the t-tail flutter mechanism revisited - MSC Software Corporation

MSC.Software 2013 Users Conference

Paper No. AM-CONF13-34

1

A DLM-BASED MSC Nastran AERODYNAMIC FLUTTER

SIMULATOR FOR AIRCRAFT LIFTING SURFACES

Emil Suciu1, Nicholas Stathopoulos

2, Martin Dickinson

3 and John Glaser

4

1Formerly with Bombardier Aerospace; Currently Loads and Dynamics Analyst with

L-3 Communications, 7500 Maehr Drive, Waco, Texas 76715, USA 2Manager, Loads & Dynamics, Bombardier Aerospace

400 Cote-Vertu Road West

Dorval, Quebec, H4S 1Y9, Canada 3Principal Engineering Specialist, Loads & Dynamics, Bombardier Aerospace

4Bombardier Aerospace (Retired)

Summary: A DLM-based aerodynamic simulator for flutter is used to identify some of the

most important aerodynamic drivers for the T-Tail flutter mechanism of a complete aircraft.

The simulator is using the Modal Descrambling Factoring Method, which permits individual

variations of each direct and each interference aerodynamic force, moment and hinge

moment independently of any other force or moment. The sensitivity of the flutter solution

to individual variations of very large numbers of direct and interference aerodynamic

derivatives can be studied with ease.

Keywords: Flutter Simulator, T-Tail, DLM, Modal Descrambling

LIST OF SYMBOLS

lmlmlmlm aaaa 4321 ,,, = lift correction factors for the modal descrambling factoring method for

direct and interference forces and moments, nml ,1,

c = reference chord; also denotes streamwise camber deformation; apparent from the context

ChEh = elevator hinge moment due to horizontal stabilizer h (roll/bending)

ChE = elevator hinge moment due to horizontal stabilizer α (torsion)

ChE(h+) = elevator hinge moment due to horizontal stabilizer h (roll/bending) AND α

(torsion)

ChEβ = elevator hinge moment due to vertical fin yaw β (torsion)

ChEδE = elevator hinge moment due to elevator rotation

ChEδR = elevator hinge moment due to rudder rotation

ChRβ = rudder hinge moment due to yaw (β or V. Fin torsion)



2

ChRδR = rudder hinge moment due to rudder rotation

CLhVF = horizontal stabilizer lift coefficient due to vertical fin lateral bending h

ClhVF = horizontal stabilizer section lift coefficient due to vertical fin lateral bending h

CL = horizontal stabilizer lift coefficient due to

Cl = horizontal stabilizer section lift coefficient due to

CLβ = horizontal stabilizer lift coefficient due to yaw (β or V. Fin torsion)

Clβ = horizontal stabilizer section lift coefficient due to yaw (β or V. Fin torsion); also

known as the top rolling moment

ClδR = horizontal stabilizer section lift due to rudder rotation (top rolling moment due to R)

Cn = normal force coefficient

CYVFβ = vertical fin side force coefficient due to yaw (β or V. Fin torsion)

DLM = Doublet Lattice Method

HS = Horizontal Stabilizer

h = heave displacement

i,j = indices

k = *c/(2*V), reduced frequency

l,m = indices

Lh, L, Lβ, L = lift due to h, , β and motions (classical notation)

L’ = factored lift

M = Mach Number

Mh, M, M, M = moment about reference axis due to h, , and motions (classical

notation)

n = number of lifting surfaces or components

NBOX = the total number of aerodynamic boxes on the aircraft

Th, T, T, T = control surface hinge moment due to h, , and motions (classical

notation)

q = (1/2)ρV2

= dynamic pressure

QHHL = List of matrices of generalized aerodynamic forces, modal set

Qh, Q, Q, Q = tab hinge moment due to h, , and motions (classical notation)

V = airspeed

Vf = flutter speed

wh, w, w, w, wc = downwash vectors due to h, , , and camber motions

(classical notation)

= pitch (or torsion) angle

= control surface rotation angle; also angle of sideslip or vertical fin torsion; will be

apparent from the context

= tab rotation angle (classical notation)

E = elevator rotation angle

R = rudder rotation angle

Cp = pressure coefficient

ρ = air density

= circular frequency of oscillation



3

1 INTRODUCTION

The current paper is a revision of the material contained in Reference [1]. The discussion

here applies to all T-Tails analyzed with linear unsteady aerodynamic methods such as the

Doublet Lattice Method (DLM, Reference [2]) with factoring.

