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The Pennsylvania State University

The Graduate School

College of Engineering

ROTORCRAFT SIMULATIONS WITH COUPLED FLIGHT

DYNAMICS, FREE WAKE, AND ACOUSTICS

A Thesis in

Aerospace Engineering

by

Umberto Saetti

©2016 Umberto Saetti

Submitted in Partial Fulﬁllment

of the Requirements

for the Degree of

Master of Science

August 2016

The thesis of Umberto Saetti was reviewed and approved∗by the following:

Joseph F. Horn

Professor of Aerospace Engineering

Thesis Advisor, Chair of Committee

Kenneth S. Brentner

Professor of Aerospace Engineering

George A. Lesieutre

Professor of Aerospace Engineering

Aerospace Engineering Department Head

∗Signatures are on ﬁle in the Graduate School.

ii

Abstract

This study presents the integration of a ﬂight simulation code (PSUHeloSim), a high

ﬁdelity rotor aeromechanics model with free wake (CHARM Rotor Module), and

an industry standard noise prediction tool (PSU-WOPWOP) into a comprehensive

noise prediction system. The ﬂight simulation uses a Dynamic Inversion autonomous

controller to follow a prescribed trajectory for both steady and maneuvering ﬂight

conditions. The aeromechanical model calculates blade loads and blade motion that

couple to the vehicle ﬂight dynamics with suitable resolution to allow high ﬁdelity

acoustics analysis (including prediction of blade-vortex interaction (BVI) noise).

The blade loads and motion data is sent to PSU-WOPWOP in a post-processing

step to predict external noise.

Particular attention is paid to the development of PSUHeloSim and to the

enhancement of the closed-loop response characteristics of the coupled simula-

tion. Speciﬁcally, is studied the use of reduced-order linear models, derived by

linearization of the coupled simulation, in the feedback linearization of the Dynamic

Inversion controller in diﬀerent ﬂight conditions. The diﬀerent reduced-order models

obtained are compared by the use of eigenvalue analysis and frequency response

in order to link their diﬀerences to physical phenomena occuring in the coupled

simulation. A validation of these reduced-order models is provided by performing

a frequency sweep of the coupled simulation. Finally their eﬀectiveness in the

feedback linearization loop is evaluated by analysing the closed loop time response

of the coupled simulation to the coupling.

The coupled analysis is being used to evaluate the inﬂuence of ﬂight path on

aircraft noise certiﬁcation metrics like EPNL and SEL for various rotorcraft in

work for the FAA. The software was used to analyze the acoustic properties of a

blade planform similar to the Blue Edge rotor blades developed by DLR and Airbus

Helicopters

−

predicting BVI noise reduction as compared to more conventional

blade geometries on the same order as that reported for the Blue Edge rotor.

iii

Table of Contents

List of Figures vii

List of Tables ix

List of Symbols x

Acknowledgments xiv

Chapter 1

Introduction 1

1.1 Motivation................................ 1

1.2 Background and Technical Barriers to Solve . . . . . . . . . . . . . 2

1.3 Objectives................................ 3

Chapter 2

Simulation Architecture 5

2.1 Helicopter Flight Dynamics Model . . . . . . . . . . . . . . . . . . . 5

2.1.1 Introduction........................... 5

2.1.2 Bell 430 Properties . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3.1 Fuselage........................ 9

2.1.3.2 Flapping........................ 10

2.1.3.3 Inﬂow ......................... 12

2.1.4 TrimAlgorithm......................... 13

2.1.5 TrimResults .......................... 14

2.1.6 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.6.1 InnerLoop ...................... 15

2.1.6.2 OuterLoop ...................... 19

2.1.6.2.1 Low Speed Mode . . . . . . . . . . . . . . 19

2.1.6.2.2 High Speed Mode . . . . . . . . . . . . . . 21

iv

2.1.6.2.3 Blending . . . . . . . . . . . . . . . . . . . 23

2.1.6.3 Error Dynamics . . . . . . . . . . . . . . . . . . . . 23

2.2 High Fidelity Rotor Module . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Introduction........................... 27

2.2.2 Reconstruction ......................... 28

2.3 Noise Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Schematic of the Simulation Process . . . . . . . . . . . . . . . . . 29

Chapter 3

Linear Model Analysis 31

3.1 Linearization .............................. 31

3.2 Linear Analysis Overview . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Reduced-Order Models . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Linearization of the coupled PSU-HeloSim/CHARM Simulation . . 42

3.5 FrequencySweep ............................ 52

Chapter 4

Simulation Results 60

4.1 Flight Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.1 Coupling Transient . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.2 Use of the Reduced-Order Models in the Coupled Simulation 62

4.1.3 Decelerated Descent . . . . . . . . . . . . . . . . . . . . . . 64

4.1.4 Coordinated Turn . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1.5 Acceleration in Ground Eﬀect . . . . . . . . . . . . . . . . . 69

4.2 AcousticResults ............................ 72

4.2.1 Introduction to Helicopter Acoustics . . . . . . . . . . . . . 72

4.2.2

Prediction of BVI Noise Reduction Using Blue Edge-like

Blades.............................. 74

Chapter 5

Conclusions and Recommendations for Future Work 82

5.1 Conclusions ............................... 82

5.2 FutureWork............................... 83

Appendix A

Structure of the Wrapper Code 86

A.1 Introduction............................... 86

A.2 InputFiles................................ 86

A.2.1 Master.txt............................ 86

A.2.2 Command.txt.......................... 86

v

A.3 OutputFiles............................... 89

A.3.1 HeloSimOut.txt......................... 89

A.3.2 PSU-WOPWOP Files . . . . . . . . . . . . . . . . . . . . . 89

Bibliography 92

vi

List of Figures

2.1

Schematic of the PSUHeloSim/CHARM/PSU-WOPWOP simula-

tionmodel. ............................... 6

2.2 An airborne Bell 430. . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Bell4303-View.............................. 7

2.4

Trim values of the controls and attitude for diﬀerent forward speed

velocities. ................................ 16

2.5 Flowchart of the simulation process. . . . . . . . . . . . . . . . . . . 30

3.1 Eigenvalues relative to the Bell 430 model in a 100 kts level ﬂight. . 37

3.2 Eigenvalues associated with the fuselage dynamics. . . . . . . . . . 38

3.3

Comparison between the eigenvalues of the full-order model and the

eighth-ordermodel............................ 39

3.4

Frequency responses of aicraft states with respect to the control

inputs for both the eigth-order and full-order models. . . . . . . . . 41

3.5

Response of the derivatives of the states to perturbations of the

statesandcontrols............................ 44

3.6

Comparison between the eigenvalues of the reduced order models

obtained from PSUHeloSim and the coupled simulation for a 120

ktslevelﬂight. ............................. 45

3.7

Comparison between the eigenvalues of the reduced order models

obtained from PSUHeloSim for the level ﬂight and descent cases. . . 48

3.8

Comparison between the eigenvalues of the reduced order models

obtained from PSUHeloSim in the level ﬂight case, and PSUH-

eloSim/CHARM coupled simulation in a 6◦descent cases. . . . . . . 49

3.9

Comparison between the eigenvalues of the reduced order models

obtained from PSUHeloSim and PSUHeloSim/CHARM coupled

simulation for both level ﬂight and the descent case. . . . . . . . . . 51

3.10Sweepinput................................ 54

3.11Sweepexcitation. ............................ 55

vii

3.12

Frequency sweep of the coupled simulation for a 120 kts level ﬂight

case. ................................... 57

3.13

On-axis Bode and correlation factor plots of the lateral sweep for a

120 kts level ﬂight case. . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1 Coupling transient: dashed line marks the start of the coupling. . . 61

4.2

Response to the coupling with the controller based on the reduced-

order models form 1) PSUHeloSim, 2) linearization of the coupled

simulation in level ﬂight, 3) linearization of the coupled simulation

in a 6◦descent. ............................. 63

4.3 Results from a 6◦decelerated descent at 0.1 g of deceleration. . . . 65

4.4

Coupled and uncoupled simulations results of a 90

◦

coordinated

turn at 100 kts 20◦ofbankangle.................... 67

4.5

Coupled simulation results of a 360

◦

coordinated turn at 60 kts and

30◦ofbankangle............................. 69

4.6

Coupled and uncoupled simulations results of an acceleration from

0 to 60 kts in ground eﬀect followed by steady climb at 12

◦

of ﬂight

pathangle. ............................... 71

4.7

Typical direction of primary radiation for various rotor noise sources.

72

4.8

Noise components and their contribultion to the OASPL predictions

for a 100 kts ﬂight case ﬂown at 150 m altitude. . . . . . . . . . . . 74

4.9

Noise components and their contribution to the PNLT for a 100 kts

ﬂight case ﬂown at 150 m of altitude. . . . . . . . . . . . . . . . . . 75

4.10

Blue Edge blade concept from Eurocopter (now Airbus Helicopters) [1].

76

4.11 Rectangular, tapered and Blue Edge-like planforms. . . . . . . . . . 77

4.12

CHARM/PSU-WOPWOP main rotor OASPL and BVISPL predic-

tions one rotor radius beneath the nominal Bell 430 rotor for three

blade geometries;

αs

= 6

◦

(aft),

µ

= 0

.

15, and

CT

= 0

.

00143. The

black circle represents the rotor tip (advaning side on the right). . . 78

4.13

BVI event as predicted by CHARM for the baseline rectangular

blade and the Blue Edge blade. . . . . . . . . . . . . . . . . . . . . 79

4.14

Contours of OASPL on a 30.48 m radius hemisphere, centered at

the Bell 430 center of grvity location. The hemisphere follows the

aircraft. OASPL contours shown are for the standard rectangular

blades). ................................. 80

4.15

Acoustic pressure time history at azimuth angle

ψ

= 125

◦

, elevation

θ

=

−

45

◦

below the rotor plane, and radius of 30.48 m from the

helicopter center of cravity (i.e. the location of the black dot in Fig.

4.14).................................... 81

viii

List of Tables

2.1 Bell 430 mass and inertia properties. . . . . . . . . . . . . . . . . . 8

2.2 Bell 430 main and tail rotor key properties. . . . . . . . . . . . . . . 8

2.3 Inner loop command ﬁlters properties. . . . . . . . . . . . . . . . . 18

2.4 Outer loop command ﬁlters properties. . . . . . . . . . . . . . . . . 19

2.5

Inner loop disturbance rejection frequencies, damping ratios, and

integratorpoles.............................. 26

2.6 Outer loop disturbance rejection frequencies and damping ratios. . . 26

2.7 Inner loop compensation gains. . . . . . . . . . . . . . . . . . . . . 26

2.8 Outer loop compensation gains. . . . . . . . . . . . . . . . . . . . . 27

3.1

Numerical values of the dutch roll frequency and damping ratio, and

of the roll subsidence eigenvalue for a 120 kts ﬂight condition. . . . 51

3.2 Description of the parameters used for the frequency sweep. . . . . 53

4.1 Charcteristics of the blade planforms. . . . . . . . . . . . . . . . . . 76

A.1 Master.txtsetup. ............................ 88

A.2 HeloSimOut.txt content. . . . . . . . . . . . . . . . . . . . . . . . . 90

A.3 Input ﬁles for PSU-WOPWOP. . . . . . . . . . . . . . . . . . . . . 91

ix

List of Symbols

All bold letters are vectors or matrices, unless otherwise speciﬁed. Any variable

with one dot overhead is the ﬁrst derivative with respect to time, two dots overhead

is the second derivative with respect to time, and so on. In some instances, a

variable may have more than one deﬁnition depending on the context. In such

cases, each deﬁnition is separated by a semicolon. All symbols, whether deﬁned

here or not, are deﬁned at the ﬁrst usage for the convenience of the reader.

Amain rotor area [ft2], sweep amplitude [%]

Asystem matrix

Af,Afs,Asfast, fast-slow and slow sub-matrices of the system matrix

ˆ

Areduced order system matrix

Bcontrol matrix

Bf,Bsfast and slow sub-matrices of the control matrix

cblade chord [in]

C,C1,C2output matrices

CM0damping matrix of the ﬂapping equation

DM0stiﬀness matrix of the ﬂapping equation

ehinge oﬀset

eerror vector

eV x, eV y , eV z

errors on the longitudinal, lateral and vertical speeds in heading

frame [ft/s]

x

e˙

φerror on the roll rate [rad/s]

ggrvitational acceleration [ft/s2]

HM0right-hand of the ﬂapping equation

Iβ

second mass moment of blade about the ﬂapping hinge [

slug −

ft2]

Ixx, Iyy, Izz principal moments of inertia [slug −ft2]

Ixz product of inertia [slug −f t2]

KP, KD, KI, KII proportional, derivative, integral, double-integral gains

KP,KD,KI,KII

proportional, derivative, integral, double-integral gain matrices

for both the inner and outer loop

Kβﬂapping stiﬀness [slug −f t2/s2]

L0

p, L0

q, etc.

roll moment derivatives normalized by moment of inertia [

rad/

(

ft−

s),1/s]

Lgain matrix

mmass of aircraft [slug]

Mβ

ﬁrst mass moment of blade about the ﬂapping hinge [

slug −f t

]

MBblade mass [slug]

Mp, Mq, etc.

pitch moment derivatives normalized by moment of inertia

[rad/(f t −s),1/s]

Mapparent mass matrix

NBnumber of blades

N0

p, N0

q, etc.

yaw moment derivatives normalized by moment of inertia

[rad/(f t −s),1/s]

p, q, r angular velocity components of fuselage in body axis [rad/s]

rcmd

commanded angular speed around the vertical body axis [

rad/s

]

sLaplace (complex) operator

xi

Tb/h rotational matrix from body to heading frame

uinput vector

uetrim input vector

u, v, w longitudinal, lateral and vertical speed in body frame [f t/s]

ue, ve, we

trim longitudinal, lateral and vertical speed in body frame

[ft/s]

vhinduced veocity at the rotor in hover [ft/s]

V, Vh, Vl

absolute airspeed and absolute airspeed identifying the transi-

tion between high and low speed ﬂight [ft/s]

Vx, Vy, Vzforward, lateral and vertical velocity in heading frame [ft/s]

