ABSTRACT
Title of dissertation: INVESTIGATION OF COUPLED
ROTOR-DRIVETRAIN-AIRFRAME-ENGINE
VIBRATIONS USING DYNAMIC
SUBSTRUCTURING
Stacy Sidle, Doctor of Philosophy, 2020
Dissertation directed by: Professor Inderjit Chopra
Department of Aerospace Engineering
Excessive vibrations continue to be a problem that rotorcraft designers are
unable to address until flight testing. Accurate prediction of vibrations throughout
the aircraft is hindered by the complexity of the models required. This dissertation
focuses on utilizing frequency based modeling and substructuring methods to enable
full coupling between the comprehensive analysis modeling of the rotor aeromechan-
ics and finite element models of the drivetrain, airframe, and engines.
A substructuring methodology is developed to analyze the coupled structural
dynamic response of an elastic airframe and engines of a helicopter in response
to main rotor and tail rotor hub loads. Transfer functions of individual compo-
nents (airframe, engine, mount struts and torque tube) are coupled together using
a sub-structuring approach to obtain consistent coupled solutions of the entire sys-
tem. Using this approach, a twin-engine, four-bladed helicopter is analyzed using
NASTRAN-based models of the airframe and engines. This ultra-efficient substruc-
turing approach is validated against the fully coupled NASTRAN model using forced
response studies. Characteristics of the mount properties, i.e., the torque tube stiff-
ness, and aft mount stiffness and damping are systematically varied to study their
effect on the engine vibration response. The fore and aft mount element properties
for minimizing the 8P engine response are identified without increasing 4P response.
A compromise between 4P and 8P response is also identified from parametric stud-
ies of rear mount properties, using just 3 parameters to represent the design space.
Using the sub-structuring approach presented here, future studies can be performed
to rapidly match airframe characteristics with available engines at approximately
1000 times the speed of running a detailed finite element model (millions of degrees
of freedom), without any reduction in accuracy.
Comprehensive vibration analysis of a rotor-airframe-engine-drivetrain system
using a time-domain modal coupling approach was conducted. Pair-wise couplings
of components were performed to isolate the contribution of each component to the
complete coupled system, and the effect of each component (airframe and drive-
train/engine) on rotor loads and hub loads was studied. The drivetrain model was
a 6-dof model consisting of inertia and torsional spring elements, while the airframe
model used was a NASTRAN superelement of a detailed finite element airframe
model. Although drivetrain coupling resulted in elastic twist of the shaft by less
than 0.02 degrees, there were noticeable reductions in the 4/rev chordwise blade
bending moments as well the 4/rev vertical hub force. The airframe coupling pro-
duced very small hub motions, less than 1?10?4 inches, and showed almost identical
trends in both the blade loads and hub loads. 8/rev hub forces and moments were
significantly affected by both the airframe and drivetrain coupling.
INVESTIGATION OF COUPLED
ROTOR-DRIVETRAIN-AIRFRAME-ENGINE
VIBRATIONS USING DYNAMIC
SUBSTRUCTURING
by
Stacy Marie Sidle
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2020
Advisory Committee:
Professor Inderjit Chopra, Chair/Advisor
Professor Anubhav Datta
Professor Olivier Bauchau
Professor James Baeder
Professor Amr Baz
?c Copyright by
Stacy Marie Sidle
2020
Dedication
This work is dedicated to my dad, William Malcolm Sidle IV. I never got to
talk helicopters with him, but I have felt his influence in everything I?ve done in this
field.
This work is also dedicated to my new son, Malcolm. His arrival definitely
threw a wrench in things, but I am proud that I can serve as an example to him of
perseverance and finishing what you start.
ii
Acknowledgments
Although every word of this dissertation was typed by me, there are countless
people who have made this work possible. I have benefitted greatly from several
mentors, not the least of whom is Dr. Inder Chopra, my thesis advisor. I know I was
not always an easy student to advise, and a lesser advisor might have given up on
me, but Dr. Chopra was persistent in encouraging me to finish, even when obstacles
and distractions kept cropping up. VT Nagaraj has also been someone whom I could
always count on to encourage me. Finally, Dr. Ananth Sridharan always impressed
and baffled me with his brilliant mind. Although at times frustrating to work with,
he was always there to answer questions and definitely made my work better.
As a highly social person, I make friends wherever I go, and the University
of Maryland has been no exception. From those who were here when I started and
showed me the ropes from day one ? I continue to be grateful for the friendship
and of Joe Schmaus, Elizabeth Ward, Will Staruk, Ben Berry, and Graham Bowen-
Davies. I?m also grateful for the my contemporaries and the newer crop of students
who allowed me to occasionally impart wisdom and (more often) interrupt their
own productivity when I needed to slack off. A non-exhaustive list of those people
includes: Field Manar, Andrew Lind, Peter Mancini, Andre Bauknecht, Tom Pills-
bury, Steve Sherman, Caitlin King, James Lankford, Dave Mayo, Fred Tsai, Bran-
dyn Phillips, Ravi Lumba, Emily Fesler, Seyhan Gul, Dan Escobar, Tyler Sinotte,
Bharath Govindarajan, Vera Klimchenko, Thomas Herrmann, Ali LeMoine, and
plenty of others. Looking back, the number of people I have included here is prob-
iii
ably one of the reasons it has taken me so long to finish, but I really wouldn?t have
it any other way. I truly hope our paths will cross again, but if they don?t, know
that you have made an impression in my life.
Finally, and most importantly, I?d like to thank my husband, Alex. We were
not married when I applied and was accepted to the program, but he did not hesitate
in telling me to accept the offer, even though it meant moving away from our beloved
Austin, TX. I?m so grateful to have had him with me on this journey, and I truly
could not have done it without him.
iv
Table of Contents
List of Figures viii
List of Tables xii
Abbreviations xiii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Rotor Vibratory Loads . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Airframe Structural Dynamics Modeling . . . . . . . . . . . . 8
1.3.3 Rotor-Airframe Coupling . . . . . . . . . . . . . . . . . . . . . 11
1.3.4 Drivetrain and Engine Studies . . . . . . . . . . . . . . . . . . 16
1.3.4.1 Rotor-Drivetrain Coupling . . . . . . . . . . . . . . . 16
1.3.4.2 Engine-Airframe Coupling . . . . . . . . . . . . . . . 18
1.3.5 Substructuring Methods . . . . . . . . . . . . . . . . . . . . . 19
1.3.5.1 Component mode synthesis . . . . . . . . . . . . . . 22
1.3.5.2 Frequency-based substructuring . . . . . . . . . . . . 25
1.4 Scope of Current Work . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.1 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . 28
2 Methodology 33
2.1 Comprehensive Analysis: PRASADUM . . . . . . . . . . . . . . . . . 33
2.1.1 Overview of Solution Method . . . . . . . . . . . . . . . . . . 34
2.1.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.2.1 Time derivatives of rotation matrices . . . . . . . . . 37
2.1.2.2 Coordinate system definitions . . . . . . . . . . . . . 38
2.1.3 Rigid Body Dynamics . . . . . . . . . . . . . . . . . . . . . . 42
2.1.4 Aerodynamics Model . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.4.1 Rotor Inflow Models . . . . . . . . . . . . . . . . . . 46
Dynamic inflow model . . . . . . . . . . . . . . . . . . 46
Free-vortex wake model . . . . . . . . . . . . . . . . . . 47
v
2.1.4.2 Fuselage and empennage aerodynamics . . . . . . . . 48
2.1.4.3 Tail rotor aerodynamics . . . . . . . . . . . . . . . . 50
2.1.5 Flexible Beam Dynamics . . . . . . . . . . . . . . . . . . . . . 50
2.1.6 Hub Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.1.6.1 Vibratory hub loads . . . . . . . . . . . . . . . . . . 52
2.1.7 Rotor-Airframe Coupling . . . . . . . . . . . . . . . . . . . . . 53
2.1.7.1 Rotor Blade Boundary Condition . . . . . . . . . . . 54
2.1.7.2 Hub Response . . . . . . . . . . . . . . . . . . . . . 56
2.1.8 Trim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.1.8.1 Basic Trim Equations . . . . . . . . . . . . . . . . . 58
2.1.8.2 Trim with Free Wake . . . . . . . . . . . . . . . . . . 60
2.1.8.3 Trim with Flexible Hub . . . . . . . . . . . . . . . . 60
2.2 Engine-Airframe Coupling . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2.1 Mathematics of Dynamical Systems . . . . . . . . . . . . . . . 63
2.2.2 Iterative Harmonic Balance Solution . . . . . . . . . . . . . . 65
2.2.2.1 Subcomponents . . . . . . . . . . . . . . . . . . . . . 65
2.2.2.2 Interaction elements . . . . . . . . . . . . . . . . . . 66
2.2.2.3 Solution procedure . . . . . . . . . . . . . . . . . . . 67
2.2.3 General Formulation of Frequency Based Substructuring . . . 71
2.2.3.1 Rigid Interface . . . . . . . . . . . . . . . . . . . . . 72
2.2.3.2 Flexible Interface . . . . . . . . . . . . . . . . . . . . 74
2.2.3.3 Rigid and flexible interfaces . . . . . . . . . . . . . . 75
2.2.3.4 Damping . . . . . . . . . . . . . . . . . . . . . . . . 76
2.2.4 Validation: Simple Test Case . . . . . . . . . . . . . . . . . . 77
3 Modeling Details 82
3.1 Baseline Helicopter Rotor Model . . . . . . . . . . . . . . . . . . . . . 82
3.2 Drivetrain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3 Airframe Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 Engine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5 Airframe-Engine Interface . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5.1 Fore mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.5.2 Aft mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.5.2.1 Strut geometry and stiffness matrices . . . . . . . . . 88
3.6 Airframe-Engine Coupling Validation . . . . . . . . . . . . . . . . . . 90
4 Engine Mount Parametric Study 115
4.1 Baseline Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 Torque Tube Stiffness Sweep . . . . . . . . . . . . . . . . . . . . . . . 119
4.3 Symmetric Strut Stiffness Sweep . . . . . . . . . . . . . . . . . . . . . 120
4.3.1 Uniform strut stiffness effects . . . . . . . . . . . . . . . . . . 121
4.3.2 Damping effects . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3.3 Individual strut stiffness effects . . . . . . . . . . . . . . . . . 122
4.4 Asymmetric Strut Stiffness Sweep . . . . . . . . . . . . . . . . . . . . 123
4.5 Minimizing Engine Response . . . . . . . . . . . . . . . . . . . . . . . 124
vi
4.5.1 Minimizing indvidual mount response . . . . . . . . . . . . . . 125
4.5.2 Minimizing maximum mount response . . . . . . . . . . . . . 126
4.5.3 Minimizing mount response at pairs of points . . . . . . . . . 127
4.5.4 Minimizing total response . . . . . . . . . . . . . . . . . . . . 128
4.5.5 Symmetric strut design . . . . . . . . . . . . . . . . . . . . . . 129
4.5.6 Asymmetric strut design . . . . . . . . . . . . . . . . . . . . . 130
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Coupled Rotor-Airframe-Drivetrain Analysis 160
5.1 Baseline Coupling Results (Prescribed Airloads) . . . . . . . . . . . . 160
5.2 Rotor-Airframe-Drivetrain Coupling . . . . . . . . . . . . . . . . . . . 162
5.2.1 Hub motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.2.2 Blade loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.2.3 Hubloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2.4 Airframe Vibrations . . . . . . . . . . . . . . . . . . . . . . . 166
5.3 Engine modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6 Summary and Conclusions 201
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.2 Key Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.2.1 Engine-Airframe Coupling . . . . . . . . . . . . . . . . . . . . 203
6.2.2 Rotor-Drivetrain-Airframe-Engine Coupling . . . . . . . . . . 204
6.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 206
Bibliography 209
vii
List of Figures
1.1 Frequencies of vibration throughout aircraft [77] . . . . . . . . . . . . 29
1.2 Vibration levels as a function of forward flight speed [78] . . . . . . . 30
1.3 Impact of vibrations on helicopter development [11] . . . . . . . . . . 31
1.4 Representation of system dynamics in three domains [59] . . . . . . . 32
2.1 PRASADUM non-rotating coordinate systems (adapted from [79]) . . 41
2.2 PRASADUM blade coordinate systems [79] . . . . . . . . . . . . . . 43
2.3 Weissinger-L lifting surface model for rotor blade . . . . . . . . . . . 49
2.4 Free-vortex wake model of rotor . . . . . . . . . . . . . . . . . . . . 49
2.5 Overview of substructuring components . . . . . . . . . . . . . . . . . 70
2.6 Simple spring-mass system with two substructure components . . . . 73
2.7 Schematic of 8-DOF mass-spring-damper system m1 = m3 = m7 =
m8 = 1, m2 = m6 = 2, m4 = m5 = 4 ki = 1, ci = 0.01, i = 1, 2, ...9 . . 79
2.8 Comparison of NASTRAN and susbtructuring results for validation
test case without any damping . . . . . . . . . . . . . . . . . . . . . . 80
2.9 Comparison of NASTRAN and susbtructuring results for validation
test case with different damping types . . . . . . . . . . . . . . . . . 81
3.1 Blade mass distribution . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.2 Blade flap bending distribution . . . . . . . . . . . . . . . . . . . . . 93
3.3 Blade lag bending distribution . . . . . . . . . . . . . . . . . . . . . . 94
3.4 Blade torsional stiffness distribution . . . . . . . . . . . . . . . . . . . 95
3.5 Rotor blade natural frequencies . . . . . . . . . . . . . . . . . . . . . 96
3.6 RSRA drivetrain model (Ref. [85]) . . . . . . . . . . . . . . . . . . . 101
3.7 Mode shapes of drivetrain model with free-free boundary condition . 103
3.8 Mode shapes of drivetrain model with fixed engines . . . . . . . . . . 104
3.9 NASTRAN FE model of airframe . . . . . . . . . . . . . . . . . . . . 105
3.10 NASTRAN FE model of GE CT7-8 engine . . . . . . . . . . . . . . . 108
3.11 Overview of Helicopter/Engine/Airframe interfaces, mounts and forc-
ing node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.12 Schematic of engine mount structures . . . . . . . . . . . . . . . . . . 112
3.13 Geometry of axial stiffness element . . . . . . . . . . . . . . . . . . . 113
viii
3.14 Comparison of substructuring and NASTRAN results for coupled
airframe-engine model: lateral and vertical translation response mag-
nitude and phase of left engine aft mount point to unit amplitude
forcing at main rotor hub . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.1 Hub load amplitudes (CW/?=0.082) . . . . . . . . . . . . . . . . . . 133
4.2 Breakdown of 4P engine mount response from individual hub load
component contributions (CW/?=0.082, ?=0.233) . . . . . . . . . . . 134
4.3 Breakdown of 8P engine mount response from individual hub load
component contributions (CW/?=0.082, ?=0.233) . . . . . . . . . . . 135
4.4 Total transverse response with forward speed (CW/?=0.082) . . . . . 136
4.5 4P transverse response of engine mount points for various torque tube
stiffnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.6 Schematic of engine mount structure . . . . . . . . . . . . . . . . . . 138
4.7 Transverse response of engine mount points for variations in all 6
strut stiffness matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.8 Transverse 4P response at engine mounts for variations in stiffness
and damping of all 6 struts. . . . . . . . . . . . . . . . . . . . . . . . 140
4.9 Transverse 8P response at engine mounts for variations in stiffness
and damping of all 6 struts. . . . . . . . . . . . . . . . . . . . . . . . 141
4.10 Transverse 4P response of engine mount points for left/right symmet-
ric variations in each individual strut holding all others at baseline
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.11 Transverse 8P response of engine mount points for left/right symmet-
ric variations in each individual strut holding all others at baseline
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.12 Effects of changes in left or right strut stiffness on 4P engine response 144
4.13 Effects of changes in left or right strut stiffness on 8P engine response 145
4.14 Effect of minimizing response at individual engine mount point on re-
sponse at other points: change in amplitude with respect to vibration
for baseline strut design . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.15 Effect of minimizing response at individual engine mount point on re-
sponse at other points: change in amplitude with respect to vibration
for baseline strut design . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.16 Effect of minimizing response of both fore or aft mount points: change
in amplitude with respect to vibration for baseline strut design . . . . 148
4.17 Effect of minimizing response of left or right engine mount points
on response at other points: change in amplitude with respect to
vibration for baseline strut design . . . . . . . . . . . . . . . . . . . . 149
4.18 Effect of minimizing engine total engine response: change in ampli-
tude with respect to vibration for baseline strut design . . . . . . . . 150
4.19 Effect of stiffer struts and stiffer torque tube on 4P and 8P engine
mount response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
ix
5.1 Effects of coupling on flap bending moments for UH-60A using mea-
sured airloads (C8534) . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2 Effects of coupling on flap bending moment peak-to-peak amplitude
and harmonics for UH-60A using measured airloads (C8534) . . . . . 170
5.3 Effects of coupling on flap bending moment higher harmonics for UH-
60A using measured airloads (C8534) . . . . . . . . . . . . . . . . . . 171
5.4 Effects of coupling on chordwise bending moments for UH-60A using
measured airloads (C8534) . . . . . . . . . . . . . . . . . . . . . . . . 172
5.5 Effects of coupling on chordwise bending moment peak-to-peak am-
plitude and harmonics for UH-60A using measured airloads (C8534) . 173
5.6 Effects of coupling on chordwise bending moment higher harmonics
for UH-60A using measured airloads (C8534) . . . . . . . . . . . . . . 174
5.7 Effects of single degree of freedom 4P hub motion on mid-span flap
bending moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.8 Effects of single degree of freedom 4P hub motion on mid-span chord-
wise bending moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.9 Effects of single degree of freedom 4P hub motion on mid-span torsion
moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.10 Effects of single degree of freedom 8P hub motion on mid-span flap
bending moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.11 Effects of single degree of freedom 8P hub motion on mid-span chord-
wise bending moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.12 Effects of single degree of freedom 4P hub motion on mid-span torsion
moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.13 1/2 Peak-to-peak hub translation resulting from coupling with air-
frame and drivetrain . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.14 1/2 Peak-to-peak hub roll and pitch rotation resulting from coupling
with airframe and drivetrain . . . . . . . . . . . . . . . . . . . . . . . 182
5.15 1/2 Peak-to-peak hub yaw rotation resulting from coupling with air-
frame and drivetrain . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.16 Effects of coupling on mid-span blade loads . . . . . . . . . . . . . . . 184
5.17 Effects of coupling on vibratory flap bending . . . . . . . . . . . . . . 185
5.18 Effects of coupling on vibratory flap bending higher harmonics . . . . 186
5.19 Effects of coupling on vibratory lag bending . . . . . . . . . . . . . . 187
5.20 Effects of coupling on vibratory lag bending higher harmonics . . . . 188
5.21 Effects of coupling on vibratory torsion moment . . . . . . . . . . . . 189
5.22 Effects of coupling on vibratory torsion momment higher harmonics . 190
5.23 Effects of coupling on 4P hub loads . . . . . . . . . . . . . . . . . . . 191
5.24 Effects of coupling on 8P hub loads . . . . . . . . . . . . . . . . . . . 192
5.25 Effects of coupling on 4P pilot/co-pilot seat vibrations . . . . . . . . 193
5.26 Effects of coupling on 8P pilot/co-pilot seat vibrations . . . . . . . . 194
5.27 Effects of engine modeling on 4P hub loads . . . . . . . . . . . . . . . 195
5.28 Effects of engine modeling on 8P hub loads . . . . . . . . . . . . . . . 196
5.29 Effects of engine modeling on 4P pilot/co-pilot seat vibrations . . . . 197
5.30 Effects of engine modeling on 8P pilot/co-pilot seat vibrations . . . . 198
x
5.31 Effects of engine modeling on 4P engine vibrations . . . . . . . . . . . 199
5.32 Effects of engine modeling on 8P engine vibrations . . . . . . . . . . . 200
xi
List of Tables
3.1 Main rotor properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Main Rotor Blade Frequencies . . . . . . . . . . . . . . . . . . . . . . 97
3.3 Airframe properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4 Tail rotor geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 Rotor blade discretization parameters . . . . . . . . . . . . . . . . . . 100
3.6 Wake discretization parameters . . . . . . . . . . . . . . . . . . . . . 100
3.7 Inertia and stiffness properties of RSRA drivetrain model (Ref. [85]) . 102
3.8 Speed ratios of drivetrain components . . . . . . . . . . . . . . . . . . 102
3.9 Natural frequencies of drivetrain model with free-free boundary con-
dition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.10 Natural frequencies of drivetrain model with fixed engines . . . . . . 104
3.11 Grid point ID numbers for NASTRAN airframe model . . . . . . . . 106
3.12 Natural frequencies for baseline airframe model . . . . . . . . . . . . 107
3.13 Grid point ID numbers for NASTRAN engine model . . . . . . . . . 109
3.14 Natural frequencies for engine model . . . . . . . . . . . . . . . . . . 110
3.15 Coordinates for aft engine struts . . . . . . . . . . . . . . . . . . . . . 113
4.1 Amplitude of 4P engine mount point displacement response (in) to
unit hub force (lb) or moment (ft-lb) . . . . . . . . . . . . . . . . . . 152
4.2 Amplitude of 8P engine mount point displacement response (in) to
unit hub force (lb) or moment (ft-lb) . . . . . . . . . . . . . . . . . . 153
4.3 Baseline vibration levels at engine mount points at 100 knots . . . . . 154
4.4 Strut stiffness factors required to achieve minimum 4P or 8P response
at each engine mount . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.5 Largest possible reduction from baseline in 4P and 8P transverse for
each mount point for symmetric and asymmetric strut stiffness values 156
4.6 Strut stiffness factors that produce minimum response for different
targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.7 Summary of parametric studies performed . . . . . . . . . . . . . . . 158
4.8 Strut stiffness factors that produce minimum response for different
targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
xii
Abbreviations ?
ayz Transverse acceleration, ayz = a2y + a
2
z
C Damping matrix
K Stiffness matrix
M Mass matrix
Mx Roll moment
My Pitching moment
NP N per rev
? Azimuth angle
Y Admittance matrix
Z Impedance matrix
 ODE residual
? Natural frequency
CBM Chordwise bending moment
CFD Computational fluid dynamics
CMS Component mode synthesis
CSD Computational structural dynamics
DAMVIBS Design Analysis Methods for Vibrations
DOF Degree of freedom
FBM Flap bending moment
FBS Frequency based substructuring
FE Finite element
FEM Finite element model
FRF Frequency response function
NASTRAN NASA Structural Analysis
PRASADUM Parallelized Rotorcraft Analysis for Simulation and Design, UMD
TM Torsion moment
xiii
1 Introduction
Vibrations have been acknowledged as a problem in helicopters even preceding
their inception, in the early days of the autogiro. Though the mechanisms responsi-
ble for helicopter vibrations are complex and have not been completely understood
by designers, the existence of helicopter vibration is likely unmistakable to anyone
who has ever heard a helicopter fly overhead. One must surely assume that the char-
acteristic ?whop-whop-whop? sound is accompanied by some significant vibrations.
