.com

Aeroservoelastic modeling and analysis of a canard-configured air-breathing hypersonic vehicles

Zeng Kaichun, Xiang Jinwu *, Li Daochun

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

Received 18 December 2012; revised 23 January 2013; accepted 16 March 2013 Available online 2 July 2013

KEYWORDS

Aeroservoelasticity; Canard;

Flight dynamics; Hypersonic vehicles; Pivot; Stability

Abstract Air-breathing hypersonic vehicles (HSVs) are typically characterized by interactions of elasticity, propulsion and rigid-body flight dynamics, which may result in intractable aeroservoelastic problem. When canard is added, this problem would be even intensified by the introduction of low-frequency canard pivot mode. This paper concerns how the aeroservoelastic stability of a canard-configured HSV is affected by the pivot stiffnesses of all-moveable horizontal tail (HT) and canard. A wing/pivot system model is developed by considering the pivot torsional flexibility, fuselage vibration, and control input. The governing equations of the aeroservoelastic system are established by combining the equations of rigid-body motion, elastic fuselage model, wing/pivot system models and actuator dynamics. An unsteady aerodynamic model is developed by steady Shock-Expansion theory with an unsteady correction using local piston theory. A baseline controller is given to provide approximate inflight characteristics of rigid-body modes. The vehicle is trimmed for equilibrium state, around which the linearized equations are derived for stability analysis. A comparative study of damping ratios, closed-loop poles and responses are conducted with varying controller gains and pivot stiffnesses. Available bandwidth for control design is discussed and feasible region for pivot stiffnesses of HT and canard is given.

© 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.

1. Introduction

Air-breathing hypersonic vehicles offer a promising technol-

ogy for both commercial and military applications. However, the integration of fuselage and propulsion causes aeroservo-

elasticity by intensive interactions of aerodynamics, inertia, elasticity, actuation, and control system dynamics.1,2 Traditionally, flight controllers are designed based on rigid-body model, and aeroservoelastic issues are dealt with by including notch filters to eliminate observability of structural modes.3 This approach may not be acceptable for hypersonic vehicles (HSVs) whose lightweight slender fuselage leads to low natural frequencies close to the rigid-body short-period mode frequency. Therefore, integration approach of aeroservoelastic modeling, analysis, and control becomes vital for HSVs.

Studies had been conducted on the aeroelasticity and aero-thermoelasticity of a tail-controlled National Aero-Space Plane (NASP) model by National Aeronautics and Space Administration (NASA) during the NASP program. An

* Corresponding author. Tel.: +86 10 82338786. E-mail addresses: zengkaichun@ase.buaa.edu.cn (Z. Kaichun), xiangjw@buaa.edu.cn (X. Jinwu), ldc@buaa.edu.cn (L. Daochun). Peer review under responsibility of Editorial Committee of CJA.

1000-9361 © 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. http://dx.doi.Org/10.1016/j.cja.2013.06.001

overview of these studies was given in Ref.4, whereas Ref.5 provided details of related analyses. The finite element method (FEM) for structure coupled with a computational fluid dynamics (CFD) code was applied to the unclassified model from tail-controlled X-30. The pitch mode of all-moveable HT due to the pivot torsional flexibility was found to have low frequency at high flight dynamic pressures. When coupling with the short-period mode, this mode might induce dynamic instability called body-freedom flutter. Similar conclusion was obtained from the aeroelastic simulation of X-43A by NASA.6

Semi-empirical approach, instead of FEM and CFD method, is preferred in the early cycle of conceptual design or control law development.7 In Ref.8, a comprehensive analytical model was developed by Frank and Schmidt, where Newtonian impact theory was employed for aerodynamics and a simple lumped-flexibility spring model for the fuselage first-order bending mode. Based on Frank model, effect of thrust coupling on aeroservoelastic stability was studied in Ref.9 Michael and David10 relaxed some assumptions of Frank's model by addressing the on-design and off-design effect of propulsion system. The steady Shock-Expansion theory was utilized to estimate the location of bow shock relative to inlet lip, whereas the flexibility of fuselage was captured by modeling the vehicle as two cantilever beams both clamped at the center-of-mass. A wide range of system analysis and control designs were conducted based on these models.11-15 Aerodynamic heating was also introduced in Ref.10, but was found to be negligible for the vehicle stability.16

