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Linear Time-Invariant Models of Rotorcraft Flight Dynamics, Vibrations, and Acoustics

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The paper discusses the development of a novel linearization algorithm to obtain high-order linear time-invariant (LTI) models of the coupled rotorcraft flight dynamics, vibrations, and acoustics. To demonstrate the methodology, the study makes use a nonlinear simulation model of a generic utility helicopter (PSU-HeloSim) that is coupled with an aeroacoustic solver based on a marching cubes algorithm. First, a revisited harmonic balance algorithm based on harmonic decomposition is applied to find the periodic equilibrium and approximate high-order LTI dynamics at 80 kts level flight. Next, the proposed output linearization scheme is applied to derive time-invariant, linearized equations of the main rotor forces and moments, and acoustics. Simulations are used to validate the response of the linearized models against that from nonlinear simulations. Additionally, the cost of linearization and potential performance benefits of employing linear models versus nonlinear simulations are assessed. The high-order LTI models thus obtained are shown to provide similar acoustic predictions compared to those of nonlinear simulations for small amplitude maneuvers, but at a fraction of the computational cost. These linear simulations are shown to run in the order of thousands of times faster than real time, and four orders of magnitude faster than nonlinear acoustic predictions based on a marching cubes algorithm.
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Linear Time-Invariant Models of Rotorcraft Flight Dynamics, Vibrations, and
Acoustics
Umberto Saetti*
Postdoctoral Fellow
Guggenheim School of Aerospace Engineering
Georgia Institute of Technology
Atlanta, GA 30332, USA
Joseph F. Horn
Professor
Kenneth S. Brentner
Professor
Department of Aerospace Engineering
The Pennsylvania State University
University Park, PA 16802, USA
ABSTRACT
The paper discusses the development of a novel linearization algorithm to obtain high-order linear time-invariant
(LTI) models of the coupled rotorcraft flight dynamics, vibrations, and acoustics. To demonstrate the methodology,
the study makes use a nonlinear simulation model of a generic utility helicopter (PSU-HeloSim) that is coupled with
an aeroacoustic solver based on a marching cubes algorithm. First, a revisited harmonic balance algorithm based
on harmonic decomposition is applied to find the periodic equilibrium and approximate high-order LTI dynamics
at 80 kts level flight. Next, the proposed output linearization scheme is applied to derive time-invariant, linearized
equations of the main rotor forces and moments, and acoustics. Simulations are used to validate the response of
the linearized models against that from nonlinear simulations. Additionally, the cost of linearization and potential
performance benefits of employing linear models versus nonlinear simulations are assessed. The high-order LTI
models thus obtained are shown to provide similar acoustic predictions compared to those of nonlinear simulations
for small amplitude maneuvers, but at a fraction of the computational cost. These linear simulations are shown to run
in the order of thousands of times faster than real time, and four orders of magnitude faster than nonlinear acoustic
predictions based on a marching cubes algorithm.
INTRODUCTION
The prediction of aerodynamically-induced noise of rotorcraft
in generalized maneuvering flight is relevant in that it can
be used to determine flight procedures that minimize noise
and impact on communities. Typical aeroacoustic predictions
for maneuvering flight make use of physics-based models in
which a noise prediction tools is coupled with a flight simula-
tion code that generates realistic trajectories and pilot control
input histories for rotorcraft maneuvers (see, e.g., Refs. 1,2).
Because the fidelity of flight simulation codes in the predic-
tion blades loads may be limited, high-fidelity aeromechanics
models (e.g., free wake, CFD) are added as a third element in
the case where the capability of predicting blade-vortex inter-
action (BVI) noise is needed (Refs. 35).
While these approaches are successful in the prediction of
aerodynamically-induced noise, they do not provide linear
systems of the coupled flight-dynamics and acoustics that can
be used in the design of control systems that alleviate noise.
Because the dynamics of helicopters in forward flight are pe-
riodic in nature, so are the blade loads and the noise produced
*Incoming Assistant Professor, Department of Aerospace Engineer-
ing, Auburn University, Auburn, AL 36849.
Presented at the VFS International 77th Annual Forum &
Technology Display, Virtual, May 10–14, 2020. Copyright © 2021
by the Vertical Flight Society. All rights reserved.
by the blades through their motion in air. More specifically,
the flight dynamics, vibrations, and acoustics are dominated
by harmonics with frequencies that are multiples of the num-
ber of blades per rotor revolution (or Nb/rev) (Ref. 6). As
such, linearizing these quantities at a desired flight condition
using conventional linearization schemes is not sufficient to
capture their higher harmonics. In fact, linear models obtained
using conventional linearization schemes only predict the ze-
roth harmonic behavior, also known as the averaged dynam-
ics (over a rotor revolution). One of the first attempts to de-
scribe the relationship between the averaged flight dynamics
and rotor noise can be found in Ref. 7. Therein, a quasi-static
acoustic mapping of helicopter BVI noise to low-order flight
dynamic models was explored to determine approach trajec-
tories that minimize BVI noise. Further, relations between
higher-harmonic actuation of blade pitch motion and rotor
acoustics were derived (i.e., the so-called “T-matrix”) in sev-
eral studies for use in the design of higher-harmonic control
(HHC) laws that minimize rotor noise (see, e.g., Refs. 811).
However, none of these studies provides a complete descrip-
tion of the relationship between the time-periodic flight dy-
namics (both states and control inputs), and the time-periodic
aerodynamically-generated noise as output. In particular, ro-
torcraft noise has yet to be included as an output of high-order
linear time-invariant (LTI) systems that account for the higher
1
harmonics of the control inputs, rigid-body and rotor states,
and of the rotorcraft noise itself. These systems could re-
lax the quasi-steady assumption of previous studies and allow
for a complete description of rotorcraft noise in maneuver-
ing flight. Further, these models could be exploited to assess
the effect of rigid-body and rotor states (and their higher har-
monics) on rotorcraft noise. This would be a novelty when
compared to the “T-matrix” approach, which only provides
information on the effect of higher-harmonic control inputs to
noise. Because these high-order LTI models include the effect
of rigid-body and rotor states on noise, constraints could be
placed on noise and transferred to the rotorcraft states through
the output matrix in the design of control laws that minimize
noise. This approach to noise abatement is similar to what
was done in Refs. 12 for the alleviation of unsteady rotor
loads. Additionally, these high-order LTI models could be
used to study the interference effects between noise-abating
HHC and the aircraft flight control system (AFCS), similarly
to what was done in Refs. 1316 for load-alleviation HHC
laws. Because of the relatively low computational effort the
simulation of these high-order LTI models require, they could
be used in real time to predict and avoid flight envelope limits
associated with high rotor noise. Examples of this approach
are provided in Refs. 17,18 for applications to the detection
and avoidance of flight envelope limits associated with struc-
tural loads.
As such, the proposed objectives of this paper are the fol-
lowing: (i) include vibration and aeroacoustic measures as an
output of the nonlinear time-periodic dynamics (NLTP) of ro-
torcraft; (ii) develop a linearization algorithm to obtain high-
order LTI systems representative of the periodic nature of the
rotorcraft flight dynamics, vibrations, and acoustics; and (iii)
validate these LTI models for vibrations and acoustics predic-
tions through batch simulations. Because this study will not
make use of high-fidelity aeromechanics models, the acous-
tics predictions will be limited to loading and thickness noise
in non-BVI conditions.