Vibrations are understood to arise from periodic motion, so the cyclic nature of the
helicopter rotor motion, unsurprisingly, gives rise to vibrations. Furthermore, the
need to reduce helicopter vibrations is not difficult to understand for anyone who
has driven a car down a bumpy road. Indeed, continued exposure to high levels of
vibrations is, at the very least, uncomfortable, if not dangerous to helicopter pas-
sengers and crew. Not only that, but the vibrations felt by the human occupants
of the helicopter are transmitted throughout the entire aircraft, leading to fatigue
and eventual failure of structural components. Finally, it appears that vibrations
can affect the performance of the engines once installed in the vehicle.
Despite the large negative effects of vibrations in helicopters, actual vibra-
tion levels inside the airframe are difficult to predict precisely, and are usually not
1
known until flight testing of prototype designs. At this post-production stage, a
full redesign is unlikely, so the problems are addressed with add-on fixes such as
absorbers or isolators. Because these additional components were not necessarily
included in the vehicle design, they carry penalties in terms of cost, weight, and
performance. Although these penalties will always exist for vibration reduction so-
lutions, if vibration prediction capabilities are improved, appropriate trade studies
could be conducted during design to mitigate them.
All of these factors highlight the importance of modeling how the rotor, air-
frame, drivetrain, and engines interact to develop the most complete understanding
of the total vibration story. This understanding will enable designers of future heli-
copter concepts to better anticipate possible vibration issues and include solutions in
the design process. The present research addresses the helicopter vibration problem
by using a few different analysis techniques to examine the ways in which com-
ponents of the coupled rotor-airframe-drivetrain-engine system interact with each
other. This chapter serves to introduce the thesis by way of explaining the back-
ground and motivation for the analysis, examining the relevant existing work in the
field and outlining the contributions of the dissertation.
1.1 Background
Vibrations in helicopters stem from a variety of sources, including the main
and tail rotors, engine, and transmission as well the interaction between the complex
rotor wake and the airframe. Figure 1.1 illustrates which characteristic frequencies
2
dominate the vibrations at various points in the aircraft. As seen in the figure, the
main source of vibrations in helicopters is the main rotor, that is, the frequencies
present in the vibrations are integer multiples of the blade passage frequency, Nb/rev,
where Nb is the number of blades on the main rotor. Frequencies associated with the
tail rotor and drivetrain are also present, though in smaller, more isolated regions
of the airframe.
To understand the mechanisms that contribute to the vibrations from the main
rotor, it is helpful to consider the trend of vibration level in helicopters as a function
of airspeed, shown in Fig. 1.2. The two regions of relatively higher vibrations point
to the leading cause of main rotor vibrations. First, at the low-speed transition
region between hover and forward flight, ? = 0.1, the rotor is operating largely in its
own wake and experiences blade-vortex interactions, where the trailed vortices from
each preceding blade are impinging on the subsequent passing blades. At moderate
increased speeds, a reduction in vibration is seen as the rotor wake is swept behind
the rotor. Finally, at higher speeds, ? > 0.3, compressibility effects are introduced
due to the transonic speeds at the blade tips, leading to further increases in the
vibration levels. As such, even if a helicopter might be capable of achieving higher
speeds, vibrations could limit the forward flight speeds a helicopter could achieve.
1.2 Motivation
As previously noted, helicopter vibrations have a number of negative impacts
for human occupants, structural components, and engine performance. As a result,
3
a variety of approaches have been developed over the years to alleviate the problem
of vibration in helicopters. The methods used to address vibration differ in their
categorization as either active or passive methods, as well as where in the sequence
of transmission from the rotor to the fuselage the method is addressing the problem,
which Loewy categorizes as ?source alleviators?, ?force attenuators?, or ?amplitude
reducers? [1]. The source alleviators are methods that address the problem of vi-
brations at the source by attempting the reduce the level of vibration produced by
the rotor. Methods that fall into this category are passive devices such as pen-
dulum [2] or bifilar absorbers [3], or active methods like higher harmonic control
(HHC) [4], individual blade control (IBC) [5], or active trailing edge flaps [6]. Force
attenuators operate between the source of the vibration, namely the main rotor,
and the airframe. They do nothing to reduce the vibrations produced by the rotor,
but instead reduce the level of vibration that is transmitted to the airframe. These
methods typically utilize flexible mount structures connecting the rotor or gear box
to the airframe [7]. Amplitude reducers are used in specific points in the airframe to
counterbalance the vibrations. This typically takes the form of vibration isolators
for specific components in the vehicle, such as seats, fuel tanks, or other equip-
ment where a low vibration environment is highly desirable. Additionally, dynamic
absorbers can be placed at different points in the airframe to effectively mitigate
vibrations. These might utilize components such as batteries as the mass in the
absorber. A number of papers exist that review the details and relative merits and
drawbacks of the various methods, the most widely cited being by Reichert [8] and
Loewy [1]. Although both papers are more than 30 years old, they represent a com-
4
prehensive view of the primary methods of vibration reduction still being used and
investigated today. Friedmann and Millott and Kesslier also provide more recent
reviews of active control strategies [9, 10].
Despite the demonstrated effectiveness of these vibration reduction strategies,
they have generally been utilized to address vibration issues in existing helicopters
when flight tests made the existence of such issues apparent. Even for helicopters
that are not yet in production, at the point of flight testing, only small changes to
the overall design are economically feasible. In 1988, Kvaternik estimated that the
weight penalty associated with with vibration reduction devices could be as much
2.5% of the gross weight, resulting in a 10-15% reduction in mission payload [11].
Further, Kvaternik illustrated the additional cost associated with vibration, repro-
duced in Fig. 1.3. From these observations, it is clear that adequately accounting
for vibrations at an early stage of the design process will be essential for efficiently
achieving close to ?jet-smooth? flight in the next generation of rotorcraft.
Accurate prediction of vibrations requires careful consideration of the three
main components involved in computing vibration levels throughout the aircraft:
(1) the vibratory forcing exerted on the aircraft, primarily by the main rotor, (2)
the vibratory response of the elastic airframe, and (3) the coupling between the main
rotor and airframe. Further, if vibration effects on the engine are to be examined,
as in this work, detailed modeling of the engines as well as the coupling between the
engines and airframe are required.
One of the difficulties associated with predicting the vibrations of the engine
due to main rotor forcing lies in the fact that the finite element models associated
5
with the engine and airframe are produced by separate entities. Due to various re-
strictions, detailed finite element models of either sub-system, or models refined by
system identification techniques based on shake test data are not readily available.
Consistent coupling of the two models requires significant disclosure of proprietary
information and often necessitates modifications to one or both mathematical mod-
els. Additionally, due to the degree of complexity of the airframe finite element
(FE) models required for accurate prediction of loads, parametric studies or opti-
mization sweeps are computationally intensive. To resolve these practical difficulties
and circumvent logistical and legal restrictions, dynamic substructuring can be used
to easily couple models together without sharing the proprietary FE models of the
constituent components.
The present research is focused on fully integrating a comprehensive rotor
analysis with a flexible drivetrain as well as coupling finite element models of the
airframe and engine. Due to the difficulty in acquiring access to full proprietary mod-
els of the airframe or engine from the respective manufacturers, a frequency-based
substructuring method was used to facilitate the coupling between the airframe and
engine models.
1.3 Literature Review
Accurate vibration prediction in helicopters is a multi-faceted and not com-
pletely resolved problem. It requires accurate vibratory loads from the rotor, de-
tailed structural dynamics models of the airframe and other sub-components, and
6
full accounting of the coupling between the rotor, airframe, and all connected com-
ponents. Prior research associated with these various facets are discussed in this
section. While not all of these aspects are directly addressed in this research, the
existing research provides necessary context. The research discussed includes the
state-of-the-art in rotor vibratory loads prediction and airframe structural model-
ing, rotor-airframe and rotor-drivetrain coupling, and the substructuring methods
that are the basis for the engine-airframe coupling examined in this work.
1.3.1 Rotor Vibratory Loads
Because the primary source of vibrations in helicopters is the main rotor,
the ability to accurately predict airframe and engine vibratory loads requires a
reliable aeromechanics model of the rotor. Great strides have been made in the past
decade at significantly improving the prediction of rotor loads. A summary of these
improvements and the current state-of-the-art in rotor loads predictions is given by
Datta, Nixon, and Chopra [12].
Periodic examination of the state-of-the-art in rotor loads prediction has been
done over the past 45 years through workshop-type settings, in which specialists
from government, industry, and academia have compared independent results for a
common problem, in an effort to share experience. Ormiston provided one of the
first attempts at assessing the state-of-the-art in rotor loads prediction [13]. For
this study, airloads, bending moments, vibratory hub shears and other results for a
hypothetical, idealized rotor model were compared among ten different participants.
7
Disagreements among the results were attributed to differences in numerical solution
methods as well as structural dynamics and aerodynamics modeling.
In the 1990s, another multi-agency study was conducted, this time comparing
vibratory hub load predictions among eight advanced aeroelastic codes with test data
for the Westland Lynx helicopter [14]. High speed flight was found to represent a
particular deficiency in vibratory loads calculations. Free wake modeling was found
to be particularly important in improving predictions at both low and high speed,
however further enhancements to the aerodynamic modeling such as fuselage upwash
and unsteady aerodynamics provided only moderate enhancements, and most codes
differed from flight test results by around 50% at the high speed flight condition.
More recent studies have focused on the inclusion of computational fluid dy-
namics (CFD) methods in rotorcraft analysis. The use of CFD drastically improves
the prediction of aerodynamic loads on the rotor, leading to significant improve-
ments in rotor loads prediction capabilities. A review article by Datta, Nixon, and
Chopra [12] provides a summary of the more recent improvements made in rotor
loads predictions, specifically due to the increased use of CFD.
1.3.2 Airframe Structural Dynamics Modeling
One way the industry sought to decrease the reliance on add-on vibration re-
duction devices was through improved modeling of the airframe structural dynamics.
In the 1980s, NASA, together with the four major helicopter manufacturers carried
out the DAMVIBS program (Design Analysis Methods for Vibrations) with a goal
8
to improve the capabilities of airframe finite element models for predicting vibratory
loads throughout the airframe [15]. Bell, Boeing, Sikorsky, and McDonnell-Douglas
all participated in the program, and their individual efforts are summarized in Ref-
erences [16], [17], [18], and [19].
The efforts of the program focused on four major technology areas: (1) finite
element modeling, (2) difficult components studies, (3) coupled rotor-airframe vi-
brations, and (4) airframe structural optimization. The bulk of work by the industry
partners was focused on the first two of these, while the final two were meant to
include contributions from government and academia subsequent to completion of
the efforts in the first two.
The goals of the finite element modeling portion of the program were not only
to develop detailed finite element models, but also to gain a better understanding
of the importance of including various components in the models. As part of this
effort, finite element models were developed for the metal airframes of the AH-
64A [20], UH-60A [21], and Model 360 [22], as well as two composite airframes (part
of the Advanced Composite Airframe Program (ACAP) of the Army) D292 [23]
and S-75 [24]. The models were correlated against vibration test data, and in all
instances, satisfactory agreement between test and analysis was found only up to
about 10 Hz, with generally poor agreement above 20 Hz. It was also found that
the effects of testing boundary conditions were important and that better modeling
of damping was needed in the finite element analyses.
The DAMVIBS difficult components studies [25,26] aimed to determine which
of the secondary structures of the aircraft have the most significant effects on the
9
prediction of the airframe natural frequencies. These structures include compo-
nents such as landing gear, canopy glass, tail rotor drive shaft, main rotor py-
lon/transmission, and engine and fuel, are difficult to model and as such, were
typically represented as lumped masses in the finite element model. Components
were systematically removed from the airframe, and ground vibration tests were
performed after each removal. Test results were compared with analysis for the
corresponding model. Overall, it was found that detailed modeling of secondary
structure panels and canopy glass as well as the tail boom was required for satisfac-
tory agreement between test and analysis, while simplified lumped mass or elastic
line representations were adequate for components such as the tail rotor drive shaft,
engines, fuel, main rotor pylon, transmission, and soft mounted black boxes. For this
work, it is worth noting that although the lumped mass engine model was deemed
adequate for predicting the overall airframe natural frequencies, a detailed engine
model would clearly be required for vibration studies on the engine itself.
Finally, each of the participating companies used their own in-house prediction
codes to conduct coupled rotor-airframe analysis of the same AH-1G airframe and
rotor models and compared results with existing vibration test data. The predicted
2/rev vibrations for this 2-bladed helicopter did not agree well with the test data for
any of the participants. Although some secondary studies suggested that aerody-
namic interactions between the rotor wake and airframe, particularly the tail boom,
could be as important as the mechanical coupling between the rotor and airframe,
this portion of the program largely ended with the conclusion that existing com-
prehensive analysis codes were not good enough for reliable vibrations prediction
10
during design. Though specific improvements to coupled rotor-airframe vibrations
prediction were not made under the DAMVIBS program, the results of those initial
studies led the way for industry and academia to perform focused research in this
area in the ensuing years. This work will be discussed in the next section.
1.3.3 Rotor-Airframe Coupling
As indicated previously, vibration analysis in helicopters has historically been
carried out by applying hub loads from a separate rotor analysis as a forcing func-
tion on the airframe model in order to determine vibration levels at various points
throughout the airframe. In this situation, the hub loads are computed with the
assumption that the hub is fixed. Any interaction between the rotor and airframe
is accounted for only by the addition of an ?equivalent? rotor mass in the airframe
model. However, this level of analysis does not adequately account for the coupling
between the rotor and the airframe, that is the effect that the flexibility of the air-
frame might have on the rotor hub loads that are being applied. This section will
provide an overview of the ways in which this coupling has been modeled.
Rotor-body coupling, a general term used to describe the act of coupling the
motion of the rotor hub to some other body, whether flexible or not, has been accom-
plished in a variety of ways. The earliest methods of coupling rotor and fuselage dy-
namics employed impedance matching techniques. The rotor or airframe impedance
is the mathematical relationship between the vibratory forces and displacements at
the hub. An impedance matching technique seeks a solution to the individual rotor
11
and airframe problems in which the impedance is the same between the two parts
? that is, the solution for the coupled rotor-airframe should converge to the same
forces and displacements at the hub. Gerstenberger and Wood [27] were the first
to describe a general analytical method for helicopter analysis in which the flexible
blade dynamics were coupled to the dynamics of the fuselage through impedance
matching. Staley and Sciarra [28] utilized an explicit impedance matching technique
in which the rotor hub loads were assumed to be composed of those calculated with
a fixed hub condition in conjunction with a correction due to small vibratory hub
motions. The airframe dynamics was represented by a mobility matrix relating
the vibratory hub motion to applied unit vibratory hub forces. Similar impedance
matching techniques have also been used in studies by Hohenemser and Yin [29],
Hsu and Peters [30], and Gabel and Sankewitsch [31]. These studies utilized very
simplified rotor models and aerodynamics, but were useful in understanding some
basic phenomena and providing support for the notion that rotor-body coupling
has a significant impact on vibratory loads and should be accounted for if accurate
vibration predictions are desired.
Other researchers have developed fully coupled analyses in which the fuselage
equations of motion are coupled to the rotor equations of motion. Many of these
analyses utilize only linear terms in the rotor analysis and again use highly simplified
aerodynamic and/or fuselage modeling. Analysis by Kunz [32] included only flapping
motion in the rotor, rigid pylon and rigid fuselage with hub-fixed as well as flexibility
in pitch allowed. Rutkowski [33] only considered vertical bending of rotor. Again,
although the numerical results of these simplified studies are of limited use, the
12
demonstration of the effects of coupling are very valuable in understanding the
fundamentals of those interactions and spurring further research.
Stephens and Peters [34] did not present any numerical results, but performed
a comparison of an iterative method to fully coupled rotor-body equations. Although
the iterative method has some disadvantages with respect to implementation and
convergence, it was demonstrated that both methods were equally rigorous, despite
implications that the iterative method was more approximate.
Given the importance of nonlinear forces in calculating rotor vibratory loads, a
coupled analysis in which the flap, lag, and torsion dynamics of the rotor blades are
fully accounted for is necessary. Warmbrodt and Friedmann [35] derived equations
representing consistent structural coupling between an elastic rotor and six rigid
body modes of the fuselage. Bir and Chopra [36] similarly developed analysis in
which an elastic rotor was coupled to a fuselage with rigid body degrees of freedom
at the hub. The analysis was developed specifically for investigating the response
of a rotor due to a three-dimensional gust, though some vibration results were also
noted.
Fledel [37] developed a coupled rotor-body formulation that accounted for dy-
namic coupling between rotor and body as well as the interactional effect of body up-
wash on the rotor. The body could undergo vertical bending and pitching/plunging
rigid body motions. Several parameters were investigated for effects on vibratory
hub loads and body vibrations. These included blade stiffness, hub location, fuse-
lage stiffness and rotor/body clearance. The rotor and fuselage stiffness as well as
rotor/body clearance were found to have a large effect on vibratory hub loads.
13
Works by Juggins [38] and Hansford [39] detail the development of Coupled
Rotor Fuselage Model (CRFM) at Westland Helicopter. This is a comprehensive
rotor analysis code using fully coupled non-linear aeroelastic equations with hub
motion. Early results showed that hub load predictions were significantly improved
with the addition of hub motion. Hansford also noted that a high fidelity NASTRAN
fuselage model would be required for accurate prediction of hub motion and airframe
vibration response.
Vellaichamy and Chopra [40, 41] performed a coupled rotor-body vibration
analysis in which a flexible stick model of an airframe was coupled to a rotor with
flexible blades undergoing elastic flap, lag, torsion, and axial deflection. The cou-
pling was accomplished through an iterative procedure. These studies emphasized
the role of blade structural nonlinearities and rotor-body coupling in the prediction
of vibration magnitudes.
Chiu and Friedmann [42] developed a coupled rotor-fuselage aeroelastic model
that incorporated active control of structural response (ACSR) in the fuselage. Their
results showed again that nonlinear couplings and fuselage motion is important for
vibration prediction, however the trim and blade elastic response were shown to be
insensitive to fuselage motion.
A study conducted by Yeo and Chopra investigated the effects of rotor-fuselage
coupling on vibration analysis [43?47]. For this analysis, a coupled rotor/fuselage vi-
bration model was developed that incorporated consistent structural, aerodynamic,
and inertial coupling effects. Initial studies utilized an elastic line model repre-
senting an AH-1G helicopter coupled with both a 4-bladed hingeless rotor [43] and
14
the 2-bladed teetering rotor associated with the AH-1G model [46]. These stud-
ies demonstrated that the flexible fuselage could have a pronounced effect on the
predicted fuselage vibrations.
A subsequent study examined the effects of refined models of the fuselage [47]
with results compared to available flight test data. Three fuselage models were
compared: an elastic line model, a 3D NASTRAN model, and a refined 3D model
that included difficult components. It was determined that the refined fuselage
model was essential in predicting high frequency modes. Their results showed good
agreement in the prediction of 2/rev magnitude and phase of pilot seat vibration,
while the 4/rev predictions were fair only in magnitude. The main rotor pylon
flexibility was shown to have significant impact on the 2/rev vibration prediction
and refined aerodynamics, especially a free wake model, were also necessary for
improved vibration prediction.
Yang and Chopra also performed a coupled rotor-fuselage vibration study for
an articulated rotor helicopter - the SH-60B, including the bifilar [48, 49]. The
analysis incorporated consistent structural, aerodynamic and inertial couplings and
was based on finite element methods in space and time. The coupled rotor, absorbers
and fuselage equations were transformed into the modal space and solved in the
fixed coordinate system. The work showed acceptable trends for magnitude and
phase with flight-test data and demonstrated the importance of modeling the bifilar
for correctly predicting fuselage vibrations. They also investigated the sensitivity
of vibration prediction to various system faults such as blade dissimilarity due to
asymmetric track/balance masses.
15
1.3.4 Drivetrain and Engine Studies
As highlighted in the previous section, inclusion of an airframe model has been
recognized as an important factor in understanding helicopter vibrations, however,
the same is not true of the drivetrain or engines. While there have been some studies
that include helicopter engines and other elements of the drive system such as gear
boxes, those studies have largely been focused on issues of stability and handling
qualities in flight dynamics. Recently, some vibration studies have begun to include
drivetrain models in the analysis. Additionally, helicopter manufacturers have long
recognized that certain vibration issues arise as a result of coupling between the
engines and airframes. This section will highlight the key findings in the area of
rotor-drivetrain coupling as well the known issues associated with engine-airframe
coupling.
1.3.4.1 Rotor-Drivetrain Coupling
As mentioned previously, the fixed hub condition often used in rotor analysis
does not adequately account for the motion of the hub due to coupling with the
rest of the aircraft. The previous section examined the coupling of the rotor with a
flexible airframe model, however, that is not the only source of motion transmitted
to the rotor. Power from the helicopter engine is transmitted to the rotor via the
drivetrain, which consists of a series of flexible rotating shafts connected through
gears. In the prediction of rotor vibratory loads, discrepancies have continued to
exist between analysis and flight test for the chordwise bending loads. It has been
16
suggested that coupling between the rotor and drivetrain could influence the lag
bending modes of the rotor, and so exclusion of a flexible drivetrain could be con-
tributing to these deficiencies. Despite this, few studies exist that investigate the
coupled rotor-drivetrain system with regard to vibratory loads.
Yeo and Potsdam recently examined coupled CFD-CSD analysis of rotor loads
in which both a simple 1-dof drivetrain model as well as a more complicated model
were included in the analysis [50, 51]. When a 1-dof drivetrain model was used,
its stiffness and inertia properties were tuned to match either the drivetrain first
or second torsion frequency typically found in rotorcraft. It was found that the
first torsion frequency affected the phase of the structural loads, while the second
torsion frequency increased both the peak-to-peak and 4/rev components of the
chord bending moment, improving the overall correlation with test data. Inclusion
of a more detailed drivetrain model resulted in very small improvements in the
correlation of the 4/rev and 5/rev harmonics of the chord bending moment.
Similar work done by Min, et. al. [52] incorporated a simplified drivetrain
model into the analysis of structural loads. It was found that the drivetrain model
did not appreciably alter the natural frequencies of the rotor, though the chordwise
and flatwise components of the mode shape for the collective lag bending mode and
first torsion mode were significantly altered. Further, the 4-6th harmonics of the
blade edgewise bending moments were significantly increased from the fixed hub
condition, resulting in improved correlation with flight test data for 4P and 6P,
although the 5P correlation was noticeably worse.