The presence of serious nonminimum phase behavior for tail-controlled configuration drove the engineers to add additional effector-the canard-to compensate undesirable ef-fects.17,18 Because the application of canard introduced an additional low-frequency mode (the canard pivot mode)5 and complex canard-HT aerodynamic interactions,19 the aeroser-voelasticity of canard-configured HSVs would be even more complicated than tail-controlled HSVs. However, until recently, few reports have focused on the aeroservoelasticity of canard-configured HSVs. This current investigation is aimed at studying the mechanism of multi-mode coupling and its constraints for designing control law of canard-configured HSVs. The modes concerned here include two wing pivot modes, fuselage elastic modes and the rigid-body short-period mode. The HSVs dynamic model developed by Michael and David10 is extended to incorporate the HT and canard pivot modes, whereas the local piston theory20 is employed for unsteady correction of the steady Shock-Expansion theory. A control law developed from rigid vehicle assumption is used to stabilize the rigid-body motion. The aeroservoelastic model is obtained by incorporating the actuator dynamics further. Stability of the system is investigated with varying controller gains and pivot stiffnesses.

2. Aeroservoelastic model of canard-configured HSV

The canard-configured HSV investigated in this paper is shown in Fig. 1, where Max represents the freestream Mach number and a the vehicle angle of attack. A body-fixed reference frame that is comprised of three orthogonal unit vectors e1, e2, e3 and three coordinates x, y, z is used to give the direction of aerodynamic forces and location of all-moveable HT

(b) Wing/pivot system"

Fig. 1 Canard-configured HSV concept.

and canard. The HT shown in Fig. 1 (b) has a root chord length of 27%L, a span of 9.5%L, a leading edge sweep of 70°, a trailing edge sweep of 15° and a surface area of 170 ft2'5 where L is the vehicle length. The canard with a surface area of 100 ft2 is designed to be geometrically similar to HT. Other parameters are illustrated in Table 1 or could be found in Ref.10

2.1. Elastic fuselage model

In the research of NASA,4 an FEM model of typical hypersonic vehicle was constructed, and mode analysis was conducted to give the mode shapes and frequencies. It was shown that the main source of fuselage flexibility was longitudinal bending, whose stiffness was much lower than the torsion and lateral bending. In Ref.10, the flexible fuselage was modeled as a pair of cantilever beams, which could only represent the first bending mode. To take high-order modes into account, a free-free Euler-Bernoulli beam model is used for elastic fuselage in the present work. Assumed mode method16 is employed to obtain its undamped frequencies rnfJ and mode shapes /(x). When the structural damping ratios f are considered, the elastic motion equations of fuselage can be written

gi + 2Cf,,«f,,gi + ®ng,- = Ni (i = 1, 2,..., Wf)

where N represents the generalized forces, g the generalized coordinates, and nf the number of retained modes. The bending displacement wf can be expressed approximately as linear combination of /¡(x), that is,

Wf(X; t) gi(t)/i(x)

where t represents the time. For each elastic mode, the generalized forces, produced by external loads including aerodynamic and propulsive forces, can be described as

N(t)= f /1(x)fz(x,t)dx + £/(Xj)Fz](t)

where fz represents the z component of distributed load, and Fzj the z component of the jth concentrated load. The vibration of fuselage affects the pressure distribution by changing the slope hf of local aerodynamic surface and inducing additional

Table 1 Vehicle geometric parameters and physical properties.

Parameter Value Parameter Value

Vehicle length L (ft) 100 Canard area Sc (ft2) 100

Vehicle width W (ft) 10 Vehicle mass m (slug) 3000

Forebody length Lf (ft) 47 Moment of inertia Iyy (slug-ft2) 5 x 106

Afterbody length La (ft) 33 Fuselage 1st mode frequency xf,i (rad/s) 18

Engine length Ln (ft) 20 HT pivot mode frequency (rad/s) 20

HT area Se (ft2) 170 Canard pivot mode frequency Xvp (rad/s) 25

Canard position xc (ft) 45 Wing area density (slug/ft2) 0.33

unsteady velocity wf. For the vehicle body, they can be represented separately as

f, 0 = -Х 4M

i—1 Щ

tVf (x, t) = X4МФМ)

2.2. Wing/pivot system model

The torsional flexibilities of the HT and canard pivots may not only produce response lag to control signals, but also resonant with each other, or with elastic modes of the fuselage. As exhibited from the ground test and finite element analysis conducted by NASA,5,6 the undamped frequencies of HT pivot mode are much lower than those of wing surface bending and torsion, even lower than those of flexible fuselage. Thus HT and canard pivot flexibilities are included besides fuselage flexibility, whereas the wing surfaces are regarded as rigid in our work. Note that the wing represents the HT or the canard in this paper.