The paper begins with a discussion of the mathematical back-
ground background behind NLTP systems linearization and an
explanation of the methodology proposed for linearizing vi-
brations and acoustic measures. The second section presents a
description of the simulation model that is used to validate the
methodology, including details about its three major compo-
nents: the flight dynamics module, the blade blade geometry
and loads calculations, and the aeroacoustics solver. The third
section demonstrates the application of the proposed method-
ology to obtain high-order LTI models capable of vibrations
and acoustics predictions for a generic utility helicopter. Sim-
ulations are used to validate the response of the linearized
models against the those from nonlinear simulations. In addi-
tion, potential performance benefits of employing linear mod-
els versus nonlinear simulations are assessed. Final remarks
summarize the overall findings of the study and identify areas
for future work.
METHODOLOGY
Mathematical Background
Consider a nonlinear time-periodic (NLTP) system in first-
order form representative of the flight dynamics of a rotor-
craft:
˙
x
x
x=f
f
f(x
x
x,u
u
u,t)(1a)
y
y
y=g
g
g(x
x
x,u
u
u,t)(1b)
where x
x
xRnis the state vector, u
u
uRmis the control in-
put vector, y
y
yRlis the output vector, and tis the dimen-
sional time in seconds. The nonlinear functions f
f
fand g
g
gare
T-periodic in tsuch that:
f
f
f(x
x
x,u
u
u,t) = f
f
f(x
x
x,u
u
u,t+T)(2a)
g
g
g(x
x
x,u
u
u,t) = g
g
g(x
x
x,u
u
u,t+T)(2b)
It is convenient to note that dimensional time can be related to
the azimuth angle ψof a reference blade, also known as non-
dimensional time, via the following relation: ψ=t, where
the main rotor angular speed in rad/s. It follows that the
fundamental period of the system is T=2π
seconds, which
corresponds to 2πradians or one rotor revolution. Let x
x
x(t)
and u
u
u(t)represent a periodic solution of the system such that
x
x
x(t) = x
x
x(t+T)and u
u
u(t) = u
u
u(t+T). Then, the NLTP sys-
tem can be linearized about the periodic solution. Consider
the case of small disturbances:
x
x
x=x
x
x+x
x
x(3a)
u
u
u=u
u
u+u
u
u(3b)
where x
x
xand u
u
uare the state and control perturbation vectors
from the periodic solution. A Taylor series expansion is per-
formed on the state derivative and output vectors. Neglecting
terms higher than first order results in the following equations:
f
f
f(x
x
x+x
x
x,u
u
u+u
u
u,t) = f
f
f(x
x
x,u
u
u,t) + F
F
F(t)x
x
x+G
G
G(t)u
u
u
(4a)
g
g
g(x
x
x+x
x
x,u
u
u+u
u
u,t) = g
g
g(x
x
x,u
u
u,t) + P
P
P(t)x
x
x+Q
Q
Q(t)u
u
u
(4b)
where:
F(t) = f(x,u)
xx,u,G(t) = f(x,u)
ux,u(5a-b)
P(t) = g(x,u)
xx,u,Q(t) = g(x,u)
ux,u(5c-d)
Note that the state-space matrices in Eq. (5) have T-periodic
coefficients. Equations (4a) and (4b) yield a linear time-
periodic (LTP) approximation of the NLTP system of Eq. (1)
as follows:
˙
x
x
x=F
F
F(t)x
x
x+G
G
G(t)u
u
u(6a)
y
y
y=P
P
P(t)x
x
x+Q
Q
Q(t)u
u
u(6b)
2
Hereafter, the notation is simplified by dropping the in front
of the linearized perturbation state and control vectors while
keeping in mind that these vectors represent perturbations
from a periodic equilibrium. The state, input, and output vec-
tors of the LTP system are decomposed into a finite number
of harmonics of the fundamental period via Fourier analysis:
x
x
x=x
x
x0+
N
i=1
x
x
xic cos2πit
T+x
x
xis sin2πit
T(7a)
u
u
u=u
u
u0+
M
j=1
u
u
ujc cos 2πjt
T+u
u
ujs sin 2πjt
T(7b)
y
y
y=y
y
y0+
L
k=1
y
y
ykc cos 2πkt
T+y
y
yks sin 2πkt
T(7c)
As shown in Ref. 14, the harmonic decomposition method-
ology can be used to transform the LTP model into an ap-
proximate higher-order linear time-invariant (LTI) model in
first-order form:
˙
X
X
X=A
A
AX
X
X+B
B
BU
U
U(8a)
Y
Y
Y=C
C
CX
X
X+D
D
DU
U
U(8b)
where the augmented state, control, and output vectors are:
X
X
XT=x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns (9a)
U
U
UT=u
u
uT
0u
u
uT
1cu
u
uT
1s... u
u
uT
Mc u
u
uT
Ms(9b)
Y
Y
YT=y
y
yT
0y
y
yT
1cy
y
yT
1s... y
y
yT
Lc y
y
yT
Ls(9c)
with A
A
ARn(2N+1)×n(2N+1),B
B
BRn(2N+1)×m(2M+1),C
C
C
Rl(2L+1)×n(2N+1), and D
D
DRl(2L+1)×m(2M+1).
Suppose that the state vector of the of the NLTP system in Eq.
(1) is given by
x
x
xT=x
x
xT
RB x
x
xT
R(10)
where x
x
xRB are the rigid-body states and x
x
xRare the rotor states,
and that the control input vector includes the typical heli-
copter controls (i.e., lateral and longitudinal cyclic, collective,
and pedals). Further, suppose that in addition to the rotor-
craft states, the output vector includes the aerodynamically-
induced noise generated by the main rotor such that:
y
y
yT=x
x
xT
RB x
x
xT
Rp
p
p0(11)
where p
p
p0Ro(with o<l) is the vector of acoustic pressures
at one or multiple spatial locations. The acoustic pressure may
be relative to points fixed in space (e.g., observers or micro-
phones on the ground) or to a points moving with the aircraft
(e.g., observers or microphones in the cockpit). The acoustic
pressure at each location can be described according to the im-
permeable emission surface formulation (Ref. 19) as follows:
4πp0(x
x
x,t) = 1
c0
tZΣρ0c0un+˜pˆ
n
n
n·ˆ
r
r
r
rΛret
dΣ+
ZΣ˜pˆ
n
n
n·ˆ
r
r
r
r2Λret
dΣ+p0
Q(x
x
x,t)(12)
where: x
x
xis the observer location,
Σis the emission surface (i.e., the surface in space-time that
emitted sound that reached the observer x
x
xat time t),
ρ0is the density of the undisturbed air,
c0is the speed of sound of the undisturbed air,
unis the rotor blade velocity normal to the blade surface,
˜p=pp0is the gauge pressure on the blade surface,
ˆ
n
n
nis the unit vector normal to the blade surface,
ˆ
r
r
r=x
x
xy
y
y
||x
x
xy
y
y|| is the emission direction,
y
y
yis the source location,
Λ=hp12Mnˆ
n
n
n·ˆ
r
r
r+M2
niret,
Mnis the local Mach number normal to the blade surface,
and
p0
Q(x
x
x,t)is the quadrupole term.
