High Order Harmonic Balance applied to an
aeroelastic T-tail model with a control surface freeplay
Sebastiano Fichera∗ and Sergio Ricci†
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale via La Masa 34, 20156 Milano Italy
This study presents a procedure to perform High Order Harmonic Balance (HOHB)
analysis on aeroelastic systems with freeplay nonlinearity. The test case used to validate
the results is a numerical/experiemtal model of an aeroelastic T-tail with a control surface
freeplay.
Nomenclature
δ
δf r
q
Cs
Ham
Ks
Ms
fa
fm
c
k
Kδ
M
p
Q
q
s
V∞
Xa
xa
surface motion
semi-amplitude of the freeplay gap
dynamic pressure, pa
generalized damping structural matrix
generalized aerodynamic forces frequency response matrix
generalized stiffness structural matrix
generalized mass structural matrix
vector of Generalized unsteady Aerodynamic Forces (GAFs)
vector representing the external force
reference chord, m
reduced frequency ωc/2V∞
penalty gain proportional to rotation
Mach number
nondimensional Laplace variable sc/2V∞
coefficients of the structural states
structural states
Lapalce variable
asymptotic velocity, m/s
coefficients of the aerodynamic states
aerodynamic states
I.
Introduction
This work is housed in the context of an experimental and numerical investigation of the behavior of a
T-tail with a control surface freeplay. For this reason, as presented in Ref.,1 a numerical (time marching
integrated) model has been designed, according to the modern aeroelastic approach, describing the dynamics
of the tail by a State Space (SS) system and the nonlinearity has a lumped element in the actuator feedback
loop; the results have been validate with those obtained form an aeroelastic wind tunnel model.
The T-tail unit considered in this study, Figure 1, is the one of the X-DIA, an aeroelastic model representative of a non-conventional three surfaces regional jet (called Target Aircraft), intensively investigated in the
last few years at the Department of Aerospace Science and Technology, Politecnico di Milano DAST-POLIMI
(see2, 3, 4 ).
∗ Ph.D.
Candidate, e-mail: fichera@aero.polimi.it, phone number: +39.02.23998310.
Professor, e-mail: sergio.ricci@polimi.it, phone number: +39.02.2399838319.
† Associate
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(a) Final model rendering
(b) Model in the wind tunnel
(c) Half rudder
Figure 1: T-tail aeroelastically scaled wind tunnel model.
II.
Aeroelastic System
The aeroelastic system is described in Eq. 1:
Ms q̈ + Cs q̇ + Ks q = qfa + fm
(1)
where Ms , Cs and Ks are respectively the mass, damping and stiffness structural matrices and fa is the
vector of Generalized unsteady Aerodynamic Forces (GAFs), and fm represents the external force. The
dimension of the model, that means the number of reduced modes, is m.
As usual, the GAFs are computed in the reduced frequency k domain by using the classical Doublet
Lattice Method (DLM), i.e.
fa = Ham (k, M∞ )q.
(2)
The model of Eq. 2 can be fitted into a state space system by using the classical Roger’s expansion,5
where ma are the number of reduced aerodynamic modes
Ham (p) ' D0 + D1 p + D2 p2 +
N
X
i=1
p
Ei
p − βi
(3)
and p = 2Vsc∞ is the non dimensional Lapalce variable. This expression can be easily transformed into a
classical state space time domain system
(
c
)ẋa = Aa xa + Ba q
2V∞
fa = Ca xa + D0 q + (
c
c 2
)D1 q̇ + (
) D2 q̈
2V∞
2V∞
and finally the aeroelastic system can be rewritten in the following form
c 2
c
(Ms − q(
) D2 )q̈ + (Cs − q(
)D1 )q̇ + (Ks − qD0 )q = qCa xa + fm .
2V∞
2V∞
A.
(4)
(5)
Aeroelastic T-tail characteristics
The T-tail SS matrices are built using the structural Finite Element model and the aerodynamic DLM, both
developed in MSC.NASTRAN. The SS model is a Reduced Order Model (ROM) with the basis made by a
rigid mode plus 9 elastic modes (m = 10). Using the Eq. 3 at the first order it leads at the same number
of the aerodynamic modes (ma = 10). The aeroelastic behavior of the model is shown in the numerical V -g
flutter diagrams computed with the free surface (see Figure 2). The first (11.63 Hz) and the second (23.22
Hz) fin bending modes cross the zero damping line respectively at 47 m/s and 78 m/s.
