Helicopter drivetrain noise and vibration refinement through optimization of
compound planetary gear set transmission error
Jose Torres
Youn Park
Zachary H. Wright
Aero Technical Lead
Head of aerospace
Engineering Manager
Romax Technology Ltd.
Romax Technology Ltd.
Romax Technology Inc.
Nottingham, United Kingdom Nottingham, United Kingdom
Boulder, CO, USA
ABSTRACT
The tools presented are focused on the development of helicopter drivetrains providing the right virtual environment
to effectively implement and test the different design options. They allow evaluation of the design criteria
(deflections and misalignment, durability, dynamics, etc.) and optimization of the gearbox at component and system
level simultaneously. As one application of these tools, an innovative method is presented to evaluate the different
combinations of planetary tooth numbers that set the phase among the planets with the objective of optimizing the
dynamic behavior of a helicopter compound planetary gearbox. Virtual testing is performed to predict the dynamic
excitations from factorizing and non-factorizing planetary arrangements and the response to those excitations at
different locations of the transmission. A gearbox system level approach has been required and applied to fully
understand the multiphysics involved that explain the important effects of something seemingly so fundamental as
the gear tooth numbers on the dynamic behavior of the entire transmission.
main factors should then be considered in the helicopter
drivetrain noise reduction: reduction of the transmission
error and improvement of the transfer path from the gears to
the gearbox housing and aircraft structure.
NOTATION
Factorising planetary gear set: Number of teeth on ring
divided by number of planets must be an integer - this
implies that the number of teeth on the sun divided by the
number of planets is also an integer.
This paper presents the application of virtual design and
testing tools to a method developed to improve the gearbox
dynamic behavior according to the points mentioned above.
This method plays with the combinations of gear tooth
numbers of a planetary gear system to set the phase among
the planets that makes the dynamic excitation generated at
the gear meshes combine in such a way that their effect on
the gearbox vibration is mitigated. This method has been
successfully applied in different projects and practical cases
by Romax Technology.
Non factorising planetary gear set: Number of teeth on ring
divided by number of planets is not an integer.
TE: Transmission Error.
INTRODUCTION 
Modern rotorcrafts must meet noise and vibration
requirements to ensure a minimum acoustic comfort inside
the cabin. In the case of helicopters, the crew and passengers
are close to the main rotor gearbox, which makes the
vibration generated in the drivetrain one of the most
important noise sources to address. An early identification
and reduction of these vibration and noise sources should
then be part of the drivetrain design process in order to find
the right design in terms of acoustic requirements and
feedback the assessment of other design criteria accordingly.
In addition, Romax Technology has developed,
than 25 years, virtual design and testing tools
development and optimization of helicopter
These tools are presented in this paper and
improve the dynamic behavior of a helicopter
planetary main gearbox.
over more
that allow
gearboxes.
applied to
compound
Regarding sidebands and the phasing effects that different
types of manufacturing and mounting errors (carrier runout,
ring gear pitch error, etc.) can introduce into the gearbox
system, further analysis is required. However this paper
covers a small portion of the sideband analysis and focuses
on the direct consequences of the different planetary
arrangements.
The most important source of noise in the helicopter
drivetrain is the transmission error or dynamic excitation
which occurs at the gear teeth when meshing. This vibration
is transmitted to the gearbox housing and then to the aircraft
structure through the housing mounts, inducing housing and
structure borne noise which consists of tonal noise
components at the frequency of the tooth mesh orders. Two
Presented at the AHS 72nd Annual Forum, West Palm
Beach, Florida, USA, May 17-19, 2016. Copyright © 2016
by the American Helicopter Society International, Inc. All
rights reserved.
1
APPROACH
AND
BACKGROUND
cause a moment that tilts the gear on the support bearing.
The tilt is reacted by the internal components of the bearing,
the carrier, and the pin. If the contact area on the tooth
surface shifts from the central position due to misalignment
of the gear, an additional moment is introduced that tends to
restore the gear to its design-intent position (See Figure 1).
For this reason, the misalignment, load sharing, and tooth
contact in an epicyclic system must be solved
simultaneously with the rest of the system, and cannot be
considered in isolation.
