The Pennsylvania State University
The Graduate School
College of Engineering
ROTORCRAFT SIMULATIONS WITH COUPLED FLIGHT
DYNAMICS, FREE WAKE, AND ACOUSTICS
A Thesis in
©2016 Umberto Saetti
Submitted in Partial Fulﬁllment
of the Requirements
for the Degree of
Master of Science
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.
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
predicting BVI noise reduction as compared to more conventional
blade geometries on the same order as that reported for the Blue Edge rotor.
Table of Contents
List of Figures vii
List of Tables ix
List of Symbols x
1.1 Motivation................................ 1
1.2 Background and Technical Barriers to Solve . . . . . . . . . . . . . 2
1.3 Objectives................................ 3
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
184.108.40.206 Fuselage........................ 9
220.127.116.11 Flapping........................ 10
18.104.22.168 Inﬂow ......................... 12
2.1.4 TrimAlgorithm......................... 13
2.1.5 TrimResults .......................... 14
2.1.6 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 15
22.214.171.124 InnerLoop ...................... 15
126.96.36.199 OuterLoop ...................... 19
188.8.131.52.1 Low Speed Mode . . . . . . . . . . . . . . 19
184.108.40.206.2 High Speed Mode . . . . . . . . . . . . . . 21
220.127.116.11.3 Blending . . . . . . . . . . . . . . . . . . . 23
18.104.22.168 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
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
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
Prediction of BVI Noise Reduction Using Blue Edge-like
Conclusions and Recommendations for Future Work 82
5.1 Conclusions ............................... 82
5.2 FutureWork............................... 83
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
A.3 OutputFiles............................... 89
A.3.1 HeloSimOut.txt......................... 89
A.3.2 PSU-WOPWOP Files . . . . . . . . . . . . . . . . . . . . . 89
List of Figures
Schematic of the PSUHeloSim/CHARM/PSU-WOPWOP simula-
tionmodel. ............................... 6
2.2 An airborne Bell 430. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Bell4303-View.............................. 7
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
Comparison between the eigenvalues of the full-order model and the
Frequency responses of aicraft states with respect to the control
inputs for both the eigth-order and full-order models. . . . . . . . . 41
Response of the derivatives of the states to perturbations of the
Comparison between the eigenvalues of the reduced order models
obtained from PSUHeloSim and the coupled simulation for a 120
ktslevelﬂight. ............................. 45
Comparison between the eigenvalues of the reduced order models
obtained from PSUHeloSim for the level ﬂight and descent cases. . . 48
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
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
3.11Sweepexcitation. ............................ 55
Frequency sweep of the coupled simulation for a 120 kts level ﬂight
case. ................................... 57
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
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
Coupled and uncoupled simulations results of a 90
turn at 100 kts 20◦ofbankangle.................... 67
Coupled simulation results of a 360
coordinated turn at 60 kts and
Coupled and uncoupled simulations results of an acceleration from
0 to 60 kts in ground eﬀect followed by steady climb at 12
pathangle. ............................... 71
Typical direction of primary radiation for various rotor noise sources.
Noise components and their contribultion to the OASPL predictions
for a 100 kts ﬂight case ﬂown at 150 m altitude. . . . . . . . . . . . 74
Noise components and their contribution to the PNLT for a 100 kts
ﬂight case ﬂown at 150 m of altitude. . . . . . . . . . . . . . . . . . 75
Blue Edge blade concept from Eurocopter (now Airbus Helicopters) .
4.11 Rectangular, tapered and Blue Edge-like planforms. . . . . . . . . . 77
CHARM/PSU-WOPWOP main rotor OASPL and BVISPL predic-
tions one rotor radius beneath the nominal Bell 430 rotor for three
black circle represents the rotor tip (advaning side on the right). . . 78
BVI event as predicted by CHARM for the baseline rectangular
blade and the Blue Edge blade. . . . . . . . . . . . . . . . . . . . . 79
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
Acoustic pressure time history at azimuth angle
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.