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25
1
2
3
4
5
6
7
Frequency
[Hz]
- 1st Fin Torsion Mode
- 1st Fin Bending Mode
st
- 1 Tailplane Bending Mode
- 1st Tailplane In-Plane Mode
- 2nd Fin Bending Mode
- 1st Fin In-Plane Mode
- 1st Tailplane Torsion Mode
9.36
11.63
13.07
18.18
23.22
36.80
38.45
Mode Shape
20
Freq. [Hz]
Mode Shape
Experimental model
Frequency
[Hz]
1 - 1st Fin Torsion Mode
2 - 1st Fin Bending Mode
3 - 1st Ballast Dummy Mode
4 - 1st Tailplane Bending Mode
5 - 1st Tailplane In-Plane Mode
6 - 2nd Fin Bending Mode
7 - 2nd Ballast Dummy Mode
8 - 3rd Ballast Dummy Mode
9 - 4rd Ballast Dummy Mode
10 - 1st Tailplane Torsion Mode
9.37
11.61
12.42
13.08
18.17
19.90
28.64
30.94
36.39
38.42
15
10
5
0
0
20
40
60
Velocity
80
100
120
0
20
40
60
Velocity
80
100
120
0.1
0.05
Damp. g
Numerical model
0
−0.05
−0.1
(a) Frequencies and mode shapes
(b) V-g V-f diagrams
Figure 2: T-tail aeroelastic characteristics.
III.
Numerical High Order Harmonic Balance Approximation
The aim of the Harmonic Balance (HB) method,67 is to approximate the response of a given nonlinear
system, that is undergoing Limit Cycle Oscillations (LCO), with a sine wave. Introducing the high order
harmonics means that the solution is represented as a Fourier series. The procedure presented below follows
outline done by Cooper et al.8, 9 in their previous work. Consider a general unforced nonlinear system of
the form
q̇ = f (q, t, w)
(6)
where q(t) is a m x 1 vector of system states, t is the time, w are system parameters and f (x, t, w) is a m
x 1 vector of nonlinear functions. Assuming that the system is undergoing self-excited LCOs, the states can
be approximated by a Fourier series
q(t) = Q0 +
N
X
Qk1 cos(kωt) + Qk2 sin(kωt)
(7)
k=1
where ω is the fundamental response frequency, Q0 , Qk1 , Qk2 are unknown coefficients and N is the order of
the approximation. The HOHB schemes refer to N > 1, otherwise, if N = 1 is chosen, the classical 1st order
HB approximation is obtained. Eq. 7 is substituted into eq. 6 and then HOHB is performed by equating
the coefficients of every sine and cosine term to zero. In other words, the total coefficient of sin(ωt), cos(ωt),
sin(2ωt), cos(2ωt), etc, as well as that of the constant term must be equal to zero, in order for the equations
of motion to be satisfied. HOHB leads to m x (2N + 1) nonlinear algebraic equations in terms of ω, Q0 ,
Qk1 , Qk2
g(Q0 , Qk1 , Qk2 , ω) = 0
(8)
where g are nonlinear functions. The solution of these equations yields a complete approximation of the LCO
behavior of the nonlinear system. Notice that there are m x (2N + 1) + 1 unknowns including the frequency
and only m x (2N + 1) equations. This problem can be overcome without loss of generality applying the so
called phase fixing where the first element of Q11 is set to zero, thus decreasing the number of unknowns to
m x (2N + 1). The main condition for this procedure to work is that the system really does undergo limit
cycle oscillations. If it does not then all the amplitudes Qk1 and Qk2 will be calculated as zero. The main
difficulty in the Harmonic Balance methodology is the solution of the nonlinear algebraic system.
A.
HOHB Methodology
In order to solve a system that undergoes self-excited oscillations, the procedure has been split in three steps
with the aim to be able to perform HOHB with a lower computational cost. In the first step the fundamental
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frequency is calculated, while in the second, the high order unknown coefficients are obtained; the last step
is a verification of the optimum of the fundamental frequency.
1.
First Step - Fundamental frequency calculation
The fundamental frequency is calculated solving, with an iterative method, the nonlinear system with an
expansion order N = 1; in other words solving the classic first order Harmonic Balance. In order to decrease
the number of the unknowns, the Q11 is set to zero. The unknowns are m x 3 as the equations. The Fourier
expansion for the first structural state is
q 1 (t) = Q10 + Q112 sin(ωt)
(9)
and for the following states is
q n (t) = Qn0 + Qn11 cos(ωt) + Qn12 sin(ωt)
n = 2, · · · , m;
(10)
the related Fourier matrix is