The undamped dynamics of the wing/pivot system shown in Fig. 1(b) is governed by

Jwphwp H Kwphwp 0

where Jwp represents rotational inertia of the wing about the pivot shaft, Kwp the pivot stiffness, and the wing rotation angle due to the flexible pivot. The undamped frequency can be determined by

For the flexible vehicle in controlled flight, one end of wing pivot, which connects with actuator, would rotate according to the actuator output hact and fuselage bending deflection hf. With the structural damping Cwp, the rotation motion is then governed by

Jwp(h wp hact + hf)

Cwphwp H" Kwphwp — Mt

where Mt is the aerodynamic torque about pivot.

As exhibited in Eq. (7), besides the aerodynamic load from the wing, inertial torques would be produced by control operation and fuselage vibration, resulting in the torsion of wing pivot. Therefore, the instantaneous pitch angle of the wing should be expressed as

Then, Eq. (7) is modified as

Jwpd + Cwp (d - hact - hf)+Kwp(d - hact - hf) = Mt (9)

2.3. Flight dynamics and feedback control system

With flat-earth assumption, the longitudinal equations of rigid-body motion are written in the body axes as8

' Hi — V sin(h - a) i — — qw — g sin h + wv — qu + ZZ + g cos h

h — q M

q — -г

where H represents the altitude, V the flight velocity, h the pitch angle, u the projection of V on x of body frame, w the projection of V on y, q the pitch rate, g the gravity acceleration, X (Z) the external load in x (z) direction, and M the pitch moment.

For small angle of attack, it is tenable to express a and V as

a ~ — i

d — hwp + hact + hf

Then the state vector of Eq. (10) is composed of five rigid body state variables H, V, a, h and q, whereas the control vector is composed of throttle set dT, deflection of HT de, and deflection of canard dc. The parameters in control vector enter implicitly into Eq. (10) through the aerodynamic/propulsive forces X, Z and moment M.

As discussed in Ref.8, the short-period mode of open-loop flight dynamics is statically unstable. To investigate the rigid-body/flexibility coupling, the stability augmentation control system is needed to tune the short-period mode characteristics.

A conceptual controller based on rigid-body assumption, composed of inner-loop and outer-loop control, is introduced, where the canard is ganged with HT using a negative gain kc = — 1.17As shown in Fig. 2, the inner-loop with gains of K1 is utilized for stability augmentation, whereas the outerloop with gains of K2 and K3 for reference command tracking. A first order linear model with time constant sact = 0.05 is included for the actuator dynamics of control surfaces, whose deflections de, dc are also constrained within ±20° by the saturation limiters.12

With elastic motion and actuator dynamics incorporated, the dynamic equations of the aeroservoelastic system can be combined as

Fig. 2 Control structure for longitudinal flight dynamics.

Hi — V sin (h - a)

U — —qw — g sin h H--

w — qu H---H g cos h

h — q M

q — T

gi + 2ffigi + xf igi — Ni (i — 1,2,..., Wf)

Jwpde + CWp de — dact + Eg

,.d/> (x)

+ KWp de — dact + £g,

@/i(x)

dact — S" (dcm — d!

JWp€c + cWp d c — d act + £ g

,.d/> (x)

+kJ dc — dact+Z g,

@/i(x)

— Mtc

dact—ML

d act — — (kcdcm — d; sact

where xe (xc) represents the pivot position of HT (canard), dcm the actuator input from the control system, and the subscript/ superscript "e" ("c") identifies the parameters of HT (canard).

2.4. Aerodynamic and propulsive forces

In Eq. (12), the rigid-body loads X, Z, M, generalized forces Ni, and the wing pivot torques M; M are determined by the pressure over the vehicle. In the current work, the pressure is calculated by qusi-steady approach, i.e. dividing the unsteady aerodynamics into steady component and an unsteady correction.

