Morphing Wing Aeroelasticity by Continuous Dynamic Simulation using Nonlinear Aerodynamic

RTO-MP-AVT-168 26 - 1

NATOUNCLASSIFIED+SWE+AUS


Morphing Wing Aeroelasticity by Continuous Dynamic Simulation using Nonlinear Aerodynamic/Nonlinear Structure Interaction (NANSI)

Methodology*

D.D. Liu1, ZONA Technology, Inc. P.C. Chen2 ZONA Technology, Inc.

M. Mignolet3, Arizona State University F. Liu4, University of California, Irvine

Ned Lindsley5, AFRL, WPAFB Phil Beran6, AFRL, WPAFB

[email protected]

ABSTRACT Our objective is to develop a nonlinear aerodynamic and nonlinear structural interaction (NANSI) methodology as an expedient aeroelastic tool to handle continuous dynamic motion of morphing vehicles/wings throughout the flight regimes of subsonic to transonic speeds. The proposed morph vehicles for method demonstration is the Lockheed Folding wing (the Folding wing). Preliminary results on Flutter/LCO of the Folding wing in continuous morph are presented. The developmental procedures of the tools needed for NANSI are presented in three sections. Section 2.0 describes the development of a nonlinear structural reduced order modeling (ROM) and morphing dynamics in terms of sub-structural formulation for the Folding wing. Section 3.0 presents the results of direct, or full-order, aeroelastic solutions in terms of flutter/LCO for the Folding wing in quasi-steady morph motion and in continuous dynamic morph motions. A CartEULER unsteady aerodynamic solver is used as an expedient, user-friendly tool for aeroelastic applications. Section 4.0 presents the establishment of aerodynamic ROM methodology based on a system identification strategy, with which an aeroelastic ROM-ROM methodology has been successfully developed. In terms of flutter/LCO solutions of the Folding wing, a selected example shows that ROM-ROM dramatically reduces the computing time by two orders of magnitude as opposed to that required by the direct method.

1.0 INTRODUCTION DoD and NASA have shown strong interests in morphing vehicle design concept and development. Recent DARPA contracts on designs of a sliding-skin concept (in-plane morph) and a folding wing concept (out-of-plane morph) are two such examples. [1 - 4]. To date, proper computational methodology remains unavailable to assess the flight dynamics and aeroelastic instability, or stability, of such type of vehicles during their morphing motion. Central in the morphing aeroelastic problem is the time/configuration-varying aerodynamics, structure, and their interaction. In fact, it appears that there is practically no computational tool available for transonic aeroelastic analysis of a vehicle morphing in a continuous dynamic manner. While their counterpart linear methods are being developed, their flutter solutions are obtained mostly in quasi-steady manner. [5 - 8]

1 President, also Professor Emeritus, ASU 2 Vice President 3 Professor, Mechanical and Aerospace Engineering Department 4 Professor, Mechanical and Aerospace Engineering Department 5 Aerospace Research Engineer 6 Principal Research Aerospace Engineer

Morphing Wing Aeroelasticity by Cont. Dynamic Simulation using (NANSI)

26 - 2 RTO-MP-AVT-168



In principle, morphing vehicle design is confined by its morphing actuator power requirement hence its morph components are likely to be lighter weight and more flexible than that of the conventional. Accordingly, the aeroelastic analysis of such type of morphing vehicles would fall in one with structural components exhibiting large deformations, hence one with geometrically nonlinear behavior. The design cruise speed of a morphing vehicle as such usually falls in transonic flight regime, hence the practice of CFD aerodynamics. These considerations call for the development of viable methodology in nonlinear aeroelasticity for morphing vehicles. Here, we present work in progress under AFOSR support focusing on the continuous dynamic simulation of nonlinear aerodynamic-nonlinear structures interactions (NANSI) methodology for morphing-wing aeroelasticity. The NANSI methodology mainly consists of two strategies: a “direct” aero-structural coupling and a reduced-order-model (ROM), or aero-and-nonlinear structural ROM, coupling. In what follows, we will discuss our NANSI methodology with these two strategies for a modeled Lockheed-Martin (LM) Folding wing (Figure 1.1) and we will address some essential technical challenges in the continuous dynamic simulation aspect of NANSI. These include: i) the geometrically nonlinear structural dynamics of a morphing wing using a sub-structuring technique and a nonlinear reduced order modeling methodology (ROM), ii) the nonlinear aerodynamics for the morphing wing using Cartesian Euler Scheme, and iii) the direct time simulation NANSI method for flutter/LCO through a full coupling of the structural dynamics and aerodynamics, and iv) further development of the reducing-order-model aeroelastic methodology or the aerodynamic ROM- structural ROM methodology (ROM-ROM) for rapid nonlinear aeroelastic computations.

2.0 MORPHING WING STRUCTURAL DYNAMICS Central to the aeroelastic analysis of such a morphing vehicle is the split of the structure into separate substructures which are assumed in our study to be hinged to one another. For such hinged configuration to morph, it is necessary that the hinge area be much stiffer than the rest of the structure and was thus assumed to be rigid. Next, a multi-body dynamic approach was undertaken to obtain the equations of motion of the wing in morphing. In this approach, each of the wing components was analyzed separately and the wing was formed through the assembly of these components while imposing compatibility equations at their interfaces, i.e. equality of displacements and forces. Then, the deflections are the sum of rigid body motions (rotation and translation) generated by the body at the left interface and deformations. These deformations are further expressed in a modal summation like form as linear combinations of selected, space varying basis functions weighted by unknown time-varying generalized coordinates. Proceeding through a weak form of the equations of finite strain elastodynamics then led to a series of ordinary differential equations to be satisfied by the generalized coordinates. This formulation builds upon previous work by the investigators on the development of nonlinear geometric reduced order models for non-morphing structures in which the selection of the basis functions and the derivations of the differential equations for the generalized coordinates are obtained from a full finite element model (e.g. Nastran) and a series of linear modal and nonlinear static solutions. Thus, the prediction of the structural dynamic behavior of the wing in morphing was accomplished in a nonlinear component mode synthesis-like strategy, i.e. by (1) decomposing the morphing wing in components and expressing the equations of motion for each

component in a multi-body dynamic framework, (2) adopting a reduced order modeling strategy to describe the wing components deformations and

deriving the corresponding differential equations for the generalized coordinates, and (3) selecting an appropriate basis for the representation of the wing deformations. Further, the validation and verification of the approach was accomplished:


RTO-MP-AVT-168 26 - 3



(4) in small deformations by comparing the natural frequencies and mode shapes of the wing in specific morphing configurations vs. full finite element predictions

(5) in large deformations by comparing the static response of the wing in specific morphing configurations vs. full finite element predictions.