Vxcmd , Vycmd , Vzcmd

commanded forward, lateral and vertical velocity in heading

frame [ft/s]

Waircraft weight [lbs]

x, y, z North, East and Down positions in NED frame [ft]

xstate vector

xetrim state vector

xf,xsfast and slow states vector

X, Y

longitudinal and lateral forces acting on the aircraft in body

frame [lbs]

Xp, Xq, etc.

longitudinal force derivatives normalized by aircraft mass [1

/s, m/

(

s−

rad)]

youtput vector

ycmd commanded trajectory [f t]

Yp, Yq, etc.

lateral force derivatives normalized by aircraft mass [1

/s, m/

(

s−

rad)]

Zp, Zq, etc.

vertical force derivatives normalized by aircraft mass [1

/s, m/

(

s−

rad)]

xii

β0, β1C, β1S

rotor blade coning, longitudinal and lateral ﬂapping angles in

multi-blade coordinates [rad]

βM

ﬂapping variables vector in Multiple Blade Coordinates (MBC)

[rad]

δ3delta 3 angle [deg]

δlat, δlong , δcoll , δped

lateral, longitudinal, collective and tail rotor collective pilot

inputs [%]

ζdamping ratio

ζDR dutch roll damping ratio

θtw blade twist [deg]

λ0, λ1C, λ1S

rotor uniform and ﬁrst harmonic inﬂow velocities in hub frame

λβmain rotor non-dimensional ﬂapping frequency

µadvance ratio

νpseudo-command vector

σDR, σRS variables associated with dutch roll and roll subsidence

ρair density [slug/f t3]

τtime constant [s]

φ, θ, ψ Euler angles [rad]

φcmd, θcmd, ψcmd commanded Euler angles [rad]

φcmdh, φcmdl

commanded roll angles relative respectively to the high and

low transition speeds [rad]

φe, θe, ψetrim Euler angles [rad]

ωDR dutch roll frequency [rad/s]

ωnnatural frequency [rad/s]

ωnDR dutch roll natural frequency [rad/s]

Ωemain rotor trim angular speed [rad/s]

xiii

Acknowledgments

First and foremost, I would like to thank my adviser, Dr. Joseph F. Horn for

his support and patience in showing me the ropes of research, and for being an

impeccable example of fairness and hard work.

I would like to thank Dr. Kenneth S. Brentner for his kind support and occasional

witty jokes, which contributed to make the graduate experience a pleasant one.

My gratitude also goes to Dr. Marco Borri, advocate of my acceptance at Penn

State and pursuit of rotorcraft studies.

These two years would not have been the same without Willca Villafana,

accidental adventure compaignon since the very ﬁrst day. The present research

eﬀort will remain as a testimony of our friendship, in the hope of coltivating more

memories together, professianally and otherwise, in the future.

The long hours spent at the VLRCOE created a great comradery among my

fellow collegues, especi Sandilya "Magic Sandy" Kambampati, Ilker Oruc, Junfeng

Yang, Yande Liu and Reed Kopp, which I thank for the help with the diﬃculties

and questions arisen along the way.

Lastly I would like to thank my family for their love and encouragment in

coltivating my passions and enthusiasm.

xiv

Chapter 1 |

Introduction

1.1 Motivation

The integration of rotorcraft simulation software with complex aeromechanical

models can provide increased ﬁdelity and functionality of the system as compared

to any of the individual tools. Continued advancement in computational resources

allows coupled codes to be executed eﬃciently and even in real-time. The prediction

of noise in generalized maneuvering ﬂight is relevant in that it can be used to

determine ﬂight procedures that minimize noise and impact on communities. This

is of particular interest to the Federal Aviation Administration, who through the

Aviation Sustainability Center of Excellence (ASCENT), is seeking to develop

noise abatement procedures. Physics-based models are particularly useful for noise

prediction when no measured data is available, such as for new rotorcraft designs

and conﬁgurations. To achieve these goals, the noise prediction should be coupled

with ﬂight simulation codes that generate realistic trajectories and pilot control

input histories for typical rotorcraft maneuvers. This could be done through either

real-time piloted simulations or through batch simulations using an autonomous

controller (that models a pilot compensation to track a desired trajectory). In

addition, such simulations should be coupled with high ﬁdelity aeromechanical

models that provide suitable blade load and blade motion predictions for acoustics

analysis (including BVI), and these aeromechanical models should be consistent with

the total forces and moments acting on the rotorcraft during the ﬂight simulation.

In this study, was continued the development of a comprehensive noise prediction

system [2], that couples a ﬂight simulation code (PSUHeloSim), a high ﬁdelity rotor

1

aeromechanics model with free wake (CHARM Rotor Module [3]), and an industry

standard noise prediction tool (PSU-WOPWOP [4] [5] [6]). All of these tools are

physics-based models that can be adapted to predict ﬂight dynamics, rotor loads,

and noise on a variety of rotorcraft conﬁgurations. In this paper, is presented the

coupling of these codes and preliminary results showing vehicle motion and noise

prediction for steady ﬂight conditions.

1.2 Background and Technical Barriers to Solve

In 2006, the GENHEL-PSU simulation code was integrated with the CHARM

free wake module and the coupled code was shown to provide improved ﬁdelity

in ﬂight dynamics [7]. Real-time operation required limitations in the rotor wake

geometry, but with use of parallel computing and the steady improvement in CPU

performance these limitations can be relaxed. Coupling of GENHEL-PSU with

Navier-Stokes CFD solutions has also been performed, with speciﬁc application to

simulation of ship airwake interactions with the helicopter [8]. These simulations are

still far slower than real-time, but scaling studies have shown that with massively

parallel processing and reduced order CFD models, real-time simulation and CFD

coupling might be possible in the near future.

GENHEL-PSU was also coupled with the acoustics prediction software PSU-

WOPWOP [9]. This coupling was a serial "one-way" coupling in that GENHEL-PSU

simulations ﬁrst calculated the helicopter motion and blade loads and then PSU-

WOPWOP used this information to predict the external acoustics. One-way

coupled simulations were reasonable since the acoustics have no impact on aircraft

dynamics. The simulations allowed predictions of rotorcraft noise in maneuvers,

whereas historically such calculations were only performed in steady-state trimmed

ﬂight. The limited ﬁdelity of the blade loads predicted by GENHEL-PSU meant

that the acoustics prediction could not account for Blade-Vortex-Interaction (BVI)

noise. However, subsequent work used a free-wake model to re-construct more

detailed blade loads for the prediction of BVI [10]. The wake model was coupled

"one-way" in the sense that the ﬂight dynamics simulation was based on a simple

blade element rotor with ﬁnite-state inﬂow and was not aﬀected by the free wake.

The free wake was used to re-construct more detailed blade loads for use only in

the acoustics prediction.

2

The research eﬀorts mentioned earlier concentrated on the coupling of codes

belonging to the disciplines of ﬂight dynamics, aeromechanics and acoustics in

diﬀerent combinations, but never all toghether. The integration of the three could

beneﬁt from the increased ﬁdelity in load prediction obtained by the integration of

a ﬂight dynamics code with a aeromechanics/free wake software, form the fact that

the use of a ﬂight dynamics code coupled with acoustics would enable predictions

in both steady and maneuvering ﬂight, and from the higher accuracy of the blade

loads that would be used for acoustic predictions. Furthermore the analysis would

require to couple a ﬂight dynamic software with a rotor module that accounts both

for detailed aeromechanics, including blade modes, and free wake. This is a new

approach since in [7] the blade motion was calculated by GENHEL. The ﬂight

dynamics simulator will then have to be simpliﬁed since there is no need to account

for vibratory terms, which will be included in the rotor module. The simulations

will also have to be integrated with a controller that is robust enough to account

for the diﬀerences between the rotor models in the ﬂight dynamics code and in

the aeromechanics code, and that achieves high precision closed-loop control of the

simulated helicopter. The helicopter will have to be able to follow an arbitrary

prescribed trajectory (within the physical limitation of the helicopter itself) in a

realistic way, thus serving as a "pilot model".

1.3 Objectives

The objective of this study is to develop a comprehensive rotorcraft acoustic

prediction tool suited for both steady and maneuvering ﬂight conditions. For this

purpose a ﬂight simulation code (PSUHeloSim), a high ﬁdelity rotor aeromechanics

model with free wake (CHARM Rotor Module), and an industry standard noise

prediction tool (PSU-WOPWOP) are be integrated.

First, fully-coupled PSUHeloSim/CHARM Rotor Module simulations are ana-

lyzed to verify the consistency of the main and tail rotor forces and moments with

speciﬁc ﬂight conditions. The performance of the dynamic inversion autonomous

controller is then addressed. Particular attention is paid to the the enhancement of

the closed-loop response characteristics of the coupled simulation. The eﬃcacy of

utilizing reduced order models obtained by linearization of the coupled simulation

in the feedback linearization loop of the dynamic inversion controller is studied,

3

especially with respect to the start up transient. Lastly, the overall capabilities

of the comprehensive tool to perform acoustic predictions are tested. Speciﬁcally

the acoustic performance of Blue Edge rotor blades for BVI noise reduction is

compared to rotor baldes with more conventional geometries. This particular test

is chosen since the prediction of BVI noise is a critical test of the utility of the

system to achieve the objective of detailed noise predictions for advanced aircraft.

Furthermore, Airbus Helicopters, the producer of the Blue Edge rotor, has reported

the noise reduction expected for this particular blade technology, allowing the

results of this study to be objectively eveluated.

4

Chapter 2 |

Simulation Architecture

2.1 Helicopter Flight Dynamics Model

2.1.1 Introduction

The ﬂight dynamics simulations were performed using the PSUHeloSim code.

This is a basic simulation tool developed at PSU to provide a generic rotorcraft

ﬂight dynamics model for research and education. PSUHeloSim is developed

in the MATLAB/Simulink environment for ease of development and adaptation

to diﬀerent rotorcraft conﬁgurations. The simulation model is constructed in

ﬁrst order state space form which makes it well suited for numerical integration,

trim, and linearization calculations. It includes the 6 DoF non-linear equations

of motion of the fuselage, second order rotor ﬂapping dynamics, and a 3-state

Pitt-Peters inﬂow [11] model, resulting in a 21-state non-linear model. It uses a

simple aerodynamic model of the fuselage and empennage based on given lift and

drag properties. A static Bailey [12] model is used for the tail rotor, and while

the main rotor includes ﬂapping dynamics, it uses linearized blade equations of

motion and simpliﬁed analytic integrations of the aero lift and drag forces along

the blade. The limitations in rotor model ﬁdelity are not signiﬁcant for the current

application, as the simple rotor model is replaced with the high-ﬁdelity CHARM

(Comprehensive Hierarchical Aeromechanics Rotorcraft Model) [13] rotor module

in the ﬁnal results. The simple rotor model is only used in the trim calculation

for initializing the simulations and in the controller design process. A general

schematic of the PSUHeloSim ﬂight dynamics model (not including the controller

described below) is shown in Fig. 2.1. The simulation is integrated with a non-

5

linear dynamic inversion control law [14]. This control law has been developed for

rotorcraft application on a number of research programs at PSU, and has recently

been used for non-real-time simulations with complex aeromechanical models [8].

The controller achieves high precision closed-loop control of the simulated helicopter

and tracks a commanded velocity vector and heading in NED frame. Engineering

simulations require a "pilot model" to regulate the helicopter (which may have

unstable dynamics) and keep it on a speciﬁc ﬂight path during maneuvers. The

NLDI controller serves this purpose.

Figure 2.1: Schematic of the PSUHeloSim/CHARM/PSU-WOPWOP simulation

model.

6

2.1.2 Bell 430 Properties

The helicopter used for the current simulation results is a Bell 430, a twin-engine

light-medium helicopter manufactured by Bell Helicopter Textron, Inc which entered

service in 1996. an illustration of it is given in Fig. 2.2 and 2.3 and a summary of

its characteristics is presented in Tables 2.1 and 2.2.

Figure 2.2: An airborne Bell 430.

Figure 2.3: Bell 430 3-View.

7

Description Variable Value Units

Weight W 8700 lbs

Moment of inertial about the

longitudinal body axis

Ixx 3462 slug −f t2

Moment of inertial about the

lateral body axis

Iyy 15362 slug −f t2

Moment of inertial about the

vertical body axis

Izz 12261 slug −f t2

Product of inertial Ixz 300 slug −f t2

Second mass moment Iβ398 slug −f t2

First mass moment Mβ37.9 lbs −f t

Blade mass MB3.61 slug

Table 2.1: Bell 430 mass and inertia properties.

Description Variable Main Rotor Tail Rotor Units

Angular speed Ω36.395 197 rad/s

Radius R 21 3.442 ft

Twist θtw -7.7 0 deg

Chord c 1.2 0.529 ft

Number of blades NB4 2 -

Hinge Oﬀset e 0.05 - -

Delta3 δ30 45 deg

Table 2.2: Bell 430 main and tail rotor key properties.

8

2.1.3 Equations of Motion

2.1.3.1 Fuselage

The basis for the the analysis proposed is the mathematical model of the aircraft

representative of of its unsteady motion. The vehicle will be divided in two major

subsystems: fuselage and main rotor. The fuselage is treated as a rigid body free

to traslate and rotate in space under the actions of gravity, aerodynamic forces and

reaction forces coming from the main rotor. The resulting degrees of freedom are

six. The Earth is considered ﬂat and stationary in the inertial space such that it

constitutes an inertial system itself, thus allowing the use of Newtoian mechanics.

The full derivation of the rigid-body equations of motion can be found in [15].

Following the approach in [16], the non-linear fuselage equations of motion are

divided in force equations 2.1, 2.2, 2.3, moment equations 2.4, 2.5, 2.6; kinematic

equations 2.7, 2.8, 2.9; and navigation equations 2.10, 2.11, 2.12.