Aerodynamic factoring is important; in Reference [3] it is shown that a flutter analysis of an

airplane under development without using factoring for the aileron and tab aerodynamic

hinge moments, when either 2-D strip theory or DLM aerodynamics is used, may not

predict flutter and in this particular case has led to an in-flight flutter incident. Intersecting

(therefore interfering) lifting surfaces pose additional problems for the flutter analyst. FAA

Advisory Circular AC No. 25.629.1A recommends that interference effects be included in

flutter analyses. Parametric variations of calculated aerodynamic forces and moments are

recommended in order to cover uncertainties in calculated values. These parametric

variations are achieved through factoring.

It is known that the T-Tail flutter mechanism (generally referred to as the vertical fin

bending/ torsion mechanism) is strongly influenced by the unsteady aerodynamic forces and

moments present on the horizontal stabilizer. See Reference [4], where the effect of steady

upload on horizontal stabilizer on T-Tail flutter speed is appended to the MSC Nastran

flutter solution (Reference [5]) and an early attempt to separate and factor differently the

horizontal stabilizer direct lift due to horizontal stabilizer pitch/torsion from the interference

lift due to vertical fin torsion on the horizontal stabilizer of a T-Tail is described.

It is also known (Reference [6]) that the unsteady aerodynamic forces and moments present

on the horizontal stabilizer of a T-Tail consist of a superposition of direct forces and

moments resulting from the rigid and elastic motions of the horizontal stabilizer and its

control surfaces and tabs and of interference forces resulting from the oscillating vertical fin

and its control surfaces and tabs. The superimposed direct and interference aerodynamic

forces and moments on the horizontal stabilizer are called stacked forces.

Thus, for general motion of a given arrangement of n lifting surfaces residing in the same

interference group, every lifting surface experiences 1 direct set of forces and moments and

n-1 sets of interference forces and moments. For n>2, each lifting surface experiences more

interference forces than direct forces!

Figure 1 shows a general (or scrambled) mode of vibration of a T-tailed airplane at the

aerodynamic surface.



4

Figure 1. Scrambled Vertical Fin Bending/Torsion/HS Roll/Elevator Rotation/Rudder Rotation Mode at

the Aerodynamic Surface. A Few Direct and Interference Aerodynamic Forces and Moments are Shown

on the Empennage.

For every mode of vibration, rigid or elastic, the Modal Descrambling factoring method

(formerly known as the General Aerodynamic Derivatives Factoring Method, Reference [6])

performs the descrambling of the general modal motion at every aerodynamic strip and

replaces each scrambled mode with 5 simpler and always the same modes: heave (or

bending), pitch (or torsion), control surface rotation, tab rotation and elastic streamwise

camber deformation. Unsteady direct and interference aerodynamic forces and moments

distributions are then calculated for the descrambled modes and they are available for

factoring separately at every aerodynamic strip (Figure 2).

ClhClClhVF

ClβClδR

ChEδE

ChEh

ChE

ChEβ

ChEδR

ChRβ

ChRδR

CYVFβ

CYVFδR



5

Figure 2. Modal Descrambling at Typical Strip of Aerodynamic Surface. Factoring Shown Only for Direct

Lift. Elastic Streamwise Camber Deformation not Shown.

Experimental evidence shows that the factors required for direct and interference

aerodynamic forces and moments are different from each other. For a DLM aerodynamic

analysis without modal descrambling the user has access only to the stacked forces and

moments for factoring. As shown in this paper, this may lead to inaccurately calculated

flutter speeds.

= =

+ +

+ +

+ +

Real 2-D Theodorsen Cp Due to General Motion; Hinge

Lines at 70% and 85% Chord; No Aerodynamic Balance

Surfaces; k=0.372; Calculate L', M', T', Q'

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1x/c

Rea

l

Cp

L' = a1*Lh + a2*L + a3*L + a4*L

Real 2-D Theodorsen Cp Due to Main Surface Heave;

k=0.372; Calculate Lh, Mh, Th, Qh

-1

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1x/c

Real

C

p

a1*Lh

Real 2-D Theodorsen Cp Due to Main Surface Pitch;

k=0.372; Calculate L, M, T, Q

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1x/c

Re

al

C

p a2*L

Real 2-D Theodorsen Cp Due to Control Surface Rotation;