These 5 steps are described in detail in the ensuing sections.

2.1 Multi-Body Dynamic Framework and Reduced Order Modelling A multi-body dynamic approach was undertaken to obtain the equations of motion of the wing in morphing. In this approach, each of the wing components is separately analyzed and the wing is formed through the assembly of these components while imposing some compatibility equations at their interfaces, see Figure. 2.1. Before addressing the derivation of the equations of motion of an arbitrary component, it was deemed important to first establish some physical requirements at the interfaces. Rotation around a hinge is only possible if the hinge line remains straight. Thus, to ensure an appropriate morphing, it will be required that the various components have rigid interfaces. It is worthwhile to recognize that rigid interfaces may also be necessary in other morphing strategies, e.g. a telescoping motion requires that the components perfectly match shape at their interfaces and thus a rigid interface condition is also appropriate in that case. Since the interfaces will be modeled as rigid sections, the force information required there reduces to a concentrated force and a concentrated moment. The kinematics of the interface is also simplified, requiring only the motion (position, velocity, and acceleration) of a single point C and an angular velocity ω at each interface. Under this condition, the loading acting on the component i is as shown in Figure 2.2. The derivation of the equations of motion for the ith component, shown in Figure 2.2, in arbitrary large elastic deformations starts from the weak form of the field equation of nonlinear elasticity. This weak form can be expressed as in the reference frame as (summation is implied over repeated indices)

( )0 0 0 0

0 00 0

t

ii i ij jk i i i i

k

vv u d X F S d X v b d X v t ds

Xρ ρ

Ω Ω Ω ∂Ω

∂+ = +

∂∫ ∫ ∫ ∫ (2.1)

where S denotes the second Piola-Kirchhoff stress tensor, 0ρ is the density in the reference configuration,

and 0b is the vector of body forces. All of these quantities are assumed to depend on the coordinates iX and are expressed in the reference configuration in which the structure occupies the domain 0Ω . The boundary, 0∂Ω , of the reference configuration domain 0Ω , is composed of two parts, 0

t∂Ω on which the tractions 0t are given (i.e. the domain on which the aerodynamics acts) and 0

u∂Ω on which the displacements are specified (i.e. the interfaces with the other components). Finally, ( )i Xν denote “trial functions” to be selected. In applying Equation (2.1) to the present problem, it is assumed that the deflection ( )iu X is the sum of rigid body motions (rotation and translation) generated by the body at the left interface and deformations. These deformations are further expressed as the sum ( ) ( )( )n

nq t U X (summation implied over n) where

( )( )nU X are appropriately selected mode shapes (basis function) and ( )nq t are the corresponding generalized coordinates. In the present context of flexible deformations appended on rigid body motions, it is appropriate to select three different sets of trial functions. The first set is ( )i ilXν δ= , i.e. a pure translational displacement field, and the corresponding equation


26 - 4 RTO-MP-AVT-168



( )

( )0

0(1) 0(1) ( ), ,

( ) ( ) ( )2

Lit

nL i R i aero G GC n

n n nn n

F F F ds m a m X m X q mX

q mX q mX mX

ω ω ω

ω ω ω ω

∂Ω

+ = + + × + × × +

⎡ ⎤+ × + × + × ×⎣ ⎦

∫ (2.2)

captures the rigid body translations aspects of the motion. The second set is ( ) ii XX =ν and permits to describe the overall rotations (rigid body) of the component, that is

( )

( ) ( )0

0(2) 0(2) ( ), ,

( )( ) ( ) ( )2 .

Li Li Lit

nL i R i aero G C C C n

nn n nn n

F F X F ds mX a I I q J

q I q I J I

ω ω ω

ω ω ω ω ω ω

∂Ω

+ = − × + × + + × −

⎡ ⎤+ + + + ×⎣ ⎦

∫ (2.3)

Describing the flexible motions is achieved by selecting the trial function as each of the mode shape in turn, i.e. ( ) ( )( )n

i iX U Xν = , to yield

( ) ( ) ( )

( ) ( ) ( )0 0

( ) ( ) ( )

( )( )( ) 0(1) 0(2) ( )

, ,

2 . . . . . .

. .

Li

Ri Rit

m m mmn n mn n mn mn n C

mnm ni

C R i C R iij jk i aerok

M q C q C G q mX a J I

U UF S d X U X F X F U F dsX s

ω ω ω ω ω ω ω

Ω ∂Ω

+ + + + − −⎡ ⎤⎣ ⎦

∂ ∂+ + + =

∂ ∂∫ ∫ (2.4)

In the above equations, the quantities LiCa ,

LiCI , m, GX denote the acceleration of the point LiC , the mass moment of inertia of the component with respect to that point, the total mass of component i, and finally the position vector of its center of mass (undeformed) from LiC . Further, ( )nmX , ( )nJ , ( )nI , mnC , and mnG denote constant vectors/tensors that involve the mass distribution in the element and the mode shape(s).

Additionally, ,L iω ω= and the term ( )0

( )ni

ij jkk

UF S d X

XΩ

∂∂∫ can be expressed as a 3rd order polynomial in the

generalized coordinate nq using a nonlinear ROM methodology (e.g., [9 - 14] and references therein) Equations (2.2)-(2.4) provide the basic equations to be solved for each component. The structure is assembled by applying conditions on the forces and the kinematics at the interfaces. For the case of a hinge, the interface forces and moments are such that 0(1) 0(1)

, , 1L i R iF F −= − and 0(2) 0(2), , 1L i R iF F −= − (on the two

direction perpendicular to the hinge line e). Further, , , 1L i R i eω ω θ−= + where θ is the relative angular velocity of the rotation around the hinge line.