˙u=rv −qw −gsin θ+X

m(2.1)

˙v=pv −ru +gsin ψcos θ+Y

m(2.2)

˙w=qu −pv +gcos φcos θ+Z

m(2.3)

˙p=1

Iyy Ixx −Ixz2hqr Iyy Izz −Izz 2−Ixz2+qpIxz (Izz +Ixx −Iyy) +

LIzz +N Ixz ](2.4)

˙q=1

Iyy hrp (Izz −Ixx) + Ixz r2−p2+Mi(2.5)

˙r=1

Iyy Ixx −Ixz2hqrIxz (Iy y −Izz −Ixx) + qp Ixz 2+Ixx2−IxxIyy +

LIxz +NIxx](2.6)

9

˙

φ=p+ tan θ(qsin φ+rcos φ)(2.7)

˙

θ=qcos φ−rsin φ(2.8)

˙

ψ=qsin φ+rcos φ

cos θ(2.9)

˙x=ucos θcos ψ+v(sin φsin θcos ψ−cos φsin ψ) +

w(cos φsin θcos ψ+ sin φsin ψ)(2.10)

˙y=ucos θsin ψ+v(sin φsin θsin ψ+ cos φcos ψ) +

w(cos φsin θsin ψ−sin φcos ψ)(2.11)

˙z=−usin θ+vsin φcos θ+wcos φcos θ(2.12)

where X, Y, Z, L, M, N are the summation of the aerodynamic, and main and tail

rotor forces and moments on and around the longitudinal, lateral and vertical body

axes. Their full derivation can be found in [17].

2.1.3.2 Flapping

PSUHeloSim models ﬂapping up to the ﬁrst harmonic and utilized a hinge oﬀset

model. The second-order dynamical model is solved in Multiple Blade Coordinates

(MBC) and it is given by

β00

M+CM0β0

M+DM0βM=H0(2.13)

where

10

βM=

β0

β1S

β1C

(2.14)

β0

M=1

Ω˙

βM(2.15)

β00

M=1

Ω2¨

βM(2.16)

The damping and stiﬀness matrixes, along with the right-hand side, are given

respectively by

CM0=γ

2

2λ2

β

γ0−k1eµ

k2µ(k3−k2e) + µ2

8+2Ω0

γ

2(λ2

β−1)

γ

02(λ2

β−1)

γ(k3−k2e)−µ2

8+2Ω0

γ

(2.17)

DM0=γ

2

k3−k2e(k2+k1e)µ

20

04

γk3−k2e

(k2+k1e)µ k3−k2e−4

γ

(2.18)

HM0=γ

2

k3+k1

2µ2!θ0+ k4+k2

2µ2!θtw+k2µθ1sw+

k2(µz−λ0) + k2

2µ(ˆphw −λ1sw)

k3+k1

4µ2θ1cw +k3(ˆqhw −λ1cw) + 2kg

γ2ˆphw + ˆq0

hw

2k2µθ0+ 2k3µθtw + k3+3k1

4µ2!θ1sw +k1µ(µz−λ0) +

k3(ˆphw−λ1sw)−2kg

γ2ˆqhw −ˆp0

hw

(2.19)

where

kg=sλ2

β−Kβ

IβΩ2(2.20)

11

k1=1−2e+e2

2(2.21)

k2=2−3e+e3

6(2.22)

k3=3−4e+e4

12 (2.23)

k3=4−5e+e5

24 (2.24)

Again, the full derivation of the ﬂapping dynamics can be found in [17].

2.1.3.3 Inﬂow

The inﬂow model is based on a Pitt-Peters dynamic inﬂlow model [11]. The inﬂow is

driven by aerodynamic normal loading harmonics up to the ﬁrst for this particular

application, as shown by Eq. 2.25

F(1) =F(1)

0+F(1)

1Ccos ψ1+F(1)

1Ssin ψ1(2.25)

It can be demonstrated that

F(1)

0

,

F(1)

1C

and

F(1)

1S

can be linked to respectively the

thrust, roll and pitch moment coeﬃcients as follows

CTa=a0s

2F(1)

0(2.26a)

CLa=−a0s

2

3

γF(1)

1S(2.26b)

CMa=−a0s

2

3

γF(1)

1C(2.26c)

The ﬁrst-order system representative of the inﬂow dynamics is given by

1

ΩM

˙

λ0

˙

λ1s

˙

λ1c

+L−1

λ0

λ1s

λ1c

=

CTa

−CLa

−CMa

(2.27)

12

where

M

is diagonal and it is called the apparent mass matrix,

L

is a gain matrix

such that

L

=

f(µ, µz, λm)

. More details on the apparennt mass and gain matrices

can be found in [11].

2.1.4 Trim Algorithm

For the purpose of this study it is important to start the simulations from an

equilibrium condition. This is dictated by the need of avoiding transients, and thus

dynamics not relevant to the research, which would add unnecessary computational

time. Therefore, a Newton-Rhapson-based [18] trimming algorithm was developed.

The aircraft system of non-linear equations of motion can generally be described by

˙

x=f(x,u)(2.28)

where the state xcan be devided in fuselage and rotor states as follows

xfT=hu v w p q r φ θ ψ x y zi(2.29a)

xrT=hβ0β1Sβ1C˙

β0˙

β1S˙

β1Cλ0λ1Sλ1Ci(2.29b)

and the controls are given by

uT=hδlat δlong δcoll δpedi(2.30)

Given a prescribed funtion in time

xe

the goal is to solve for a subset of

x

and

u

subject to

˙

xe=f(xe,ue)(2.31)

Since there are twenty-ﬁve variables to solve for but just twenty-one constraints,

the values of the

x

,

y

,

z

, and

ψ

are arbitrarily set since the position does not aﬀect

the equilibrium and the aircraft is trimmed with no sideslip angle. This can be be

done by Eq. 2.49, where the projection of the longitudinal axis of the helicopter on

the

xy

plane has the same orientation of the projection of velocity vector on the

same plane.

ψ= arctan ˙y

˙x(2.32)

13

Now let xsbe the subset of states to solve for

xs=hu v w p q r φ θ β0β1Sβ1C˙

β0˙

β1S˙

β1Cλ0λ1Sλ1Ci(2.33)

so that the vector of trim variables is

ν=

xs

u

(2.34)

The trim problem to be solved numerically is therefore given by

f∗(ν) = ˙

xe−f(xs,ue) = 0 (2.35)

˙

xe

is the state derivative target vector of which, for the current applications, the

variables are prescribed as follows

˙ue

˙ve

˙we

˙pe

˙qe

˙re

˙

φe

˙

θe

˙

ψe

˙xe

˙ye

˙ze

˙

xre

=

0

0

0

0

0

0

0

0

0

Vx

Vy

Vz

0

(2.36)

where

˙

xre

is the rotor state derivative target vector and

Vx

,

Vy

and

Vz

are respectively

the forward, lateral and vertical velocities in NED frame.

2.1.5 Trim Results

Fig 2.4 shows the capability of the aircraft trim algorithm at diﬀerent forward

speeds. It is interesting to note how the collective pitch mimics a typical plot of the

14

required power and the fact that the aircraft pitches down as the speed increases.

Also, consistently with the decrease in pitch, the forward cyclic pitch decreases

as well at higher speeds. The heading is not shown since the aircraft is always

trimmed with zero sideslip angle.

2.1.6 Controller Design

2.1.6.1 Inner Loop

The control architecture used is Nonlinear Dynamic Inversion, a popular method

among aircraft controls due its ability of making a dynamical system follow a desired

response. It has been used for a number of rotorcraft applications at Penn State,

including [19]. This section, along with the next two, describes the structure of

such controller. Let the linearized dynamics of the helicopter around an operating

point be described in state-space form by

˙

x=Ax +Bu (2.37a)

y=Cx (2.37b)

where A and B are the eigth-order order system and control matrices describing

the rigid fuselage motion of the aircraft. The state vector is given by the aircraft

rigid body states (excluding the heading, ψ)

xT=hu v w p q r φ θi(2.38)

and the controls by Eq. 2.30. Given a desired reference trajectory

ycmd

(

t

), the

interest lays in controlling the output

y

(

t

)so that it follows the command. In this

particular application the reference trajectory and the output are given respectiely

by Eq. 2.39 and 2.40

ycmd =

φcmd

θcmd

Vzcmd

rcmd

(2.39)

15

(a) Controls

(b) Attitude

Figure 2.4: Trim values of the controls and attitude for diﬀerent forward speed

velocities.

16

y=

φ

θ

Vz

r

(2.40)

where

φ

,

θ

, and

r

,

Vz

are the roll attitude, pitch attitude, roll rate, and vertical

speed (positive up) respectively. The output matrix C that identiﬁes the controlled

states is given by

C=

C1

C2

(2.41)

where

C1=

00000010

00000001

(2.42a)

C2=

0 0 −1 0 0 0 0 0

0 0 0 0 0 1 0 0

(2.42b)

C1

corresponds to the roll and pitch attitudes whereas

C2

is related to the yaw rate

and vertical speed. This partitioning is due to the fact that the output equation

has to be diﬀerentiated two times to see the controls in the output equation while

the same procedure has to be done just once for rand Vzas given in Eq. 2.43.

¨

φ

¨

θ

˙

Vz

˙r

=

C1A2x+C1ABu

C2Ax +C2Bu

(2.43)

Filtering is applied to command a smooth trajectory for the controlled state and its

derivative. Second-order ﬁlters in the form of 2.44 are used for roll and pitch, for

which the ﬁrst time derivatives of roll and pitch rate are needed for feed forward.

First-order ﬁlters are utilized for vertical speed and yaw rate, for which the ﬁrst

time derivatives are needed for feed forward. The general form of the ﬁlters is given

by Eq. 2.44 and 2.45.

17

Command ωnζ τ

Roll Attitude 2.5 0.8 -

Pitch Attitude 2.5 0.8 -

Vertical Speed - - 2.0

Yaw Rate - - 0.4

Table 2.3: Inner loop command ﬁlters properties.

G(s) = ωn2

s2+ 2ζωn+ωn2(2.44)

G(s) = 1

τs + 1 (2.45)

The ﬁlter properties were chosen to meet Level 1 speciﬁcations when refering to

ADS-33E [20] for small amplitude response to pilot inputs on the three body axes.

Table 2.3 reports the values used for the command ﬁlters.

PID/PII/PI controllers are used to reject external disturbances and to compen-

sate for diﬀerencies between the inversion model described in the next section and

the non-linear dynamics. The dynamic inversion control law is thus given by

u=

C1AB

C2A

−1

ν−

C1A2

C2A

x

(2.46)

where

ν

is the pseudo-command vector and

e

is the error as given respectively in

2.47 and 2.48.

νφ

νθ

νVz

νr

=

¨

φcmd

¨

θcmd

˙

Vzcmd

˙rcmd

+

eφ

eθ

eVz

er

KP+

˙eφ

˙eθ

0

0

KD+

Reφdt

Reθdt

ReVzdt

Rerdt

KI+

0

0

RR eVzdt2

0

KII (2.47)

e=ycmd −y;(2.48)

KP

,

KD

,

KI

,

KII

are 4-by-4 diagonal matrices identifying respectively the propor-

tional, derivative, integral, and double-integral gain matrices. The gain values are

18

Command τ

Forward Speed 2.0

Lateral Speed 2.0

Yaw Rate 0.5

Table 2.4: Outer loop command ﬁlters properties.

discussed in more detail in section 2.1.6.3

2.1.6.2 Outer Loop

2.1.6.2.1 Low Speed Mode

In low speed mode, the outer loop controller

tracks forward, lateral, and vertical velocity in the heading frame. The heading

frame is a vehicle carried frame where the x-axis is aligned with the current aircraft

heading, the z-axis is positive up in the inertial frame, and the y-axis is to the right,

forming a left-handed orthogonal coordinate system. Eq. 2.49 shows the rotation

from body to the heading frame

Th/b =

cos θsin φsin θcos φsin θ

0 cos φ−sin φ

sin θ−sin φcos θ−cos φcos θ

(2.49)

so that the velocities in the heading frame are given by

Vx

Vy

Vz

=Th/b

u

v

w

(2.50)

In the outer loop also the commanded lateral and forward velocities

Vx

and

Vy

go

throught ﬁrst-order ﬁlters of the form of Eq. 2.45, of which the time constants are

given in Table 2.4 and chosen to meet the desired rise time properties for TRC in

ADS-33 [20]. The same can be said for the heading

ψ

in low speed mode. The

ﬁltered velocities and heading are then subtracted from the measured values in

order to ﬁnd the error, which goes throught PI controller. Feedforward coming out

from the ﬁlter is added as well, hence leading to the desired commands in Eq. 2.51.

The gains design is discussed in the next section.

19

νVx

νVy

νψ

=

˙

Vxcmd

˙

Vycmd

˙

ψcmd

+

eVx

eVy

eψ

KP+

ReVxdt

ReVydt

Reψdt

KI(2.51)

In low speed ﬂight (i.e. speeds not exceeding 40 kts), the lateral velocity is

controlled by acting on the roll angle

φ

through a linear relationship with the

forward acceleration in heading frame. Starting from the lateral velocity equation

of motion

˙v=−ru +pw +gsin φcos θ+Y

m(2.52)

a number of assumptions can be taken. The ﬁrst step consists in the approximation

of the lateral body speed with the lateral speed in heading frame.

˙v'νVy(2.53)

The second step is to consider both the products of the forward speed and the yaw

rate, and the roll rate and the vertical speed small.

ru '0(2.54)

pw '0(2.55)

The third step is to assume the term given by the lateral body force divided by the

mass of the aircraft, which in fact is the lateral acceleration is small as well.