1

0
  



q 0
.
q̇ =  ..

 
 
0
q̈


0
0
cos(ωt)
sin(ωt)
−ω sin(ωt)
ω cos(ωt)
−ω 2 cos(ωt) −ω 2 sin(ωt)
..
..
.
.
0
0
0
0
0
0
···
···
···
..
.
0
0
0
..
.
···
···
···
1
0
0


1

Q


 0





0 




0
0

1 


Q

12 



0
0


2


 
Q


0


0
0

Qq n o

 Q2 
11
 ˙ 

..
..
=


Qq  Q . (11)
.
.
Q212 



¨



Qq
cos(ωt)
sin(ωt)  
 .. 


.




 m
−ω sin(ωt)
ω cos(ωt)  





2
2

 Q0 
−ω cos(ωt) −ω sin(ωt) 



m

Q11 



 m
Q12
For the aerodynamic states the Fourier expansion follows the same scheme
xna a (t) = Xa n0 a + Xa n11a cos(ωt) + Xa n12a sin(ωt)
na = 1, · · · , ma ;
(12)
and the related Fourier matrix is

1
  
0

xa 
 
.

x˙a =  ..

 
 
x¨a
0
0
cos(ωt)
sin(ωt) · · ·
−ω sin(ωt) ω cos(ωt) · · ·
..
..
..
.
.
.
0
0
..
.
···
···
1
0
0
0
0
0


1

X
a


0



1 



X
a 11 





1 


X


a
12




0
0
2

 


X
 a0 



0
0

X
x
2 
a
a


n o
 Xa 11
 ˙ 
..
..

=

X
x

Xa ; (13)
2
a
a
.
.
 Xa 12 


¨


Xa xa
 .. 

cos(ωt)
sin(ωt)  

. 






