Development of a Spacial Dynamic Handling and Securing ...
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Development of a Spacial Dynamic Handling and Securing Model for Shipboard Helicopters by Michael J. L´ eveill´ e A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfilment of the requirements for the degree of Master of Applied Science in Aerospace Engineering Ottawa-Carleton Institute for Mechanical and Aerospace Engineering Department of Mechanical and Aerospace Engineering Carleton University Ottawa, Ontario, Canada September 2013 Copyright c 2013 - Michael J. L´ eveill´ e
Transcript of Development of a Spacial Dynamic Handling and Securing ...
Development of a Spacial Dynamic Handling and
Securing Model for Shipboard Helicopters
by
Michael J. Leveille
A thesis submitted to
the Faculty of Graduate and Postdoctoral Affairs
in partial fulfilment of
the requirements for the degree of
Master of Applied Science
in
Aerospace Engineering
Ottawa-Carleton Institute for Mechanical and Aerospace Engineering
Department of Mechanical and Aerospace Engineering
Carleton University
Ottawa, Ontario, Canada
September 2013
Copyright c©
2013 - Michael J. Leveille
The undersigned recommend to
the Faculty of Graduate and Postdoctural Affairs
acceptance of the thesis
Development of a Spacial Dynamic Handling and Securing
Model for Shipboard Helicopters
Submitted by Michael J. Leveille
in partial fulfilment of the requirements for the degree of
Master of Applied Science
Dr. R. G. LangloisThesis Supervisor
Dr. M. YarasDepartment Chair
Carleton University
2013
ii
Abstract
Maritime helicopters have the ability to greatly increase the range of influence and
utility of naval vessels. For smaller ships in rough seas, flight deck motion becomes
considerable and additional infrastructure is required for securing and traversing he-
licopters while on-deck. The current state-of-the-art in dynamic modelling of the
on-deck helicopter/ship dynamic interface includes a fully spacial securing simula-
tion and one capable of modelling planar traversing and manoeuvring operations.
A fully spacial securing and manoeuvring simulation named SSMASH (Spacial Se-
curing and Manoeuvring Analysis for Shipboard Helicopters) has been developed to
provide complete analysis capability of the on-deck helicopter/ship dynamic interface.
Specifically, a new capability to model helicopter response to manoeuvring events in
the presence of flight deck motion is realized. The SSMASH simulation is modeled
with twelve degrees-of-freedom and mass coupling between the helicopter body and
wheel carriages. A five degree-of-freedom tire model is used to extend the model
capability to include ground handling characteristics. The model has been validated
against the state-of-the-art securing simulation Dynaface R© and experimental data
from land-based manoeuvring trials. Excellent correlation with the Dynaface R© sim-
ulation is achieved, while good correlation with the experimental data is observed
as well. Improvements to the modelling of grip-limited tire behaviour would likely
improve agreement with experimental helicopter responses for some manoeuvres.
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Dedicated to my family,
for teaching me that anything is possible.
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Acknowledgments
I would first like to thank my thesis supervisor, Dr. Robert Langlois, for his invaluable
guidance, insight, and support for this project. His eagerness to share knowledge and
ideas has been exceptional, while his dedication to maintaining a high quality of work
has been a great source of inspiration.
I would also like to thank my fellow researchers of the Applied Dynamics Group.
Whether it meant reviewing and critiquing my modeling approach, troubleshooting
coding problems, sharing LATEX knowledge, or simply lending an ear, they were always
available and willing to help. I wish you all the best for your own research efforts and
careers. The relentless support, encouragement, and patience from my family and
the love of my life, Natasha Skanes, has helped me work through the long nights and
difficult times. I am grateful to have such people in my life.
Finally, I would like to thank the Ontario Graduate Scholarship Program and the
Department of Mechanical and Aerospace Engineering at Carleton University for the
financial support that has allowed me to pursue graduate studies and complete this
project.
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Table of Contents
Abstract iii
Acknowledgments v
Table of Contents vi
List of Tables viii
List of Figures ix
List of Symbols xii
1 Introduction 1
1.1 Shipboard Helicopter Modelling . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Securing and Manoeuvring Models . . . . . . . . . . . . . . . 4
1.1.2 Tire Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Aerodynamic Modelling . . . . . . . . . . . . . . . . . . . . . 10
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Dynamic Modelling and Simulation 13
2.1 Conventions and Mathematical Identities . . . . . . . . . . . . . . . . 13
2.2 System Description and Modelling Coordinates . . . . . . . . . . . . 19
vi
2.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Force-Generating Elements . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.1 Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.2 Steering Axis Friction . . . . . . . . . . . . . . . . . . . . . . 62
2.5.3 Securing Probe Model . . . . . . . . . . . . . . . . . . . . . . 64
2.5.4 Aerodynamic Body Forces . . . . . . . . . . . . . . . . . . . . 71
2.5.5 Oleo Pneumatic Strut Stiffness and Damping Model . . . . . . 73
2.5.6 Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . . 80
2.5.7 Compiling the Generalized Active Force Vector . . . . . . . . 81
2.6 Swing Arm Suspension Configurations . . . . . . . . . . . . . . . . . 82
2.7 Computational Simulation Environment . . . . . . . . . . . . . . . . 86
2.7.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 87
2.7.2 Programming Languages . . . . . . . . . . . . . . . . . . . . . 88
2.7.3 Simulation Structure . . . . . . . . . . . . . . . . . . . . . . . 89
3 Verification and Validation 91
3.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2.1 Spacial Dynamics Validation . . . . . . . . . . . . . . . . . . . 95
3.2.2 Planar Handling and Manoeuvring Validation . . . . . . . . . 109
4 Conclusions and Recommendations 125
4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
List of References 130
Appendix A 133
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List of Tables
3.1 Qualitative and quantitative results from the SSMASH verification trials. 92
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List of Figures
1.1 Illustration of the dynamic nature of on-deck helicopter securing . . . 2
1.2 Illustration of the ASIST securing and traversing system . . . . . . . 3
1.3 Schematic representation of Blackwell’s on-deck helicopter model . . . 5
1.4 Forces and moments considered in the Dynaface R© simulation . . . . 6
1.5 Schematic representation of the HeliMan dynamic model . . . . . . . 7
2.1 Illustration of (a) a bang-bang friction model and (b) a modified fric-
tion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Definition of the inertial, ship, and helicopter frames of reference. . . 20
2.3 Definition of the oleo and wheel carriage frames of reference. . . . . . 21
2.4 Definition of the tire frame of reference. . . . . . . . . . . . . . . . . . 50
2.5 Illustration of the tire deformations. . . . . . . . . . . . . . . . . . . . 52
2.6 Illustration of the modelling coordinates and vectors used to describe
the securing probe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.7 Compression regions used to model the oleo spring force. . . . . . . . 74
2.8 Illustration of the pneumatic oleo damping force. . . . . . . . . . . . 77
2.9 Illustration of a typical swing arm suspension station configuration. . 83
2.10 Input and output structure of the SSMASH simulation. . . . . . . . . 90
3.1 Helicopter centre of gravity z position versus time for validation case
1A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.2 Oleo strut extensions versus time for validation case 1A. . . . . . . . 98
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3.3 Helicopter pitch angle versus time for validation case 1A. . . . . . . . 99
3.4 Helicopter centre of gravity z position versus time for validation case
1B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 Oleo strut extensions versus time for validation case 1B. . . . . . . . 101
3.6 Helicopter roll angle versus time for validation case 1B. . . . . . . . . 102
3.7 Helicopter pitch angle versus time for validation case 1B. . . . . . . . 102
3.8 Helicopter roll angle versus time for validation case 1C. . . . . . . . . 103
3.9 Oleo strut extensions versus time for validation case 1C. . . . . . . . 104
3.10 Helicopter pitch angle versus time for validation case 1C. . . . . . . . 105
3.11 Helicopter roll angle versus time for validation case 1D. . . . . . . . . 106
3.12 Helicopter yaw angle versus time for validation case 1D. . . . . . . . . 106
3.13 Helicopter pitch angle versus time for validation case 1D. . . . . . . . 107
3.14 Oleo strut extensions versus time for validation case 1D. . . . . . . . 108
3.15 The Indal Technologies Inc. DLTV . . . . . . . . . . . . . . . . . . . 109
3.16 RSD claw position versus time for validation case 2A. . . . . . . . . . 111
3.17 Securing probe x-force versus time for validation case 2A. . . . . . . . 111
3.18 Securing probe y-force versus time for validation case 2A. . . . . . . . 111
3.19 Helicopter yaw angle versus time for validation case 2A. . . . . . . . . 113
3.20 Main landing gear tire lateral deflection versus time for validation case
2A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.21 Main landing gear tire vertical load versus time for validation case 2A. 114
3.22 RSD claw position versus time for validation case 2B. . . . . . . . . . 115
3.23 Securing probe x-force versus time for validation case 2B. . . . . . . . 116
3.24 Securing probe y-force versus time for validation case 2B. . . . . . . . 116
3.25 Helicopter yaw angle versus time for validation case 2B. . . . . . . . . 117
3.26 Tail wheel steer angle versus time for validation case 2B. . . . . . . . 117
x
3.27 Main landing gear tire lateral deflection versus time for validation case
2B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.28 Main landing gear tire vertical load versus time for validation case 2B. 119
3.29 RSD claw position versus time for validation case 2C. . . . . . . . . . 119
3.30 Securing probe x-force versus time for validation case 2C. . . . . . . . 120
3.31 Securing probe y-force versus time for validation case 2C. . . . . . . . 121
3.32 Main landing gear tire lateral deflection versus time for validation case
2C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.33 Helicopter yaw angle versus time for validation case 2C. . . . . . . . . 122
3.34 Tail wheel steer angle versus time for validation case 2C. . . . . . . . 123
3.35 Main landing gear tire vertical load versus time for validation case 2C. 123
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List of Symbols
Superscripts and Operators:
˙ First time derivative
¨ Second time derivative
~ Vector
[T b←a] Transformation matrix from the a to b frame of reference
ddt
( ) Time derivative of expression
˜ Skew symmetric matrix
| | Absolute value (scalar) or magnitude (vector)
ˆ Unit vector
Reference Frames:
N Inertial frame of reference
S Ship frame of reference
H Helicopter frame of reference
O Oleo/suspension mount frame of reference
W Wheel carriage frame of reference
T Tire frame of reference
Variables:
~Aeq Equivalent drag area vector of the helicopter body
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A Oleo piston area
a Attachment point of the securing probe to the helicopter fuselage
b Intersection point of a vector along the undeflected securing probe
axis with the securing claw plane
bt Width of the tire/ground contact interface
CeffP Effective in-plane probe damping constant
CPx Probe damping constant in the x direction
CPy Probe damping constant in the y direction
CPz Probe damping constant in the axial direction
Ctz Vertical tire deflection constant
Cx Longitudinal tire damping constant
Cy Lateral tire damping constant
Cz Vertical tire damping constant
c Point corresponding to the tip on an undeflected securing probe
DeffPS
Effective in-plane probe stiffness transition displacement
DPS,x Probe stiffness transition displacement in the x direction
DPS,y Probe stiffness transition displacement in the y direction
DPS,z Probe axial stiffness transition displacement
db Securing probe length factor
dt Unloaded tire diameter
ei ith Euler parameter
~F Pplane,dIn-plane probe force vector due to probe tip deflection
~F Pplane,vIn-plane probe force vector due to probe tip deflection rate
Fd Oleo damping force component
Fd,rel1, Fd,rel2 Primary and secondary oleo damping force relief stages, respectively
Ff Total oleo friction force in the axial direction
Foleo Total oleo force
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FPz,d Axial probe force due to probe extension
FPz,v Axial probe force due to probe extension rate
Fr rth generalized active force
F ∗r rth generalized inertial force
Fs Oleo spring force component
Fseal Oleo seal drag force
Fssy Steady state lateral tire force
Ftrr Tire rolling resistance force
Ftx Longitudinal tire force
Fty Lateral tire force
Ftz Vertical tire force
Fµ Nominal friction force; oleo bearing friction force
Fµ,axle Axle rolling friction force
Fµ,mod Modified friction force
IA Tire inclination angle relative to the ground plane[kI
k]
Inertia matrix for the kth body in its local frame of reference
KeffPS
Effective in-plane probe stiffness for small deflections
KeffPL
Effective in-plane probe stiffness for large deflections
KPS,x Small displacement probe stiffness in the x direction
KPS,y Small displacement probe stiffness in the y direction
KPL,x Large displacement probe stiffness in the x direction
KPL,y Large displacement probe stiffness in the y direction
KPS,z Small displacement axial probe stiffness
KPL,z Large displacement axial probe stiffness
Kty Lateral tire stiffness
Kttwist Rotational tire stiffness
Ktx Longitudinal tire stiffness
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k Oleo extension stiffness
[L] Matrix of Euler parameters used to determine Euler parameter
derivatives from angular velocity components
Lh Half-length of the tire/ground contact interface
Lu relaxation length for a tire rolling with zero yaw angle
Ly Relaxation length for a tire rolling with yaw
b1lb2 Instantaneous distance between the upper and lower oleo bearings
b1lb20 Distance between the upper and lower oleo bearings at full droop
mk Mass of the kth body
Mssy Steady state tire twisting moment
Mtx Tire contact patch x moment
Mty Tire contact patch y moment
Mtz Tire twisting moment
Mµ,steer Friction moment at a steering joint, about the steering axis
N Normal load reacted by the upper oleo bearing
Nt Tire cornering power parameter
P1 Attachment point of the oleo to the helicopter fuselage for a swing arm
suspension configuration
P2 Attachment point of the oleo to the swing arm for a swing arm
suspension configuration
P Instantaneous tire pressure; uncompressed oleo gas pressure
Po Unloaded tire inflation pressure (gauge)
Po,abs Unloaded tire inflation pressure (absolute)
Pr Rated tire inflation pressure
~p Vector of Euler parameters, 4× 1
qi ith generalized coordinate
~Rk
Active force on the kth body
xv
~Rk∗
Inertial force on the kth body
raxle Wheel axle radius
~Tk
Active torque on the kth body
~Tk∗
Inertial torque on the kth body
~U Vector of all generalized speeds
ui ith generalized speed
[V ] Partial linear velocity matrix
~vr rth partial linear velocity vector
V Uncompressed oleo volume
v Oleo compression velocity
voleo Oleo extension velocity for a swing arm suspension station
vwind,x Longitudinal wind velocity component, expressed in the inertial frame
vwind,y Lateral wind velocity component, expressed in the inertial frame
[W ] Partial angular velocity matrix
~wr rth partial angular velocity vector
wt Unloaded tire width
x Oleo compression distance
~Y H Auxiliary terms in the angular acceleration expression for the
helicopter body
~Y WiAuxiliary terms in the angular acceleration expression for the ith
wheel carriage
~ZH Auxiliary terms in the linear acceleration expression for the
helicopter body
~ZWiAuxiliary terms in the linear acceleration expression for the ith
wheel carriage
αt Tire yaw, or slip angle
β Friction smoothing decay constant
xvi
∆δy Change in lateral tire deflection due to rolling
∆P Change in tire pressure due to vertical loading
∆twist Change in tire twist angle due to rolling
RSD~δb
Securing probe deformation vector
δplane Magnitude of in-plane probe deflection
δssy Steady state lateral tire deflection
δx Tire longitudinal deflection; securing probe longitudinal deflection
δxo Initial longitudinal tire deflection
δx,max Maximum longitudinal tire deflection
δy Tire lateral deflection; securing probe lateral deflection
δyo Initial lateral tire deflection
δy,max Maximum lateral tire deflection
δz Tire vertical deflection; securing probe axial extension
θ, γ Angles used to define the tire frame relative to the wheel carriage
frame; specific heat ratio of oleo gas (γ)
µaxle Axle rolling friction coefficient
µbearing Oleo bearing friction coefficient
µrr Tire rolling resistance coefficient
µsteer Steering joint friction coefficient
µsteer,eff Effective steering joint friction coefficient
ξd Direction angle of in-plane probe deflection
ξv Direction angle of in-plane probe deflection rate
ρ Air density
φ Tire slip angle parameter
ψ tire twist angle
ψo Initial tire twist angle
ψss Steady state tire twist angle
xvii
Chapter 1
Introduction
Maritime helicopters form an integral part of modern naval fleets. They have the abil-
ity to greatly increase the area of influence of large and small vessels alike, increasing
the utility of helicopter-equipped ships. Vertical takeoff and landing capabilities mean
that helicopters can be operated from relatively small ships in comparison to vessels
designed to carry fixed-wing aircraft. However, small vessels experience greater flight
deck motion which can complicate helicopter launch, retrieval, and storage. A brief
history of naval helicopter operations from small ships is covered by Linn [1]. Here, an
overview of current technologies developed to facilitate operating maritime helicopters
from small ships is presented.
As can be seen in Figure 1.1, the dynamic effect of ship deck motion on an em-
barked helicopter can be severe. Such motion can exceed the aircraft’s own capability
to remain positioned on the flight deck, so various systems have been developed to
provide additional means of securing. Harsh weather environments and other opera-
tional considerations require that helicopters be manoeuvred to and from a storage
hangar, so modern securing systems are designed to facilitate this process.
Securing systems are often classified as active or passive in nature. An active sys-
tem applies a constant vertical load on the helicopter body to augment its traction
1
2
Figure 1.1: Illustration of the dynamic nature of on-deck helicopter securing [2].
with the ship deck. A passive system achieves helicopter securing by positive reten-
tion of a rigid probe that is built into the aircraft structure. The passive Recovery
Assist, Secure and Traverse (RAST) and Aircraft Ship Integrated Secure and Traverse
(ASIST) systems developed by Indal Technologies Inc. have been widely adopted by
navies worldwide [3]. Both of these systems use a Rapid Securing Device (RSD) to
capture and retain a helicopter securing probe, though the means of aligning the RSD
claw and securing probe during landing differ between the two systems.
The RAST system uses a tensioned haul-down cable to guide the helicopter to-
wards the claw trap, whereas the ASIST system actively tracks the securing probe
location during final helicopter descent and aligns itself for capture upon touchdown.
An illustration of the ASIST system is shown in Figure 1.2. The Twin Claw or
TC-ASIST system is a derivative of the ASIST system that is designed for use with
probe-less shipboard helicopters. The system operation is similar to ASIST, though
securing is achieved by capture of the main landing gear wheel spurs.
The capability to accurately model various phases of at-sea helicopter operation
3
Figure 1.2: Illustration of the ASIST securing and traversing system [4].
is essential to facilitate engineering analysis of the systems involved. Such analysis
can be used to determine safe operating limits, define operating procedures, and
determine peak loads on the securing equipment and aircraft structure. The topic of
developing mathematical models and simulations for this purpose is discussed in the
following section.
1.1 Shipboard Helicopter Modelling
A shipboard helicopter represents a complex, non-linear, multibody system that is
affected by various mechanical and aerodynamic phenomena. It follows that modelling
of shipboard helicopters has been an active area of research in the fields of multibody
dynamics, aerodynamics, and flight mechanics for some time. Study of the effects
of ship motion on embarked helicopters is termed ‘dynamic interface analysis’ by
Langlois and Tadros [5]. Various models have been developed which attempt to
characterize a particular aspect of the dynamic interface, be it hover and landing,
securing, or handling and manoeuvring. These models are often implemented in
4
computer simulations due to their associated mathematical complexity.
Dynamic interface analysis pertaining to helicopter operations can be considered in
two distinct regimes: in-flight and on-deck. Computational fluid dynamics based flight
models such as that developed by Alpman et al [6] model fully coupled interactions
of the helicopter with local flow fields around the ship structure while the helicopter
is in hover. Flight-simulator based modelling has also been done by Ferrier et al [7]
with emphasis on Ship-Helicopter Operational Limit (SHOL) determination.
Several models have been developed to characterize the on-deck dynamic interface.
These are of particular interest to this work and are discussed in the following section.
1.1.1 Securing and Manoeuvring Models
Some of the earliest work in on-deck modelling of an entire helicopter system was done
by Blackwell and Feik [8]. A mathematical model was developed to simulate helicopter
response to ship deck motion. The helicopter system was modelled as a single body
with forcing element representations of the suspension and tires. No attempt was
made to model helicopter securing devices, aerodynamic forces, or helicopter handling
behaviour. A schematic of Blackwell’s model is shown in Figure 1.3.
A more complex simulation of the on-deck helicopter dynamic interface named
Dynaface R© was developed by Langlois et al [9] at Indal Technologies Inc.. This
simulation models on-deck securing. The helicopter system is again modelled as
a single body, though effects from the helicopter suspension, tires, securing devices,
lashing cables, and aerodynamics are modelled. This simulation represents the current
state-of-the-art in on-deck securing analysis [5]. No attempt at modelling helicopter
handling characteristics is made by Dynaface R©, so traversing manoeuvres cannot be
simulated by this software. Figure 1.4 shows the forces considered by Dynaface R© in
determining the helicopter response.
A simulation software named HeliMan was developed by Linn [1] for analysis of
5
Figure 1.3: Schematic representation of Blackwell’s on-deck helicopter model [8].
shipboard helicopter handling and manoeuvring. The helicopter is modelled as a
planar system with degrees of freedom corresponding to planar translation, yaw rota-
tion, and steering rotation of a castor wheel assembly. A passive probe type securing
device is incorporated so that helicopter traversing can be simulated. The securing
probe claw position can be input from predefined files or real-time user joy-stick in-
puts. Since the HeliMan simulation uses planar equations of motion to describe the
helicopter system, it cannot simulate response to ship motion. A schematic represen-
tation of this model is shown in Figure 1.5.
Though the HeliMan simulation does not model helicopter response to ship mo-
tion, a one-tenth scaled experimental motion apparatus was developed for this pur-
pose by Feldman [11] as part of the same research effort. The goal if this work was
to demonstrate the effects of ship motion on helicopter response during traversing
6
Figure 1.4: Forces and moments considered in the Dynaface R© simulation [10].
manoeuvres and investigate the possibility for autonomous manoeuvring. The ex-
perimental apparatus uses a reconfigurable one-tenth scale Dead Load Test Vehicle
(DLTV) mounted atop a motion platform. Ship deck motion is provided by a single
degree of actuation to achieve combinations of surge, sway, heave, roll, and pitch mo-
tions. It was observed that ship deck motion and traversing manoeuvers had additive
effects on securing probe loads and model response. This highlights the need for si-
multaneous modelling of on-deck helicopter response to ship motion and manoeuvring
events. Such a capability is not possessed by any of the models reviewed.
1.1.2 Tire Modelling
Pneumatic tires represent the single most complex force generating element of an on-
deck helicopter model, an observation shared by Langlois et al [9]. This complexity
7
Figure 1.5: Schematic representation of the HeliMan dynamic model [1].
arises from two main areas: characterization of the rubber interaction with the ground
surface and elastic deformation mechanics of the tire carcass.
Behaviour of the rubber/ground interface depends on the chemical composition of
the tread rubber, ground surface roughness on a range of length scales, surface con-
taminants, and heterogeneity of the ground surface. Studies of these effects on rubber
friction from a contact mechanics point of view include work by Schallamach [12] and
Persson [13]. Though such work provides detailed investigation into the nature of
friction development at the contact interface, no effort is made by Schallamach to
model a whole tire. A two-dimensional model is developed by Persson, though only a
simple mass-spring representation of the tire carcass is employed. As such, this type
of work does not yet form a suitable basis upon which to develop a tire model for use
in on-deck helicopter simulation.