The lack of available studies into the effects of drivetrain modeling on rotor
17
vibrations points to a need for further study into this area.
1.3.4.2 Engine-Airframe Coupling
Understanding the engine response to vibratory hub loads is of particular
importance given the tight tolerances in the turbomachinery. Additionally, excessive
vibrations in the engine could have an adverse impact on engine performance during
flight. To date, there have been very few studies in the literature looking at how
main rotor hub vibrations are transmitted to the engines. The DAMVIBS program
concluded that lumped mass models of the engines were adequate for predicting
natural frequencies of the fuselage up to 20 Hz. However, a more detailed structural
model of the engine and its coupling to the airframe is clearly required for evaluating
the transmission of vibratory loads into the engine itself.
In 1970, Balke [53] reviewed the existing vibration criteria that had been es-
tablished by the various engine manufacturers. He highlighted the lack of standard-
ization of vibration criteria, not only in terms of the vibration limits themselves, but
also in how those vibration levels are even defined. He stressed the need for research
into determining typical vibration environments present in VTOL aircraft, as well
as improved communication among the various engine manufacturers and between
engine and airframe manufacturers to establish improved and standardized design
and test procedures throughout the industry.
Reports from 1978 by Sikorsky, Bell and Hughes, Kaman, and Boeing [54?
56] outlined each company?s experiences with problems related to the structural
dynamic interactions between the rotor, airframe, engine, and drivetrain. Sikorsky
18
documented a total of 18 problems encountered over 25 years of development of
their various platforms [54]. Of these 18 issues, 6 were related to forced vibrations
of the installed engine due to main rotor aerodynamic excitations at nP for the
specific helicopter. Among the problems identified by Bell [55] was vibrations on the
installed engine in excess of the engine vibration limits. In addition to deficiencies
in ground test methodology, they specifically pointed to the inability to predict
dynamic loads from the main rotor to the engine mounts as a key shortcoming
contributing to this issue. The Kaman report [57] concluded that the types of
compatibility issues documented in their report would not be eliminated through
improved analytical methods and instead recommended that future research focus
on improved dynamic testing methods for more efficient identification of the nature
of such problems once encountered.
Though each of the manufacturers stressed the need for further research into
these areas, limited information has been put forth in the literature since. Some
work has been done to include engine dynamics in academic modeling efforts; how-
ever, these efforts typically utilize highly simplified engine models and focus on
issues related to flight dynamics, such as handling qualities and stability, without
addressing vibrations [58].
1.3.5 Substructuring Methods
As computing resources have become increasingly powerful and efficient over
time, the finite element structural models developed by civil, mechanical, and aerospace
19
engineers have become more detailed and complex; helicopter airframe models are
no exception. Despite their improved availability, powerful computing resources are
still at a premium, so techniques that can effectively reduce the size of these large
models while still providing accurate results are necessary. Dynamic substructuring
is a means of modeling and analyzing a complicated mechanical system by way of
its individual subcomponents and has been a powerful tool for analysis, especially
of large structures.
Substructuring is easily understood conceptually as a strategy of ?divide and
conquer?, that is breaking a large, complicated problem into smaller sub-problems.
The solution of each of the sub-problems can then be utilized to achieve a solution
to the larger original problem. de Klerk, Rixen, and Voormeeren [59] provide an
overview of the historical development of formal substructuring methods from initial
work in the field of domain decomposition in 1890 and the component mode synthesis
methods of the 1960s that are most associated with the concept of substructuring.
A variety of techniques have been developed since that time, but they all involve
the same basic steps:
1. Division of the structure into subcomponents
2. Definition of connections between subcomponents
3. Analysis of individual subcomponents
4. Assembly of subcomponent analyses into the complete structure
Although the various methods that have been developed over the years differ in the
20
details of the individual steps, primarily steps 3 and 4, they all offer some shared
advantages. First, they allow more efficient use of computing resources. Analysis
of the subcomponents typically involves a reduction in the number of degrees of
freedom, hence a reduction in total computational time. Further, unless changes are
made in the modeling of a particular subcomponent, the individual analyses need
only be done once. Substructuring also facilitates easier ability to share models
between separate entities. Often the modeling of large structures is divided among
different individuals or groups within an organization or even between different
organizations entirely. Due to the proprietary nature of these models, sharing of
modeling details can be difficult. In the substructuring process, engineers can focus
the analysis on the details necessary to other partners and avoid sharing details that
could be deemed proprietary.
As described by de Klerk, Rixen and Voormeeren, substructuring techniques
can be classified by whether the subcomponent analysis is done in the physical,
modal, or frequency domain, as well as whether the interface displacements or forces
are treated as unknowns (primal or dual assembly). This is illustrated in Fig. 1.4.
Substructuring techniques that analyze the subcomponents in the modal domain,
that is, through a transformation from physical degrees of freedom to modal degrees
of freedom, are classified as component mode synthesis (CMS) methods. Techniques
that analyze subcomponents in the frequency domain through Fourier transforms or
frequency response functions (FRFs), are classified as frequency based substructur-
ing (FBS) methods. These two methods will be described in the following sections.
At this point, it is helpful to briefly define some mathematical relationships.
21
Typically, subcomponent models exist in the physical domain and are characterized
by the mass, stiffness, damping matrices, M, K, and C associated with a set of
physical degrees of freedom, u, related the equation
Mu?(t) + Cu?(t) + Ku(t) = f(t) (1.1)
In the frequency domain, the equation of motion can be expressed as
[??2M + i?C + K]U(i?) = F(i?) (1.2)
or, more compactly,
ZU = F (1.3)
This equation can be solved for the nodal displacements and rewritten as
U = YF (1.4)
In equation (1.3), Z is commonly referred to as the structure?s impedance ma-
trix, while Y in equation (1.4) is called the admittance matrix. It is straightforward
to observe that the impedance and admittance matrices are inverses of each other.
1.3.5.1 Component mode synthesis
For the most part, the term dynamic substructuring has referred to the class of
techniques called component mode synthesis (CMS). In CMS methods, the physical
displacement coordinates are represented by a set of generalized coordinates related
to a set of normalized displacement modes, or component modes. A coordinate
transformation utilizing a matrix of mode shapes, or basis vectors, converts between
the physical and generalized coordinates. A variety of methods that fall into the
22
CMS category have been developed and refined over the years, and they differ, in
part, in the variety of definitions for sets of component modes. A general overview
of the methods is given here, however, a more thorough overview including some
mathematical details and examples is presented in Reference [60].
In general, the mode sets utilized in CMS methods fall into one of four cate-
gories: normal modes, constraint modes, rigid body modes, and attachment modes.
These sets are defined as:
(i) Normal modes are the eigenvectors of the system and are classified according to
the boundary condition imposed on the component. The boundary conditions
can be free, fixed, or loaded interface, or a combination among the boundary
nodes.
(ii) Constraint modes represent the deformation of the structure that occurs when
one of the specified number of constraint coordinates is given a unit displace-
ment, while holding the remaining constraint coordinates fixed.
(iii) Rigid-body modes represent the undeformed motion of the component.
(iv) Attachment modes represent the deformation of the structure due to a unit
force applied to a single node.
The earliest version of a CMS method was developed in the 1960s by Hurty
[61, 62]. Hurty?s method utilized fixed-interface normal modes, constraint modes,
and rigid-body modes. In Hurty?s method, a subset of boundary nodes was selected
to yield a set of statically determinate constraints, from which the rigid-body modes
23
were computed. The remaining boundary nodes were designated at redundant con-
straints, from which the constraint modes were computed. Craig and Bampton
built on and generalized Hurty?s work [63]. The Craig-Bampton method represents
the motion of a subcomponent with fixed-interface normal modes and constraint
modes, demonstrating that the use of rigid body modes is unnecessary if all bound-
ary nodes are treated in the same manner as Hurty?s redundant constraints. This
modification simplified the overall formulation and programming, likely leading the
Craig-Bampton method to become essentially synonymous with component mode
synthesis.
Following the development of the Craig-Bampton method, CMS methods re-
ceived continued attention as more researchers contributed their own modifications
to the existing formulations. Methods developed by Goldman [64] and Hou [65]
were formulated using only free-interface normal modes without any static modes.
Benfield and Hruda [66] introduced the concept of branch modes, in which free-free
boundary conditions were used on the identified primary structure and the branch
modes were calculated for the remaining structures with a fixed boundary condition
at the structure interface. Guyan reduction [67] was also used to determine interface
loading from the attached substructures.
MacNeal [68] developed a hybrid method which utilized both fixed and free
interface modes. He also suggested the inclusion of statically determined deflection
influence coefficients. The addition of the these coefficients, termed ?residual flexi-
bility?, offers improved accuracy by accounting for the static effects of the neglected
modes. Rubin [69] extended this idea adding residual inertial and dissipative effects
24
to further improve accuracy. Craig and Chang [70, 71] generalized the methods of
MacNeal and Rubin as well as developed a new method employing residual attach-
ment modes, which produced results comparable to or better than those obtained
by other methods.
1.3.5.2 Frequency-based substructuring
The component mode synthesis techniques for substructure analysis of the
prior section are formulated for use in the time domain; however, as stated pre-
viously, vibrations in helicopters exist primarily at only a few frequencies, namely
integer multiples of the blade passage frequency, pNb per rev, where Nb is the number
of blades. Therefore, studies of helicopter vibrations are ideally suited to methods
formulated in the frequency domain, so-called frequency-based substructuring (FBS)
methods.
As illustrated in Fig. 1.4, the frequency domain representation of a subcompo-
nent can by generated through a Fourier transform of the physical model or through
frequency response functions (FRFs) obtained either analytically from the full model
or the modal reduction or experimentally. Unless a modal representation is all that
is available, FBS methods offer the distinct advantage that all elements of the system
dynamics are embedded in the FRFs.
The classical method of frequency-based substructuring (FBS) was developed
in 1988 by Jetmundsen, Bielawa, and Flannelly [72] which utilized partitioned
impedance matrices for the substructure components that were then connected us-
ing graph theory. The method was originally conceived to analyze the effects of
25
structural changes in helicopter airframes, specifically utilizing experimentally de-
rived mobilities [73]. Gordis, Bielawa, and Flannelly [74] generalized the Jetmundsen
formulation and Ren and Beards [75] proposed a similar formulation. Another mod-
ification to the method includes reformulation such that the coupling of structures
is done in a ?dual? manner, in contrast to the ?primal? assembly that is typically
utilized. The essential difference between these two assembly methods is that the
primal assembly is a coupling through interface displacements, while the dual as-
sembly is a coupling through interface forces. What arises from the dual assembly is
a method known as Lagrange multiplier frequency-based substructuring (LM FBS),
formalized by de Klerk, Rixen, and de Jong [76]. This method has the advantage
of utilizing the admittance matrices of the system directly. The admittance ma-
trices consist of the frequency response relationships of the structure, and so are
more easily acquired through experimental testing. For the purposes of the present
work, even though the structural models used are based on the full finite element
models of the structures, the proprietary nature of the data made transfer of fre-
quency response information the most viable option for acquiring model information.
Therefore, the LM FBS method of Ref. [76] forms the basis of the substructuring
methodology used in this work, the mathematical details of which will be explained
in Chapter 2.
Though not explored in this work, it is worth noting that a key advantage
of FBS methods lies in the ease with which the methods can be applied using
experimental data. More recent work in the field of dynamic substructuring has
focused on improving the accuracy of the methods, both CMS and FBS, when
26
applied to experimental data. Some of the difficulties associated with applying
substructuring analysis to experimental data include difficulty in modeling rotational
degrees of freedom, truncation and experimental errors, the continuity of interfaces
in contrast to the discrete nature of measurements, measurement of rigid body
modes, and nonlinear effects arising from friction in joints. These issues are discussed
further in Reference [59].
1.4 Scope of Current Work
The prediction of vibrations in helicopters is a multi-faceted problem. The
goals of the present research are to examine the various dynamic interactions that
exist within the helicopter in order to better understand the nature of those inter-
actions and their impact on vibrations throughout the helicopter. To that end, the
technical approach of the research is separated into two primary tasks:
1. Engine-airframe coupling using dynamic substructuring. Analysis was
developed to allow coupling of airframe and engine finite element models devel-
oped by different entities. The analysis utilizes frequency-based dynamic sub-
structuring techniques to produce an accurate coupled model while requiring
only limited levels of information exchange for each of the models. This analy-
sis produces results commensurate with those generated using NASTRAN on
the full models in a fraction of the computing time, allowing for parametric
studies of engine mount properties to be carried out.
27
2. Rotor coupling to fixed-frame components within comprehensive
analysis. Existing comprehensive analysis capabilities were expanded to ac-
count for a flexible hub condition, allowing the rotor to be coupled to a flexible
drivetrain and/or airframe model. The effects of these couplings on the rotor
blade structural loads, fixed frame hub loads, pilot seat and engine vibrations
were examined across a range of forward flight speeds.
1.4.1 Overview of Dissertation
The remainder of the dissertation consists of five chapters. Chapter 2 describes
the methodology that was used for this work. This includes the existing comprehen-
sive analysis and modifications made for rotor-airframe and rotor-drivetrain coupling
as well as the substructuring framework utilized for the coupled airframe-engine
analysis. Chapter 3 describes the modeling details of the rotor, airframe, engine,
and engine mount structures. Chapter 4 presents the results of a parametric study
performed on the properties of the engine mount structures for reduced vibrations.
Chapter 5 examines the coupled rotor-airframe-drivetrain system. Finally, Chapter
6 serves to summarize the major conclusions of the current work as well as suggest
directions for future study.
28
Figure 1.1: Frequencies of vibration throughout aircraft [77]
29
Figure 1.2: Vibration levels as a function of forward flight speed [78]
30
Figure 1.3: Impact of vibrations on helicopter development [11]
31
Figure 1.4: Representation of system dynamics in three domains [59]
32
2 Methodology
This chapter presents the details of various aspects of the mathematical mod-
eling utilized in this work. An overview of the baseline comprehensive analysis is
given first, including the blade structural and aerodynamic modeling and vehicle
trim procedures. The detailed mathematical framework for coupling the rotor is to
the fixed frame drivetrain and airframe components is described next. Finally, de-
tails of the frequency-based substructuring framework used for the engine-airframe
coupling are given.
2.1 Comprehensive Analysis: PRASADUM
The baseline rotordynamic analysis for this work was performed using PRASADUM
(Parallelized Rotorcraft Analysis for Simulation and Design, University of Mary-
land), developed by Ananth Sridharan at the University of Maryland and based on
the work in Reference [79]. PRASADUM is used to model the rotor aeromechanics
and vehicle flight dynamics. The analysis in this work required several modifications
to the baseline code to allow for efficient coupling of the rotor with a flexible airframe
and/or drivetrain. For completeness and clarity as it pertains to these modifications,
a summary of the major components of the methodology is presented here, with de-
33
tailed descriptions available in Ref. [79]. For the sake of brevity, some of the detailed
mathematical equations have not been reproduced, particularly for aspects of the
analysis not directly affected by the rotor-structure coupling. However, complete
descriptions, including mathematical formulations, of all modifications made to the
analysis for this coupling are included within this section.
2.1.1 Overview of Solution Method
The equations of motion governing the system dynamics are formulated in a
state-space form as a system of first-order non-linear coupled ODEs of the form
f(y, y?,u, t) =  = 0 (2.1)
Here t is the current time, u is a vector of control inputs consisting of the main rotor
collective and cyclic controls and tail rotor collective, and y is a vector of system
states. The state vector is subdivided into yF, y?, and yrotor, defined as
? yF represents the 12 airframe rigid body states
? y? represents the induced inflow coefficients for the main rotor and tail rotor
? yrotor represents the vector of deflection states for all blades. In the presence
of a flexible hub, this vector will also consist of states representing the hub
motion.
Similarly, the residual vector, , is assembled from F, ?, and rotor defined as
? F are the residuals corresponding to the non-linear equations enforcing force
and moment equilibrium for the fuselage rigid body motions and kinematic
34
compatibility between time derivatives of Euler angles and body-axes angular
rates.
? ? are the residuals corresponding to the dynamic inflow equations for the
main and tail rotors
? rotor are the residuals corresponding to the mode-weighted Euler-Bernoulli
beam equations for the rotor blades. In the case of a flexible hub, rotor will
also contain the residuals for the hub motion equations.
For the rotorcraft trim problem related to in this work, the governing differ-
ential equations are numerically reduced to a set of non-linear algebraic equations,
represented as
F(X) = trim = 0 (2.2)
In this case X is a vector of unknown trim variables, including the pilot controls,
vehicle attitudes, rotor induced inflow ratios, and rotor response. The algebraic
equation residuals, trim, are calculated through numerical manipulation of the ODE
residuals from Eqn. 2.1. HYBRD, a non-linear system solver available through the
open-source library, NETLIB, is used to solve the system algebraic equations to
within a user-specified threshold, ?trim for an error metric e(|trim|).
2.1.2 Coordinate Systems
PRASADUM makes use of a variety of reference frames depending on what is
most relevant at a given stage in the analysis. Forces and moments are resolved into
the body axes to ensure equilibrium for trimmed flight. Rotating axes for the blade
35
deformations are also utilized. Coordinate transformations are used to transfer loads
and displacements from one system to another. The coordinate transformations take
the form of a rotation matrix representing a three Euler angle rotations, occurring
in the following order:
? Yaw rotation about the z-axis through an angle ?
? Pitch rotation about the intermediate y-axis through an angle ?
? Roll rotation about the intermediate z-axis through an angle ?
The rotation matrices for each indivi?dual rotation are g?iven by
??? cos? sin? 0?? ?
?
T? = ??? sin? cos? 0???
?
? (2.3)
0 0 1
? ?
???cos ? 0 ? sin ????
T? = ???? 0 1 0 ???? (2.4)
?sin ? 0 cos ? ?
???1 0 0 ???
T ?? = ???0 cos? sin????? (2.5)
0 ? sin? cos?
The full rotation matrix from coordinate system A to coordinate system B, denoted
TBA, consists of the three individual rotation matrices multiplied as
TBA = T? T? T? (2.6)
36
The reverse transformation, from coordinate system B to coordinate system A,
denoted TBA, is achieved through the reverse sequence of opposite rotations. It
can be easily demonstrated with trigonometric identities and matrix properties that
these two transformations are the transpose of each other. That is,
TAB = T?? T
>
?? T?? = TBA (2.7)
2.1.2.1 Time derivatives of rotation matrices
Because the various rotation angles can be time-dependent and the coordinate
transformations are used to transform time-dependent quantities and their deriva-
tives, it is also necessary to know the time-derivatives of these rotation matrices.
The first and second time derivativ?es each of the sequen?tial rotations are given by
???? sin? cos? 0? ?T?? = ???? cos? ? sin? 0?
??
??? ?? (2.8)
0 0 0
? ?
???? sin ? 0 ? cos ?
T? = ????
?
0 0 0 ??
?
? ??? ?? (2.9)
? cos ? 0 ? sin ??
???0 0 0 ???
T?? = ????0 ? sin? cos? ???? ?? (2.10)
0 ? cos? ? sin?
37
?? ? ? ???? sin? cos? 0??? ???? cos? ? sin? 0
T? = ????
?
? ? cos? ? sin? 0???? ?? + ???? sin? ? cos? 0?
?
???? ??
2 (2.11)
? 0 0 0? ? 0 0 0? ???? sin ? 0 ? cos ???? ???? cos ? 0 sin ?
T? = ???? 0 0 0 ??
?
? ? ?? ??? + ????
2
0 0 0 ???? ?? (2.12)
? cos ? 0 ? sin ?? ?? sin ? 0 ? cos ?
???0 0 0 ??? ??
?
?0 0 0 ???
T?? = ??? 2?0 ? sin? cos? ???? ?? + ????0 ? cos? ? sin????? ?? (2.13)
0 ? cos? ? sin? 0 sin? ? cos?
The individual time derivatives can then be used to construct the time deriva-
tives of the full transformation matrix by considering the expanded derivatives of
the sequential multiplication of the individual rotation matrices. These are given by
d d
T?AB = (TAB) = (T? T? T?)
dt dt
= T?? T? T? + T? T?? T? + T? T? T?? (2.14)
d2 d ( )
T?AB = (TAB) = T?AB
dt2 dt
= (T?? T? T? + T? T?? T? + T? T? T?? )+
2 T?? T?? T? + T?? T? T?? + T? T?? T?? (2.15)
2.1.2.2 Coordinate system definitions
Figure 2.1 (a) shows the general set of non-rotating coordinate systems used
when a fixed hub condition is assumed. These are:
38
? Earth-fixed axes, or gravity axes, (?iG, j?G, k?G) represent an inertial reference
frame used to track the motion of the helicopter in space.
? Helicopter body-fixed axes (?iB, j?B, k?B) are obtained from the earth-fixed axes
through a translation of the helicopter CG and three Euler rotations ?F , ?F ,
?F , representing nose-right yaw, pitch up, and roll right motions, respectively.
Rotation from earth-fixed to body-fixed axes is done with TBG
? Helicopter hub non-rotating axes (?iH , j?H , k?H) are obtained from the helicopter
body axes through a translation of the origin to the location of the hub relative
to the CG, followed by two Euler rotations, ?S, ?S, representing the tilt of
the shaft. There is also a 180? rotation about the intermediate y-axis. This
transformation is represented by the matrix THB.
Figure 2.1 (b) shows the additional non-rotating coordinate systems used when
the rotor is coupled to a flexible airframe model. These are:
? NASTRAN axes (?iN , j?N , k?N) are similar to the body-fixed axes but are ro-
tated by 180? about the y-axis to align with the NASTRAN convention that
the z-axis points up. Since the airframe model used in the analysis was gener-
ated in NASTRAN, the eigenvectors or hub frequency response functions used
to compute the response of the hub are given in the NASTRAN coordinate
system. Since the transformation from the body-fixed axes to the hub-fixed
axes is already defined by THB, it is simplest to flip the signs of the x- and
z-translations and rotations of the hub in the NASTRAN axes, bringing them
39
into the body-fixed axes and then apply THB for transformation to hub-fixed
coordinates.
? Deformed hub axes (?iH2 , j?H2 , k?H2) arise as a result of the motion of the
hub relative to the airframe. Specifically, the application of hub forces and
moments on the airframe will produce displacements, xhub, yhub, zhub, and pitch
and roll rotations, ?y and ?x , respectively. Similarly, the hub torque ishub hub
applied to the drivetrain and produces the ?z rotation. Thub H2H is the rotation
matrix for the transformation from the hub-fixed axes to the deformed hub
axes.
For completeness, it is worth listing the remaining coordinate transformations
from rotor hub frame to the deformed blade axes. The coordinate systems are
illustrated in Fig. 2.2.