Shock-Expansion theory is employed to compute the steady component of pressure. Although the surfaces of undeformed vehicle are all flat in the present model, they would be curved by structural deformations. Thus the basic procedure for pressures calculation should consist of (A) determining the flow conditions at the nose using freestream Mach number Mam and flow deflection angle sN, and (B) using Prandtl-Meyer expansion from nose along the downstream to take the pressure change into account.21 Then the steady pressure can be expressed as

Ps(x) — Pn

MaNAs(x)

where p represents pressure, Ma Mach number, y the ratio of specific heat, Ds(x) the difference between local panel inclination s(x) and sN, and subscript "N" the parameter at the nose. For the vehicle body, Ds(x) produced by fuselage bending can be given as

As(x) — sgn(n ■ e3) I —

dwf dx

dwf dx

where n is the surface normal vector, and e3 = [00 1] .

As the wing surfaces of HT and canard are regarded to be rigid, they are both treated as flat plate. The aerodynamic interaction of canard to HT is also considered by introducing HT effectiveness ratio given in Ref.19

The steady flight condition may be interrupted by control input or aerodynamic perturbations. In these situations, unsteady pressure correction is necessary. Local piston theory is used here and expressed as20

Pus(x) — Ps(x)

C — 1 Vn(x)

where as is the local sound speed determined by Shock-Expansion theory. For fuselage, vn is the normal component of the unsteady velocities including vehicle rigid-body pitching and fuselage vibration velocity, that is

Vn — (wfe3 + qef x r)

where r is the vector of local point in vehicle body, and e2 = [01 0]T.

As to the HT and canard, besides vehicle pitching and fuselage vibration, deflection motion of the wing surface due to control operation or elasticity also contributes to vn. Thus,

|wfe3 + qef x(r„p + rlocal ) + def x rtocal]

where rwp represents the position vector of the wing pivot relative to center-of-mass, and rlocal the vector of a point fixed on the wing surface relative to the wing pivot.

The one-dimensional scramjet engine model developed by Michael and David10 is employed and the thrust force T is given as

T = ma( Ve - Vi)+(Pe - Pi)Ae -(i - pJA> (18)

where ma represents the engine inflow mass rate, Ve the exit velocity, Vi the entry velocity, pe the exit pressure, pi the entry pressure, p1 the freestream pressure, Ae the engine exit area, and Ai the inlet area. JP-8 is chosen as the fuel and fuel-equivalence-ratio is used as thrust control parameter dT. The interested reader could refer to Ref.10 for details and Ref.22 for experimental/CFD results.

Fig. 3 Flowchart for aeroservoelastic analysis.

3. Trim and linearization of aeroservoelastic system

The flowchart for aeroservoelastic analysis of the vehicle is presented in Fig. 3. All the linearization analysis and time domain simulations are based on steady level flight conditions that are obtained by trimming the state variables of the aero-servoelastic system.

As the elastic dynamics of fuselage can be represented as several generalized modes, an undamped modal analysis is needed to give the mode frequencies and shapes. For each mode concerned, generalized displacement is trimmed according to the associated generalized force. The detailed processes of trimming and linearization are described as below.

3.1. Solution of trim state

With aerodynamic and propulsive forces determined, the vehicle can be trimmed for steady level flight condition. Steady level flight implies that the unsteady state variables and their derivatives to time are identically zero in Eq. (12). In addition, the pitch angle h equals the angle of attack a when the wind speed is zero. Within these conditions, Eq. (12) is reduced to

GA-SQP algorithm23 is used for solving the equations above. The trim variables contain steady state parameters a, gi, de, dc, and control inputs dcm, dT that satisfy Eq. (19).

3.2. Small perturbation equation for stability analysis

Once the trim state is obtained, Eq. (12) can be linearized into state space form under the small perturbation assumption. Define the state vector xs, the control vector us, and the force vector Fs as

xs — [XA Us — [Adcm Adx]T

Fs — [AX AZ AM A Ni AMt AM^]1

where xrb denotes the state vector of rigid-body motion, xf the state vector of fuselage bending motion, eH the integral error of altitude, and eV the integral error of velocity. Then, according to Eq. (12), the linearized equations of the aeroservoelastic system can be given as

X s — T1Xs + T2Fs + T3Us (21)

where T1, T2, and T3 are the coefficient matrices.