METHODOLOGY
A helicopter main gearbox is a complex mechanical system,
which includes planetary gear trains, shafts, bearings and a
gearbox housing. These components interface with each
other through gear meshes, bearing mountings, and other
connections. As such, modifying one design parameter can
alter the performance of seemingly unrelated system
components. In the process of transmission design, it is
imperative that one accounts for the impact of each
modification on the entire system. Because of the complex
interactions within a transmission, it is difficult to achieve a
given set of targets using conventional transmission analysis
methods. Traditionally, the gears and the housing are
analyzed separately, and the influence of housing flexibility
on gear mesh misalignment is obtained through a series of
approximations. These approximations, for example, linear
bearings and rigid housings, result in a considerable
reduction in accuracy. An improved approach is required to
suit the advanced state of design and manufacturing in the
automotive industry.
Figure 1. Restoring moment caused by shift of loaded
contact area on planet gear.
The approach presented here comprises a software package
that allows modeling and analysis of a complete helicopter
driveline in which gears, epicyclic systems, bearings, and
shafts are modeled as functional analysis objects with
correlation to validated tests. The software can calculate all
of the gear meshing points, forces, and load distribution, and
takes into account all of the assigned boundary conditions.
The planetary carriers and housing are meshed in a
commercial FE package, imported into the model, and
coupled with the internal transmission through the bearing
nodes. The transmission system model built by this approach
is very compact compared to conventional finite element
models and much more sophisticated and accurate than only
analytical solutions. Since the gears, shafts and bearings are
all defined as objects, it is much easier to develop these 3D
components in the model space. In essence, the user simply
needs to select the appropriate parts for the system, position
them, and assign their respective attributes.
Mesh Misalignment
When loads are applied to the gearbox, the gears become
misaligned due to deflections of the shafts, bearings, and
housing. This misalignment must be included in the
transmission error calculation. The mesh misalignment is
found by resolving the displacements at the gear mesh across
the face width along the line of action (LOA) (see Figure 2).
The line of action is the imaginary line that connects the two
base circles, and contact always takes place in this direction,
normal to the tooth surface.
Before proceeding with a noise and vibration case study, a
brief review of gearbox operating attributes associated with
transmission error should be given. There are several topics
discussed below: tooth contact, mesh misalignment, bearing
deflections, shaft deflections, housing deflections and planet
load sharing.
Tooth Contact
Transmission error is defined as the rotational error between
the input and output of a gear pair, taking into account gear
ratio. It is often expressed as a linear error along the line of
action. For an epicyclic system, it can be for an individual
gear pair or can represent the total rotational error between
any two members, such as the sun and carrier, sun and ring,
or ring and carrier. The forces acting on the planet gear
Figure 2. Deflections resolved along the line of action.
2
Bearing Deflections
FEA packages, or FE mesh data that can be solved within
the model.
The stiffness of a rolling element bearing is non-linear and
generally increases with applied load. The stiffness submatrix for a rolling element bearing, linking the
displacements and tilts of the inner and outer races, is
obtained as the slope of the force versus deflection curve
near the bearing’s operating displacement. The stiffness
terms are obtained from detailed bearing models, which
include the contact of the rolling elements with the
raceways. The non-linear effects of internal clearance and
pre-load, along with centrifugal effects in high-speed
bearings, are effectively modeled (see Figure 3).
Planet Load Sharing
For epicyclic gear trains, load sharing between planets is an
important phenomenon to consider. A linear analogy is
suitable to exemplify how the load sharing can be unequal
between the planets. Imagine two flat plates that are
separated by springs in parallel. The springs are non-linear
and represent the combined shaft, bearing, and gear stiffness,
and the separation of the plate from the springs represents
the backlash. If the initial backlash is not equal across the
plate, either due to tooth thickness tolerance or pin positional
error, one of the springs will start to take up load before the
others. Also, if one spring is stiffer than others, it will carry
more of the load for a given displacement. See the
illustration in Figure 5.
Figure 3. Non-linear bearing stiffness and radial internal
clearance.
The total displacement of the gears includes contributions
from the bearing stiffness and radial internal clearances,
shaft deflections, and housing deflection.
Figure 5. Load sharing between planets is analogous to a
flat plate supported by springs.
Shaft Deflections
The uniform slender shafts are modeled using Timoshenko
beam elements, and FE mesh data is used for shaft
components that are more complex, such as the epicyclic
carrier or the ring gear. One of the FE carriers used is
illustrated in Figure 4 below.
MODELING TRANSMISSION ERROR
TE is the predominant excitation at the gear meshes. When
considering the calculation of TE for a single gear mesh, the
following must be considered: torque for the given load
case, gear tooth modifications, errors, and alignment
conditions (see Figure 6).
Assembly Error
Gear Manufacturing Error
Pre-loads
Loads
Profile Modifications
Misalignment
Tooth Effects
Tolerance
s
Tooth Flexibility
TRANSMISSION
ERROR
Figure 4. Epicyclic carrier modeled using FE.