Y

m'0(2.56)

The last step consists in applying the small angle assumption to the sine and cosine

functions, obtaining:

gsin φcos θ'gφ (2.57)

Eq. 2.52 thus reduces to

φcmd =νVy

g(2.58)

The forward speed is controlled by acting on the pitch attitude

θ

instead. The

20

derivation of the inversion law is similar to the one carried out above. Let’s consider

the equation of motion relative to the forward speed

˙u=rv −qw −gsin θ+X

m(2.59)

By assuming the ﬁrst, second and third terms on the right hand side small, by

applying the small angles approximation, and by assuming the forward speed in

body frame equal to the one in heading frame

rv '0(2.60)

qw '0(2.61)

X

m'0(2.62)

sin θ'θ(2.63)

˙u'νVx(2.64)

Eq. 2.59 reduces to

θcmd =−νVx

g(2.65)

The yaw rate

r

command is simply given by

νψ

, which is obtained by extraction

from the ﬁrst order ﬁlter the yaw command goes through.

rcmd =νψ(2.66)

2.1.6.2.2 High Speed Mode

In high speed ﬂight (i.e. speeds exceeding 60

kts), because of the possibility of high speed turns the term containing the yaw rate

r

in Eq. 2.52 cannot be neglected anymore. In a coordinated turn we no longer

control

Vy

since it is assumed

Vy

= 0 in order to have the heading coinciding with

the longitudinal speed in heading frame. We can then proceed to the following

simpliﬁcations:

˙v'0(2.67)

21

Y

m'0(2.68)

pw '0(2.69)

The forward speed in body axis

u

can be approximated with the absolute velocity

V, leading to an expression for the yaw rate commanded to the inner loop

rcmd =gsin φcosθ

V(2.70)

The forward speed is controlled the same way as in the low speed ﬂight case,

whereas the inversion law for the roll angle is derived by combining the pitch and

yaw rate equations of motion. Starting from the pitch rate equation

˙

θ=qcos φ−rsin φ(2.71)

the pitch rate can be neglected

˙

θ'0(2.72)

leading to

q=rtan φ(2.73)

Let’s now consider the yaw rate equation

νψ=(qsin φ+rcos θ)

cos θ(2.74)

substituting 2.73 in it leads to

νψ=r(sin2φ+ cos2θ)

cos θcos φ

=r

cos θcos φ

(2.75)

Substituting now Eq. 2.70 into 2.75 and carrying out the necessary simpliﬁcations

we obtain

22

νψ=g

Vtan φ(2.76)

which rearranged gives the equation for bank angle necessary for a coordinated

turn

φcmd = arctan V

gνψ!(2.77)

2.1.6.2.3 Blending

For cases in which the absolute velocity lays between 40

and 60 kts (i.e.

Vl

and

Vh

), a mixed approach is applied: the commanded roll

attitude and yaw rate are given by weighting the low and high speed approach with

respect to the absolute velocity. Speciﬁcally

φcmd =V−Vl

Vh−Vl

(φcmdh−φcmdl) + φcmdl(2.78)

rcmd =V−Vl

Vh−Vl

(rcmdh−rcmdl) + rcmdl(2.79)

where the subscripts hand lidentify respectively the high and low speed cases.

Equations 2.80 and 2.81 provide a summary of the commanded roll attitude

and yaw rate

φcmd =

˙

Vycmd

gV < Vl

V−Vl

Vh−Vl(φcmdh−φcmdl) + φcmdlVl< V < Vh

arctan V

gνψV > Vh

(2.80)

rcmd =

νψV < Vl

V−Vl

Vh−Vl(rcmdh−rcmdl) + rcmdlVl< V < Vh

gsin φcosθ

VV > Vh

(2.81)

2.1.6.3 Error Dynamics

Feedback compensation is needed both in the inner and the outer loop because of

two main reasons. Firstly, we use approximations and the inversion is not exact.

Secondly, there are external disturbances to the system. Similarly as in [16] it can

23

be demonstrated that for a Dynamic Inversion controller

e(n)=ν−y(n)

cmd (2.82)

where

n

is the number of times the output equation has to be derived in order for

the controls to appear explicitely in the output equation. For the states for which

this has to be done twice, a PID control strategy applied to the pseudo-command

vector is given by

ν= ¨ycmd(t) + KD˙e(t) + KPe(t) + KIZt

0e(τ)dτ (2.83)

Substituting 2.83 into 2.82, we obtain the closed-loop error dynamics

¨e(t) + KD˙e(t) + KPe(t) + KIZt

0e(τ)dτ = 0 (2.84)

The gains can be chosen so that the frequencies of the error dynamics are of the

same order as the command ﬁlters, ensuring that the bandwidth of the response to

disturbancies is comparable to the one of an input given by a pilot. By taking the

Laplace transform and therefore switching to frequency domain the error dynamics

becomes

e(s)s2+KDs+sKP+1

sKI= 0 (2.85)

or equivalently

e(s)s3+KDs2+KP+KI= 0 (2.86)

In order to obtain gains that would guarantee a desired response, the error dynamics

can be set equal to the third-order system given in Eq. 2.87.

(s2+ 2ζωns+ωn2)(s+p)=0 (2.87)

Developing the product between the polynomials leads to

s3+ (p+ 2ζωn)s2+ (2ζωnp+ωn2)s+ωn2p= 0 (2.88)

Let’s now set the coeﬃcients of the polynimial equal to the the gains of Eq. 2.86

24

KD= 2ζωn+p(2.89a)

KP= 2ζωnp+ωn2(2.89b)

KI=ωn2p(2.89c)

Speciﬁcally this approach is used for φand θin the inner loop.

Similarly, for those stases for which a PI/PII compensation strategy is applied,

the pseudo-command vector is given by

ν= ˙ycmd (t) + KPe(t) + KIZt

0e(τ)dτ +KII Zt

0Zt

0e(τ)dτ (2.90)

which leads to the following closed-loop error dynamics

˙e(t) + KPe(t) + KIZt

0e(τ)dτ +KII Zt

0Zt

0e(τ)dτ = 0 (2.91)

and, therefore, to

s+KP+1

sKI+1

s2KII = 0 (2.92)

In case of a PII controller the closed-loop error dynamics is set equal to a third-order

system as before. The resulting gains are

KP= 2ζωn+p(2.93a)

KI= 2ζωnp+ωn2(2.93b)

KII =ωn2p(2.93c)

This particular type of compensation is applied to the vertical velocity

Vz

in the

inner loop. In case of a PI controller,

p

and

kII

are set to zero in order to have

second-order error dynamics. The resulting gains are

KP= 2ζωn(2.94a)

KI=ωn2(2.94b)

25

ωnζp

Roll Attitude 2.0 1.0 0.75

Pitch Attitude 2.0 1.0 0.75

Vertical Speed 2.5 1.0 0.1

Yaw Rate 2.5 1.0 -

Table 2.5: Inner loop disturbance rejection frequencies, damping ratios, and inte-

grator poles.

ωnζ

Forward Speed 0.5 1.0

Lateral Speed 0.5 1.0

Heading 0.2236 0.559

Table 2.6: Outer loop disturbance rejection frequencies and damping ratios.

This type of compensation is applied to

Vx

,

Vy

, and

ψ

in the outer loop and

r

in the

inner loop. Table 2.5 and 2.6 show the natural frequencies, damping ratios, time

constants, and the integrator pole values, respectively, fot the inner and the outer

loop. Note that the integrator pole

p

is usually chosen to be one-ﬁfth of the natural

frequency and that outer loop error dynamics needs to be at a lower frequency than

the equivalent inner loop dynamics (e.g. for the longitudinal velocity

ωn

= 0

.

5,

which is less than ωn= 2.0of the pitch attitude error dynamics)

The numerical value of the resulting gains are shown in Table 2.7 and Table 2.8.

KDKPKIKII

Roll Attitude 4.75 7.0 3.0 -

Pitch Attitude 4.75 7.0 3.0 -

Vertical Speed - 2.1 1.2 0.1

Yaw rate - 7.0 3.0 -

Table 2.7: Inner loop compensation gains.

26

KPKI

Forward Speed 1.0 0.5

Lateral Speed 1.0 0.5

Heading 0.25 0.05

Table 2.8: Outer loop compensation gains.

2.2 High Fidelity Rotor Module

2.2.1 Introduction

PSUHeloSim is integrated with a high-ﬁdelity rotor module for fully-coupled or

one-way-coupled simulations. The CHARM Rotor Module uses a Constant Vorticity

Contour (CVC) full-span free-vortex wake model, combined with a vortex lattice,

lifting surface blade model [21]. The module calculates blade motion including

structural modes in the blade dynamics. This module runs as a separate code

obtaining the state, state derivatives and controls from PSUHeloSim at each time

step of the simulation and returning the forces, moments, and ﬂapping coeﬃcients

of the rotor systems. In the one-way coupled mode, the blade loads are stored for

use in acoustic prediction, but are not used by the PSUHeloSim ﬂight dynamics

model. In the fully-coupled mode, the forces and moments calculated by CHARM

are used as inputs for the PSUHeloSim code. Thus in the fully coupled mode,

CHARM acts as the main rotor module and/or tail rotor module of the simulation

(it replaces the simple built-in rotor models in PSUHeloSim). In either mode,

CHARM is able to produce loading ﬁles that are then used by PSU-WOPWOP to

determine the aerodynamically induced noise. One can choose to couple the main

rotor, the tail rotor or both. The acoustic prediction is able to operate with more

than one rotor at a time. The only present limitation is that the loading output for

acoustic prediction is limited to a single rotor revolution which is assumed to be

periodic. This means that acoustic analysis can be performed just for steady or

quasi-steady ﬂight conditions.

27

2.2.2 Reconstruction

When simulating main rotor and tail rotor physics with the CHARM rotor module,

the simulation time step is driven by the largest allowable tail rotor blade sweep per

time step (since the tail rotor has a larger RPM). For example, 15

◦

blade sweep per

time step is usually considered the largest acceptable time step for blade element

rotor simulations in ﬂight dynamics. Consequently the main rotor (which turns

slower) will have a smaller blade sweep. One of the unique features of this system

is the ability to capture relevant physics for the acoustics. In particular, the free

wake needs a suﬃcient number of elements for accurate blade loading, which in BVI

conditions should be as ﬁne as 1

◦

azimuthal resolution. With such high temporal

(azimuthal) and corresponding spatial resolution requirements, it is impossible to

perform real-time analysis with the free wake and could take substantial computa-

tional power to be useful in the computation of realistic maneuvers

−

tens of hours

on a single processor. Fortunately, this issue has been addressed in the CHARM

rotor module through "reconstruction" of the rotor wake in post-processing. In this

approach, a low resolution wake and larger time step is used in the ﬂight simulation

step, which is acceptable for ﬂight dynamics modeling. Then, for the regions of

the maneuver where acoustics are of interest, a higher resolution wake and blade

loading is reconstructed in the CHARM rotor module [22]. These high resolution

blade loads are then used by PSU-WOPWOP to predict BVI-dominated noise. In

recent work, this method was used to perform real-time, BVI-noise predictions [23].

Currently reconstruction can only be applied to one rotor.

2.3 Noise Prediction Model

The noise prediction model used in this work, PSU-WOPWOP [4] [5] [6], is a

numerical implementation of Farassat’s Formulation 1A [21] of the Ffowcs Williams-

Hawkings (FW-H) equation [24]. Formulation 1A is used to predict the discrete

frequency noise prediction (thickness, loading, BVI, etc.) from ﬁrst principles

when provided with the aircraft and rotor blades position, motions, and blade

loading. PSU-WOPWOP predicts the acoustic pressure time history for either

stationary or moving observers and the code is also able to convert the output

signals into acoustic spectra, such as 1/3rd octave bands and multiple types of noise

28

metrics relevant to noise certiﬁcation and community annoyance (PNL, PNLT, SEL,

EPNL, and OASPL, etc.). The broadband noise is computed in PSU-WOPWOP

by implementing an empirical prediction developed by Pegg [25] that predicts the

broadband noise in 1/3rd octave bands. This is then combined with the discrete

frequency noise for a total noise prediction. In a recent research eﬀort, it has been

demonstrated that a ﬂight simulation coupled with CHARM and PSU-WOPWOP

can predict the noise in "real time". The system developed in this work is somewhat

diﬀerent, but it is still reasonably fast.

2.4 Schematic of the Simulation Process

The simulation process consists of three main steps: 1) solving trim for the prescribed

ﬂight condition, 2) running a PSUHeloSim /CHARM coupled simulation, and 3)

performing an acoustic prediction with PSU-WOPWOP based on the results of

the simulation. A Newton-Raphson based trimming algorithm is used to ﬁnd an

equilibrium condition for the state and the controls. Note that this trim solution

is based only on the base PSUHeloSim model. Once trim is achieved, the trim

state and control solution is used as initial conditions of the coupled simulation

(both for PSUHeloSim and the CHARM rotor modules). During the simulation,

the time history of velocity and heading commands are fed to the dynamic inverse

controller in the PSUHeloSim code. The controller calculates the control input

based on the tracking error and feedforward signals as deﬁned by the control law.

The sim code updates the state, state derivatives, and swashplate inputs, which

are then used as inputs for the CHARM rotor module. The resulting main rotor

forces and moments calculated by CHARM are either saved as output (in one

way coupled mode) or fed back into the simulation model in fully-coupled mode.

When performing fully-coupled simulations, the full coupling is not initiated until

three seconds of simulation have passed. This allows the free-wake model time

to develop and initialize. After the simulation is completed, the PSU-WOPWOP

acoustics analysis is performed using the aircraft state and loading ﬁles generated

by CHARM. Fig. 2.5 shows the ﬂowchart of the simulation process.

29

Figure 2.5: Flowchart of the simulation process.

30

Chapter 3 |

Linear Model Analysis

3.1 Linearization

A general non-linear time-invariant system is described by

˙

x=f(x,u)(3.1a)

y=g(x,u)(3.1b)

where

x

is the state variable vector of dimension

n

,

u

the input vector of dimension

m

and

y

is the output vector of dimension

p

. A non-linear system in an equilibrium

condition is described as follows

˙

xe=f(xe,ue)(3.2a)

ye=g(xe,ue)(3.2b)

where

xe

and

ue

are the equilibrium state and input vector

˙

xe

such tat

˙

xe

is constant.

Let’s now consider the case of small disturbances on the state, controls and output

x=xe+∆x (3.3a)

u=ue+∆u (3.3b)

y=ye+∆y (3.3c)

31

A Taylor series expansion can now be performed on the state vector time derivative

f(xe+∆x,ue+∆u) = f(xe,ue) + ∂f(x,u)

∂xxe,ue

∆x +∂f(x,u)

∂uxe,ue

∆u

+O∆x2,∆u2(3.4)

By reorganizing and neglecting the terms of second order and higher we obtain

∆ ˙x =∂f(x,u)

∂xxe,ue

∆x +∂f(x,u)

∂uxe,ue

∆u (3.5)

where

∂f(x,u)

∂xxe,ue

=

∂f1(x,u)

∂x1·· · ∂ f1(x,u)

∂xn

.