Utilizing aerodynamics and propulsion model, stability and control derivatives of the vehicle can be determined. Then Fs can be written as

Fs — Xs Us

Substituting Eqs. (22) and (21), the small perturbation equations can be expressed as

xs — Axs -I- Bus where

A — T + T @Fs

B — T + T3

When the controller gains K are given, the closed-loop system can be described as

xs — Axs H Bus y — Cxs

Us — Ky

Once the linearized equations are obtained, the stability of aeroservoelastic system can be investigated by solving the eigenvalues problem of state matrices (A for the open-loop, whereas A + BKC for the closed-loop).

'X — mg sin h — 0 mg cos h + Z — 0 M — 0

4i — xr (i — 1; 2..... «f)

de — d cm

-/i(x)

dc — kcdcm + X 4;

4. Results and discussion

To investigate the stability of multi-coupling dynamical system, the vehicle is trimmed at Mam = 8 and H = 85000 ft in steady-level flight. The trimmed states, control variables and elastic deformations are presented in Table 2 for both the rigid and flexible cases, where h^^h^^) represents the wing rotation angle of HT (canard) due to the flexible pivot. A baseline controller designed based on rigid vehicle assumption is given in Table 3 to provide approximate inflight characteristics of rigid-body flight dynamics.

Table 2 Trim results for steady-level flight.

Variable Flexible vehicle Rigid vehicle

a (°) 2.93 3.03

dcm (°) 10.0 7.5

¿T 0.337 0.310

hWp (°) -1.94 0

hWp (°) 1.49 0

gi 3 0

g2 0.126 0

g3 2.13 x 10-9 0

Fuselage 2nd bending mode 0 o Zeros HX pivot mode x Poles K ^x Ji Fuselage 1st bending mode ^ Canard pivot • mode S

............... y 0 / Actuator dynamics " Short-period mode £ ® :

-25 -20 -15 -10 -5 Real

(a) Flexible vehicle

Table 3 Baseline controller gains.

Gain Value

For angle of attack ka 7

For pitch rate kq 2

For flight velocity kV -0.01

For flight path angle kc 33.5

For altitude kH 0.001

For integral error of velocity keV 9.0 x 10-4

For integral error of altitude keH 1.0 x 10-4

4.1. Baseline vehicle stability analysis

Based on the trimmed state, linearization is performed for the flexible vehicle. The open-loop poles/zeros are presented in Fig. 4. It is indicated that the short-period mode is unstable, exhibiting an exponential divergence behavior, whereas the phugoid mode is a lightly-damped oscillation.

In order to investigate the baseline controller performance, the closed-loop poles/zeros and responses of the rigid and flexible vehicles are given in Figs. 5 and 6. The responses are obtained by giving an initial perturbation of q = 0.1 rad/s to the trimmed state. With the controller, the rigid vehicle becomes stable and exhibits good dynamic performance by recovering rapidly from the perturbation. However, the same controller fails to stabilize the flexible vehicle as the canard pivot mode poles are observed to move across the imaginary axis. This indicates that a well-designed controller based on the rigid vehicle assumption does not work on the flexible vehicle.

Fuselage 2nd bending mode o Zeros Canard pivot mode x Poles * o ** Fuselage 1 st bending mode ^ HT pivot mode A

Actuator dynamics 0 Short-period ^ mode ®

-25 -20 -15 -10 -5 0 5 10 Real

Fig. 4 Poles and zeros of linearized open-loop aeroservoelastic system.

X \ Short-period mode o Zeros x Poles

A ° y Actuator / dynamics / X

-10 -4 2 8

(b) Rigid vehicle

Fig. 5 Poles and zeros of linearized closed-loop aeroservoelastic system.

Fig. 6 Simulation results for rigid/flexible vehicle under a initial perturbation (q = 0.1 rad/s).

As shown in Fig. 5(a), one of the wing pivot modes becomes unstable when the baseline controller is incorporated. From the equations of wing/pivot system shown in Eq. (7), it can be seen that the controller affects the wing pivot modes by producing additional aerodynamic and inertial torques. To investigate the influence degree of these torques, frequency responses of control input to HT and canard wing pivot modes are given in Fig. 7. It is shown that response magnitudes of wing pivot modes are high in a wide bandwidth near their natural frequencies. Therefore, these additional torques are inneg-ligible, and may be the reason for wing pivot modes being prone to dynamic instability when the controller is incorporated.