Gear Tolerances
Housing Deflections
Figure 6. The variety of input parameters used to
calculate TE for a single gear mesh.
The outer raceway of the bearings are connected to the
gearbox housing. The housing data can be either a reduced
stiffness matrix, which can be created by most commercial
3
For a single gear mesh, it can be assumed that the alignment
of the mesh stays constant for a constant speed and torque.
RomaxDESIGNER will run a full quasi-static system
deflection analysis to determine the boundary conditions of
speed, torque, and misalignment required for the TE analysis
(see Figure 7).
transmission error of each gear mesh, including all phase
information. This logic is illustrated in Figure 8.
Figure 8. Planetary gear TE calculation steps.
So for helicopter transmissions with epicyclic gears, the
route to TE involves an iterative quasistatic simulation in
which the gear contact behaviour is co-simulated with the
static deflection behaviour to yield the TE – this method is
called “gearbox TE” (Figure 8).
Figure 7. Single gear mesh TE calculation steps.
For a planetary system, the interactions between the different
gear meshes within the assembly invalidate the assumption
of constant boundary conditions that is used for the single
mesh case. In order to look at planetary TE, a full-system
quasi-static analysis is performed at each rotational position
of the transmission. This full-system analysis includes the
effects of:




In this method to obtain the TE, a useful side effect of these
static analysis methods is that the values of the non-linear
component stiffnesses at the specified loading condition are
automatically calculated at the same time. These values can
then be used to linearise these non-linear components.
Time-varying gear mesh stiffness and load
point on the gear face, calculated by
analyzing the gear tooth contact condition
and taking into account the gear microgeometry.
Time-varying misalignment due to shaft,
bearing, and housing deflections.
Load (torque) sharing between planets. A
calculation of how the torque is shared
between the planets is important; this also
changes with time and is dependent on
many factors, including the backlash
(perhaps due to manufacturing errors) and
stiffness of the gear mesh.
Relative phasing of planetary gear meshes,
including that between the various sunplanet meshes, the various ring-planet
meshes, and the ring-planet and sun-planet
meshes of a given planet.
These linearised representations of non-linear components
like gears, bearings, clearances etc. can then be used to build
yet another model – the dynamic model. As a linear
representation of the complete gearbox, this can be solved
using standard eigenvalue analysis methods to get a
frequency domain linear dynamic model valid for a specified
torque loading condition. Fortunately all this complexity is
largely hidden from the user, who simply presses a button
and waits a short time.
Combining the TE excitation with this frequency domain
model using the gear whine analysis postprocessing tools
provides the user with results familiar to any noise &
vibration engineer such as vibration waterfall plots and order
cuts due to gear whine excitation. A range of other tools in
the post-processing user interface provide useful information
to help in the identification of problem areas and provide
insight into ways of improving the design to reduce noise
and vibration. These include: modal energy distributions,
modal response contributions, mode shapes, operating
deflection shapes, frequency response functions and bearing
dynamic force predictions.
A key point is that the analyses are solved simultaneously in
this approach. This is because it is not proper to solve the
shaft-bearing system to predict the mesh misalignment and
subsequently use the calculated misalignment to predict the
transmission error at the gear mesh. The character of the
tooth contact and the misalignment are inextricably linked.
Thanks to cleverly optimised algorithms and coding
combined with some one-time-only up-front calculations on
3D FE components, these accurate vibration predictions
across the whole frequency range are available in a matter of
seconds from the moment the engineer presses the noise &
vibration analysis button.
By interrogating the system analysis results in the software,
it is possible to extract the time-varying misalignment and
4
Analysis Results:
Excitations (e.g. transmission
error or user-defined
excitation)
Natural frequencies,
mode shapes
Transfer functions, FRFs
1. SPATIAL
MODEL
2. MODAL
MODEL
Description of drivetrain
structure; mass and stiffness
matrices
Drivetrain described by
‘modal’ mass, stiffness and
damping
Figure 9. Noise and vibration analysis process and results
5
Whole system response
due to excitations
CASE STUDY
This case study focuses on the planetary gear stages of a
typical helicopter main gearbox with a compound planetary
system. In particular this paper will show how the
methodology explained above is applied to design, analyze
and optimize the planetary gear sets to improve the dynamic
behavior of the gearbox.
The tools used are able to analyze the whole helicopter
powertrain (see Figure 10). All the drivetrain components
(bearings, gears, shafts and housings) are modelled
according to the approach and methodology explained in the
previous section.