Tire deformations from applied load and moment are highly non-linear and vary
with tire construction, size, and operating conditions. Extensive work was done by
8
Smiley and Horne [14] in 1958 to characterize relevant aspects of tire elasticity for the
purpose of developing a model for bias-ply aircraft tires. This work provides equations
that model the transient and steady state tire response to longitudinal, lateral, and
vertical tire deformation as well as tire twist. Response of a tire rolling with yaw
and camber angles is also modelled. Rubber interaction with the ground surface is
characterized by peak longitudinal and lateral friction coefficients.
Work to update a subset of the Smiley and Horne model was done by Daugh-
erty [15] and Tanner et al [16] in 2003 and 2005, respectively. The goal of that work
was to better characterize radial ply aircraft tires which have become standard on
modern aircraft, with focus on higher speed behaviour pertinent to wheel shimmy
analysis. Though some differences in tire behaviour were identified, only minor dif-
ferences were found that relate to aspects of the Smiley and Horne model as it applies
to low velocity manoevring. Furthermore, analytic expressions to model the tire be-
haviour were not developed in a way suitable for inclusion in a computer simulation.
Further, no attempt was made to characterize the transient response of radial tires.
Tire models have also been developed that are largely empirical in nature, such as
the so called Magic Formula by Pacejka [17]. Models such as this rely on extensive
testing to generate sufficient data to model tire response within the tested operating
regime. Some of the Magic Formula modelling methods were used by Tremblay [18] to
develop a semi-empirical tire model for use with the HeliMan simulation by Linn [1].
Experimental testing was done to obtain the steady state response characteristics
of a McCreary Airtrac 6.00-6 8 ply tire. Similarity methods developed by Pacejka
were used to estimate tire behaviour at normal loads other than those tested. A
single contact point transient model was used to characterize aspects of transient tire
response.
The performance of Tremblay’s semi-empirical model was evaluated against the
Smiley and Horne model by comparing HeliMan simulations run with each tire model
9
to experimental data. The experimental data was gathered using a Dead Load Test
Vehicle (DLTV) representative of a Sikorsky S-70B Seahawk. Though some different
behaviour was observed, there was no marked improvement in the accuracy of the
simulation using the Tremblay tire model. This could have been due to limitations
from the empirical nature of the new model; tire testing with the exact tires used on
the DLTV may have improved the results. Furthermore, Tremblay cites the need to
perform additional tire testing at various normal loads to ensure accurate modelling
of the steady state tire response. This would be critical for applications involving
ship motion.
Other notable tire models pertinent to aircraft modelling include a low parameter
tire model by Wood et al [19], another empirical model used by Verzichelli [20], and
an analytic model developed by Kilner [21] to characterize tire vertical and drag loads
from surface irregularities. The tire model by Wood is sufficiently comprehensive to
model runway handling of aircraft, though it does not model transient tire response.
The same can be said for the empirical model used by Verzichelli. Though the work
from Kilner could be incorporated in a more comprehensive tire model, it does not
characterize sufficiently broad tire response to form the basis of a complete tire model
for on-deck modelling of maritime helicopters.
In the on-deck shipboard helicopter models discussed in Section 1.1.1, the following
tire models are used. Blackwell uses a three-dimensional spring-damper representa-
tion of the tire, while Dynaface R© uses a similar approach with optional tire stiffness
modelling from the Smiley and Horne equations. Neither simulation models rolling
response of the tire. The HeliMan simulation by Linn uses a three degree-of-freedom
subset of the Smiley and Horne tire model in its native configuration, though it has
also been adapted to use the semi-empirical model from Tremblay.
From this review, the Smiley and Horne tire model appears best suited for use
in an on-deck helicopter securing and manoeuvring simulation. Advantages of this
10
tire model include comprehensive characterization of transient and steady state tire
response. Furthermore, it can be used without extensive testing of particular tires as
would be required by Magic Formula based empirical tire models.
1.1.3 Aerodynamic Modelling
Shipboard helicopters experience aerodynamic forces due to ship motion, wind veloc-
ity, and to some extent helicopter motion. Modelling these effects can be a complex
aerodynamic problem as the local flow fields are affected by the ship superstructure.
These local flow effects are considered by Alpman [6], Ferrier [7], and Advani [22] in
modelling the dynamic interface during hover and landing. An unsteady ship airwake
model from McKillip [23] has also been developed for this purpose. The hover phase
has also been analyzed by Zan [24] and Lee [25] to determine the effect of ship airwake
on rotor thrust and fuselage loads, respectively. More recently, Wall et al have char-
acterized the unsteady airwake and studied its effects on the helicopter blade sailing
phenomenon in [26] and [27].
The author is not aware of any on-deck helicopter models that consider ship struc-
ture influence on local flow fields. It is possible that such effects would be secondary
in comparison to the securing forces, tire behaviour, and nominal aerodynamic loads.
The on-deck shipboard helicopter models discussed in Section 1.1.1 either make no
attempt at modelling aerodynamic effects on the helicopter or use a simple approach.
Blackwell does not characterize aerodynamic forces. The Dynaface R© simulation mod-
els aerodynamic forces in the helicopter x and y directions using equivalent drag areas
of the helicopter body and data lookup tables for aerodynamic rotor forces. No in-
teraction of the global wind with the ship superstructure is considered. Linn uses a
similar approach with HeliMan, though aerodynamic rotor forces are not considered.
11
1.2 Objectives
The objective of this work is to develop and validate a state-of-the-art spacial securing
and manoeuvring simulation for shipboard helicopters. The simulation core should
possess the following characteristics:
• Inclusion of surge, sway, heave, roll, pitch, and yaw ship deck motion;
• Fully three-dimensional modelling of on-deck helicopter dynamics;
• Coupled representation of helicopter sprung and un-sprung masses;
• Flexible configuration to model a wide range of modern maritime helicopters;
• RSD securing system with a single passive securing probe; and
• Transient and steady state tire modelling for static and rolling conditions.
This simulation will fill a void in current on-deck dynamic interface modelling capa-
bility. That is, it will model the helicopter response to ship motion during traversing
manoeuvres. It will effectively have the combined capability of the Dynaface R© and
HeliMan simulations, and be fully spacial.
The current work is aimed at developing the core simulation structure. Future
work can be done to expand the utility of the simulation through development of
additional system sub-models and analysis tools. The simulation can also be incor-
porated with real-time software for the purposes of ground operator training, flight
deck monitoring, and real-time determination of safe operating windows.
1.3 Thesis Overview
This chapter has presented an introduction to shipboard helicopter operations, re-
viewed the current state-of-the-art in on-deck helicopter modelling, and identified
objectives for the thesis work. A thorough description of the modelling techniques
12
used to satisfy the thesis objectives is presented in Chapter 2, while this work is ver-
ified and validated in Chapter 3. Finally, concluding remarks and recommendations
for future work are made in Chapter 4.
Chapter 2
Dynamic Modelling and Simulation
A simulation model for Spacial Securing and Manoeuvring Analysis of Shipboard
Helicopters, or SSMASH, has been created. This chapter outlines the work completed
to develop the mathematical models used by the SSMASH simulation, which is broken
down into several categories.
First, the conventions and mathematical identities used in the model development
are explained in Section 2.1, after which a more detailed description of the helicopter
system is given in Section 2.2. A comprehensive development of the system kinematics
and dynamics is presented in Sections 2.3 and 2.4 so that SSMASH can be used to
form the basis of future work in the areas of mission planning, aircraft design, securing
system design, and operator training. The force-generating elements of the system are
characterized in Section 2.5, and adaptation of the model for swing arm suspension
configurations is developed in Section 2.6. Finally, implementation of the dynamic
model in a computer simulation is described in Section 2.7.
2.1 Conventions and Mathematical Identities
This section outlines various naming conventions and mathematical identities that
are used throughout the development of the mathematical models.
13
14
Vector Naming Conventions
Common naming conventions are used for vector quantities. These conventions apply
to either kinematic vectors which represent a body’s displacement, velocity, or accel-
eration, or kinetic vectors representing such quantities as applied forces or moments
on a body. The two conventions are defined as follows.
Kinematic Vectors
Consider the vector quantity cd~a
b. The entries a through d for kinematic vectors
are defined as follows.
• a is the type of vector quantity where “r”, “v”, “a”, and “g” signify displace-
ment, velocity, acceleration, and acceleration due to gravity, respectively. Sim-
ilarly, “w” and “α” signify angular velocity and angular acceleration, respec-
tively.
• b is the body that the vector is describing.
• c is the body or reference frame relative to which the vector quantity is de-
scribed.
• d is the frame of reference in which the vector quantity is expressed.
For example, cd~rb is read as the position of body b relative to body c, expressed
in frame of reference d. Note that the b entry is not required in gravitational
acceleration expressions, and is left absent in such cases.
Kinetic Vectors
Consider again the vector quantity cd~a
b. The entries a through d for kinetic
vectors are defined as follows.
• a is the type of vector quantity where “F”, and “M” signify force and moment
vectors, respectively.
15
• b is the forcing element.
• c is the body upon which the force or moment is acting.
• d is the frame of reference in which the force or moment is expressed.
For example, cd~Fb
is read as the force of element b on body c, expressed in frame
of reference d.
For all kinematic and kinetic vector quantities that are 3x1 in size, the first, second,
and third row entries are referred to as the x, y, and z components respectively.
Time Derivatives of Vector Quantities
Consider the following time derivative of a vector quantity.
d
dt
(cd~a
b)
(2.1)
Throughout this work, the differentiation is always carried out in the frame in
which the parenthesized vector is expressed. As such, it is referred to here as a locally-
evaluated derivative. This is significant in that the frame in which the differentiation
is carried out affects the nature of the result. For example, the velocity of body b
relative to any stationary or moving body c, apparent to an observer in the inertial
frame, can only be obtained when the relative position vector is differentiated in the
inertial frame. As such, the relative position vector must be expressed in the inertial
frame prior to carrying out this local differentiation. The frame in which the vector
to be differentiated is expressed determines the frame in which an observer would
witness the differentiated vector quantity.
16
Cross Product Evaluation
Cross products must often be evaluated in the kinematic and dynamic development to
follow. The dynamic method used in this work requires that the cross product be ex-
pressed as a matrix multiplication operation. This is accomplished by Nikravesh [28]
by representing the first vector in the cross product expression as a skew symmetric
matrix as follows:
~a × ~b = [~a]~b (2.2)
The skew symmetric matrix for vector ~a having components a1, a2, and a3 in the x,
y, and z directions respectively is given by:
[~a] =
0 − a3 a2a3 0 − a1−a2 a1 0
Transformation Matrices
Transformation matrices are used to transform vector expressions from one frame of
reference to another. In this work, Euler parameters are used exclusively to define
the relative orientation of two such reference frames. Following the convention used
by Nikravesh [28], the Euler parameters are defined as follows for a rotation of angle
φ about an axis with unit vector ~u:
eo = cos
(φ
2
)(2.3)
~e = ~u sin
(φ
2
)(2.4)
17
with the components of ~e being the Euler parameters e1, e2, and e3. The correspond-
ing transformation matrix is defined by Nikravesh [28] as:
[T b←a] = 2
e02 + e12 − 1
2e1e2 − e0e3 e1e3 + e0e2
e1e2 + e0e3 e02 + e2
2 − 12
e2e3 + e0e1e1e3 + e0e2 e2e3 + e0e1 e0
2 + e32 − 1
2
(2.5)
where [T b←a] identifies the transformation matrix from the rotated frame of reference,
a, to the non-rotated frame of reference, b, and e0 through e3 are the four Euler param-
eters that define the relative orientation of frame a with respect to frame b. The use
of Euler parameters is advantageous in that the transformations are singularity-free
and a computational advantage is achieved since the transformation matrix elements
are polynomial expressions rather than trigonometric functions.
When successive transformations are required, the expression may be compressed
from its full form as follows:
[T c←b] [T b←a] = [T c←a] (2.6)
where [T c←a] is evaluated as the matrix multiplication product [T c←b] [T b←a].
Time Derivatives of Transformation Matrices
The time derivative of a transformation matrix,[T b←a
], must be determined when
evaluating the time derivatives of vector expressions containing transformation ma-
trices. This is accomplished using the following relation from Nikravesh [28]:
[T b←a
]=[bb ~w
a]
[T b←a] = [T b←a] [ba ~wa] (2.7)
18
Friction Smoothing
When modelling the friction force of contacting surfaces, a bang-bang friction model
as shown in Figure 2.1 can be used to describe a friction force that opposes the
direction of movement. The discontinuity at v = 0 of a bang-bang friction model
renders it unsuitable for inclusion in a numerical simulation, so a modified friction
model used successfully by Langlois [9] and Linn [1] is implemented. A smoothing
function is applied to the steady state friction force as:
Fµ,mod = sign(v)Fµ(1− e−β|v|
)(2.8)
where Fµ,mod is the modified friction force, v is the relative velocity of the contact-
ing surfaces, sign(v) is a function that determines the sign of v, Fµ is the nominal
magnitude of the friction force, and β is a decay constant for the smoothing function.
The decay constant is chosen such that most of the nominal friction force is achieved
for a suitably small value of v. Larger values for β result in a friction model that
closer resembles the bang-bang representation, while smaller values for β increase
the numerical efficiency of the solver. A suitable compromise for β is assessed on a
case-by-case basis. The effect of the smoothing function is illustrated in Figure 2.1.
v
Fμ
(a)
v
Fμ
(b)
Figure 2.1: Illustration of (a) a bang-bang friction model and (b) a modified frictionmodel.
19
2.2 System Description and Modelling Coordi-
nates
This work will detail the development of a three-dimensional, four-body dynamic
model representative of a shipboard helicopter with a single probe-type securing sys-
tem. A full tire model is incorporated so that handling characteristics are modelled
when the wheels are allowed to rotate. The native configuration of the helicopter
model is one with three oleo pneumatic strut suspension stations, though provisions
are made to model swing arm type suspension stations as well. Each suspension
station is modelled with a degree of rotational freedom to characterize wheel steer,
though this degree of freedom can be locked on each suspension station as required.
The four mass bodies are the helicopter and its three wheel carriages. As such, mass
coupling effects between the wheel carriages and the helicopter body are captured.
The resulting model includes 18 Degrees of Freedom (DOF) which are summarized
as follows:
• 6 DOF ship motion
– surge, sway, heave, roll, pitch, yaw
• 6 DOF helicopter motion, relative to the ship
– surge, sway, heave, roll, pitch, yaw
• 6 DOF helicopter suspension
– 2 DOF per suspension station
– suspension compression, wheel steer
The system is modelled using nine reference frames, three of which are shown in
Figure 2.2. These frames of reference include the inertial frame, N , the ship frame,
S, the helicopter frame, H, three oleo frames, Oi, and three wheel carriage frames,
Wi. The helicopter frame origin is located at the helicopter centre of gravity, with
20
the positive x axis oriented forward along the longitudinal axis of the helicopter and
the positive y axis oriented towards the starboard side of the helicopter. Each oleo
frame origin is located at the joint between a suspension station and the helicopter
body, with its z axis oriented along the axis of the suspension strut or swing arm. Its
y axis is normal to the plane of a road wheel on the corresponding wheel carriage.
Each wheel carriage frame is located at the centre of gravity of a suspended wheel
carriage, which is assumed to lie along the rolling axle line. The native definition of
the oleo and wheel carriage frames is illustrated in Figure 2.3.
N
S
H
1, 2, 3
4
u
q1, 2, 3
u6
uu5
q8, 9, 10
7, 8, 9u
u11
u10
u12
nx
nz
ny
sx sy
sz
h^x h
^y
h^z
Figure 2.2: Definition of the inertial, ship, and helicopter frames of reference.
In the present work, the wheel frame is always oriented parallel to its corresponding
oleo frame so that the transformation matrix between the two frames is always an
identity matrix. However, these transformation matrices are included in the kinematic
and dynamic development so that a constant orientation difference between the two
frames can be accounted for in future work.
The generalized coordinates, qi, and speeds, ui, are chosen such that they provide
a complete description of the ship-helicopter system described above. Furthermore,
they are consistently specified relative to the previous body in the kinematic chain.
21
Hh^x
h^z
Oox
oz
Wwx
wz
q , uk j
uj+1
h^y
oy
wy
Oleo Strut
Figure 2.3: Definition of the oleo and wheel carriage frames of reference.
This approach simplifies the implementation and modification of motion constraints
between bodies and, as will become evident, is well suited to using experimentally-
gathered ship motion data for dynamic analysis involving prescribed ship deck motion.
Inclusion of generalized coordinates and speeds to describe the ship state will render
the dynamic equations of motion overdetermined, though an approach to isolate the
full-rank equations of motion is presented in Section 2.4.
The position of the ship frame of reference relative to the inertial frame, expressed
in the inertial frame, is
NN~r
S =
q1q2q3 (2.9)
while the orientation of the ship frame relative to the inertial frame is defined by the
22
following Euler parameters:
N~pS =
e0e1e2e3
=
q4q5q6q7
(2.10)
The position of the helicopter centre of gravity relative to the ship frame, expressed
in the ship frame, is
SS~r
H =
q8q9q10
(2.11)
while the orientation of the helicopter frame relative to the ship frame is defined by
the following Euler parameters:
S~pH =
e0e1e2e3
=
q11q12q13q14
(2.12)
Though the location and orientation of each suspension station is fully general,
naming conventions are chosen to suit two standard helicopter configurations. The
auxiliary suspension station refers to either the nose location of a tricycle configura-
tion helicopter or the tail location of a tail wheel configuration. It follows that the
right and left suspension stations refer to the right and left sides of the main landing
gear for either helicopter configuration, respectively. The remaining generalized co-
ordinates specify the state of the three helicopter suspension stations. The distance
from the auxiliary oleo frame to its corresponding wheel carriage frame, along the
axis of the auxiliary oleo is {O1O1~rW1
}z
={q15}
(2.13)
while the x and y components of O1O1~rW1 are constant offset values. The orientation
of the auxiliary oleo frame relative to the helicopter frame is defined by the following
23
Euler parameters:
H~pO1 =
e0e1e2e3
=
q16q17q18q19
(2.14)
The distance from the right main oleo frame to the right main wheel carriage frame,
along the axis of the right main oleo is
{O2O2~rW2
}z
={q20}
(2.15)
while the x and y components of O2O2~rW2 are constant offset values. The orientation of
the right main oleo frame relative to the helicopter frame is defined by the following
Euler parameters:
H~pO2 =
e0e1e2e3
=
q21q22q23q24
(2.16)
The distance from the left main oleo frame to the left main wheel carriage frame,
along the axis of the left main oleo, is
{O3O3~rW3
}z
={q25}
(2.17)
while the x and y components of O3O3~rW3 are constant offset values. The orientation of
the left main oleo frame relative to the helicopter frame is defined by the following
Euler parameters:
H~pO3 =
e0e1e2e3
=
q26q27q28q29
(2.18)
The generalized speeds are chosen to be the time derivatives for all translational
generalized coordinates, and the relevant entries in the angular velocity vectors for
rotational degrees of freedom. The velocity of the ship frame relative to the inertial
24
frame, evaluated and expressed in the inertial frame, is
NN~v
S = NN~r
S =
q1q2q3 =
u1u2u3 (2.19)
while the angular velocity of the ship frame relative to the inertial frame, expressed
in the ship frame is:
NS ~w
S =
u4u5u6 (2.20)
The velocity of the helicopter frame relative to the ship frame, evaluated and expressed
in the ship frame, is
SS~v
H = SS~r
H =
q8q9q10
=
u7u8u9 (2.21)
while the angular velocity of the helicopter frame relative to the ship frame, expressed
in the helicopter frame is:
SH ~w
H =
u10u11u12
(2.22)
The velocity of the auxiliary wheel carriage frame relative to the auxiliary oleo frame,
evaluated and expressed in the auxiliary oleo frame, is
O1O1~vW1 = O1
O1~rW1 =
00q15
=
00u13
(2.23)
while the angular velocity of the auxiliary oleo frame relative to the helicopter frame
is:
HO1~wO1 =
00u14
(2.24)
Note that there is no relative angular velocity between any of the wheel carriage
frames and their corresponding oleo frame. The velocity of the right wheel carriage
25
frame relative to the right oleo frame, evaluated and expressed in the right oleo frame
is
O2O2~vW2 = O2
O2~rW2 =
00q20
=
00u15
(2.25)
while the angular velocity of the right oleo frame relative to the helicopter frame is:
HO2~wO2 =
00u16
(2.26)
The velocity of the left wheel carriage frame relative to the left oleo frame, evaluated
and expressed in the left oleo frame is
O3O3~vW3 = O3
O3~rW3 =
00q25
=
00u17
(2.27)
while the angular velocity of the left oleo frame relative to the helicopter frame is:
HO3~wO3 =
00u18
(2.28)
To relate the time derivatives of the Euler parameters used to describe the various
frame orientations to their appropriate body frame angular velocities, it is shown by
Nikravesh [28] that
~p =1
2[L]T ~w (2.29)
where ~p is the vector of Euler parameter time derivatives, ~w is the body angular
velocity expressed in its local frame, and
[L]T =
−e1 −e2 −e3e0 −e3 e2e3 e0 −e1−e2 e1 e0
(2.30)
26
where e0 through e3 are the four Euler parameters that make up ~p. Equation 2.29 is
applied to the ship, helicopter, and oleo frame Euler parameters to determine their
respective time derivatives as follows:
N~pS =
q4q5q6q7
=1
2
−q5 −q6 −q7q4 −q7 q6q7 q4 −q5−q6 q5 q4
NS ~w
S (2.31)
S~pH =
q11q12q13q14
=1
2
−q12 −q13 −q14q11 −q14 q13q14 q11 −q12−q13 q12 q11
SH ~w
H (2.32)
H~pO1 =
q16q17q18q19
=1
2
−q17 −q18 −q19q16 −q19 q18q19 q16 −q17−q18 q17 q16
HO1~wO1 (2.33)
H~pO2 =
q21q22q23q24
=1
2
−q22 −q23 −q24q21 −q24 q23q24 q21 −q22−q23 q22 q21
HO2~wO2 (2.34)
H~pO3 =
q26q27q28q29
=1
2
−q27 −q28 −q29q26 −q29 q28q29 q26 −q27−q28 q27 q26
HO3~wO3 (2.35)
At this point in the model development, all generalized coordinates and speeds
have been specified. The following section will derive the kinematic equations that
describe the system motion in light of the appropriate degrees of freedom and con-
straints.
27
2.3 Kinematics
The kinematic equations that describe the system motion are developed with two
main requirements. The first requirement is to determine the partial linear and
angular velocity matrices that are needed for Kane’s dynamic equations of motion,
presented in Section 2.4. The partial velocity matrices summarize the contribution of
each generalized speed to the total velocity expression for a given body. As such, they
are determined by inspection of the translational and angular velocity expressions for
the body of interest. The second requirement is to express the body accelerations in
a form that allows the generalized speed derivative vector to be isolated, facilitating
solution of the dynamic equations of motion. The equations of motion for each body
will be evaluated in its respective body frame, so the partial velocity matrices and
acceleration expressions must ultimately be described in the appropriate body frames
as well.
Since the ship motion is prescribed for the present work, the dynamic response
of the ship need not be determined by the dynamic model. As a result, the partial
velocity matrices and acceleration expressions are not developed for the ship. How-
ever, the ship acceleration and angular velocity will contribute to the partial velocity
matrices and acceleration expressions for the helicopter and wheel carriages.
The kinematic development starts with the helicopter body, and works along the
kinematic chain to the oleo frames and subsequently to the wheel carriage bodies.
