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All content in this area was uploaded by Umberto Saetti on May 10, 2021

Content may be subject to copyright.

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. 3–5).

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. 8–11).

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. 13–16 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

x∈Rnis the state vector, u

u

u∈Rmis the control in-

put vector, y

y

y∈Rlis 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=Ωt, where

Ωthe main rotor angular speed in rad/s. It follows that the

fundamental period of the system is T=2π

Ωseconds, 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

A∈Rn(2N+1)×n(2N+1),B

B

B∈Rn(2N+1)×m(2M+1),C

C

C∈

Rl(2L+1)×n(2N+1), and D

D

D∈Rl(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

p0∈Ro(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=p−p0is 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

x−y

y

y

||x

x

x−y

y

y|| is the emission direction,

y

y

yis the source location,

Λ=hp1−2Mnˆ

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=t−r

c0

(13)

where r=||x

x

x−y

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. 21–23), har-

monic balance (Refs. 24–26), 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

X∗T

k=x

x

x∗T

k0x

x

x∗T

k1cx

x

x∗T

k1s... x

x

x∗T

kNc x

x

x∗T

kNs (16a)

U

U

U∗T

k=u

u

u∗T

k0u

u

u∗T

k1cu

u

u∗T

k1s... u

u

u∗T

kNc u

u

u∗T

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

X∗T

kU

U

U∗T

ki(17)

where Θ

Θ

Θk∈Rn(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

ek∈Rn(2N+1)and W

W

W∈Rn(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

Ak∈Rn(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

Jk∈Rn(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

Θk−ˆ

J

J

J−1

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 kekk∞becomes 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

2∆Xjhp0+

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

0.20.2969√xc−0.3516x2

c+0.2843x3

c−0.1015x4

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

0c(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=

1−v

V+∆vα

VCL2

upper surface

1−v

V−∆vα

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/V∆vα/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,t−r/c0)

rΛret

dΣ

(40a)

I2=ZΣ˜pˆ

n

n

n·ˆ

r

r

r

r2Λret

dΣ=ZΣQ2(y

y

y,t−r/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,t−r/c0)

rΛret

∆Σi(41a)

I2≈

Ntri

∑

i=1Q2(y

y

y,t−r/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(ti−1)

2∆t(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/Ω. Since the nominal angular speed of

the main rotor for the utility helicopter is Ω=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

1e−7. 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 2.000e−4 1/49

Aeroacoustics 1.809e−1 19

Flight Dynamics + Aeroacoustics 1.812e−1 19

Linearized Flight Dynamics + Aeroacoustics 5.3e−6 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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