Hinge Line at 70% Chord; No Aerodynamic Balance

Surface; k=0.372; Calculate L, M, T, Q

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1x/c

Re

al

C

p a3*L

Real 2-D Theodorsen Cp Due to Tab Rotation; Hinge Line

at 85% Chord; No Aerodynamic Balance Surface; k=0.372;

Calculate L, M, T, Q

0

3

6

9

12

15

0 0.2 0.4 0.6 0.8 1x/c

Re

al

C

p a4*L

Reference Plane

h

Reference Plane

h

Reference Axis

Reference Plane

Reference Plane

Reference Plane



6

The MSC Nastran implementation of the DLM (Reference [5]) together with the MSC

Nastran SOL145 with the PKNL flutter solution method chosen and program LSP3G

(Lifting Surface Program 3 General) which implements the Modal Descrambling factoring

method (Reference [6]) and is in effect an aerodynamic flutter simulator are used for

calculating all the results presented here. MSC.PATRAN is used to display and animate

mode shapes.

2 MORE ON THE MODAL DESCRAMBLING FACTORING METHOD

Figure 2 illustrates the summary of the original Modal Descrambling factoring method

showing the descrambling of a general mode of vibration for a lifting surface strip into

simpler motions: heave h, pitch α, control surface rotation β and tab rotation δ and factoring

of the lift. Elastic streamwise camber deformation is not shown in this figure. Elastic

streamwise camber deformation and its effect on flutter calculations are discussed in detail

in Reference [7].

The downwash vectors based on descrambled modal displacements (h, α, β, δ, c) for the

entire aircraft for any elastic or rigid mode look like:

(1)

with the scrambled downwash jw being the sum of the descrambled downwash vectors as

defined for the modal descrambling factoring scheme (Figure 2). The pressure coefficients

and integrated forces and moments can then be obtained for the scrambled or descrambled

modes:

(2)

with i,j=1,NBOX.

If Equation 2 operates on the scrambled modes, we have the typical aerodynamic analysis

which calculates stacked aerodynamic forces and moments and only the stacked forces will

be available for factoring; it will be later seen that this is incorrect. If Equation (2) operates

on the descrambled modes of Equation (1), we have access to the descrambled aerodynamic

forces and moments, but not yet to the interference aerodynamic forces and moments.

In order to calculate the interference aerodynamic forces and moments, the entire aircraft

descrambled downwash vectors are then partitioned into n components. This number is

generally fairly small, such as 4 for the airplane analyzed: (1) wing with associated control

surfaces and tabs, (2) horizontal stabilizer with associated control surfaces and tabs, (3)

vertical fin with associated control surfaces and tabs and (4) engines, pylons, interference

body and ventral fins.

cjjjjhjj wwwwww

jjipi wAC1



7

The effect of interference is calculated as follows: for any mode, we start with component

(or lifting surface) 1 and we only permit its motion in bending (or h); we then calculate all

direct forces for component 1 due to component 1 bending and the interference forces due to

component 1 bending on the rest of the components, all n-1 of them. Then we only permit

motion in torsion (or α) for component 1 and calculate all direct forces for component 1 due

to component 1 torsion and the interference forces due to component 1 torsion on the rest of

the components; the same sets of direct and interference forces are calculated due to

component 1 moving in the descrambled modes consisting of control surface rotation β,

then tab rotation δ, then elastic streamwise camber deformation c. Once we are done with

component 1, we repeat the process with every component. Reference [6] contains the

mathematical description of this process.

For an aircraft with 4 components, for each mode of vibration descrambled into 5

elementary modes, now we will have to do the processing a total of 20 times vs. one time

for the scrambled mode.

See Figure 2 for an example of the application of correction factors to strip direct lift.

The stacked unfactored lift at any aerodynamic strip of component l is:

(3)

where nml ,1, .

The stacked factored lift at any aerodynamic strip of component l is:

(4)

where as before, nml ,1, and lmlmlmlm aaaa 4321 ,,, are the direct and interference factors on

the lift, respectively. For the time being, the forces and moments due to camber deformation

are not factored, due to a lack of experimental or CFD guidance. Also for the time being,

the real and imaginary forces and moment distributions are factored by the same real factors

derived from steady state wind tunnel and CFD calculations. If unsteady experimental or

CFD guidance is available, the imaginary part can easily be factored differently from the

real part.