The position of the helicopter frame relative to the ship fame is
NN
→rH
=NN
→rS
+SN
→rH
(2.36)
while the velocity of the helicopter frame relative to the inertial frame is determined
28
by differentiating Equation 2.36 as follows:
NN~v
H =d
dt
(NN~r
H)
=d
dt
(NN~r
S)
+d
dt
(SN~r
H)
=d
dt
(NN~r
S)
+d
dt
([TN←S] SS~r
H)
= NN~r
S +[TN←S
]SS~r
H + [TN←S] SS~rH
= NN~r
S + [TN←S][NS ~w
S]SS~r
H + [TN←S] SS~rH
(2.37)
Equation 2.37 is then expressed in the helicopter frame as
NH~v
H = [TH←N ]NN~vH
= [TH←N ]
(NN~r
S + [TN←S][NS ~w
S]SS~r
H + [TN←S] SS~rH
)
= [TH←N ]NN~rS + [TH←S]
[NS ~w
S]SS~r
H + [TH←S] SS~rH
= [TH←N ]NN~rS − [TH←S]
[SS~r
H]NS ~w
S + [TH←S] SS~rH
(2.38)
where it is ensured that NS ~w
S is expressed as a vector so that the partial velocity
matrix for the helicopter can be constructed later on. In general, no vectors containing
generalized speeds shall appear as entries in skew symmetric matrices in the velocity
expressions that will be used to construct partial velocity matrices. This refers to
all expressions that describe the velocity of a body relative to the inertial frame,
expressed in the body frame. Next, the acceleration of the helicopter relative to the
29
inertial frame is determined by differentiating Equation 2.37 to obtain
NN~a
H =d
dt
(NN~v
H)
=d
dt
(NN~r
S + [TN←S][NS ~w
S]SS~r
H + [TN←S] SS~rH
)
=NN~r
S +[TN←S
] [NS ~w
S]SS~r
H + [TN←S][NS ~w
S]SS~r
H
+ [TN←S][NS ~w
S]SS~r
H +[TN←S
]SS~r
H + [TN←S] SS~rH
=NN~r
S + [TN←S][NS ~w
S][
NS ~w
S]SS~r
H + [TN←S][NS ~w
S]SS~r
H
+ [TN←S][NS ~w
S]SS~r
H + [TN←S][NS ~w
S]SS~r
H + [TN←S] SS~rH
=NN~r
S + [TN←S][NS ~w
S][
NS ~w
S]SS~r
H + [TN←S][NS ~w
S]SS~r
H
+ 2 [TN←S][NS ~w
S]SS~r
H + [TN←S] SS~rH
(2.39)
which, when expressed in the helicopter frame becomes:
NH~a
H = [TH←N ]NN~aH
= [TH←N ]NN~rS + [TH←S]
[NS ~w
S][
NS ~w
S]SS~r
H + [TH←S][NS ~w
S]SS~r
H
+ 2 [TH←S][NS ~w
S]SS~r
H + [TH←S] SS~rH
(2.40)
It will be required to have all time derivatives of the generalized speeds expressed as
vectors rather than entries in skew symmetric matrices. Rearranging Equation 2.40
to obtain NS ~w
S as a vector gives:
NH~a
H = [TH←N ]NN~rS + [TH←S]
[NS ~w
S][
NS ~w
S]SS~r
H − [TH←S][SS~r
H]NS ~w
S
+ 2 [TH←S][NS ~w
S]SS~r
H + [TH←S] SS~rH
(2.41)
30
The velocity of the ith oleo frame origin is
NN~v
Oi =NN~v
H + HN~v
Oi (2.42)
with
HN~v
Oi =d
dt
(HN~r
Oi)
=d
dt
([TN←S] [T S←H ]HH~r
Oi)
=[TN←S
][T S←H ]HN~r
Oi + [TN←S][T S←H
]HN~r
Oi + [TN←S] [T S←H ]HN~rOi
= [TN←S][NS ~w
S]
[T S←H ]HN~rOi + [TN←S] [T S←H ]
˜[SH ~w
H]HN~r
Oi + 0
(2.43)
where i = 1...3 for the auxiliary, right, and left oleo frames respectively. The accel-
eration of the ith oleo frame relative to the helicopter frame is then determined by
31
differentiation of Equation 2.43 as follows:
HN~a
Oi =d
dt
(HN~v
Oi)
=d
dt
([TN←S]
[NS ~w
S]
[T S←H ]HH~rOi + [TN←S] [T S←H ]
˜[SH ~w
H]HH~r
Oi
)
=[TN←S
] [NS ~w
S]
[T S←H ]HH~rOi + [TN←S]
[NS ~w
S]
[T S←H ]HH~rOi
+ [TN←S][NS ~w
S] [T S←H
]HH~r
Oi + [TN←S][NS ~w
S]
[T S←H ]���*0
HH~r
Oi
+[TN←S
][T S←H ]
˜[SH ~w
H]HH~r
Oi + [TN←S][T S←H
] ˜[SH ~w
H]HH~r
Oi
+ [TN←S] [T S←H ]˜[SH ~w
H]HH~r
Oi + [TN←S] [T S←H ]˜[SH ~w
H]���*
0HH~r
Oi
= [TN←S][NS ~w
S][
NS ~w
S]
[T S←H ]HH~rOi + [TN←S]
[NS ~w
S]
[T S←H ]HH~rOi
+ [TN←S][NS ~w
S]
[T S←H ]˜[SH ~w
H]HH~r
Oi
+ [TN←S][NS ~w
S]
[T S←H ]˜[SH ~w
H]HH~r
Oi + [TN←S] [T S←H ]˜[SH ~w
H] ˜[
SH ~w
H]HH~r
Oi
+ [TN←S] [T S←H ]˜[SH ~w
H]HH~r
Oi
= [TN←S][NS ~w
S]
[T S←H ]HH~rOi + [TN←S] [T S←H ]
˜[SH ~w
H]HH~r
Oi
+ [TN←S][NS ~w
S][
NS ~w
S]
[T S←H ]HH~rOi + [TN←S] [T S←H ]
˜[SH ~w
H] ˜[
SH ~w
H]HH~r
Oi
+ 2 [TN←S][NS ~w
S]
[T S←H ]˜[SH ~w
H]HH~r
Oi
(2.44)
The acceleration expressions for the oleo frames are not transformed to their respective
local frames since they are used only for definition of the system state and do not
correspond to bodies of mass.
Continuing along the kinematic chain, the velocity of the ith wheel carriage relative
32
to the inertial frame is
NN~v
Wi = NN~v
H + HN~v
Oi + OiN ~v
Wi (2.45)
where the velocity of the ith wheel carriage relative to the ith oleo frame is given by
Equation 2.46.
OiN ~v
Wi =d
dt
(OiN ~r
Wi
)=
d
dt
([TN←S] [T S←H ] [TH←Oi ]
OiOi~rWi
)=[TN←S
][T S←H ] [TH←Oi ]
OiOi~rWi + [TN←S]
[T S←H
][TH←Oi ]
OiOi~rWi
+ [TN←S] [T S←H ][TH←Oi
]OiOi~rWi + [TN←S] [T S←H ] [TH←Oi ]
OiOi~rWi
= [TN←S][NS ~w
S]
[T S←Oi ]OiOi~rWi + [TN←S] [T S←H ]
˜[SH ~w
H]
[TH←Oi ]OiOi~rWi
+ [TN←H ] [TH←Oi ]˜[HOi~wOi
]OiOi~rWi + [TN←Oi ]
OiOi~rWi
(2.46)
Inserting Equations 2.37, 2.43, and 2.46 into Equation 2.45, expressing angular ve-
locities as vectors, and transforming to the ith wheel carriage frame gives:
NWi~vWi = [TH←N ]NN~r
S − [TH←S][SS~r
H]NS ~w
S + [TH←S] SS~rH
− [TN←S]˜[
[T S←H ]HN~rOi]NS ~w
S − [TN←S] [T S←H ][HN~r
Oi]SH ~w
H
− [TN←S]˜[
[T S←H ] [TH←Oi ]OiOi~rWi
]NS ~w
S − [TN←H ]˜[
[TH←Oi ]OiOi~rWi
]SH ~w
H
− [TN←S] [T S←H ] [TH←Oi ]˜[OiOi~rWi
]HOi~wOi + [TN←Oi ]
OiOi~rWi
(2.47)
33
The acceleration of the ith wheel carriage relative to the the inertial frame is
NN~a
Wi = NN~a
H + HN~a
Oi + OiN ~a
Wi (2.48)
with the acceleration of the ith wheel carriage frame relative to the ith oleo frame
being determined by differentiation of Equation 2.46 as follows:
OiN ~a
Wi =d
dt
(OiN ~v
Wi
)=[TN←S
] [NS ~w
S]
[T S←H ] [TH←Oi ]OiOi~rWi + [TN←S]
[NS ~w
S]
[T S←H ] [TH←Oi ]OiOi~rWi
+ [TN←S][NS ~w
S] [T S←H
][TH←Oi ]
OiOi~rWi + [TN←S]
[NS ~w
S]
[T S←H ][TH←Oi
]OiOi~rWi
+ [TN←S][NS ~w
S]
[T S←H ] [TH←Oi ]OiOi~rWi +
[TN←S
][T S←H ]
˜[SH ~w
H]
[TH←Oi ]OiOi~rWi
+ [TN←S][T S←H
] ˜[SH ~w
H]
[TH←Oi ]OiOi~rWi + [TN←S] [T S←H ]
˜[SH ~w
H]
[TH←Oi ]OiOi~rWi
+ [TN←S] [T S←H ]˜[SH ~w
H] [TH←Oi
]OiOi~rWi + [TN←S] [T S←H ]
˜[SH ~w
H]
[TH←Oi ]OiOi~rWi
+[TN←S
][T S←H ] [TH←Oi ]
˜[HOi~wOi
]OiOi~rWi + [TN←S]
[T S←H
][TH←Oi ]
˜[HOi~wOi
]OiOi~rWi
+ [TN←S] [T S←H ][TH←Oi
] ˜[HOi~wOi
]OiOi~rWi + [TN←S] [T S←H ] [TH←Oi ]
˜[HOi~wOi
]OiOi~rWi
+ [TN←S] [T S←H ] [TH←Oi ]˜[HOi~wOi
]OiOi~rWi +
[TN←S
][T S←H ] [TH←Oi ]
OiOi~rWi
+ [TN←S][T S←H
][TH←Oi ]
OiOi~rWi + [TN←S] [T S←H ]
[TH←Oi
]OiOi~rWi
+ [TN←S] [T S←H ] [TH←Oi ]OiOi~rWi
(2.49)
Evaluating the transformation matrix time derivatives, combining like terms, and
34
rearranging gives:
OiN ~a
Wi = [TN←S][NS ~w
S]
[T S←Oi ]OiOi~rWi + [TN←S] [T S←H ]
˜[SH ~w
H]
[TH←Oi ]OiOi~rWi
+ [TN←H ] [TH←Oi ]˜[HOi~wOi
]OiOi~rWi + [TN←S]
[NS ~w
S][
NS ~w
S]
[T S←Oi ]OiOi~rWi
+ [TN←S] [T S←H ]˜[SH ~w
H] ˜[
SH ~w
H]
[TH←Oi ]OiOi~rWi
+ [TN←H ] [TH←Oi ]˜[HOi~wOi
] ˜[HOi~wOi
]OiOi~rWi
+ 2 [TN←S][NS ~w
S]
[T S←H ]˜[SH ~w
H]
[TH←Oi ]OiOi~rWi
+ 2 [TN←S][NS ~w
S]
[T S←H ] [TH←Oi ]˜[HOi~wOi
]OiOi~rWi
+ 2 [TN←S] [T S←H ]˜[SH ~w
H]
[TH←Oi ]˜[HOi~wOi
]OiOi~rWi + 2 [TN←S]
[NS ~w
S]
[T S←Oi ]OiOi~rWi
+ 2 [TN←S] [T S←H ]˜[SH ~w
H]
[TH←Oi ]OiOi~rWi + 2 [TN←H ] [TH←Oi ]
˜[HOi~wOi
]OiOi~rWi
+ [TN←S] [T S←H ] [TH←Oi ]OiOi~rWi
(2.50)
Inserting Equations 2.39, 2.44, and 2.50 into Equation 2.48, expressing all generalized
speed derivatives as vectors rather than skew symmetric matrices, and transforming
35
to the ith wheel carriage frame gives:
NWi~aWi = [TWi←N ]NN~r
S + [TWi←S][NS ~w
S][
NS ~w
S]SS~r
H − [TWi←S][SS~r
H]NS ~w
S
+ 2 [TWi←S][NS ~w
S]SS~r
H + [TWi←S] SS~rH
− [TWi←S]˜[
[T S←H ]HH~rOi]NS ~w
S − [TWi←H ][HH~r
Oi]SH ~w
H
+ [TWi←S][NS ~w
S][
NS ~w
S]
[T S←H ]HH~rOi + [TWi←H ]
˜[SH ~w
H] ˜[
SH ~w
H]HH~r
Oi
+ 2 [TWi←S][NS ~w
S]
[T S←H ]˜[SH ~w
H]HH~r
Oi
− [TWi←S]˜[
[T S←H ] [TH←Oi ]OiOi~rWi
]NS ~w
S −[TW (i←H
] ˜[[TH←Oi ]
OiOi~rWi
]SH ~w
H
− [TWi←Oi ]˜[OiOi~rWi
]HOi~wOi + [TWi←S]
[NS ~w
S][
NS ~w
S]
[T S←H ] [TH←Oi ]OiOi~rWi
+ [TWi←H ]˜[SH ~w
H] ˜[
SH ~w
H]
[TH←Oi ]OiOi~rWi + [TWi←Oi ]
˜[HOi~wOi
] ˜[HOi~wOi
]OiOi~rWi
+ 2 [TWi←S][NS ~w
S]
[T S←H ]˜[SH ~w
H]
[TH←Oi ]OiOi~rWi
+ 2 [TWi←S][NS ~w
S]
[T S←H ] [TH←Oi ]˜[HOi~wOi
]OiOi~rWi
+ 2 [TWi←H ]˜[SH ~w
H]
[TH←Oi ]˜[HOi~wOi
]OiOi~rWi + 2 [TWi←S]
[NS ~w
S]
[T S←Oi ]OiOi~rWi
+ 2 [TWi←H ]˜[SH ~w
H]
[TH←Oi ]OiOi~rWi + 2 [TWi←Oi ]
˜[HOi~wOi
]OiOi~rWi
+ [TWi←Oi ]OiOi~rWi
(2.51)
With the linear velocity and acceleration expressions developed, the next task
is to determine the angular velocity and angular acceleration expressions for the
helicopter and wheel carriage bodies. First, the angular velocity of the ship relative
to the inertial frame is
NN ~w
S = [TN←S]NS ~wS (2.52)
36
and the angular acceleration of the ship relative to the inertial frame is determined
by differentiating Equation 2.52 as follows:
NN ~α
S =d
dt
(NN ~w
S)
=d
dt
([TN←S]NS ~w
S)
=[TN←S
]NS ~w
S + [TN←S]NS ~wS
= [TN←S]�������*
0[NS ~w
S]NS ~w
S + [TN←S]NS ~wS
= [TN←S]NS ~wS
(2.53)
The angular velocity of the helicopter relative to the inertial frame is
NN ~w
H = NN ~w
S + SN ~w
H (2.54)
with
SN ~w
H = [TN←S] [T S←H ] SH ~wH (2.55)
Inserting Equations 2.52 and 2.55 into Equation 2.54 and transforming into the heli-
copter frame gives:
NH ~w
H = [TH←S]NS ~wS + S
H ~wH (2.56)
The angular acceleration of the helicopter relative to the ship, expressed in the inertial
37
frame is determined by differentiating Equation 2.55 as follows:
SN ~α
H =d
dt
([TN←S] [T S←H ] SH ~w
H)
=[TN←S
][T S←H ] SH ~w
H + [TN←S][T S←H
]SH ~w
H + [TN←S] [T S←H ] SH ~wH
= [TN←S][NS ~w
S]
[T S←H ] SH ~wH + [TN←S] [T S←H ]
���
����*0
˜[SH ~w
H]SH ~w
H
+ [TN←S] [T S←H ] SH ~wH
= [TN←S][NS ~w
S]
[T S←H ] SH ~wH + [TN←S] [T S←H ] SH ~w
H
(2.57)
Combining Equations 2.53 and 2.57 and transforming to the helicopter frame gives
the expression for the helicopter angular acceleration relative to the inertial frame,
expressed in the helicopter frame as:
NH ~α
H =NH ~α
S + SH ~α
H
= [TH←S]NS ~wS + [TH←S]
[NS ~w
S]
[T S←H ] SH ~wH + S
H ~wH
(2.58)
The angular velocity of the ith wheel carriage is
NN ~w
Wi = NN ~w
H + HN ~w
Oi +����*
0Oi ~wWi (2.59)
with
HN ~w
Oi = [TN←S] [T S←H ] [TH←Oi ]HOi~wOi (2.60)
Inserting Equations 2.56 and 2.60 into Equation 2.59 and expressing in the ith wheel
38
frame gives:
NWi~wWi = [TWi←S]NS ~w
S + [TWi←H ] SH ~wH + [TWi←Oi ]
HOi~wOi (2.61)
The angular acceleration of the ith oleo frame relative to the helicopter, evaluated
and expressed in the inertial frame is:
HN ~α
Oi =d
dt
([TN←S] [T S←H ] [TH←Oi ]
HOi~wOi
)=[TN←S
][T S←H ] [TH←Oi ]
HOi~wOi + [TN←S]
[T S←H
][TH←Oi ]
HOi~wOi
+ [TN←S] [T S←H ][TH←Oi
]HOi~wOi + [TN←S] [T S←H ] [TH←Oi ]
HOi~wOi
= [TN←S][NS ~w
S]
[T S←Oi ]HOi~wOi + [TN←S] [T S←H ]
˜[SH ~w
H]
[TH←Oi ]HOi~wOi
+ [TN←H ] [TH←Oi ]˜[HOi~wOi
]HOi~wOi + [TN←S] [T S←H ] [TH←Oi ]
HOi~wOi
= [TN←S][NS ~w
S]
[T S←Oi ]HOi~wOi + [TN←S] [T S←H ]
˜[SH ~w
H]
[TH←Oi ]HOi~wOi
+ [TN←S] [T S←H ] [TH←Oi ]HOi~wOi
(2.62)
Combining Equations 2.58 and 2.62 transformed into the ith wheel carriage frame,
the angular acceleration of the ith wheel carriage relative to the inertial frame is:
NWi~αWi =N
Wi~αH + H
Wi~αOi +��
��*0
Oi~αWi
= [TWi←S]NS ~wS + [TWi←S]
[NS ~w
S]
[T S←H ] SH ~wH + [TWi←H ] SH ~w
H
+ [TWi←S][NS ~w
S]
[T S←Oi ]HOi~wOi + [TWi←H ]
˜[SH ~w
H]
[TH←Oi ]HOi~wOi
+ [TWi←Oi ]HOi~wOi
(2.63)
At this point in the kinematic development, all velocity expressions needed to
compile the partial velocity matrices have been determined. The linear partial velocity
39
matrix for the helicopter is constructed such that
NH~v
H =[NHV
H]~U (2.64)
where ~U is the vector of all generalized speeds and[NHV
H]
is the linear partial
velocity matrix for the helicopter. Similarly, the angular partial velocity matrix for
the helicopter,[NHW
H], is constructed such that:
NH ~w
H =[NHW
H]~U (2.65)
From Equations 2.64 and 2.65 and by observing the expressions for NH~v
H and NH ~w
H it
is found that:
[NHV
H]
=
[[TH←N ] − [TH←S]
[SS~r
H]
[TH←S] [0]3x3 [0]3x3 [0]3x3
]3x18
(2.66)
and
[NHW
H]
=
[[0]3x3 [TH←S] [0]3x3 [I]3x3 [0]3x3 [0]3x3
]3x18
(2.67)
Similarly, the linear and angular partial velocity matrices for the auxiliary wheel
carriage are found to be:
[NW1V W1
]=[
[TW1←N ] [TW1←S]
[−[SS~r
H]
+˜[
[T S←H ]HH~rO1
]+
˜[[T S←O1 ] O1
O1~rW1
]]
[TW1←S] [TW1←H ]
[−[HH~r
O1
]+
˜[[TH←O1 ] O1
O1~rW1
]][TW1←O1 ] [0 0 1]T
− [TW1←O1 ]˜[O1O1~rW1
][0 0 1]T [0]3x4
]3x18
(2.68)
40
and
[NW1WW1
]=[
[0]3x3 [TW1←S] [0]3x3 [TW1←H ] [0]3x1 [TW1←O1 ] [0 0 1]T [0]3x4
]3x18
(2.69)
Likewise, the linear and angular partial velocity matrices for the right wheel carriage
frame are found to be:
[NW2V W2
]=[
[TW2←N ] [TW2←S]
[−[SS~r
H]
+˜[
[T S←H ]HH~rO2
]+
˜[[T S←O2 ] O2
O2~rW2
]]
[TW2←S] [TW2←H ]
[−[HH~r
O2
]+
˜[[TH←O2 ] O2
O2~rW2
]][0]3x2 [TW2←O2 ] [0 0 1]T
− [TW2←O2 ]˜[O2O2~rW2
][0 0 1]T [0]3x2
]3x18
(2.70)
and
[NW2WW2
]=[
[0]3x3 [TW2←S] [0]3x3 [TW2←H ] [0]3x3 [TW2←O2 ] [0 0 1]T [0]3x2
]3x18
(2.71)
Finally, the linear and angular partial velocity matrices for the left wheel carriage
41
frame are found to be:
[NW3V W3
]=[
[TW3←N ] [TW3←S]
[−[SS~r
H]
+˜[
[T S←H ]HH~rO3
]+
˜[[T S←O3 ] O3
O3~rW3
]]
[TW3←S] [TW3←H ]
[−[HH~r
O3
]+
˜[[TH←O3 ] O3
O3~rW3
]][0]3x4 [TW3←O3 ] [0 0 1]T
− [TW3←O3 ]˜[O3O3~rW3
][0 0 1]T
]3x18
(2.72)
and
[NW3WW3
]=[
[0]3x3 [TW3←S] [0]3x3 [TW3←H ] [0]3x5 [TW2←O3 ] [0 0 1]T]3x18
(2.73)
With the first requirement of the kinematic development satisfied, the remaining
work in this section will describe the angular acceleration expressions for the helicopter
and wheel carriages in a suitable form. Specifically, it must be possible to isolate
the generalized speed derivatives. Noting that the locally evaluated second time
derivatives of the relative position vectors are
NN~r
S =d
dt
(NN~v
S)
=d
dt
u1u2u3 =
u1u2u3 (2.74)
SS~r
H =d
dt
(SS~v
H)
=d
dt
u7u8u9 =
u7u8u9 (2.75)
O1O1~rW1 =
d
dt
(O1O1~vW1
)=
d
dt
00u13
=
00u13
(2.76)
42
O2O2~rW2 =
d
dt
(O2O2~vW2
)=
d
dt
00u15
=
00u15
(2.77)
O3O3~rW3 =
d
dt
(O3O3~vW3
)=
d
dt
00u17
=
00u17
(2.78)
and that the locally evaluated time derivatives of the relative angular velocities are
NS ~w
S =d
dt
(NS ~w
S)
=d
dt
u4u5u6 =
u4u5u6 (2.79)
SH ~w
H =d
dt
(SH ~w
H)
=d
dt
u10u11u12
=
u10u11u12
(2.80)
HO1~wO1 =
d
dt
(HO1~wO1
)=
d
dt
00u14
=
00u14
(2.81)
HO2~wO2 =
d
dt
(HO2~wO2
)=
d
dt
00u16
=
00u16
(2.82)
HO3~wO3 =
d
dt
(HO3~wO3
)=
d
dt
00u18
=
00u18
(2.83)
the linear and angular acceleration expressions for the helicopter and wheel carriage
frames given in Equations 2.39, 2.58, 2.51, and 2.63 can be simplified to
NH~a
H =[NHV
H]~U + ~ZH (2.84)
NH ~α
H =[NHW
H]~U + ~Y H (2.85)
NWi~aWi =
[NWiV Wi
]~U + ~ZWi
(2.86)
NWi~αWi =
[NWiWWi
]~U + ~Y Wi
(2.87)
respectively where ~ZH , ~Y H ,~ZWi, and ~Y Wi
represent all terms of the acceleration
43
expressions which do not contain a generalized speed derivative. They are:
~ZH = [TH←S][NS ~w
S][
NS ~w
S]SS~r
H + 2 [TH←S][NS ~w
S]SS~r
H (2.88)
~Y H = [TH←S][NS ~w
S]
[T S←H ] SH ~wH (2.89)
~ZWi= [TWi←S]
[NS ~w
S][
NS ~w
S]SS~r
H + 2 [TWi←S][NS ~w
S]SS~r
H
+ [TWi←S][NS ~w
S][
NS ~w
S]
[T S←H ]HH~rOi + [TWi←H ]
˜[SH ~w
H] ˜[
SH ~w
H]HH~r
Oi
+ 2 [TWi←S][NS ~w
S]
[T S←H ]˜[SH ~w
H]HH~r
Oi
+ [TWi←S][NS ~w
S][
NS ~w
S]
[T S←Oi ]OiOi~rWi
+ [TWi←H ]˜[SH ~w
H] ˜[
SH ~w
H]
[TH←Oi ]OiOi~rWi + [TWi←Oi ]
˜[HOi~wOi
] ˜[HOi~wOi
]OiOi~rWi
+ 2 [TWi←S][NS ~w
S]
[T S←H ]˜[SH ~w
H]
[TH←Oi ]OiOi~rWi
+ 2 [TWi←S][NS ~w
S]
[T S←H ] [TH←Oi ]˜[HOi~wOi
]OiOi~rWi
+ 2 [TWi←H ]˜[SH ~w
H]
[TH←Oi ]˜[HOi~wOi
]OiOi~rWi
+ 2 [TWi←S][NS ~w
S]
[T S←H ] [TH←Oi ]OiOi~rWi
+ 2 [TWi←H ]˜[SH ~w
H]
[TH←Oi ]OiOi~rWi + 2 [TWi←Oi ]
˜[HOi~wOi
]OiOi~rWi (2.90)
~Y Wi= [TWi←S]
[NS ~w
S]
[T S←H ] SH ~wH
+ [TWi←S][NS ~w
S]
[T S←Oi ]HOi~wOi + [TWi←H ]
˜[SH ~w
H]
[TH←Oi ]HOi~wOi (2.91)
With the simplified acceleration expressions shown in Equations 2.84 through 2.87
determined, sufficient kinematic development to apply Kane’s dynamic equations of
44
motion has been completed.