The integrands in Eq. (12) are evaluated at the retarded time,
which is defined as:
τ=tr
c0
(13)
where r=||x
x
xy
y
y|| is the distance between the observer loca-
tion x
x
xand the source location y
y
ywhen the sound was emitted. It
is worth noting that the quadrupole term p0
Q(x
x
x,t)was included
in Eq. (12) for the sake of generality but will not be considered
in this preliminary investigation. The inclusion of the acous-
tic pressure as an output of a NLTP system, and the successive
linearizaion and decomposition into harmonics of the main ro-
tor angular speed, enables prediction of the aerodynamically-
generated noise via a set of LTI ordinary differential equations
(ODE). Because the solution of the integrals in Eq. (12) is
generally computationally expensive (cannot be performed in
real time and likely requires parallel computing for an array
of observers), the capability of approximating such integrals
with an LTI system would drastically abate the computational
cost of rotorcraft noise prediction and possibly enable real-
time prediction. Further, the complete description of rotor-
craft noise through LTI systems enables the use of LTI system
theory for the design of flight control laws that minimize noise
in generalized maneuvering flight.
Periodic Trim Solution Algorithm
A necessary step toward the approximation of the NLTP rotor-
craft dynamics with LTP systems is the determination the pe-
riodic orbit about which the NLTP system is linearized, which
involves computing the states and controls that result in a pe-
riodic equilibrium (i.e., trimming a vehicle about a periodic
orbit). Assuming that a periodic solution x
x
x(t) = x
x
x(t+T)
and u
u
u(t) = u
u
u(t+T)exists for the system in Eq. (1), then
the balance problem is stated as follows: determine x
x
x(t)and
u
u
u(t)such that:
˙
x
x
x=f
f
f(x
x
x,u
u
u,t)(14)
where ˙
x
x
x(t) = ˙
x
x
x(t+T). In other words, the balance problem
consists of determining the periodic state and control vectors
such that the system dynamics are periodic. Several solutions
exist for trimming a vehicle about a periodic orbit, namely:
3
averaged approximate trim (Refs. 12,13), time marching trim,
autopilot trim (Ref. 20), periodic shooting (Refs. 2123), har-
monic balance (Refs. 2426), and modified harmonic balance
(Ref. 27). In this study, the latter is used as it incorporates
three major innovations when compared to other techniques:
it is based on harmonic decomposition and thus does not rely
on state transition matrices, it simultaneously solves for the
approximate higher-order LTI dynamics about the periodic so-
lution, and it can be used to compute the high-harmonic con-
trol inputs that attenuate arbitrary state harmonics.
The modified harmonic balance algorithm begins with as-
suming that the fundamental period Tof the nonlinear time-
periodic system is known. Note that this solution strategy is
iterative in nature, in that a candidate solution is refined over
a series of computational steps until a convergence criteria is
reached. Consider the candidate periodic solution at iteration
kof the algorithm: x
x
x
k(t)and u
u
u
k(t). One iteration of the algo-
rithm begins with approximating the candidate periodic solu-
tion using a Fourier series with a finite number of harmonics:
x
x
x
k=x
x
x
k0+
N
i=1
x
x
x
kic cos2πit
T+x
x
x
kis sin2πit
T(15a)
u
u
u
k=u
u
u
k0+
M
j=1
u
u
u
kjc cos 2πjt
T+u
u
u
kjs sin 2πjt
T(15b)
As such, the candidate periodic solution is re-written in terms
of its respective Fourier coefficients:
X
X
XT
k=x
x
xT
k0x
x
xT
k1cx
x
xT
k1s... x
x
xT
kNc x
x
xT
kNs (16a)
U
U
UT
k=u
u
uT
k0u
u
uT
k1cu
u
uT
k1s... u
u
uT
kNc u
u
uT
kNs (16b)
Since the balance problem simultaneously solves for the pe-
riodic solution and the necessary control inputs that ensure it,
the harmonic realization of the candidate periodic solution of
Eq. (16a) is augmented with the harmonic realization of the
candidate control inputs of Eq. (16b) to form the vector of
unknowns at iteration k:
Θ
Θ
ΘT
k=hX
X
XT
kU
U
UT
ki(17)
where Θ
Θ
ΘkRn(2N+1)+m(2M+1).
Next, the state derivative vector calculated along the candidate
periodic solution over a single periodic orbit is decomposed
into a finite number of harmonics via Fourier analysis:
˙
x
x
x
k=˙
x
x
x
k0+
N
i=1
˙
x
x
x
kic cos2πit
T+˙
x
x
x
kis sin2πit
T(18)
Note that the number of state derivative harmonics that are
retained in Eq. (18) is equal to the number of state harmonics
retained in Eq. (15a) (i.e.,N). Consider differentiating the
candidate periodic solution of Eq. (15a):
˙
x
x
x
k=d
dt x
x
x
k0
| {z }
˙
x
x
x
k0
+
N
i=1d
dt x
x
x
kic +2πi
Tx
x
x
kis
| {z }
˙
x
x
x
kic
cos2πit
T
+d
dt x
x
x
kis 2πi
Tx
x
x
kic
| {z }
˙
x
x
x
kis
sin2πit
T(19)
Since at equilibrium the Fourier coefficients of the system dy-
namics are constant (i.e., their time derivative is zero), the fol-
lowing integral relations are true:
˙
x
x
x
0=0
0
0 (20a)
˙
x
x
x
ic =2πi
Tx
x
x
is (20b)
˙
x
x
x
is =2πi
Tx
x
x
ic (20c)
A total of n(2N+1)constraints are formed by requiring that
the state derivative Fourier coefficients in Eq. (18) and the
state Fourier coefficients in Eq. (15a) satisfy the integral re-
lations in Eq. (20). This leads to the definition of the error
vector at the iteration kas:
e
e
eT
k=W
W
W"˙
x
x
x
k0T˙
x
x
x
kic 2πi
Tx
x
x
kis T˙
x
x
x
kis +2πi
Tx
x
x
kic T#
(21)
where e
e
ekRn(2N+1)and W
W
WRn(2N+1)×n(2N+1)is a diagonal
scaling matrix to make all elements of the error vector ap-
proximately the same order of magnitude (e.g., 1 deg error is
equivalent to 1 ft error).
Next, the NLTP system is linearized at incremental time steps
along the candidate periodic solution, yielding the following
LTP system:
˙
x
x
x=F
F
Fk(t)x
x
x+G
G
Gk(t)u
u
u(22)
Hereafter, the notation is simplified by dropping the in front
of the linearized perturbation state and control vectors while
keeping in mind that these vectors represent perturbations
from a periodic equilibrium. Next, the state and output vec-
tors of the LTP systems are decomposed into a finite number
of harmonics via Fourier analysis:
x
x
x=x
x
x0+
N
i=1
x
x
xic cos2πit
T+x
x
xis sin2πit
T(23a)
u
u
u=u
u
u0+
M
j=1
u
u
ujc cos 2πjt
T+u
u
ujs sin 2πjt
T(23b)
Note that the number of state harmonics retained in Eq. (23a)
is the same as in Eqs. (15a) and (18) (i.e.,N), whereas the
number of control input harmonics retained in Eq. (23b) is the
same as in Eq. (15b). As shown in Ref. 14, the LTP model can
be approximated by a higher-order linear time-invariant (LTI)
model in first-order form through the harmonic decomposition
methodology:
˙
X
X
X=A
A
AkX
X
X+B
B
BkU
U
U(24)
4
where the augmented state and control vectors are:
X
X
XT=x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns (25a)
U
U
UT=u
u
uT
0u
u
uT
1cu
u
uT
1s... u
u
uT
Mc u
u
uT
Ms(25b)
and where A
A
AkRn(2N+1)×n(2N+1)and B
B
Bk
Rn(2N+1)×m(2M+1)are the LTI system and control ma-
trices. Closed-form expressions can be found in Ref. 14. It is
worth noting that harmonic decomposition does not rely on
state transition matrices, which makes the methodology more
computationally efficient and less numerically sensitive than
other suitable approaches such as the Lyapounov-Floquet
method (Ref. 28) and frequency lifting methods (Ref. 29).