Compi

Compi+1

Hinged

Figure 2.1: Separation of the Wing into Components with Interfaces


RTO-MP-AVT-168 26 - 5



Left

RightFaero

Compi

)2(,

)1(, , o

iLo

iL FF

icLa ,

iL,ω

iR,ω

CRia )2(,

)1(,

oiR

oiR FF

Figure 2.2: Forces and Kinematics of Component i

2.2 Basis Selection for the Reduction Order Modelling During morphing, the angle between the inboard and outboard wings will vary and thus even a rigid body dynamic analysis of the system requires the separate consideration of these two components. The characterization of the structural deformations in the entire folding wing must similarly proceed with separate descriptions of the elastic displacements in the inboard and outboard wings and then the reassembly of the entire wing. Procedures to perform these tasks are usually referred to as dynamic sub-structuring or component mode synthesis and have been investigated in details in the case of linear structures. Before discussing the available options, note that the total displacement fields of the inboard and outboard wings will be expressed as

( ) ( ) ( )( ) ( ) ( ), , , , , , , , ,i i iR Eu x y z t u x y z t u x y z t= + (2.5)

and ( ) ( ) ( )( ) ( ) ( ), , , , , , , , ,o o o

R Eu x y z t u x y z t u x y z t= + (2.6)

where the superscripts (i) and (o) refer to inboard and outboard while the subscripts R and E relate to rigid body and elastic, respectively. The rigid body displacements of the outboard wing, ( )( ) , , ,o

Ru x y z t , will be deduced from the motions of the rigid hinge line ( ), , ,Hu x y z t . Then, in the decomposition of Equation (1.6), the elastic motions

( )( ) , , ,oEu x y z t will vanish at all times on the hinge line. Accordingly, it is appropriate to express

( )( ) , , ,oEu x y z t in terms of the cantilevered modes of the outboard wing with the hinge line clamped.

As for the outboard wing, the rigid body displacements of the inboard wing, ( )( ) , , ,i

Ru x y z t , are associated with the rotation around the rigid root of the wing and thus the elastic displacements ( )( ) , , ,i

Eu x y z t must vanish there at all times. With this observation, one can formulate the three following dynamic sub-structuring (or component mode synthesis) strategies for the expansion of ( )( ) , , ,i

Eu x y z t : (i) express ( )( ) , , ,i

Eu x y z t in terms of the free interface modes (i.e. cantilevered modes of the inboard, clamped at the root, free at the hinge line) and of attachment modes at the hinge line following for example the procedures of McNeal, Rubin, or Craig and Chang (see [15] for review)

(ii) express ( )( ) , , ,iEu x y z t in terms of the fixed interface modes (i.e. modes of the inboard, clamped at both

the root and the hinge line) and of constraint modes at the hinge line following a Craig-Bampton format

(iii) express ( )( ) , , ,iEu x y z t in terms of the loaded interface modes following the procedure of Benfield and

Hruda.


26 - 6 RTO-MP-AVT-168



The first two procedures involve generalized coordinates associated with the modes of the structure (either free or fixed interface modes) but also with “modes” corresponding to the boundary, either forces at the boundary (the attachment modes of approach (i)) or its displacements (the constraint modes of approach (ii)). In the linear case, the inclusion of a handful of extra degrees-of-freedom is not a significant penalty. In nonlinear problems, however, the computational cost and complexity increase significantly with increasing number of generalized coordinates and thus an approach that is thrifty in generalized coordinates is desired. At the contrary of the strategies (i) and (ii), the approach of Benfield and Hruda, option (iii) above, satisfies this requirement and thus will be adopted here. The loaded interface modes of the Benfield and Hruda approach correspond to the clamped inboard wing exhibiting additional mass and stiffness matrices on the degrees-of-freedom at the hinge which together form the loading of the interface. These mass and stiffness matrices result from the motions of the outboard wing induced by the allowed displacements of the interface, i.e. the hinge. For this specific problem, the only allowed displacements of the interface induce the rigid body motions of the outboard wing. Accordingly, there is no stiffness generated and thus no stiffness loading of the inboard wing. Further, the mass loading is associated with the rigid motions of the outboard and thus can be accounted for directly by considering the inboard with the outboard attached as a heavy (i.e. not massless) but rigid component. Thus, the loaded interface modes of the inboard wing are those corresponding to the flexible inboard to which is appended the outboard wing rigid at Γ =0, both components having their respective mass distributions. According to approach (iii), the elastic deformations of the inboard will thus be expressed directly as a linear combination of these loaded interface modes.

2.3 Validation in Small Deformations: Linear Natural Frequencies and Mode Shapes The first validation/verification effort focused on the comparison of the natural frequencies and mode shapes of the folding wing, at constant, both zero and nonzero values of Γ, as obtained by the present approach (referred to as MORPH) and by Nastran analysis of the corresponding one piece structure. These computations validate/verify the terms (1)

ij jM q and (1)ij jK q . Shown in Figures 2.3 and 2.4 are the

prediction errors of the MORPH estimates of the natural frequencies (Figure 2.3) and mode shapes (Figure 1.4) vs. their counterparts from Nastran. Note that these errors are all less than 1% for the first six modes thereby demonstrating the adequacy of the Benfield and Hruda approach, even at large angles Γ.

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 2 3 4 5 6Mode Number

Rel

ativ

e Er

ror o

f Nat

. Fre

q. (%

)

0 degree10 degree20 degree30 degree40 degree

Figure 2.3: Relative Errors of Natural Frequencies Obtained from the Sub-structuring Approach vs. Full

Nastran


RTO-MP-AVT-168 26 - 7



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6

Mode Number

Rel

ativ

e E

rror

Nor

m o

f Mod

es (

%)

0 degree10 degree20 degree30 degree40 degree

Figure 2.4: Relative Errors of Mode Shapes from Sub-structuring Approach vs. Full Nastran

2.4 Validation in Large Deformations: Nonlinear Static Deflections

As discussed above, the term ( )0

( )ni

ij jkk

UF S d X

XΩ

∂∂∫ was expressed as a 3rd order polynomial in the

generalized coordinate nq using a nonlinear ROM methodology, more specifically the one (referred to here as ELSTEP) based on a set of linear and “dual” modes (see [9]) with the modifications suggested in [10] for cantilevered systems. The validation of the nonlinear ROM was accomplished in two steps, the ROMs of each wings were first validated alone, and then the ROM of the entire folding wing model was validated at different morphing angles.

2.4.1 Assessment on the Inboard and Outboard Wings Alone

To approximately simulate the morphing conditions, the full (inboard + outboard) wing model at Γ= 0 degree was first subjected to a transverse loading (equal forces on each of the nodes of the full wing tip) and the reactions at the hinge line determined by Nastran. These forces were then applied to the inboard wing and the corresponding deflections were estimated from both Nastran and the inboard ROM model consisting of its first 9 transverse modes and the corresponding 9 duals (generation 2, see [9]). Shown in Figure 2.5 are the transverse displacements obtained at the tip of the inboard wing by both Nastran and the reduced order model (“ROM (9t9d)”). Note the excellent agreement of these results and that a similar matching holds for the rotations. The validation of the decondensed model of the outboard was achieved by comparing the tip displacements of that wing under the given loading, both in the transverse and inplane directions, as predicted by the Nastran and the ROM (first 4 transverse modes and corresponding duals). These results are presented vs. the applied load in Figures 1.6-1.7. The excellent matching of these results demonstrates again the well foundedness of the reduced order modeling approach.