.

.....

.

.

∂fn(x,u)

∂x1·· · ∂ fn(x,u)

∂xn

xe,ue

=A

(3.6)

∂f(x,u)

∂uxe,ue

=

∂f1(x,u)

∂u1·· · ∂ f1(x,u)

∂um

.

.

.....

.

.

∂fn(x,u)

∂u1·· · ∂ fn(x,u)

∂um

xe,ue

=B

(3.7)

A Taylor series expansion can be performed also on the output

g(xe+∆x,ue+∆u) = g(xe,ue) + ∂g (x,u)

∂xxe,ue

∆x +∂g(x,u)

∂uxe,ue

∆u

+O∆x2,∆u2(3.8)

By reorganizing and neglecting the terms of second order and higher we obtain

∆y =∂g(x,u)

∂xxe,ue

∆x +∂g(x,u)

∂uxe,ue

∆u (3.9)

32

where

∂g(x,u)

∂xxe,ue

=

∂g1(xe,ue)

∂x1·· · ∂ g1(xe,ue)

∂xn

.

.

.....

.

.

∂gp(xe,ue)

∂x1·· · ∂ gp(xe,ue)

∂xn

xe,ue

=C

(3.10)

∂g(x,u)

∂uxe,ue

=

∂g1(xe,ue)

∂u1·· · ∂ g1(xe,ue)

∂um

.

.

.....

.

.

∂gp(xe,ue)

∂u1·· · ∂ gp(xe,ue)

∂um

xe,ue

=D

(3.11)

The linearized system can therefore be written as follows

∆ ˙x =A∆x +B∆u (3.12a)

∆y =C∆x +D∆u (3.12b)

3.2 Linear Analysis Overview

Referring to Eq. 3.12, which are the perturbations of states, controls and outputs,

and eﬀectively our new states, control and output (but keeping in mind that they

still are perturbations) we can write the linearized system as

˙

x=Ax +Bu (3.13a)

y=Cx +Du (3.13b)

If the matrices

A

,

B

,

C

,

D

are constant with time, we have a linear time invariant

system (LTI) expressed in the so-called state space form. Being

x

=

x(t)

,

u

=

u(t)

and

y

=

y(t)

a closed form solution to a perturbation from initial conditions (i.e.

equilibrium) is given by

33

x(t) = eAtx0+Zt

t0

eA(t−τ)Bu(τ)dτ (3.14)

where

eAt=I+At+A2t2

2! +A3t3

3! +··· (3.15)

In order to understand the time response properties, which are going to be given

by the state matrix, the eigenvalue problem has to be introduced. Let’s consider

the equation

Avi=λivi, i = 1, . . . , n (3.16)

where the vector

v

is called eigenvector and the scalar

λ

is the eigenvalue. It be

noted that the eigenvector can either be a line or row vector of dimension

n

; in the

ﬁrst case it will be called "left eigenvector" whereas in the latter "right eigenvector".

The two cases are denoted respectively by

v

and

w

. It can be demonstrated that if

there are

n

distinct eigenvalues of

A

, where

A

a square matrix of dimension

n

, the

following euqation holds true

eAtx0=

n

X

i=1

wix0eλitvi(3.17)

It follows that the response to non-zero initial conditions, meaning

x0

(0)

6

= 0 and

u(t)=0, will be given by

x(t) =

n

X

i=1

wix0eλitvi(3.18)

Let’s now analyze the case of a response to zero initial condition but non-zero input,

meaning x0(0) = 0 and u(t)6= 0. The response is given by

x(t) =

n

X

i=1

viZt

t0

eλi(t−τ)wiBu(τ)dτ (3.19)

The eigenvectors and eigenvalues can be both real-valued or complex, the ﬁrst

leading to subsidence or divergence mode, the latter to oscilatory modes which can

either be stable or unstable. The convergence properties are dictated by the sign of

the real part of the eigenvalue: if negative the response is stable, if positive the

response in unstable. This can easily be understood by noticing that the responce is

34

driven by a summation of exponentials with the exponent being the the eigenvalue

associated with the respective mode times the time. Since the response of the linear

system is given by a summation of the responses of each mode, all of them have to

be stable to guarantee stability. This leads to the conclusion that the real parts of

the eigenvalues of a system, for it to be stable, have to be all strictly negative.

In case of complex eigenvalues, they can be expressed in the following form

λi=σ+iω (3.20)

where the eigenvalue’s complex conjugate is given by

¯

λi=σ−iω (3.21)

The response to non-zero initial conditions can therefore be written as follows

x(t) =

n

X

i=1 wix0e¯

λitvi+

n

X

i=1

¯

wix0eλit¯

vi!(3.22)

or equevalently, by using complex number properties

x(t) =

n

X

i=1

Aeσitcos (ωit+φi(x0)) (3.23)

The real and imaginary parts of the eigenvalues,

σi

and

ωi

, are also known as

damping and frequency of oscillation of their respective modes. Some useful

parameters known as period of oscillation, time to half/double amplitude, natural

frequency and damping ratio are deﬁned as

T=2π

ω(3.24)

t1/2=ln 2

|σ|(3.25)

ωn=√ω2+σ2(3.26)

ζ=−σ

ωn

(3.27)

where

35

|ζ|<1(3.28)

3.3 Reduced-Order Models

Because of the diﬃculty of measuring the states associated with ﬂapping and inﬂow

dynamics, and the fact that it may be cumbersome to perform a good estimation

on them, this research proposes a reduced order model approach for feedback

control design. The problem is approached by the use of two methods: truncation

and residualization. Truncation consists in the removal of the rows and columns

associated with the states that are either de-coupled or that are assumed to be

constant. In our model this is speciﬁcally done for the states associated with

position

x, y, z

and heading

ψ

, which are de-coupled from the other states. The

state vector therefore reduces to:

xT=hu v w p q r φ θ β0β1Sβ1C˙

β0˙

β1S˙

β1Cλ0λ1Sλ1Ci.(3.29)

Residualization is based on singular perturbation theory [26] and accurately models

low frequency and steady state but neglects high frequency. It assumes that some

modes are "fast" in the means that they quickly reach steady state

˙

x

= 0, which

can readily be applied to the rotor inﬂow and ﬂapping states. Fig. 3.1 shows the

eigenvalues for the Bell 430 model in a 100 kts level ﬂight. It is quite evident how

the modes associated with the rotor states are at higher frequency then the ones

associated with the fuselage dynamics. Fig. 3.2 shows the eigenvalues associated

with the fuselage dynamics more into detail.

In light of the frequency division between rotor and fuselage eigenvalues, the

state vector is divided into fast and slow components:

x=

xs

xf

(3.30)

where

xsT=hu v w p q r φ θi(3.31)

xfT=hβ0β1Cβ1S˙

β0˙

β1S˙

β1Cλ0λ1Sλ1Ci(3.32)

36

Figure 3.1: Eigenvalues relative to the Bell 430 model in a 100 kts level ﬂight.

The dynamical system can then be re-written in the following form

˙

xs

˙

xf

=

AsAsf

Afs Af

xs

xf

+

Bs

Bf

u(3.33)

By assuming that the fast states reach steady state quickly, the algebraic constraint

˙

xf= 0 can be imposed. It follows that

Afsxs+Af sxf+Bfu= 0 (3.34)

Solving for the fast states leads to

xs=Af

−1(−Afsxs−Bfu)(3.35)

By substituting the latter result into 3.33, a new expression for the slow states can

be found:

˙

xs=ˆ

Axs+ˆ

Bu (3.36)

where

37

Figure 3.2: Eigenvalues associated with the fuselage dynamics.

ˆ

A=As−Asf Af

−1Afs (3.37a)

ˆ

B=Bs−BsAf

−1Bf(3.37b)

The resulting model is of eigth order and its eigenvalues closely resemble the ones

relative to the fuselage dynamics of the full-order model, as shown in Fig. 3.3.

The frequency response of some selected combinations of outputs and inputs

can give good insight into the accuracy of the reduced order mode. By looking

at the Bode plots in Fig. 3.4 we can see that the reduced order model is a good

approximation for dynamics with frequencies of less than 5 rad/s. This can be

appreciated both in the frequency response of pitch attitude to a longitudinal cyclic

input, and of roll attitude to lateral cyclic. Also the Bode plots relative to vertical

speed response due to collective perturbations indicate discrepancies between the

full and reduced-order model at high frequencies. The response of yaw rate to the

pedal seems to be accurate at all frequencies in consideration. This is because the

main rotor states have small inﬂuence on yaw dynamics, which are dominated by

the tail rotor and the airframe aerodynamic loads.

38

Figure 3.3: Comparison between the eigenvalues of the full-order model and the

eighth-order model.

39

(a) Frequency response of the pitch attitude with respect to a

logitudinal cyclic input.

(b) Frequency response of the vertical body velocity with respect

to a collective input.

40

(c) Frequency response of the roll attitude with respect to a lateral

cyclic input.

(d) Frequency response of the yaw rate with respect to a tail rotor

collective input.

Figure 3.4: Frequency responses of aicraft states with respect to the control inputs

for both the eigth-order and full-order models.

41

3.4 Linearization of the coupled PSU-HeloSim/CHARM

Simulation

In order to derive a reduced order model that better reﬂects the aircraft dynamics

it has been performed a linearization of the fully-coupled simulation. The process

consists of ﬁrst ﬁnding a trim, after the coupling transient, for the coupled simulation

in a speciﬁed ﬂight condition; subsequently another fully-coupled simulation is

run starting from that trim. The second simulation is run by not integrating

the fuselage states in the so called "freeze mode" and with the dynamic inversion

controller disabled. This is done to keep the helicopter model still while scheduled

disturbances, in the form of doublets, are given to the fuselage states and controls.

The response of these disturbances on the fusalage state derivative is measured to

compute a state space form of the helicopter dynamics which serves as a reduced

order model. Given that the system is stable, the rotor state derivatives reach

steady-state in a short time

lim

t→∞ ˙

xf= 0 (3.38)

where

˙

xf=f(xse,xf,u)(3.39)

Consequently, also the fuselage state derivatives will reach a steady-state

˙

xs=f(xse,xfss ,u)(3.40)

This way the linear model obtained accounts for main and tail rotor eﬀects and is

conceptually the same as the reduced order models obtained by proper residualiza-

tion in the previous section. As shown in Fig. 3.5, the disturbances are given in

the form of doublets of two seconds of perdiod and of 1

◦

and 1

ft/s

respectively

for the angular variables and the speeds. The wake is initialized for 10 seconds

before starting the linearization process so that the linearization takes 34 seconds

of simulation time. Since the response of the fuselage state derivative is oscillatory,

it is averaged over a set period of time.

42

(a) Body velocities derivatives

(b) Angular rates derivatives

43

(c) Euler Angles derivatives

Figure 3.5: Response of the derivatives of the states to perturbations of the states

and controls.

44

Fig. 3.6 shows a comparison between the eigenvalues of the reduced order model

obtained from PSUHeloSim and the one derived from the PSUHeloSim/CHARM

coupled simulation for a 120 kts level ﬂight condition. It is evident that whereas

the eigenvalues representative of the phugoid, short period, and dutch roll modes

are similar between the two models, the roll subsidence is substantially diﬀerent.

Speciﬁcally the frequencies associated with the mode are about 6 rad/s apart, the

one obtained by linearization of the coupled simulation being the lowest one.

Figure 3.6: Comparison between the eigenvalues of the reduced order models

obtained from PSUHeloSim and the coupled simulation for a 120 kts level ﬂight.

To have a better understanding of this behavior, let’s now consider Eq. 3.41

which, according to [27], analytically describes the eight-order linear dynamics of a

helicopter. Note that the trim values of roll, pitch and yaw have been ommitted

since they are set to zero in trim.

45

XuXvXwXpXq−weXr+ve

YuYvYwYp+weYqYr−ue

ZuZvZwZp−veZq+ueZr

L0

uL0

vL0

wL0

pL0

qL0

r

MuMvMwMpMqMr

N0

uN0

vN0

wN0

pN0

qN0

r

0 0 0 1 sin φetan θecos φetan θe

0 0 0 0 cos θe−sin θe

0−gcos θe

gcos φecos θe−gsin φesin θe

−gsin φecos θe−gcos φesin θe

0 0

0 0

0 0

0 Ω sec θe

−Ω cos θe0

(3.41)

Eq. 3.42 and 3.43 report the system matrix A of respectively the reduced-order

model obtained with PSUHeloSim and the one obtained by linearization of the

PSUHeloSim/CHARM Rotor Module simulation.

−0.0360 −0.0017 0.0340 −0.8643 7.7170 0.2820 0.0000 −32.150

0.0031 −0.1274 −0.0108 −7.9939 −0.6895 −202.70 32.120 −0.0401

0.0072 −0.0159 −0.8567 −0.1813 200.05 0.0015 1.3000 0.9915

−0.0009 −0.0530 0.0242 −5.9040 −0.8745 0.4859 0.0000 0.0000

0.0037 0.0009 −0.0137 0.1403 −1.0230 −0.0029 0.0000 0.0000

−0.0023 0.0258 0.0002 −0.1549 0.1163 −0.6701 0.0000 0.0000

0.0000 0.0000 0.0000 1.0000 0.0012 −0.0308 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.9991 0.0405 0.0000 0.0000

(3.42)

46

−0.0583 0.0108 0.0080 −1.0950 8.3480 3.5300 0.0189 −32.160

0.0094 −0.1270 −0.0301 −7.4700 −1.3710 −201.40 32.3300 −0.2319

0.0035 −0.0127 −0.9447 −5.0680 200.00 2.2560 1.0850 0.7751

0.0041 −0.0342 0.0209 −1.8540 −0.9089 −0.0072 0.0814 −0.0878

0.0074 −0.0041 −0.0109 0.1237 −0.8105 0.0351 −0.0258 −0.0074

−0.0073 0.0284 0.0166 −0.3352 0.3839 −0.7143 −0.0809 0.0967

0.0000 0.0000 0.0000 1.0000 0.0015 −0.0345 0.0000 −0.0005

0.0000 0.0000 0.0000 0.0000 0.9990 0.0430 0.0005 0.0000

(3.43)

It is evident that the stability derivative

Lp

,

A4,4

, is substantially smaller in the

model derived by linearization of the coupled simulation, which would explain the

lower frequency of the roll subsidence mode, given the approximation

λRS 'Lp(3.44)

There is no clear evidence that links this behavior to one particular physical

phenomena going on in the coupled simulation. CHARM Rotor Module is a much

more complex model then the simple rotor model adopted in PSUHeloSim and a

multitude of things could be happening due to the more complex aerodynamics

and ﬂexible blade dynamics. However, this is an indication that the linearization of

the coupled simulation is a viable way to a better design of feedback control since

it captures some of the highly nonlinear behaviors not modelled by a simple rotor

model.