2.4 Dynamics
This section shows the development of the governing equations of motion for the
shipboard helicopter system using Kane’s dynamic method [29]. Kane’s dynamical
equations state that
Fr + F ∗r = 0 for r = 1...n (2.92)
with n being the number of generalized speeds that describe the system. For this
application, n = 18 including the prescribed ship motion. The generalized active and
inertial forces for each body, Fr and F ∗r respectively are defined as:
Fr =m∑k=1
(N ~vr
k · ~Rk
+ N ~wrk · ~T
k)
(2.93)
F ∗r =m∑k=1
(N ~vr
k · ~Rk∗
+ N ~wrk · ~T
k∗)(2.94)
where m is the number of bodies in the system, ~Rk
and ~Tk
are the active force and
torque acting on the kth body, respectively, and ~Rk∗
and ~Tk∗
are the inertial force
and torque acting on the kth body, respectively. Finally, N ~vrk and N ~wr
k are the rth
partial linear and angular velocities for body k, respectively.
The above expressions can be expressed using vector notation and the partial
velocity matrices obtained in the kinematic development to show:
~F =m∑k=1
([Nk V
k]T ~Rk
+[Nk W
k]T ~T k
)(2.95)
~F∗
=m∑k=1
([Nk V
k]T ~Rk∗
+[Nk W
k]T ~T k∗)
(2.96)
45
For the system at hand, the generalized active and inertial force vectors are then:
~F =[NHV
H]T ~RH
+[NHW
H]T ~TH
+3∑i=1
([NWiV Wi
]T ~RWi+[NWiWWi
]T ~TWi)(2.97)
~F∗
=[NHV
H]T ~RH∗
+[NHW
H]T ~TH∗
+3∑i=1
([NWiV Wi
]T ~RWi∗+[NWiWWi
]T ~TWi∗)
(2.98)
where i signifies the ith wheel carriage. The active force and torque vectors affecting
the helicopter and wheel carriages are determined using the system submodels in
Section 2.5, while the inertial forces and torques are determined from Kane [29] as:
~Rk∗
=−mk · Nk ~ak (2.99)
~Tk∗
=−[kI
k]Nk ~α
k − Nk ~w
k ×([
kIk]Nk ~w
k)
=−[kI
k]Nk ~α
k −[Nk ~w
k] [
kIk]Nk ~w
k (2.100)
where k again describes the body upon which the inertial force or torque is applied.
Equations 2.99 and 2.100 can be expanded using the kinematic expressions for the
body accelerations from Equations 2.84 through 2.87. For the helicopter body:
~RH∗
=−mH · NH~aH
=−mH
[NHV
H]~U −mH · ~ZH (2.101)
46
and
~TH∗
=−[HI
H]NH ~α
H −˜[NH ~w
H] [
HIH]NH ~w
H
=−[HI
H] [
NHW
H]~U −
[HI
H]~Y H −
˜[NH ~w
H] [
HIH]NH ~w
H (2.102)
Similarly, the inertial forces and torques acting on the ith wheel carriage are:
~RWi∗
=−mWi· NWi
~aWi
=−mWi
[NWiV Wi
]~U −mWi
· ~ZWi(2.103)
and
~TWi∗
=−[WiIWi]NWi~αWi −
˜[NWi~wWi
] [WiIWi]NWi~wWi
=−[WiIWi] [
NWiWWi
]~U −
[WiIWi]~Y Wi−
˜[NWi~wWi
] [WiIWi]NWi~wWi (2.104)
Now the generalized inertial force vector for the shipboard helicopter system can be
expressed as:
~F∗
=−mH
[NHV
H]T [N
HVH]~U −mH
[NHV
H]T ~ZH
−[NHW
H]T [
HIH] [
NHW
H]~U −
[NHW
H]T [
HIH]~Y H
−[NHW
H]T ˜[
NH ~w
H] [
HIH]NH ~w
H
+3∑i=1
(−mWi
[NWiV Wi
]T [NWiV Wi
]~U −mWi
[NWiV Wi
]T ~ZWi
−[NWiWWi
]T [WiIWi] [
NWiWWi
]~U −
[NWiWWi
]T [WiIWi]~Y Wi
−[NWiWWi
]T ˜[NWi~wWi
] [WiIWi]NWi~wWi
)(2.105)
47
To facilitate solution for the vector of generalized speed derivatives, ~U , Equation 2.105
is written in the form:
~F∗
= [A] ~U + ~B (2.106)
with
[A] =−mH
[NHV
H]T [N
HVH]−[NHW
H]T [
HIH] [
NHW
H]
+3∑i=1
(−mWi
[NWiV Wi
]T [NWiV Wi
]−[NWiWWi
]T [WiIWi] [
NWiWWi
]) (2.107)
and
~B =−mH
[NHV
H]T ~ZH −
[NHW
H]T [
HIH]~Y H −
[NHW
H]T ˜[
NH ~w
H] [
HIH]NH ~w
H
+3∑i=1
(−mWi
[NWiV Wi
]T ~ZWi−[NWiWWi
]T [WiIWi]~Y Wi
−[NWiWWi
]T ˜[NWi~wWi
] [WiIWi]NWi~wWi
)(2.108)
Now, Kane’s equations of motion can be solved as follows:
~F + ~F∗
=0
~F + [A] ~U + ~B =0
[A] ~U =− ~B − ~F
[A] ~U = ~C
(2.109)
with
~C = − ~B − ~F (2.110)
For the system at hand, [A] is an 18x18 matrix and ~U and ~C are 18x1 column
vectors. With prescribed ship motion, the first six elements of ~U are known and
48
the system appears to be overdetermined. However, this is required for solution of
Equation 2.109 given that [A] is rank deficient by 6 since the active and inertial forces
affecting the ship have not been included in the dynamic formulation. Equation 2.109
is thus broken down as follows to obtain the full rank equations of motion for the
helicopter system. First, consider an expanded form of Equation 2.109:
[A] ~U = ~C[D]6x18
[E ]6x12 [M ]12x12
[~UShip
]6x1[
~UHeli
]12x1
=
[L]6x1
[N ]12x1
(2.111)
From Equation 2.111, it is apparent that the full rank equations of motion for the
helicopter system are:
[M ] ~UHeli = [N ]− [E ] ~UShip (2.112)
where the prescribed ship motion, ~UShip, has been moved to the right hand side.
Because relative position vectors have been used to describe the system state, the
prescribed ship motion appears here only as its linear and angular accelerations u1
through u6. Though the ship orientation and angular velocity is needed to evaluate
matrices M , N , and E in Equation 2.112, the ship position and linear velocity relative
to the inertial frame are not required here. This could be advantageous if such
information is not immediately available from experimental data.
All that remains is to determine the generalized active force vector, ~F , for the
shipboard helicopter system. This vector will describe the active forces and torques
affecting the helicopter and each of the three wheel carriages. These forces are deter-
mined using submodels that characterize the various forcing elements in the system
such as the tires, suspension oleos, securing probe, body aerodynamics, and gravity
49
forces.
2.5 Force-Generating Elements
This section describes the submodels that are used to characterize the various forcing
elements in the shipboard helicopter system. At its completion, the generalized active
force vector, ~F , is compiled.
2.5.1 Tire Model
The tire response is modelled with five degrees-of-freedom corresponding to longitu-
dinal, lateral, and vertical deflections as well as camber and twist angles. The Smiley
and Horne work [14] is used to characterize the tire behaviour in these degrees of
freedom. The modelling equations developed by Smiley and Horne are well suited to
this work since they adequately model tire response in low speed, high deflection op-
erating conditions where steady state may not be achieved. Such conditions describe
much of the operating regime for shipboard helicopter manoeuvring.
A new reference frame, called the Tire Frame, is necessary to match the reference
frame in which the tire characteristics are described by Smiley and Horne. Tire forces
are determined in this frame prior to transformation to the appropriate wheel carriage
frame. Also, the definition of the Tire Frame is used to determine the inclination angle
of the tire relative to the ship deck, a parameter used by the tire model. The tire
frame is illustrated in Figure 2.4.
The Tire Frame is defined by two successive rotations from its corresponding wheel
carriage frame. First, the W frame is rotated by angle θ about its wy axis (axle axis)
to form the T ′ frame. The magnitude of this rotation is such that the t′x axis is
parallel to the ship deck, or t′x · sz = 0. Finally, the T ′ frame is rotated by angle γ
about the tx axis to form the T frame. The magnitude of this rotation is such that
50
Ssx
sz
sy
W, T'
wx
wz
w , t'^y y
Ground Plane
^
t'^
x
θ
t'^
z
Ssx
sz
sy
T', T
t^
z
t'^
z
t^
y
t'^
y
t' , t^
x x
^ γ
Figure 2.4: Definition of the tire frame of reference.
the ty axis is also parallel to the ship deck, or ty · sz = 0. In the resulting T frame,
both axes tx and ty are parallel to the ship deck. This quality is expressed by:
tx × ty = sz (2.113)
From the above definition of the T frame, it is apparent that the negative of the angle
γ used in its definition is equal to the inclination angle of the tire, or:
IA = −γ (2.114)
51
The transformation matrix from the wheel carriage frame to the tire frame is then:
[T T←W ] = [T T←T ′ ] [T T ′←W ]
=
1 0 00 cos γ sin γ0 − sin γ cos γ
cos θ 0 sin θ0 1 0
− sin θ 0 cos θ
=
cos θ 0 sin θ− sin γ sin θ cos γ sin γ cos θ− cos γ sin θ − sin γ cos γ cos θ
(2.115)
The position and velocity of a tire centre with respect to the ship frame can be
determined directly from the system state vector, and is naturally expressed in the
ship frame. However, the relative position and velocity between a tire centre and
the ship deck must be expressed in the tire frame for use with the equations that
characterize the tire model. The known transformation matrices from the ship to the
wheel carriage frame and wheel carriage frame to tire frame are used to determine
the transformation matrix from the ship frame to the tire frame as:
[T T←S] = [T T←W ] [TW←S] (2.116)
The tire centre positions and velocities relative to the ship deck are tracked throughout
the simulation, and Equation 2.116 is used to express these vectors in the tire frame
as ST~r
T and ST~v
T . These vectors are used to track the tire deformation, velocity, and
yaw or slip angle. The yaw angle of the tire is determined from the ratio of its lateral
to forward velocity by:
αt = tan−1
{ST~v
T}y∣∣∣{ST~vT}x
∣∣∣ (2.117)
52
The following equations characterize the tire response to deformations shown in Fig-
ure 2.5.
Ground Plane
Ss^ x
s^ z
s^ y
Ss^ x
s^ z
s^ y
Tt^x
t^z
Tt^x
t^z
t^y
t^y
δx
δzδy
Tt^x t
^z
t^y
ψ
Figure 2.5: Illustration of the tire deformations.
Instantaneous Tire Pressure and Vertical Force
The following characterization of the tire response is based on the work of Smiley
and Horne [14]. Determination of the tire vertical force begins with determining the
instantaneous gas pressure within the tire, P , as:
P = Po + ∆P (2.118)
53
with Po being the initial gauge inflation pressure and the change in pressure, ∆P , is
∆P =n St Po,abs
(δzwt
)2
(2.119)
St =1.5wt
dt(2.120)
n =1 (isothermal) (2.121)
where n characterizes the nature of the compression process, Po,abs is the initial ab-
solute inflation pressure, δz is the vertical deflection, wt is the undeflected tire width,
and dt is the undeflected tire diameter. Next, the vertical tire force is determined as:
Ftz = [P + 0.08Pr] wt
√wt dt f1
(δzwt
)+ Cz
{ST~v
T}
3(2.122)
with
f1
(δzwt
)=
0.96 δzwt
+ 0.216 δzwt
2
Ctzfor
δzwt
≤ 10Ctz3
(2.123)
= 2.4
(δzwt
− Ctz)
forδzwt
>10Ctz
3(2.124)
Ctz = 0.02 for type I
= 0.03 for type III and VII (2.125)
where Pr is the rated pressure of the tire, Cz is the tire vertical damping constant,
and Ctz is an empirical constant that depends on the standard tire type designation.
54
Half Footprint Length
The half footprint length of the tire contact patch will be used to calculate its cor-
nering power and subsequently its steady state cornering force. It is calculated as:
Lh = 0.85 dt
√(δzdt
)−(δzdt
)2
(2.126)
Footprint Width
The footprint width of the tire contact patch will be used to determine the tire
response to rolling with a non-zero camber angle. It is determined as:
bt = 1.7wt
√δzwt
− 2.5
(δzwt
)4
+ 1.5
(δzwt
)6
(2.127)
Rolling Relaxation Lengths
The rolling relaxation lengths are used to determine the decay rate of lateral force or
twisting moment of a rolling tire when no additional lateral deflection or tire twist is
incurred. The un-yawed relaxation length applies to a tire that is rolling without a
yaw, or slip angle. It is given by:
Lu =
(2.8− 0.8P
Pr
)(1.0− 4.5δz
dt
)wt (2.128)
55
The yawed relaxation length applies to a tire that is rolling with non-zero yaw. It is
given by:
Ly =11δz2rt
(2.8− 0.8P
Pr
)wt for
δzdt≤ 0.053
=
(64δzdt− 500
(δzdt
)2
− 1.4045
)(2.8− 0.8P
Pr
)wt for 0.053 <
δz2dt≤ 0.068
=
(0.9075− 4δz
dt
)(2.8− 0.8P
Pr
)wt for
δzdt> 0.068
(2.129)
Lateral Tire Stiffness
The lateral tire stiffness is used to determine the tire response to lateral deflection.
It is determined as:
Kty = τ wt (P + 0.24Pr)
[1− 0.7δz
wt
](2.130)
with
τ = 3 for type I
τ = 2 for type III and VII
It was found in the development of Dynaface R© [9] that a minimum lateral stiffness
was required for large vertical tire deflections. This is done here by ensuring that
the lateral stiffness does not fall below 20% of the lateral stiffness of a tire in its
vertically-undeflected state. This is accomplished as:
Kty ,min = 0.2 τ wt (P + 0.24Pr) (2.131)
56
Rotational Tire Stiffness
The rotational stiffness of the tire is used to determine its response to twist deforma-
tion. It is determined as follows for tires of type I:
Ktψ =475
(δzdt
)2
(P + 0.8Pr) w3t for
δzdt≤ 0.02
=19
[δzdt− 0.01
](P + 0.8Pr) w3
t forδzdt> 0.02
(2.132)
and as follows for tires of types III and VII:
Ktψ =250
(δzdt
)2
(P + 0.8Pr) w3t for
δzdt≤ 0.03
=15
[δzdt− 0.015
](P + 0.8Pr) w3
t forδzdt> 0.03
(2.133)
Longitudinal Tire Stiffness
The longitudinal tire stiffness is used to determine the tire response to longitudinal
deflection when it is not allowed to roll. It is evaluated using the expression,
Ktx = 0.8dt (P + 4Pr)3
√δzdt
(2.134)
This expression represents an average for several tire types and constructions as the
longitudinal stiffness is highly dependent on the carcass construction of a particular
tire [14].
Cornering Power and Yaw Angle Parameters
The cornering power and yaw angle parameters are used to determine the steady
state lateral force that a tire will develop due to rolling with a non-zero yaw angle.
57
The cornering power is given by
Nt = (Ly + Lh)Kty (2.135)
while the slip angle parameter is:
φ =
(Nt
µFtz
)αt (2.136)
Steady State Lateral Force and Deflection
Here, the steady state lateral force generated by a tire rolling with non-zero yaw angle
and possibly having a non-zero camber angle is determined. First, the steady state
yawed rolling lateral force is
Fssy =− µFtz(φ− 4φ3
27
)for φ ≤ 1.5
=− µFtz(φ
|φ|
)for φ > 1.5
(2.137)
and the steady state camber thrust is
Fγy = KtγIA (2.138)
where the value of Ktγ depends on whether the tire is stationary or rolling. For a
stationary tire,
Ktγ ' 0.57Ftz (2.139)
and for a rolling tire,
Ktγ ' −(
0.57Ftz +ξLhLKty
rt
)' 1.0Ftz (2.140)
58
Now, the steady state lateral deflection of the tire can be determined as:
δssy =Fssy + Fγy
Kty
(2.141)
Steady State Twisting Moment and Twist Angle
Here, the steady state twisting moment generated by a tire rolling with non-zero yaw
angle and/or non-zero camber angle is determined. First, the steady state yawed
rolling twisting moment is
Mssz =0.8µFtzLhφ for φ ≤ 0.1
=µFtzLh(φ− φ2 − 0.01
)for 0.1 < φ ≤ 0.55
=µFtzLh (0.2925− 0.1φ) for φ > 0.55
(2.142)
while the contribution due to rolling with non-zero camber is:
Mγz = 0.01b2KtxIA (2.143)
Now, the steady state tire twist can be determined as:
ψss =Mssz +Mγz
Ktα
(2.144)
The steady state values for lateral force and twisting moment represent the values
that would be reached by the tire if it were allowed to roll for a large distance with
its vertical load, yaw, and camber angles held constant. When rapid changes in yaw
and camber angles are accompanied by low tire rolling velocities, the actual response
of the tire will not match such steady state values. In the following sections, the
instantaneous tire response is described.
59
Lateral Deflection and Lateral Force
Given an initial lateral tire deflection and the longitudinal and lateral distances trav-
elled by the tire centre since the time of the initial condition, the instantaneous lateral
deflection and resulting lateral force can be calculated by considering the relaxation
lengths developed in Equations 2.128 and 2.129. First, the change in lateral deflection
since the time of the initial condition is
∆δy =δssy −(δssy − δyo
)e−|{∆ST~r
T}x|
Lu for αt ' 0
=δssy −(δssy − δyo
)e−|{∆ST~r
T}x|
Ly for αt 6= 0
(2.145)
which is then used to determine the instantaneous lateral tire deflection as
δy = ∆δy −{
∆ST~r
T}y
(2.146)
where{
∆ST~r
T}y
is the lateral distance travelled by the tire centre since the time
of the initial condition. If the lateral deflection is too great to be sustained by the
available tire grip, or Ktyδy > µFtz , then the value is lowered to the maximum value
that can be sustained:
δy,max =µFtzKty
(2.147)
Finally, the resulting lateral force is determined including the damping contribution
as:
Fty = Ktyδy − Cyvty (2.148)
Above, Cy is the lateral damping constant and vty represents the y component of
the relative velocity between the tire centre and its contact patch centre. This is
approximated as{
∆ST~v
T}y, the change in the tire frame y velocity relative to the ship
frame since the time of the initial condition. This ensures that a constant damping
60
force is not generated for a tire rolling with constant lateral velocity, as would be the
case with any non-zero yaw angle.
Yawed Rolling Twist and Twisting Moment
Given an initial tire twist, the longitudinal distance travelled by the tire centre since
the initial condition time, and the twist gained by the tire frame since the initial con-
dition time, the instantaneous tire twist and resulting twisting moment can be calcu-
lated by considering the relaxation lengths developed in Equations 2.128 and 2.129.
First, the change in twist since the time of the initial condition is
∆ψ =ψss − (ψss − ψo) e−|{∆ST~r
T}x|
Lu for α ' 0
=ψss − (ψss − ψo) e−|{∆ST~r
T}x|
Ly for α 6= 0
(2.149)
which is used to determine the instantaneous tire twist as:
ψ = ∆ψ − ψgain (2.150)
where ψgain is the twist angle gained by the tire frame since the time of the initial
condition. Finally, the resulting twisting moment is determined as:
Mtz = Ktψψ (2.151)
Longitudinal Deflection and Longitudinal Force
For a tire that is constrained from rolling, the longitudinal deflection and force are
evaluated as follows. First, the change in longitudinal deflection is determined as
δx = δxo −{
∆ST~r
T}x
(2.152)
61
where{
∆ST~r
T}x
is the longitudinal distance travelled by the tire centre since the
time of the initial condition. The maximum value is again limited by the available
grip such that:
δx,max =µFtzKtx
(2.153)
Then the longitudinal force developed by a tire that is constrained from rolling is
Ftx = Ktxδx − Cx{ST~v
T}x
(2.154)
where{ST~v
T}x
is the longitudinal velocity of the tire centre. If the tire is allowed
to roll, then the longitudinal force is comprised of the tire rolling resistance, Ftrr ,
and the axle friction due to side load, Fµ,axle. The rolling resistance of the tire acts
opposite the direction of motion and is expressed as:
Ftrr = µrrFtz
(1− e−β|{
ST ~v
T}x|)
(2.155)
The axle friction also acts opposite the direction of motion and was found by Linn [1]
to be well represented by:
Fµ,axle = 2µaxle
(raxlewt
)Fty
(1− e−β|{
ST ~v
T}x|)
(2.156)
The resultant longitudinal force for a rolling tire is then:
Ftx = (Ftrr + Fµ,axle)(
1− e−β|{ST ~v
T}x|)
(2.157)
Contact Patch Moments
The x and y tire moments acting at the tire contact patch can now be evaluated. Smi-
ley and Horne [14] give expressions which account for shifts of the contact patch centre
62
of pressure due to lateral and longitudinal deflections of the tire. These expressions
are used to determine the x and y moments respectively as follows:
Mtx =− 0.8δyFtz (2.158)
Mty =− 0.25δxFtz for a constrained tire
' 0 for a rolling tire (2.159)
Equipolent Forces and Moments
Ultimately, the equipolent tire forces and moments about the appropriate wheel car-
riage frame origin are needed for inclusion in the dynamic equations of motion. The
expressions developed in the tire model characterize these forces and moments at the
contact patch with the ground surface, so the following transformations are needed:
WW~Ft
= [TW←T ]
FtxFty−Ftz
(2.160)
WW~M
t= [TW←T ]
Mtx
Mty
Mtz
+
˜[WW~r
T +
{0 0
dt2
}T][TW←T ]
FtxFty−Ftz
(2.161)
Note that a positive vertical force acts upwards in the tire model equations, which is
reflected by the negative multiplier of Ftz in Equations 2.160 and 2.161 to align with
the positive z-axis of the wheel carriage frame.