Every one of the direct and interference aerodynamic forces and moments for all the lifting

surfaces can be independently factored with pin-point accuracy, including far-fetched

interference effects such as the vertical fin side force due to wing tip aileron tab deflection

or the elevator hinge moment due to wing camber, etc.

Thus, the Modal Descrambling factoring method and simulator program gives the user

unprecedented control over all direct and all interference forces for factoring, on any lifting

surface of the aircraft.

n

m lmlmlmlmlmlmlmlmlm

factoredstacked

l

cLaLaLaLahL

chL

1 4321

,

0.1*)(*)(*)(*)(*)(

)('

n

m lmlmlmlmlm

unfactoredstacked

l cLLLLhLchL1

, )()()()()()(



8

Experimental data and common sense along with analysis will tell us which interference

effects are important.

Because of linearity, the following is true for the DLM:

(5)

It is also known that for unfactored unsteady DLM analyses,

(6)

However, is the following true for the factored DLM including interference effects?

(7)

where L’ in Equation (7) is given by Equation (4).

Future wind tunnel tests or unsteady CFD calculations will establish the level of

approximation expressed in Equation (7).

3 NUMERICAL RESULTS

Three configurations of a Bombardier stick model of a twin engine airplane with manual

control surfaces are analyzed: Model No. 1 has “nominal” vertical fin bending and torsional

stiffnesses; Model No. 2 has 25% higher vertical fin bending and torsional stiffnesses than

the nominal and Model No. 3 has 50% higher vertical fin bending and torsional stiffnesses

than the nominal configuration. As such, all three models are versions with much reduced

vertical fin stiffnesses of a real airplane model.

Each aileron has a geared tab, the left aileron has a trim tab and the rudder also has a trim

tab. The ailerons, elevators and the rudder all have aerodynamic balance surfaces. Mass

balance is present on all three control surfaces. The control surfaces mechanical circuits are

modeled to include the mass and inertia properties of the control wheels, control columns,

quadrants, cables and pushrods and the elastic properties of the cables and pushrods.

The aerodynamic model of the airplane under consideration is partitioned into four

components as follows: component 1: wing with its ailerons and tabs; component 2:

horizontal stabilizer with its elevators; component 3: vertical fin with its rudder and tab and

the fourth component consists of the pylons, engines and ventral fins. Aerodynamic

correction factors are obtained from steady state wind tunnel data and CFD calculations.

The controls-free configuration is analyzed at one transonic Mach Number. The

aerodynamic model is common for all three structural dynamic models. The effects on

calculated flutter speed of vertical fin bending and torsion stiffness increases and of

variations of aerodynamic derivatives can thus be studied and displayed in a compact format

in parametric fashion.

)()()()()()( cLLLLhLchL DLM

REALITYDLM chLchL )()(

REALITYDLM chLchL )()('



9

On all the plots of aerodynamic forces the direct forces are plotted in blue and the

interference forces are plotted in red. The same applies to the aerodynamic factors. See

Figures 3 and 4, where all real and imaginary unfactored direct and interference lifts are

plotted on the horizontal stabilizer for a k value of 0.700 at the analysis Mach Number.

Figure 3. Real Parts of Unfactored Descrambled Direct and Interference Cn on Horizontal Stabilizer of the

T-Tail for General Mode of Vibration Shown in Figure 1. k=0.700.

Figure 4. Imaginary Parts of Unfactored Descrambled Direct and Interference Cn on Horizontal Stabilizer

of the T-Tail for General Mode of Vibration Shown in Figure 1; k=0.700.