The LTI system coefficient matrices are used to define the Ja-
cobian matrix of the harmonic balancing algorithm at iteration
k:
J
J
Jk= [A
A
AkB
B
Bk](26)
where J
J
JkRn(2N+1)×[n(2N+1)+m(2M+1)].
The Jacobian matrix is used in each algorithm iteration to
compute a candidate periodic solution and controls update
(i.e., the vector of unknowns) given the error vector at that iter-
ation via a Newton-Rhapson scheme (Ref. 30). It is clear that
the Jacobian matrix is not square because the number of con-
straints in Eq. (20) is less than the number of unknowns in Eq.
(17). In fact, the number of constraints is n(2N+1)whereas
the number of unknowns is n(2N+1) + m(2M+1). This
leads to an under-determined problem which does not have
a unique solution. To make the problem square such that the
solution is unique, m(2M+1)conditions have to be specified.
These are the trim conditions. Because typical periodically-
forced flight vehicles such as helicopters only utilize con-
trol input bandwidths significantly lower than the forcing fre-
quency (that is, if HHC is not utilized), one can safely assume
the control input harmonics higher than the zeroth to be zero.
This corresponds to imposing 2Mm conditions, which brings
down the number of unknowns to n(2N+1) + m. It follows
that mconditions still need to be specified. Note that if the
minputs are given and the corresponding equilibrium solu-
tion is required, then the problem in consideration becomes a
closed system. On the other hand, in the case where one or
more (possibly all) of the mcontrol inputs is unknown, then
each input is used to ensure some desired condition (e.g., trim
equation). For periodically-forced aerospace vehicles such as
helicopters, for which the vehicle dynamics are invariant with
respect to position and heading (see, e.g., Ref. 31), the ze-
roth harmonic of the position and heading can be arbitrarily
assigned and removed from the vector of unknowns. Since
these vehicles typically employ control about four axes (i.e.,
roll, pitch, yaw, and heave) leading to four control inputs, fix-
ing the three components of the zeroth harmonic of the po-
sition (x0,y0,z0) and heading (ψ0) at equilibrium leads to a
square problem. Hence, Newton Rhapson is used to find a
candidate periodic solution update (in harmonic form):
ˆ
Θ
Θ
Θk+1=ˆ
Θ
Θ
Θkˆ
J
J
J1
ke
e
ek(27)
where ˆ
Θ
Θ
Θkand ˆ
J
J
Jkare the vector of unknowns and the Jacobian
matrix deprived of the unknowns that were fixed, respectively.
As a final step, the new candidate periodic solution is recon-
structed in the time domain:
x
x
x
k+1=x
x
x
k+10+
N
i=1
x
x
x
k+1ic cos2πit
T+x
x
x
k+1is sin2πit
T
(28a)
u
u
u
k+1=u
u
u
k0+
M
j=1
u
u
u
k+1jc cos 2πjt
T+u
u
u
k+1js sin 2πjt
T
(28b)
The next iteration of the algorithm then proceeds with this
new candidate solution, starting from Eqs. (15a) and (15b).
The algorithm is stopped when kekkbecomes less than an
arbitrary tolerance. It is worth noting that the algorithm re-
quires a first guess of the periodic solution over one periodic
orbit. A flowchart of the algorithm is shown in Fig. 1.
An added benefit of the algorithm is that, to update the solu-
tion, a higher-order LTI approximation of the NLTP system
is computed at each iteration along the candidate periodic so-
lution. Thus, the algorithm not only solves for the periodic
solution of NLTP systems, but also simultaneously constructs
a higher-order LTI approximation of the NLTP system.
Output Linearization Algorithm
Once the high-order linearized dynamics (i.e., the A
A
Aand C
C
Cco-
efficient matrices) and the periodic orbit are obtained through
the modified harmonic balance algorithm, the remaining out-
put and feed-through coefficient matrices (i.e.,C
C
Cand D
D
D) can
be found by linearizing the output defined in Eq. (1b) at incre-
mental azimuthal steps over one rotor revolution, and by sub-
sequently applying harmonic decomposition. However, this
is approach is only valid for those measures for which per-
turbations can be applied at each azimuthal step over a single
rotor revolution. An example of these measures are the ro-
torcraft states or the main rotor forces and moments, which
are typically calculated as part of the flight dynamics calcu-
lations. Instead, for those output measures that necessitate
the solution of partial differential equations (PDE’s) and thus
require several time steps to be computed, or for measures
for which, in general, perturbations cannot be performed on a
per-time step basis but rather on a per-revolution basis, the ap-
proach described above does not work. An example of these
output measures is rotor noise, as it involves the solution of
PDE’s and typically requires at least at least 1/Nbrotor revo-
lutions to be computed. As such, a novel numerical approach
is required to include the aerodynamically-induced noise as
an output of the high-order LTI system. More specifically,
a novel approach is required for computing those elements
of the high-order output and feed-through matrices associated
with rotor noise.
The proposed procedure for the extraction of high-order LTI
models represents a generalization of the method in Ref. 32 to
arbitrary harmonics and is articulated in the following steps:
1. Find the periodic equilibrium over one rotor revolution
of the rotorcraft flight dynamics only. To achieve this,
5
Figure 1: Periodic trim solution algorithm flowchart.
the modified harmonic balance scheme of Ref. 27 can be
used to yield the periodic solution x(t)and u(t)at the
desired flight condition.
2. Run a flight dynamics simulation without integrating the
states but forcing them to follow the periodic trajec-
tory above while recording data needed for the aeroa-
coustic calculations. Then, perturbations are applied
to each of the coefficients of the periodic state and
control trajectory (i.e.,x
x
x
0,x
x
x
1c,x
x
x
1s,..., x
x
x
Nc ,x
x
x
Ns , and
u
u
u
0,u
u
u
1c,u
u
u
1s,..., u
u
u
Mc,u
u
u
Ms) one by one. The duration of
each perturbation is one rotor revolution. The perturba-
tions are applied first in the positive then in the negative
direction. Store the time history of the data necessary for
the aeroacoustic calculations.
3. Process the data with the aeroacoustic solver to obtain
the perturbation time history of the chosen composite
acoustic measure (e.g., acoustic pressure).
4. Compute the time-varying partial derivatives using cen-
tral difference approximations. The derivative corre-
sponding to the ith composite acoustic measure relative
to the perturbation in the jt h element of the augmented
state vector of of Eq. (9a):
p0
i
Xj
(ψ) = 1
2Xjhp0+
i j (ψ)p0
i j (ψ)i(29)
where is Xjis the size of the perturbation in the jth aug-
mented state vector. These derivatives together constitute
an interim time-periodic matrix ˆ
P
P
P(ψ)Ro×n(2N+1).