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
Inner loop disturbance rejection frequencies, damping ratios, and
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
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
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 [%]
Af,Afs,Asfast, fast-slow and slow sub-matrices of the system matrix
Areduced order system matrix
Bf,Bsfast and slow sub-matrices of the control matrix
cblade chord [in]
CM0damping matrix of the ﬂapping equation
DM0stiﬀness matrix of the ﬂapping equation
eV x, eV y , eV z
errors on the longitudinal, lateral and vertical speeds in heading
φerror on the roll rate [rad/s]
ggrvitational acceleration [ft/s2]
HM0right-hand of the ﬂapping equation
second mass moment of blade about the ﬂapping hinge [
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
proportional, derivative, integral, double-integral gain matrices
for both the inner and outer loop
Kβﬂapping stiﬀness [slug −f t2/s2]
roll moment derivatives normalized by moment of inertia [
mmass of aircraft [slug]
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
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]
commanded angular speed around the vertical body axis [
sLaplace (complex) operator
Tb/h rotational matrix from body to heading frame
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
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
Waircraft weight [lbs]
x, y, z North, East and Down positions in NED frame [ft]
xetrim state vector
xf,xsfast and slow states vector
longitudinal and lateral forces acting on the aircraft in body
Xp, Xq, etc.
longitudinal force derivatives normalized by aircraft mass [1
ycmd commanded trajectory [f t]
Yp, Yq, etc.
lateral force derivatives normalized by aircraft mass [1
Zp, Zq, etc.
vertical force derivatives normalized by aircraft mass [1
β0, β1C, β1S
rotor blade coning, longitudinal and lateral ﬂapping angles in
multi-blade coordinates [rad]
ﬂapping variables vector in Multiple Blade Coordinates (MBC)
δ3delta 3 angle [deg]
δlat, δlong , δcoll , δped
lateral, longitudinal, collective and tail rotor collective pilot
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
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]
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]
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.
Chapter 1 |
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 , that couples 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   ). 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 . 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 . 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 . 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 . 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.
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  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".
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,
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.
Chapter 2 |
2.1 Helicopter Flight Dynamics Model
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  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  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)  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-
linear dynamic inversion control law . 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 .
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
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.
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.
2.1.3 Equations of Motion
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 .
Following the approach in , 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
˙v=pv −ru +gsin ψcos θ+Y
˙w=qu −pv +gcos φcos θ+Z
Iyy Ixx −Ixz2hqr Iyy Izz −Izz 2−Ixz2+qpIxz (Izz +Ixx −Iyy) +
LIzz +N Ixz ](2.4)
Iyy hrp (Izz −Ixx) + Ixz r2−p2+Mi(2.5)
Iyy Ixx −Ixz2hqrIxz (Iy y −Izz −Ixx) + qp Ixz 2+Ixx2−IxxIyy +
φ=p+ tan θ(qsin φ+rcos φ)(2.7)
θ=qcos φ−rsin φ(2.8)
ψ=qsin φ+rcos φ
˙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 .
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
The damping and stiﬀness matrixes, along with the right-hand side, are given
k2µ(k3−k2e) + µ2
k2(µz−λ0) + k2
4µ2θ1cw +k3(ˆqhw −λ1cw) + 2kg
γ2ˆphw + ˆq0
2k2µθ0+ 2k3µθtw + k3+3k1
4µ2!θ1sw +k1µ(µz−λ0) +
Again, the full derivation of the ﬂapping dynamics can be found in .
The inﬂow model is based on a Pitt-Peters dynamic inﬂlow model . The inﬂow is
driven by aerodynamic normal loading harmonics up to the ﬁrst for this particular
application, as shown by Eq. 2.25
It can be demonstrated that
can be linked to respectively the
thrust, roll and pitch moment coeﬃcients as follows
The ﬁrst-order system representative of the inﬂow dynamics is given by
is diagonal and it is called the apparent mass matrix,
is a gain matrix
f(µ, µz, λm)
. More details on the apparennt mass and gain matrices
can be found in .
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  trimming algorithm was developed.
The aircraft system of non-linear equations of motion can generally be described by
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)
and the controls are given by
uT=hδlat δlong δcoll δpedi(2.30)
Given a prescribed funtion in time
the goal is to solve for a subset of
Since there are twenty-ﬁve variables to solve for but just twenty-one constraints,
the values of the
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
plane has the same orientation of the projection of velocity vector on the
ψ= arctan ˙y
Now let xsbe the subset of states to solve for
xs=hu v w p q r φ θ β0β1Sβ1C˙
so that the vector of trim variables is
The trim problem to be solved numerically is therefore given by
f∗(ν) = ˙
xe−f(xs,ue) = 0 (2.35)
is the state derivative target vector of which, for the current applications, the
variables are prescribed as follows
is the rotor state derivative target vector and
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
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
22.214.171.124 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 . 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)
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
interest lays in controlling the output
)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
Figure 2.4: Trim values of the controls and attitude for diﬀerent forward speed
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
0 0 −1 0 0 0 0 0
0 0 0 0 0 1 0 0
corresponds to the roll and pitch attitudes whereas
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.