26 - 8 RTO-MP-AVT-168



0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 100 200 300 400 500

Tip Load Magnitude (each - lbs)

Tip

Dis

plac

emen

t/Win

g Sp

an

NastranROM (9t9d)

Figure 2.5. Comparison of Inboard Transverse Tip Displacement Obtained by NASTRAN and by the ROM

(Normalized by the Wing Span of the Entire Wing).

0

0.04

0.08

0.12

0.16

0.2

0 100 200 300 400 500


Tip

Dis

plac

emen

t/Win

g Sp

an

Nastran

ROM (4t4d)

Figure 2.6: Comparison of Outboard Transverse Tip Displacement Obtained by NASTRAN and by the

Decondensed ROM (Normalized by the Wing Span of the Entire Wing).

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 100 200 300 400 500


Tip

Dis

plac

emen

t/Win

g Sp

an

Nastran

ROM (4t4d)

Figure 2.7: Comparison of Outboard Inplane Tip Displacement (Along Y, the Largest Component) Obtained by

NASTRAN and by the ROM (Normalized by the Wing Span of the Entire Wing).


RTO-MP-AVT-168 26 - 9



2.4.2 Assessment on the Folding Wing through the Multibody Formulation

The final assessment of the reduced order modeling was performed by assembling the inboard and outboard wing following the multibody approach, i.e. with the code MORPH. Shown in Figures 2.8-2.10 are the prediction of the transverse displacement (Z) at the tip of the combined wing (node 288) and the displacements in the Y and Z directions at the middle of the hinge (node 95), all for different values of Γ. In these comparisons, the wing (complete, inboard and outboard) was assumed to be subjected to concentrated forces of equal magnitudes at every node and acting in the Z direction (i.e. perpendicular to the wing at Γ=0 degree). The very good agreement shown between Nastran and the ROMs/MORPH package demonstrates the applicability of the proposed nonlinear structural dynamic formulation and implementation.

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 10 20 30 40

Force Magnitude (each - lbs)

Def

lect

ions

/Win

g Sp

an

T3, 288 ROM T3, 288 NastranT2, 95 ROM T3, 95 ROMT2, 95 Nastran T3, 95 Nastran

Figure 2.8 Comparisons of Z Displacements at the Wing Tip (node 288) and Y and Z Displacements at the

Middle of the Hinge (node 95), Γ=0 Degree.

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 10 20 30 40


Def

lect

ions

/Win

g Sp

an


Figure 2.9: Comparisons of Z Displacements at the Wing Tip (node 288) and Y and Z Displacements at the



26 - 10 RTO-MP-AVT-168



-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 10 20 30 40


Def

lect

ions

/Win

g Sp

an


Figure 2.10: Comparisons of Z Displacements at the Wing Tip (node 288) and Y and Z Displacements at the


3.0 NONLINEAR AEROELASTICITY OF CONTINUOUS MORPHING FOLDING WING BY FULL-ORDER APPROACH

The folding wing considered in the current effort has inboard and outboard wing components which are connected by a rigid hinge line so that the outboard wing can rotate with regard to the inboard one. In our proposed morphing motion, the morphing angle of the inboard wing is changing with time while the outboard wing rotates accordingly around the hinge line in order to remain horizontal. The one-piece model has 6 modes whose modal shapes are defined in the global frame and thus are different for different morphing angles. The one-piece model is only for fixed morphing angle simulations. The two-piece model has 13 modes for both inboard and outboard components, and the outboard wing has additional 6 modes related to the rigid-body motion. The modes from both components are combined to build up a model with 32 modes whose modal shapes are defined in the local wing plane so that one set of modal shapes serve for all morphing angles. The ROM solver for the two-piece model requires input of morphing angle as well as morphing angular velocity and acceleration of the inboard wing component to account for continuous morphing motion.

3.1 Validation of CartEULER Using Skewed-Cartesian Grid For the folding wing of one-piece model with non-zero morphing angle, a skewed-Cartesian grid has to be used for CartEULER. Shown in (Fig 3.1) are the spanwise airloads of the folding wing at several morph angles Γ and Mach numbers (all at AoA = 0° except the lower case on the right). Solutions obtained by skewed-Cartesian/Euler (S- CartEULER) are seen to agree reasonably well with that of CFL3D/Euler with body-fitted grid. In order to further verify the unsteady aerodynamics generated by S-CartEULER, flutter simulations were performed for the folding wing of one-piece model at different morph angles and Mach numbers. Shown in (Fig 3.2) are two cases of flutter/LCO solutions at M=0.95 for the folding wing of morph angles at Γ= 0°, 30°. The LCO results can be translated in parallel to the “hump-mode” flutter cases in terms of the “q” versus “g” dependency. In fact, we have obtained a total of 6 such cases of transonic flutter/LCO results for the folding wing of one-piece model. Shown in Table 3.1 is a comparison of the transonic flutter results obtained here using S- CartEULER and those using the transonic module (ZTRAN) of ZAERO.


RTO-MP-AVT-168 26 - 11



5.012.0

45.0

1.75

7.04.5

X Y

Z

Γ = 30o

Y

L

0 2 4 6 8 10 12-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12CFL3DCartEuler

M = 0.95, Γ = 10o

Y

L

0 2 4 6 8 10 12

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1CFL3DCartEuler

M = 0.60, Γ = 10o

Y

L

0 2 4 6 8 10 12

0

0.1

0.2

0.3

0.4CFL3DCartEuler

M = 0.95, Γ = 30o

Y

Y

L

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2 CFL3DCartEulerZAERO

M = 0.60, α = 2 o, Γ = 10o

Y

Figure 3.1. LM Modeled Folding wing, Skewed-Cartesian grid, and Spanwise airloads at various Mach numbers ( M = 0.6, 0.95 ) , Angles of attack (α = 0°,2°) and Morph angles ( Г= 10°,30°):

Validation of S- CartEULER versus CFL3D/Euler

Time (s)

Tip

Dis

p/S

pan

b

0 1 2 3 4 5 6-0.001

-0.0005

0

0.0005

0.001

F = 4.6265, g = - 0.0420

M = 0.95, q ∞ = 614.7 psfTime (s)

Tip

Dis

p/S

pan

b

0 1 2 3 4 5 6-0.001

-0.0005

0

0.0005

0.001

F = 5.4155, g = - 0.0162

M = 0.95, q ∞ = 919.2 psf

Time (s)