−ω sin(ωt) ω cos(ωt)  m 

 Xa 0 







Xa m



11

 m

Xa 12
substituting the expansion 11 and 13 in the Eq. 5 yields
c 2
¨ {Q} + (Cs − q( c )D1 )[Qq]
˙ {Q} + (Ks − qD0 )[Qq]{Q} = qCa [Xa xa ]{Xa } + fm
(Ms − q(
) D2 )[Qq]
|
{z
}
2V∞
2V∞
|
{z
}
|
{z
}
K
M
C
(14)
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c
((
)[Xa˙xa ] − Aa [Xa xa ]){Xa } = Ba [Qq]{Q}
2V∞
{z
}
|
(15)
(M + C + K){Q} = qCa [Xa xa ]A−1 Ba [Qq]{Q} + fm
|
{z
}
(16)
(M + C + K − KX){Q} = fm
(17)
A
KX
m
m
Eq. 17 has m x 3 equations and the same number of unknowns that are ω, Q10 , Q112 , . . . , Qm
0 , Q11 , Q12 . As
said before, the system is solved using a Newton-Raphson method.
2.
Second Step - High order unknown coefficients calculation
The goal of the first step is to obtain the fundamental frequency necessary to solve the system, Eq. 17, as
a linear system with only Qn0 , Qnk1 , Qnk2 with k = 1, · · · , N and n = 1, · · · , m as unknowns. The complete
Fourier expansion for the structural states is
q n (t) = Qn0 +
N
X
Qnk1 cos(kωt) + Qnk2 sin(kωt)
k = 1, · · · , N ;
n = 1, · · · , m;
(18)
na = 1, · · · , ma ;
(19)
k=1
and for the aerodynamic states
xna a (t) = Xa n0 a + Xa nk1a cos(ωt) + Xa nk2a sin(ωt)
k = 1, · · · , N ;
the Fourier matrices have the same structure as the ones in Eq. 11 and Eq. 13 but the size now is m x
(2N + 1). Eq. 21 is the final form of the equation for the HOHB
(M + C + K − KX){Q} = fm
{z
}
|
(20)
{Q} = Z−1 fm .
(21)
Z
The Fourier coefficients of the nonlinear function have been evaluated using the Discrete Fourier Transform (DFT). In order to solve numerically the system, the constant terms, the cosine and sine coefficients are
˙ [Qq],
¨ etc which dimension are (m, m(2N +1)),
equated to zero. For this reason the Fourier matrices [Qq], [Qq],
are expanded to dimension (m(2N + 1), m(2N + 1)). The system matrices too are expanded spreading the
elements along the diagonal in order to match the dimension of the HO model and allowing his solution.
3.
Nonlinear force
One of the main issue of the HOHB is the representation of the nonlinear force fm . The idea is to describe
this function as a Fourier series
f (t) = F0 +
N
X
Fk1 cos(kωt) + Fk2 sin(kωt)
(22)
k=1
and use the same procedure as before to obtain the Fourier matrices
h
f (t) = 1
cos(ωt)
sin(ωt) · · ·
cos(kωt)


F0 









F
11






 h in o
i F12 
= Ff F ;
sin(kωt)
.

 .. 






Fk1 







Fk2
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(23)
the force is properly introduced in the aeroelastic system by the Bs that spreads the nodal force on the
modal dof, as shown in Eq. 24
fm = Bs f
(24)
fm = Bs [F f ]{F }
(25)
substituting 25 in 21, the whole HOHB system is built
{Q} = Z−1 Bs [F f ]{F }.
(26)
The nonlinear force is a consequence, in this case, of the freeplay motion δ of the control surface. For
this reason is necessary to define the correlation between δ and f .
As the ones before, the displacement δ has been expanded by a Fourier series
δ(t) = ∆0 +
N
X
∆k1 cos(kωt) + ∆k2 sin(kωt)
(27)
k=1
h
δ(t) = 1
cos(ωt)
sin(ωt) · · ·
cos(kωt)


∆0 









∆
11





 h in o
i ∆12 
= ∆δ ∆
sin(kωt)
.
.. 










∆ 




 k1 
∆k2
the relation between the rudder motion and the force is represented by Eq. 29