? TRH is the rotation matrix from the non-rotating hub axes to the rotating
un-preconed blade axes. This is a rotation about the non-rotating hub z-axis
through angle ?j for blade j, where ?j is the azimuth angle of blade j.
? TUR is the rotation matrix from the rotating unpreconed blade axes to the
undeformed blade preconed axes. This transformation consists of a rotation
through angle ??p about the unpreconed y-axis. Because of the definition of
the unpreconed axes, ?p is negative for vertically upward motion of the blade
tip.
? TDU is the rotation matrix from the undeformed preconed blade axes to the
40
(a) Fixed hub coordinate systems
(b) Flexible hub coordinate systems
Figure 2.1: PRASADUM non-rotating coordinate systems (adapted from [79])
41
deformed blade axes. This transformation is unique to every point on the blade
elastic axis. The transformation consists of three translations along the the
undeformed axes, representing hinge offset, axial fore-shortening due to bend-
ing, spanwise position of the beam cross-section, lead-lag displacement and
flap-bending displacement. The translations are followed by three rotations
arising from the flap and lag bending and elastic twist of the beam.
2.1.3 Rigid Body Dynamics
For the purposes of calculating the vehicle flight dynamics, the helicopter
fuselage is assumed to be rigid. The airframe linear and angular velocities are
obtained from the partition of the system state vector containing the fuselage states,
given by
{ }>
yF = uF vF wF pF qF rF ?F ?F ?F xCG yCG zCG (2.16)
where the terms (uF, vF, wF, pF, qF, rF) are the linear and angular velocity compo-
nents of the helicopter CG in the body-fixed axes, (?F, ?F, ?F) are the Euler angles
defining the fuselage orientation with respect to earth-fixed axes, and (xCG, yCG, zCG)
represents the position of the helicopter CG in earth-fixed axes.
In the fuselage body-fixed axes, the force and moment equilibrium equations
42
(a) Rotating blade unpreconed axes
(b) Blade preconed axes
(c) Undeformed and deformed blade axes
Figure 2.2: PRASADUM blade coordinate systems [79]
43
are
X = mF(u?F + qFwF ? rFvF + g sin ?F) (2.17)
Y = mF(v?F + rFuF ? pFwF ? g sin?F cos ?F) (2.18)
Z = mF(w?F + pFvF ? qFuF ? g cos?F cos ?F) (2.19)
L = Ixxp?F ? Ixy(q?F ? pFrF)? Ixz(r?F + pFqF)? Iyz(q2F ? r2F)? (Iyy ? Izz)qFrF
(2.20)
M = Iyy q?F ? Iyz(r?F ? qFpF)? Iyx(p?F + qFrF)? I 2zx(rF ? p2F)? (Izz ? Ixx)rFpF
(2.21)
N = Izz r?F ? Izx(p? 2 2F ? rFqF)? Izy(q?F + rFpF)? Ixy(pF ? qF)? (Ixx ? Iyy)pFqF
(2.22)
These equations represent half of the equations used to compute the residuals for
the fuselage rigid body motions, F. The remaining six arise from the equivalence
of the linear velocity of the fuselage and the time derivative of the CG position and
the kinematic compatibility between time derivatives of Euler angles and body-axes
angular rates. These are represented by the following equations:
uF = x?CG (2.23)
vF = y?CG (2.24)
wF = z?CG (2.25)
44
pF = ??F ? ??F sin ?F (2.26)
qF = ??F cos?F + ??F cos ?F sin?F (2.27)
rF = ???F sin?F + ??F cos ?F cos?F (2.28)
The terms on the left-hand side of Eqns. 2.17-2.22 represent the total applied
forces and moments about the center of gravity from the main rotor (MR), tail rotor
(TR), horizontal and vertical tails (HT, VT), and fuselage aerodynamics, given by
X = XMR +XTR +XHT +XVT +XF (2.29)
Y = YMR + YTR + YHT + YVT + YF (2.30)
Z = ZMR + ZTR + ZHT + ZVT + ZF (2.31)
L = LMR + LTR + LHT + LVT + LF (2.32)
M = MMR +MTR +MHT +MVT +MF (2.33)
N = NMR +NTR +NHT +NVT +NF (2.34)
The following sections contain further details on the calculations of each of these
forces.
2.1.4 Aerodynamics Model
The aerodynamic loads are primarily the result of aerodynamic forces on indi-
vidual blade sections. These forces are calculated from look-up tables based on the
sectional airfoil and local velocity and Mach number at that section. The velocity
at a blade section is a combination of the motion of the blade through the air and
the inflow through the rotor. The rotor inflow can be calculated using a dynamic
45
inflow model or free wake model, both of which are briefly described in this section.
In addition, the contributions of the helicopter fuselage, empennage, and tail rotor
to the aerodynamic forces and moments on the vehicle are discussed.
2.1.4.1 Rotor Inflow Models
The inflow at the main rotor is calculated in two ways, depending on the
level of fidelity desired in the solution, either with a dynamic inflow model or free-
vortex wake model. This section provides general descriptions of both inflow models.
The descriptions given in this section are meant only to summarize the main rotor
aerodynamic models used in PRASADUM and are largely qualitative. For further
mathematical details, refer to Ref. [79].
Dynamic inflow model
A dynamic inflow model is designed to approximately capture the time-variation
of the inflow over the rotor disk. In general, the inflow for specific time at a point
on the rotor disk (r, ?) is given by
?(r, ?) = ?0 + r (?1c cos? + ?1s sin?) (2.35)
where ?0 represents the uniform inflow, ?1c is the longitudinal inflow, and ?1s is
the lateral inflow. PRASADUM utilizes the Peters-He model [80] for the inflow,
which consists of a set of coupled first order differential equations for the inflow
parameters, ?0, ?1c, and ?1s.
46
Free-vortex wake model
For vibrations analysis, particularly at flight conditions between hover (? = 0)
and transition speeds (? ? 0.1), where blade-vortex interactions are significant, a
dynamic inflow model in inadequate. Instead, an in-house free-vortex wake model,
known as Maryland Free Wake is used to more accurately capture the effects of the
rotor wake on the rotor aerodynamics.
The wake calculations are carried out in two parts, near wake and far wake
models. In the near wake, a Weissinger-L lifting surface model is used, illustrated
in Fig. 2.3. Each blade is represented using a distribution of horseshoe vortices
with a bound vortex at the 1/4-chord. The strength of the vortex trailers is found
using Helmholtz?s laws for vorticity conservation. These trailers make up the near
wake of the blade, which is truncated for ?? = 30? behind the blade. After that
point, this vortex sheet is assumed to have completely rolled up and its circulation
is concentrated in the free trailers from the blade tips. This is the far wake.
The tip vortices trail from the blade tips in a curved path that is discretized
into straight line segments, as illustrated in Fig. 2.4. The vortices are free to convect
due to the influence of the local velocity and instantaneous vorticity. The motion of
the vortices is tracked using a time-accurate two-step backward predictor-corrector
scheme developed by Bhagwat and Leishman [81]. The velocity field induced by the
wake at any location is computed by application of the Biot-Savart law, followed by
numerical integration of the induced velocity contributed by each vortex element in
the flow field.
47
2.1.4.2 Fuselage and empennage aerodynamics
The aerodynamic forces and moments on the body of the fuselage are com-
puted based on the flow velocity at a reference point on the fuselage, defined relative
to the vehicle center of gravity. The flow velocity components comprise the compo-
nents of the vehicle translational velocity and rigid body angular velocity as well as
components of an interference velocity. The interference velocity is calculated from
the main rotor downwash, longitudinal tilt of the rotor tip path plane and wake
skew angle. From the components of the flow velocity at the reference point, the
flow incidence angles are calculated, and with the dynamic pressure at the reference
point, the lift, drag and moment coefficients are obtained from look-up tables based
on wind-tunnel measurements [82].
Similarly, the aerodynamic loads on the vehicle empennage (horizontal and
vertical tail surfaces) are computed based on the flow velocity at a reference point on
these surfaces. The flow velocity again comprises an interference velocity, rigid body
angular velocity, and vehicle translational velocity, this time with an empirically
determined factor to model the loss of dynamic pressure at the tail surfaces due
to the wake of the fuselage. The lift and drag coefficients are then obtained using
look-up tables (based on wind tunnel measurements) and the incidence angles and
dynamic pressure at each surface. Full mathematical details of the fuselage and
empennage aerodynamics can be found in Ref. [79].
48
Figure 2.3: Weissinger-L lifting surface model for rotor blade
Figure 2.4: Free-vortex wake model of rotor
49
2.1.4.3 Tail rotor aerodynamics
The induced inflow of the tail rotor is modeled with a 1-state Pitt-Peters
dynamic inflow model [83] and is assumed to be uniform over the rotor disk. The
governing ODE for the tail rotor is given in Ref. [79].
The tail rotor thrust and torque are calculated in the shaft axes of the tail
rotor based on the free-stream velocity and induced inflow. The relationship between
these quantities is based on the closed-form solution for a lifting rotor in forward
flight given in [84]. The free stream velocity at the tail rotor comprises the vehicle
translational velocity, the rigid body angular velocity, and the induced velocity due
to the wake of the main rotor and fuselage, obtained from look-up tables based on
the wake skew angle and tilt of the tip-path plane. The tail rotor thrust equation
also includes a factor to account for the blockage effects of the vertical fin.
2.1.5 Flexible Beam Dynamics
The rotor blades are modeled as flexible isotropic nonlinear Euler-Bernoulli
beams with flap and lag bending and elastic torsion using a geometrically exact
formulations. The rotor motions are represented by the partition of the state vector
given by {
y > > > > > >
}>
rotor = ??1 ?1 ??2 ?2 ? ? ? ??Nm ?Nm (2.36)
where ?j is a vector of generalized displacements for the j-th mode.
These beam models are discretized into spatial finite elements, and the un-
derlying partial differential equations (PDEs) are reduced to ordinary differential
50
equations (ODEs) using a numerical implementation of the Galerkin method. Modal
reduction is also applied to reduce the number of degrees of freedom while preserving
the dominant blade motions of interest.
Contributions to the beam equation residuals include the internal structural
loads as well as the aerodynamic, gravitational, and inertial loads. The components
of the strain tensor are derived from the displacement field. The strain and material
properties are used to calculate the beam stresses, which are integrated over the
cross-section to give the sectional elastic forces and moments.
The external loads on the rotor blades consist of the inertial and gravitational
loads, and the aerodynamic loads, or fluid forces, due to the relative motion between
the blade and the air. To calculate these fluid forces, the absolute velocity of a point
in the cross section of the blade is found by adding contributions of the vehicle
motion, hub motion, undeformed rotating blade motion and flexible blade motion.
Velocity is transformed into deformed beam axes and broken down into com-
ponents tangential and perpendicular to the airfoil as well as along the span of the
blade. The velocity components lead to the sectional angle of attack used to calcu-
late the lift and drag forces and pitching moment on the airfoil. The aerodynamic
forces comprise both contributions from circulatory forces as well as non-circulatory
components arising from the pitch rate and plunge acceleration of the airfoil.
51
2.1.6 Hub Loads
The contributions of all of the rotor forces and moments to the hub is necessary
both for achieving a trimmed flight condition (in which rotor and body forces are all
balanced) as well as calculating the vibratory response of the airframe or drivetrain
needed for this work. The force and moment components per unit span, represent-
ing the sum of the inertial and aerodynamic loads, are calculated in the rotating
undeformed blade axes. The blade loads for each individual blade are then resolved
along the hub non-rotating axes and added together for the total hub loads. The
hub loads are converted to the helicopter body axes to give the main rotor contri-
bution to vehicle force and moment equilibrium equations. They are also converted
to the NASTRAN axes for calculating the response of the flexible airframe at the
main rotor hub.
2.1.6.1 Vibratory hub loads
For computation of the airframe vibratory response from the hub loads, it is
necessary to compute the steady vibratory components of the hub loads, generally
the pNb/rev components. To do this, the time history of the hub loads over one
rotor revolution is assumed to be represented by a Fourier series as
?N ?N
sN(?) = a0 + ans sin(n?) + anc cos(n?) (2.37)
n=1 n=1
where a0, ans, and anc are the Fourier coefficients. For a rotor with Nb identical
blades, the hub loads consists only of a steady component, represented by a0 and
52
harmonics for which n = pNb. In general, N is theoretically infinite, though for the
purposes of this analysis, only Nb/rev and 2Nb/rev harmonics are considered, i.e.
4/rev and 8/rev for a 4-bladed rotor.
The Fourier coefficients are calculated from the time histories of the individual
hub load components by numerically calculating the following integrals
?
1 2?
a0 = ? f(?) d? (2.38)2? 0
1 2?
ans = ? f(?) sin(n?) d? (2.39)? 0
1 2?
anc = f(?) cos(n?) d? (2.40)
? 0
From the sine and cosine coefficients, the magnitude and phase the vibratory hub
loads are computed as
?
f = a2 2mag ns + anc (2.41)
f ?1phs = tan (anc/ans) (2.42)
2.1.7 Rotor-Airframe Coupling
Most often, the helicopter airframe is included in rotor analysis as a rigid body,
in which the rotor hub is rigidly attached to the fuselage at a fixed position relative
to the CG. However, as previously discussed, the inclusion of flexible hub motion
has been shown to have an impact on the rotor loads, so it is necessary to couple
the analysis of the rotor with the flexible airframe. This section will detail how the
motion of the hub is incorporated into the rotor analysis as well as how the response
of the hub on the airframe is calculated from the rotor hub loads.
53
2.1.7.1 Rotor Blade Boundary Condition
To calculate the aerodynamic and inertial loads on the rotor blades, it is
necessary to know the velocity and acceleration of the blade sections. Consider a
point within the cross-section of a blade. The position of the point relative to the
ground, rp is given by
rp = rhub + rea + rcs (2.43)
where rhub is the position of the hub with respect to earth, rea is the position of
the elastic axis in the undeformed blade frame, and rcs is the position of the point
within the cross-section with respect to the deformed elastic axis.
Helicopter rotor blades are mounted to a hub that is positioned above the
vehicle center of gravity. In previous iterations of PRASADUM, the connections
from body to shaft, shaft to hub, and hub to blade were assumed to be rigid. In the
present analysis, the connection from the shaft to the hub and hub to blade are still
considered rigid, but the flexible airframe means that the body to shaft connection
is not rigid, and will therefore affect the boundary condition of the blade.
With regard to Eqns. 2.43, the addition of the flexible hub condition does not
directly affect the position of either the elastic axis in the undeformed blade frame
or the position of the point in the cross-section with respect to the deformed elastic
axis. As such, the details of those terms will not be reproduced here, but can be
found in Ref. [79]. On the other hand, rhub is affected by the flexible hub boundary
condition.
54
At the most basic level, the position of the hub is given by
r >hub = {xhub yhub zhub ?x ? ? } (2.44)hub yhub zhub
where xhub, yhub, zhub, ?x , ?y , ?z are the translations and rotations of thehub hub hub
hub in the NASTRAN frame, defined in Section 2.1.2.2. The angular motion of the
hub, represented by ?x , ?y , ?z and their time derivatives are used to computehub hub hub
the transformation matrices from the fixed hub frame to the deformed hub frame,
TH2H . The full transformation from the deformed hub frame to earth-fixed axes are
then given by
TGH2 = TGHTH2H (2.45)
T?GH2 = T?GHTH2H + TGHT?H2H (2.46)
T?GH2 = T?GHTH2H + 2T?GHT?H2H + TGHT?H2H (2.47)
where TGH is the transformation from the hub-fixed axes to the earth-fixed axes
and is based only on the rigid body motions of the airframe.
The components of the hub velocity and acceleration that are used in the
calculation of the aerodynamic and inertial loads of the blade sections are also
modified by the addition of elastic displacements of the hub relative to the vehicle
CG. Consider that the fixed position of the hub relative to the vehicle CG is given
by R = rCG and that the angular rotation of the fixed hub about the CG is given
by ?. The flexible hub contributes to the offset from the CG as
R = rCG + rhub (2.48)
55
The velocity of the hub relative to the CG is then given by
r?hub = vhub + ??R (2.49)
which is added to the existing rigid body translational velocity of the hub to give
the total velocity of the hub in the earth-fixed axes. Similarly, the components of
the hub acceleration are given by
r?hub = ahub + ???R + ?? r?hub (2.50)
Again, this contribution of the acceleration due to the motion of the hub about the
CG is added to the rigid body acceleration to give the total acceleration of the hub
in the earth-fixed frame. These adjusted velocity and acceleration terms for the hub
are then used in the calculation of the aerodynamic and inertial loads of the blade
sections.
2.1.7.2 Hub Response
Consider that the rotor is attached to a flexible structure, such as a test stand,
airframe, or drivetrain. The dynamics of this structure may be expressed as a linear
system of the form
Mx?s + Cx?s + Kxs = Fs (2.51)
where xs represents the set of structural degrees of freedom, and M, K, and C
represent the mass, stiffness, and damping matrices of the structure, respectively.
Fs represents the forcing vector on each of the hub nodal degrees of freedom.
These matrices can be condensed to a modal representation in the usual way.
In modal space, each mode is uncoupled from the other modes, and the scalar
56
governing equation for each mode is given by
??i + 2?i?
2
n ??i + ?n ?i = fi (2.52)i i
The modal forcing at the hub on the airframe or drivetrain is computed from the
dot product of the hub forcing with the shape of the eigenvector at the hub node as
?6
fi = VijFj (2.53)
j=1
where the subscript i represents a specific mode and the subscript j represents the
hub degree of freedom. In general, there are six hub degrees of freedom. However,
the hub torque is transmitted through the drivetrain only. Therefore, the indicated
sum includes only the hub torque (j = 6) for drivetrain coupling, and includes
the three hub forces and hub pitch and roll moments (j = 1, 2, 3, 4, 5) for airframe
coupling.
For a given modal forcing and set of modal coefficients satisfying Eqn. 2.52,
the response at the hub can be reconstructed as
?Nm
qj = Vi,j?i (2.54)
i=1
where Nm is the number of airframe or drivetrain modes included in the analysis.
This hub motion is then used as the hub boundary condition as described in the
previous section.
2.1.8 Trim
The rotorcraft trim problem, known as coupled or propulsive trim, combines
the concept of general aircraft trim (as applied to fixed wing aircraft) with the con-
57
cept of rotor trim. For a rotor to be trimmed, the blade response in level flight
should be periodic over one rotor revolution. In other words, when the forces and
moments are averaged over one rotor revolution, steady-state values are achieved
between cycles. For the aircraft to be trimmed, the combined forces and moments
on the aircraft must be in equilibrium. In general, the forces and moments on a
rotorcraft are oscillatory in nature. However, it has been shown that the overall
vehicle trim state is insensitive to the vibratory loads on the aircraft, so the time-
averaged forces and moments generated by the trimmed rotor with periodic response
are sufficient for establishing the overall aircraft trim condition. In general, a ro-
torcraft trim problem is formulated for a steady coordinated helical climbing turn
of constant radius. This condition is satisfied for constant flight speed, V , constant
flight path angle, ?, and constant turn rate, ??. For this work, only steady level
flight is considered, which is the special case of a coordinated turn in which ? = 0
and ?? = 0.
2.1.8.1 Basic Trim Equations
This section lists the trim variables and associated trim equations that must
be satisfied for rotorcraft trim:
? The first six trim variables are the four pilot controls of main rotor collective
and lateral/longitudinal cyclic and tail rotor pedal and the fuselage pitch and
roll attitudes. The associated trim equations ensure that the body forces and
moments are in equilibrium. That is, the sum of all the rotor and body forces
and moments about the vehicle CG is zero.
58
? Dynamic inflow: The trim variables associated with the main rotor and tail
rotor inflow are
X? = {?0,MR ?1c,MR ?1s,MR ?TR} (2.55)
The corresponding trim equations ensure that the inflow ratios are time-
invariant when averaged over one revolution of the main rotor, as in
? T
? =0,MR ? ??0 dt (2.56)MR0T
? = ?? dt (2.57)1c,MR ? 1cMR0T
? = ? ??1s dt (2.58)1s,MR MR0T
? = ??TR dt (2.59)TR
0
? Rotor response: The trim variables for the rotor response represent the
steady and harmonic components of the rotor response when a Galerkin method
with harmonic balance is used to obtain the time resolution of the rotating
blade modes.
{ (R) (R) (R)Xrotor = . . . ?j,0 ?j,kc ?j,ks . . . } (2.60)
where j goes from 1 to the number of blade modes, Nm, and k goes from 1 to
the number of harmonics to be included in the solution, Nh. The residuals for
59
the steady, sine, and cosine components of the blade motions are given by
? T
steady = ? blade 1(t) dt (2.61)0T
cos,k = ? blade 1(t) cos k?t dt (2.62)0T
sin,k = blade 1(t) sin k?t dt (2.63)
0
(2.64)
2.1.8.2 Trim with Free Wake
When a free wake model is used in trim, a loose-coupling procedure is used to
separately converge the aerodynamics and rotor/flight dynamics. As a first step, the
vehicle is trimmed as described above using dynamic inflow, generating a starting
guess for the controls, fuselage orientation, and rotor response. With this initial
trim solution, the free wake solution is computed until a converged wake solution
is achieved. The converged wake geometry is then used to compute the inflow
over the rotor disk, and the trim solver is called again with this inflow to solve
for an updated set of trim variables, excluding the dynamic inflow variables. This
procedure is iterated until a converged solution is achieved for both the wake and
trim variables to within a given threshold.
2.1.8.3 Trim with Flexible Hub
When the rotor is coupled to a flexible airframe or drivetrain, the trim pro-
cedure is modified to account for this coupling with the addition of trim variables
associated with the flexible hub motion and the associated residuals.
60
The vector of trim variables for the hub motion is given by
(h) (h) (h) (h)
Xhub = {..., ?i,4c, ?i,4s, ?i,8c, ?i,8s, ...} (2.65)
representing the 4/rev and 8/rev sine and cosine components of the airframe or
drivetrain response for each mode included in the analysis. In general, this set of
trim variables would include all harmonics up to the total number of harmonics
included in the analysis, however, since only 4/rev and 8/rev components of the
vibratory hub loads are transmitted to airframe, the other coefficients are assumed
to be zero are omitted from the set of trim variables.