Similarly, the partial derivatives computed from the
perturbations in the augmented control vector of Eq.
Eq. (9b) form an interim time-period matrix ˆ
Q
Q
Q(ψ)
Ro×m(2M+1).
5. Perform a Fourier decomposition of each element of the
interim matrices ˆ
P
P
P(ψ)and ˆ
Q
Q
Q(ψ):
ˆ
P
P
P=ˆ
P
P
P0+
L
k=1
ˆ
P
P
Plc cos lψ+ˆ
P
P
Pls sin lψ(30a)
ˆ
Q
Q
Q=ˆ
Q
Q
Q0+
L
k=1
ˆ
Q
Q
Qlc cos lψ+ˆ
Q
Q
Qls sin lψ(30b)
Then, those portions of the high-order time-invariant ma-
trices C
C
Cand D
D
Dof Eq. (8b) relative to vector of acoustic
pressures are given by:
C
C
CT
p=hˆ
P
P
PT
0ˆ
P
P
PT
1cˆ
P
P
PT
1s... ˆ
P
P
PT
Lc ˆ
P
P
PT
Lsi(31a)
D
D
DT
p=hˆ
Q
Q
QT
0ˆ
Q
Q
QT
1cˆ
Q
Q
QT
1s... ˆ
Q
Q
QT
Lc ˆ
Q
Q
QT
Lsi(31b)
It is worth noting that Step 2 and 3 require performing np=
2nψ[n(2N+1) + m(2M+1)] evaluations of the rotorcraft
flight dynamics and aeroacoustics, where nψis the number
of azimuthal steps. Or, equivalently, nrev =2[n(2N+1) +
m(2M+1)] rotor revolutions are required for the output lin-
earization.
SIMULATION MODEL
Flight Dynamics
In this work, the helicopter model used to simulate the sys-
tem dynamics is PSUHeloSim (Ref. 33), a MATLAB®imple-
mentation of the General Helicopter (GenHel) flight dynamics
simulation model (Ref. 34) with improved trimming and lin-
earization routines. PSUHeloSim is representative of a utility
helicopter similar to a UH-60. It is worth noting that while the
flight dynamics model resembles the UH-60, the rotor is dif-
ferent from that of a UH-60 in detail, hence the acoustics are
6
just for a utility helicopter. The model contains a 6-degree-of-
freedom rigid-body dynamic model of the fuselage, nonlinear
aerodynamic lookup tables for the fuselage, rotor blades, and
empennage, rigid flap and lead-lag rotor blade dynamics, a
three-state Pitt-Peters inflow model (Ref. 35), and a Bailey
tail rotor model (Ref. 36). The state vector is:
x
x
xT=u v w p q r φ θ ψ x y z β0β1cβ1sβ0D˙
β0˙
β1c˙
β1s˙
β0D
ζ0ζ1cζ1sζ0D˙
ζ0˙
ζ1c˙
ζ1s˙
ζ0Dλ0λ1cλ1sλ0T(32)
where:
u,v,ware the longitudinal, lateral, and vertical velocities in
the body-fixed frame,
p,q,rare the roll, pitch, and yaw rates,
φ,θ,ψare the Euler angles,
x,y,zare the positions in the North-East-Down (NED) frame,
β0,β1c,β1s,β0Dare the flapping angles in multi-blade coor-
dinates,
ζ0,ζ1c,ζ1s,ζ0Dare the lead-lag angles in multi-blade coor-
dinates,
λ0,λ1c,λ1s,are the main rotor induced inflow ratio harmon-
ics, and
λ0Tis the tail rotor induced inflow ratio.
The control vector is:
u
u
uT=δlat δlon δcol δped(33)
where δlat and δlon are the lateral and longitudinal cyclic in-
puts, δcol is the collective input, and δped is the pedal input.
Blade Geometry and Loads
While for flight dynamics predictions it is sufficient to calcu-
late the blade loads based on a discretization of the blade in
the sole spanwise direction, aeroacoustic calculations require
increased resolution of the blade loads distribution over the
blade surface. In addition, a detailed blade geometry is re-
quired to calculate the unit vectors perpendicular to the blade
surface, as well as the velocity vector of each blade surface
panel. As such, the upper and lower blade surfaces are mod-
eled with a discrete number of panels both in the spanwise and
chordwise directions. Note that this approach is used solely
for aeroacoustic calculations and does not directly affect any
calculation used for the flight dynamics.
The main rotor airfoil section adopted for this study is a
NACA 0012. The thickness distribution is described by the
following equation (Ref. 37):
zt(xc) = ¯
t
0.20.2969xc0.3516x2
c+0.2843x3
c0.1015x4
c
(34)
where ¯
tis the maximum thickness of the blade section
expressed as a fraction of the chord, and xcis the non-
dimensional chordwise coordinate. Note that xc=0 corre-
sponds to the leading edge and xc=1 corresponds to the trail-
ing edge. The maximum thickness of the blade section is
assumed to be 12% of the blade chord. Since the airfoil in
consideration is symmetric, Eq. (34) is used for describing
the sectional geometry of both the upper and lower blade sur-
faces. The chordwise panels are distributed unevenly across
the blade chord. In fact, the chordwise panels distribution is
proportional to the square of the non-dimensional chordwise
coordinate, such that the chordwise panels are concentrated
toward the leading edge of the section. Consider a twist an-
gle θ(y,t)function of the spanwise location yand of time t,
given by the summation of inherent blade twist distribution,
blade pitch input, and dynamic twist. Then, the longitudinal
and vertical position of an arbitrary blade surface element in
blade frame axes is given by the following equation:
x
z=cosθsinθ
sinθcosθ xc
±tzt(xc)1/4
0c(35)
where t=0.2076 ft is the maximum thickness of the blade
section and cis the blade chord. In the current study, the
blade chord is assumed constant and equal to 1.73 ft. The
blade frame axes are defined such that their origin lies at the
intersection of the quarter-chord line, the mean line, and the
blade root. The xaxis is along the chordwise direction, point-
ing from the leading to the trailing edge; the yaxis along in the
spanwise direction, pointing from the blade root to the blade
tip; and the zaxis is perpendicular to xand y, pointing from
the lower to the upper surface (for zero twist). It is worth not-
ing that the following relation exists between the blade frame
convention used in the analysis above and that used in PSUH-
eloSim and GenHel:
xBF
yBF
zBF
=
x
y+e0
z
(36)
where e0is the spar length (i.e., the distance between the
flap/lag hinge and the blade root). The PSUHeloSim/GenHel
convention is such that the origin lies at the intersection of the
quarter-chord line, the mean line, and the flap/ag hinge. The
xBF axis is along the chordwise direction, pointing from the
trailing to leading edge; the yBF axis is in the spanwise direc-
tion, pointing from the blade root to the blade tip; and the zBF
axis is perpendicular to xand y, pointing from the upper to
the lower surface. Note that because PSUHeloSim includes
dynamic twist as a state of the system and because the blade
pitch input may change with time, the blade geometry is re-
calculated at each time step based on the current blade twist
and blade pitch input. Figure 2shows the discretized geome-
try of a generic utility helicopter rotor blade with 10 spanwise
panels and 10 chordwise panels. Based on this geometry, the
unit vectors normal to each blade element are calculate and
stored for the undeformed blade geometry, and rotated accord-
ing to the transformation matrix in Eq. (35) at each time step
based on the current blade pitch at each spanwise location. For
aeroacoustic calculations, the blade surface locations, veloc-
ities, and unit vectors normal to the surface are transformed
from the blade to the inertial frame via the transformations
defined in Ref. 34.