Tip

Dis

p/S

pan

b

0 1 2 3 4 5 6-0.6

-0.4

-0.2

0

0.2

0.4

0.6F = 6.5169, g = 0.1255

M = 0.95, q ∞ = 1336.3 psf

Time (s)

Tip

Dis

p/S

pan

b

0 1 2 3 4 5 6-0.6

-0.4

-0.2

0

0.2

0.4

0.6F = 7.4213, g = - 0.0711

M = 0.95, q ∞ = 1895.9 psf

Time (s)

Tip

Dis

p/S

pan

b

0 1 2 3 4 5 6-0.02

-0.01

0

0.01

0.02F = 7.3925, g = - 0.1039

M = 0.95, q ∞ = 2632.9 psfTime (s)

Tip

Dis

p/S

pan

b

0 1 2 3 4 5 6-0.02

-0.01

0

0.01

0.02F = 8.5068, g = - 0.0882

M = 0.95, q ∞ = 3586.3 psf

Time (s)

Tip

Dis

p/S

pan

b

0 1 2 3 4 5 6-0.02

-0.01

0

0.01

0.02F1 = 11.3279, g1 = 0.2291; F2 = 17.0840, g2 = 0.0063

M = 0.95, q ∞ = 4039.0 psfTime (s)

Tip

Dis

p/S

pan

b

0 1 2 3 4 5 6-0.2

-0.1

0

0.1

0.2F1 = 9.3347, g1 = - 0.1697; F2 = 16.8583, g2 = 0.0804

M = 0.95, q ∞ = 4282.0 psf

Figure 3.2 Fullter/LCO results obtained by Direct aerolastic coupling of NANSI Methodology :

Left: 4-plots: Folding Wing Flutter/LCO at (Γ = 0°, M = 0.95) Right: 4-plots: Folding Wing Flutter/LCO at (Γ = 30°, M = 0.95)

Table 3.1. Comparison of Folding Wing Flutter Solutions: skewed-CartEULER vs. ZTRAN

Machnumber

Flutter Dynamic Pressure (psf)

Flutter Frequency (Hz)

CartEuler+ ZTRAN* CartEuler+ ZTRAN*

Γ = 0o 0.90 1210.1 1140.0 6.48 6.170.95 966.9 923.3 5.54 5.29

Γ = 10o 0.90 2567.0 2092.0 8.03 7.440.95 3058.5 2525.0 8.24 7.53

Γ = 30o 0.90 3770.0 3890.0 17.71 17.200.95 4009.1 4127.0 17.08 16.83

* ZTRAN: Transonic Field-Panel method using steady CFL3D Euler solution as the background flow + CartEULER = Skewed-CartEULER


26 - 12 RTO-MP-AVT-168



3.2 Validation of the Substructure Model: Two-piece vs. One-piece Wings ASU has assured the modal shapes and frequencies of the linear substructure two-piece model at different morphing angles matched well with those of NASTRAN one-piece model. A full-order (direct) CFD/Structures simulation is performed for both models to further verify the equivalence of the two models at fixed morphing angles. The comparison of aeroelastic responses in terms of wing tip displacement are shown in Figures. 3.3-3.8. For all 6 cases with morphing angles of

0 ,10 , 20 ,30 , 40 , 60o o o o o oΓ = and at 4 different altitudes (H = 0 Kft, 10Kft, 20Kft, 30Kft), the time histories of wing tip displacement of a (locked) two-piece model match well with those of one-piece model. All solutions were obtained for Mach number 0.95 and at angle of attack 0 degree. The initial kick is to apply a small velocity for all the structure grid points. All results showed proper LCO and convergent solutions which were found to confirm correct physics.

T im e

Tip

Dis

plac

emen

t/S

pan

0 2 4 6 8-1

-0 .5

0

0 .5

1

O n e -p ie c e M o d e lT w o -p ie c e M o d e l

H = 0 k ft

T im e

Tip

Dis

plac

emen

t/S

pan

0 5 1 0 1 5 2 0-1

-0 .5

0

0 .5

1

O n e P ie c e M o d e lT w o P ie c e M o d e l

H = 1 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 2 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 3 0 k ft

Figure 3.3: Validation of Two-Piece Model at Dihedral Angle 0o


RTO-MP-AVT-168 26 - 13



T im e

Tip

Dis

plac

emen

t/S

pan

0 2 4 6 8-1

-0 .5

0

0 .5

1


H = 0 k ft

T im e

Tip

Dis

plac

emen

t/S

pan

0 5 1 0 1 5 2 0-1

-0 .5

0

0 .5

1


H = 1 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 2 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 3 0 k ft


T im e

Tip

Dis

plac

emen

t/S

pan

0 2 4 6 8-1

-0 .5

0

0 .5

1


H = 0 k ft

T im e

Tip

Dis

plac

emen

t/S

pan

0 5 1 0 1 5 2 0-1

-0 .5

0

0 .5

1


H = 1 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 2 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 3 0 k ft



26 - 14 RTO-MP-AVT-168



T im e

Tip

Dis

plac

emen

t/S

pan

0 2 4 6 8-1

-0 .5

0

0 .5

1


H = 0 k ft

T im e

Tip

Dis

plac

emen

t/S

pan

0 5 1 0 1 5 2 0-1

-0 .5

0

0 .5

1


H = 1 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 2 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 3 0 k ft


T im e

Tip

Dis

plac

emen

t/S

pan

0 2 4 6 8-1

-0 .5

0

0 .5

1


H = 0 k ft

T im e

Tip

Dis

plac

emen

t/S

pan

0 5 1 0 1 5 2 0-1

-0 .5

0

0 .5

1


H = 1 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 2 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 3 0 k ft

Figure 3.7: Validation of Two-piece Model at Dihedral Angle 40o


RTO-MP-AVT-168 26 - 15



T im e

Tip

Dis

plac

emen

t/S

pan

0 2 4 6 8-1

-0 .5

0

0 .5

1


H = 0 k ft

T im e

Tip

Dis

plac

emen

t/S

pan

0 2 4 6 8-0 .0 5

-0 .0 4

-0 .0 3

-0 .0 2

-0 .0 1

0

0 .0 1

0 .0 2

0 .0 3

0 .0 4

0 .0 5


H = 1 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 2 0 k ft

T im e

Tip

Dis

plac

emen

t/

Spa

n

0 2 4 6 8-0 .0 0 5

0

0 .0 0 5

0 .0 1


H = 3 0 k ft

Figure 3.8: Validation of two-piece Model at Dihedral angle 60o

3.3 Quasi-steady Flutter Boundary: Folding Wing There exists two time scales in a physical wing morphing problem. While the wing is morphing at one speed, the wing undergoes a different time scale as measured by its aeroelastic “vibration frequency”. When the morphing time scale is much larger (or larger) than the aeroelastic time, then as an approximation the aeroelastic analysis can be performed at a fixed morphing position. This is known as the quasi-steady for morphing. When these two time scales are comparable, transient aeroelastic behavior is of interest. We call the latter “continuous dynamic morphing”.