The modal response of the hub node for the airframe or drivetrain can be
calculated by ? ( )
(h) (h) (h)
?i (?) = ?i,kc cos k? + ?i,ks sin k? (2.66)
k=4,8
Recall that the modal equations of the airframe/drivetrain are given by
mi??i + ci??i + ki?i = fi (2.67)
where fi is the modal forcing given by
?6
fi = VijFj (2.68)
j=1
The residual for each airframe/drivetrain mode at a given time is then given by
i = mi??i + ci??i + ki?i ? fi (2.69)
The weighted residuals for the hub motion are then calculated for each mode and
61
each harmonic as
? T
i,kc = ? i cos k?t dt (2.70)0T
(h)
i,ks = i sin k?t dt (2.71)
0
(2.72)
62
2.2 Engine-Airframe Coupling
The engine and airframe coupling can be taken care of in several ways. First,
the engine model can be included directly into the airframe analysis. That is, the
engine is included in the FE model used to generate either the airframe modes or
the frequency response used in either of the rotor-airframe coupling procedures de-
scribed in the previous sections. In the second approach, the engine and airframe are
modeled separately, for the purposes of studying the effects of different engine mount
structure parameters on the engine response. A coupling framework for coupling the
airframe and engine FE models was developed and implemented utilizing frequency
response functions of separate components. The general terminology for the method
is frequency-based substructuring (FBS). Two versions of this methodology will be
described in this section. The first is an iterative harmonic balance method. The
second is a more mathematically formal method built from a framework introduced
by deKlerk, Rixen, and Voormeeren [59].
2.2.1 Mathematics of Dynamical Systems
The dynamics of each component structure can be described by the structure?s
mass, stiffness, and damping matrices. The equations of motion of a substructure
can then be written as
M(s)u?(s)(t) + C(s)u?(s)(t) + K(s)u(s)(t) = f (s)(t) (2.73)
63
where M(s), C(s), and K(s) are the generalized mass, damping, and stiffness matrices
of the substructure, u(s) is the vector of the substructure degrees of freedom, and f (s)
is the vector of applied forces. In general, these forces may be externally applied,
as in the case of main rotor hub loads, or may be interaction forces between two
subcomponents.
In the frequency domain, the equation of motion can be expressed as
[??2M(s) + i?C(s) + K(s)]U(s)(i?) = F(s)(i?) (2.74)
or, more compactly,
Z(s)(i?)U(s)(i?) = F(s)(i?) (2.75)
where Z(s)(i?) is the dynamic stiffness matrix or, as it?s commonly referred to in
literature, the displacement impedance matrix, and is defined as
Z(s)(i?) = [??2M(s) + i?C(s) + K(s)] (2.76)
This equation can be solved for the nodal displacements and rewritten as
U(s) = Y(s)F(s) (2.77)
where Y is the mobility matrix or the frequency response transfer function and is
given by
Y(s) = [Z(s)]?1 (2.78)
The transfer function relates the output displacement response to an input excitation
force, and in this form is complex-valued and frequency-dependent. Individual terms
in the Y matrix, Yij, represent the response of point i (or more generally, DOF i),
64
Ui, due to a unit oscillatory force (or moment) at DOF j, Fj. In other words,
Yij = ?Ui/?Fj (2.79)
2.2.2 Iterative Harmonic Balance Solution
Using substructuring to couple individual components together begins with
identifying all of the separate structures that will be coupled, or subcomponents,
and the interactions between the subcomponents. Figure 2.5 shows these elements
schematically, as well as an example assembly of two subcomponents.
2.2.2.1 Subcomponents
Each subcomponent in the analysis is a dynamical system modeled with free-
free boundary conditions, as shown in Fig. 2.5(a). Each subcomponent might have
hundreds or even thousands of nodes included in the full finite element model,
however, for the purposes of this analysis, only a small subset of ?interface? nodes
are important to the overall substructure assembly. Those nodes are:
1. Nodes where external forces are applied
2. Nodes connected to nodes of other subcomponents
3. Nodes where specific output is desired
There may be some overlap among these categories, but identification of the full set
(s)
of Nn interface nodes for each subcomponent is the first step in the substructur-
ing procedure. In general, every node is assumed to have 6 degrees of freedom, 3
65
translational and 3 rotational. The frequency response transfer functions, Y(s), are
computed with unit magnitude sinusoidal forcing individually applied for each of
the 6 Nn degrees of freedom with responses recorded at all 6 Nn degrees of freedom,
for a total of 36 N2n frequency response transfer functions for a single subcomponent.
These 36 N2n FRFs are computed for every frequency of interest in the analysis and
generally consist of a magnitude and phase or are complex-valued. For the analy-
sis in this work, the frequency response transfer functions are computed using the
NASTRAN SOL 108 solution sequence, which computes direct frequency response
of the full model, without any modal reduction. With this solution method, the
contributions of every model degree of freedom is included in the response, even
though only a subset of responses is recorded.
The forces acting on a given subcomponent may be a combination of both
externally applied forces, Fe, and interaction forces, Fm, due to connection with
other subcomponents (explained the the following section). The response of the a
given node is then calculated by
Un(s) = ?mYnm(s)Fm(s) + ?eYneFe (2.80)
Thus, interface forces and externally applied loads are treated in an identical manner
in the analysis to compute the response at the key nodes using linear superposition.
2.2.2.2 Interaction elements
Interaction elements, shown in Fig. 2.5(b) refer to massless springs and/or
dampers that connect two or more subcomponents together. These elements are
66
the counterparts of subcomponents: while the dynamical systems have input forcing
and output displacement, interface elements are used to compute output forcing for
input displacements.
To simplify the analysis, each interface element is assumed to connect two
distinct interface nodes, each located on a different subcomponent. Km and Cm
are matrices that contain the spring and damper coefficients (including any cross-
coupling between degrees of freedom) at an interface and are used to compute the
interface loads using the resultant spring compression and/or damper velocity. These
quantities are obtained from the relative response of the nodes at either end of the
interface, U(A)n and U
(B)
n . The total interface force due to both springs and dampers
is given by ( )
Fm = (Km + i?C
(B) (A)
m) Un ?Um (2.81)
2.2.2.3 Solution procedure
A schematic representation of simple system is shown in Fig. 2.5(c). In this
system, two subcomponents are connected between a pair of nodes with an inter-
action element. A force is applied on subcomponent A and the response is desired
from subcomponent B. The known quantities for the system consist of the frequency
response transfer functions of each of the subcomponents, the stiffness and damping
matrices for the interaction element, and the magnitude and phase of the external
forcing. The response displacements for the nodes of the subcomponents are the
unknowns. In the following procedure, an implicit harmonic balance formulation is
used in which the generally complex-valued forces and responses are recast using
67
sine and cosine components. This somewhat complicates the equations used to cal-
culate the responses, and for the sake of brevity, the details of those calculations are
omitted here.
1. Let the response of a given nodal degree of freedom be given by
Ui(t) = Xi,c cos?t+ Xi,s sin?t (2.82)
Xi,c and Xi,s are the unknown harmonic coefficients for each nodal degree of
freedom. The vector of unknowns for the full system is then given by
X = {X1,c,X1,s...XN,c,XN,s} (2.83)
where N is the total of all interface nodes for all subcomponents. Assign an
initial guess of X0 = 0.
2. Compute the interface forces Fm using Eqn. 2.81 for the given set X.
3. Compute the output nodal response Xn using Eqn. 2.80, and extract the har-
monic coefficients Xnc and Xns
4. Calculate the difference between the guess value and output value of the nodal
responses as
 = |X0 ?Xn| (2.84)
If the guess values X0 are exactly equal to the calculated response coefficients
Xn, then the initial guess is a consistent solution. Otherwise, the initial guess
is updated using Newton-Raphson iterations until the error, , reduces below
a pre-determined threshold.
68
This process is repeated for each of the external forcing frequencies.
69
(a) Substructuring subcomponents
(b) Substructuring interaction elements
(c) Interaction of sub-
components and interac-
tion elements
Figure 2.5: Overview of substructuring components
70
2.2.3 General Formulation of Frequency Based Substruc-
turing
Although the procedure outlined in the previous section produces accurate re-
sults, the implementation of the procedure is somewhat confusing and the solution
procedure can be lengthy and somewhat slow. As such, the analysis was reformu-
lated to take advantage of a more compact mathematical structure introduced by
deKlerk, Rixen, and Voormeeren [59]. The reformulation has the advantages of
being much simpler to code and faster to run. In addition, it allows specifically
for rigid connections between subcomponents. In the previous formulation, a rigid
connection can only be approximated with a very stiff spring.
It is instructive at this point to consider a very simple system to understand
the mathematical formulation. A more complicated system will be used later in
this section to validate the methodology before finally applying the methodology to
the airframe-engine system in the next chapter. Figure 2.6 shows a simple system
consisting of two sets of three masses connected with springs, each set representing
one substructure component. The two components are connected by either a rigid
or flexible connection to form one coupled system. Clearly, this system could be
easily modeled without the use of substructuring, but the details of the modeling
are more easily understood for a simple system first.
As before, the equations of motion for each subcomponent can be written as
U(s) = Y(s)[F(s) + G(s)] (2.85)
71
where now the applied forces have been specifically separated into externally applied
forces, F(s), and interface forces, G(s). The procedure for the coupling of n individ-
ual substructures begins with assembling the individual substructure equations of
motion?in a block-diagonal format as???? ????? ?? ???? ?? ?? ?????U(1)??? ???Y(1) ???????? ? ?? ?????
????F(1)?????? ??????G(1)????????
? .?? ..? ???? = ??
. . . ??????
?
? .?? .
. ????? +?? ????
..
? . ???????
?? (2.86)
U(n)?? Y(n) ??F(n)?? ??G(n)??
U = Y[F + G] (2.87)
This system of equations represents all of the substructures of the assembled system.
In this system, generally the system mobility matrix Y and external forces F are
known, while the nodal displacements U and the interface forces G are unknowns.
The interface forces can be separated into rigid and flexible interface forces, Gr and
Gf , respectively. The constraint equations defining the rigid and flexible interfaces
between the substructures will be derived in the following sections.
2.2.3.1 Rigid Interface
At a rigid interface between substructures, two conditions must be met: in-
terface compatibility and force equilibrium. Interface compatibility requires that
displacements on either side of the connection be equal, while force equilibrium
requires that the sum of the interface forces be zero. Consider a coupled system
with N total degrees of freedom and Nr rigid interface degrees of freedom. The
72
(a) Rigid interface
(b) Flexible interface
Figure 2.6: Simple spring-mass system with two substructure components
compatibility condition for a rigid interface can be expressed by
BrU = 0 (2.88)
where Br matrix is a signed Boolean matrix (entries are either 0, 1, or -1) having
dimensions Nr ? N which expresses the condition that pairs of matching interface
degrees of freedom have the same displacement. For example, in the simple mass-
spring system is shown in Fig. 2.6, the vector of displacements for the system would
be
U = {UA1 , UA A2 , U3 , UB, UB, UB T1 2 3 } (2.89)
The condition of the rigid interface between mass 3 from subsystem A and mass 1
from subcomponent B is UA = UB or UA ? UB3 1 4 1 = 0, which can be expressed as
BrU = 0, where [ ]
Br = 0 0 0 1 ?1 0 0 0 (2.90)
73
The quantity BrU can be viewed as a vector of relative displacements for all interface
degrees of freedom, which will be useful in the consideration of flexible interfaces.
The force at the rigid interface between the two substructures can be written as
Gr = ?B Tr ? (2.91)
where ? is a vector of length Nr where each element represents the unknown mag-
nitude of the force at each rigid interface. Returning to the sample problem, we see
that
?? ??? 0 ???
?? 0 ???? ?? ?? ?????
0 ?
????
???
?
G = ?BT ?r r ? = ?? ? (2.92)?? ? ?
???
????
0
??
? ??????? 0 ??
0
where the equal magnitude and opposite sign elements ensure force equilibrium at
the interface.
2.2.3.2 Flexible Interface
For a coupled system with a flexible interface, as shown in Fig. 2.6(b), the
condition of interface compatibility required for a rigid interface is no longer neces-
sary, though force equilibrium must still be ensured. As before, the interface force
74
vector is given by
Gf = B
T
f?f (2.93)
where Bf matrix is a signed Boolean matrix (entries are either 0, 1, or -1) having
dimensions Nf?N defined identically to the Br matrix for the rigid interface. Nf in
this instance is the number of flexible interface connections that exist in the system.
The previous equation 2.92 shows that force equilibrium is achieved.
Because interface compatibility is not necessary for a flexible connection, BfU 6=
0. Instead, BfU represents the relative displacement between the interface nodes,
and thus can be used with the interface stiffness to calculate the interface force
magnitude of the interface forces, ?f , as follows
?f = KBfU (2.94)
Here, K is the stiffness matrix representing the stiffness between subcomponents.
2.2.3.3 Rigid and flexible interfaces
For a coupled system with both rigid and flexible interfaces, the results of the
previous two sections are combined. Combining equations 2.87, 2.88, 2.91, 2.93, and
2.94 the fully coupled system with a rigid and flexible connections is represented by
the equations
U = Y[F?BTr ?r ?BTf?f ] (2.95)
?f = KBfU (2.96)
BrU = 0 (2.97)
75
Equation 2.95 is a system of equations representing the N total degrees of freedom
in the uncoupled system. Equations 2.96 and 2.97 are the constraint equations for
the Nf flexible connections and Nr rigid connections, respectively, resulting in a
total of N ?Nr ?Nf degrees of freedom for the coupled system.
2.2.3.4 Damping
In general, damping can be included in a model in a number of ways. There
may be individual viscous dampers in the structure. If modal reduction is used, a
modal damping factor can be specified for each of the normal modes. It is also pos-
sible that a structural damping factor is applied to the entire structure. Structural
damping is given as an imaginary component to the FE model?s stiffness matrix.
Any or all of these damping methods may be applied in a given structure. If
damping is included in a component model when the frequency response is calcu-
lated, it will be embedded in the resulting FRF and does not need to be considered
separately. The addition of damping at the interface is the only damping that needs
to be accounted for. Whether a viscous damper or structural damping factor is
applied at the interface, the damping force is added to the magnitude of the flexible
interface force defined in Eqn. 2.96.
For a viscous damper, the damping coefficients are defined in a matrix C,
and the force is proportional to velocity, U?. In the frequency domain, velocity is
related to displacement by U? = i?U so that the addition of damping to the flexible
interface force is given by
?f = [K + i?C]BfU (2.98)
76
When structural damping is applied at the interface, a structural damping
coefficient, ?, is applied as an imaginary component to modify structure?s stiffness
matrix such that the stiffness matrix is (1 + i?)K. This leads to the interface force
being given as
?f = (1 + i?)KBfU (2.99)
Comparing Eqns. 2.98 and 2.99, it is straightforward to see that structural damping
can replaced with an equivalent viscous damper with damping matrix given by
?
[C] = [K] (2.100)
?
This is especially useful when utilizing the iterative harmonic balance solution pro-
cedure described in the previous section.
2.2.4 Validation: Simple Test Case
A simple 8-dof spring-mass-damper system is used to validate the analysis. A
schematic of this system is shown in Fig. 2.7. The full system is partitioned into
two substructures, as indicated by the dotted lines. Free-free frequency response
transfer functions are generated for nodes 1 and 4 in substructure A and nodes 5
and 8 in substructure B. The two substructures are then connected via an interface
stiffness/damping element between nodes 4 and 5. Finally, a unit harmonic force is
applied at node 1 and the resulting response of node 8 is examined.
Systematic validation of the modeling was done to ensure that all interface
and damping options were accounted for. Figures 2.8 and 2.9 show the comparison
of the magnitude and phase of the response between the NASTRAN calculation
77
and the developed substructuring analysis. The following cases are present in the
results:
? No damping, rigid interface
? No damping, flexible interface
? Flexible interface, viscous dampers
? Flexible interface, structural damping, ? = 0.04
? Flexible interface, viscous and structural damping, ? = 0.04
The results show complete agreement between the substructuring analysis and NAS-
TRAN analysis for this example system. This agreement provides confidence in ap-
plication of the substructuring methodology to a complex coupled airframe-engine
model. Validation of that modeling is presented in the following chapter.
78
Figure 2.7: Schematic of 8-DOF mass-spring-damper system
m1 = m3 = m7 = m8 = 1, m2 = m6 = 2, m4 = m5 = 4
ki = 1, ci = 0.01, i = 1, 2, ...9
79
(a) Rigid interface
(b) Flexible interface
Figure 2.8: Comparison of NASTRAN and susbtructuring results for validation test
case without any damping
80
(a) Viscous damping
(b) Structural damping
(c) Viscous and structural damping
Figure 2.9: Comparison of NASTRAN and susbtructuring results for validation test
case with different damping types
81
3 Modeling Details
Several aspects of the modeling details are described in this chapter. First, the
baseline models used in the comprehensive analysis are given, including the details
of the rotor model, blade structural and aerodynamic properties, and fuselage aero-
dynamic and inertial properties. Next, the drivetrain model is described. Finally,
the detailed finite element models of the engine and airframe models are described,
along with details of the connections between the engine and airframe and validation
of the coupled engine/airframe analysis.
3.1 Baseline Helicopter Rotor Model
The helicopter modeled in this analysis is based on the UH-60A Blackhawk.
Significant parameters related to the main rotor geometry are given in Table 3.1.
The blade structural properties are based on values given in Ref. 85. Spanwise dis-
tribution of the blade mass, m, flap bending stiffness, EIy, lag bending stiffness,
EIz, and torsion stiffness, GJ , are given in Figs. 3.1-3.4. A fan plot representing
the the rotating natural frequencies as a function of RPM for this rotor is shown
in Fig. 3.5 and agrees well with fan plots for this rotor generated by other compre-
hensive analysis codes. The frequencies for the first ten blade modes are given in
82
Table 3.2.
The primary airfoil for the main rotor is the SC1095. Details of the airfoil
aerodynamic properties are given in Ref. [82]. Similarly, the aerodynamic properties
of the fuselage and empennage are taken from Ref. [82]. Parameters used for the
tail rotor modeling are given in Table 3.4.
Finally, a summary of the parameters used for the rotor and wake discretization
are given in Tables 3.5 and 3.6.
3.2 Drivetrain Model
The drivetrain model used in this study is the Rotor Systems Research Aircraft
(RSRA) drivetrain model [82]. The schematic of this model has been reproduced in
Fig. 3.6. This model represents an equivalent single speed representation of the ac-
tual geared drivetrain. The model consists of torsional inertia and stiffness elements
representing the rotating masses and flexible shafts of the complete drivetrain. The
gears in the drivetrain impose constraints on the shaft speeds of each station, which
modify the effective stiffness and inertia values of the model. The effective stiffness
and inertia properties, K? and J?, are defined based on the ratio of the rotational
speed of the component to that of the main rotor, r = ?/?MR, and are related the
actual stiffness and inertia properties, K and J , as J? = r2J and K? = r2K [86].
The values of these parameters are shown in Tables 3.7 and 3.8.
For coupling with the rotor as described in the previous chapter, a modal
representation of the drivetrain is obtained. If the rotation of the individual inertia
83
elements, ??, are the degrees of freedom of the drivetrain, the equations of motion
for the model are
[M ]{???} + [K]{??} = {? ?} (3.1)
where [M] is the diagonal mass matrix consisting of the inertia elements J?, J?1 2 , J
?
3 ,
J?, and ?J?4 5 . The stiffness matrix, [K], is given by? ??? K? ?K?? 12 12
0 0 0 0 ?
? ?? ???K? K? +K?12 12 23 ?K? ?? 23
0 0 0 ?
??? 0 ?K? K? ? ? ? ? ? ?? 23 23
+K34 +K35 +K36 ?K34 ?K35 ?K36???
?
[K] = ?? (3.2)
??? 0 0 ?K?34 K? 0 0? 34?? ?
????? 0 0 ?K? 0 K?35 35 0 ???
0 0 ?K? ?36 0 0 K36
The resulting natural frequencies for a free-free boundary condition are shown in
Table 3.9 and the mode shapes are shown in Fig. 3.7. The first elastic mode
consists entirely of anti-symmetric engine motion, and so does not contribute to the
rotor shaft rotation. Similarly, mode 5 shows negligible hub participation and was
excluded from the solution. Therefore, only modes 2-4 were included in the analysis.
3.3 Airframe Model
The airframe used in this study is a representative medium lift helicopter.
The full finite element model, shown in Fig. 3.9 was developed by the airframe
manufacturer, and consists of a total of 15732 nodal grid points with connections
between them modeling the complete airframe structure. A superelement model [87]
84
for the airframe was generated by the OEM from the full finite element model and
provided for this work. The superelement reduces the full airframe model down to
388 nodes. Comparisons between the full model and the superelement model show
excellent agreement between the first 100 natural frequencies for the two models (up
to 69.1 Hz or 16/rev).
Frequency response functions (FRFs) for the airframe finite element model
were generated using NASTRAN?s SOL 108 with a structural damping factor of
? = 0.04 (2%). SOL 108 computes the direct frequency response of the structure,
meaning no modal reduction is used to compute the FRFs. As described in the
preceding chapter, the FRFs give the response of individual nodal degrees of freedom
due to individual unit sinusoidal forcing components at those same nodes. The FRFs
were generated for a wide range of frequencies, including the specific 4/rev and 8/rev
frequencies for the UH-60A rotor, 17.34 Hz and 34.68 Hz, respectively. The grid
points used are listed in Table 3.11.
In addition to the FRFs, natural frequencies and mode shapes were calculated
from the provided superelement model. The first ten natural frequency values and
modes shape descriptions for the airframe are given in Table 3.12 to give some basic
information about the system.
3.4 Engine Model
A NASTRAN finite element model of the turboshaft engine used in the pro-
vided airframe was provided by the engine manufacturer and is shown in Fig. 3.10.
85
The model consists of a total of 478 grid points. Since the model is one-dimensional
(all nodes on the engine axis), off-axis points were added to the physical connection
points on the engine at the rear mount plane to facilitate connection to the top of
the struts connecting to the engine (fully detailed in the following section). Addi-
tionally, the model was replicated with non-repeated grid points so two identical
engines could be included with the airframe model.
The translations and rotations of the rearmost node on the engine axis were
used to obtain the motions of the mounting nodes on the engines. The vertical
plane containing the mounting nodes was assumed to only undergo rigid-body rota-
tions. This configuration allows for accurate modeling of the load transfer from the
rear mount struts to the engine. As with the airframe, FRFs for the engine were
generated using NASTRAN?s direct frequency response solution (SOL 108). Unit
forces and moments were applied at the fore mount point and the two off-axis rear
mount points. The resulting displacements were obtained at the fore and aft mount
locations on the engine axis. The engine grid points are listed in Table 3.13.
The first few natural frequencies of the engine mode are given in Table 3.14.
It should be noted that the engine model is axially symmetric, so the natural fre-
quencies occur in pairs for the two directions transverse to the engine axis.