The surface pressure distribution for each blade spanwise seg-
ment is calculated based on the local lift coefficient, and on
7
Figure 2: Discretized geometry of the undeformed generic
utility helicopter rotor blade with 10 spanwise panels and 10
chordwise panels.
NACA 0012 lookup tables based on thin airfoil theory. These
lookup tables provide the velocity ratio v/Vand velocity in-
crement ratio vα/Vas functions of the normalized chord-
wise coordinate. The lookup table used in this study is shown
quantitatively in Table 1and are taken from Ref. 37. The
pressure coefficient chordwise distribution on each spanwise
segment is given by the following equation:
CP=
1v
V+vα
VCL2
upper surface
1v
Vvα
VCL2
lower surface
(37)
where CLis the lift coefficient of the spanwise segment. The
blade gauge pressure chordwise distribution at each spanwise
segment is then found by:
˜p=1
2ρu2
T+u2
PCP(38)
where uTand uPare respectively the tangential and perpen-
dicular velocities of the oncoming flow to the blade spanwise
segment in consideration. The lift coefficient, and tangential
and perpendicular velocities are calculated in PSUHeloSim
according to Ref. 34. Figure 3shows the pressure distribution
on the rotor blades of a generic utility helicopter trimmed at
80 kts forward and level flight and for a non-dimensional time
of ψ=210 deg.
Aeroacoustic Solver
This section provides an overview of the MATLAB® based
acoustic solver that is used to compute the noise generated
aerodynamically by the main rotor blades. The solver is based
on Ref. 38 and adopts a marching cubes algorithm to find the
impermeable Ffowcs Williams-Hawkings surface (Ref. 39).
Table 1: NACA 0012 velocity and velocity increment ratios
vs. normalized chordwise location.
Normalized Chordwise
Location, xc
v/Vvα/V
0 0 1.988
0.5 0.800 1.475
1.25 1.005 1.199
2.5 1.114 0.934
5.0 1.174 0.685
7.5 1.184 0.558
10 1.188 0.479
15 1.188 0.381
20 1.183 0.319
25 1.174 0.273
30 1.162 0.239
40 1.135 0.187
50 1.108 0.149
60 1.080 0.118
70 1.053 0.092
80 1.022 0.068
90 0.978 0.044
95 0.952 0.029
100 0.900 0
The acoustic pressure is then calculated based on an emission
surface formulation of the Ffowcs Williams-Hawkings equa-
tions shown in Eq. (12). It is worth noting that the aeroacous-
tic code does not currently solve for the quadrupole term.
The first step of the marching cubes method consists of defin-
ing a structured grid on the control surface (i.e., the rotor
blades surface) along with successive slices for each source
time instant, such that a 3D structured grid of cubes is con-
structed. Note that the emission surface is by definition an
isosurface of observer time in this 3D field. The grid points on
control surface are identified by the indices iand j, whereas
the source time slices are identified by the index k. As such,
each cube vertex is defined by the indices i,j, and k. Next,
upon the calculation of the observer time t=τ+r(x
x
x,y
y
y,τ)
c0for
each grid point, the isosurface is computed for each time step
ti. To do so, each vertex is assigned a value of 0 if t<ti,
1 otherwise. The isosurface will intersect a cube at t=tiif
and only if the edges of that cube have at least one value that
differs from that of the other edges. Then, for these cubes,
the location of the intersection points between the isosurface
and the cubes edges is computed via linear interpolation and
lookup tables. Lastly, the isosurface is reconstructed with the
triangles formed by the intersection points. The triangles ver-
tices are used to readily calculate the area of each triangle,
∆Σi. Additionally, the unit vector normal to the isosurface, ˆ
n
n
n,
the surface velocity u, and the gauge pressure ˜pare linearly in-
terpolated at the intersections and subsequently averaged over
the three vertices of each triangle. This way, each triangle is
assigned with a unique set of these quantities.
8
Figure 3: Gauge pressure distribution on the rotor blades of a
generic utility helicopter at 80 kts level flight at a
non-dimensional time of ψ=210 deg.
Consider re-writing Eq. (12) as:
4πp0(x
x
x,t) = 1
c0
tI1+I2(39)
where:
I1=ZΣρ0c0vn+˜pˆ
n
n
n·ˆ
r
r
r
rΛret
dΣ=ZΣQ1(y
y
y,tr/c0)
rΛret
dΣ
(40a)
I2=ZΣ˜pˆ
n
n
n·ˆ
r
r
r
r2Λret
dΣ=ZΣQ2(y
y
y,tr/c0)
r2Λret
dΣ(40b)
Note that the quadrupole term was neglected. Then, the in-
tegrals contained in the terms I1and I2can be approximated
numerically by direct summation over all triangles that com-
pose the emission surface as:
I1
Ntri
i=1Q1(y
y
y,tr/c0)
rΛret
∆Σi(41a)
I2
Ntri
i=1Q2(y
y
y,tr/c0)
r2Λret
∆Σi(41b)
(41c)
where Ntri is the total number of triangles. Note that the con-
trol surface is described as the sum of eight total surfaces, i.e.,
the upper and lower surfaces of each of the four blades of the
main rotor. Note that in this preliminary study, no end caps at
the tip and root of the blade were used. The time derivative
tI1is obtained via a centered finite difference scheme over
the observer time evaluations:
tI1(ti)I1(ti+1)I1(ti1)
2t(42)
Figure 4shows the isosurface calculated using the marching
cubes for an observer that moves with the aircraft, is located
three rotor radii in front on the main rotor, and lies in the plane
of rotation of the rotor. The isosurface is relative to a 80 kts
level flight condition at a non-dimensional time of ψ=210
deg. The figure also shows the interpolated gauge pressure
distribution on the isosurface.
Figure 4: Interpolated gauge pressure distribution on the
isosurface computed with the marching cubes algorithm for
an observer moving with the aircraft.
RESULTS
Periodic Trim
The dynamics of helicopters in forward flight are dominated
by harmonics with frequencies that are multiples of the num-
ber of blades per revolution (or Nb/rev) (Ref. 6). This is be-
cause the main rotor aerodynamic forces and moments are pe-
riodic with respect to the main rotor azimuth angle ψ. Under
the assumption that the angular speed of the main rotor is con-
stant, the fundamental period of the helicopter flight dynamics
is defined as T=2π/. Since the nominal angular speed of
the main rotor for the utility helicopter is =27 rad/s, the
resulting fundamental period is approximately T=0.23 sec-
onds. The objective of the periodic trim algorithm is to find
the periodic solution for the state vector in level forward flight
at a forward speed of ˙x=80 kts with constant control setting.