2.5.2 Steering Axis Friction
Similar to the axle rolling friction incorporated in the longitudinal force for a rolling
tire, a friction torque is applied to the oleo axis of a steerable suspension station.
Due to lateral and/or longitudinal loads acting at the tires of a suspension station, a
63
torque must be reacted between the oleo shaft and its housing. This creates friction
that counteracts axial compression and extension as well as rotation about the oleo
axis. The linear friction force is accounted for within the oleo model, so here an
expression for the steer friction torque is developed. It is desirable to model the
steering friction independently of the oleo model so that steering friction can be
captured for mechanisms that do not steer about the axis of a gas oleo. The magnitude
of the friction torque could be modelled by the following equation:
Mµ,steer = µsteer (R1 +R2) roleo (2.162)
where µsteer is the friction coefficient at the reaction interface within the oleo, R1 and
R2 are the upper and lower reaction forces respectively, and roleo is the radial distance
from the steer axis to the friction interface within the oleo. In general, R1 and R2
can be determined from the distance between the moment reaction locations and the
location and magnitude of the applied load at the tire contact point. However, such
relative positions are not easily determined and will vary with oleo compression. The
values for µsteer and roleo are also difficult to determine and would likely require tuning
to match experimental data. Therefore, the oleo steer friction is modelled using an
effective steer friction coefficient as follows:
Mµ,steer = µsteer,eff
√F 2tx + F 2
ty
(1− e−β|{
HO ~w
O}z|)
(2.163)
with the value of µsteer,eff having units of length. To obtain a vector form of the steer
friction moment, a convention is adopted in which a positive steer moment about the
oleo axis imparts a positive torque about the wheel carriage frame z axis.
WO~M
µ,steer=
00
Mµ,steer
(2.164)
64
2.5.3 Securing Probe Model
The effect of the securing probe and RSD system on the helicopter body is char-
acterized in this section. The kinematic development of the model determines the
deflection and deflection rate of the probe tip as a result of its capture in the RSD
claw, after which the resulting forces at the probe tip are determined.
Ss^ x
s^ z
s^ y
Deck Plane
Hh^x
h^z
h^y
Claw Plane
Claw
bRSD
b
a
cz
Probe
Figure 2.6: Illustration of the modelling coordinates and vectors used to describethe securing probe.
Probe Kinematics
The kinematic development of the securing probe model will make repeated use of
three points: a, b, and c that are shown in Figure 2.6. Point a represents the point of
attachment of the securing probe to the helicopter body and point c represents the
location of an undeflected probe tip. Point b lies at the intersection of the line from
a to c with the plane of the securing device claw. It follows that the probe deflection
65
can be characterized by the distance between point b and the centre of the securing
device claw. First, the position of point b must be determined. Representing point b
as a point along the line from a to c gives:
SH~r
b = dbaH~r
c +(
[TH←S] SS~rH + H
H~ra)
(2.165)
while expressing b as a point on the plane of the securing device gives:
(SH~r
b − [TH←S] SS~rRSD)·
[TH←S]
001
= 0 (2.166)
Combining Equations 2.165 and 2.166 and solving for the vector length factor, db,
gives:
db =
([TH←S] SS~r
RSD − [TH←S] SS~rH − H
H~ra)·
[TH←S]
001
aH~r
c ·
[TH←S]
001
(2.167)
Then, SH~rb is determined from Equation 2.165 using the result of Equation 2.167. The
deformation vector of the securing probe can then be determined as:
RSDH
~δb
= [TH←S] SS~rRSD − S
H~rb (2.168)
The in-plane deformation components of the probe are defined using this vector as:
δx ={
RSDH
~δb}x
(2.169)
δy ={
RSDH
~δb}y
(2.170)
66
The axial deformation of the probe is determined using the length factor, db, and the
undeformed length of the probe as follows:
δz = (db − 1) |aH~rc| (2.171)
Note that the convention is positive for probe extension.
The rate of probe deformation is also required so that the damping force can be
calculated. To this end, the relative velocity between the probe intersection point
with the securing device plane, SS~rb, and the securing claw centre, SS~r
RSD, is evaluated
in the ship frame and transformed to the helicopter frame as follows:
RSDH
~δb = [TH←S] RSDS
~δb
= [TH←S] SS~rRSD − [TH←S] SS~r
b(2.172)
with
SS~r
b =d
dt
(SS~r
b)
=d
dt
(db [T S←H ] aH~r
c + SS~r
H + [T S←H ]HH~ra)
=db [T S←H ] aH~rc + db
[T S←H
]aH~r
c + db [T S←H ]���>
0aH~r
c
+ SS~r
H +[T S←H
]HH~r
a + [T S←H ]���>
0HH~r
a
=db [T S←H ] aH~rc + db [T S←H ]
˜[SH ~w
H]aH~r
c + SS~r
H + [T S←H ]˜[SH ~w
H]HH~r
a
(2.173)
All terms in the final form of Equation 2.173 are known from the kinematic develop-
ment presented in Section 2.3, except for the rate of change of the length factor, db.
67
Recall Equation 2.167, and consider its form when expressed in the ship frame:
db =
(SS~r
RSD − SS~r
H − [T S←H ]HH~ra)·
001
[T S←H ] aH~r
c ·
001
=f
g(2.174)
Then the time derivative of the length factor, evaluated in the ship frame is:
db =f g − fgg2
(2.175)
with
f =
(SS~r
RSD − SS~r
H − [T S←H ]˜[SH ~w
H]HH~r
a
)·
001
(2.176)
g =
([T S←H ]
˜[SH ~w
H]aH~r
c
)·
001
(2.177)
Finally, the in-plane deflection rates of the probe are:
δx ={
RSDH
~δb}x
(2.178)
δy ={
RSDH
~δb}y
(2.179)
and the axial extension rate is:
δz = db |aH~rc| (2.180)
Probe Forces
The probe forces are calculated in two components, being the in-plane force and the
axial force. The in-plane force acts in the helicopter frame x-y plane, while the axial
68
force acts along the axis of the undeflected probe. Both force components are assumed
to act at the securing device claw centre.
In-plane Probe Force
Characterization of the in-plane probe force is largely derived from work by Lan-
glois [9] and Linn [1] and begins with identifying the in-plane deflection components
of the securing probe. The in-plane deflection magnitude and direction, respectively,
are determined using the following expressions:
δplane =√δx
2 + δy2 (2.181)
ξd = tan−1(δyδx
)(2.182)
The magnitude of the in-plane force is then determined for two regions. The first
region corresponds to small probe displacements and accounts for nearly free motion
of the securing probe within the securing device claw. The second region is much
more stiff and corresponds to elastic deformation of the securing probe, such that
∣∣∣~F Pplane,d
∣∣∣ =δplaneKeffPS
for δplane ≤ DeffPS
=DeffPSKeffPS
+(δplane −Deff
PS
)KeffPL
for δplane > DeffPS
(2.183)
where KeffPS
is the effective small displacement probe stiffness, KeffPL
is the effective
large displacement probe stiffness, and DeffPS
is the effective transition displacement.
69
These terms are determined by the following expressions:
KeffPS
=1√(
cos2 ξdKPS,x
)2+(
sin2 ξdKPS,y
)2 (2.184)
KeffPL
=1√(
cos2 ξdKPL,x
)2+(
sin2 ξdKPL,y
)2 (2.185)
DeffPS
=1√(
cos2 ξdDPS,x
)2+(
sin2 ξdDPS,y
)2 (2.186)
where KPS,x and KPS,y are the small displacement probe stiffnesses in the x and y
directions respectively, KPL,x and KPL,y are the large displacement probe stiffnesses
in the x and y directions respectively, and DPS,x and DPS,y are the transition displace-
ments in the x and y directions respectively.
The in-plane damping force is calculated similarly to the deflection force, though
with only a single stage. The in-plane deflection rate magnitude and direction, re-
spectively, are:
δplane =
√δx
2+ δy
2(2.187)
ξv = tan−1
(δy
δx
)(2.188)
so that the in-plane damping force magnitude is
∣∣∣~F Pplane,v
∣∣∣ = δplaneCeffP (2.189)
where the effective damping constant of the probe, CeffP , is determined from its x and
70
y components CPx and CPy , respectively, as:
CeffP =
1√(cos2 ξvCPx
)2+(
sin2 ξvCPy
)2 (2.190)
Axial Probe Force
The axial force component is determined from the extension and extension rate of
the securing probe given by Equations 2.171 and 2.180, respectively. The direction
of the axial probe force is taken to be along the undeflected probe axis. Two regions
are used to characterize the extension force component as follows:
FPz,d =δzKPS,z for 0 < δz ≤ DPS,z
=DPS,zKPS,z +(δz −DPS,z
)KPL,z for δz > DPS,z
(2.191)
where KPS,z is the small extension probe axial stiffness, KPL,z is the large extension
probe axial stiffness, and DPS,z is the axial displacement transition. The probe is
considered as a tension only component in its axial direction, so the displacement
force is zero for a state of compression, or:
FPz,d = 0 for δz ≤ 0 (2.192)
The axial damping force is calculated as:
FPz,v = δzCPz (2.193)
where CPz is the axial damping constant of the probe.
71
Equipolent Force and Moment
The development of the in-plane and axial securing probe forces has determined the
force components acting at the probe tip, whereas the combined equipolent forces and
moments acting about the helicopter centre of gravity are required. The combined
probe force vector acting on the helicopter, expressed in the helicopter frame is then:
HH~FP
=
∣∣∣~F Pplane,d
∣∣∣ cos ξd +∣∣∣~F Pplane,v
∣∣∣ cos ξv∣∣∣~F Pplane,d
∣∣∣ sin ξd +∣∣∣~F Pplane,v
∣∣∣ sin ξvFPz,d + FPz,v
(2.194)
and the equipolent moment is determined by:
HH~M
P=HH~r
RSD × HH~FP
=(SH~r
RSD − SH~r
H)× H
H~FP
=˜[
[TH←S](SS~r
RSD − SS~r
H)]
HH~FP
(2.195)
2.5.4 Aerodynamic Body Forces
This section outlines the method used to characterize the aerodynamic forces on
the helicopter body due to relative wind. An apparent wind speed can be observed
by the helicopter due to a combination of wind velocity in the inertial frame, ship
translational and angular velocity, and velocity of the helicopter relative to the ship.
The kinematic modelling presented here is sufficiently general to capture all of these
effects. It is not within the scope of the present work to attempt characterization
of the flow fields around the ship structure, nor any movement of the aerodynamic
centre of pressure (CP) of the helicopter body due to changes in flow attitude.
The aerodynamic forces on the helicopter body are modelled using equivalent
72
drag areas projected onto planes normal to the helicopter frame x, y, and z axes.
The equivalent drag area represents the equivalent reference area of a body with a
drag coefficient of unity. The aerodynamic forces are taken to act at the helicopter
body centre of pressure. As such, determination of the relative wind velocity at this
point is required. Consider the global wind velocity to be:
NN~v
wind =
vwind,x
vwind,y
0
(2.196)
It is the relative wind velocity at the helicopter centre of pressure that is required,
so first the velocity of the helicopter centre of pressure with respect to the inertial
frame is determined. For this work, the centre of pressure is assumed to be at a fixed
location with respect to the helicopter centre of gravity. This allows its velocity to
be expressed as:
NN~v
CP = NN~v
H + HN~v
CP (2.197)
with
HN~v
CP =d
dt
(HN~r
CP)
=d
dt
([TN←S] [T S←H ]HH~r
CP)
=[TN←S
][T S←H ]HH~r
CP + [TN←S][T S←H
]HH~r
CP + [TN←S] [T S←H ]����*
0HH~r
CP
= [TN←S][NS ~w
S]
[T S←H ]HH~rCP + [TN←S] [T S←H ]
˜[SH ~w
H]HH~r
CP
(2.198)
More useful, however, is the velocity of the centre of pressure with respect to the
inertial frame, expressed in the helicopter frame. To this end, Equation 2.198 is
73
transformed into the helicopter frame.
HH~v
CP = [TH←N ]HN~vCP
= [TH←S][NS ~w
S]
[T S←H ]HH~rCP +
˜[SH ~w
H]HH~r
CP
(2.199)
The velocity of the helicopter centre of pressure with respect to the inertial frame,
expressed in the helicopter frame is then:
NH~v
CP = NH~v
H + HH~v
CP (2.200)
Now the relative wind velocity can be determined as:
CPH ~vwind = [TH←N ]NN~v
wind − NH~v
CP (2.201)
From this, the equivalent drag area vector of the helicopter can be used to deter-
mine the aerodynamic force vector acting on the helicopter body at its aerodynamic
centre of pressure. In the following equation, element by element multiplication is
carried out to obtain a 3x1 vector of force components.
HH~F
wind=
1
2ρ ~Aeq
CPH ~vwind
∣∣∣CPH ~vwind∣∣∣ (2.202)
2.5.5 Oleo Pneumatic Strut Stiffness and Damping Model
The suspension struts are modelled using a gas oleo model from Langlois [9] that char-
acterizes the response to stroke travel and compression rate. The total strut response
is comprised of three primary force components being the spring force, damping force,
and friction force. The convention used throughout the oleo model is that compressive
displacements and velocities are considered positive.
74
Oleo Spring Force
The spring force behaviour is modelled in 5 distinct regions depicted in Figure 2.7.
The regions are described as follows for various oleo compression distances, x:
1. Extension past full droop
2. Small compression, x < x0
3. Ideal gas compression, x0 < x < x1
4. Cubic compression, x1 < x < x2
5. Cubic compression, x > x2
Compression
Force
x x x0 1 21
2 3 4 5
Figure 2.7: Compression regions used to model the oleo spring force.
Region 1
For the extension region past the full droop state of the oleo, its spring force is largely
a function of its installed structural stiffness. As such, a stiff linear relationship is
75
used to model the oleo spring force in this region.
Fs = kx (2.203)
where k is the extension spring constant and x is the oleo compression. Note that
the oleo compression value is negative in this region and a negative spring force is
developed.
Region 2
The first compression region models the force required to begin compressive displace-
ment of the oleo. It represents the force needed to overcome the internal gas pressure
of the oleo and initiate compression travel. This region should also provide a contin-
uously differentiable transition from regions 1 to 3. As such, a stiff cubic equation is
employed as follows:
Fs = a + bx+ cx2 + dx3 (2.204)
Region 3
The second compression region models the oleo behaviour as compression of an ideal
gas based on the initial oleo volume, piston area, initial pressure, and the specific
heat ratio of the working gas. The spring force in this region is characterized by
Fs =PV γA
(V − Ax)γ(2.205)
where P is the initial oleo pressure, V is the initial oleo volume, γ is the specific heat
ratio of the working gas, and A is the oleo piston area.
76
Region 4
The third compression region is modelled as a cubic polynomial. This region accom-
modates the possible inclusion of a pressure relief valve which is triggered at x = x1.
The spring force in this region is characterized by
Fs =PV γA
(V − Ax1)γ+ e1δ + f1δ
2 + g1δ3 (2.206)
with δ = x− x1.
Region 5
The final compression region is again modelled as a cubic polynomial representing
the oleo response to an additional relief valve triggered at x = x2. The spring force
in this region is characterized by
Fs =PV γA
(V − Ax1)γ+ e1δ1 + f1δ
21 + g1δ
31 + e2δ2 + f2δ
22 + g2δ
32 (2.207)
with δ1 = x2 − x1 and δ2 = x− x2.
To obtain a smooth transition between regions 1 and 3, the constants a, b, c, and d
used in region 2 are determined from the following analysis. First, noting that at zero
oleo compression Fs(0) = 0, it is apparent that a = 0. Also at zero oleo compression
dFsdx
(0) = k so it follows that b = k. From the ideal gas model used in region 3, the
spring force and its derivative at x = x0 are given by the following relations:
Fs(x0) =PV γA
(V − Ax0)γ(2.208)
dFsdx
(x0) =PV γA2γ
(V − Ax0)γ+1 (2.209)
77
These are used to solve for the two remaining constants in region 2 as:
c =PV γA (3 (V − Ax0)− Aγx0)− 2kx0 (V − Ax0)γ+1
(V − Ax0)γ+1 x20(2.210)
d =PV γA (Aγx0 − 2 (V − Ax0)) + kx0 (V − Ax0)γ+1
(V − Ax0)γ+1 x30(2.211)
Oleo Damping Force
The oleo damping force is modelled using four sets of quadratic polynomials. One set
is used to model rebound damping, while the remaining three are used for compres-
sion. At certain values of damping force in compression, the model can be triggered
to follow the second or third set of quadratic polynomials as shown in Figure 2.8.
This represents the actuation of pressure relief valves in the oleo piston orifices. In
the transition region between polynomials, the damping force is held constant at the
appropriate trigger value. If multistage compression damping is not featured in a
particular oleo, the trigger damping forces can be set to sufficiently large values such
that transitions do not occur in the operational range of the oleo.
Compression Velocity
Damping
Force
Fd,rel1
Fd,rel2
Figure 2.8: Illustration of the pneumatic oleo damping force.
78
In rebound, the damping force is represented by the following polynomial:
Fd = −(c0,1 + c0,2 |v|+ c0,3v
2)
(2.212)
where c0,1, c0,2, and c0,3 are the rebound damping constants and v is the oleo com-
pression velocity. The magnitude of the compression velocity is used in the linear
term to remain consistent with the convention of negative damping force for rebound
oleo velocity. In the first, second, and third compression ranges the oleo damping
force is represented by the following polynomials:
Fd =c1 + c2v + c3v2 for 0 ≤ Fd ≤ Fd,rel1
Fd =c4 + c5v + c6v2 for Fd,rel1 < Fd ≤ Fd,rel2
Fd =c7 + c8v + c9v2 for Fd > Fd,rel2
(2.213)
where c1 though c9 are compression damping constants and Fd,rel1 and Fd,rel2 are the
primary and secondary damping relief forces, respectively.
Oleo Friction Force
The axial friction force developed within the oleo is modelled as two components:
1. Seal drag force, Fseal
2. Bearing friction force, Fµ
The seal drag force is a constant parameter while the bearing friction force is a
function of the applied moment normal to the oleo axis. The applied moment is
determined using the oleo and wheel carriage geometry in conjunction with the forces
and moments acting on the wheel carriage. This applied moment is resisted by normal
loads in the oleo shaft bearings and a corresponding friction force is developed. The
79
applied moment from the wheel carriage loads, about the lower oleo shaft bearing,
expressed in the oleo frame is determined as follows:
b1O~M
W=b1O~r
W × WO~F + W
O~M
=(OO~r
W − OO~r
b1)× W
O~F + W
O~M
=˜[
OO~r
W − OO~r
b1]WO~F + W
O~M
(2.214)
The component of b1O~M
Walong the oleo axis is not reacted by normal loads in the oleo
shaft bearings, so only its first two components are used to determine the resultant
normal load at the upper shaft bearing, N , as:
N =
∣∣∣[b1O ~MW
xb1O~M
W
y 0]∣∣∣
b1lb2(2.215)
where the separation distance between the upper and lower oleo shaft bearings, b1lb2 ,
are determined by
b1lb2 =b1 lb20 + x (2.216)
and b1lb20 is the bearing separation at full droop. The magnitude of the oleo bearing
friction force, Fµ, is then determined as:
Fµ = µbearingN (2.217)
The total friction force in the oleo, Ff , is then determined and implemented with a
smoothing function for numerical efficiency such that
Ff =v
|v|(Fseal + Fµ)
(1− e−β|v|
)(2.218)
80
For suspension configurations in which there is no applied moment on the oleo, such
as a trailing arm configuration, the calculated bearing friction force will be zero and
the resulting oleo friction force will be comprised of only the constant seal drag force.
Total Oleo Response
The total response force of the oleo is then a summation of the spring, damping, and
friction force components
Foleo = Fs + Fd + Ff (2.219)
To obtain a vector expression of the oleo force, a convention is adopted in which a
compressive (positive) oleo force will exert a downward (positive) force on its appro-
priate wheel carriage
WO~FO
=
00
Fs + Fd + Ff
(2.220)
2.5.6 Gravitational Force
The final active forces that need to be determined are the gravitational forces on the
helicopter and wheel carriage masses. These forces are determined by the following
equations:
HH~g = [TH←N ]
00
g ·mH
(2.221)
WiWi~g = [TWi←N ]
00
g ·mWi
for i = 1...3 (2.222)
where mH and mWiare the masses of the helicopter and ith wheel frame, respectively,
and g is the acceleration due to gravity.
81
2.5.7 Compiling the Generalized Active Force Vector
Recall that the objective of the submodels is to facilitate determination of the gener-
alized active force vector for the dynamic system. Its definition from Equation 2.97
is:
~F =[NHV
H]T
H~RH
+[NHW
H]T
H~TH
+3∑i=1
([NWiV Wi
]TWi~RWi
+[NWiWWi
]TWi~TWi)
where H~RH
and Wi~RWi
are the active forces on the helicopter and ith wheel carriage,
respectively, while H~TH
and Wi~TWi
are the active torques. All active force and torque
vectors are 3x1 in size and are expressed in the local body frame of reference. These
vectors are obtained using appropriate summation of the submodel outputs as follows:
H~RH
=HH~g + H
H~FO1
+ HH~FO2
+ HH~FO3
+ HH~F
wind+ H
H~FP
=HH~g − [TH←O1 ]W1
O1~FO1 − [TH←O2 ]W2
O2~FO2 − [TH←O3 ]W3
O3~FO3
+ HH~F
wind
+ HH~FP
(2.223)
H~TH
=HH~M
O1+ H
H~M
O2+ H
H~M
O3+ H
H~M
wind+ H
H~M
P+ H
H~M
mu,steer
=−[HH~r
O1
][TH←O1 ]W1
O1~FO1 −
[HH~r
O2
][TH←O2 ]W2
O2~FO2
−[HH~r
O3
][TH←O3 ]W3
O3~FO3
+˜[HH~r
CP]HH~F
wind+ H
H~M
P
− [TH←O1 ]W1O1
~Mµ,steer
− [TH←O2 ]W2O2
~Mµ,steer
− [TH←O3 ]W3O3
~Mµ,steer
(2.224)
Wi~RWi
= [TWi←Oi ]WiOi~FOi
+ WiWi
~Ft+ Wi
Wi~g for i = 1...3 (2.225)
82
Wi~TWi
=−˜[OiOi~rWi
]WiOi~FOi
+ WiWi
~Mt+ [TWi←Oi ]
W3O3
~Mµ,steer
for i = 1...3
(2.226)
Note that, in general, there may be more than one tire acting on each wheel carriage.