Figure 11. Main gearbox
The main gearbox is further simplified into the model shown
in Figure 12. It consists only of the compound planetary
system, with two stages of gear planetary sets, the first one
with 4 planets and the second one with 8 planets. This model
is going to be used for an initial check of the different
changes in planetary arrangements. However the design and
dynamic virtual testing of the gearbox will be completed
with the model show in Figure 11.
Figure 10. Full helicopter drivetrain model
The picture above shows the complete model of the
helicopter drivetrain that is used for this case study. As the
main objective is to improve the design of the planetary
systems from a dynamic point of view, the main gearbox is
isolated for more detail analysis.
In Figure 11 this main gearbox is displayed. It contains the
housing and planetary carriers specified as FE components,
while the other components are defined by analytical
solutions (non-linear bearing stiffness model, 6 DOF gear
contact model, etc.)
Figure 12. Simplified model of main gearbox
This simple model is axisymmetric in terms of geometry and
loading conditions and it will be used just as a reference.
The loading conditions applied to this main gearbox are
6600 rpm input speed and 1 MW input power. Then 271 kW
is transferred to the tail rotor while the main rotor takes 729
kW.
6
PLANETARY ARRANGEMENTS –
FACTORISING & NON FACTORIZING
The options when selecting the gear tooth numbers on a
planetary set are at first instance to keep the planets equally
or unequally spaced. For an equally spaced arrangement, if
the planet gears mesh in phase their number of teeth follows
a factorizing arrangement. If the planet gears are out of
phase the number of teeth are set up according to a nonfactorizing arrangement.
A key aspect in the planetary transmission error is how the
transmission errors from the different gear meshes are
combined into a compound total excitation. With the
capability to calculate the total compound transmission
error, the planetary gear set can be considered as a whole
system that can be optimized in order to improve the
dynamic behavior of the gearbox.
Properties of the main gearbox planetary stages
1st stage
In that sense, different parameters of the gear planetary
system can be changed for that objective, one of the most
important is the phase among the planets.
2nd stage
Sun Planet Ring Sun Planet Ring
Number
56
20
96
of teeth
Factorising Mesh
frequency
1458.9
(Hz)
Number
55
21 97
of teeth
Non
Mesh
factorising
frequency
1447.8
(Hz)
When the planet gears are meshing in a planetary system,
they can be in phase (the planets start meshing at the same
time and mesh simultaneously over the mesh cycle) or out of
phase (the planets do not mesh simultaneously at the same
meshing point over the mesh cycle). This phasing property
will have a large influence on the compound total
transmission error coming out of the planetary system as a
dynamic excitation that is transmitted to the rest of the
system.
56
20
96
537.5
55
21
97
523.9
In Figure 13 a factorising planetary gear set is displayed next
to a non-factorising one. As can be seen in the factorising
arrangement the mesh points of the gears are symmetric and
meshing at the same roll angle simultaneously over the
meshing cycle. That is not the case in the non-factorising
arrangement where the gear meshes are not in phase as the
planetary system rotates.
The phase among the planets depends at an initial stage on
the number of teeth of the planetary gear set, so by changing
this parameter it is possible to highly influence the dynamic
behavior of the gearbox.
Factorising (equally spaced)
Non factorising (equally spaced)
Figure 13. Factorizing / non-factorizing planetary arrangements
7
In particular for this case study, the factorizing and nonfactorising arrangements are applied and evaluated in a
virtual testing environment both in terms of total compound
dynamic excitation and gearbox system response.
Also 4 traces of the TE calculated at each gear mesh are
dispayed.
At each gear mesh there is transmission error as the
planetary system rotates. As explained, the TE is a dynamic
excitation generated at the gear meshes. Because this simple
model does not have any additional effects on the TE, the
TE signal includes only the TE harmonics.
Initially the calculation of TE is performed for the simple
model (figure 12) from which the first results and
conclusions are obtained. In Figure 14 a top view of the
second stage of the main gearbox planetary system is
displayed.
Figure 14. 2nd Planetary stage and excitation at each gear
mesh
8
How the TE generated at each gear mesh is combined into a
total compound TE is an important point that heavily
depends on the phase among the planets.
Both factorizing and non-factorising gear arrangements are
integrated into the main gearbox model in order to evaluate
the effect of each of them.
Figure 15 shows for the factorizing arrangement the start and
end of active profile of the planetary gear pairs when
meshing. As can be seen, the sun-planet and the ring-planet
gear meshes are perfectly aligned, which effectively means
that they go through the same meshing point simultaneously
over the meshing cycle. This means that the planets are in
phase, which will determine how the previous periodic TE
signals (figure 14) are going to combine into a total
compound TE in the factorizing case.