Wing Folding Angle Γ (deg)

Flut

terD

yn.P

ress

ure

(psf

)

0 20 40 60 80200

400

600

800

1000

1200

1400

1600

1800

2000ZAERO Result for M = 0.80CartEULER Result for M = 0.95

Figure 3.9: Morphing Angle Effects on the Flutter Boundary of the Folding Wing


26 - 16 RTO-MP-AVT-168



Employing Nastran/Linear and ZAERO, Lee and Weisshaar [16] found that the flutter trend of a folding wing of two-piece substructures is one in which the flutter dynamic pressure increases first and then decreases with increasing morphing angle from 0 to 90 degrees. Their flutter solutions were obtained in frequency-domain using ZAERO and the wing morphs in a quasi-steady manner (Figure 3.9). Present study also adopts quasi-steady wing-morph approach by applying CartEULER and NL Structural ROM (of two-piece substructures) in time-domain to a different folding wing. The present finding also indicates a slight increase in flutter dynamic pressure with increasing morphing angle (Figure 3.9).

3.4 Continuous Morphing Simulation: Folding Wing Therefore for continuous morphing simulation, we define two types of morphing schedules (Equations (3.1) and (3.2)) as follows:

Constant Speed: 1 00 0

1 0

( - )( ) ( )t t t

t tΓ Γ

Γ = Γ + −−

(3.1)

Exponential: 1

0 1

( )

1 1 0 - ( - )

1-( ) ( )1-

t t

t t

ete

λ

λ

− −

Γ = Γ − Γ −Γ (3.2)

Note that the duration of morphing is from t = to to t = t1 when the morphing angles are 0Γ and 1Γ , respectively. Shown in Figure 3.10 is time variations of morphing angle Γ for three different morphing schedules. Figure 3.11 presents the comparison of aeroelastic responses from the above-mentioned three different morphing schedules. The flight conditions are M=0.95, AOA=0o and H=10Kft.

Time (s)

Γ(d

eg)

0 5 10 15 20 250

20

40

60

801. M orph (no accel.), eq. (1)2. M orph (λ = 0.6), eq. (2)3. M orph (λ = - 0.4), eq. (2)

Figure 3.10: Three Different Morphing Schedules


RTO-MP-AVT-168 26 - 17



T im e

Tip

Dis

plac

emen

t/

Spa

n

0 5 1 0 1 5 2 0 2 5-0 .3

-0 .2

-0 .1

0

0 .1

0 .2

0 .31 . M o rp h ( n o a c c e l.)2 . M o rp h (λ = 0 .6 )3 . M o rp h (λ = - 0 .4 )

Γ fro m 0 o to 6 0 o

T im e ( s )

Dam

ping

g

0 5 1 0 1 5 2 0 2 5-0 .0 1

0

0 .0 1

0 .0 2

0 .0 3

0 .0 4L in e a rλ = 0 .6λ = - 0 .4

Figure 3.11: Continuous Morphing Response I

To better understand the differences of time histories due to different morphing schedules, the continuous morphing responses are shown together with the fixed morphing angle responses for 0oΓ = and 60oΓ = at the same flight conditions in Figure 3.12. For fixed 0oΓ = , it will develop into LCO, while it is small amplitude close to neutral response for 60oΓ = . As the morphing schedule with 0.4λ = − delays the effective increase of morphing angle until late stage of the morphing period, the response of this kind of morphing schedule is mostly similar to the one for fixed 0oΓ = before 15t s= . On the other hand, the morphing schedule with 0.6λ= increases the morphing angle very fast at the earlier stage of the morphing period, and thus the response is all the way small amplitude oscillation similar to the one for fixed 60oΓ = . It comes without surprise that the response for the constant speed morphing case is somewhat in-between the other two. The flutter boundary of the folding wing can be roughly measured to be at altitude H=18Kft for morphing angle 0o, and H=10Kft for morphing angle 60o. The direct simulation of a continuous morphing case is performed at H=14Kft which is in-between the flutter boundaries of morphing angles 0o and 60o. The folding wing is scheduled to morph from 0o and 60o during the time period of 12s ~ 16s. The wing tip displacement and damping time histories are shown in Figure 3.13. The aeroelastic response demonstrates a first diverging and then converging phenomenon. If the morphing schedule is reversed, i.e. morphing from 60o and 0o, the aeroelastic response shows the reverse in trend. Shown in Figure 3.14 a divergent oscillating persists up to 40S, presumably it will approach LCO of the Γ=0° case asymmetrically in time. For these two cases, a much larger velocity is applied on all structure grid points as the initial kick.


26 - 18 RTO-MP-AVT-168



T im e

Tip

Dis

plac

emen

t/

Spa

n

0 5 1 0 1 5 2 0 2 5- 0 . 3

- 0 . 2

- 0 . 1

0

0 . 1

0 . 2

0 . 31 . M o r p h ( n o a c c e l . )2 . M o r p h ( λ = 0 . 6 )3 . M o r p h ( λ = - 0 . 4 )

Γ f r o m 0 o t o 6 0 o

(a)

Time

Tip

Dis

plac

emen

t/S

pan

0 5 10 15 20 25-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Γ = 0o

(b)

Time

Tip

Dis

plac

emen

t/S

pan

0 5 10 15 20 25-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Γ = 60o

(c)

Figure 3.12: Comparison of Aeroelastic Responses


RTO-MP-AVT-168 26 - 19



T i m e ( s )

Tip

Dis

plac

emen

t/

Spa

n

0 1 0 2 0 3 0 4 0- 0 . 5

- 0 . 4

- 0 . 3

- 0 . 2

- 0 . 1

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

Γ f r o m 0 o t o 6 0 o

M = 0 . 9 5 , A O A = 0 o , H = 1 4 k f t

c o n v e r g i n g a t Γ = 6 0 od i v e r g i n g a t Γ = 0 o

T i m e ( s )

Dam

ping

g

0 1 0 2 0 3 0 4 0- 0 . 0 1

- 0 . 0 0 5

0

0 . 0 0 5

0 . 0 1

Figure 3.13: Continuous Morphing Response II

T i m e ( s )