In such cases, WiWi
~Ft
and WiWi
~Mt
represent a summation of the force and moment
contributions, respectively, from each tire on the wheel carriage.
2.6 Swing Arm Suspension Configurations
Thus far, a derivation for a helicopter model with three prismatic oleo suspension
stations has been presented. The development of this model has been sufficiently
general that adaptation to accommodate a trailing or leading arm type suspension
at any of the suspension stations is accomplished with only minor changes to the
equations of motion. The configuration of a swing arm suspension station is illustrated
in Figure 2.9.
Recall that for a prismatic oleo suspension station, the angular velocity of the
oleo frame with respect to the helicopter frame has only one non-zero entry which
represents steer rotation about the oleo axis. This definition is given by
HOi~wOi =
00
uj+1
(2.227)
for the ith oleo frame with uj+1 being the generalized speed which represents the wheel
steer velocity at the ith suspension station. Also recall that the locally-evaluated linear
83
Hh^x
h^z
Oox oz
W
wx
wzq
k
uj+1
h^y
oy
wy
Oleo Strut uj
p1
p2
Figure 2.9: Illustration of a typical swing arm suspension station configuration.
velocity of the ith wheel frame relative to the ith oleo frame is
HOi~vOi =
00uj
(2.228)
with uj being the generalized speed which represents the extension velocity along the
ith oleo axis.
For a swing arm type suspension, the locally evaluated velocity of the ith wheel
frame relative to the ith oleo frame contains only zero entries:
HOi~vOi =
000
(2.229)
The jth generalized speed is now used as an additional entry in the relative angular
84
velocity expression so that rotation about the oleo frame y axis is possible.
HOi~wOi =
0ujuj+1
(2.230)
Accordingly, changes must be made to the partial linear and angular velocity matrices
for the ith wheel frame. The jth column of the partial linear velocity matrix must be
changed from
[TWi←Oi ] [0 0 1]T (2.231)
for a prismatic oleo configuration, to
− [TWi←Oi ]˜[OiOi~rWi
][0 1 0]T (2.232)
for a swing arm configuration. Similarly, the jth column of the partial angular velocity
matrix must be changed from
[0 0 0]T (2.233)
for a prismatic oleo configuration, to
[TWi←Oi ] [0 1 0]T (2.234)
for a swing arm configuration.
The final change required to the equations of motion involves setting the time
derivative of the generalized coordinate used to describe the axial distance from the
ith oleo frame to the ith wheel carriage to zero. As an example, for the auxiliary
suspension station, this requires that q15 = 0 rather than q15 = uj for a prismatic oleo
configuration.
Additional changes required at the sub-model level are outlined below. These
changes are required to accurately model the oleo response and active forces and
85
moments on the wheel carriages and helicopter. In the following analysis, the vector
OiOi~rWi remains a description of the relative displacement between the ith wheel carriage
and the attachment point of the ith suspension station to the helicopter body. In the
case of a swing arm configuration, this attachment point refers to the swing arm to
helicopter revolute joint centre. Two additional points, P1 and P2, are defined to
describe the oleo mounting points to the helicopter and the swing arm, respectively.
When P1 is described in the helicopter frame and P2 in the oleo frame, both vectors
are constant. The following kinematic analysis is used to determine the compression
distance and velocity of the oleo or spring element in a swing arm suspension station.
First, consider the relative position vector from P1 to P2 which is given by the
following expression:
P1H ~r
P2 = HH~r
Oi + [TH←Oi ]OiOi~rP2 − H
H~rP1 (2.235)
The difference between the full droop magnitude of P1H ~r
P2 and its magnitude for a
particular system state gives the oleo compression. The following time derivative is
evaluated to facilitate determination of the axial oleo velocity.
P1H ~v
P2 =d
dt
(P1H ~r
P2
)=
d
dt
(HH~r
Oi + [TH←Oi ]OiOi~rP2 − H
H~rP1
)=��
�*0HH~r
Oi + [TH←Oi ]˜[HOi~wOi
]OiOi~rP2 + [TH←Oi ]��
��*0OiOi~rP2 −��
�*0HH~r
P1
= [TH←Oi ]˜[HOi~wOi
]OiOi~rP2
(2.236)
Then, the extension velocity of the oleo is the projection of P1H ~v
P2 onto a unit vector
in the direction of P1H ~r
P2 :
voleo = P1H ~v
P2 ·P1H ~r
P2∣∣∣P1H ~r
P2
∣∣∣ (2.237)
86
The above analysis determines the inputs required for the oleo model presented
earlier. For a swing arm suspension geometry, the active force and torque contribu-
tions by the oleo on the wheel carriage are
WiWi
~FOi
= [TWi←H ]Foleo
P1H ~r
P2∣∣∣P1H ~r
P2
∣∣∣ (2.238)
WiWi
~MOi
=˜[WiWi~rP2
]WiWi
~FOi
=˜[
OiOi~rP2 − Oi
Oi~rWi
]WiWi
~FOi
(2.239)
The corresponding active force and torque contributions by the oleo on the helicopter
body are now
HH~FOi
=− [TH←Wi]WiWi
~FOi
(2.240)
HH~M
Oi=−
[HH~r
P1
][TH←Wi
]WiWi
~FOi
(2.241)
for a swing arm suspension station. Equations 2.238 through 2.241 are used with
Equations 2.223 through 2.226 for any swing arm suspension stations.
2.7 Computational Simulation Environment
This section describes the implementation of the dynamic model in a computer simu-
lation. The numerical methods used to solve the equations of motion are presented in
Section 2.7.1, followed in Section 2.7.2 by a brief description of the coding languages
used. Finally, the simulation architecture of SSMASH is presented in Section 2.7.3.
87
2.7.1 Numerical Methods
With the generalized active force vector, ~F , determined it is now possible to solve the
dynamic equations of motion shown in Equation 2.112. The kinematic relationships
between the generalized coordinates and speeds developed in Section 2.2 are also used
to complete the system of first order differential equations. For computer simulation,
it is desirable to describe the entire system of equations in the form:
[K ] ~y = ~x (2.242)
so that ~y can be determined by solving a set of linear equations and a numerical
integrator can be used to propagate the solution state vector, ~y, in time from a set of
initial conditions. For the system at hand, the three components of Equation 2.242
are:
[K ] =
[M ]12x12 [0]12x22
[0]22x12 [I ]22x22
(2.243)
~y =
[~UHeli
]12x1
[~q8:29
]22x1
(2.244)
~x =
[[N]−[E]~UShip
]12x1
[Q]22x1
(2.245)
88
with
~q8:29 = [Q]22x1 =
u7u8u9
12
[SLH
]T SH ~w
H
u13
12
[HLO1
]T HO1~wO1
u15
12
[HLO2
]T HO2~wO2
u17
12
[HLO3
]T HO3~wO3
22x1
(2.246)
Equation 2.242 is solved for the system state derivative vector, ~y, using an algorithm
from Kahaner [30] employing upper and lower factorization and partial pivoting. The
solution is then propagated in time using another algorithm from Kahaner [30] which
employs a variable order, variable time step, Adam’s method integration technique.
2.7.2 Programming Languages
Initial coding of the the SSMASH simulation was done in the MATLAB R© computing
environment to facilitate initial verification of the equations of motion and sub-model
functionality. The extensive data analysis and plotting capability of the intrinsic
MATLAB R© functions enabled assess-modify-check loops to be completed rapidly
during this phase. Once a working simulation was achieved, the model was re-coded
in Fortran to increase the speed of the computations.
Fortran has proven well suited to this purpose due to its array storage and ma-
nipulation methods. Run times for the final version of the SSMASH simulation
89
coded in Fortran and compiled with Compaq Visual Fortran are typically an order of
magnitude less than an equivalent simulation run with the MATLAB R© code. Such
a reduction in simulation run time is a significant advantage in that it crosses the
threshold of real-time computation on the machines used for development. Since a
variable time step integrator is used, the computation speed of the simulation varies
with the type of motion that is experienced by the helicopter system. Typical run-
times for the simulation coded in Fortran range from one to four seconds per ten
seconds of simulated time, whereas equivalent simulations executed in MATLAB R©
require thirty seconds or more to complete. These run-times are typical for a machine
using an Intel R© CoreTM
i7 CPU.
2.7.3 Simulation Structure
SSMASH employs a simple structure of input and output data files shown in Fig-
ure 2.10. The input files contain information required to set up the solution run, solver
preferences, all parameters required to characterize the helicopter system and its ini-
tial conditions, and the prescribed motion of the ship and securing device claw. The
output files contain the time history of the system state vector and auxiliary terms
such as securing probe loads, tire deflections, and tire loads. A detailed description
of the contents of the input and output files is given in Appendix A.
90
heli.inp
helicopter input
parameters
initials.inp solution.inp
rsdmo.inpshipmo.inp
prescribed.inp
run.out
state.out
aux.out
solver controls
and solution
time span
securing device
claw position
prescribed ship
motion data
controls for
prescribed inputs
initial conditions
for the helicopter
system
run time and
output information
time history of the
generalized
coordinates and
speeds
time histories of
probe loads, tire
deformation, and
tire loads
SSMASH
main simulation
Figure 2.10: Input and output structure of the SSMASH simulation.
Chapter 3
Verification and Validation
The SSMASH simulation has been scrutinized to ensure that the mathematical mod-
els are correctly implemented in the software and that the model represents the phys-
ical system with sufficient accuracy. Correct implementation of the mathematical
models is verified in Section 3.1 and the simulation is validated against other soft-
ware and experimental data in Section 3.2.
3.1 Verification
Due to the complex and highly non-linear nature of the dynamic models used in the
SSMASH simulation, direct comparison of the simulated helicopter response to hand
calculations is feasible for only the most simple cases. Proper implementation of the
equations of motion and functionality of the system sub-models was verified through-
out the coding process by ensuring that the response matches trends expected from
certain input parameters and/or changes to these parameters. Comparison between
the simulation as coded in MATLAB R© and Fortran has also served to strengthen
such verification.
A series of simulations using the final version of SSMASH have been run with a
series of modifications to the input parameters describing the helicopter configuration,
91
92
ship orientation, securing probe status, and wind speed. The intent of this work has
been to verify that the system behaviour matches expected trends. Numerical values
of system response characteristics are also compared when possible. A generic tricycle
configuration helicopter with three oleo strut suspension stations is used for this study.
Modifications are always made from a reference configuration corresponding to the
steady-state response from a drop test onto a flat ship deck with no wind and all
steering and tire rotations constrained. The results from these verification trials are
shown in Table 3.1. Unless stated otherwise, the units of measure in Table 3.1 are
kilonewtons for force, meters for length, and degrees for angles of rotation.
Table 3.1: Qualitative and quantitative results from the SSMASH verification trials.
Case Description System Response Verified
1Referencestate
Variable Nose Load Right Load Sum Load Pitch
XExpected 35.004 33.755 102.514 small
Final 34.999 33.758 102.514 0.436
2
Increasehelicoptermass by1000 kg
Variable z-pos. Nose Load Right Load Sum Load
XInitial -2.292 34.999 33.758 102.514
Expected increase increase increase 112.324
Final -2.272 38.338 36.993 112.324
3HelicopterC.G. movedforward
Variable Pitch Nose Load Right Load Sum Load
XInitial 0.436 34.999 33.758 102.514
Expected negative increase decrease 102.514
Final -0.354 44.000 29.257 102.514
Continued on following page
93
Continued from previous page
4Positive shippitch 5◦
Variable Pitch Nose Load Right Load Sum Load
XInitial 0.436 34.999 33.758 102.514
Expected increase decrease increase 102.124
Final 0.8084 31.174 34.475 102.124
5Positive shippitch, tiresfree to roll
Variable x-pos. Yaw y-pos.
XInitial 0.000 0.000 0.000
Expected negative 0.0 0.0
t = 5 s -10.613 0.000 0.000
6
Positive shippitch, tiresfree to roll,probe active
Variable x-pos. Yaw x-force y-force
XInitial 0.000 0.000 0.000 0.000
Expected small neg. 0.0 8.935 0.0
Final -0.022 0.000 8.936 0.000
7Positive shiproll 5◦
Variable Roll Right Load Left Load Main Sum
XInitial 0.000 33.758 33.758 67.518
Expected positive increase decrease 67.261
Final 2.023 41.185 25.984 67.168
8
Positive shiproll 10◦,unlock tires,enable nosewheel steer
Variable Roll Yaw Steer y-pos.
XInitial 0.000 0.000 0.000 0.000
Expected 0.0 90 0 positive
t = 10 s 0.004 92.4 -0.871 9.685
9
Positive shiproll 5◦,negative shippitch 5◦,unlock tires
Variable x-pos. Tire Def. Yaw y-pos.
XInitial 0.000 0.000 0.000 0.000
Expected positive small neg. small pos. small pos.
t = 20 s positive -0.002 1.727 0.155
Continued on following page
94
Continued from previous page
10
Ship roll 5◦,ship pitch 5◦,unlock tires,unlock nosewheel steering
Variable Yaw Steer Tire Def.
XInitial 0.000 0.000 0.000
Expected 45 0 0
t = 20 s 45.6 -0.406 0.000
11
Positivex-directionwind velocity,unlock tires
Variable Pitch x-pos.
XInitial 0.436 0.000
Expected decrease positive
Final 0.259 13.726
12Positivey-directionwind velocity
Variable Roll Tire Def. Right Load Yaw
XInitial 0.000 0.000 33.758 0.000
Expected positive negative increase small pos.
Final 1.7499 -0.0048 40.550 0.0011
13
Increase noseoleo gaspressure1000 kPa
Variable Nose Oleo Compression
XInitial 0.277
Expected decrease
Final 0.211
14Increase noseoleo volume0.5 L
Variable Nose Oleo Compression
XInitial 0.277
Expected increase
Final 0.347
15Increase noseoleo pistonarea 20 cm2
Variable Nose Oleo Compression
XInitial 0.277
Expected decrease
Final 0.134
95
3.2 Validation
The validation process for SSMASH has been developed to reflect the complex nature
of the simulation in that spacial dynamic response to ship motion and helicopter
manoeuvring are modelled. The validation process must make use of readily available
data; gathering new experimental data for this work is both cost and time prohibitive.
Limited experimental testing of shipboard helicopter manoeuvring in the presence of
ship motion has been conducted to date, and the data for such trials are not available
for comparison at the time of writing. As such, a two-stage validation process has
been carried out.
The spacial dynamic modelling of SSMASH is validated by comparison to the
proprietary Dynaface R© software in Section 3.2.1. Dynaface R© is a well-validated
software in its own right for shipboard helicopter securing analysis, though it does
not attempt to characterize helicopter handling. As such, manoeuvring events can
not be compared with Dynaface R©. Experimental data from full-scale Deal Load Test
Vehicle (DLTV) land-based trials are used to validate the manoeuvring aspects of the
SSMASH simulation in Section 3.2.2.
3.2.1 Spacial Dynamics Validation
A series of comparisons between SSMASH and Dynaface R© simulation results are
shown here. Identical initial conditions and motion inputs (when applicable) are
supplied to both simulation software, facilitating direct comparison of the predicted
system response. All tires are locked in rotation and all steering degrees of freedom
are constrained in the SSMASH simulation. A tricycle-configuration, nose-wheeled
helicopter model somewhat representative of the Sikorsky CH-148 Cyclone is used for
these comparisons.
96
CASE 1A: Flat Drop
In this validation case, the helicopter is dropped from a height of 2.7 meters above the
ship deck. This distance refers to the height of the helicopter centre of gravity, and
results in approximately 0.2 meters of tire ground clearance. All suspension stations
are initially at full-droop and all initial velocities are zero. The ship deck is statically
oriented parallel to the inertial frame and the securing probe is disabled. Figure 3.1
shows the time history of the helicopter z position for the SSMASH and Dynaface R©
simulations.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0-2.7
-2.6
-2.5
-2.4
-2.3
-2.2
-2.1
-2.0
Time [s]
Pos
itio
n[m
]
SSMASH
Dynaface R©
Figure 3.1: Helicopter centre of gravity z position versus time for validation case1A.
Extremely good correlation is observed between the two softwares with sub-
centimeter differences in predicted height of the helicopter centre of gravity. Such
small differences can be accounted for by small differences in input values used to
describe the helicopter characteristics. Dynaface R© simulations have been run us-
ing Imperial units, while the SSMASH simulation uses SI units. Small truncation
and rounding errors during unit conversion has likely rendered the inputs to each
simulation software slightly different.
Expectedly, the time history of the oleo strut extensions shown in Figure 3.2 shows
similar correlation. Both simulations predict the same touchdown time and peak oleo
97
compressions. Furthermore, the motion is nearly perfectly in phase with the timing
of the peaks being separated by less than one tenth of a second.
The final pertinent aspect of the helicopter response for this validation case is
shown in Figure 3.3. Here, the helicopter pitch angle is shown. Though the magnitude
of the pitching motion is very small, the trends are significant. Both simulations show
the same magnitude of pitching motion, and are generally in phase with each other.
Small differences in peak magnitudes and a small phase shift in the second, third, and
fourth oscillation peaks become visible at this scale. Such variation is expected from
differences in the tire modelling being used, and to some extent from small differences
in friction modelling in the oleo struts.
The SSMASH simulation uses a tire model derived completely from the Smiley
and Horne [14] work while Dynaface R© uses a combined approach that incorporates
user-input stiffness values. It is likely that small differences in instantaneous tire
vertical stiffness are becoming apparent at the fine resolution visible in Figure 3.3.
Differences in lateral and longitudinal tire stiffness may also be present, an observation
that will be further developed in the remaining spacial validation cases.
CASE 1B: Inclined Drop
The initial conditions for this validation case are similar to those from case 1A, though
the ship is rolled five degrees to starboard. This elicits a more significant dynamic
response from the helicopter. The securing probe remains disabled. Figure 3.4 shows
the time history of the helicopter centre of gravity height. It is apparent that a small
offset of approximately 2.5 centimeters is present, though the timing and relative peak
magnitudes appear to have excellent correlation as seen in the previous validation
case.
It is possible that a difference in the ship rotation centre is accounting for this
offset since the plotted values are shown relative to the inertial frame of reference.
98
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.5
0.7
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]
Right Main Gear
SSMASH
Dynaface R©
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.5
0.7
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]Nose Gear
SSMASH
Dynaface R©
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.5
0.7
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]
Left Main Gear
SSMASH
Dynaface R©
Figure 3.2: Oleo strut extensions versus time for validation case 1A.
99
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.00.10.20.30.40.50.60.70.80.9
Time [s]
Pit
chA
ngl
e[d
eg] SSMASH
Dynaface R©
Figure 3.3: Helicopter pitch angle versus time for validation case 1A.
0 1 2 3 4 5 6−3−2.9−2.8−2.7−2.6−2.5−2.4−2.3−2.2−2.1
Time [s]
Pos
itio
n[m
]
SSMASH
Dynaface R©
Figure 3.4: Helicopter centre of gravity z position versus time for validation case1B.
100
If the ship rotation centre in the Dynaface R© simulation is significantly lower than
the same in the SSMASH simulation, the five degree roll rotation will also lower
the ship-helicopter interface. This would explain a small constant offset in helicopter
centre of gravity heights. If the above observation is correct, the oleo extension time
history should not show such an offset. The oleo extensions for this validation case
are shown in Figure 3.5. Indeed, much less difference between the two simulations
is evident in Figure 3.5 which supports the previous observations regarding the z
position offset. Sub-centimeter differences here are likely due to an emerging trait
difference between Dynaface R© and the SSMASH simulation. The friction modelling
in Dynaface R© appears to have a more significant stick-slip nature evidenced by flat
segments in the various oleo extension traces. This could allow the oleos to settle
with significant residual axial friction force. The small differences in settling position
between the right and left oleos imply that a steady state roll difference of approx-
imately 0.14 degrees should exist between the two simulations. The helicopter roll
angle relative to the inertial frame is shown in Figure 3.6.
As expected, a settling roll angle difference of approximately 0.2 degrees is evident
in Figure 3.6. Excellent correlation is again achieved in the roll response with small
differences largely accounted for by oleo extension differences and possibly differences
in vertical tire stiffness. Differences in tire modelling may be contributing larger
sources of deviation between the two simulations now that greater vertical loads and
large lateral loads are being reacted by the tires. Finally, the helicopter pitch response
for this validation case is shown in Figure 3.7.
The pitch response is again small in absolute magnitude, though good correlation
between the two simulations is evident. A greater nose-down pitch motion is predicted
by the SSMASH simulation near 1.5 seconds. This peak corresponds to the peak
difference in the left main oleo extension from Figure 3.5 at the same time. The
SSMASH simulation predicts greater oleo extension here, implying a nose down pitch
101
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.5
0.7
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]
Right Main Gear
SSMASH
Dynaface R©
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.5
0.7
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]Nose Gear
SSMASH
Dynaface R©
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.5
0.7
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]
Left Main Gear
SSMASH
Dynaface R©
Figure 3.5: Oleo strut extensions versus time for validation case 1B.
102
0 1 2 3 4 5 6−2
0
2
4
6
8
10
12
Time [s]
Rol
lA
ngl
e[d
eg]
SSMASH
Dynaface R©
Figure 3.6: Helicopter roll angle versus time for validation case 1B.
0 1 2 3 4 5 6-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Time [s]
Pit
chA
ngl
e[d
eg]
SSMASH
Dynaface R©
Figure 3.7: Helicopter pitch angle versus time for validation case 1B.
103
motion as is apparent in Figure 3.7 at 1.5 seconds.
CASE 1C: Sinusoidal Ship Roll
This validation case compares the helicopter response when the ship deck follows a
sinusoidal roll motion with an amplitude of five degrees and a period of ten seconds.
The initial system state for this simulation is the steady state response from validation
case 1A. The securing probe remains disabled. The time history of the helicopter roll
angle relative to the inertial frame is shown in Figure 3.8.
0 5 10 15−8−6−4−2
02468
Time [s]
Rol
lA
ngl
e[d
eg]
SSMASH
Dynaface R©
Figure 3.8: Helicopter roll angle versus time for validation case 1C.
As expected, the amplitude of the helicopter roll response is slightly greater than
the ship deck excitation amplitude of five degrees due to inertial effects. Excellent
peak roll angle correlation between the two simulations is achieved. It appears that
the roll response predicted by Dynaface R© due to ship roll direction change is slightly
delayed, after which a greater roll rate is evident. Figure 3.9 shows that a greater
stick-slip effect in the Dynaface R© simulation is likely responsible for the more erratic
roll rate.
Both simulations predict nearly identical peak oleo extension and compression
values, though the Dynaface R© simulation holds these peak values for longer periods
of time. This observation is in harmony with those made previously about the greater
104
0 5 10 150.6
0.7
0.8
0.9
1.0
Time [s]
Ole
oE
xte
nsi
on[m
]
Right Main Gear
SSMASH
Dynaface R©
0 5 10 150.6
0.7
0.8
0.9
1.0
Time [s]
Ole
oE
xte
nsi
on[m
]Nose Gear
SSMASH
Dynaface R©
0 5 10 150.6
0.7
0.8
0.9
1.0
Time [s]
Ole
oE
xte
nsi
on[m
]
Left Main Gear
SSMASH
Dynaface R©
Figure 3.9: Oleo strut extensions versus time for validation case 1C.
105
stick-slip nature of the Dynaface R© response. The helicopter pitch response is shown
in Figure 3.10.