f = −Kδ |δ − δf r | if δ < −δf r
f =0



f = Kδ |δ − δf r |
if
− δf r ≤ δ ≤ δf r
if δ > δf r
Torque
and shown in Figure 3
0.0
2 fr
0.0
Rudder Displacement, 
Figure 3: Freeplay stiffness.
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(28)
(29)
4.
Step three - Verification of the optimum of the fundamental frequency
The inner HOHB procedure can be summarized in the following steps:
1. choose the airspeed and the initial guess signal δ and ω (and calculate the coefficients ∆0 , ∆11 and
∆12 ),
2. calculate the guess force (and the coefficients F0 , F11 and F12 ) evaluating the guess signal with the
nonlinear function associate,
3. solve the eq. 17 that yields the fundamental frequency ωf ,
4. choose an initial guess signal δ with ωf (and calculate the coefficients ∆0 , ∆k1 and ∆k2 ),
5. calculate the guess force (and the coefficients F0 , Fk1 and Fk2 ) evaluating the guess signal with the
nonlinear function associate,
6. solve the eq. 26 that yields to the new value of δ,
7. calculate the residual as R
"
#
Qq
δi (t) = [Cs ]
{Q}
˙
Qq
h
R = ∆20 − ∆10
∆211 − ∆111
∆222 − ∆122
···
(30)
∆2k1 − ∆1k1
∆2k2 − ∆1k2
i
(31)
8. form the Jacobean and calculate an improved guess (J−1 R). Iterate until the residual R is small.
In order to verify the validity of the fundamental frequency calculated at point 3, an outer step is
necessary. The value of ωf is ranged close to the calculated one and the residual is evaluated. As the
residual shows a optimum point, the fundamental frequency is updated and the inner loop iterated.
IV.
Results
The procedure here presented allows to use different number of harmonics; for these study, N = 15 has
been chosen in order to be able to catch the significant superharmonics shown by the experimental/numerical
integration results.
Figure 4 shows the LCO amplitude and frequency trend increasing the airspeed. The HOHB shows, as
the numerical integrated model and the experimental dataa , two regions: the first is the consequence of the
flutter of the first bending mode, while the second is a combination of the first and the second mode flutters.
Waveforms comparison, portraits diagram and FFT at different speeds (50, 75 and 85 m/s) are shown in
the following figures. It is possible to see a good agreement of the trends even if the HOHB is not completely
able to catch the peaks due to the foldings of freeplay stiffness (Figure 3).
The FFTs in Figure 6 show a discrepancy of the second superharmonic between the experimental and
the numerical integration/HOHB results. It is due to a not perfect tuning of the first ballast dummy mode
of the FE model with experimental one.
Figure 7 shows the FFT’s diagrams for the LCO transition region: the numerical integration result shows
both the frequency of the first and the second bending mode, with different amplitudes; the HOHB results
show different behaviors led by the guess fundamental frequency: if the guess ω is colse to the low LCO, the
HOHB falls on it, otherwise it found the convergency on the high. Increasing the airspeed, the HOHB leads
only to the high LCO frequency.
a The experimental data are available only in the low region of the airspeed tested due to maximum freestream speed of the
wind tunnel.
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1.7
25
1.6
LCO fundamental frequency (Hz)
Experimental data
HB15 low
HB15 high
Numerical integration
LCO amplitude (grad)
1.5
1.4
1.3
1.2
Experimental data
HB15 low
HB15 high
Numerical integration
20
15
1.1
1
40
50
60
70
Airspeed (m/s)
80
90
10
40
100
50
(a) Amplitude - Velocity
60
70
Airspeed (m/s)
80
90
100
1
1.5
(b) Frequency - Velocity
Figure 4: Experimental, Numerical Integration and HOHB comparison
250
HB15
Numerical integration
Experimental data
1
0
HB15
Numerical integration
Experimental data
150
−1
−2
100
0
0.01
0.02
0.03
0.04
0.05
Time (s)
0.06
0.07
0.08
0.09
400
δ velocity
200
δ velocity
δ amplitude
2
HB15
Numerical integration
Experimental data
200
50
0
−50
−100
0
−200
−150
0
0.01
0.02
0.03
0.04
0.05
Time (s)
0.06
0.07
0.08
−200
−1.5
0.09
(a) one period
