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transmission error is possible by selecting suitable profile modifications. Extensive
research work has been carried out by many researchers on the analysis of errors, dynamic
response, and noise and vibration reduction in single planetary gears. They are using the
various methods of reducing the vibration and noise in planetary gear by changing the
number of teeth and by using the analytical as well as FEM for reducing the vibration and
noise.
Planetary gears are very popular due to their advantages such as high power density,
compactness, and multiple and large compact gear ratios and load sharing among planets.
Gearing arrangement is comprised of four different elements that produce a wide range of
speed ratios in compact layout. These elements are, (1) Sun gear, an externally toothed ring
gear co-axial with the gear train (2) Annulus, an internally toothed ring gear coaxial with
the gear train (3) Planets, externally toothed gears which mesh with the sun and annulus,
and (4) Planet Carrier, a support structure for planets, co-axial with the train. Planetary gear
system as shown in Figure 1 is typically used to perform speed reduction due to several
advantages over conventional parallel shaft gear systems. Planetary gears are also used to
obtain high power density, large reduction in small volume, pure torsional reactions and
multiple shafting. Another advantage of the planetary gearbox arrangement is load
distribution. If the number of planets in the system are more the ability of load shearing is
greater and the higher the torque density. The planetary gearbox arrangement also creates
greater stability due to the even distribution of mass and increased rotational stiffness.
In recent years, enhancement of interior quietness in passenger cars, Automobiles
is an important factor for influencing occupant comfort. Planetary gear sets are essential
components of automatic transmissions because of their compact size and wide gear ratio
range. They produce high speed reductions in compact spaces, greater load sharing, higher
torque to weight ratio, diminished bearing loads and reduced noise and vibration. A Despite
their advantage, the noise induced by the vibration of planetary gear systems remains a key
concern. Planetary gears have received considerably less research attention than single
mesh gear pairs. This paper focus on the study of two PGTs with different phasing (angular
positions) while keeping every individual set unchanged.
2
Figure 1: Basic layout of planetary gear box
This figure shows that the basic layout planetary gear train in which there is one
Sun gear, Three Planet gear and one ring gear. They can produce the high speed reduction
in compact space and having greater load shearing capacity & high torque to weight ratio.
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1.1 Problem Statement
The vibration occurs due to the improper meshing between the gear and continuous
wear of gear that occurs because of continuous machining operation & Defects in
manufacturing. Due to the vibrations in the system noise is generated and also it impacts
on the system performance.
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1.2 Objectives
The main objective of project is to reduce the vibration and noise in the planetary
gear train.
The objectives are as follows
1. Development of experimental setup for reduce the Vibration and Noise in planetary gear
train.
2. Measure the vibration with single PGT arrangement.
3. Measure the vibration & Noise in PGT with phasing arrangement.
4. Measure the vibration & Noise in PGT without phasing arrangement.
5. Comparison of results between phasing and without phasing arrangement.
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1.3 Scope
Vibration and noise generated in the planetary gear train is reduced by phasing
arrangement between two planetary gear sets. The phasing angle which is obtained from
the number of teeth is provided between two gear sets for measuring the vibration and noise
in gear train. The phasing angle to be used in planetary gear train should be as per calculated
from number of teeth.
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1.4 Methodology
After developing the experimental setup the vibrations and noise are measured by using
FFT analyzer and sound measuring instrument. In the first part the vibration and noise are
measured by without phasing arrangement of planetary gears after that the vibration and
noise are measured by phasing arrangement between the planetary gear pair. After taking
the results of both compare the results of noise and vibration with and without phasing
arrangement.
1.4.1 Method of Phasing Gears
To control the vibrations in tooth gearings effectively, one should have an adequate
knowledge of the physical nature of what causes vibrations in planetary gear pair with
imprecise and deformed teeth. Vibrations in gearing is caused by an internal excitations, as
it occurs at the contact of two compressed elastic bodies (teeth) during their relative motion
and acts on both bodies with the same intensity but in opposite directions.
Because the variation of tooth mesh stiffness during meshing as a principal source
of internal excitation force and vibration, modifications of the optimal tooth shape and
contact ratio (CR) have been studied as ways of reducing the variation in mesh
stiffness. Major variations in stiffness are caused by changes in meshing pair numbers,
usually in the range 1.0-2.0 for normal spur gears. It is impossible to avoid this variation
due to the integer numbers of gear teeth.
If another meshed and phased gear pair is added to reverse the stiffness functions
of the two pairs, these phasing gears will complement the primary gears and reduce the
mesh stiffness variation. The phasing gear pair is made up of two gears half the width and
half the pitch phasing of the primary gears. The conceptual model of phasing gears is shown
in Figure 2.