(Key for Figure 15 and 16)
Figure 15. Phase among gear meshes of factorizing
planetary arrangement
Figure 16. Phase among gear meshes of non-factorizing
arrangement
On the other hand, Figure 16 shows the planet phasing of the
non-factorising arrangement. The planet meshes are not
meshing at the same roll angle at the same time, so the
planets are out of phase which will also determine how the
individual mesh TE signals are going to combine into a total
compound TE.
9
How the TE from different individual gear meshes combines
into a total compound planetary TE is an important point
that will determine in the cases of factorizing and nonfactorizing arrangements which planetary design is more
suitable to improve the gearbox dynamic behavior. So that, a
more detail analysis in terms of the dynamic excitations
generated in the factorizing and non-factorizing
arrangements is required.
In the non-factorising case (right side of Figure 17), since
the planet gears are out of mesh, the TEs at individual gear
meshes are going to generate for each planet dynamic forces
that will add up in radial direction, while the dynamic
excitations are going to be cancelled out in the output
torsional direction.
The helicopter main gearbox has two planetary stages with
the ring gear fixed and directly connected to the housing and
the carriers are the output of each planetary stage. With this
layout the important dynamic excitations to evaluate are the
resultant radial forces on the ring gear (that are then
transmitted to the housing), and the total compound TE at
the output of the carrier shafts, which is going to be
transferred to the rest of the gearbox system through the
bearings and shaft supports.
Figure 17 shows for the factorizing and non-factorizing
cases how the forces involved in the TE are going to
combine within the planetary system.
Due to the phasing effects, in the case of a factorising
arrangement (left side of Figure 17), the TEs at individual
gear meshes are going to generate for each planet dynamic
forces that will add up the excitations in the output torsional
direction, while the resultanat forces in the radial direction
are going to be cancelled out.
So the planetary gear set can now be evaluated as a system
that generates dynamic excitations that are applied to the
gearbox system. The objective is to evaluate the effects of
the factorising /non-factorising planetary arrangements on
the dynamic behavior of the main gearbox.
Factorizing
Non Factorizing
Sun-Planet Forces
Ring-Planet Forces
Figure 17. TE Forces Factorizing / non factorizing
planetary arrangements
10
Planet Resultant
Forces
Resultant Forces
The total compound TE at the torsional output for the
factorising (left) and non-factorising (right) is shown in
Figure 18.
COMPOUND TRANSMISSION ERROR RESULTS
The factorising and non-factorising arrangements are
evaluated firstly using the simple model shown in figure 12.
Under the assumptions of this simple model, the only
excitation is the transmission error. The resultant dynamic
forces in the ring gear and the total compound TE at the
carrier output are calculated.
As can be seen in the factorizing arrangement the compound
TE in the torsional output is much bigger that the one
obtained from the non-factorizing arrangement. The single
gear mesh TEs are added up in the factorizing arrangement
while they are cancelled out in the non-factorising one (the
FFT of each trace is shown below the trace in question).
.
Non Factorizing
Factorizing
Figure 18. Total compound transmission for factorising/non-factorising cases – simple
model
11
The dynamic radial forces on the ring gear in the factorizing
(left side of Figure 19) and non-factorising (right side of
Figure 19) cases are also calculated.
These radial forces are cancelled out in the factorising
planetary arrangement and are added up in the nonfactorising case according to the results.
The first harmonic of these dynamic radial forces on the ring
gear happens at the planetary mesh frequency (the higher
harmonics happen at multiples of the mesh frequency).
Factorizing
Non Factorizing
Figure 19. Dynamic radial forces on ring gear for factorising (left) /non-factorising (right)
arrangements – simple model
12
Transmission error results from full detail model
The planetary arrangements that have been considered so far
are now evaluated in the real model with all the components
fully defined (the model shown in figure 11 where the
housing is included and the planetary carrier and ring shafts
are fully defined with their complete geometry and
stiffness). This model is used now to calculate the results
under realistic assumptions of geometrical complexity and
loading and boundary conditions. This means that in the
compound planetary system each stage now can deform in a
different way under complex loads (the forces at the bevel
gear mesh are non axisymmetrical, the flexibility of housing
will act as common support of the ring gears ,etc.).
The total compound planetary TE and resultant radial forces
are calculated with this complete model for the factorising
and non-factorising planetary arrangements. The calculation
of these results is now performed for 4 full rotations of the
main rotor so that, according to the gearbox reduction ratios,
the full system rotates a full cycle (final relative positions
between the planetary stages is the same as initial one).