−1
−0.5
0
δ amplitude
0.5
(b) portraits
Figure 5: Experimental, Numerical Integration and HOHB comparison at 50 m/s
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1
HB15
Numerical integration
Experimental data
0.9
0.8
Amplitude
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
Frequency (Hz)
Figure 6: Experimental, Numerical Integration and HOHB FFT comparison at 50 m/s
1
HB15 low
HB15 high
Numerical integration
0.9
0.8
Amplitude
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
Frequency (Hz)
Figure 7: Numerical Integration and HOHB FFT comparison at 75 m/s
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400
HB15 high
Numerical integration
1
300
0
200
−1
−2
100
0
0.005
0.01
0.015
0.02 0.025
Time (s)
0.03
0.035
0.04
0.045
500
δ velocity
δ amplitude
2
0
−100
−300
0
−400
HB15 high
Numerical integration
−500
0
0.005
0.01
0.015
0.02 0.025
Time (s)
0.03
0.035
0.04
0.045
HB15 high
Numerical integration
−500
−1.5
−1
(a) one period
−0.5
0
δ amplitude
0.5
(b) portraits
Figure 8: Numerical Integration and HOHB comparison at 85 m/s
1
HB15 high
Numerical integration
0.9
0.8
0.7
Amplitude
δ velocity
−200
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
Frequency (Hz)
Figure 9: Numerical Integration and HOHB FFT comparison at 85 m/s
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1.5
V.
Conclusions
This paper has presented an HOHB procedure compared with experimental and numerical integration
results for a T-tail in presence of freeplay nonlinearity in the rudder’s control chain. The method has been
shown to be able to find the correct solution for the nonlinear system; results are in agreement with those
found in previous work. As a future work, the stepped HOHB procedure will be tested with different test
cases and different types nonlinearity.
Acknowledgments
The author would like to thank Dr. Earl H. Dowell for his generous advices during the development
of this work and for his hospitality at the Mechanical Department of Duke University (NC); prof. Paolo
Mantegazza and Ph.D. Mauro Manetti for their suggestions.
References
1 Fichera, S., Quaranta, G., and Ricci, S., “Experimental and numerical investigation of the behavior of a T-tail with control
surface freeplay,” AIAA/ASME/ASCE/AHS/ASC 53rd Structures, Structural Dynamics, and Materials Conferences, Honolulu
- Hawaii, USA, 23/26 April 2012.
2 Ricci, S., Scotti, A., and Zanotti, D., “Control of an all-movable foreplane for a three surfaces aircraft wind tunnel model,”
Mechanical Systems and Signal Processing, Vol. 20, July 2006, pp. 1044–1066.
3 Ricci, S., Scotti, A., De Gaspari, A., and Riccobene, L., “Active Aeroelastic Control over a Multisurface Wing: Modelling
and Wind-Tunnel Testing,” AIAA Journal, Vol. 47, No. 9, 2009, pp. 1995–2010, doi: 10.2514/1.34649.
4 Mattaboni, M., Quaranta, G., and Mantegazza, P., “Active Flutter Suppression for a Three Surface Transport Aircraft
by Recurrent Neural Networks,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 4, 2009, pp. 1295–1307.
5 Roger, K. L., “Airplane Math Modeling Methods for Active Control Design,” Tech. Rep. CP–228, AGARD, August 1977.
6 Kim, Y. B. and Noah, S. T., “Stability and bifurcation analysis of oscillators with piecewise-linear characteristics. A
general approach,” Journal of Applied Mechanics, Transactions ASME , Vol. 58, No. 2, 1991, pp. 545–553, Cited By (since
1996): 80.
7 Liu, L. and Dowell, E. H., “Harmonic balance approach for an airfoil with a freeplay control surface,” AIAA Journal,
Vol. 43, No. 4, 2005, pp. 802–815, Cited By (since 1996): 25.
8 Dimitriadis, G., Vio, G. A., and Cooper, J. E., “Application of higher-order harmonic balance to non-linear aeroelastic
systems,” Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials
Conference, Vol. 7, 2006, pp. 5176–5187.
9 Vio, G. A., Dimitriadis, G., and Cooper, J. E., “Improved implementation of the harmonic balance method,” Collection of
Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Vol. 8, 2007,
pp. 7879–7890.
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