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.
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
G(s) = 1
τs + 1 (2.45)
The ﬁlter properties were chosen to meet Level 1 speciﬁcations when refering to
ADS-33E  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
is the pseudo-command vector and
is the error as given respectively in
2.47 and 2.48.
are 4-by-4 diagonal matrices identifying respectively the propor-
tional, derivative, integral, and double-integral gain matrices. The gain values are
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 126.96.36.199
188.8.131.52 Outer Loop
184.108.40.206.1 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
cos θsin φsin θcos φsin θ
0 cos φ−sin φ
sin θ−sin φcos θ−cos φcos θ
so that the velocities in the heading frame are given by
In the outer loop also the commanded lateral and forward velocities
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 . The same can be said for the heading
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.
In low speed ﬂight (i.e. speeds not exceeding 40 kts), the lateral velocity is
controlled by acting on the roll angle
through a linear relationship with the
forward acceleration in heading frame. Starting from the lateral velocity equation
˙v=−ru +pw +gsin φcos θ+Y
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.
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.
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.
The last step consists in applying the small angle assumption to the sine and cosine
gsin φcos θ'gφ (2.57)
Eq. 2.52 thus reduces to
The forward speed is controlled by acting on the pitch attitude
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
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
Eq. 2.59 reduces to
The yaw rate
command is simply given by
, which is obtained by extraction
from the ﬁrst order ﬁlter the yaw command goes through.
220.127.116.11.2 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
in Eq. 2.52 cannot be neglected anymore. In a coordinated turn we no longer
since it is assumed
= 0 in order to have the heading coinciding with
the longitudinal speed in heading frame. We can then proceed to the following
The forward speed in body axis
can be approximated with the absolute velocity
V, leading to an expression for the yaw rate commanded to the inner loop
rcmd =gsin φcosθ
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
Let’s now consider the yaw rate equation
νψ=(qsin φ+rcos θ)
substituting 2.73 in it leads to
cos θcos φ
cos θcos φ
Substituting now Eq. 2.70 into 2.75 and carrying out the necessary simpliﬁcations
which rearranged gives the equation for bank angle necessary for a coordinated
φcmd = arctan V
For cases in which the absolute velocity lays between 40
and 60 kts (i.e.
), 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
(φcmdh−φcmdl) + φcmdl(2.78)
(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
gV < Vl
Vh−Vl(φcmdh−φcmdl) + φcmdlVl< V < Vh
gνψV > Vh
νψV < Vl
Vh−Vl(rcmdh−rcmdl) + rcmdlVl< V < Vh
VV > Vh
18.104.22.168 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  it can
be demonstrated that for a Dynamic Inversion controller
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
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
sKI= 0 (2.85)
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
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
which leads to the following closed-loop error dynamics
˙e(t) + KPe(t) + KIZt
0e(τ)dτ +KII Zt
0e(τ)dτ = 0 (2.91)
and, therefore, to
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
This particular type of compensation is applied to the vertical velocity
inner loop. In case of a PI controller,
are set to zero in order to have
second-order error dynamics. The resulting gains are
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-
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
in the outer loop and
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
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
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.
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.
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
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 . 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.
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 . 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 .
Currently reconstruction can only be applied to one rotor.
2.3 Noise Prediction Model
The noise prediction model used in this work, PSU-WOPWOP   , is a
numerical implementation of Farassat’s Formulation 1A  of the Ffowcs Williams-
Hawkings (FW-H) equation . 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
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  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.
Figure 2.5: Flowchart of the simulation process.