Tip

Dis

plac

emen

t/

Spa

n

1 0 2 0 3 0 4 0- 0 . 2

- 0 . 1

0

0 . 1

0 . 2

Γ f r o m 6 0 o t o 0 o

M = 0 . 9 5 , A O A = 0 o , H = 1 4 k f t

d i v e r g i n g a t Γ = 0 oc o n v e r g i n ga t Γ = 6 0 o

T i m e ( s )

Dam

ping

g

0 1 0 2 0 3 0 4 0- 0 . 0 2

- 0 . 0 1

0

0 . 0 1

0 . 0 2

Figure 3.14: Continuous Morphing Response III


26 - 20 RTO-MP-AVT-168



4.0 AERODYNAMIC ROM/AEROELASTIC ROM-ROM METHODOLOGY FOR FOLDING WING

Figure 4.1: Aeroelastic ROM-ROM Roadmap

As shown in the road map of Figure 4.1, ROM-ROM has three parts: Training, Validation and ROM-ROM Applications. - Training is to build the AeroROM from (1) to (4) using system ID approach. We select a proper Input

such as generalized coordinate in terms of an impulse function called filtered impulse model (FIM). Here we treat a given CFD solver as a black box. We get an output in terms of GAF. We then build up a response relation between input and output. The linear relation is ARMA, nonlinear is NeuralNet.

- Validation is to assure the AeroROM works. So we select a new input say a simple harmonic signal. It goes through both ways and we validate the ROM result with the full order, or direct solution from CFD.

- ROM-ROM Application is build on the ROM-ROM equation where its RHS is the GAF of the neural net (or ARMA) and the LHS is the nonlinear Structural ROM. Now we solve for the response in general coordinate “q”.

4.1 Aerodynamic ROM Methodology We have developed time-domain aerodynamic reduced order modeling (ROM) based on the CFD solver, CartEULER System identification technique is applied to find the simplified mathematical model (ROM) between the response of the aeroelastic system, i.e., the modal coordinates (or generalized coordinates) and the generalized aerodynamic forces (GAFs). The parameters defining the model structure are found through fitting a set of recorded (or measured) input/output data from the dynamic system; a process which usually is called training. Here, we implement the filtered impulse method (FIM) signals as the training excitation inputs. A FIM signal is given by:


RTO-MP-AVT-168 26 - 21



( ) ( ) ( )2

0 00 0

0

sin when 0 when

a t tu t Ae t t t tt t

ω ω π ω ω− −= − ≥

= < (4.1)

A staggered sequence (one after another) of FIM input of modal coordinates is employed for training. Each mode uses its own natural frequency as the ω in Equation (4.1). The reason behind this choice is that it is believed that each mode would be excited around its own natural frequency. With the prescribed motion (staggered FIMs), the CFD solution by CartEULER is carried out and the time responses of the GAF are recorded so that the complete training data set in terms of the pair of inputs (modal coordinates) and outputs (GAFs) is obtained. Note that, the GAFs should be normalized by the dynamic pressure. Two types of discrete time modeling structures are used to represent the ROM between generalized coordinates and normalized GAFs. The first one is the autoregressive moving average (ARMA) model. By denoting y as the output, and u as the input vector, we have

( ) ( ) ( )1 1

1a bn n

i ji j

y t a y t i t j= =

= − + − +∑ ∑b u (4.2)

where an and bn represent the order of the model.

Σ

Σ

Σ

Σ

ƒ

ƒ

ƒ

ƒ

......

......

U

yp

n1(1)

n2(1)

nS(1)

a1(1)

a2(1)

aS(1)

n(2) a(2)

Inputs Input Layer Output Layer

wij(1,1)

wi(2,1)

...

wij(1,2)

bS(1)

b(2)

b2(1)

b1(1)

Figure 4.2: Two-layer Feed-Forward Neural Network

The second is the feed-forward neural network (NNet). A two layer neural network is shown in Figure 4.2, which consists of an input layer and an output layer, to represent the forward dynamics of the plant. The input layer shown in Figure 4.2 includes vector U consisting of current and previous input signals, and, vector py consisting of an previous plant outputs, and S neurons. The size of vector U would be the product of number of components (or channels) of input signal and the delay order bn . The modeled plant output at time t by the neural network would be given in a concise notation as

( ) ( ) ( ) ( ) ( ) ( )( ) ( )2 2,1 1,1 1,2 1 2tanh py t a W W U W y b b= = ⋅ ⋅ + ⋅ + + (4.3)

When the neural network is given appropriate weights and biases, it provides the desired outputs. The process to get the proper set of the weight matrixes and the biases is called the training or learning of the network. It can be seen that the ARMA model can be represented by one neuron model with linear transfer function. We always start with the ARMA model to determine the proper delay order an and bn . Using the training data, an optimization procedure is implemented to search for the best parameters ia , ib


26 - 22 RTO-MP-AVT-168



in Equation (3.2) and ( ) ( )1,1 1, W b , ( )2,1W and ( )2b in Equation (3.3) by minimizing the mean square of the error between model output and targeted output (mse) or the generalized mean square error (msereg).

4.2 LCO Solutions for the Folding Wing using ROM-ROM The ROM-ROM procedure is to solve the time consuming issue of coupling the nonlinear structural dynamic equations with the generalized aerodynamic forces iF directly computed by CFD solver, e.g., CartEULER. With the aerodynamic ROM in lieu of the CFD solver, e.g., using the neural network model, the nonlinear aeroelastic equation is recast as

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )1 2 3 2,1 1,1 1,2 1 221 tanh 2ij j ij j ij j ijl j l ijlp j l p i i i i p i iM q D q K q K q q K q q q V W W U W F b bρ ∞+ + + + = ⋅ ⋅ + ⋅ + + (4.4)

Thus, Equation (4.4) is a ROM-ROM approach ready to be time integrated. It can be seen that the time savings by using the ROM-ROM procedure could be dramatic. Following ROMROM approach, the aerodynamic ROMs are firstly sought before solving Equation (4.4). The inputs (prescribed staggered motion of modal coordinates), and the normalized GAF solutions by CartEULER and the trained ROM are shown in Figure 4.3 for the folding wing at zero morphing angle (as a flat plate). After the aerodynamic ROMs are obtained through training, they can be validated by comparing the normalized GAF predictions by the aerodynamic ROMs with the direct full-order CFD solutions where any modal coordinate is prescribed by a sinusoidal excitation. Figure 4.4 presents such a validation comparison at which the first modal coordinate is given a sinusoidal excitation while all others are kept zero. We can clearly see that the aerodynamic ROMs have done an excellent job of prediction.