DESCRAMBLED UNFACTORED DIRECT AND INTERFERENCE REAL Cn ON

HORIZONTAL STABILIZER OF T-TAIL FOR GENERAL (SCRAMBLED) MODE OF

VIBRATION; k=0.700

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

semispan η

Rea

l C

n

CN DUE TO HS ROLL/BENDING

CN DUE TO HS TORSION

CN DUE TO ELEVATOR ROTATION

CN DUE TO HS CAMBER

CN DUE TO V. FIN LAT BENDING

CN DUE TO V. FIN TORSION

CN DUE TO RUDDER ROTATION

CN DUE TO RUDDER TAB

ROTATION

CN DUE TO V. FIN CAMBER

DESCRAMBLED UNFACTORED DIRECT AND INTERFERENCE IMAGINARY Cn

ON HORIZONTAL STABILIZER OF T-TAIL FOR GENERAL (SCRAMBLED) MODE

OF VIBRATION; k=0.700

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

semispan η

Imag

inary

Cn

CN DUE TO HS ROLL/BENDING

CN DUE TO HS TORSION

CN DUE TO ELEVATOR ROTATION

CN DUE TO HS CAMBER

CN DUE TO V. FIN LAT BENDING

CN DUE TO V. FIN TORSION

CN DUE TO RUDDER ROTATION

CN DUE TO RUDDER TAB

ROTATION

CN DUE TO V. FIN CAMBER



10

Note the high visibility of the direct lift due to horizontal stabilizer roll/bending and elevator

rotation and of the interference lift due to vertical fin bending and torsion and rudder

rotation. The effect of the vertical fin elastic streamwise camber deformation is visible but

small on the horizontal stabilizer; the rudder trim tab has very high rotational stiffness and at

the low frequency of this mode it rotates negligibly relative to the rudder; the rudder trim

tab has no visible effect at the low frequency of the empennage mode shown in Figure 1.

At each aerodynamic strip of component l , a total of 4 matrices of aerodynamic correction

factors, each 4X4 in size are required, for a total of 64 factors per strip. One 4X4 matrix of

correction factors is the direct set and the other three are the interference sets of matrices of

correction factors for the lift, moment, control surface hinge moment and tab hinge moment.

For 209 strips at one Mach Number, we thus have a total of 13376 (64 factors/strip*209

strips) individual factors to derive and potentially study the effect of each on the flutter

solution.

In Figure 5, a comparison between the Clβ distribution on the horizontal stabilizer as

calculated with the DLM and with a CFD code is shown at the Mach Number analyzed.

Figure 5. Typical DLM vs. CFD Horizontal Stabilizer Spanwise Clβ Distribution. k=0.000.

The factor for the Clβ at every spanwise strip is obtained by dividing the CFD-calculated

value by the DLM-calculated value. At the Mach Number analyzed, the CFD prediction is

larger than the DLM prediction everywhere along the span.

Figure 6 shows a typical variation with Mach Number of the total horizontal stabilizer CLβ

(interference) factor and of the CLα (direct) factor.

DLM vs. CFD Clβ Distribution on Horizontal Stabilizer;

k=0.000

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

semispan, η

Cn*

c/c a

ve

CFD

DLMUnit Rigid Yaw Mode



11

Figure 6. Typical CLα and CLβ Total Factors vs. Mach Number for the Horizontal Stabilizer; k=0.000

At all Mach Numbers, the CLα factor is smaller than unity. At low Mach Numbers, the CLβ

factor is typically greater than unity (and greater than the CLα factor) and it decreases with

increasing Mach Number until it crosses the CLα factor and continues to decrease below the

CLα factor. For the Mach Number analyzed here, the CLβ factor is greater than the CLα

factor.

Figures 7 and 8 show the factored and unfactored, real and imaginary, total direct, total

interference and the stacked factored and unfactored real and imaginary lifts on the

horizontal stabilizer at k=0.700. Note the different shapes of the related factored and

unfactored lifts, as well as the relative magnitudes of the real and imaginary parts.

Figure 7. Real Parts of Total Unfactored and Factored Descrambled Direct and Interference Stacked Cn

on Horizontal Stabilizer of the T-Tail for General Mode of Vibration Consisting of Vertical Fin

Bending/Torsion/HS Roll/Elevator Rotation/Rudder Rotation Shown in Figure 1. k=0.700.

Typical Horizontal Stabilizer CLα and CLβ Factors

vs. Mach Number; k=0.000

Mach Number M

Aer

od

yn

am

ic F

act

ors

CLα Factor vs. M

CLβ Factor vs. M

Analysis CLβ Factor

Analysis CLα Factor

1.000

CLα Factor<1.000

CLβ Factor>1.000

CLβ Factor<1.000

TOTAL UNFACTORED AND FACTORED DIRECT, INTERFERENCE AND

STACKED REAL Cn ON HORIZONTAL STABILIZER OF T-TAIL FOR GENERAL

(SCRAMBLED) MODE OF VIBRATION ; k=0.700

-0.010

-0.005

0.000

0.005

0.010

0.015

0.0 0.2 0.4 0.6 0.8 1.0

semispan η

Real

Cn

UNFACTORED TOTAL DIRECT LIFT

FACTORED TOTAL DIRECT LIFT

UNFACTORED INTERFERENCE LIFT

FACTORED INTERFERENCE LIFT

UNFACTORED STACKED

FACTORED STACKED



12

Figure 8. Imaginary Parts of Total Unfactored and Factored Descrambled Direct and Interference

Stacked Cn on Horizontal Stabilizer of the T-Tail for General Mode of Vibration Shown in Figure 1.

k=0.700.