0 5 10 150.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
Time [s]
Pit
chA
ngl
e[d
eg] SSMASH
Dynaface R©
Figure 3.10: Helicopter pitch angle versus time for validation case 1C.
Both simulations show a small amount of coupling between the helicopter roll and
pitch motion with similar peak pitch angles.
CASE 1D: General Ship Motion
The helicopter response to a general ship motion profile is compared in this validation
case. This type of flight deck motion involves combinations of roll, pitch, yaw, surge,
sway, and heave motions and can be determined from experimental trials or simulated
ship response to wave patterns. For this validation case, the flight deck motion was
obtained from simulated ship response, and is representative of a typical naval frigate
in rough seas. The initial conditions once again reflect the steady state response from
case 1A, and the securing probe is activated. As evidenced by the peak helicopter roll
angle magnitudes shown in Figure 3.11, this ship motion profile creates a significant
dynamic response from the helicopter system.
Excellent correlation between the two simulations is evident from similar peak
roll angles and similar small time scale roll oscillations. Figure 3.12 shows similarly
close correlation in the helicopter yaw response, while the pitch response is shown in
106
0 5 10 15 20 25 30−40
−30
−20
−10
0
10
20
30
Time [s]
Rol
lA
ngl
e[d
eg]
SSMASH
Dynaface R©
Figure 3.11: Helicopter roll angle versus time for validation case 1D.
Figure 3.13.
0 5 10 15 20 25 30−8−6−4−2
02468
10
Time [s]
Yaw
Angl
e[d
eg] SSMASH
Dynaface R©
Figure 3.12: Helicopter yaw angle versus time for validation case 1D.
The peak helicopter pitch angles are again nearly identically predicted by the two
simulations. The small time scale pitch oscillations are also very similar in nature,
magnitude, and timing. The extent of the helicopter excitation due to the prescribed
ship motion is best evidenced by the oleo extensions shown in Figure 3.14. As ex-
pected, the response predicted by the two simulations is in close agreement with more
stick-slip motion evident in the Dynaface R© response. Both simulations predict sev-
eral events where one of the main landing gear oleos is at or near full extension. This
implies that tire liftoff from the ship deck is either occurring or imminent at these
107
0 5 10 15 20 25 30-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5
Time [s]
Pit
chA
ngl
e[d
eg]
SSMASH
Dynaface R©
Figure 3.13: Helicopter pitch angle versus time for validation case 1D.
times.
Spacial Validation Summary
Validation of the spacial dynamic modelling in the SSMASH simulation has been
completed by comparison to simulated helicopter response data from Dynaface R©.
Excellent correlation between the two software packages is evident in all cases tested
with only small differences in the predicted helicopter response to various dynamic
events.
It is expected that small differences in the oleo friction modelling account for
most of the variation. The Dynaface R© simulation consistently exhibits more stick-
slip behaviour in the oleo struts, so it seems likely that a more aggressive decay
constant is used in the axial friction smoothing function. Higher values here can be
used with the SSMASH simulation at the cost of computation speed. The tire models
used by the two simulations are also slightly different. It is possible that, particularly
with large vertical and lateral deflections, differences in linear tire stiffnesses have
influenced the dynamic response.
The largest difference in modelling approach between the SSMASH simulation
and Dynaface R© is that mass coupling effects between the helicopter body and wheel
108
0 5 10 15 20 25 300.6
0.7
0.8
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]
Right Main Gear
SSMASH
Dynaface R©
0 5 10 15 20 25 300.6
0.7
0.8
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]Nose Gear
SSMASH
Dynaface R©
0 5 10 15 20 25 300.6
0.7
0.8
0.9
1.0
1.1
Time [s]
Ole
oE
xte
nsi
on[m
]
Left Main Gear
SSMASH
Dynaface R©
Figure 3.14: Oleo strut extensions versus time for validation case 1D.
109
carriages are captured by SSMASH whereas they are not captured by Dynaface R©.
Mass coupling effects are only expected to be significant in high-frequency motion,
so it is possible that some of the smaller time scale system response differences are
accounted for by this difference in modelling approach.
3.2.2 Planar Handling and Manoeuvring Validation
The manoeuvring analysis capabilities of the SSMASH simulation are validated by
comparison to experimental data gathered by Indal Technologies Inc. using a full-
scale instrumented DLTV representative of a Sikorsky S-70B Seahawk, as shown in
Figure 3.15. Parameters recorded continuously during the DLTV trials include the
position of the securing device claw, the x and y components of the securing probe
force in the helicopter frame, the main tire lateral deflections, and the tail wheel steer
angle. The helicopter orientation was also recorded at discrete times.
Figure 3.15: The Indal Technologies Inc. DLTV [31].
Since the DLTV used in the experimental trials was designed to mimic the S-70B
110
Seahawk, input parameters representative of this aircraft are used for the SSMASH
simulation. Some parameters require additional consideration. Linn [1] carried out
several comparisons with simple experimental manoeuvres to select appropriate values
for the probe stiffness and tire rolling resistance. These values are also used for
the SSMASH simulation in this validation. Unknown parameters that describe the
joint and axle friction have been estimated by tuning the simulated response to the
experimental DLTV data.
Three pertinent manoeuvring events from the DLTV trials are used for comparison
with the SSMASH simulation:
• Traversing while Yawed
• Helicopter Yaw to 90 Degrees
• Tail Steer to 180 Degrees
Detailed comparison plots of the system response and corresponding discussions
are presented in the following sections.
CASE 2A: Traversing while Yawed
In this validation case the helicopter is pulled forward by the RSD claw while at a
yaw angle of approximately 9 degrees. The tail wheel is not allowed to rotate, so this
orientation is maintained during the manoeuvre. The manoeuvre terminates with the
RSD claw moving rearward to its approximate starting position. Figure 3.16 shows
the x and y positions of the RSD claw in the ship frame of reference.
This manoeuvre assesses the accuracy of the SSMASH simulation of tire and probe
stiffness, the tire relaxation length, the tire rolling resistance, and the modelling of
the axle rolling friction due to tire lateral load. Figures 3.17 and 3.18 show the x and
y probe forces, respectively.
111
0 5 10 15 20 25 30 35 40-0.10.00.10.20.30.40.50.60.7
Time [s]
Pos
itio
n[m
]x-positiony-position
Figure 3.16: RSD claw position versus time for validation case 2A.
0 5 10 15 20 25 30 35 40−6−4−2
02468
1012
Time [s]
For
ce[k
N]
SSMASHDLTV
Figure 3.17: Securing probe x-force versus time for validation case 2A.
0 5 10 15 20 25 30 35 40−25
−20
−15
−10
−5
0
5
10
Time [s]
For
ce[k
N]
SSMASHDLTV
Figure 3.18: Securing probe y-force versus time for validation case 2A.
112
Figure 3.17 shows that the rise of longitudinal force required to pull the heli-
copter forward is well predicted by the SSMASH simulation. The experimental and
simulated data show significant force oscillations at approximately 1 Hz which likely
correspond to like-frequency noise in the prescribed RSD claw x position. Since the
securing probe is a very stiff system, small oscillations in the claw position can cause
large fluctuations in the probe force and corresponding disturbances to the helicopter
body.
It is believed that the oscillations become too large in the simulated case due to
small amounts of RSD claw position measurement error in the experimental test. It is
not known how measurements of the RSD claw position were made, but it is plausible
that the physical oscillations in the claw position were not as significant or sharply
defined as they are in the gathered data. As such, the simulated helicopter would
be subjected to higher amplitude claw position noise than was the DLTV. When the
oscillation in the simulated probe x force becomes too large, the appropriate response
trend is lost.
Agreement between the simulated and experimental probe y force shown in Fig-
ure 3.18 is very good with nearly identical load increase rate and peak load. Further-
more, small-scale oscillation events are well captured by the SSMASH simulation.
Though the decreasing load trends are similar, the rate of load release while the he-
licopter is traversed rearwards is slightly greater in the experimental data. This may
be explained by the slight differences in the helicopter yaw angle during this phase of
the manoeuvre.
The helicopter yaw angle is shown in Figure 3.19. The yaw trend of the helicopter
is well predicted by the simulation, though the final experimental value is approx-
imately 0.5 degrees higher than the peak simulated yaw angle. The departure of
the experimental data seems to occur around the time of load reversal on the probe.
While this may be a result of measurement error, it could also correspond to some
113
0 5 10 15 20 25 30 35 408.8
9.0
9.2
9.4
9.6
9.8
10.0
Time [s]
Yaw
Angl
e[d
eg] SSMASH
DLTV
Figure 3.19: Helicopter yaw angle versus time for validation case 2A.
free steering play in the DLTV tail suspension that is not modelled in the simulation.
Such an effect could also explain the difference in load relaxation rate observed in
Figure 3.18.
Lateral tire deflections during the yawed traversing manoeuvre are shown in Fig-
ure 3.20. Apart from an apparent offset in the experimental data, the agreement here
is excellent in terms of peak tire deformation and small-scale oscillation effects that
are captured by the simulation. The absence of large fluctuations here strengthens
arguments that the stiff nature of the securing probe system is itself responsible for
fluctuations in the probe x force observed in Figure 3.17.
Finally, the main tire vertical loads for this manoeuvre are shown in Figure 3.21 to
show the importance of modelling out-of-plane effects during planar manoeuvres. The
changing vertical load here confirms that out-of-plane helicopter response is present
for this manoeuvre.
CASE 2B: Helicopter Yaw to 90 Degrees
In this validation case the helicopter is yawed to nearly 90 degrees from an initial
angle of approximately 9 degrees. This is accomplished by traversing the RSD claw
forward approximately 4.5 meters with the tail wheel free to rotate. The RSD claw
114
0 5 10 15 20 25 30 35 40−10
0
10
20
30
Time [s]
Defl
ecti
on[m
m]
Right Main Tire
SSMASHDLTV
0 5 10 15 20 25 30 35 40−10
0
10
20
30
Time [s]
Defl
ecti
on[m
m]
Left Main Tire
SSMASHDLTV
Figure 3.20: Main landing gear tire lateral deflection versus time for validation case2A.
0 5 10 15 20 25 30 35 4025
30
35
Time [s]
Ver
tica
lL
oad
[kN
] RightLeft
Figure 3.21: Main landing gear tire vertical load versus time for validation case 2A.
115
position history is shown in Figure 3.22. Note that this signal contains visible noise,
so large fluctuations in the probe x force are expected.
0 10 20 30 40 50 60 70 80 900.00.51.01.52.02.53.03.54.04.5
Time [s]
Pos
itio
n[m
]
x-positiony-position
Figure 3.22: RSD claw position versus time for validation case 2B.
Figure 3.23 shows a comparison between the simulated and experimental probe
x force for this manoeuvre. As expected, there is significant oscillation in both data
sets, though good agreement is observed for the initial 30 seconds or so. After this
point, the experimental longitudinal probe force begins a steady decline, ultimately
becoming negative. It is not physically plausible for the probe x force to become
negative during this manoeuvre because the RSD claw is continuously traversing
forward. This indicates potential drift in the experimental data or rotation of the
securing probe in its housing.
The experimental securing probe forces were measured using strain gauges at-
tached to the probe body. It is possible for the securing probe to rotate about its
axis. Such a rotation would effectively rotate the coordinate system in which the
probe forces were recorded. Any rotation of the securing probe was not recorded
during the DLTV experiments, so this effect cannot be duplicated with any certainty
during analysis of the simulated data.
The probe y force shown in Figure 3.24 shows similar trends as the probe x force
with good agreement between simulation and experiment in the first 30 seconds of
116
0 10 20 30 40 50 60 70 80 90−4
−2
0
2
4
6
8
Time [s]
For
ce[k
N]
SSMASHDLTV
Figure 3.23: Securing probe x-force versus time for validation case 2B.
the manoeuvre. After this time the data sets show opposite trends.
0 10 20 30 40 50 60 70 80 90−16−14−12−10−8−6−4−2
02
Time [s]
For
ce[k
N]
SSMASHDLTV
Figure 3.24: Securing probe y-force versus time for validation case 2B.
Better correlation is achieved between the experimental and simulated helicopter
yaw and tail wheel steer angles which are shown in Figures 3.25 and 3.26, respectively.
The simulated and experimentally-measured main tire lateral deflections for this
manoeuvre are shown in Figure 3.27. Excellent correlation is achieved for the left
tire, while moderate correlation for the first 60 seconds of the manoeuvre is achieved
for the right tire once the initial offset is accounted for. It is expected that similar
deformation would be observed from both main tires, though this is not the case in the
DLTV data past 60 seconds of manoeuvring time. This implies a marked difference
117
0 10 20 30 40 50 60 70 80 900
102030405060708090
Time [s]
Yaw
Angl
e[d
eg] SSMASH
DLTV
Figure 3.25: Helicopter yaw angle versus time for validation case 2B.
0 10 20 30 40 50 60 70 80 90−100
−80
−60
−40
−20
0
20
Time [s]
Angl
e[d
eg]
SSMASHDLTV
Figure 3.26: Tail wheel steer angle versus time for validation case 2B.
118
in the operating environment of the two main tires.
0 10 20 30 40 50 60 70 80 90−10
0
10
20
Time [s]
Defl
ecti
on[m
m]
Right Main Tire
SSMASHDLTV
0 10 20 30 40 50 60 70 80 90−10
0
10
20
Time [s]
Defl
ecti
on[m
m]
Left Main Tire
SSMASHDLTV
Figure 3.27: Main landing gear tire lateral deflection versus time for validation case2B.
It is possible that the right side tire encountered an area of ground with a much
reduced friction coefficient or other surface irregularity around 60 seconds into the ma-
noeuvre. The ground surface is completely homogeneous in the SSMASH simulation
so such events cannot be captured in the simulated data.
The main tire vertical loads for this manoeuvre are shown in Figure 3.28 to again
highlight the presence of out-of-plane effects.
119
0 10 20 30 40 50 60 70 80 9025
30
35
Time [s]
Ver
tica
lL
oad
[kN
]
RightLeft
Figure 3.28: Main landing gear tire vertical load versus time for validation case 2B.
CASE 2C: Tail Steer to 180 Degrees
In this validation case the helicopter is maneuvered to achieve nearly 180 degrees of
tail wheel steer rotation. This is accomplished by a combination of longitudinal and
lateral RSD claw movements shown in Figure 3.29.
0 10 20 30 40 50 60 70 80-0.2-0.10.00.10.20.30.40.50.6
Time [s]
Pos
itio
n[m
]
x-positiony-position
Figure 3.29: RSD claw position versus time for validation case 2C.
The experimental and simulated probe x force data are shown in Figure 3.30.
Poor agreement is seen here. When comparing the probe x force between 10 and 20
seconds with the prescribed RSD claw position, it seems physically inconsistent for
the experimental probe force to be negative because the probe is pulling the helicopter
forward. A negative probe x force would imply that the helicopter is being forced in
120
its negative x direction. Such disconnect between the experimental forces here and
the expected trends could be due to loss of strain gauge calibration, securing probe
rotation, or other unmeasured events during the DLTV trial.
0 10 20 30 40 50 60 70 80−20
−15
−10
−5
0
5
10
15
Time [s]
For
ce[k
N]
SSMASHDLTV
Figure 3.30: Securing probe x-force versus time for validation case 2C.
The probe y force shown in Figure 3.31 shows better correlation between the
experimental and simulated data. Good correlation is evident up to 20 seconds into
the manoeuvre, after which the data sets diverge for a short time of approximately
10 seconds. This diversion is partially corrected at 30 seconds into the manoeuvre,
after which the trends of the data sets are in close agreement. It seems that the
system response is well modelled by the simulation from 30 seconds onward, though
it is offset due to the short divergence. This observation is supported by the main
tire lateral deflections shown in Figure 3.32.
The right and left tire lateral deflections are well predicted by the simulation
except during the period from 20 to 30 seconds into the manoeuvre. During this time
the lateral deflections of the main landing gear tires in the DLTV trial are allowed
to relax, whereas they further increase in the simulated response. This explains the
marked departure between the experimental and simulated probe y force during this
time span. It is possible that irregularities in the ground surface of the DLTV test
facility allowed the tires to slide along its surface, or that some other unmodelled
121
0 10 20 30 40 50 60 70 80−40
−30
−20
−10
0
10
20
30
Time [s]
For
ce[k
N]
SSMASHDLTV
Figure 3.31: Securing probe y-force versus time for validation case 2C.
−10 0 10 20 30 40 50 60 70 80−20
0
20
40
Time [s]
Defl
ecti
on[m
m]
Right Main Tire
SSMASHDLTV
−10 0 10 20 30 40 50 60 70 80−20
0
20
40
Time [s]
Defl
ecti
on[m
m]
Left Main Tire
SSMASHDLTV
Figure 3.32: Main landing gear tire lateral deflection versus time for validation case2C.
122
effect is allowing the tires to relax.
Though the tire model used in the SSMASH simulation allows lateral and longi-
tudinal tire sliding when the available grip is exceeded, contributions from tire twist
loading are not considered in determining when sliding can occur. This behaviour is
outside the scope of the Smiley and Horne modelling equations. Figure 3.33 shows
that high yaw rates of the DLTV are achieved in this trial. This indicates that signif-
icant twist could be developing in the tires, possibly to the extent that large portions
of the contact patch are sliding along the ground surface and cannot sustain any
additional lateral force. Such an event would explain a decrease in the experimen-
tal lateral tire deflection and corresponding lateral load that is not captured in the
simulation.
The simulated lateral tire deflection is finally relaxed around 32 seconds into the
manoeuvre. This timing aligns with the final period of high helicopter yaw rate.
After this time, the tire lateral deflection is well predicted by the simulation once a
small offset is considered. This offset likely accounts for the offset in the probe y force
during the same time period.
0 10 20 30 40 50 60 70 8010121416182022242628
Time [s]
Yaw
Angl
e[d
eg]
SSMASHDLTV
Figure 3.33: Helicopter yaw angle versus time for validation case 2C.
The tail wheel steer angle for this manoeuvre is shown in Figure 3.34. Again,
excellent correlation between simulation and experiment is observed for all but the
123
time period between 20 and 30 seconds.
0 10 20 30 40 50 60 70 80−180−160−140−120−100−80−60−40−20
0
Time [s]
Angl
e[d
eg]
SSMASHDLTV
Figure 3.34: Tail wheel steer angle versus time for validation case 2C.
Finally, the main tire vertical loads are shown in Figure 3.35 to highlight the
presence of out-of-plane effects during this manoeuvre as well.
0 10 20 30 40 50 60 70 8020
25
30
35
40
Time [s]
Ver
tica
lL
oad
[kN
] RightLeft
Figure 3.35: Main landing gear tire vertical load versus time for validation case 2C.
Planar Validation Summary
Validation of the planar manoeuvring capabilities of the SSMASH simulation has
been carried out by comparison to full scale experimental data. Excellent correlation
is consistently achieved between the experimental and simulated helicopter state re-
sponse, while agreement between the securing probe forces is good at times but less
124
consistent. In general, the level of agreement with the experimental data achieved
by the SSMASH simulation is comparable to that achieved by the HeliMan planar
handling simulation developed by Linn [1].
Some of the discrepancies between the simulated probe forces and those measured
in the DLTV trials may be explained by probe rotation about its axis. This would
change the reference frame in which the forces are described and cannot be accounted
for when analyzing the data because any such rotation was not measured during
the trials. Regardless of possible probe rotation, simulation of the securing probe
system is difficult due to its stiff nature when modelled as a forcing element. Small
irregularities in the prescribed securing claw position create large oscillations in the
securing probe force. This effect is witnessed in the experimental data as well, though
to a slightly lesser extent.
There have been several instances where the lateral deformation of a tire on the
DLTV main landing gear is more readily relaxed than in simulation. These events may
be due to encounters with irregularities in the ground surface of the DLTV test facility
or un-modelled tire relaxation effects. Large values of tire twist are not accounted for
in determining when tire sliding can occur since this behaviour is beyond the scope
of the Smiley and Horne tire modelling equations. It is believed that twist-induced
lateral tire relaxation accounts for tire behaviour discrepancies when large helicopter
yaw gain is accompanied by relatively little forward traverse.
Chapter 4
Conclusions and Recommendations
4.1 Conclusions
The objective to develop and validate a state-of-the-art spacial securing and manoeu-
vring simulation for shipboard helicopters has been met by the SSMASH simulation.
This work has filled a void in the analysis capability of the on-deck helicopter dy-
namic interface. That is, helicopter response to spacial ship motion and traversing
manoeuvres can now be concurrently modelled in a simulation environment. Some
conclusions reached during this work are listed below.
• The SSMASH simulation has been successfully modelled in the Fortran coding
language for faster-than-real-time program execution on an available Intel R©
CoreTM
i7 CPU. Typical run-times for the simulation coded in Fortran range
from one to four seconds per ten seconds of simulated time. Simulations run
using the Fortran version of the model typically run an order of magnitude faster
than identical simulations run in the MATLAB R© computing environment.
• The dynamic simulation has full capability to model spacial helicopter response
to a wide range of dynamic events. High impact and high frequency events such
as free drops onto the ship deck can be analyzed in addition to longer-time-scale
responses to general ship deck motion.
125
126
• The simulation is sufficiently flexible in terms of helicopter configuration to
model most common maritime helicopters. Tricycle and tail wheel helicopter
landing gear configurations are accommodated. Each suspension station can be
modelled as a prismatic oleo strut or swing arm with one or two tires on each
wheel carriage.
• A passive securing mechanism representative of the RAST and ASIST systems
is successfully integrated in the SSMASH simulation. The securing claw can
be traversed in the ship frame of reference, allowing the simulation to model
dynamic manoeuvring events.
• A five degree-of-freedom tire model is implemented using the Smiley and Horne
equations to characterize tire response. Transient and steady state tire response
is adequately modelled. Minimal tire parameters are required as inputs and no
experimental testing is needed to characterize a particular tire.
• The spacial dynamic modelling aspect of the SSMASH simulation has been
validated by comparison to a software from industry called Dynaface R©. The
Dynaface R© simulation software is extensively validated in its own right against
experimental data. Excellent correlation between SSMASH and Dynaface R© is
achieved for all cases studied. These validation cases include free drops onto
the ship deck, unsecured response to ship roll, and secured response to a severe
general ship motion profile.
• The dynamic manoeuvring aspect of the SSMASH simulation has been val-
idated by comparison to experimental data from full-scale, land-based trials.
Excellent correlation is achieved regarding prediction of the system state re-
sponse, including helicopter yaw angle and tail wheel steer rotations. Good
correlation is evident for some probe force and tire deformation predictions
made by the SSMASH simulation, though less-consistent agreement is observed
127
here. It is believed that unmodelled aspects of the experimental environment in-
cluding ground surface heterogeneity, free play in mechanical mechanisms, and
unmeasured system configuration change account for some of the discrepancy.
Limitations in limit tire friction modelling are also believed to be a source of
error. Naturally, the non-linear nature of the helicopter system in combination
with a stiff securing probe make high-accuracy characterization of the helicopter
response very difficult for complex manoeuvres. In general, the level of agree-
ment between the SSMASH simulation and experimental data is comparable
to that of the planar HeliMan simulation.
4.2 Recommendations
The SSMASH simulation represents a vast core dynamic modelling capability. Some
refinement of the existing simulation is warranted, after which significant development
can be done to increase its utility. Some aspects that should be considered for future
work are listed below.
• Modeling of the tire friction limit should be extended to include localized sliding
from large amounts of tire twist deformation. It is believed that this work
will greatly improve the simulation accuracy for manoeuvres involving large
helicopter yaw gains accompanied by little forward traverse.
• Though the SSMASH simulation runs faster than real-time on the machines
used in development, further reduction in program run time would be desirable.