Because the short-period mode frequency is close to those of elastic modes, its designed characteristics determine the degree of elastic/rigid-body modes coupling. To investigate how the system stability is impacted by the controller gains, a variable feedback gain amplifier ksp is introduced to represent different designed characteristics of short-period mode. Fig. 8 shows the root-locus of closed-loop system as ksp increases from 0 to 1.50. In addition to stabilizing the aeroservoelasticity system when ksp is low, this loop also drives the poles of elastic modes to approach the imaginary axis. This phenomenon indicates that elastic/rigid-body mode interactions are enhanced as the separation of structural vibration frequencies to those of rigid-body shrinks. Dynamic instability called body-freedom

flutter resulting from the coupling of short-period mode with wing pivot modes is firstly observed when ksp increases to 0.98. Therefore, the occurrence of body-freedom flutter clearly limits the utilization of high gain controller, i.e. limits the available bandwidth.

4.2. Effects of wing pivot stiffnesses

To increase the available bandwidth for control system design, modification of the baseline vehicle is necessary besides improving the control design approach. Because the wing pivot modes may involve in the body-freedom flutter, generalized stiffnesses of wing pivot modes are critical factors. Analysis of the aeroservoelastic system with varying pivot stiffnesses of wings is conducted to study how the stability is impacted by the wing pivot modes. Generalized stiffnesses represented by undamped frequencies of wing pivot modes (o>wp and Хф) are used in our work. Although the generalized stiffness could be improved not only by strengthening the pivot but also by adopting new support schemes, it will be still referred as ''pivot stiffness'' for convenience.

4.2.1. Wing pivot/rigid-body modes interaction Because both HT and canard pivot elasticity cause response lag of control surface deflection to control input, the stability augmentation system based on rigid-body assumption will deviate from its designed performance. As shown in Fig. 9, the damping ratio of short-period mode decreases when either xwp or tt>wp is reduced. It indicates that the elasticity of wing pivots results in degrading stability of rigid-body mode.

Fig. 10 show the effects of pivot stiffnesses on damping ratios of HT pivot mode and canard pivot mode (fp f^). It can be seen that there is a complex relationship between the damping ratios and stiffnesses of wing pivots. However, the general trend for decreasing the stiffness of either pivot is to reduce damping ratio of the corresponding wing pivot mode. Body-freedom flutter indicated by a sharply dropping damping ratio is observed when either HT pivot stiffness or canard pivot stiffness is further reduced. It is also shown that the canard and HT pivot modes do not apparently affect each other, except for the situation that their natural frequencies are close.

Fig. 7 Frequency response from control input to torsion angels of HT and canard pivots.

Fig. 8 Root-locus of closed-loop system with feedback gain amplifier ksp of short-period mode (ksp = 0-1.50).

Fig. 9 Contour map of short-period mode damping ratio vs pivot stiffnesses.

(b) HT pivot mode Fig. 10 Damping ratio vs pivot stiffnesses.

4.2.2. Elastic modes resonance

(1) Wing pivot/fuselage modes resonance. The elastic motion of canard or HT will produce additional aerodynamic forces acting on itself. By contributing to the generalized forces of fuselage elastic mode, these forces will also induce vibration of the fuselage. At the same time, the fuselage vibration also affects the wing pitch as indicated in Eq. (8). Therefore, the wing pivot modes and the fuselage modes are highly coupled, and pivot/fuselage modes resonance may happen when any one of the wing pivot mode frequencies is close to those of fuselage bending modes. As exhibited in Ref.5, the first bending mode is the most important among all the elastic modes of fuselage. Coupling of this mode with wing pivot modes is concerned.

As shown in Fig. 11, the damping ratio ff1 of the first fuselage bending mode drops as the canard pivot mode frequency «wp approaches œf1. However, ff1 increases as the HT pivot mode frequency X,p approaches œf>1 and reaches a peak when œf>1 equals œ>wp. This opposite effect of the two wing pivot modes on fuselage first bending mode is the result of different locations of HT and canard on fuselage, because when looking at the first bending mode shape, we can see that the pitch of HT and canard caused by this mode will always have opposite directions.