Under these conditions the system deflections of the main
gearbox can be different for each angular position of the
planetary stages as the system rotates, and this also has an
effect on the resultant dynamic excitations.
As can be seen there are several low frequency signals that
make the high frequency signals to fluctuate (fluctuation as
the high frequency signals move up and down, but no
modulation is introduce). The high frequency ones
correspond to the TE at the mesh order and its harmonics.
According to these results the TEs from individual gear
meshes are added up in the torsional output. Indeed
comparing the amplitudes of the TE harmonics to the results
obtained from the previous simple model, it is clear that the
combination of the single mesh TEs occurs in the same way.
The low frequency signals correspond to effects related to
the carrier deflections, ring gear deflections, unequal load
sharing, etc. over the rotation of the planetary system.
Looking more in detail into the main gearbox system
deflections, as can be seen in Figure 21, the input torque
through the bevel gears creates some non-asymmetrical
deflections that are transferred through the planetary
systems. This load in addition to the main and tail rotor
output torques cause the main gearbox to deflect in a nonaxisymmetric way. As the planetary stages rotate they are
under different deflections at each angular position, which
are order of excitations as can be seen in the total compound
planetary TE and resultant radial forces results.
In Figure 20, the total compound planetary TE at carrier
output of the second stage is displayed for the factorizing
arrangement.
Figure 20. Total compound transmission error factorising arrangement – real
model
13
However the high frequency excitations corresponding to the
gear mesh TE are cancelled out among the planet gear
meshes, as this is a non-factorising arrangement.
The dynamic radial forces on the ring gear are also
calculated using the main gearbox model with realistic set
up. Figure 23 shows these results (the charts of trace have
been cut off at certain point, however the FFTs contain all
the information of the full periodic function).
As can be seen the low frequency signals caused by the
gearbox system deflections are also present in these results.
The relative angular positions of the planetary stages as the
whole system rotates is also related to this effect. The local
deformations on the ring gear shafts occur at different
relative positions between the planetary stages as they rotate,
and these local deformations are transferred to the housing
that support both ring gears making them linked and affected
by the deflections of each other.
Figure 21. Gearbox system deflections
In Figure 22 the results of total compound planetary TE of
the non-factorising gear arrangement are displayed.
These results show the same effect related to the low
frequency signals caused by the gearbox system deflections.
As can be seen in figure 23, in the factorising case the radial
forces on the ring gear are almost cancelled out regarding
the mesh order excitation and its harmonics (amplitude of 15
N) compared to the non-factorising gear arrangement where
these forces are added up (amplitude of 350 N at mesh
order) in line with the conclusions obtained so far.
Figure 22. Total compound transmission error non-factorising arrangement – real
model
14
Factorizing arrangement
Non-factorising arrangement
Figure 23. Dynamic radial forces on ring gear for factorising (left) /non-factorising (right) arrangements – real
model
The dynamic excitations calculated for the factorizing and
non-factorising planetary arrangements are transferred to the
gearbox housing and from there to the rest of the helicopter
frame.
However none of the planetary arrangements proposed is
better design by itself in terms of excitations as their
suitability depends on the gearbox system and how sensitive
is to each kind of excitation.
15
Sideband analysis
Looking at the single gear mesh TEs more in detail, the
results are as follows for the second planetary stage in the
non-factorising case (see Figure 24):
Ring -> planet 1
Sun -> planet 1
Figure 24. Single gear mesh TE ring->planet 1 (left) and sun->planet1 (right) of non-factorising arrangement –
real model
As can be seen there are very small sidebands around the
mesh order that are displayed in more detail in Figure 25:
16
Ring -> planet 1
Sun -> planet 1
X= 523.9 Hz
Y= 3.08um
X= 523.9 Hz
Y= 4.60 um
X= 529.3 Hz
Y=0.10 um
X= 529.3 Hz
Y=0.15 um
X= 518.5 Hz
Y= 0.10 um
X= 5.4 Hz
Y= 0.70 um
X= 518.5 Hz
Y= 0.15 um
X= 5.4 Hz
Y= 0.89 um
Figure 25. Sideband details of single gear mesh TE ring->planet 1 (left) and sun->planet1 (right) of non-factorizing
arrangement – real model
The difference in frequency between the upper and lower
sidebands from the mesh frequency is 5.4 Hz respectively.
The frequency of the 2nd stage carrier rotation is 5.4 Hz.