Flutter solutions for each stiffness model have been run using the common aerodynamic

model. First, the flutter solutions with no aerodynamic factoring have been performed.

Then, the flutter solutions with “nominal” aerodynamic factoring have been run. Then,

flutter solutions are run with parametric variations of various aerodynamic derivatives

factors from the nominal values. One derivative factor at a time is varied from nominal and

its effect on the flutter solution is noted.

Table 1 shows all the flutter solutions calculated with and without aerodynamic factoring

and all the variations of the aerodynamic derivatives factoring. The primed derivatives

indicate nominal factored values. The magnitude of the up-or-down excursions from

nominal derivatives values is between 20% and 50% as indicated in Table I.

Cases Ran

(Derivatives factoring)

%(Vf -VfREF)/VfREF

Model 1

%(Vf -VfREF)/VfREF

Model 2

%(Vf -VfREF)/VfREF

Model 3

No Factoring 65.979 109.716 149.243

Nominal Factoring 0.000 11.313 126.452

Clβ factors=Clα factors 65.456 137.462 163.584

ChEh’*0.5 0.826 14.368 116.020

ChE(h+α)’*0.5 3.220 21.002 97.578

ChEδE’*1.2 1.899 14.588 92.678

ClδR Theoretical 7.570 23.066 139.994

ChRβ’*0.5 -1.596 15.001 134.132

ChEδR’*0.5 10.322 36.113 131.489

ChRδR’*1.2 -1.817 10.542 135.701

ClhVF Theoretical 0.000 11.891 130.003

ChEβ’*1.2 2.119 16.598 128.269

Clα Theoretical 2.835 16.846 124.993

Nominal, camber discarded 2.009 15.029 125.048

CYVFβ’*1.2 -1.294 11.616 125.048

Table 1. Cases Ran; VfREF Is for the Nominal Factoring Case, Model #1; Primed Derivatives Are Factored

Nominal Values; Derivatives Factoring Variations Are from Nominal Values.

TOTAL UNFACTORED AND FACTORED DIRECT, INTERFERENCE AND

STACKED IMAGINARY Cn ON HORIZONTAL STABILIZER OF T-TAIL FOR

(SCRAMBLED) GENERAL MODE OF VIBRATION ; k=0.700

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.0 0.2 0.4 0.6 0.8 1.0

semispan η

Imag

inary

Cn

UNFACTORED TOTAL DIRECT LIFT

FACTORED TOTAL DIRECT LIFT

UNFACTORED INTERFERENCE LIFT

FACTORED INTERFERENCE LIFT

UNFACTORED STACKED

FACTORED STACKED



13

When a derivative is listed such as “ClδR Theoretical” in Table 1, it means that the

unfactored DLM-calculated value is used.

Figure 9 shows in graphical form the effect of both stiffness (and therefore mode shape)

variation and aerodynamic derivatives variation on the flutter mechanism of the T-Tail of all

three models analyzed. For clarity, not all the cases listed in Table 1 are plotted in Figure 9.

Figure 9. Effect of Vertical Fin Stiffness Variations and of Aerodynamic Factoring Variations, One

Derivative at a Time from Nominal on Flutter Speed of T-Tailed Aircraft. Primed Derivatives Are

Factored Nominal Values. Reference Speed Is for Nominal Factoring, Model 1.

It is immediately apparent that the change in the flutter solution due to changes in the Clβ

factors is dramatic; at the Mach Number analyzed, the T-Tail flutter speeds calculated with

the Clβ factors = Clα factors are decidedly optimistic (see Table 1 and Figures 6 and 9).

Figures 10 and 11 show the V-g and V-f plots for the complete aircraft for the nominal

factoring case (with the Clβ factors > Clα factors) and then for the nominal factors but with

the Clβ factors being equal to the Clα factors for Model No. 1. Again, the change in the T-

Tail flutter speed due to changes in the Clβ factors is dramatic.