Two typical LCO solutions with the ROM-ROM approach are shown in Figure 4.5 and Figure 4.6 along with the solutions obtained with CartEULER directly providing the aerodynamic forces. In both cases of the attitude 0 Kft and -10 Kft, LCO amplitudes predicted by the ROMROM approach agree very well with the direct CFD approach, but phase shifting occurs as the LCO develops.


RTO-MP-AVT-168 26 - 23



0 0.5 1 1.5-2

0

2

q

q1q2q3q4q5q6

0 0.5 1 1.5-1

0

1

2

GA

F 1 Direct

ROM Sim.

0 0.5 1 1.5-0.5

0

0.5

GA

F 2 Direct

ROM Sim.

0 0.5 1 1.5-0.2

0

0.2

GA

F 3 Direct

ROM Sim.

0 0.5 1 1.5-0.5

0

0.5

GA

F 4 Direct

ROM Sim.

0 0.5 1 1.5-0.5

0

0.5

GA

F 5 Direct

ROM Sim.

0 0.5 1 1.5-0.5

0

0.5

GA

F 6

t

DirectROM Sim.

Figure 4.3: Aerodynamic ROM Training (Mach=0.95, AoA=0): System Inputs (staggered FIM Excitations for the

Modal Coordinates); System Outputs: Normalized GAFs


26 - 24 RTO-MP-AVT-168



0 1 2 3 4 5 6 7-1

0

1

t(s)

q

q1q2q3q4q5q6

0 1 2 3 4 5 6 7-1

0

1

GA

F 1 Direct

Aero ROM

0 1 2 3 4 5 6 7-0.5

0

0.5

GA

F 2 Direct

Aero ROM

0 1 2 3 4 5 6 7-0.1

0

0.1

GA

F 3 Direct

Aero ROM

0 1 2 3 4 5 6 7-0.5

0

0.5

GA

F 4 Direct

Aero ROM

0 1 2 3 4 5 6 7-0.5

0

0.5

GA

F 5 Direct

Aero ROM

0 1 2 3 4 5 6 7-0.5

0

0.5

GA

F 6

t(s)

DirectAero ROM

Figure 4.4: Validation of the Aerodynamic ROMs (Mach=0.95, AoA=0): Comparison between the GAF

Predictions by the Aerodynamic ROMs and the Direct Full-Order CFD Solutions (Mode 1 is given a Sinusoidal Excitation)


RTO-MP-AVT-168 26 - 25



0 1 2 3 4 5 6 7 8 9 10-1

0

1

q1

NL Structure + CartEulerNL Structure + Aerodynamic ROM

0 1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

q2

0 1 2 3 4 5 6 7 8 9 10-0.01

0

0.01

q3

0 1 2 3 4 5 6 7 8 9 10-0.02

0

0.02

q4

0 1 2 3 4 5 6 7 8 9 10-0.01

0

0.01

q5

0 1 2 3 4 5 6 7 8 9 10-5

0

5x 10

-3

q6

0 1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

Tip

Dis

p/S

pan

t(s) Figure 4.5: Nonlinear Aerodynamics/Nonlinear Structural Analyses for the Folding Wing (Zero Morphing

Angle) at M=0.95, H=0kft; LCO: ROM-ROM (Red),vs Direct (Blue)


26 - 26 RTO-MP-AVT-168



0 1 2 3 4 5 6 7 8 9 10-2

0

2

q1

NL Structure + CartEulerNL Structure + Aerodynamic ROM

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

q2

0 1 2 3 4 5 6 7 8 9 10-0.05

0

0.05

q3

0 1 2 3 4 5 6 7 8 9 10-0.05

0

0.05

q4

0 1 2 3 4 5 6 7 8 9 10-0.02

0

0.02

q5

0 1 2 3 4 5 6 7 8 9 10-0.01

0

0.01

q6

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

Tip

Dis

p/S

pan

t(s) Figure 4.6: Nonlinear Aerodynamics/Nonlinear Structural Analyses for the Folding Wing (Zero Morphing

Angle) at M=0.95, H=-10kft; LCO: ROM-ROM (Red),vs Direct (Blue)


RTO-MP-AVT-168 26 - 27



5.0 CONCLUSIONS In this paper, we focused on the validation of the proposed dynamic sub-structuring (or component mode synthesis) approach of the folding wing into inboard and outboard components, especially in the context of a geometrically nonlinear response. Improvements in the nonlinear reduced order modeling scheme, most notably the inclusion of inplane displacements in the model, were found to be necessary and were performed through a novel strategy for the determination of the ROM stiffness coefficients. With this modified approach, a very good prediction of the nonlinear response of the entire folding wing was achieved for tip deflections up to 1/4 of the span. With the accomplishment of NL/Linear Structural reduced order modeling (ROM) procedure for substructure models (one-piece/two-piece) of the folding wing, a full-order (direct) CFD/Structures simulation has been achieved by closely coupling an unsteady EULER solver called CartEULER with the NL/Linear Structural ROM procedure. The two-piece wing model including morphing dynamics thus can be used for either quasi-steady (morphing time only) or continuous morphing simulations. First, quasi-steady results are presented showing validation of the two-piece wing model is done by comparing the aeroelastic responses from both one-piece and two-piece linear models at fixed morphing angles. Next, continuous morphing results are obtained using the two-piece model ROM coupled with CartEULER. An Euler-based code with boundary layer option constructed on Cartesian grid, CartEULER is capable of generating time-accurate unsteady aerodynamic forces for the folding wing in continuous morphing motion. CartEULER has been validated with CFL3D on the Lockheed folding wing up to an morph angle of 60 degrees, and with transonic flutter solutions with ZTRAN. Aeroelastic ROM-ROM is aimed for rapid predictions of the aeroelastic instability by solving the nonlinear structural ROM and Aerodynamic ROM jointly. Aerodynamic ROM is not new, Various approaches were initiated previously by Silva, Dowell, Beran, and others. ZONA adopts a different approach which uses the CFD solver as a black box, hence a non-intrusive ROM method. ZONA aero ROM methodology in fact adopts a system ID approach, which is a non-intrusive procedure. To implement the system identification method, it must be handled in time domain. It has been shown that a successful aerodynamic ROM is the key to achieve an accurate and rapid aeroelastic analysis. We presented the establishment of aerodynamic ROM methodology which yields ROM results in excellent agreement with that of full-order for a folding wing at fixed morph angle In fact, we see that ROM-ROM dramatically reduces the computing time by two orders of magnitude for aeroelastic analysis applied to the Lockheed folding wing. .

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Morphing Wing Aeroelasticity by Continuous Dynamic Simulation using Nonlinear Aerodynamic