3.5 Airframe-Engine Interface
With respect to the substructuring analysis used to couple the engine and air-
frame, there are five separate substructure components: the airframe, two engines,
86
and two engine mount torque tubes. These components are connected using a com-
bination of rigid connections and interface stiffness and damping elements. Care was
taken to ensure that the substructuring methodology matched the relevant features
of the fully coupled NASTRAN model. The connections between the airframe, en-
gine, and torque tube are shown schematically in Fig. 3.11 and the details of these
connections are discussed in the following sections.
3.5.1 Fore mount
The fore end of the engine is connected to the transmission input by a torque
tube. The torque tube consists of 9 beam elements with axial, bending, and tor-
sion degrees of freedom. The mass and stiffness matrices for the torque tube were
provided by the manufacturer and frequency response functions were generated for
the responses to unit load excitation at each of the end nodes. The aft end of the
torque tube is connected rigidly to the fore engine mount point in all 6 degrees of
freedom. The fore end of the torque tube is rigidly connected to the airframe in the
three translation degrees of freedom (i.e., gimbal).
3.5.2 Aft mount
The aft mount of each engine consists of three struts. As illustrated in Fig.
3.12 (a), the aft end of the engine (A1, A2) is connected by a rigid bar to the top of
the struts (S1a-b, S2a-b), and the bottom of the struts are connected directly to the
airframe (S1c-e, S2c-e). One outboard strut and two inboard, shown schematically
87
in Fig. 3.12(b) comprise the aft mount structure. The struts all lie in the same
vertical plane and are symmetric between the left and right engines. The struts are
represented by axially extensible linear bar elements. Based on the geometry and
material properties of these elements, stiffness matrices are computed to use as the
interface between the engine and airframe transfer function models.
3.5.2.1 Strut geometry and stiffness matrices
In order to properly resolve the transmission of forces between the engine and
airframe, the geometry of the struts must be accounted for. The coordinates of the
end points for each strut are given in Table 3.15. Note that, since the six struts
are symmetric, only the right side set of struts is listed. The left side struts have
identical x- and z-coordinates and opposite signed y-coordinates.
The struts lie in the vertical y ? z plane, so the geometry of the problem
is simplified to two dimensions. The struts are modeled as axially extensible rod
elements. The force transmitted along the length of a rod is given by
?L
F = EA = keff ?L (3.3)
L
where E is the Young?s modulus, A is the cross-sectional area, L is the length of
the rod, and ?L is the change in length of the rod due to compression or extension.
The physical characteristics of the rod can be combined to give an effective axial
stiffness, keff = EA/L.
Consider an axial stiffness element oriented arbitrarily in a two-dimensional
plane as illustrated in Fig. 3.13. Implementation of the strut stiffness into the
88
substructuring methodology requires properly resolving the force along the rod into
into y-and z-components, Fy and Fz, resulting from the y-and z-components of the
rod?s change in length, ?y and ?z. The rod?s change in length is
?L = (?y cos ? + ?z sin ?) (3.4)
The force in rod can be decomposed into the horizontal and vertical Fy and
Fz as
Fy =F cos ? = keff ?L cos ? (3.5)
Fz =F sin ? = keff ?L sin ? (3.6)
Combining Eqns. 3.4-3.6 gives
Fy = keff (?y cos
2 ? + ?z sin ? cos ?) (3.7)
Fz = keff (?y sin ? cos ? + ?z sin
2 ?) (3.8)
Using the relations sin ? = Lz and cos ? = Ly gives
L L
keff
Fy = (L
2
y?y + LyLz?z) (3.9)L2
keff
Fz = (LyLz?y + L
2
z?z) (3.10)L3
These equatio?ns c?an be combin?ed and rewrit?te?n in ?matrix form? ? ???Fy??? k L2 L L ?y ?yeff ??? y y z?????? ??? ? ?= = [K]?? ?? (3.11)L2
F L L L2z y z z ?z ?z
with the stiffness matrix for the strut re?presented by?
k ??? L2 Leff y yLz[K] = ??? (3.12)L2
LyL
2
z Lz
89
Since the struts only allow axial extension in the y ? z plane, the remaining
elements of a full 6? 6 stiffness matrix [K] are all zero.
3.6 Airframe-Engine Coupling Validation
To validate the substructuring analysis of the baseline coupled airframe-engine
model, FRFs of the engine response at the mount points due to forcing at the main
rotor hub were calculated and compared to the results based on the full coupled
NASTRAN model. The comparisons for the transfer function magnitudes of the
response of one engine mount are shown in Fig. 3.14. The substructuring approach
completely captures the behavior of the NASTRAN model for various frequencies
from 0 to 40 Hz, including the magnitude and phase over a range of frequencies.
90
Table 3.1: Main rotor properties
Property Value
Rotor type articulated
Number of blades 4
Radius, R 26.83 ft
Rotor speed, ? 258 RPM
Chord, c 1.73 ft
Blade weight 256.9 lb
Shaft tilt (positive aft) -3?
Primary airfoil SC1095
Hinge offset 0.0466R
Root cutout 0.2R
Location of swept tip 0.929R
Tip sweep angle 20?
Pitch link stiffness 67900 ft-lb/rad
Solidity, ? 0.0832
Lock number, ? 6.33
91
Figure 3.1: Blade mass distribution
92
Figure 3.2: Blade flap bending distribution
93
Figure 3.3: Blade lag bending distribution
94
Figure 3.4: Blade torsional stiffness distribution
95
Figure 3.5: Rotor blade natural frequencies
96
Table 3.2: Main Rotor Blade Frequencies
Mode Freq (/rev)
1 0.267
2 1.035
3 2.837
4 4.600
5 4.710
6 5.222
7 7.923
8 11.567
9 12.426
10 14.214
97
Table 3.3: Airframe properties
Property Value
Roll inertia, Ixx 4658 slug-ft
2
Pitch inertia, Iyy 38512 slug-ft
2
Yaw inertia, Izz 36796 slug-ft
2
Roll-yaw coupling inertia, Ixz 1882 slug-ft
2
Fuselage CG location (360, 0, 243)
Main rotor shaft location (341.215, 0, 300)
98
Table 3.4: Tail rotor geometry
Property Value
Number of blades 4
Radius 5.5 ft
Rotation speed 124.62 rad/s
Section lift-curve slope 5.73 /rad
Chord 0.81 ft
Cant angle (from vertical) 20?
99
Table 3.5: Rotor blade discretization parameters
Parameter Value
Azimuthal samples for trim 40
Flexible finite elements 20
Quadrature points per element 8
Number of blade modes 6
Blade harmonics per mode 9
Table 3.6: Wake discretization parameters
Parameter Value
Wake discretization 10 degrees
Near wake 30 degrees
Blade bound vortex segments 40
Number of wake turns 6
100
Figure 3.6: RSRA drivetrain model (Ref. [85])
101
Table 3.7: Inertia and stiffness properties of RSRA drivetrain model (Ref. [85])
Description Symbol Value
Inertia values (in-lb-sec2)
Main rotor hub (MR) J?1 75
Transmission (TRAN) J?2 909
Coupling gear box (GB) J?3 1044
Power turbine (EN1) J?4 6494
Power turbine (EN2) J?5 6494
Tail rotor (TR) J?6 4724
Stiffness values (in-lb/rad)
Main rotor shaft K?12 42.95E+06
Transmission, coupling shaft K?23 1679E+06
Engine shaft K?34 1184E+06
Engine shaft K?35 1184E+06
Tail rotor shaft K?36 4797E+06
Table 3.8: Speed ratios of drivetrain components
Component RPM Scale, r
Main rotor 258 1
Engine 20900 81
Tail rotor 1214 4.1
102
Table 3.9: Natural frequencies of drivetrain model with free-free boundary condition
Mode Frequency
rad/s Hz /rev
1 427.0 68.0 15.8
2 637.7 101.5 23.6
3 751.6 119.6 27.8
4 1305.1 207.2 48.3
5 3075.9 489.5 113.5
Figure 3.7: Mode shapes of drivetrain model with free-free boundary condition
103
Table 3.10: Natural frequencies of drivetrain model with fixed engines
Mode Frequency
rad/s Hz /rev
1 520.2 82.8 19.3
2 749.8 119.3 27.8
3 1303.7 207.5 48.3
4 3069.4 488.5 113.7
Figure 3.8: Mode shapes of drivetrain model with fixed engines
104
Figure 3.9: NASTRAN FE model of airframe
105
Table 3.11: Grid point ID numbers for NASTRAN airframe model
Grid ID Description
690 Pilot floor (left)
698 Co-pilot floor (right)
1805 Inner strut connection (left)
1806 Middle strut connection (left)
1809 Outer strut connection (left)
1847 Outer strut connection (right)
1850 Middle strut connection (right)
1851 Inner strut connection (right)
16401 Torque tube connection (right)
17401 Torque tube connection (left)
115910 Main rotor hub
160004 Tail rotor hub
106
Table 3.12: Natural frequencies for baseline airframe model
Mode Freq (/rev) Description of mode
1-6 0.0 Rigid body modes
7 1.29 Vertical bending
8 1.50 Lateral bending
9 2.13 Tail boom torsion
10 2.28 Horizontal tail vertical bending
11 2.42 Transmission pitch
12 2.92 Fuselage torsion
13 3.07 Landing gear (symmetric)
14 3.13 Landing gear (asymmetric)
15 3.34 Horizontal tail/landing gear
16 3.41 Fuselage torsion
107
Figure 3.10: NASTRAN FE model of GE CT7-8 engine
108
Table 3.13: Grid point ID numbers for NASTRAN engine model
Grid ID Description
16403 Aft off-axis inboard (right)
16405 Aft off-axis outboard (right)
17403 Aft off-axis inboard (left)
17405 Aft off-axis outboard (left)
9500044 Aft on-axis point (left)
9500048 Fore on-axis point (left)
9600044 Aft on-axis point (right)
9600048 Fore on-axis point (right)
109
Table 3.14: Natural frequencies for engine model
Mode Freq (/rev)
1-6 0.0
7 15.52
8 15.52
9 18.29
10 18.29
11 22.33
12 22.33
13 27.00
14 27.00
15 33.39
16 33.39
110
(a) Location of the four engine mount points and main rotor
hub
(b) Interfaces between airframe, engine, and torque tube models
Figure 3.11: Overview of Helicopter/Engine/Airframe interfaces, mounts and forcing
node
111
(a) Engine-airframe connections (right engine)
(b) Rear engine mount struts
Figure 3.12: Schematic of engine mount structures
112
Figure 3.13: Geometry of axial stiffness element
Table 3.15: Coordinates for aft engine struts
Strut position Coordinate 1 Coordinate 2
Inner (380.7, 16.5, 268.5) (380.691, 20.3, 279.3)
Middle (380.7, 21.6, 267.7) (380.691, 20.3, 279.3)
Outer (380.7, 37.6, 258.5) (380.691, 39.7, 279.3)
113
(a) Fx in, y out (b) Fx in, z out
(c) Fy in, y out (d) Fy in, z out
(e) Fz in, y out (f) Fz in, z out
Figure 3.14: Comparison of substructuring and NASTRAN results for coupled
airframe-engine model: lateral and vertical translation response magnitude and
phase of left engine aft mount point to unit amplitude forcing at main rotor hub
114
4 Engine Mount Parametric Study
Having developed and validated the frequency-based substructuring method-
ology for coupling a helicopter airframe model with the associated engine model,
parametric studies were conducted to determine the effects of changing stiffness and
damping properties of the engine mount structures on the vibration levels experi-
enced at different points on the engines. This chapter begins with a description of
the baseline analysis results, followed by discussion of the effects of changing the
stiffness and damping of the torque tube and engine struts on the baseline.
4.1 Baseline Results
Following the validation of the coupled model, 4P and 8P hub loads were ap-
plied to the coupled model frequency response functions (FRFs) and the resulting
(steady-state) response magnitudes and phases at the four engine mounts were cal-
culated to yield FRFs for the entire structure for a particular set of engine mount
properties. These global FRFs represent the response (magnitude and phase) of the
engine mount point along a particular direction due to unit amplitude vibratory hub
loads. Tables 4.1 and 4.2 show the 4P and 8P response displacement magnitudes
for each of the engine mounts due to individual hub load components.
115
For the results presented here, PRASADUM was used to calculate hub loads
for the UH-60A rotor described previously using 6 coupled blade modes in con-
junction with harmonic balance method up to 9P. Hub forces and moments were
calculated at a density altitude of 5,000 ft, and steady level flight at speeds of 55,
80, 100, 125 and 150 knots, representing advance ratios of 0.128, 0.187, 0.233, 0.291
and 0.349, respectively. Figure 4.1 shows the magnitudes of the 4P and 8P hubload
components for these flight conditions.
Scaling the FRF magnitudes by a single hub load amplitude (and with appro-
priate phase shift) for a specific flight condition provides the contribution of that
hub forcing component to the total response. Vector addition of the contributions
from all six hub loads yields the total engine response. Using this technique, multi-
ple flight conditions can be analyzed using the same set of FRFs for a particular set
of mount properties, further reducing the number of calculations required for the
analysis. Since vibrations normal to the engine axis are thought to be more detri-
mental to the engine than those along the engine shaft, for this analysis, particular
attention was paid to the y and z degrees of freedom, i.e., the transverse response.
Figure 4.2 shows the lateral and vertical 4P response acceleration of each
engine mount point, ay and az, due to each of the hub load components at 100 knots,
as well as the total response from 0 to 90 degrees of rotor azimuth. Examining the
breakdown of the engine response in this way allows for the identification of the
hub load components that contribute dominantly to the total engine vibration. For
instance, at the fore mount point of the right engine, the 4P component of the
pitching moment (My forcing) contributes most to the lateral response (y), while
116
the hub roll moment (Mx forcing) is the biggest contributor to the vertical response
(z). The hub forces and moments have similar magnitudes at 4P and comparable
sensitivities, so the contribution of each is directly reflected in the response levels.
It should be noted that although the rotor torque, Mz, is included in the hub loads
and transfer functions presented here, the analysis does not include contributions
from Mz, since the torque load is transmitted through the drivetrain, not directly
to the airframe as the other forces and moments.
The contributions of each hub load component to the 8P response of each
engine mount point are shown in Fig. 4.3, also for 100 knots flight condition. For
8P, the hub roll moment, Mx, is the primary contributor to the lateral response of
each mount point and the vertical response of the aft mount points. Both hub roll
and pitch moments, Mx and My contribute significantly to the vertical 8P response
of the fore mount points.
To capture the inherent axisymmetry in the engine and quantify vibrations
with a common metric, the ay and az compon?ents of the acceleration were condensed
into the total transverse response, ayz = a2y + a
2
z using vector addition of the
accelerations along the y and z coordinate directions. Figure 4.4 shows the total
4P and 8P transverse acceleration for each of the engine mounts for each of the
speed cases considered. The responses were calculated using the procedure described
above, and represent baseline values for possible minimization. The following trends
were noted:
1. There is an overall minimum in engine vibratory response at 80 knots (? =
117
0.187), though this effect is driven by reduced vibratory loading from the main
rotor as the wake is swept away and tip compressibility drag is minimal.
2. There is a high degree of asymmetry between the left and right engines in
this helicopter. The 4P transverse response of the left engine has similar
magnitudes for the fore and aft mounts, but for the right engine, the fore
and aft mounts vibrate at distinctly different amplitudes (up to 100% more
vibration at the intake compared to exhaust end).
3. The transverse response of all four mounts is driven primarily by the 8P rolling
moment. The peaks in response occur at 55 and 100 knots, following the trend
in Mx moments shown in Fig. 4.1(a).
4. The left and right engines show comparable 8P response levels with the fore
mount of each showing transverse response magnitudes close to double those
of the aft mounts.
5. Overall, the peak accelerations due to 8P excitation are about one order of
magnitude greater than the corresponding response levels due to 4P excitation.
This information reveals the possibility of targeting specific loads as a means of
reducing the engine response. It also highlights the need for accurate vibratory loads
prediction in terms of both magnitude and phase in addition to accurate structural
modeling. While structural models can provide a measure of the sensitivity of certain
response degrees of freedom to each hub load component, the inclusion of the phase
of the response and loads can lead to either an increase or decrease in the overall
118
engine mount vibration.
After establishing these baseline levels of vibration, the substructuring analysis
was used to perform parametric studies that would be untenable with traditional
NASTRAN analysis of the fully coupled assembly. Since the 100 knots airspeed case
represents a relatively high engine vibration case for both 4P and 8P, the remaining
results were generated using 100 knots hub vibratory loads. Table 4.3 lists the
baseline 4P and 8P transverse accelerations for the four engine mount points. The
stiffness and damping values for the torque tube and struts were varied from the
baseline values to identify the sensitivity of these vibration levels to each design
parameter. The results of these parametric studies are discussed in the remaining
sections.
4.2 Torque Tube Stiffness Sweep
A parametric sweep was conducted to determine the effect of changing the
stiffness properties of the torque tube on vibratory response of the engine at the
mount points. The stiffness matrix for the torque tube was scaled by a factor ranging
from 0.25 to 4, corresponding to possible changes in the torque tube material and/or
cross-sectional dimensions. The 2% structural damping of the baseline model was
preserved for all components, and the rear mount strut properties were preserved
at their baseline values. The 4P and 8P responses at both mounts of the left and
right engines for various torque tube stiffnesses are shown in Fig. 4.5.
The right and left engine response sensitivity to varying torque tube stiffness
119
show similar trends for both 4P and 8P response. Changing the torque tube stiffness
has very little effect on the 4P response of either engine at the aft mount point.
Similarly, the aft mount response at 8P is relatively less sensitive to the torque tube
stiffness. However, the 8P fore mount response reduces by about half for 200%
torque tube stiffness. Decreasing the torque tube stiffness shows a marked increase
in both 4P and 8P response at the fore mount locations. Reducing the torque tube
stiffness to 50% results in a 65% increase in the 8P response and marginal increase
in the 4P response.
4.3 Symmetric Strut Stiffness Sweep
A parametric sweep was performed by scaling the axial stiffness of the aft
mount struts with a factor ranging from 0.25 to 4. A schematic of the six mount
struts is shown in Fig. 3.12. The mount configurations for the left and right engines
are symmetric, with each engine supported by two inboard struts (labeled inner
and middle) and one outboard strut (labeled outer). For this study, the stiffness
values of the pairs of inner, middle, and outer struts were varied independently,
maintaining the symmetry between the left and right engine mounts. The effects
of strut stiffness on the 4P and 8P transverse response levels are examined in the
remainder of this section.
120
4.3.1 Uniform strut stiffness effects
For this study, the effects of collectively varying the stiffness of the struts were
analyzed. Figure 4.7 shows the 4P and 8P transverse response of all four engine
mount points due to changing the stiffness of all six struts.
Decreasing all strut stiffnesses uniformly results in increased 4P response at all
but one mount point. At 50% strut stiffness, both mount points of the right engine
and the aft mount of the left engine show significant increases in response from the
baseline. In particular, the aft right mount increases from the 0.4g baseline to 1.2g.
However, the fore mount of the left engine shows a decrease from the 0.25g baseline
to a minimum of 0.16g. On the other hand, increasing the stiffness of all the struts
beyond the baseline decreases the 4P response of the aft right mount significantly
from 0.4g to 0.25g at 400% strut stiffness. The effects on the 4P response of the
other mounts are less noticeable, though all but the aft left mount show a decrease
with increasing stiffness.
The trends in the 8P response are the same for all four engine mount points,
with differences only in magnitude. Increasing strut stiffness beyond the baseline
results in decreased response, with each engine mount point achieving a reduction
of about 0.3g when the strut stiffness is increased to 400% of the baseline. Alter-
natively, decreasing the strut stiffness results in increased transverse response, with
the peak response level occurring around 35% baseline stiffness.
121
4.3.2 Damping effects
The effects structural damping were also examined. Structural damping fac-
tors of ? = 0.04, 0.1 and 0.2 corresponding to 2%, 5%, and 10% proportional damp-
ing, respectively, were applied for each case of the uniform stiffness sweep. Figures
4.8 and 4.9 shows the results of this study. Structural damping in the struts has a
significant effect on the 4P response, particularly in the right engine aft mount, with
reduction of up to 40% at 50% strut stiffness. The impact on the 8P response is less
pronounced, though increased damping does result in up to 20% reduction of 8P
response for 20-50% strut stiffness. However, for the increased stiffness values that
achieve reduced response levels, damping has almost no effect. Therefore, damping
was maintained at the baseline 2% for the remainder of the study.
4.3.3 Individual strut stiffness effects
Figures 4.10 and 4.11 show the variations in transverse 4P and 8P response,
respectively, at all engine mount points due to independent variations of the inner,
middle, or outboard strut stiffness. The stiffness effects of the two inboard struts
(inner and middle) on the 4P response show the same trends of the uniform stiffness
sweep, while the outboard strut stiffness has no effect on the 4P response. The
results of varying the stiffness of both pairs of inboard struts simultaneously are not
shown here, but reveal that either of the two inboard struts could be changed to
achieve nearly identical results as changing both.
In contrast with the 4P results, a uniform change in the 8P response magni-
122
tudes at all 4 mount points appears to be driven by the stiffness of the outboard
struts. Changes in the outboard strut stiffness produce the same results as changing
the stiffness of all three pairs of struts. If the inner strut is stiffened or softened, the
acceleration increases on the fore mounts and decreases on the aft mounts, while the
middle strut mainly affects acceleration of the fore mounts of both engines, though
both effects are small.
4.4 Asymmetric Strut Stiffness Sweep
The parametric studies of the previous sections were conducted while main-
taining symmetry between the left and right struts. This is largely due to consid-
erations of ease of manufacturing and installation. However, asymmetry has been
noted in the response between the left and right engines, indicating that an asym-
metric solution study might provide improved results. Another parametric sweep
was conducted, independently scaling each of the six mounts struts with a factor
ranging from 0.25 to 4.
Figures 4.12 and 4.13 show the effects on the 4P and 8P transverse response due
to simultaneously varying only the left or right set of struts while keeping the other
set at the baseline. Reduction of the left strut stiffness to 50% baseline results in an
increased response at all four mount points though only the aft left mount increase
is significant. Increasing only the left mount stiffness has negligible effects on the
4P response. Increasing the stiffness of just the right struts results in decreases in
the right mount response with no effect on the left mounts. Decreasing the right
123
strut stiffness also leads to a significant increase in the response of the right engine,
especially at the aft mount, but results in a slight decrease in the response of the
left engine.
Changing the strut stiffness on one side has negligible effects on the 8P response
of the engine mounts on the other side. On the other hand, the 8P response of one
engine is affected by the stiffness of the struts on that side. Increases in stiffness
lead to decreased response and decreases in stiffness lead to increased response, with
the right strut stiffness having the greatest effect.