MOST FLUTTER ANALYSES ARE HERE

SOME ARE HERE

ALL SHOULD BE HERE



14

Figure 10. V-g and V-f Plots of Model No. 1 T-Tail Flutter Solution for Nominal Aerodynamic Factoring;

Clβ factors > Clα factors

Modal Damping; T-Tail Aircraft Model #1; Nominal Aerodynamics Factoring;


-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

V (KEAS)/Vmax

da

mp

ing

, g

Vertical Fin Bending

Vertical Fin Torsion

Rudder Rotation

Modal Frequency; T-Tail Aircraft Model #1; Nominal Aerodynamics Factoring;


0

5

10

15

20

25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

V (KEAS)/Vmax

Fre

qu

en

cy

(H

z)

Vertical Fin Bending Vertical Fin Torsion

Rudder Rotation



15

Figure 11. V-g and V-f Plots of Model No. 1 T-Tail Flutter Solution for Nominal Aerodynamic Factoring;

Clβ factors = Clα factors.

Modal Damping; T-Tail Aircraft Model #1; Nominal Aerodynamics Factoring;

Clβ factors = Clα factors

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

V (KEAS)/Vmax

da

mp

ing

, g

Vertical Fin Bending

Vertical Fin TorsionRudder Rotation

Modal Frequency; T-Tail Aircraft Model #1; Nominal Aerodynamics Factoring;

Clβ factors = Clα factors

0

5

10

15

20

25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

V (KEAS)/Vmax

Fre

qu

en

cy

(H

z)

Vertical Fin Bending Vertical Fin Torsion

Rudder Rotation



16

Finally, in Figure 12, a graphical summary of the unsteady aerodynamic forces and their

phases on the horizontal stabilizer which affect T-Tail flutter are shown. Control surfaces

are not included. Positive yaw is assumed. The interference section lift Clβ is always

present. The geometric angle of attack at the horizontal stabilizer in yaw due to dihedral and

therefore the phase and magnitude of the top rolling moment can be modified by choosing

positive, negative or no dihedral. For steady state uplift on the horizontal stabilizer, the

unsteady Queijo lift (References [8], [6]) is present and produces a top rolling moment in

phase with the interference lift and positive dihedral to produce the worst case for flutter.

Figure 12. A Summary of the Aerodynamic Forces and their Phases on the Horizontal Stabilizer Affecting

T-Tail Flutter; No Control Surfaces; Positive Yaw Is Assumed.

4 CONCLUSIONS

The scrambled vertical fin bending/torsion/horizontal stabilizer roll/elevator rotation/rudder

rotation mode at the aerodynamic surface (Figure 1) illustrates the complexity of the modal

motion participating in the T-Tail flutter mechanism.

The T-Tail flutter mechanism speed is strongly influenced by the bending and torsional

stiffness levels and motions of the vertical fin; this effect is known and is included here for

completeness. From the aerodynamic standpoint, the T-Tail flutter mechanism has been

shown to be dependent on a large number of aerodynamic forces, moments and control

surface hinge moments present on the horizontal stabilizer for the bending and torsional

modes of the vertical fin. Among these aerodynamic forces and moments, the Clβ

interference factor is the main aerodynamic driver of the T-Tail flutter mechanism.

Interference Lif t Clβ on

H.S.; Always Present

Lif t on H.S. Due

to Dihedral

Queijo Lif t on H.S.

for Steady Upload

Side Force on V. Fin

Due to Positive Yaw

The Worst Case for

FlutterBetter The Best



17

The second most important driver is the elevator hinge moment due to elevator rotation,

then it is the elevator hinge moment due to horizontal stabilizer roll/bending and torsion,

broken down into about 1/3rd

due to horizontal stabilizer roll/bending and 2/3rd

due to

pitch/torsion. The factors on the ClδR and ChEδR also have a strong effect on the calculated

speed of the T-Tail flutter mechanism. The flutter speed sensitivity due to variations of the

horizontal stabilizer lift due to vertical fin lateral bending is relatively small (see Table 1).

Vertical fin direct derivatives variations have small but measurable effects on calculated T-

Tail flutter speeds for all three airplane models. The elastic streamwise camber deformation

on the entire aircraft also has a small effect on the T-Tail flutter speeds for all three airplane

models (Table 1); the camber effect is more pronounced on other aircraft lifting surfaces

calculated flutter speeds (see also Reference [7]).