The controller is scheduled on the forward velocity and it is also assumed to

be working properly also in conditions where other components of the velocity

might be signiﬁcant. It is thereforw worthwile to consider a condition of steady

descending ﬂight and analize if and how the controller design could beneﬁt from

the scheduling of the model with respect to both the horizontal and vertical speed

instead of the absolute velocity alone. By ﬁrst taking a look at Fig. 3.7, which

shows the eigenvalues of the reduced-order models of the Bell 430 in level and 6

◦

descending ﬂight both at 120 kts of absolute airspeed, no substantial diﬀerences

are evident.

However, by comparing the eigenvalues of the reduced order in forward ﬂight

47

Figure 3.7: Comparison between the eigenvalues of the reduced order models

obtained from PSUHeloSim for the level ﬂight and descent cases.

and the ones of a 6

◦

descending ﬂight obtained by linearization of the coupled

simulation, as shown if Fig. 3.8, it is clear that the latest are diﬀerent in the means

of roll subsidence and dutch roll modes.

While the diﬀerence in roll subsidence was present already in the level ﬂight

case, the one concerning the dutch roll mode seem to be exclusive of the descent

case. To shed some light on this behavior, let’s take a look to the eigth-order

A-matrix derived by linearization of the coupled simulation in descent ﬂight.

48

Figure 3.8: Comparison between the eigenvalues of the reduced order models

obtained from PSUHeloSim in the level ﬂight case, and PSUHeloSim/CHARM

coupled simulation in a 6◦descent cases.

−0.0306 0.0073 −0.0349 −0.8813 −10.380 3.7580 0.0631 −32.060

−0.0056 −0.1882 0.0378 12.360 −1.3060 −203.50 31.169 −0.3928

0.0279 0.0388 −0.9518 −6.1350 202.10 3.0520 0.7928 1.9170

−0.0039 −0.0342 0.0380 −1.991 −0.8091 −0.0031 −0.0764 −0.0803

0.0014 0.0010 −0.0061 0.1261 −0.8409 0.0209 −0.0028 0.0089

0.0045 0.0559 −0.0226 −0.7685 0.5541 −0.2820 0.2178 0.1737

0.0000 0.0000 0.0000 1.0000 0.0009 −0.0625 0.0000 −0.0006

0.0000 0.0000 0.0000 0.0000 0.9998 0.0145 0.0006 0.0000

(3.45)

Let’s also consider an approximation, found in [27] and derived from 3.41, of the

frequency of the dutch roll mode

ωDR2'ueNv+σDR

1−σDRLr

Lp−ue

(3.46)

49

where

σDR =g−Npue

Lp

(3.47)

Even though being a small term in magnitude (

O

(10

−2

)),

Nv

(

A6,2

) is multiplied

by the forward trim velocity, which is of order

O

(10

2

); this means that even small

changes would change the nature of the frequency. In fact in the linear model

obtained from the coupled simulation a greater

Nv

, along with

Lp

and

Np

, which

are present in the damping term, drive the frequency to a higher value with respect

to the model obtained with PSUHeloSim. Let’s now take a look at the damping

ratio

ζDR ' −Nr+Yv+σDR Lr

Ue−Lv

Lp

2ωDR 1−σDR Lr

LpUe(3.48)

It is more tricky here, when compared to the natural frequency, to understand

which derivatives actually contribute to the decrease in damping ratio.

Lp

as well as

Lr

are less negative (smaller in magnitude) thus contributing to a greater numerator

and a smaller term in the parenthesis in the denominator. The denominator though

also contains the frequency, which is greater for the for the model obtained by

lineariation of the coupled sim, and has the biggest eﬀect in driving the damping

ratio to a smaller value for the case previously mentioned. The natural frequency

is ﬁnally given by

ωnDR =ωDR

q1−ζ2

DR

(3.49)

which for a higher frequency and higher damping ration will increase. A summary

of the numerical values of the dutch roll frequency and damping ratio, and of the

roll subsidence eigenvalue is given in Table 3.1

For the sake of clarity, the eigenvalues of the reduced-order models both in level

and descending ﬂight, and derived from PSUHeloSim and by linearization of the

coupled simulation, are shown together in Fig. 3.9

50

λsωnDR ζDR

PSU HeloSim Level Flight −5.904 1.967 0.266

Coupled Sim Level Flight −1.854 2.102 0.254

Coupled Sim 6 deg Descent −1.991 3.717 0.182

Table 3.1: Numerical values of the dutch roll frequency and damping ratio, and of

the roll subsidence eigenvalue for a 120 kts ﬂight condition.

Figure 3.9: Comparison between the eigenvalues of the reduced order models

obtained from PSUHeloSim and PSUHeloSim/CHARM coupled simulation for both

level ﬂight and the descent case.

51

3.5 Frequency Sweep

System identiﬁcation methods can be used to experimentally derive frequency

resposes by injecting iputs at frequencies of interest and observing the outputs. In

order to validate the ﬁndings of the previous sections a frequency sweep of the

coupled simulation is performed on the lateral cyclic, thus on lateral axis, for both

the level ﬂight and descent cases. The results are then compared to the frequency

response of the reduced-order models obtained by linearization of the coupled

simulation. The procedure adopted follows the one developed by [28], where an

exponentially increasing frequency is used. The sweep function, as shown in Fig.

3.10, is given by

δsweep =

0 0 < t < ts

A(t−ts)

tisin (ωmin(t−ts)) ts< t < (ts+ti)

Rt

ts+tiω(τ)dτ (ts+ti)< t < (ts+ti+T)

0t > (ts+ti+T)

(3.50)

where tis the time of the simulation and

ω(τ) = ωmin +C2eC1τ

T−1(ωmax −ωmin)(3.51)

The description and the numerical values of the various parameters used in Eq.

3.50 are given in Table 3.2.

52

Parameter Description Value Units

ωmin Lowet frequency 0.3rad/s

ωmax Highest Frequency 13 rad/s

ts

Time at which the

sweep starts

40 s

ti

Period of the lowest

frequency

2π

ωmin s

T

Total time of the

sweep

90 s

C1Constant 4

C2Constant 0.0187

AAmplitude 5%

Table 3.2: Description of the parameters used for the frequency sweep.

53

White noise, processed through a low-pass ﬁlter that abaits frequencies higher

than the maximum one used for the sweep, is added to the sweep signal in order to

enrich the spectral content of the excitation.

δexcitation =δsweep +δwhite noise (3.52)

where

δwhite noise :σ= 0.05A(3.53)

The excitation, which can be seen in Fig. 3.11, is then added to the lateral cyclic

input as follows

θ1Cexcitation =θ1C+δexcitation (3.54)

Fig. 3.12 shows the controls, body velocities, angular rates and attitude plots of a

frequency sweep of the coupled simulation for a 120 kts level ﬂight case. Of interest

the fact that towards the high frequencies a resonance is triggered.

Figure 3.10: Sweep input.

54

Figure 3.11: Sweep excitation.

55

(a) Controls

(b) Body velocities

56

(c) Angular Rates

(d) Attitude

Figure 3.12: Frequency sweep of the coupled simulation for a 120 kts level ﬂight

case.

57

The data from the frequency sweep is the processed in CIFER

®

[29] in order

to obtain the frequency respose plots so that they can be compared to the one

obtained from the reduced-order model derived from the linearization of the coupled

simulation. Fig. 3.13 shows the on-axis Bode and coherence factor plots of the

lateral sweep for a 120 kts level ﬂight case. Note that the correlation factor at

low and high frequencies is below 0

.

6, which is the standard value that indicates

wether the data is acceptable or not. The low value of the coherence factor for

frequencies ranging from 6 to 10

rad/s

is to be attributed to the fact that modes

associated with rotor dynamics come into play. However, for those frequencies for

which the correlation factor exceeds 0.6 the magnitude Bode plot obtained from the

frequency sweep follows more closely the one of the eigth-order model derived by

linearizing the coupled simulation rather than the one derived from PSUHeloSim.

This would conﬁrm the analysis of the previous sections. The phase plot from the

frequency sweep, even though being again more similar to the one of the eigth-order

model derived by linearizing the coupled simulation at low frequencies, does not

ﬁnd a match for the high frequencies, where the phase has a signiﬁcant drop.

This is expected since the reduced-order models, even the ones not obtained by

linearization of the coupled simulation, do not model the main rotor ﬂapping and

inﬂow dynamics.

58

(a) p

θ1C

(b) φ

θ1C

Figure 3.13: On-axis Bode and correlation factor plots of the lateral sweep for a

120 kts level ﬂight case.

59

Chapter 4 |

Simulation Results

4.1 Flight Simulation Results

A number of basic maneuvers were simulated using the standard PSUHeloSim

model and the fully-coupled simulation with the CHARM rotor module. The

simulations are used to verify that the fully-coupled simulations follow the expected

behavior and that the NLDI controller can adequately stabilize and control the

coupled model.

4.1.1 Coupling Transient

When using the one-way coupled simulation or the stand-alone PSUHeloSim model,

the CHARM rotor module is not used in the ﬂight dynamics solution. This means

that the dynamic simulation involves just the base PSUHeloSim and the Dynamic

Inversion based controller. The DI controller is designed around linearized models

of the PSUHeloSim, which leads to very accurate tracking of controller commands.

The trim solver is also based on the PSUHeloSim model and results in near perfect

initialization. This is seen in the red line plotted in Fig. 4.1, which shows the

attitude response when the commanded trajectory simply holds a 100 kts level

ﬂight trim condition. It can be seen that there is no deviation from trim. In the

fully-coupled case, the main rotor forces, moments calculated by the CHARM

rotor module are fed back into the dynamic simulation, changing the nature of

the nonlinear model. So when the coupling is turned on, after three seconds of

simulation, the helicopter goes through a transient due to the diﬀerences of forces

and moments between CHARM and the PSUHeloSim model. The controller is

60

robust enough to restore the trim, causing the aircraft to converge to a steady state

after a period of time. The new equilibrium is usually slightly diﬀerent from the

initial trim. This is partly due to diﬀerences in trim of the CHARM rotor model

and the simple rotor model in PSUHeloSim. In addition, a helicopter can trim

with diﬀerent combinations of roll attitude and sideslip angle. When trimming

PSUHeloSim the yaw attitude / sideslip are set to zero, but after coupling is

initiated the system settles into a slightly diﬀerent steady state.

Figure 4.1: Coupling transient: dashed line marks the start of the coupling.

61

4.1.2 Use of the Reduced-Order Models in the Coupled Simu-

lation

Diﬀerent reduced-order models have been used in the feedback linearization of

the Dynamic Inversion controller to verify the ﬁndings described in the previous

chapter. The simulations run consist in a 6

◦

degree descent at 120 kts of absolute

airspeed. The plots in Fig. 4.2 show how the controllers based on the diﬀerent

models respond to the coupling and thus how the transients change from case

to case. It can be readily noticed by looking at the roll attitude plot that the

controllers based on the linearized model from the coupled simulation give smaller

oscillations in amplitude compared to the ones of the original controller. Speciﬁcally,

the reduction in the magnitude of the oscillations given by the linearized model from

the coupled simulation with forward-speed-based trim is attributed to the better

modelling of the roll subsidence. A further decrease in amplitude of vibrations

resulting by the use of the linearized model from the coupled simulation with

forward-and-vertical-speed-based trim is the consequence of the better prediction

of the dutch roll mode in descending ﬂight. Similar improvements can also be seen

in the pitch response, where the latter two models provide a better approximation

of the short period mode, especially in damping. Of remark is the fact that the

oscillations given by using the third method tend to dampen out more quickly thus

reducing the time the system needs to reach steady state. The use of linearized

models obtained by the coupled simulation also give high-frequency oscillation

in the controls as is apparent by looking at the lateral cyclic plot. This is to be

attributed to the fact that the frequency at which the roll compensator excites the

ﬂap progressive mode. In fact, by looking at the lateral cyclic plot, we can deduce

that the frequency of the vibrations is about 8 rad/s which is very close to the

frequency of the the ﬂap regressive mode (shown in Fig. 3.1).

62

(a) Attitude

(b) Controls

Figure 4.2: Response to the coupling with the controller based on the reduced-order

models form 1) PSUHeloSim, 2) linearization of the coupled simulation in level

ﬂight, 3) linearization of the coupled simulation in a 6◦descent.

63

4.1.3 Decelerated Descent

A decelerating descent maneuver was simulated with and without coupling. The

results are shown in Fig. 4.3. With coupling, the simulation is initially ﬂown in

steady 100 knot level ﬂight for a period of time to allow the helicopter to return

to trim after the transient at initialization. Figure 4.3 shows the response after

the initial wait period. The maneuver consists of a 6 degrees decelerated descent

from 100 to 60 kts at 0.1 g of deceleration. Figure 4 compares responses of the

"de-coupled" baseline PSUHeloSim and the fully coupled model with CHARM. In

both cases, the vehicle response follows the command after the initial transient,

and the responses are similar for both models. The velocity and altitude proﬁles

are essentially identical, which is expected since these are tracked by the controller.

There are slight diﬀerences in attitudes due to diﬀerences in the two rotor models.

Note that there is some deviation of the lateral cross track (y-position), but the

lateral drift is only about 4 ft after 900 ft down range motion.

(a) Position

64

(b) Absolute velocity and acceleration

(c) Attutide

Figure 4.3: Results from a 6◦decelerated descent at 0.1 g of deceleration.