Therefore as the carrier rotates there are some nonaxisymmetric effects that generate an excitation at the carrier
order and sidebands around the mesh order.
As can be seen the unequal load sharing accords with the
non-axisymmetric carrier deflections, so it is possible to
conclude that the sidebands are generated by unequal load
sharing among the planets.
However these sidebands are very small (their amplitude is
30 times lower than the amplitude of the first TE harmonic),
so they are not going to be considered further.
Looking at the carrier deflections in Figure 26 indeed there
are non-axisymmetric deformations in the carrier that the
planets see as the carrier rotates, generating the effects
mentioned above.
These deflections of the carrier are in line with the gearbox
system deflections mentioned above (figure 21).
It is also important to analyze the unequal load sharing
among the planets (Figure 27).
Figure 26. 2nd stage carrier deflections - non-factorizing
arrangement
17
Figure 27. Unequal load sharing results – nonfactorising arrangement real model
The objective, by calculating the dynamic response on these
points, is to evaluate on the one hand the vibration
transmitted to the rest of the helicopter frame through the top
connection and bottom housing supports and on the other
hand evaluate the air borne noise radiated from the housing
as the housing walls vibrate.
SYSTEM DYNAMIC RESPONSE
In order to evaluate how the factorizing and non-factorising
planetary arrangements influence the gearbox dynamic
behavior and obtain further conclusions regarding the
suitability of each planetary arrangement, a more detailed
analysis of the dynamic response is required.
One of the key points is the transfer path of the vibration
from the point of the gearbox system where the TE is
generated (gear meshes) to the point where the response
happens. Deeper analysis regarding how sensitive the
transfer path is to torsional excitations (factorizing) or radial
forces (non factorizing) is performed.
According to the excitation results presented, the factorizing
planetary arrangement generates much higher excitation in
the output torsional direction, while the non-factorising one
generates much higher radial dynamic forces. To evaluate
the effect of this excitations the first harmonic of all single
mesh TEs are applied into the real model shown in figure 11.
Also regarding the vibration resonance at different points a
more detailed analysis is carried out in terms of the
dominant mode influence and the actual vibrational
deflection at those points.
The dynamic response is calculated at the following
locations on the gearbox housing (Figure 28) for the
factorising and non-factorising cases to check the differences
between the responses generated in each case.
Response at the top connection
The response in terms of velocity at the point of the top
connection is shown in Figure 29 for the factorizing
arrangement on the left hand side and for the non-factorising
one on the right hand side. Below these charts there is a
single chart displaying both signals.
The effects of each planetary arrangement on the response
depends in this case on the frequency range. In particular the
highest peak showing resonance in the factorizing case is at
740 Hz while the resonance peak in the non-factorising case
occurs at 3100 Hz.
A more detailed analysis of the dominant modes at those
critical frequencies is done according to the mode shapes
shown in Figures 30 and figure 31.
Top connection
The mode shape at 739 Hz (figure 30), which is the
dominant mode driving the resonance at 740 Hz in the
factorizing case, shows that the vibrational deflection travels
from the planetary systems through the carrier output shafts
into the main rotor shaft. Through the main rotor shaft
excites the top part of the housing where the response point
is located. This is a torsional mode that is more easily
excited in the factorizing case since the total compound TE
at the torsional carrier output is reinforced.
Housing wall
Housing supports
On the other hand the mode shape at 3089 Hz (figure 31),
which is the dominant mode driving the resonance at 3100
Hz in the non-factorizing case, shows a different dynamic
excitation transfer path.
Figure 28. Response nodes
In this case the high response at the top connection of the
housing happens due to the high vibration of the housing
walls around this top section. This mode is more easily
excited in the non-factorising case as the dynamic radial
forces transferred from the ring gear to the housing make the
housing walls vibrate, which in this case makes the top part
of the housing vibrate.
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Response at top connection – factorising/non factorising
Response at top connection - factorizing
740 Hz
3100 Hz
Figure 29. Response at top connection – nonfactorizing
19
Dominant mode at 739 Hz
corresponding to peak at 740 Hz
in the factorising arrangement
Dominant mode at 3089 Hz
corresponding to peak at 3100 Hz in the
non-factorising arrangement
peak at 3100 Hz
Figure 30. Mode shape
at 739 Hz factorising
arrangement
Figure 31. Mode shape at
3089 Hz non-factorising
arrangement
Response at the point located on housing wall
In Figure 32 the response on the housing surface node is
displayed for both factorizing and non-factorising cases. As
can be seen the resonance to the dynamic excitations
depends on the modes that are excited at different
frequencies depending on how sensitive these modes are to
torsional or radial excitations.