4.5 Minimizing Engine Response
In the previous sections, the sensitivity of the 4P and 8P engine response to
changes in stiffness of one or more struts was examined, holding the other stiff-
ness values constant at the baseline. For the large parameter space that is created
by changing each of six stiffness value independently, the?best? solution might be
missed by analyzing the results in this way. That is, a minimum vibration level may
be achieved by a combination of stiffness values not included in the analysis. To this
end, the full set of parametric sweeps for the strut stiffness scaling was examined to
determine the effect of targeting different metrics for minimization. This section will
examine several metrics as targets for minimization and discuss the overall changes
in vibrations for each of those targets. For most of the results presented, symmetric
struts were assumed, with the impact of utilizing an asymmetric design given at the
end of the section. The minimization metrics that will be discussed are:
124
1. 4P/8P response of each of the four individual mount points
2. Average 4P/8P response of only the fore or aft mount points
3. Average 4P/8P response of only the left or right mount points
4. Maximum 4P/8P response
5. Average 4P/8P response of all mount points
6. Average combined 4P and 8P response of all mount points
4.5.1 Minimizing indvidual mount response
For this study, the strut stiffness factors that result in minimum transverse
response for each of the individual 4 mount points were identified and are listed in
Table 4.4. These results are given for both symmetric and asymmetric struts. The
corresponding reductions in 4P and 8P response levels at the mount points are shown
in Table 4.5. These values represent the best possible reduction in vibration for a
single mount point at a single frequency for a single flight condition, i.e. these struts
are point designs, with no consideration for the response at the other 3 mounts or at
other excitation frequencies. These point designs may not be practical to construct.
However, the maximum reduction in vibration establishes a baseline to gauge the
level of compromise when the response of all 4 mounts at both 4P and 8P excitation
are considered.
Fig. 4.14 shows how the responses of each mount point is affected by the choice
of mount stiffness corresponding to a minimum for an individual mount stiffness. For
instance, targeting a minimum 4P response at the fore left mount yields a reduction
125
in that response of almost 0.16g or 60%. However, the 4P response of the two
aft mounts are increased by 100% or more. Similarly, targeting the minimum 8P
response of the aft left mount gives at 0.27g reduction (19%) in that response while
increasing the 4P response of that mount by 0.29g (120%). These results indicate
that the choice of a point design may adversely affect engine vibration at other mount
points, or even the same mount point but at a different excitation frequency. For a
practical design, it is necessary to consider the engine mount vibration at multiple
points, and multiple frequencies. The following sections systematically expand the
scope of the metric involved in identifying a good strut design.
4.5.2 Minimizing maximum mount response
For this study, the strut stiffness factors that produced the minimum largest 4P
or 8P response among the four mount points (maximum response) were determined.
In other words, for each combination of mount stiffness values, the largest response
among the 4 engine mounts was determined. From this set of response values, the
smallest response was identified, and that combination of stiffness factors was used
to calculate the other response levels.
Figure 4.15 shows the effects of minimizing the maximum overall 4P or 8P
response on the response of all the mount points. This is similar metric to mini-
mizing an individual mount response, although it mitigates instances where a large
reduction in one mount response might be accompanied by a significant increase in
the response of another mount at that frequency. When minimizing the largest 4P
126
response, the 4P response of all but one mount is reduced by 9-31%, while the 8P
response is increased by 17-28%.
Minimizing the largest 8P response results in 5-15% reductions in 8P response.
The design results in 40% reduction in the 4P response of the fore left mount, and
19-125% increase in the 4P response of the other three mounts. Therefore, the
response at all points and one frequency may still result in increased response at
select points at other excitation frequencies.
4.5.3 Minimizing mount response at pairs of points
This section considers the minimum average response at pairs of engine points,
one frequency at a time. The four pairs considered are fore mounts for the two
engines, aft mounts for the two engines, left engine mounts, and right engine mounts.
As an example, minimizing the average 8P response of the fore mounts results in
decreased 8P response at all mounts at 8P. However, this design also results in
almost 100% increase in 4P response at the aft mount.
Minimizing the average response for the left engine mounts (and separately
for the right engine) was also studied, at a single frequency. For 4P excitation, 56%
reduction of the one of the targeted mounts corresponds to 67% and 245% increase
on the two points not considered. For 8P excitation, the vibration at the target
points can be decreased through strut design at that frequency, at the cost of more
vibration at 4P at the same points. Further, targeting the 8P response of either the
left or right engine mounts results in reductions in the target responses as well as
127
the 4P response on the opposite side, but leads to large increases in the 8P response
on the opposite side and the 4P response on the same side.
These cases again illustrate the nature of the design compromise. Favorable
results can indeed be achieved for the targeted locations in the elastic structure at a
single frequency. However, the response for the alternate excitation frequency may
increase. Further, the set of mounts not targeted for response reduction may exhibit
more vibration at either 4P or 8P.
4.5.4 Minimizing total response
The results in the preceding section have shown that targeting the individual
mounts, or pairs of mounts for vibration reduction can lead to adverse effects on
the remaining mounts, and so the average of the response magnitudes from all
four engine mounts were considered. Table 4.8 shows the stiffness values which
produce the minimum transverse response averaged over the four mount points,
for 4P excitation and 8P excitation. The last row shows the strut stiffness ratios
(relative to the baseline) obtained when the amplitude of average 4P and average
8P vibration is used as the metric for reduction. Both symmetric and asymmetric
strut designs (with respect to left vs. right engine) are considered in this study.
The results for symmetric strut design are considered first, and then the results for
asymmetric strut design are discussed.
128
4.5.5 Symmetric strut design
Figure 4.18 shows the resulting change from the baseline response for each
mount when the symmetric stiffness values are used. When targeting the average
4P response, the fore left mount response is increased by 30%, and all other mounts
exhibit vibration reduction of 24-43%. For this design (first row of Table 4.8), the 8P
response decreases marginally (2-3.5%) at the fore mounts and increases marginally
(2-5%) at the aft mounts.
When targeting the average 8P response (second row of Table 4.8), all four
mounts show a decrease of 9-16% at 8P. For 4P excitation, the response increases
by as much as 125% for three of the mounts. The 4P response of the fore left mount
exhibits qualitatively opposite behavior to the other three mount response levels,
i.e., while the other three mounts exhibit more 4P vibration, the fore left mount
exhibits reduction of amplitude (in this case, by 40%).
Targeting only the 4P or 8P response (averaged over all 4 mount points) could
lead to adverse effects on the response for the frequency not targeted. Therefore,
it may be necessary to target the average response at all mounts and at both fre-
quencies. The vibration reductions obtained by tuning the strut stiffnesses for this
criterion are shown in Fig. 4.18(c).
Vibration reductions at all mount points and both frequencies are obtained,
except the fore mount of the left engine. In particular, the 4P response of the fore
left mount increases by 7%, while the other three decrease by 11-39%. The 8P
response is reduced by 7-18% at all mounts. This behavior may result from the
129
asymmetry of the airframe structure; a few airframe mode shapes (in the vicinity
of 4P) exhibiting opposite signs for the eigenvector displacement components at
the fore mount of the left engine could drive the opposite qualitative behavior for
that one mount. This result indicates that the airframe structure may require local
redesign to extract additional vibration reduction at the engine mounts. However,
such an endeavor must be weighed against other constraints for airframe design such
as weight and vibration reduction at other stations.
4.5.6 Asymmetric strut design
When the strut stiffness is allowed to vary between the left an right engine
mounts, slightly different vibration trends are noted. The outbooard strut design is
unchanged when asymmetry is allowed. Further, only the left engine struts change
for minimum 4P, and only the right struts change for targeting 8P response. When
minimizing the average combined 4P and 8P response (Table 4.8), all 6 struts should
be stiffened to 400% baseline for a symmetric design. With symmetric struts, a
reduction of 1.18g in combined 4P and 8P response magnitude is achieved. If asym-
metry is allowed, softening the left inner strut to 50% provides a marginal benefit:
the total reduction increases to 1.22g.
4.6 Summary
The results of the stiffness sweep study show that overall reduction in both 4P
and 8P response is possible if the stiffness of all struts is increased above the baseline.
130
Both the symmetric and asymmetric stiffness sweep results indicate that the largest
cumulative percent reduction in the 4P and 8P response of all four mounts can be
achieved if all six struts have their stiffness increased to 400% of the baseline.
Further, results of the study of torque tube stiffness indicate that reductions in
both 4P and 8P response at the fore mounts is possible if the torque tube stiffness
is increased to 200% of its baseline. A final symmetric strut stiffness sweep was
conducted with 200% torque tube stiffness. The goal of this sweep was to determine
whether changing the torque tube design results in a different strut design, providing
a qualitative measure of the cross-coupling between fore mount and aft mount design.
It was found that increasing the stiffness of all six struts to 400% of the baseline
gives the best improvement in 4P and 8P vibration reduction. In both cases it was
found that the large reductions were retained when increasing the strut stiffness
to only 200% and only the aft right mount response was further improved by the
400% stiffness, achieving an additional 10% reduction. Trade studies of vibration
reduction and the possible weight/cost penalties of increasing the strut stiffness are
necessary to determine whether additional stiffening is advantageous on the whole.
Based on these findings, the transverse response at the four engine mount
points were calculated for two improved designs. The first improved design is when
only the rear mount struts are stiffened to 200%. The second improved design is
when both the struts and torque tube are stiffened to 200%. 4P and 8P hub loads
were applied at the main rotor hub based on the four level cruise flight speeds
previously considered: 55, 80, 100, 125 and 150 knots. Figures 4.19(a) and 4.19(b)
show the 4P and 8P response levels for the baseline and the two improved designs.
131
This speed sweep demonstrates the cumulative effects of increasing the stiffness
of both the fore mount torque tube and aft mount struts. Overall, reductions in the
4P and 8P transverse response are achieved at all four mounts and flight conditions
with one exception. The 4P response of the fore left mount shows increases of 5-
13% when only the strut stiffness is increased. This detrimental effect is mitigated
to 0.3-8% increase for the second improved design. At all other mounts, enormous
reductions of 11-58% are achieved at both 4P and 8P.
Specifically, the stiffer aft mounts reduce the 4P response by 30% for the aft
right mount and 10-20% for the fore right and aft left mounts. The effect of stiffer
struts on reduction of 8P response at all mounts is less significant, but still noticeable
(5-8%). A stiffer torque tube results in significant reduction in the 8P response (up
to 50%), while having a small benefit on the 4P response at the fore mount points
for both engines.
132
(a) 4P hub load components
(b) 8P hub load components
Figure 4.1: Hub load amplitudes (CW/?=0.082)
133
(a) Fore left lateral response (b) Fore left vertical response
(c) Fore right lateral response (d) Fore right vertical response
(e) Aft left lateral response (f) Aft left vertical response
(g) Aft right lateral response (h) Aft right vertical response
Figure 4.2: Breakdown of 4P engine mount response from individual hub load com-
ponent contributions (CW/?=0.082, ?=0.233)
134
(a) Fore left lateral response (b) Fore left vertical response
(c) Fore right lateral response (d) Fore right vertical response
(e) Aft left lateral response (f) Aft left vertical response
(g) Aft right lateral response (h) Aft right vertical response
Figure 4.3: Breakdown of 8P engine mount response from individual hub load com-
ponent contributions (CW/?=0.082, ?=0.233)
135
(a) Total 4P transverse response at various flight speeds
(b) Total 8P transverse response at various flight speeds
Figure 4.4: Total transverse response with forward speed (CW/?=0.082)
136
(a) 4P response
(b) 8P response
Figure 4.5: 4P transverse response of engine mount points for various torque tube
stiffnesses
137
Figure 4.6: Schematic of engine mount structure
138
(a) 4P response
(b) 8P response
Figure 4.7: Transverse response of engine mount points for variations in all 6 strut
stiffness matrices.
139
(a) Fore left
(b) Fore right
(c) Aft left
(d) Aft right
Figure 4.8: Transverse 4P response at engine mounts for variations in stiffness and
damping of all 6 struts.
140
(a) Fore left
(b) Fore right
(c) Aft left
(d) Aft right
Figure 4.9: Transverse 8P response at engine mounts for variations in stiffness and
damping of all 6 struts.
141
(a) Inner strut effects
(b) Middle strut effects
(c) Outer strut effects
Figure 4.10: Transverse 4P response of engine mount points for left/right symmetric
variations in each individual strut holding all others at baseline values.
142
(a) Inner strut effects
(b) Middle strut effects
(c) Outer strut effects
Figure 4.11: Transverse 8P response of engine mount points for left/right symmetric
variations in each individual strut holding all others at baseline values.
143
(a) Left strut stiffness effects
(b) Right strut stiffness effects
Figure 4.12: Effects of changes in left or right strut stiffness on 4P engine response
144
(a) Left strut stiffness effects
(b) Right strut stiffness effects
Figure 4.13: Effects of changes in left or right strut stiffness on 8P engine response
145
(a) Minimizing individual 4P response
(b) Minimizing individual 8P response
Figure 4.14: Effect of minimizing response at individual engine mount point on
response at other points: change in amplitude with respect to vibration for baseline
strut design
146
(a) Minimizing maximum 4P response
(b) Minimizing maximum 8P response
Figure 4.15: Effect of minimizing response at individual engine mount point on
response at other points: change in amplitude with respect to vibration for baseline
strut design
147
(a) Minimizing fore or aft 4P response
(b) Minimizing fore or aft 8P response
Figure 4.16: Effect of minimizing response of both fore or aft mount points: change
in amplitude with respect to vibration for baseline strut design
148
(a) Minimizing left or right 4P response
(b) Minimizing left or right 8P response
Figure 4.17: Effect of minimizing response of left or right engine mount points on
response at other points: change in amplitude with respect to vibration for baseline
strut design
149
(a) Minimizing average 4P response
(b) Minimizing average 8P response
(c) Minimizing average combined 4P and 8P response
Figure 4.18: Effect of minimizing engine total engine response: change in amplitude
with respect to vibration for baseline strut design
150
(a) 4P engine mount response: various airspeeds
(b) 8P engine mount response: various airspeeds
Figure 4.19: Effect of stiffer struts and stiffer torque tube on 4P and 8P engine
mount response
151
Table 4.1: Amplitude of 4P engine mount point displacement response (in) to unit
hub force (lb) or moment (ft-lb)
Hub force component
Fx Fy Fz Mx My Mz
x 1.37E-05 5.73E-06 4.52E-06 8.32E-08 3.63E-07 1.92E-07
fore left y 4.32E-06 3.92E-06 3.72E-06 8.17E-08 1.37E-07 3.72E-08
z 3.95E-06 1.38E-05 8.68E-06 5.88E-07 7.34E-08 1.66E-07
x 1.38E-05 2.61E-06 8.92E-06 7.60E-08 5.45E-07 2.98E-07
fore right y 5.17E-06 4.41E-06 7.55E-06 5.08E-08 1.99E-07 8.33E-08
z 3.36E-06 1.20E-05 6.43E-06 4.57E-07 8.60E-08 6.50E-08
x 1.35E-05 6.16E-06 4.36E-06 1.00E-07 3.55E-07 1.89E-07
aft left y 6.50E-06 1.41E-06 7.74E-06 3.70E-07 1.63E-07 9.44E-08
z 4.03E-06 2.78E-06 4.25E-06 5.40E-08 2.11E-07 5.12E-08
x 1.36E-05 2.22E-06 9.09E-06 6.19E-08 5.39E-07 2.99E-07
aft right y 8.30E-06 2.41E-06 1.75E-05 3.10E-07 3.82E-07 2.11E-07
z 3.89E-06 1.74E-06 2.57E-06 3.43E-08 1.24E-07 1.39E-08
152
Table 4.2: Amplitude of 8P engine mount point displacement response (in) to unit
hub force (lb) or moment (ft-lb)
Hub Force Component
Fx Fy Fz Mx My Mz
x 3.43E-08 3.75E-07 8.15E-08 4.88E-08 9.08E-09 1.72E-08
fore left y 5.79E-07 2.67E-06 1.73E-06 3.62E-07 6.72E-08 3.15E-08
z 3.74E-06 1.36E-06 6.81E-07 1.00E-07 3.73E-07 4.24E-08
x 6.43E-08 3.91E-07 6.42E-08 5.19E-08 1.08E-08 1.55E-08
fore right y 9.42E-07 2.90E-06 1.74E-06 3.96E-07 1.30E-07 2.76E-08
z 3.66E-06 1.25E-06 4.66E-07 8.45E-08 3.63E-07 4.47E-08
x 1.51E-07 2.90E-07 1.76E-07 4.06E-08 2.01E-08 1.72E-08
aft left y 3.75E-07 3.67E-07 4.51E-07 5.82E-08 3.64E-08 7.73E-09
z 1.96E-07 1.68E-06 3.71E-06 2.19E-07 2.36E-08 4.50E-08
x 1.57E-07 3.09E-07 2.05E-07 4.36E-08 1.78E-08 1.63E-08
aft right y 4.42E-07 3.88E-07 5.46E-07 6.43E-08 4.41E-08 4.22E-09
z 2.22E-07 1.74E-06 4.80E-06 2.34E-07 2.56E-08 3.27E-08
153
Table 4.3: Baseline vibration levels at engine mount points at 100 knots
4P 8P
Fore left 0.26g 2.31g
Fore right 0.29g 2.57g
Aft left 0.23g 1.59g
Aft right 0.40g 1.31g
154
Table 4.4: Strut stiffness factors required to achieve minimum 4P or 8P response at
each engine mount
Inner Middle Outer
Left Right Left Right Left Right
Asymmetric 25% 200%
Aft left 25% 400%
Symmetric 25%
Asymmetric 200% 50% 25% 50% 25% 25%
Fore left
Symmetric 400% 25% 200%
4P
Asymmetric 100% 25% 50% 25% 25% 400%
Aft right
Symmetric 25% 25% 400%
Asymmetric 200% 50%
Fore right 25% 25%
Symmetric 42%
Asymmetric 400% 25% 400% 25%
Aft left 25%
Symmetric 400% 400%
Asymmetric 400% 25% 400% 25%
Fore left 25%
Symmetric 400% 400%
8P
Asymmetric 25% 400% 25% 400%
Aft right 25%
Symmetric 400% 400%
Asymmetric 25% 400% 25% 400%
Fore right 25%
Symmetric 400% 400%
155
Table 4.5: Largest possible reduction from baseline in 4P and 8P transverse for each
mount point for symmetric and asymmetric strut stiffness values
Asymmetric Symmetric
Aft left -0.10g -44.90% -0.06g -24.34%
Fore left -0.23g -85.66% -0.16g -60.26%
4P
Aft right -0.20g -50.24% -0.17g -42.93%
Fore right -0.11g -36.55% -0.09g -29.31%
Aft left -0.50g -36.70% -0.27g -19.44%
Fore left -0.46g -19.24% -0.36g -14.89%
8P
Aft right -0.64g -38.75% -0.32g -19.50%
Fore right -0.48g -17.75% -0.44g -16.44%
156
Table 4.6: Strut stiffness factors that produce minimum response for different targets
Inner Middle Outer
Left Right Left Right Left Right
Symmetric 25% 25%
4P total 400%
Asymmetric 400% 25% 400% 25%
Symmetric 25% 400%
8P total 400%
Asymmetric 25% 400% 400% 25%
Symmetric 400%
4P + 8P 400% 400%
Asymmetric 50% 400%
157
Table 4.7: Summary of parametric studies performed
Parameter Range Summary
strut stiffness 25-400% baseline 4P response unaffected by stiffness of
outboard struts; 8P response
unaffected stiffness of inboard struts
strut damping 2%, 5%, 10% Response unaffected by damping at
high stiffness
torque tube stiffness 25-400% baseline 8P response of fore mount most
affected; increasing stiffness decreases
response
158
Table 4.8: Strut stiffness factors that produce minimum response for different targets
Inner Middle Outer
Left Right Left Right Left Right
Symmetric 25% 25%
4P total 400%
Asymmetric 400% 25% 400% 25%
Symmetric 25% 400%
8P total 400%
Asymmetric 25% 400% 400% 25%
Symmetric 400%
4P + 8P 400% 400%
Asymmetric 50% 400%
159
5 Coupled Rotor-Airframe-Drivetrain
Analysis
A coupled analysis was undertaken to examine the effects of the various fixed
frame components on the rotor loads and hub loads. The effects of coupling on
the vibratory loads experienced by the engines were also investigated. Prescribed
airloads were used to compare results with existing test data as well as work per-
formed by other researchers. Free-flight analysis was performed on an expanded
flight envelope to examine the importance of these effects at different flight condi-
tions. Results were obtained for CW/? = 0.084 and steady level flight conditions
with forward speeds of 50-175 knots (advance ratios ? = 0.12? 0.41).
5.1 Baseline Coupling Results (Prescribed Air-
loads)
The results in this section utilize measured airload and lag damper data from
the NASA/Army UH-60A Airloads Flight Test Program [88] for a high speed flight
condition (C8534). A baseline, uncoupled analysis (fixed hub condition) is performed
160
and compared with results of coupling with the RSRA drivetrain model and the
airframe model described in Chapter 3. The coupled and uncoupled results are
compared to existing flight test data.
Figure 5.1 shows the waveform for the blade flap bending moments at 30%,
50%, and 70% radial stations. These plots show little discernible effect of drivetrain
coupling on the flap bending moments, while the airframe coupling has a more
noticeable, though still small effect. These effects can be better demonstrated when
the waveform is broken down by harmonic content, as shown in Figs. 5.2 and
5.3. In particular, the drivetrain coupling has only a small effect on the 4/rev
flap bending moment. On the other hand, the airframe coupling has the most
noticeable effect on the 4/rev and 5/rev flap bending moments. Further, although
the effects of airframe coupling are noticeable, there is not a marked improvement
in the correlation between the prediction and test data.
The chordwise bending moments are shown in Fig. 5.4, with the harmonic
content shown in Figs. 5.5 and 5.6. The drivetrain coupling has a more noticeable
effect on the chordwise bending than observed for the flap bending moments. In
particular, there is a slight increase in the 1/2 peak-to-peak amplitude and 4/rev
component, resulting in an overall improvement of the correlation with flight test
data. The airframe coupling also had a larger effect on the chordwise bending,
particularly decreasing the 5/rev and increasing the 6/rev content, again leading to
slightly improved correlation.
Overall, these results show that the flap bending moments are not much af-
fected by either the drivetrain or airframe coupling, though the airframe coupling
161
has a noticeable small effect on the 4/rev and 5/rev components. On the other
hand, the chordwise bending moments do show effects from both the drivetrain and
airframe coupling. These results agree qualitatively with similar studies [51,52].
5.2 Rotor-Airframe-Drivetrain Coupling
The results in this section are obtained for free flight from 50?175 knots using
Maryland Free Wake. A baseline case was computed with a hub fixed rigidly to
the aircraft, and coupled results were obtained for the rotor coupled separately with
the drivetrain and airframe described in Sections 3.2 and 3.3, as well as for the
rotor coupled with both the drivetrain and airframe. The flexible engine model was
included in the airframe model for these results.