65

4.1.4 Coordinated Turn

Figure 4.4 shows a 90 degrees turn maneuver at 100 kts forward airspeed. Once

again, time was allotted to allow the controller to stabilize the aircraft after the

coupling transient. Once again we see very similar ﬂight path with both the coupled

and de-coupled PSUHeloSim. Note that the velocity ﬂuctuations are quite small.

The accelerations seen are largely in the lateral axis due to Dutch Roll oscillations

since this mode appears to be less damped with the coupled model.

(a) Trajectory

(b) Absolute velocity and acceleration

66

(c) Attutide

Figure 4.4: Coupled and uncoupled simulations results of a 90

◦

coordinated turn at

100 kts 20◦of bank angle.

Fig. 4.5 shows the simulation results of a steady circular turn, of particular

interest in terms of noise predictions. The simulation has been cut such that both

the coupling transient and the one following the turn command are not shown. It

can be noticed that after stabilizing, the altitude, absolute velocity and roll are

very close to constant whereas the pitch angle slightly increases of about one degree

during the turn; this change can be considered negligible. The heading is seen to

increase with a constants slope and the trajectory is almost perfectly circular: the

diﬀerence between the major and minor axis is about 3 ft. The interaction of the

tail rotor with the main rotor wake causes some oscillation, especially on the roll

angle. The same noise can be seen also by looking at the absolute acceleration plot.

67

(a) Trajectory

(b) Absolute velocity and acceleration

68

(c) Attutide

Figure 4.5: Coupled simulation results of a 360

◦

coordinated turn at 60 kts and

30◦of bank angle.

4.1.5 Acceleration in Ground Eﬀect

Fig. 4.6 shows the plots from both a coupled and uncoupled simulation of an

acceleration in ground eﬀect and a subsequent climb. The initial altitude of the

helicopter is 10 ft and the acceleration command is given at the tenth second of

simulation. The acceleration has a magnitude of 0.05 g and it is used to make the

helicopter transition from hover to the speed for best climb, which happens to be 60

kts for the Bell 430. It can be noticed that for the uncoupled case the acceleration

stays constant and equal to the commanded value. However the coupled simulation

suﬀers from higher absolute accelerations, mostly due to the interaction between

the rotor and the wake that gets reingested in it because of ground eﬀect. Note

that the ground eﬀect physics is not modelled in the reduced order model used in

the feedback linearization. As the helicopter reaches higher speeds, the absolute

acceleration becomes more and more similar to the commanded one since the

ground eﬀect disappears as the wake is skewed back by the oncoming ﬂow. As

reported in [30], ground eﬀect can usually be considered negligible when the speed

69

with respect to the oncoming ﬂow is two times greater than the induced velocity in

hover (for advance ratios grater than 0.1)

V∞>2vh(4.1)

where

vh=sW

2ρA = 21.89kts (4.2)

By looking at the acceleration plot, we can notice that the absolute acceleration

has a strong decrease when the helicopter reaches about 45 knots. This ﬁnding

agrees with the theory shown above, which predicts a transition speed of 43.78 kts.

The high vibrations seen in the Euler angles for the coupled case are also due to

the coupling of the controller with the wake.

(a) Trajectory

70

(b) Absolute velocity and acceleration

(c) Attutide

Figure 4.6: Coupled and uncoupled simulations results of an acceleration from 0 to

60 kts in ground eﬀect followed by steady climb at 12◦of ﬂight path angle.

71

4.2 Acoustic Results

4.2.1 Introduction to Helicopter Acoustics

Helicopter rotor noise consists of several noise sources including discrete frequency

noise (thickness, loading, and blade-vortex-interaction (BVI) noise), broadband

noise, and high-speed-impulsive (HSI) noise (HSI noise only occurs in high-speed

forward ﬂight). Each of these noise sources has a unique directivity, as shown in

Fig. 4.7. Thickness noise is dominant in the plane of the rotor, so it is the primary

noise heard as a distant helicopter approaches. Only the motion of the rotor blades

and the aircraft, along with the geometry of the rotor blades, is needed to compute

the thickness noise; hence, the ﬂight simulation code is readily able to provide this

information (at very low computational cost). High-speed impulsive noise has the

same directivity as thickness noise, but it only occurs in high-speed ﬂight (and will

not be addressed here).

Figure 4.7: Typical direction of primary radiation for various rotor noise sources.

Loading noise is another important source of rotor noise, which is typically

directed below the rotor

−

so loading noise is important as the aircraft is overhead.

There are two important types of loading noise that are generally dealt with

separately: BVI noise and broadband noise. BVI noise is the dominant noise source

when it occurs. It has a very impulsive nature and originates from a nearly parallel

72

close interaction between a blade and the tip vortex of a previous blade. BVI noise

is highly directional and depends strongly on the vortex strength, miss distance,

and interaction angle. This is the reason that a high-ﬁdelity rotor and wake model

are needed in this system to predict BVI noise accurately. Broadband noise is

another type of loading noise that is a result of stochastic loading due to various

airfoil self-noise sources, turbulence ingested into the rotor, or turbulence entrained

by tip vortices (when they are not quite close enough to cause signiﬁcant BVI

noise). The empirical model derived by Pegg [25] is used in PSU-WOPWOP to

predict the broadband noise and the input data is relatively easy to obtain from

the ﬂight simulation system. Fig. 4.8 shows a small contribution to each of the

noise components to the Overall Sound Pressure Level (OASPL) as function of

the uprange/downrange distance for the Bell 430 helicopter ﬂying at 100kts at an

altitude of 150m. At

x

= 0, the helicopter is directly overhead. OASPL is not

weighted by frequency and hence tends to reﬂect the large amplitude of the low

frequency components of the rotor noise. Notice that as the aircraft approaches

(negative distances) the thickness noise is the dominant source of noise. This is

because the thickness noise directivity is in the plane of rotor; hence, the observer

hears it ﬁrst. The loading noise becomes begins takes over as the dominant noise

source as the aircraft passes overhead and continues downrange (positive distances).

The broadband noise makes only a contribution to OASPL, so it is not shown in

Fig. 4.8.

Fig. 4.9 shows a similar plot of the noise components, but in this case the

tone corrected, perceived noise level (PNLT) is plotted as a function of the up-

range/downrange distance from the target observer location and the aircraft is

directly overhead at

x

= 0. PNLT uses a frequency weighting that is intended to be

representative of human annoyance; hence, higher frequencies are more important.

The relative importance of the various noise sources is quite diﬀerent in this case.

The thickness noise is still the dominant noise as the aircraft is approaching (larger

negative distances), but the broadband noise is signiﬁcant as the aircraft approaches

the overhead condition and dominant for all downrange positions (positive dis-

tances). The loading noise also increases overhead and decreases downrange, but is

signiﬁcantly lower in this ﬂight condition than broadband noise. This is because

the loading noise in level ﬂight has fairly low frequency content at this is a level

ﬂight condition. For level ﬂight BVI noise is not expected, but if there had been

73

Figure 4.8: Noise components and their contribultion to the OASPL predictions

for a 100 kts ﬂight case ﬂown at 150 m altitude.

BVI noise the loading noise levels would have been substantially higher.

4.2.2 Prediction of BVI Noise Reduction Using Blue Edge-like

Blades

An important attribute of this work is that the coupled system can accurately predict

the acoustic characteristics of dominant noise sources without reliance on test data.

During approach and landing, blade-vortex interaction (BVI) noise is a dominant

out-of-plane noise source responsible for much of the ground noise. In order to

provide a tool for evaluating the impact of modiﬁcations to ﬂight path and rotor

design on ground noise exposure during landing, it is necessary to demonstrate that

the model can accurately predict BVI noise for BVI-dominant ﬂight conditions. The

ability of the CHARM/WOPWOP and subsequently CHARM/PSU-WOPWOP

solutions to predict main rotor BVI noise in these ﬂight conditions for conventional

rotors (and tiltrotors) was demonstrated in prior work [22] [23]. In the current work,

this demonstration was extended to an advanced blade design known to reduce BVI

74

Figure 4.9: Noise components and their contribution to the PNLT for a 100 kts

ﬂight case ﬂown at 150 m of altitude.

noise (Fig. 4.10) [1], Airbus Helicopter’s "Blue Edge" blade. The concept behind

this design is described in [31] as: "With a standard blade, air coming oﬀ the end

of the blade causes a vortex around the tip. Under certain ﬂight conditions the

advancing blade then hits the vortex of the preceding blade. This causes a sudden

change in the relative angle of attack and thus a change in pressure on the surface

of the blade. This BVI causes the slapping sound ubiquitous to helicopter operations.

With Blue Edge technology, the blade tip is swept forward, then aft. This causes the

advancing blade tip to hit the previous blade’s vortex at an oblique angle, reducing

the noise level by 3 to 4 EPNdB."

Calculations were performed to demonstrate the ability of the new analysis

system to predict the reduction in BVI noise obtained using a Blue Edge-like plan-

form. Three blade planforms were compared operating on a Bell 430 rotor/aircraft

conﬁguration: 1) conventional rectangular blades

−

nominally the current Bell 430

blade; 2) tapered blades; 3) Blue Edge-like planform with taper and forward/aft

sweep. The planform characteristics of each of these three blade sets are provided

75

Figure 4.10: Blue Edge blade concept from Eurocopter (now Airbus Helicopters) [1].

Rectaangular Tapered Blue Edge

Root Cutout 0.1 0.1 0.1

Chord [ft] 1.2 1.8 to 1.0 1.8 to 1.0

Anhedral Tip None None None

Swept Tip None None

12

◦

fwd @ r=0.6,

34.4◦aft @ r=0.85

Root Airfoil NACA 0012 NACA 0012 NACA 0012

Tip Airfoil NACA 0012 NACA 0009 NACA 0009

Table 4.1: Charcteristics of the blade planforms.

in Table 4.1. Figure 4.11 compares the tapered and Blue Edge-like planforms.

No optimization of the tapered and Blue Edge-like planforms was performed to

minimize noise. The Blue Edge-like planform forward/aft sweep schedule is roughly

comparable to photographs of the Airbus Blue Edge blade, capturing the key

feature of reducing the "parallel" nature of the BVI.

The ﬂight condition studied was a descent at low speed (

µ

= 0

.

15) with the

rotor tilted back 6 degrees relative to the ﬂight path. The sound pressure level

was determined in a plane one rotor radius beneath the rotor plane. The CHARM

solution was performed with an azimuthal resolution of ∆

ψ

= 15

◦

and then

reconstructed to a resolution of ∆

ψ

= 1

◦

using the method described in [22]. The

76

Figure 4.11: Rectangular, tapered and Blue Edge-like planforms.

blade aerodynamics and acoustic solution at 187 observer points was completed in

3 minutes on a single core of an oﬀ-the-shelf CPU.

Fig. 4.12 shows predictions of both the overall sound pressure level (OASPL)

and the BVI sound pressure level (BVISPL) (harmonics 6-40 of blade passage

frequency) for this conﬁguration. The magnitude and directionality predicted is

characteristic of the results seen for BVI-noise dominated descent ﬂight conditions.

The analysis predicts that the taper reduces the peak BVISPL by roughly 2dB and

the Blue Edge-like planform further reduces the peak BVISPL by another 3dB for

a total reduction of peak BVISPL of 5dB, capturing the documented beneﬁt of the

Blue Edge planform.

Fig. 4.13 shows the CHARM model of the rotor wake sheet and tip vortices

for advancing-side BVI for the Blue Edge planform compared with a rectangular

blade as predicted by the CHARM code. Notice in the ﬁgure that the tip vortex

(the red curved line) is nearly parallel to the entire length of the blade for the

rectangular blade (left), while the shape of the Blue Edge planform (right) results

in an interaction that occurs over a wider range of rotor azimuth angles; hence, it

is a much less impulsive interaction.

The measurement plane shown in Fig. 4.12 reveals that the main rotor BVI

noise is signiﬁcantly reduced by the Blue Edge-like rotor planform, but a more

typical noise prediction for a complete rotorcraft is made either on a hemisphere

or a ground plane. To demonstrate the fully-coupled system this BVI noise was

predicted for the full helicopter conﬁguration. Here the aircraft ﬂight condition

is a forward speed of 68 kts and a 6

◦

descent ﬂight proﬁle

−

providing the same

77

Figure 4.12: CHARM/PSU-WOPWOP main rotor OASPL and BVISPL predictions

one rotor radius beneath the nominal Bell 430 rotor for three blade geometries;

αs

= 6

◦

(aft),

µ

= 0

.

15, and

CT

= 0

.

00143. The black circle represents the rotor

tip (advaning side on the right).

main rotor operating condition as shown in Fig. 4.12 and 4.13. Fig. 4.14 shows the

OASPL of the Bell 430 helicopter (with rectangular main rotor blades and the tail

rotor included). Notice in the ﬁgure that the focused region of BVI noise is still

clearly evident on the hemisphere surface.

Figure 4.15 shows the acoustic pressure time histories for each of the main rotor

planforms at a point located on the hemisphere at an azimuth of 125

◦

and down

45

◦

from the main rotor tip-path plane (indicated by a small black dot in Figure

13). Notice in the ﬁgure, for each blade geometry there are four very narrow and

high amplitude pressure spikes (or group of spikes). These are the BVI from each

of the four blades on the main rotor. The thickness and loading noise of the main

and tail rotor also occur the same time, but at lower amplitude at this location.

The smaller more frequent pulses are the tail rotor thickness noise. Comparison

of the three diﬀerent rotor blade geometries shows how the BVI acoustic pressure

spikes amplitude is greatly reduced for the case of the Blue Edge-like rotor. The

78

(a) Rectangular

(b) Blue Edge

Figure 4.13: BVI event as predicted by CHARM for the baseline rectangular blade

and the Blue Edge blade.

79

Figure 4.14: Contours of OASPL on a 30.48 m radius hemisphere, centered at the

Bell 430 center of grvity location. The hemisphere follows the aircraft. OASPL

contours shown are for the standard rectangular blades).

tapered blade also has a small reduction in BVI spike amplitude, primarily seen on

the positive part of the pressure spike. The other features, i.e., the tail rotor noise,

is essentially unchanged.