The presence of the wing aerodynamic surface with or without factoring also influences the

T-Tail flutter mechanism. Whether the wing is or is not in the same interference group with

the T-Tail lifting surfaces has little effect on the calculated T-Tail flutter speed; the effect is

mostly dynamic, transmitted through the fuselage.

The effects on the calculated T-Tail flutter mechanism speed of individual variations of all

these aerodynamic forces, moments and hinge moments have been explored (Figure 9)

through the use of the flutter simulator program implementing the Modal Descrambling

factoring method.

A good knowledge of measured or CFD-calculated direct and interference aerodynamic

derivatives is therefore essential for the accurate calculation of the T-Tail flutter mechanism

speed using the DLM or similar methods and the Modal Descrambling Factoring Method.

Other applications of the Flutter Simulator program implementing the Modal Descrambling

Factoring Method are for the flutter analysis of the wing-horizontal stabilizer in close

vertical proximity and for the wing-engine nacelle flutter analysis; aerodynamic interference

is important for these cases. Gust loads analyses could also benefit from the use of the

Modal Descrambling Factoring Method. Last but not least, the more accurate set of

generalized aerodynamic forces matrices QHHL generated by the Modal Descrambling

Factoring method can be used in nonlinear calculations of limit cycle oscillations caused by

control surface free play (Reference [9]).

The flutter simulator program gives the user unprecedented control over all direct and all

interference aerodynamic forces, moments and control surfaces and tabs hinge moments for

factoring on any lifting surface of the aircraft.

5 ACKNOWLEDGEMENTS

Messrs. Frederic Bradley and Jason Bensimhon of the Bombardier Dynamics Group have

contributed with the preparation of the structural dynamic model and with the voluminous

aerodynamic data compilation required for this project.



18

6 REFERENCES

[1] Suciu, E., Stathopoulos, N., Dickinson, M. and Glaser, J.,”The T-Tail Flutter Mechanism

Revisited”, Paper No. IFASD-2011-121, Presented at the International Forum for

Aeroelasticity and Structural Dynamics, Paris, France, June 26-30, 2011.

[2] Giesing, J.P., Kalman, T.P., and Rodden, W.P., “Subsonic Unsteady Aerodynamics for

General Configurations, Part I, Vol. I – Direct Application of the Nonplanar Doublet

Lattice Method”. Air Force Flight Dynamics Laboratory Report No. AFFDL-TR-71-5

Part I, Vol. I, 1971.

[3] French, R.M., Noll, T.,Cooley, D.E., Moore, R. and Zapata, F.; "Flutter Prediction

Involving Trailing Edge Control Surfaces"; Journal of Aircraft, May 1988, Vol 25, No 6.

[4] Suciu, E., “MSC/NASTRAN Flutter Analyses of T-Tails Including Horizontal Stabilizer

Static Lift Effects and T-Tail Transonic Dip”, Presented at the 1996 MSC/NASTRAN

World Users’ Conference, Newport Beach, CA, June 3-7, 1996.

[5] Rodden, W.P., and Johnson, E.H., “User’s Guide V68, MSC Nastran Aeroelastic

Analysis”, The MacNeal-Schwendler Corporation, Los Angeles, CA, 1994.

[6] Suciu, E., “A General Aerodynamic Derivatives Factoring Method for the MSC Nastran

DLM Capable of Controlling All Lifting Surfaces Aerodynamic Forces and Moments,

Including All Interference Effects”, Paper No. 2003-39, MSC.Software 2003 Virtual

Product Development Conference, October 13 - 15, 2003, Dearborn, Michigan, USA.

[7] Panza, J.L., and Suciu, E., “A Closer Look at the Elastic Streamwise Camber

Deformation of Swept and Unswept Wings”, Presented at the International Forum on

Aeroelasticity and Structural Dynamics 2005 as Paper No. IF-090, Munich, Germany,

June 28 – July 1, 2005.

[8] Queijo, M.J., “Theoretical Span Load Distributions and Rolling Moments for

Sideslipping Wings of Arbitrary Planform in Incompressible Flow”, NACA Report

1269, 1956.

[9] Kholodar, D., and Dickinson, M., “Aileron Freeplay”, International Forum on

Aeroelasticity and Structural Dynamics Paper No. IFASD-2009-158, Seattle, WA, June

21-24, 2009.

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the t-tail flutter mechanism revisited - MSC Software Corporation