The choice of 80 kts level flight is justified by the fact that
readily-available flight test data exists at this particular flight
condition such that comparisons can be made with the numeri-
cal solution. Since the helicopter under consideration has four
rotor blades, the first four state harmonics are retained in the
numerical solution of the periodic motion (N=4). Only the
zeroth harmonic is retained for the control input (M=0). The
vector of n(2N+1) + m(2M+1) = 292 unknowns is:
Θ
Θ
ΘT=x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns u
u
uT
0(43)
The n(2N+1) = 288 constraints are given by Eq. (20), with
the exception of the zeroth harmonic of the derivative of the
9
xposition state which is set to the desired forward speed (i.e.,
˙x0=80 kts). Because there are m(2M+1) = 4 unknowns
more than there are constraints, the zeroth harmonics of the
position states (x,y,z)and yaw angle ψ, denoted as x0,y0,
z0, and ψ0, are removed from the problem and set to arbi-
trary values. This choice is justified by the fact that the zeroth
harmonic of the position and yaw angle do not affect the dy-
namics of the helicopter. This way, the number of unknowns
decreases to 288 such that the problem is square. The modi-
fied vector of unknowns is denoted as,
ˆ
Θ
Θ
ΘT=ˆ
x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns u
u
uT
0(44)
where ˆ
x
x
x0is the zeroth-harmonic state vector without the po-
sition and yaw angle states included. In this example, the az-
imuthal resolution is ψ=1 deg, such that the number of time
steps per revolution is nψ=360. The initial guess of the algo-
rithm is found by trimming the helicopter at incremental time
instants (as if it was a NLTI system) over one rotor revolution
using standard NLTI trimming techniques, and averaging the
trim solution thus found.
Figure 5shows the periodic angular rates obtained with the
proposed algorithm (solid line) using an error tolerance of
1e7. As expected, since the helicopter in consideration has
four main rotor blades with equal mass and since the state har-
monics retained in the solution are four, the higher-harmonic
content of the periodic solution is limited to the fourth sine
and cosine harmonics. Additional harmonics which are mul-
tiples of four (i.e., 8/rev, 12/rev, 16/rev, etc.) can readily be
captured by increasing the number of state harmonics retained
in the solution (i.e.,N=8, N=12, N=16, etc.). It should
be noted that the algorithm is able to capture harmonics in the
periodic solution that are not only multiples of the Nb/rev har-
monics. For instance, if one rotor blade had a different mass
with respect to the others such that the rotor was imbalanced,
then the algorithm would capture the 1/rev component in the
resulting periodic solution, provided that the first harmonic is
retained in the solution.
To validate the results, the vertical acceleration is computed
along the periodic solution and compared with flight test data
of the U.S. Army Rotorcraft Aircrew System Concepts Air-
borne Laboratory (RASCAL) JUH-60A helicopter as shown
in Fig. 6. This figure shows good agreement between the
numerical solution of the periodic motion and the flight test
data. Particularly, the phase of the 4/rev component of the
vertical acceleration closely matches that of the flight test
data, whereas the amplitude of the signal is slightly under-
predicted. This may be due to limitations of the flight dynam-
ics model, specifically the use of a low-order inflow model
and reliance on only the rigid flap and lead-lag motions. The
algorithm takes on average 40 seconds per iteration. It fol-
lows that the approximate computation time associated with
the solution in Fig. 7is 4 minutes.
Main Rotor Forces and Moments
In this section, the proposed output linearization algorithm is
used to to obtain the high-order linearized output equations
0/4 /2 3/2 5/4 3/2 7/4 2
-0.01
0
0.01
p [deg/s]
Initial guess Solution
0/4 /2 3/2 5/4 3/2 7/4 2
-1
0
1
q [deg/s]
10-3
0/4 /2 3/2 5/4 3/2 7/4 2
Azimuth angle, [deg]
-5
0
5
r [deg/s]
10-4
Figure 5: Comparison between the numerical solution and
initial guess of the angular rates periodic motion of a
helicopter.
0/4 /2 3/2 5/4 3/2 7/4 2
Azimuth angle, [deg]
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Vertical acceleration, az [g]
Solution Flight data
Figure 6: Comparison of the vertical acceleration along the
periodic solution obtained numerically with RASCAL
JUH-60A flight-test data at 80 kts level flight.
of the main rotor forces and moments expressed in body-
fixed axes referenced at the aircraft center of gravity. The
main rotor forces and moments are chosen for this first ex-
ample, rather than the acoustic pressure or other composite
noise measures, because their linearized equations can also
be computed via harmonic decomposition. This allows for a
direct comparison between harmonic decomposition and the
proposed output linearization scheme, such that the latter can
be validated. In this example, the output vector of the nonlin-
ear time-periodic dynamics is:
y
y
yT= [X Y Z L M N](45)
where X,Y,Zare the main rotor forces and L,M,Nare the
main rotor moments. The flight condition in consideration
is 80 kts forward and level flight. The harmonics retained
10
123456
Iteration
10-8
10-6
10-4
10-2
100
102
104
||e||
Figure 7: Trim iteration error vs. number of iterations.
in the state, control input, and output vectors are up to the
fourth (i.e.,N=M=L=4) for both harmonic decomposi-
tion and the proposed output linearization method. The az-
imuthal resolution utilized is ψ=1 deg. Figure 8shows the
forces and moments following a 10% longitudinal cyclic stick
doublet. In this figure, it is shown how the high-order LTI
system obtained via the proposed method yields identical re-
sults to those derived with harmonic decomposition. Further,
the main rotor forces and moments obtained using the high-
order LTI models (both new method and harmonic decompo-
sition) are very close to those given by the nonlinear flight
dynamics model (i.e., PSUHeloSim) for the relatively small
amplitude maneuver in consideration. Because a simple lin-
ear lag damper is used in the simulation model, which tends
to underpredict the lag damping, the progressive lag mode
is lightly damped. This corresponds to the oscillation seen
in the response. Note that each force/moment perturbation
response from the linear models is reconstructed from that
force/moment harmonic coefficients time history using Eq.
(7c). To compare the response of the linear models with that
of the nonlinear time-periodic dynamics, the periodic trim is
subsequently added to the perturbation response. More study
is required to understand over what range of maneuver ampli-
tudes the linearized models will be accurate.
Acoustics
In this section, the proposed output linearization algorithm is
used to obtain the high-order linearized output equations of
the acoustic pressure at a single observer location moving with
the helicopter. The observer is fixed in the helicopter body
frame. The observer is located three rotor radii in front on
the main rotor and lies in the plane of rotation of the rotor
(i.e., approximately 5.72 ft along the water line). The observer
position is shown qualitatively in Fig. 9. The chosen flight
condition is 80 kts forward flight.