20
Figure 32. Dynamic response at location on housing wall – factorizing/non factorizing
Response on housing supports
In this case, the factorizing case shows significantly higher
response than the non-factorising one, so it can be concluded
that, regarding the response at the housing supports, the nonfactorising arrangement is better in term of gearbox dynamic
behavior.
The structure borne path by the 4 housing supports is
addressed in terms of energy (Figure 33).
Figure 33. Dynamic response at housing supports – factorizing/non factorizing
21
The highest peak of vibration or resonance happens at 1660
Hz. In Figure 37 the vibration (operating deflected shape) at
1660 Hz with factorizing planetary arrangement is
displayed.
This transfer path is sensitive to torsional excitations, so the
gearbox system gets excited in this way in the factorizing
case, because in the total torsional output the TE from the
different gear meshes are added up.
Concerning the transfer path that the excitation follows from
the gear meshes to the housing supports, it is possible to see
that the planetary carrier output shafts are highly excited and
this vibration travels to the drive shafts going to the rotors
(main rotor drive shaft going up, tail rotor one going down
in the picture). From the tail rotor drive shaft the vibration is
transmitted to the bottom part of the housing through the
bearings supporting the shaft. This makes the housing
supports located in this bottom region of the housing vibrate,
as shown.
Housing support ODS at highest peak at
1660 Hz in the factorising arrangement
Figure 37. Operating deflected shape at
1660 Hz – factorizing arrangement
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CONCLUSIONS
REFERENCES
The combination of planetary gear tooth numbers set the
phase among the gear meshes and has an important effect in
the gearbox dynamic behavior. By using the right tools it is
possible to predict this effect, which allows the engineer to
take decisions at very early stage in the development process
on the optimum gear design that improves the gearbox
dynamic performance.
Robert H. Badgley, Thomas Chiang, “Investigation of
Gearbox Design Modifications for Reducing Helicopter
Gearbox Noise,” AD 742735 USAAMRDL Technical report
72-6, 1972.
1
Rajendra Singh, Teik C. Lim, “Vibration transmission
through rolling element bearings in geared rotor systems,”
AD-A231 325 NASA contractor report 4334, 1990.
2
The case study presented about the dynamic effects of
factorising and non-factorising planetary arrangements
makes clear that these planetary gear systems do not
improve anything by themselves but just provide different
types of excitations. These possible improvements depend
indeed on the gearbox system sensitivity to the different
kinds of excitations generated, so a system level analysis
that also captures the influences between the different
physics involved is required.
David P. Fleming, “Vibration Transmission Through
Bearings With Application to Gearboxes,” NASA/TM—
2007-214954, 2007.
3
4
Riza Jamaluddin, Brian K. Wilson and Edmund
Stilwell, “Boundary Conditions Affecting Gear Whine of a
Gearbox Housing Acting as a Structural Member,” 2009-012031 Society of Automotive Engineers, 2009.
5
Barry James and Mike Douglas, “Development of a
Gear Whine Model for the Complete Transmission System,”
2002-01-0700 Society of Automotive Engineers, Inc., 2002.
The tools developed by Romax Technology and presented in
this paper provide the right virtual environment to design
and test the different design options at the required system
level (planetary gear sets, housing, etc. and the different
components integrated) and multiphysics level (influence of
system deflections on transmission error excitation, effect of
gear planetary architecture on the helicopter main gearbox
dynamics, etc.)
Barry James, Dr. Owen Harris, “Predicting unequal
planetary load sharing due to manufacturing errors and
system deflections, with validation against test data,” 200201-0699 Society of Automotive Engineers, Inc., 2002.
6
From the analysis with the simple model with no housing
and simple loading conditions, it was possible to conclude
the differences in terms of dynamic excitations between the
factorising and non-factorising planetary arrangements.
However these results did not provide the interaction
between the planetary gear sets and the rest of the system,
which is the key point to analyse.
Then using the same tools another model was defined with
real layout and realistic boundary and loading conditions to
perform virtual testing of the helicopter main gearbox in the
different scenarios of factorising and non-factorising gear
arrangements. This was the right virtual environment to
evaluate how this gear arrangements affect the gearbox
dynamic behaviour since it was possible to understand at a
system level the transfer path of the excitations from the
gear meshes to the response locations on the housing,
providing as well the magnitude of these responses.
Using the tools and methods presented the engineer can
understand the helicopter gearbox behavior and improve it
according to the noise and vibration design criteria and
requirements.
Author contact: Jose Torres jose.torres@romaxtech.com
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