Chapter 3 |
Linear Model Analysis
A general non-linear time-invariant system is described by
is the state variable vector of dimension
the input vector of dimension
is the output vector of dimension
. A non-linear system in an equilibrium
condition is described as follows
are the equilibrium state and input vector
Let’s now consider the case of small disturbances on the state, controls and output
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)
By reorganizing and neglecting the terms of second order and higher we obtain
∆ ˙x =∂f(x,u)
∂x1·· · ∂ f1(x,u)
∂x1·· · ∂ fn(x,u)
∂u1·· · ∂ f1(x,u)
∂u1·· · ∂ fn(x,u)
A Taylor series expansion can be performed also on the output
g(xe+∆x,ue+∆u) = g(xe,ue) + ∂g (x,u)
By reorganizing and neglecting the terms of second order and higher we obtain
∂x1·· · ∂ g1(xe,ue)
∂x1·· · ∂ gp(xe,ue)
∂u1·· · ∂ g1(xe,ue)
∂u1·· · ∂ gp(xe,ue)
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
are constant with time, we have a linear time invariant
system (LTI) expressed in the so-called state space form. Being
a closed form solution to a perturbation from initial conditions (i.e.
equilibrium) is given by
x(t) = eAtx0+Zt
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
Avi=λivi, i = 1, . . . , n (3.16)
where the vector
is called eigenvector and the scalar
is the eigenvalue. It be
noted that the eigenvector can either be a line or row vector of dimension
; in the
ﬁrst case it will be called "left eigenvector" whereas in the latter "right eigenvector".
The two cases are denoted respectively by
. It can be demonstrated that if
distinct eigenvalues of
a square matrix of dimension
following euqation holds true
It follows that the response to non-zero initial conditions, meaning
= 0 and
u(t)=0, will be given by
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
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
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
where the eigenvalue’s complex conjugate is given by
The response to non-zero initial conditions can therefore be written as follows
or equevalently, by using complex number properties
Aeσitcos (ωit+φi(x0)) (3.23)
The real and imaginary parts of the eigenvalues,
, 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
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
x, y, z
, which are de-coupled from the other states. The
state vector therefore reduces to:
xT=hu v w p q r φ θ β0β1Sβ1C˙
Residualization is based on singular perturbation theory  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
= 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:
xsT=hu v w p q r φ θi(3.31)
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
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
By substituting the latter result into 3.33, a new expression for the slow states can
Figure 3.2: Eigenvalues associated with the fuselage dynamics.
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.
Figure 3.3: Comparison between the eigenvalues of the full-order model and the
(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.
(c) Frequency response of the roll attitude with respect to a lateral
(d) Frequency response of the yaw rate with respect to a tail rotor
Figure 3.4: Frequency responses of aicraft states with respect to the control inputs
for both the eigth-order and full-order models.
3.4 Linearization of the coupled PSU-HeloSim/CHARM
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
xf= 0 (3.38)
Consequently, also the fuselage state derivatives will reach a steady-state
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
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.
(a) Body velocities derivatives
(b) Angular rates derivatives
(c) Euler Angles derivatives
Figure 3.5: Response of the derivatives of the states to perturbations of the states
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 , 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.
0 0 0 1 sin φetan θecos φetan θe
0 0 0 0 cos θe−sin θe
gcos φecos θe−gsin φesin θe
−gsin φecos θe−gcos φesin θe
0 Ω sec θe
−Ω cos θe0
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
−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
It is evident that the stability derivative
, 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
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
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
However, by comparing the eigenvalues of the reduced order in forward ﬂight
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.
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
Let’s also consider an approximation, found in  and derived from 3.41, of the
frequency of the dutch roll mode
Even though being a small term in magnitude (
) is multiplied
by the forward trim velocity, which is of order
); 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
, along with
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
ζDR ' −Nr+Yv+σDR Lr
2ωDR 1−σDR Lr
It is more tricky here, when compared to the natural frequency, to understand
which derivatives actually contribute to the decrease in damping ratio.
as well as
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
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
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.
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 , where an
exponentially increasing frequency is used. The sweep function, as shown in Fig.
3.10, is given by
0 0 < t < ts
tisin (ωmin(t−ts)) ts< t < (ts+ti)
ts+tiω(τ)dτ (ts+ti)< t < (ts+ti+T)
0t > (ts+ti+T)
where tis the time of the simulation and
ω(τ) = ωmin +C2eC1τ
The description and the numerical values of the various parameters used in Eq.
3.50 are given in Table 3.2.
Parameter Description Value Units
ωmin Lowet frequency 0.3rad/s
ωmax Highest Frequency 13 rad/s
Time at which the
Period of the lowest
Total time of the
Table 3.2: Description of the parameters used for the frequency sweep.
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)
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.
Figure 3.11: Sweep excitation.
(b) Body velocities
(c) Angular Rates
Figure 3.12: Frequency sweep of the coupled simulation for a 120 kts level ﬂight
The data from the frequency sweep is the processed in CIFER
 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
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
Figure 3.13: On-axis Bode and correlation factor plots of the lateral sweep for a
120 kts level ﬂight case.