(2) Wing pivot modes resonance. As mentioned previously, there is a leap of and as «wp and Xwp approach each other. This indicates that resonance is likely to happen when

Fig. 11 Damping ratio of fuselage 1st bending mode vs pivot stiffnesses.

the pivot stiffnesses of HT and canard are close. To verify this situation, two test cases are given. In Case 1, xewp and xcwp are both taken as 40 rad/s, whereas in Case 2, taken as 40 rad/s, 30 rad/s separately. Numerical simulations are performed for both cases within an initial perturbation of q = 0.1 rad/s from the trimmed state. Comparison of the results is shown in Fig. 12. Although the canard pivot in Case 1 is less flexible than that of Case 2, its vibration has a larger magnitude and reaches equilibrium more slowly. These results indicate that the wing pivot modes resonance will cause intensive vibration of vehicle body. Therefore, sufficient separation should be given to avoid its occurrence.

4.2.3. Stiffnesses design of HT and canard pivots The instability of baseline HSV aeroservoelastic system as shown in Fig. 5 involves a coalescence of the vehicle short-period and the wing pivot modes, which could be alleviated by increasing the gap between wing pivot modes frequencies (<»wp and Xwp) and short-period mode frequency œsp. One way for stabilizing the system is to keep œsp low; nevertheless it may lead to an unacceptable controller with large tracking errors. A more feasible way is to increase the two wing pivot modes frequencies by strengthening the pivot shafts or adopting new support scheme.

To obtain the feasible stiffnesses of wing pivots, a designing space defined in stiffnesses of canard and HT pivots is given. As the body-freedom flutter is indicated by the occurrence of zero damping, the flutter boundary in pivot stiffnesses

Fig. 12 Simulation results of canard pivot torsion angle for two cases.

12 18 24 30 36 42 48

eo'^ (rad/s)

Fig. 13 Body-freedom flutter boundary in designing space of pivot stiffnesses.

necessary besides improving the control design approach. For the wing pivot modes, measures should be taken to increase their natural frequencies, such as strengthening the pivot or adopting new support scheme.

(3) Elastic modes resonance will induce intensive vibration with large response magnitude and long recovery time when the vehicle is under a disturbance or commanded maneuver. The resonance is likely to happen when two elastic modes have close natural frequencies. Thus, sufficient separation should be given to the designed natural frequencies of HT pivot mode, canard pivot mode, and fuselage elastic modes.

Acknowledgments

Fig. 14 Feasible region for pivot stiffnesses design.

designing space can be given as Fig. 13. It reveals that the value of X,p should not be less than 19.8 rad/s, 15.8 rad/s separately to stabilize the aeroservoelastic system with the baseline controller. Note that sufficient separation between œWp and œWp is also necessary to keep the two wing pivot modes from coupling with each other.

An example feasible region in design space of œWp and œWp is given in Fig. 14, where all of the elastic modes have a damping ratio no less than a critical value of 0.01, which may be determined from flight quality requirements. It is shown that an isolated narrow zone exists where damping ratios of both two pivot modes are greater than 0.01. However, in the same zone, the damping ratio of fuselage 1st bending mode drops sharply below 0.01. This phenomenon can be explained by the resonance of elastic modes, which has been discussed in Section 4.2.2.

5. Conclusions

(1) From the aeroservoelastic stability analysis, the canard-configured HSV is prone to body-freedom flutter that involves a coalescence of the rigid-body flight dynamics and the elastic modes. This phenomenon apparently limits the available bandwidth of control law design.

(2) The wing pivot modes are found to be critical for the occurrence of body-freedom flutter as the pivots are relatively flexible. Modification of the vehicle structure is

This study was co-supported by the National Natural Science Foundation of China (Nos. 90916006, 91116019 and 91216102).

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Zeng Kaichun received the B.S. degree in School of Aeronautic Science and Engineering from Beihang University in 2008, and is working for a Ph.D. degree there. His main research interests are aeroelasticity, flight dynamics and control.

Xiang Jinwu is a professor and Ph.D. supervisor in School of Aeronautic Science and Engineering, Beihang University, Beijing, China. His area of research includes aircraft design, aeroelasticity and structural dynamics, etc.

Li Daochun is an associate professor in School of Aeronautic Science and Engineering, Beihang University. His current research interests are aircraft design, aeroelasticity and structural dynamics.