The noise comparisons shown in this section demonstrate the utility of the ﬂight

simulation, high-ﬁdelity wake, noise prediction system that has been developed here.

Furthermore, design changes to reduce BVI noise

−

one of the more challenging

components of the noise to predict −show the expected noise reduction trends.

80

(a) Rectangular (b) Tapered

(c) BlueEdge

Figure 4.15: Acoustic pressure time history at azimuth angle

ψ

= 125

◦

, elevation

θ

=

−

45

◦

below the rotor plane, and radius of 30.48 m from the helicopter center

of cravity (i.e. the location of the black dot in Fig. 4.14).

81

Chapter 5 |

Conclusions and Recommenda-

tions for Future Work

5.1 Conclusions

It has been demonstrated that the integrated simulation was capable of predicting

realistic maneuvers when coupling the CHARM rotor module and PSUHeloSim

simulation. It is crucial that the simulation include a robust ﬂight controller, to

handle the transients and change in aeromechanics upon coupling with the higher

ﬁdelity main rotor and tail rotor models.

The dynamic inversion controller proved to work well when using redued-order

models in the feedback linearization. Some of the limitations introduced by the fact

that the feedback linearization relies on model scheduling based on the absolute

velocity and the PSU-HeloSim code were relaxed by introducing Vx-Vz scheduling

and by obtaining the reduced order models by linearization of the coupled PSU-

HeloSim/CHARM simulation. These approaches proved to be eﬀective in all the

ﬂight conditions studied. In level ﬂight the linearized model derived from the coupled

simulation accounts for a lower roll damping as compared to the one obtained with

PSU-HeloSim alone. In descending ﬂight the linearized model derived from the

coupled simulation also proved to model the dutch roll mode more accurately, thus

improving the overall performance of the controller both in terms of amplitude of

oscillations and time to achieve steady state.

The system identiﬁcation performed by a frequency sweep of the the lateral cyclic

proved to validate the accuracy of reduced-order models derived by linearization

82

of the coupled simulation, up to frequencies of about 5

rad/s

. This is expected

since the reduced-order models, even the ones not obtained by linearization of the

coupled simulation, are not well suited for frequencies exceeding 5rad/s.

The closed-loop response of the system, even though being stable, is not satis-

factory in oﬀ-nominal conditions such as ground eﬀect. The coupled simulation

suﬀers from higher absolute accelerations, mostly due to the interaction between

theunsteady nature of the wake in ground eﬀect and the controller. To address

this problem, the reduced-order model could be obtained by linearization of the

coupled simulation in such condition and used in the feedback linearization loop.

As the helicopter reaches higher speeds, the ground eﬀect disappears as the wake is

skewed back by the oncoming ﬂow.

The use of the CHARM rotor module signiﬁcantly enhances the ﬁdelity level of

the simulation, by adding free wake and nonlinear dynamics of ﬂexible blades (as

opposed to 3-state inﬂow, and a rotor disk model with linearized ﬂapping dynamics).

While this level of ﬁdelity is not necessarily required for ﬂight simulation, the

CHARM rotor module captures higher resolution blade loading needed for acoustics

calculations. One of the motivations for full coupling (feedback of CHARM rotor

forces to the vehicle dynamics) is to ensure consistency of the rotor force output

with the ﬂight trajectory ﬂown.

The CHARM rotor module successfully captured the behavior of the "Blue

Edge" blade in terms of blade vortex interaction thus underlining its strength in

comparison to other more classic blade geometries. This case also demonstrates

the predictive capability of the entire system.

The coupling of the simulation and CHARM rotor module results in a coupling

transient. The transient is a simulation artifact and not relevant to the physics

of interest. Some additional processing time is required to allow the controller to

stabilize and re-trim the aircraft before performing the maneuver of interest. As

mentioned earlier, the controller improvements reduced this transient both in time

and amplitude of oscillations thus improving the eﬃciency of the tool.

5.2 Future Work

One area of focus of the work for the next future will be the study of how the reduced

order models obtained by linearization of the coupled PSUHeloSim/CHARM rotor

83

module simalation, when used in the feedback linearization, will work in ﬂight

conditions not considered so far. Descending ﬂight at low speeds and high descent

rates will be studied since it is particularly keen to strong blade-vortex interactions

driven by the re-ingestion of the wake into the rotor. Another condition to be

analyzed would be ground eﬀect, where the wake does not contract as it should thus

generating steady and unsteady behaviors not predcted by the simple Pitt-Peters

model. The scheduling of the feedback linearization is also planned to be expanded

to other variables which possibly include the lateral velocity

Vy

and the atmospheric

density.

Another certain area of focus for the prospective eﬀorts would be the imple-

mentation of an atmospheric model into PSUHeloSim and CHARM rotor module

(PSU-WOPWOP already has it) so that the simulations, particularly the maneu-

vers concerning substantial altitude change such as descents and climbs, would

be modeled more accurately and could possibly lead to new prospectives in noise

reduction procedures.

In the longer run two main problems will have to be addressed: the extension

of the code to perform noise prediction of unsteady maneuvers, as compared

to prevailing capabilities of dealing only with quasi-steady cases, and the full

automation and parallelization of the code. The ﬁrst item will require CHARM

rotor module to modify the strategy of generating the loading ﬁles which are

currently based on one entire rotor revolution and thus based on a periodicity

assumption. The second item instead will need the development of the wrapper

code that orchestrates the PSUHeloSim/CHARM rotor module/PSU-WOPWOP

such that it can run on diﬀerent clusters.

The author’s hope is that when the previously mentioned improvements will

have been eﬀectively implemented, the research will procede to the study of

an optimization code based on evolutionary algorithms which would "teach" the

helicopter how to ﬂy silently in arbitrary ﬂight conditions. The possible parameters

to be optimized would strictly be related to how and where the helicopter ﬂies,

making it a problem strictly related to ﬂight dynamics ; no optimization would

be conduced on material parts of the aircraft. Evolutionary algorithms would

particularly be apt for this task because of their capability of dealing with multi-

objective optimization and with strongly non-linear systems such as helicopters

themselves and noise prediction in particular.

84

Another fascinating prospect of this research would be to derive some simpliﬁed

laws of how ﬂight dynamics alters the noise produced and consequently build an

online controller which would account for variables related to noise. The approach

to this problem would probably involve neural networks as they are popular in

subjects related to adaptive controls.

85

Appendix A|

Structure of the Wrapper Code

A.1 Introduction

The wrapper code is written in FORTRAN/C++ and it is developed in the Visual

Studio environment. Its function is to orchestrate the interaction of PSUHeloSim,

written in MATLAB/Simulink and eﬀectively used as a dynamic library, and

CHARM and PSU-WOPWOP, both written in FORTRAN. The detailed structure

of the code is not reported, the appendix focuses instead on the setup of the input

ﬁles necessary to perform the coupled simulation. The structure of the output ﬁles

is also reported.

A.2 Input Files

A.2.1 Master.txt

The ﬁle to be setup is called master.txt and can be found in the HeloSim/HeloSim

folder. All the items below have to be numerically speciﬁed in the ﬁle in the order

in which they are presented in Table A.1.

A.2.2 Command.txt

Command.txt is the ﬁle to be set up in case the user wishes to simulate a maneuver

which is diﬀerent from climb/descent, acceleration/deceleration, or a combination

of the two. In order to use a user-deﬁned version of it, the user will have to set

86

Variable Units Description

NAME OF AIRCRAFT

Current aircrafts: B430, S76,

BK117, EC145, H145.

ABSOLUTE VELOCITY kts Allowable values: 0 to 140.

CLIMB/DESCENT RATE

ft/min

Climb positive, descent negative.

Allowable values: -1519 to 1519.

ALTITUDE ft

Initial altitude. No atmospheric

model implemented yet.

ABSOLUTE VELOCITY INCRE-

MENT/DECREMENT

kts

Has to be always positive, it’s the

sign of ACCELERATION that de-

termines the increment or decre-

ment. Not considered if ACCEL-

ERATION = 0.

FINAL TIME s

Disregarded if ACCELERATION

6

=0. In this case the ﬁnal time

will be given by the time to make

the helicopter reach the speciﬁed

ABSOLUTE VELOCITY INCRE-

MENT/DECREMENT.

ACCELERATION g

Suggested not to go beyond

±

0

.

1.

Note that the acceleration acts on

the absolute velocity so that in

case of climb/descent the ﬂight

path angle remains constant.

CUSTOM COMMAND FILE

0: no user-deﬁned command ﬁle,

1: use user-deﬁned command ﬁle.

Use if maneuver is diﬀerent from

an acceleration or deceleration or

climb/descent. An example Com-

mandExample.txt is provided in

this same folder. For more info on

how to set up the command.txt

ﬁle see next paragraph.

FREEZE MODE

0: not engaged, 1: engaged. If

engaged fuselage states do not get

integrated.

MAIN ROTOR COUPLING

0: uncoupled, 1: coupled. If equal

to 0, Simulink simulation will be

run without running CHARM.

87

TAIL ROTOR COUPLING

0: uncoupled, 1: coupled. If

MAIN ROTOR COUPLING = 0

this ﬂag will be disregarded.

Y CONTROLLER

0: uncoupled, 1: coupled. Should

always be set equal to 1 in case of

forward ﬂight. Should be desabled

for turns.

PSU-WOPWOP MODE

0: generate PSU-WOPWOP in-

put ﬁles at the end of simula-

tion, 1: generate PSU-WOPWOP

input ﬁles at arbitrary times, 2:

generate PSU-WOPWOP input

ﬁles each PSU-WOPWOP DT sec-

onds.

PSU-WOPWOP TIMES VEC-

TOR LENGTH

Deﬁnes the number of times steps

at which the PSU-WOPWOP ﬁles

are generated. Valid if PSU-

WOPWOP MODE = 1.

PSU-WOPWOP TIMES VEC-

TOR

s

Vector of times at which PSU-

WOPWOP ﬁles are generated.

They should be speciﬁed on a row.

It must be of the same length iden-

tiﬁed by PSU-WOPWOP TIMES

VECTOR LENGTH.

PSU-WOPWOP DT s

Time interval between the

generation of subsequent PSU-

WOPWOP input ﬁles. Valid if

PSU-WOPWOP MODE = 2.

PSU-WOPWOP START TIME s

Time at which PSU-WOPWOP

input ﬁles start being generated

with PSU-WOPWOP MODE =

2.

FILE PATH

File path of where the HeloSim2.0

folder (up to the folder before).

Table A.1: Master.txt setup.

88

CUSTOM COMMAND FILE = 1 in the master.txt ﬁle. Command.txt can be

found in the HeloSim/HeloSim folder; Table 2. Shows the structure of it.

Note that the four diﬀerent commands correspond to variations from trim

respectively of forward, sideward and vertical speed in heading frame, and heading.

The ﬁrst three commands have units of ft/s whereas the fourth has units of radians.

A.3 Output Files

A.3.1 HeloSimOut.txt

HeloSimOut.txt contains the time history of all the variables of interest. Table A.2

explains the content of each column.

A.3.2 PSU-WOPWOP Files

CHARM produces a number of ﬁles that are then used by PSUWOPWOP to

perform the acoustic prediction in a post-processing step. periodic loading. Table

A.3 describes the diﬀerent ﬁles.

89

Column Description Variable Units

1 Time t s

2, 3, 4 Body velocities u,v,w ft/s

5, 6, 7 Roll, pitch, and yaw rates p,q,r rad/s

8, 9, 10 Euler angles φ,θ,ψrad

11, 12, 13 Position x, y, z ft

14, 15, 16 MBC ﬂapping states β0,β1S,β1Crad

17, 18, 19 MBC ﬂapping derivates ˙

β0,˙

β1S,˙

β1Crad/s

20, 21, 22 Inﬂow states ˙

λ0,˙

λ1S,˙

λ1C

23, 24, 25 Body velocities derivatives ˙u,˙v,˙w f t/s2

26, 27, 28

Roll, pitch, and yaw rates

derivatives

˙p,˙q,˙r rad/s2

29, 30, 31

Euler angles time derivatives

˙

φ,˙

θ,˙

ψrad/s

32, 33, 34 NED velocities ˙x,˙y,˙zft/s

35, 36, 37 MBC ﬂapping derivates ˙

β0,˙

β1S,˙

β1Crad/s

38, 39, 40

MBC ﬂapping second deriva-

tives

¨

β0,¨

β1S,¨

β1Crad/s2

41, 42, 43 Inﬂow derivatives ˙

λ0,˙

λ1S,˙

λ1C1/s

44, 45, 46, 47

Controls: lateral and lon-

gitudinal cyclic, collective,

and TR collective

θ1S,θ1C,θ0,θ0Trad

48, 49, 50, 51 Rotor Coeﬃcients CT,CL,CM,CQ

52, 53, 54 Main rotor forces XM R ,YMR,ZM R lbs

55, 56, 57 Main rotor moments LM R ,MM R,NM R lbs-ft

58, 59, 60 Tail Rotor Forces XT R,YT R ,ZT R lbs

61, 62, 63 Tail Rotor Forces XT R,YT R ,ZT R lbs-ft

64, 65, 66

CHARM MBC ﬂapping

states

β0CH ,β1SC H ,β1CCH rad

Table A.2: HeloSimOut.txt content.

90

Name Index 1 Index 2 Description

loading.dat

Number of ﬁle

in chronological

history

Identiﬁes which

rotor it refers

to. 1: MR, 2:

TR.

Contains the loading

on the diﬀerent blades

of the rotor it refers to

avareged over one ro-

tor revolution

geometry.dat

number of ﬁle

in chronological

history

Identiﬁes which

rotor it refers

to. 1: MR, 2:

TR.

Contains the blades

position

charm.dat

Number of ﬁle

in chronological

history

Contains the setup for

the PSU-WOPWOP

noise prediction,

including the heli-

copter’s position,

velocities and Euler

angles at the time in

the simulation it is

produces.

bsurface.dat

number of ﬁle

in chronological

history

identiﬁes which

rotor it refers

to. 1: MR, 2:

TR

Table A.3: Input ﬁles for PSU-WOPWOP.

91

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