5.2.1 Hub motion
The 1/2 peak-to-peak amplitude of the hub motion relative to the airframe
resulting from coupling with either the flexible airframe, drivetrain, or both is ex-
amined in this section. Figure 5.13 shows the three degrees of freedom of hub trans-
lation due to coupling with the airframe only as well as the airframe and drivetrain
together. These results show that the coupling produces very small hub motions,
generally less that 0.05 in. The in-plane hub motions remain relatively consistent
for flight speeds from 50 to 125 knots. At 175 knots, the longitudinal hub motion
increases by approximately three times compared to the lower speeds, while the lat-
eral hub motion increases by four times the lower speeds. The vertical hub motion is
162
highest at 50 and 175 knots, with a minimum around 100 knots, while being overall
lower than the in-plane motion by almost an order of magnitude. The pitch and
roll rotation of the hub, shown in Fig. 5.14, average around 0.03 degrees and show
similar trends to the in-plane hub translations, with substantial increases at 175
knots.
Figures 5.13 and 5.14 also reveal that the addition of the drivetrain coupling
does not significantly change the hub translation or pitch/roll rotations beyond the
effects of the airframe coupling. This is not surprising, given that the drivetrain
is only coupled to the torsional degree of freedom of the rotor, and so should not
significantly affect the other degrees of freedom. On the other hand, the airframe
does have a significant effect on the torsional motion of the rotor hub beyond that of
the drivetrain only, as shown in Fig. 5.15. Generally, the addition of the airframe to
the rotor-drivetrain coupling serves to decrease the peak-to-peak torsional motion
of the rotor by 25-35%.
5.2.2 Blade loads
Figure 5.16 shows the uncoupled and coupled mid-span flap and lag bending
and torsion moment for the 75 knots flight condition. From the waveforms, it is
evident that the effects of coupling are smallest for the flap bending moment, with
the drivetrain coupling having the greatest effect. The torsion moment is largely
unaffected by the drivetrain coupling alone, while noticeably affected by the presence
of the airframe model. The lag bending moment appears significantly affected by
163
the presence of the airframe, while also affected to a lesser degree by the drivetrain
coupling.
To better quantify the effects of the various couplings on the bladeloads, it is
instructive to examine the peak-to-peak and vibratory components of the waveforms
over the entire span of the blade. Figures 5.17 and 5.18 show the half peak-to-peak
and vibratory flap bending components. These results show that the drivetrain has
very little effect on the flap bending moment overall. There is a minor reduction in
the peak-to-peak flap bending at the root and mid-span. There are also noticeable
reductions in higher harmonics (7, 8, 9/rev), with the 8/rev reduction being partic-
ularly significant, though these components fairly small. The airframe coupling had
a more noticeable, though still small, effect on the flap bending, with a significant
impact on the 5/rev flap bending load, reducing the 5/rev component by 50% or
more across the entire blade span.
Figures 5.19 and 5.20 show the half peak-to-peak and vibratory flap bending
components, and confirm that the coupled analysis has a greater impact on these
loads than the flap bending loads. The drivetrain has a large effect on the higher
harmonics, particularly the 8/rev component, which is increased by as much as
200% at about 0.7R. On the other hand, the airframe coupling has very little effect
on most on the lag bending components, except for the 5/rev component, which is
reduced by about 50% at the mid-span.
Figures 5.21 and 5.22 show the half peak-to-peak and vibratory torsion mo-
ment components. The drivetrain significantly reduces the 8/rev torsion moment,
while having very little effect on the other vibratory components. The airframe
164
coupling has the biggest impact on the 5/rev and 6/rev components, though all are
somewhat affected.
5.2.3 Hubloads
Although the magnitude of the hub motion that results when the rotor is
coupled to a flexible component is very small and the overall effect on the blade loads
is generally small, as shown in the previous sections, the change in the vibratory
hubloads that results from this motion can be significant. Figures 5.23 and 5.24
show the 4P and 8P hub forces and moments for the uncoupled rotor as well as
the rotor coupled with either the drivetrain, airframe or both. The drivetrain alone
has very little effect on the 4P in-plane force, and a more noticeable, though small
effect on the 4P vertical force and in-plane moments, primarily at 150 knots. On the
other hand, the 8P hubloads are more impacted by the presence of the drivetrain,
particularly the vertical force at 50-125 knots and and in-plane moment at 50 and
125 knots.
Coupling with the airframe alone decreases the 4P vertical force from the
uncoupled baseline. This effect is small and increases with increasing flight speed,
resulting in about a 10% reduction in 4P vertical force at 175 knots. On the other
hand, the airframe coupling increases the 4P in-plane forces and moments, as well
as all 8P hub loads across all flight speeds. The effect is most pronounced for the
in-plane force. The 4P and 8P in-plane forces are increased by 60-100%
Finally, the addition of the drivetrain to the rotor-airframe coupling follows
165
similar trends as the addition of the drivetrain compared to the uncoupled case.
For instance, the drivetrain increases the 4P vertical force by about 5% from the
uncoupled case. Similarly, the airframe/drivetrain coupling increases the 4P vertical
force by about 5% from the airframe coupling alone.
5.2.4 Airframe Vibrations
Although identifying the effects of various couplings on the structural loads
in the blades and fixed frame hubloads is important to deepening our overall un-
derstanding of the physics at work, from a practical standpoint, it is the resulting
vibrations experienced throughout the airframe, particularly at the pilot and copilot
seat, that are of greatest interest to helicopter users. Figures 5.25 and 5.26 show the
transverse and vertical 4P and 8P acceleration at grid points corresponding to the
pilot and co-pilot seats of the airframe. Vibration levels for the uncoupled case were
calculated by applying the calculated hub loads from the uncoupled analysis to the
airframe finite element model. These results show that the coupled model generally
predicts lower 4P vibration and higher 8P vibration at the pilot/co-pilot seats than
the uncoupled analysis, though the difference between the uncoupled and coupled
analysis is much more significant for the 4P vibrations. The addition of the drive-
train model to the airframe coupling has little impact on the 4P vertical vibration
and no discernible impact on the 4P transverse acceleration. The 8P vibrations are
much more impacted by the presence of the drivetrain in addition to the airframe.
166
5.3 Engine modeling
The fully coupled results shown in the previous section include a detailed flexi-
ble engine model coupled to the airframe. The turboshaft engines used in helicopters
have heavy rotors spinning at very high speeds ( 20,000 RPM) which produce gyro-
scopic forces that are transmitted through the airframe. The detailed NASTRAN
engine model provided for this work included gyroscopic elements that can be turned
on and off within NASTRAN. For the analysis presented thus far, these gyroscopic
effects were neglected. The following results will examine the impact of the complex-
ity of the engine model on the analysis, including the effects of including gyroscopic
forces. Four cases are considered:
1. flexible airframe without any engine model
2. flexible airframe with rigid engine mass
3. flexible airframe with flexible engine
4. flexible airframe with flexible engine with gyroscopic forces.
Figures 5.27 and 5.28 show the 4P and 8P vertical hub forces and in-plane
hub forces and moments for the uncoupled baseline and the four coupled cases.
These results show that the presence of an engine model of any complexity has the
greatest impact on the in-plane hub forces. There is very little difference between
the rigid, flexible, and gyroscopic engine in the prediction of 4P hub loads. The
8P hub loads are more sensitive to engine modeling, particularly at high speeds,
167
though the gyroscopic forces do not a discernible impact when compared to the
flexible engine without gyroscopic forces.
Figures 5.29 and 5.30 compare the effects of the engine modeling on the 4P
and 8P accelerations calculated at the pilot and co-pilot seats. Since the vibratory
hub loads driving the vibrations throughout the airframe are not much influenced by
the differences in engine modeling, the differences in vibrations at the pilot/co-pilot
seats are due to differences in the models.
Figures 5.31 and 5.32 compare the effects of the engine modeling on the 4P
and 8P accelerations calculated at the four engine mount points identified in Chap-
ter 3. At 4P, the flexibility of the engine model increases the vibration predicted
on the left engine and decreases the predicted vibration on the right engine when
compared to the rigid engine model. Further, the gyroscopic forces have no notice-
able effect on the 4P vibrations. On the other hand, the 8P engine vibrations are
highly susceptible to differences in engine models. The gyroscopic engine modeling
drastically increases the 8P vibrations on the left engine compared to to the flexible
engine, while decreasing the 8P vibrations on the right engine.
168
(a) 30%R
(b) 50%R
(c) 70%R
Figure 5.1: Effects of coupling on flap bending moments for UH-60A using measured
airloads (C8534)
169
(a) 1/2 peak-to-peak (b) 3/rev
(c) 4/rev (d) 5/rev
Figure 5.2: Effects of coupling on flap bending moment peak-to-peak amplitude and
harmonics for UH-60A using measured airloads (C8534)
170
(a) 6/rev (b) 7/rev
(c) 8/rev (d) 9/rev
Figure 5.3: Effects of coupling on flap bending moment higher harmonics for UH-
60A using measured airloads (C8534)
171
(a) 30%R
(b) 50%R
(c) 70%R
Figure 5.4: Effects of coupling on chordwise bending moments for UH-60A using
measured airloads (C8534)
172
(a) 1/2 peak-to-peak (b) 3/rev
(c) 4/rev (d) 5/rev
Figure 5.5: Effects of coupling on chordwise bending moment peak-to-peak ampli-
tude and harmonics for UH-60A using measured airloads (C8534)
173
(a) 6/rev (b) 7/rev
(c) 8/rev (d) 9/rev
Figure 5.6: Effects of coupling on chordwise bending moment higher harmonics for
UH-60A using measured airloads (C8534)
174
(a) x motion (b) ?x motion
(c) y motion (d) ?y motion
(e) z motion (f) ?z motion
Figure 5.7: Effects of single degree of freedom 4P hub motion on mid-span flap
bending moment
175
(a) x motion (b) ?x motion
(c) y motion (d) ?y motion
(e) z motion (f) ?z motion
Figure 5.8: Effects of single degree of freedom 4P hub motion on mid-span chordwise
bending moment
176
(a) x motion (b) ?x motion
(c) y motion (d) ?y motion
(e) z motion (f) ?z motion
Figure 5.9: Effects of single degree of freedom 4P hub motion on mid-span torsion
moment
177
(a) x motion (b) ?x motion
(c) y motion (d) ?y motion
(e) z motion (f) ?z motion
Figure 5.10: Effects of single degree of freedom 8P hub motion on mid-span flap
bending moment
178
(a) x motion (b) ?x motion
(c) y motion (d) ?y motion
(e) z motion (f) ?z motion
Figure 5.11: Effects of single degree of freedom 8P hub motion on mid-span chord-
wise bending moment
179
(a) x motion (b) ?x motion
(c) y motion (d) ?y motion
(e) z motion (f) ?z motion
Figure 5.12: Effects of single degree of freedom 4P hub motion on mid-span torsion
moment
180
(a) Longitudinal hub motion
(b) Lateral hub motion
(c) Vertical hub motion
Figure 5.13: 1/2 Peak-to-peak hub translation resulting from coupling with airframe
and drivetrain 181
(a) Roll hub motion
(b) Pitch hub motion
Figure 5.14: 1/2 Peak-to-peak hub roll and pitch rotation resulting from coupling
with airframe and drivetrain
182
(a) Roll hub motion
Figure 5.15: 1/2 Peak-to-peak hub yaw rotation resulting from coupling with air-
frame and drivetrain
183
(a) Flap bending moment
(b) Lag bending moment
(c) Torsion moment
Figure 5.16: Effects of coupling on mid-span blade loads
184
(a) 1/2 peak-to-peak (b) 3/rev
(c) 4/rev (d) 5/rev
Figure 5.17: Effects of coupling on vibratory flap bending
185
(a) 6/rev (b) 7/rev
(c) 8/rev (d) 9/rev
Figure 5.18: Effects of coupling on vibratory flap bending higher harmonics
186
(a) 1/2 peak-to-peak (b) 3/rev
(c) 4/rev (d) 5/rev
Figure 5.19: Effects of coupling on vibratory lag bending
187
(a) 6/rev (b) 7/rev
(c) 8/rev (d) 9/rev
Figure 5.20: Effects of coupling on vibratory lag bending higher harmonics
188
(a) 1/2 peak-to-peak (b) 3/rev
(c) 4/rev (d) 5/rev
Figure 5.21: Effects of coupling on vibratory torsion moment
189
(a) 6/rev (b) 7/rev
(c) 8/rev (d) 9/rev
Figure 5.22: Effects of coupling on vibratory torsion momment higher harmonics
190
(a) Vertical hub force
(b) In-plane hub force
(c) In-plane hub moment
Figure 5.23: Effects of coupling on 4P hub loads
191
(a) Vertical hub force
(b) In-plane hub force
(c) In-plane hub moment
Figure 5.24: Effects of coupling on 8P hub loads
192
(a) Pilot transverse acceleration (b) Pilot vertical acceleration
(c) Co-pilot transverse acceleration (d) Co-pilot vertical acceleration
Figure 5.25: Effects of coupling on 4P pilot/co-pilot seat vibrations
193
(a) Pilot transverse acceleration (b) Pilot vertical acceleration
(c) Co-pilot transverse acceleration (d) Co-pilot vertical acceleration
Figure 5.26: Effects of coupling on 8P pilot/co-pilot seat vibrations
194
(a) Vertical hub force
(b) In-plane hub force
(c) In-plane hub moment
Figure 5.27: Effects of engine modeling on 4P hub loads
195
(a) Vertical hub force
(b) In-plane hub force
(c) In-plane hub moment
Figure 5.28: Effects of engine modeling on 8P hub loads
196
(a) Pilot transverse acceleration (b) Pilot vertical acceleration
(c) Co-pilot transverse acceleration (d) Co-pilot vertical acceleration
Figure 5.29: Effects of engine modeling on 4P pilot/co-pilot seat vibrations
197
(a) Pilot transverse acceleration (b) Pilot vertical acceleration
(c) Co-pilot transverse acceleration (d) Co-pilot vertical acceleration
Figure 5.30: Effects of engine modeling on 8P pilot/co-pilot seat vibrations
198
(a) Left fore engine mount (b) Right fore engine mount
(c) Left aft engine mount (d) Right aft engine mount
Figure 5.31: Effects of engine modeling on 4P engine vibrations
199
(a) Left fore engine mount (b) Right fore engine mount
(c) Left aft engine mount (d) Right aft engine mount
Figure 5.32: Effects of engine modeling on 8P engine vibrations
200
6 Summary and Conclusions
This chapter summarizes the primary results of this dissertation, identifies the
key conclusions of the work and provides suggestions for future work in this area.
6.1 Summary
The work presented in this thesis consists of two main parts: the coupling
between the finite element models of an engine and a helicopter airframe and the
coupling of the aeromechanical model of a helicopter rotor to models representing a
flexible drivetrain and airframe.
A fixed-frame frequency domain sub-structuring approach was developed to
couple engine and airframe models in order to analyze vibration levels at engine
supports of a helicopter. This methodology has several key advantages over conven-
tional full finite element models:
1. Transfer functions can be generated from full finite element models without the
necessity of modal approximation or truncation. The various types of damping
(combination of complex stiffness, viscous damping and modal damping) can
also be included in the analysis without any loss of accuracy.
201
2. Millions of finite element DOF can be reduced to only the set of interface nodes,
external forcing nodes, and any additional nodes where one is interested in the
response. Because only the small subset of nodes are involved in the solution,
the solution process is computationally efficient and up to 1000 times faster
than full finite element models.
3. The method may be extended to nonlinear interface stiffness and/or damping
as well as experimentally acquired transfer functions for individual compo-
nents.
The computational efficiency of the developed substructuring methodology
allowed for parametric studies on the engine mount structures for a S-92 helicopter
airframe coupled with the associated GE turboshaft engines. The conclusions from
those studies are summarized in the following section.
Existing comprehensive analysis methodology was expanded to account for
a flexible hub condition, allowing the rotor to be coupled to components in the
fixed frame, namely a flexible airframe and drivetrain. This coupling leverages the
previous frequency domain substructuring methodology, using frequency domain
models of the fixed frame components and coupling them with the rotor analysis,
and the impact of these couplings on the blade structural loads, fixed frame hubloads,
and vibrations at the pilot seat and engines were examined. Further, the effects of
the complexity of the engine modeling within the airframe were also examined. The
primary conclusions of this work are detailed in the next section.
202
6.2 Key Conclusions
6.2.1 Engine-Airframe Coupling
1. The transverse 4P engine response is sensitive to changes in either or both
inboard strut stiffness, but is unaffected by changes in the outboard struts.
Conversely, the transverse 8P engine response is sensitive to changes in the
outboard strut stiffness while being generally unaffected by changes in either
or both pairs of inboard struts.
2. Doubling the stiffness of the torque tube reduced the 8P transverse response
of the fore mount points of the engine by 50%, while having little effect on the
aft mount point response or the 4P response at any mount point.
3. The 4P response at the aft mount of the right engine may be high if not
specifically targeted for vibration reduction. If ignored, the vibration may
increase by 100% or more of the baseline value.
4. In most cases, strut structural damping does not offer significant reduction in
vibratory response at the engine mounts.
5. Changing the stiffness of the torque tube does not affect the 4P response and
reduces the fore mount response for both engines by 20%.
6. The 8P response is most affected by asymmetry in the mount struts stiffness.
7. When the stiffness of the aft mount struts is varied, the 4P response of the left
203
engine fore mount increases when the response of the other three mounts de-
crease. This represents a fundamental compromise in overall engine vibration
minimization.
8. Stiffening the torque tube and each of the aft mount struts to 200% of their
baseline values results in significant reductions of 4P and 8P response at all
engine mounts for most flight conditions considered. On average, the 4P re-
sponse was reduced by 14% and the 8P response was reduced by 37%.
9. The high level of asymmetry present in the system as well as the lack of a
consistent solution to minimize vibrations at each mount point simultaneously
indicate that the problem of engine vibrations is likely best addressed through
an active vibration control strategy.
6.2.2 Rotor-Drivetrain-Airframe-Engine Coupling
A baseline UH-60 rotor model was individually coupled with representative
drivetrain and airframe models. The following trends were observed:
1. The addition of a flexible drivetrain has very little impact on the blade struc-
tural loads. When examining the vibratory components, the higher harmonics,
particularly 8P, are the most affected, with the lag bending increased, while
the flap bending and torsion are decreased. On the other hand, the airframe
coupling are seen most noticeably in the 5P blade loads.
2. Despite small impacts on the blade loads, the fixed frame hub loads can be
204
significantly changed by the airframe and drivetrain coupling, which has im-
plications for the prediction of vibrations throughout the airframe. The 4P in-
plane forces and moments are significantly increased by the airframe coupling,
but not by the drivetrain coupling. The 4P vertical force is decreased with
airframe coupling, and increased with drivetrain coupling, while the opposite
is true for all the 8P hub forces and moments. As a result, when compared
to the uncoupled case, 4P vibrations computed at the pilot/co-pilot seats are
lower for the coupled cases, while 8P vibrations are higher.
3. The presence of an engine mass within the airframe model is important in
predicting the hub loads, particularly the in-plane forces. However, the com-
plexity of the engine model does not have much influence. In particular, the
4P hub loads are largely unchanged by a rigid engine mass versus a flexible
engine with or without gyroscopic effects. The 8P are generally decreased by a
flexible versus rigid engine, and although this effect is small, it is significant for
the 8P in-plane forces at higher vibration conditions. The engine gyroscopic
forces have negligible influence on the hub loads.
4. The vertical 4P vibration at the pilot/co-pilot seat is significantly impacted
by the engine flexibility, though asymmetrically. That is, the flexible engine
results in a decrease of the predicted vibration at the pilot seat and an increase
at the co-pilot seat. The 8P vibration prediction is generally decreased by the
flexible engine and increased with the addition of gyroscopic effects.
5. The predicted 4P engine vibration is asymmetrically affected by the flexible
205
engine model in that vibrations on the left engine are increased while vibra-
tions on the right are decreased by the flexible model when compared with the
rigid model. The gyroscopic forces have negligible impact on the 4P engine vi-
brations. The 8P engine vibration predictions are significantly by both engine
flexibility and gyroscopic forces. The flexible engine generally results in an
increase in the predicted 8P vibration over the rigid engine Gyroscopic forces
increase the predicted vibration on the left engine while decreasing them on
the right.
6.3 Recommendations for Future Work
The research presented in this dissertation have illustrated the impact that
coupling of rotor analysis with airframe and drivetrain models can have on the pre-
diction of vibrations at the main rotor hub and throughout the airframe. However,
much work is still required before analysis tools can be used reliably for vibration
prediction during design. To that end, the overall effectiveness of the analysis in this
work needs to be benchmarked against flight test data. This level of validation re-
quires compatibility between the rotor, airframe, engine, and drivetrain models with
respect to the actual components used in testing. It is also important to have access
to a wide range of test data, including accurate measurements of blade loads, hub
loads, and vibrations throughout the airframe, specifically at the pilot seat and on
the engines. The ability of this type of validation to be conducted at the university
level will require a very high level of cooperation with the industry and government
206
entities who produce the components and models and perform the flight tests.
In addition to further validating the analysis against available data, there are
additions the the analysis that are likely to have great impact on the vibration
predictions presented here. The first of these is the incorporation of CFD aerody-
namics into the comprehensive analysis. The accuracy of aerodynamic loads and
blade structural loads have been shown to be significantly improved by the use of
CFD over free wake modeling. Previously, coupled CFD/CSD analysis has been
done with the PRASADUM code used in this analysis, and access to high perfor-
mance computing resources should make this type of analysis relatively simple to
incorporate.
In addition, sources of vibratory loading on the airframe beyond the main
rotor need to be accounted for. The first and simplest of these to account for is the
tail rotor. As described in section 2.1, a dynamic inflow model was used to calculate
the thrust and torque at the tail rotor, however, a structural model of the tail rotor
blades should be incorporated to gain understanding of the vibratory loads that are
produced at the tail rotor and their impact on vibrations throughout the airframe.
The interactional aerodynamics between the rotor and airframe can also be ex-
amined. Though the primary source of the 4P and 8P loading on the airframe come
through the main rotor hub, the rotor wake impinges on the airframe, particularly
at low speeds, thus provides another source of vibratory loading that has not been
accounted for. Addressing the interactional aerodynamics can include incorporation
of panel methods as well as full CFD modeling including the airframe.
Finally, a major component of this work was focused on reduction of engine
207
vibrations by tuning engine mount structural properties. For the purposes of this
work, those properties were assumed to be linearly elastic. The analysis can easily be
adapted to allow for non-linear properties, as those found in elastomerics, allowing
for the possibility of further reduction of engine vibrations.
208
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