Chapter 4 |
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
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
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.
4.1.2 Use of the Reduced-Order Models in the Coupled Simu-
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).
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.
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.
(b) Absolute velocity and acceleration
Figure 4.3: Results from a 6◦decelerated descent at 0.1 g of deceleration.
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.
(b) Absolute velocity and acceleration
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.
(b) Absolute velocity and acceleration
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 , ground eﬀect can usually be considered negligible when the speed
with respect to the oncoming ﬂow is two times greater than the induced velocity in
hover (for advance ratios grater than 0.1)
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.
(b) Absolute velocity and acceleration
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.
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
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  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
= 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. 4.9 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
= 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
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
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  . In the current work,
this demonstration was extended to an advanced blade design known to reduce BVI
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) , Airbus Helicopter’s "Blue Edge" blade. The concept behind
this design is described in  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
Figure 4.10: Blue Edge blade concept from Eurocopter (now Airbus Helicopters) .
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
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 (
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 ∆
reconstructed to a resolution of ∆
using the method described in . The
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
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;
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
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
(b) Blue Edge
Figure 4.13: BVI event as predicted by CHARM for the baseline rectangular blade
and the Blue Edge blade.
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.
(a) Rectangular (b) Tapered
Figure 4.15: Acoustic pressure time history at azimuth angle
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).
Chapter 5 |
Conclusions and Recommenda-
tions for Future Work
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
of the coupled simulation, up to frequencies of about 5
. 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
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
and the atmospheric
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
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.
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.
Structure of the Wrapper Code
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
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.
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
Variable Units Description
NAME OF AIRCRAFT
Current aircrafts: B430, S76,
BK117, EC145, H145.
ABSOLUTE VELOCITY kts Allowable values: 0 to 140.
Climb positive, descent negative.
Allowable values: -1519 to 1519.
Initial altitude. No atmospheric
model implemented yet.
ABSOLUTE VELOCITY INCRE-
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
=0. In this case the ﬁnal time
will be given by the time to make
the helicopter reach the speciﬁed
ABSOLUTE VELOCITY INCRE-
Suggested not to go beyond
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.
0: not engaged, 1: engaged. If
engaged fuselage states do not get
MAIN ROTOR COUPLING
0: uncoupled, 1: coupled. If equal
to 0, Simulink simulation will be
run without running CHARM.
TAIL ROTOR COUPLING
0: uncoupled, 1: coupled. If
MAIN ROTOR COUPLING = 0
this ﬂag will be disregarded.
0: uncoupled, 1: coupled. Should
always be set equal to 1 in case of
forward ﬂight. Should be desabled
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-
PSU-WOPWOP TIMES VEC-
Deﬁnes the number of times steps
at which the PSU-WOPWOP ﬁles
are generated. Valid if PSU-
WOPWOP MODE = 1.
PSU-WOPWOP TIMES VEC-
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
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 =
File path of where the HeloSim2.0
folder (up to the folder before).
Table A.1: Master.txt setup.
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
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.
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 ˙
20, 21, 22 Inﬂow states ˙
23, 24, 25 Body velocities derivatives ˙u,˙v,˙w f t/s2
26, 27, 28
Roll, pitch, and yaw rates
29, 30, 31
Euler angles time derivatives
32, 33, 34 NED velocities ˙x,˙y,˙zft/s
35, 36, 37 MBC ﬂapping derivates ˙
38, 39, 40
MBC ﬂapping second deriva-
41, 42, 43 Inﬂow derivatives ˙
44, 45, 46, 47
Controls: lateral and lon-
gitudinal cyclic, collective,
and TR collective
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
β0CH ,β1SC H ,β1CCH rad
Table A.2: HeloSimOut.txt content.
Name Index 1 Index 2 Description
Number of ﬁle
rotor it refers
to. 1: MR, 2:
Contains the loading
on the diﬀerent blades
of the rotor it refers to
avareged over one ro-
number of ﬁle
rotor it refers
to. 1: MR, 2:
Contains the blades
Number of ﬁle
Contains the setup for
including the heli-
velocities and Euler
angles at the time in
the simulation it is
number of ﬁle
rotor it refers
to. 1: MR, 2:
Table A.3: Input ﬁles for PSU-WOPWOP.
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