The azimuthal resolution used for the output linearization is
ψ=15 deg, such that the number of time steps per rev-
012345
0
500
1000
1500
2000
X [lb]
012345
-1000
-500
0
500
Y [lb]
012345
Time [s]
-2
-1.8
-1.6
-1.4
-1.2
Z [lb]
104
012345
-10000
-5000
0
5000
L [lb-ft]
012345
-2
-1
0
1
2
M [lb-ft]
104
012345
Time [s]
1.8
1.9
2
2.1
2.2
N [lb-ft]
104
Nonlinear Harmonic Decomposition New Method
Figure 8: Comparison between the main rotor forces and
moments obtained using the nonlinear flight dynamics model
(i.e., PSUHeloSim), harmonic decomposition, and the
proposed output linearization method.
olution is nψ=24. The harmonics retained in the state,
control input, and output vectors are up to the fourth (i.e.,
N=M=L=4). As such, the linearization algorithm requires
nrev =2[n(2N+1) + m(2M+1)] = 212 rotor revolutions,
equivalent to np=2nψ[n(2N+1)+ m(2M+1)] = 5088 flight
dynamics and aeroacoustic evaluations. Using a MATLAB®
implementation of the algorithm on a computer equipped with
an Intel®CoreTM 8th Gen i7-8650U processor, the lineariza-
tion algorithm takes approximately 15 minutes. It is worth
noting that when performing the position and Euler angles
perturbations in the linearization scheme, the perturbations
are also applied to the observer position. This way, there is
no relative motion between the helicopter body frame and the
helicopter position during linearization. The spatial resolution
used to discretize each rotor blade surface is 10 chordwise
panels and 10 spanwise panels, as suggested in Re. 1. Note
that the blade upper and lower surfaces are considered as two
different surfaces. As such, 800 panels are used to character-
ize the rotor blade surface. Figure 10 shows the acoustic pres-
sure at the observer location described above for a 10% lon-
gitudinal stick doublet. The high-order LTI model is shown
to provide a similar prediction of the acoustic pressure com-
pared to that of the nonlinear model for the relatively small
amplitude maneuver in consideration. Note that the acoustic
pressure perturbation response from the linear model is recon-
structed from its harmonic coefficients time history using Eq.
(7c). To compare the response of the linear model with that
of the nonlinear time-periodic dynamics, the periodic trim is
11
Figure 9: Location of an observer fixed in the helicopter
body frame.
subsequently added to the perturbation response. Figure 11
shows a comparison between breakdown of the components
of the aerodynamically-induced noise as obtained with the
nonlinear and linear simulations. In this figure, the thickness
noise, pTis shown to remain essentially unchanged through-
out the maneuver. The change in loading noise relative to
the monopole, pL1, throughout the maneuver is shown to not
be described accurately by the linear model. The causes of
this mismatch will need to be understood and addressed in
the future work. Finally, the variations of the loading noise
contribution from the dipole, pL2, throughout the maneuver
are shown to be captured accurately by the linearized model.
These preliminary results suggest that the nonlinear simula-
tion of the coupled flight dynamics and acoustics runs approx-
imately 19 times slower than real-time. On the other hand, the
linear simulation of the flight dynamics and acoustic that uses
the high-order LTI model runs approximately 1826 faster than
real time. These results are reported in Table 2, along with the
time associate with the computation of a time step for each
simulation strategy.
It is worth noting that with increasing number of state, input,
and output harmonics retained in the linearization process,
and thus with increasing order of the approximate LTI system,
the cost of simulating the LTI system would increase as well.
This would directly affect the computational time required for
each simulation time step. Further, increasing number of har-
monics would dictate a finer time step to ensure stable numer-
ical integration and at least twice the amount of time steps per
the highest harmonic to respect the Nyquist–Shannon sam-
pling. As such, the results showed in this section are specific
to the number of harmonics retained and time step chosen. It
is also worth noting that while the cost of the nonlinear simu-
lation process would nearly double for each observer added
(as the computational time associated with flight dynamics
is negligible compared to the acoustic calculations), the cost
of the linear simulation would not change significantly. It
follows that the proposed methodology provides an increas-
ing computational advantage relative to nonlinear simulations
with increasing number of observers. However, the cost of lin-
earization would still roughly double for each observer added.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time [s]
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Acoustic Pressure [Pa]
Nonlinear
LTI
Figure 10: Acoustic pressure: nonlinear model vs. high-order
LTI obtained through the proposed output linearization
method.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-2
0
2
pT [Pa]
Nonlinear LTI
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1
0
1
pL1
[Pa]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time [s]
-2
-1
0
pL2
[Pa]
Figure 11: Acoustic pressure components: nonlinear model
vs. high-order LTI obtained through the proposed output
linearization method.
CONCLUSIONS
A novel linearization scheme was developed to obtain high-
order, time-invariant, linearized models of the rotorcraft flight
dynamics, vibrations, and acoustics. The proposed methodol-
ogy is demonstrated through an example involving a generic
utility helicopter in non-BVI conditions. First, a modified har-
monic balance algorithm based on harmonic decomposition is
applied to find the periodic equilibrium and approximate high-
order linear time-invariant dynamics about that equilibrium.
Next, the proposed output linearization scheme is applied to
derive time-invariant, linearized equations for the rotor vibra-
tions and acoustics. Batch simulations are used to validate
the response of the linearized models against the that from
nonlinear simulations. Additionally, the cost of linearization
12
Table 2: Simulations time performance with a time step of ψ=15 deg, 10 blade chordwise elements, and 10 blade spanwise
elements.
Simulation Type 1 Time Step [sec] ×Real Time
Flight Dynamics 2.000e4 1/49
Aeroacoustics 1.809e1 19
Flight Dynamics + Aeroacoustics 1.812e1 19
Linearized Flight Dynamics + Aeroacoustics 5.3e6 1/1826
and potential performance benefits of employing linear mod-
els versus nonlinear simulations are assessed. Based on the
current work, the following conclusions can be reached:
1. The proposed output linearization method yields nearly
identical results compared to harmonic decomposition
when the two methods are applied to the main rotor
forces and moments. The method is also shown to pre-
dict accurately the nonlinear response for the case shown
and for small amplitude maneuvers. More study is re-
quired to understand over what amplitude range the lin-
ear models will be accurate.
2. High-order linear time-invariant models obtained with
the proposed method are shown to provide similar pre-
dictions of the acoustic pressure compared to nonlinear
acoustic predictions for small amplitude maneuvers, but
at a fraction of the computational cost. Based on the
current number of state, input, and output harmonics re-
tained in the linearized models, linear simulations of the
coupled flight dynamics and acoustics run approximately
1826 faster than real time, and four orders of magni-
tude faster than nonlinear acoustic predictions based on
a marching cubes algorithm.
3. While the cost of the nonlinear simulations would nearly
double for each observer added (as the computational
time associated with flight dynamics is negligible com-
pared to the acoustic calculations), the cost of linear sim-
ulation would not change significantly. As such, the
proposed methodology provides an increasing computa-
tional advantage relative to nonlinear simulations with
increasing number of observers. However, the cost of lin-
earization would still roughly double for each observer
added.
Future studies will concentrate on obtaining linearized mod-
els using increased azimuthal resolution, number of state, in-
put, and output harmonics, and observer locations. Paramet-
ric studies will be performed to assess model accuracy against
cost of simulation. In addition, high-fidelity aeromechanic
models will be incorporated in the nonlinear simulation model
to allow for the derivation of linearized models suited for
acoustic predictions in BVI conditions. Finally, linearized
models will be derived across the flight envelope and used to
construct stitched simulations capable of real-time predictions
of vibratory loads and acoustics (Ref. 40).
ACKNOWLEDGEMENTS
The Rotorcraft Aircrew System Concepts Airborne Labora-
tory (RASCAL) flight data used in this research/investigation
was provided to the Pennsylvania State University under the
U.S. Army/Navy/NASA Vertical Lift Research Center of Ex-
cellence, Agreement No. W911W6-17-2-0003.
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