Run times are much slower than those achieved by the Dynaface R© simulation,
a consideration that may be critical for large batch analysis studies. Improve-
ments in computation speed may be realized through refinement of the code
to optimize array storage and access operations, variable transfer to and from
sub-functions, and by minimizing file input/output operations.
128
• Aerodynamic rotor lift, drag, and side loads should be modelled. This is accom-
plished by Dynaface R© using an assortment of manufacturer provided look-up
tables. This would be a simple implementation within the SSMASH simula-
tion, facilitating analysis of capture immediately following touch down while
rotor loads are still significant.
• A graphical user interface (GUI) to manage input and output data would be
a valuable tool to simplify interaction with the SSMASH simulation. A 3D
graphical display of helicopter response would also be a valuable tool for inter-
pretation of output data.
• The capability to accept real-time joystick input for the securing device move-
ment would facilitate direct human interaction with the simulation. This is
available in the HeliMan simulation.
• Additional securing mechanisms such as a secondary probe, cable lashings, land-
ing gear tie-downs, and manoeuvring cables should be modelled to reflect a more
complete assortment of securing and manoeuvring equipment used in practice.
• Additional passive securing systems such as the TC-ASIST system should be
modelled. Active securing systems could also be modelled, facilitating compar-
ison studies of the relative merits for passive and active securing.
• SSMASH could form the core dynamics model of a real-time simulation for
ground operator training.
• SSMASH could form the core dynamics model of a real-time flight deck moni-
toring system to predict and identify safe operational windows. Such an ability
might safely extend operational limits of naval helicopters to higher sea states
when greater flight deck motion is expected.
• The dynamic formulation used in this work allows extension of the governing
equations to model coupled dynamics between the ship and embarked helicopter
129
without significant re-derivation. Though coupled effects between maritime
helicopters and relatively massive naval vessels are expected to be insignificant,
such modelling capability may be of benefit to other systems. Dynamic coupling
effects may be significant for applications involving small craft with embarked
vehicles of similar mass.
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dynamic response. In AERO 2007 Conference, Toronto, Canada, April 2007.
[28] Parviz E. Nikravesh. Computer-Aided Analysis of Mechanical Systems. Prentice
Hall, Englewood Cliffs, New Jersey, United States of America, 1988.
[29] T. R. Kane, P. W. Likins, and D. A. Levinson. Spacecraft Dynamics. McGraw-
Hill, 1983.
[30] D. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice
Hall, Englewood Cliffs, New Jersey, United States of America, 1988.
[31] D. R. Linn and R. G. Langlois. Development and experimental validation of a
shipboard helicopter on-deck maneuvering simulation. Journal of Aircraft, 43(4),
August 2006.
Appendix A
Simulation Input Parameters, Initial
Conditions, and Output Data
This appendix describes the input parameters required by the SSMASH simulation
to fully define the helicopter system, its initial conditions, and the motion inputs for
the flight deck and securing device claw. The compositions of the output data files
are also described.
Contents of the HELI.INP input file
The HELI.INP input file contains all parameters required to define the helicopter
system characteristics. The contents of each line of this input file are described
below.
Line 1 The helicopter mass [kg].
Line 2 The first row of the helicopter inertia matrix: Ixx Ixy Ixz [kg m2].
Line 3 The second row of the helicopter inertia matrix: Iyx Iyy Iyz [kg m2].
Line 4 The third row of the helicopter inertia matrix: Izx Izy Izz [kg m2].
Line 5 The auxiliary wheel carriage mass [kg].
Line 6 The first row of the auxiliary wheel carriage inertia matrix: Ixx Ixy Ixz [kg m2].
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134
Line 7 The second row of the auxiliary wheel carriage inertia matrix: Iyx Iyy Iyz
[kg m2].
Line 8 The third row of the auxiliary wheel carriage inertia matrix: Izx Izy Izz [kg m2].
Line 9 The right wheel carriage mass [kg].
Line 10 The first row of the right wheel carriage inertia matrix: Ixx Ixy Ixz [kg m2].
Line 11 The second row of the right wheel carriage inertia matrix: Iyx Iyy Iyz [kg m2].
Line 12 The third row of the right wheel carriage inertia matrix: Izx Izy Izz [kg m2].
Line 13 The left wheel carriage mass [kg].
Line 14 The first row of the left wheel carriage inertia matrix: Ixx Ixy Ixz [kg m2].
Line 15 The second row of the left wheel carriage inertia matrix: Iyx Iyy Iyz [kg m2].
Line 16 The third row of the left wheel carriage inertia matrix: Izx Izy Izz [kg m2].
Line 17 The coordinates of the mounting point of the auxiliary suspension to the
helicopter body, in the helicopter frame: x y z [m].
Line 18 The coordinates of the mounting point of the right suspension to the heli-
copter body, in the helicopter frame: x y z [m].
Line 19 The coordinates of the mounting point of the left suspension to the helicopter
body, in the helicopter frame: x y z [m].
Line 20 The full droop vector displacement components from the auxiliary oleo frame
origin to the auxiliary wheel carriage frame origin, expressed in the auxiliary
oleo frame: x y z [m].
Line 21 The full droop vector displacement components from the right oleo frame
origin to the right wheel carriage frame origin, expressed in the right oleo
frame: x y z [m].
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Line 22 The full droop vector displacement components from the left oleo frame origin
to the left wheel carriage frame origin, expressed in the left oleo frame: x y z
[m].
Line 23 The full droop auxiliary swing arm orientation relative to the helicopter body,
used for swing arm type suspension only, Euler parameters: e0 e1 e2 e3.
Line 24 The full droop right swing arm orientation relative to the helicopter body,
used for swing arm type suspension only, Euler parameters: e0 e1 e2 e3.
Line 25 The full droop left swing arm orientation relative to the helicopter body, used
for swing arm type suspension only, Euler parameters: e0 e1 e2 e3.
Line 26 The mounting coordinates of the oleo to the helicopter body for the auxiliary
suspension station, used for swing arm type suspension only, expressed in the
helicopter frame: x y z [m].
Line 27 The mounting coordinates of the oleo to the helicopter body for the right
suspension station, used for swing arm type suspension only, expressed in
the helicopter frame: x y z [m].
Line 28 The mounting coordinates of the oleo to the helicopter body for the left
suspension station, used for swing arm type suspension only, expressed in
the helicopter frame: x y z [m].
Line 29 The mounting coordinates of the oleo to the swing arm for the auxiliary
suspension station, used for swing arm type suspension only, expressed in
the auxiliary oleo frame: x y z [m].
Line 30 The mounting coordinates of the oleo to the swing arm for the right suspen-
sion station, used for swing arm type suspension only, expressed in the right
oleo frame: x y z [m].
Line 31 The mounting coordinates of the oleo to the swing arm for the left suspension
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station, used for swing arm type suspension only, expressed in the left oleo
frame: x y z [m].
Line 32 The suspension station type: auxiliary right left [1=prismatic oleo, 2=swing
arm].
Line 33 The suspension station steering lock status: auxiliary right left [1=locked,
0=free].
Line 34 The suspension station axle rolling lock status: auxiliary right left [1=locked,
0=free].
Line 35 The effective steer friction moment coefficient for all suspension stations,
µsteer,eff [m].
Line 36 The steer damping moment coefficient for all suspension stations [N s].
Line 37 The coordinates of the helicopter body centre of pressure, expressed in the
helicopter frame: x y z [m].
Line 38 The equivalent drag areas of the helicopter body, ~Aeq: x y z [m2].
Line 39 The wind velocity components in the inertial frame: x y z [m/s].
Line 40 The securing probe activation status [1=active, 0=disabled].
Line 41 The securing probe initial capture status [1=precaptured, 0=uncaptured].
Line 42 The mounting coordinates of the securing probe to the helicopter body, ex-
pressed in the helicopter frame: x y z [m].
Line 43 The displacement vector components from the securing probe mount to the
undeflected probe tip, expressed in the helicopter frame: x y z [m].
Line 44 The small displacement probe stiffness in the x direction, KPS,x [N/m].
Line 45 The small displacement probe stiffness in the y direction, KPS,y [N/m].
Line 46 The large displacement probe stiffness in the x direction, KPL,x [N/m].
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Line 47 The large displacement probe stiffness in the y direction, KPL,y [N/m].
Line 48 The probe stiffness transition displacement in the x direction, DPS,x [m].
Line 49 The probe stiffness transition displacement in the y direction, DPS,y [m].
Line 50 The probe damping rate in the x direction, CPx [N s/m].
Line 51 The probe damping rate in the y direction, CPy [N s/m].
Line 52 The probe damping rate in the z direction, CPz [N s/m].
Line 53 The probe stiffness transition displacement in the axial direction, DPS,z [m].
Line 54 The small displacement probe stiffness in the axial direction, KPS,z [N/m].
Line 55 The large displacement probe stiffness in the axial direction, KPL,z [N/m].
Line 56 The properties of the first tire on the auxiliary suspension station.
Line 57 The properties of the second tire on the auxiliary suspension station.
Line 58 The properties of the first tire on the right suspension station.
Line 59 The properties of the second tire on the right suspension station.
Line 60 The properties of the first tire on the left suspension station.
Line 61 The properties of the second tire on the left suspension station.
Line 62 The type of force-generating element for each suspension station: auxiliary
right left [1=linear spring/damper, 2=gas oleo].
Line 63 The linear spring/damping rates for the auxiliary suspension station.
Line 64 The linear spring/damping rates for the right suspension station.
Line 65 The linear spring/damping rates for the left suspension station.
Line 66 The gas oleo properties for the auxiliary suspension station.
Line 67 The gas oleo properties for the right suspension station.
Line 68 The gas oleo properties for the left suspension station.
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Tire parameters
The parameters that define each tire are detailed below.
Column 1 The displacement vector component from the wheel carriage frame origin
to the tire centre, expressed in the wheel carriage frame, in the x direction
[m].
Column 2 The displacement vector component from the wheel carriage frame origin
to the tire centre, expressed in the wheel carriage frame, in the y direction
[m].
Column 3 The displacement vector component from the wheel carriage frame origin
to the tire centre, expressed in the wheel carriage frame, in the z direction
[m].
Column 4 The unloaded tire diameter, dt [m].
Column 5 The nominal tire width, wt [m].
Column 6 The radius of the wheel axle, raxle [m].
Column 7 The unloaded inflation pressure of the tire, Po [Pa].
Column 8 The rated inflation pressure of the tire, Pr [Pa].
Column 9 The standard tire type designation.
Column 10 The maximum coefficient of friction with the ground surface, µ.
Column 11 The coefficient of rolling resistance, µrr.
Column 12 The axle friction coefficient, µaxle.
Column 13 The tire vertical deflection constant, Ctz .
Column 14 The longitudinal tire damping rate, Cx [N s/m].
Column 15 The lateral tire damping rate, Cy [N s/m].
Column 16 The vertical tire damping rate Cz [N s/m].
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Gas oleo parameters
The parameters that define each gas oleo are detailed below.
Column 1 The transition compression distance x0 [m].
Column 2 The transition compression distance x1 [m].
Column 3 The transition compression distance x2 [m].
Column 4 The extension spring rate of the oleo past full droop, k [N/m].
Column 5 The uncompressed internal gas pressure, P [Pa].
Column 6 The uncompressed internal gas volume, V [m3].
Column 7 The piston area, A [m2].
Column 8 The specific heat ratio of the oleo gas, γ.
Column 9 The liner spring force constant for region 4, e1 [N/m].
Column 10 The liner spring force constant for region 5, e2 [N/m].
Column 11 The quadratic spring force constant for region 4, f1 [N/m2].
Column 12 The quadratic spring force constant for region 5, f2 [N/m2].
Column 13 The cubic spring force constant for region 4, g1 [N/m3].
Column 14 The cubic spring force constant for region 5, g2 [N/m3].
Column 15 The constant damping force constant for extension, c0,1 [N].
Column 16 The linear damping force constant for extension, c0,2 [N s/m].
Column 17 The quadratic damping force constant for extension, c0,3 [N s2/m2].
Column 18 The constant damping force constant for the first compression curve,
c1 [N].
Column 19 The linear damping force constant for the first compression curve,
c2 [N s/m].
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Column 20 The quadratic damping force constant for the first compression curve, c3
[N s2/m2].
Column 21 The constant damping force constant for the second compression curve,
c4 [N].
Column 22 The linear damping force constant for the second compression curve, c5
[N s/m].
Column 23 The quadratic damping force constant for the second compression curve,
c6 [N s2/m2].
Column 24 The constant damping force constant for the third compression curve, c7
[N].
Column 25 The linear damping force constant for the third compression curve,
c8 [N s/m].
Column 26 The quadratic damping force constant for the third compression curve,
c9 [N s2/m2].
Column 27 The first compression damping relief force, Fd,rel1 [N].
Column 28 The second compression damping relief force, Fd,rel2 [N].
Column 29 The seal drag force, Fseal [N].
Column 30 The distance of the lower oleo shaft bearing from the oleo frame origin,
along the oleo frame z axis [m].
Column 31 The length between the upper and lower oleo shaft bearings at full droop,
b1lb20 [m].
Column 32 The oleo shaft bearing friction coefficient, µbearing.
Column 33 The decay constant for the oleo friction force smoothing function, β.
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Contents of the INITIALS.INP input file
The INITIALS.INP input file describes the initial conditions of the helicopter system.
The contents of each line of this input file are described below.
Line 1 The helicopter centre of gravity position relative to the ship frame origin,
expressed in the ship frame: x y z [m].
Line 2 The helicopter orientation relative to the ship frame, Euler parameters:
e0 e1 e2 e3.
Line 3 The distances from the oleo frame origins to the wheel carriage frame origins
along the respective oleo frame z axes: auxiliary right left [m].
Line 4 The orientation of the auxiliary prismatic oleo or swing arm relative to the
helicopter frame, Euler parameters: e0 e1 e2 e3.
Line 5 The orientation of the right prismatic oleo or swing arm relative to the heli-
copter frame, Euler parameters: e0 e1 e2 e3.
Line 6 The orientation of the left prismatic oleo or swing arm relative to the heli-
copter frame, Euler parameters: e0 e1 e2 e3.
Line 7 The helicopter translational velocity components relative to the ship frame,
expressed in the ship frame: x y z [m/s].
Line 8 The helicopter angular velocity components relative to the ship frame, ex-
pressed in the helicopter frame: x y z [rad/s].
Line 9 For prismatic oleo suspension stations: the translational velocity component
of the wheel carriages along their respective oleo frame z axes, relative to
the oleo frame. For swing arm suspension stations: the angular velocity
component of the wheel carriages about their respective oleo frame y axes:
auxiliary right left [m/s] or [rad/s].
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Line 10 The angular velocity component of the wheel carriages about their respective
oleo frame z axes: auxiliary right left [rad/s].
Contents of the SOLUTION.INP input file
The SOLUTION.INP input file describes the desired solution time span, specifies how
often to print results to file, and sets the numerical integrator options. The contents
of each line of this input file are described below.
Line 1 The solution time span for the simulation: start end [s].
Line 2 The time step between calls to the numerical integrator [s].
Line 3 The number of rows of the system state solution to store in memory before
printing to output file [integer].
Line 4 The integration status [1=integrate past solution time, -1=evaluate at solu-
tion time]
Line 5 The number of equations for which the roots are desired [0].
Line 6 The integration method [1=Adams, 2=Gear, 3=automatic].
Line 7 The requested relative accuracy in all solution components [double].
Line 8 The smallest physically meaningful state vector solution value [double].
Contents of the PRESCRIBED.INP input file
The PRESCRIBED.INP input file specifies the type of prescribed motion to be used
for the ship and securing device. The contents of each line of this input file are
described below.
Line 1 The method used to describe the RSD claw position [1=from file, 2=con-
stant].
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Line 2 The constant prescribed RSD claw position relative to the ship frame origin,
expressed in the ship frame: x y z [m].
Line 3 The method used to describe the ship motion [1=from file, 2=specified here].
Line 4 The type of sinusoidal ship motion [0=off, 1=roll, 2=pitch, 3=yaw, 4=surge,
5=sway, 6=heave].
Line 5 The sinusoidal ship motion properties: amplitude [degrees or m], period [s].
Line 6 The ship frame orientation relative to the inertial frame, x-y-z Euler angles:
x y z [degrees].
Contents of the RSD POS.INP input file
The RSD POS.INP input file describes the motion trajectory of the securing device
claw. This information is used if the PRESCRIBED.INP file identifies that the se-
curing device motion should be read from file. The contents of each column of the
RSD POS.INP file are described below.
Column 1 The simulation time [s].
Column 2 The securing claw x position relative to the ship frame origin, expressed
in the ship frame [m].
Column 3 The securing claw y position relative to the ship frame origin, expressed
in the ship frame [m].
Column 4 The securing claw z position relative to the ship frame origin, expressed
in the ship frame [m].
Contents of the SHIPMO.INP input file
The SHIPMO.INP input file describes the motion of the flight deck. This information
is used if the PRESCRIBED.INP file identifies that the ship motion should be read
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from file. The contents of each column of the SHIPMO.INP file are described below.
Column 1 The simulation time [s].
Column 2 The ship roll angle relative to the inertial frame [degrees].
Column 3 The ship pitch angle relative to the inertial frame [degrees].
Column 4 The ship yaw angle relative to the inertial frame [degrees].
Column 5 The translational velocity of the ship relative to the inertial frame, ex-
pressed in the inertial frame, in the x direction [m/s].
Column 6 The translational velocity of the ship relative to the inertial frame, ex-
pressed in the inertial frame, in the y direction [m/s].
Column 7 The translational velocity of the ship relative to the inertial frame, ex-
pressed in the inertial frame, in the z direction [m/s].
Column 8 The ship roll rate relative to the inertial frame [deg/s].
Column 9 The ship pitch rate relative to the inertial frame [deg/s].
Column 10 The ship yaw rate relative to the inertial frame [deg/s].
Column 11 The translational acceleration of the ship relative to the inertial frame,
expressed in the inertial frame, in the x direction [m/s2].
Column 12 The translational acceleration of the ship relative to the inertial frame,
expressed in the inertial frame, in the y direction [m/s2].
Column 13 The translational acceleration of the ship relative to the inertial frame,
expressed in the inertial frame, in the z direction [m/s2].
Column 14 The ship roll acceleration relative to the inertial frame [deg/s2].
Column 15 The ship pitch acceleration relative to the inertial frame [deg/s2].
Column 16 The ship yaw acceleration relative to the inertial frame [deg/s2].
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Contents of the RUN.OUT output file
The RUN.OUT output file summarizes general information about the simulation run.
The contents of each line of this output file are described below.
Line 1 The run-time of the simulation [s].
Line 2 The total number of main loop iterations completed.
Line 3 The number of times that the solution was printed to file.
Contents of the STATE.OUT output file
The STATE.OUT output file documents the time history of the helicopter system
state vector. Columns 2 through 13 are comprised of the generalized speeds u7 though
u18, while columns 14 through 35 are comprised of the generalized coordinates q8
through q29. The contents of each column are described in greater detail below.
Column 1 The simulation time [s].
Column 2 The x component of the helicopter translational velocity relative to the
ship frame, expressed in the ship frame [m/s].
Column 3 The y component of the helicopter translational velocity relative to the
ship frame, expressed in the ship frame [m/s].
Column 4 The z component of the helicopter translational velocity relative to the
ship frame, expressed in the ship frame [m/s].
Column 5 The helicopter roll rate relative to the ship frame, expressed in the heli-
copter frame [rad/s].
Column 6 The helicopter pitch rate relative to the ship frame, expressed in the
helicopter frame [rad/s].
Column 7 The helicopter yaw rate relative to the ship frame, expressed in the heli-
copter frame [rad/s].
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Column 8 For a prismatic oleo suspension: the translational velocity component of
the auxiliary wheel carriage relative to the auxiliary oleo frame along the
auxiliary oleo frame z axis, expressed in the auxiliary oleo frame [m/s].
For a swing arm suspension station: the angular velocity of the auxiliary
swing arm about the auxiliary oleo frame y axis [rad/s].
Column 9 The angular velocity component of the auxiliary wheel carriage about
the auxiliary oleo frame z axis [rad/s].
Column 10 For a prismatic oleo suspension: the translational velocity component of
the right wheel carriage relative to the right oleo frame along the right
oleo frame z axis, expressed in the right oleo frame [m/s]. For a swing
arm suspension station: the angular velocity of the right swing arm about
the right oleo frame y axis [rad/s].
Column 11 The angular velocity component of the right wheel carriage about the
right oleo frame z axis [rad/s].
Column 12 For a prismatic oleo suspension: the translational velocity component of
the left wheel carriage relative to the left oleo frame along the left oleo
frame z axis, expressed in the left oleo frame [m/s]. For a swing arm
suspension station: the angular velocity of the left swing arm about the
left oleo frame y axis [rad/s].
Column 13 The angular velocity component of the left wheel carriage about the left
oleo frame z axis [rad/s].
Column 14 The helicopter x position relative to the ship frame, expressed in the ship
frame [m].
Column 15 The helicopter y position relative to the ship frame, expressed in the ship
frame [m].
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Column 16 The helicopter z position relative to the ship frame, expressed in the ship
frame [m].
Column 17 The helicopter orientation relative to the ship frame, Euler parameter e0.
Column 18 The helicopter orientation relative to the ship frame, Euler parameter e1.
Column 19 The helicopter orientation relative to the ship frame, Euler parameter e2.
Column 20 The helicopter orientation relative to the ship frame, Euler parameter e3.
Column 21 The distance from the auxiliary oleo frame to the auxiliary wheel carriage
along the auxiliary oleo frame z axis [m].
Column 22 The auxiliary oleo frame orientation relative to the helicopter, Euler
parameter e0.
Column 23 The auxiliary oleo frame orientation relative to the helicopter, Euler
parameter e1.
Column 24 The auxiliary oleo frame orientation relative to the helicopter, Euler
parameter e2.
Column 25 The auxiliary oleo frame orientation relative to the helicopter, Euler
parameter e3.
Column 26 The distance from the right oleo frame to the right wheel carriage along
the right oleo frame z axis [m].
Column 27 The right oleo frame orientation relative to the helicopter, Euler
parameter e0.
Column 28 The right oleo frame orientation relative to the helicopter, Euler
parameter e1.
Column 29 The right oleo frame orientation relative to the helicopter, Euler
parameter e2.
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Column 30 The right oleo frame orientation relative to the helicopter, Euler
parameter e3.
Column 31 The distance from the left oleo frame to the left wheel carriage along the
left oleo frame z axis [m].
Column 32 The left oleo frame orientation relative to the helicopter, Euler
parameter e0.
Column 33 The left oleo frame orientation relative to the helicopter, Euler
parameter e1.
Column 34 The left oleo frame orientation relative to the helicopter, Euler
parameter e2.
Column 35 The left oleo frame orientation relative to the helicopter, Euler
parameter e3.
Contents of the AUX.OUT output file
The AUX.OUT output file documents the time history of additional information that
describes the helicopter response during the simulation. The contents of each column
are described below.
Column 1 The simulation time [s].
Column 2 The securing probe force component in the x direction, expressed in the
helicopter frame [N].
Column 3 The securing probe force component in the y direction, expressed in the
helicopter frame [N].
Column 4 The securing probe force component in the z direction, expressed in the
helicopter frame [N].
Column 5 The auxiliary tire lateral deflection [m].
149
Column 6 The right tire lateral deflection [m].
Column 7 The left tire lateral deflection [m].
Column 8 The auxiliary tire vertical load [N].
Column 9 The right tire vertical load [N].
Column 10 The left tire vertical load [N].