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Figure 2: Conceptual model of phasing of gear pair
In this gear pair the angle is provided between the two teeth is depend on the number
of teeth, is the number of teeth varies the angle between the two gears will be change. This
figure shows the inclination of one gear pair by keeping another gear fixed.
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1.5 Organization of the Study
Organization of this Study includes:
1) 1st chapter is related to the introduction of planetary gear in which the advantages of PGT
are discussed and collecting the problems found in gear pair and also includes the Problem
statement, objectives, Scope, Methodology.
2) 2nd chapter is related to the Literature Survey.
3) 3rd chapter is related to the Experimental Validation in this chapter the information of
experimental setup is given.
4) 4th chapter is related to Results and discussion.
5) 5th chapter is related to the conclusion and future scope.
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CHAPTER 2
LITERATURE REVIEW
R.G.Parkar et al. (2000) studied the dynamic response of planetary gear was of the
fundamental importance in helicopters, automotive transmissions, aircraft engines, and a
variety of industrial machinery. The complex, dynamic forces at the sun, planet and ring
planet meshes were the source of the vibration. Modeling of the dynamic tooth forces
remains an important issue that has not been resolved even for single-mesh gear pairs. The
multiple meshes of planetary gears further complicate the dynamic modeling.
Consequently, dynamic analyses of planetary gears were less developed than for other
single-mesh gear configurations. In particular, experimental frication of the existing
analytical models was especially limited.
As a result, design options to minimize noise and loads in planetary gears have
developed empirically without strong analytical or experimental foundation. Their design
strategies include tooth-shape medications, gear geometry adjustments (pitch, contact ratio,
etc.), reduction of manufacturing tolerances, use of sun, ring, or carrier components, and
vibration isolation concepts. A particular strategy was the use of planet phasing, where the
planet configuration and tooth numbers were chosen such that self-equilibration of the
mesh forces reduces the net forces and torques on the sun, ring, and carrier, thereby
reducing vibration. Figure 3 shows the schematic layout of planetary gear in which the
mesh force was acting on the planet gear. (5)
Figure 3: Planetary gear schematic. Fi denotes the mesh force at the
i th sun
planet mesh
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C.Gill-Jeong et al. (2010) studied that gearing assembly is one of the major vibration
source in power transmission system especially used in automotive, aerospace, marine and
industrial applications. Their study presents a novel means of reducing gear vibrations
using a simple 1:1 ratio spur gear pair using a method of phasing. Variation in the gear
mesh stiffness over a mesh cycle which depends on the number of pairs of teeth in contact
was one of the principal causes of vibrations and instabilities and has a strong influence on
the overall dynamics of the geared system.
Their method was based on reducing the variation in gear mesh stiffness by adding
another pair of gears with phasing. Because of added phasing gear, the numbers of pairs of
teeth in contacts were increased which reduces the variation in mesh stiffness. A simple
spur gear model with rectangular-wave-type mesh stiffness was assumed and mesh
stiffness variation was obtained numerically using MATLAB 7.5 software and was
comparable in both cases i.e. normal and phasing gears.
Their numerical result of analysis shows the reduction in mesh stiffness variation
and the possibility of reduction in vibration in simple spur gear pair using the proposed
method. (6)
Majid Mehrabi, Dr. V. P. Singh (2013) works on planetary gearboxes were usually used
in a wide variety of machinery such as automobiles, helicopters and aircraft engines .Their
numerous advantages are high speed reductions in compact spaces, high torque/weight
ratio, greater load sharing, diminished bearing loads and reduced noise and vibration. A
typical simple planetary gear set consists of a sun gear, a ring gear and a number of identical
planet gears (typically 36) meshing both with the sun and ring gears.
They were well known for their symmetrical structure which allows an equal share
of the total external torque applied between the planetary gears, the sun and the ring.
However non stationary conditions of system such as overload conditions, torque
fluctuation may affect the dynamic behavior of a planetary gear transmission.
The inequality of the load distribution however arises on each planet gear because
of random errors of manufacture, assembly and operating conditions. Their results in noise
and vibration which were key concerns in their applications and drop in efficiency of
planetary gear system. In some helicopters planetary gear vibration was the primary source
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of cabin noise that can exceed 100dB. Before 1990, their literature on analytical planetary
Gear dynamics was scare. They studied the Eigen value problem for a thirteen degree of
freedom system and identified the natural frequencies and vibration modes. (21)
A. Kahraman, R. Singh (1990) Discussed on the dynamic response of gears which is
related to the noise generation and dynamic loads. Prior studies have yielded a vast
literature on their topic and, in particular, a remarkable variety of mathematical models as
discussed in reference. More recent studies were cited in their comprehensive bibliography
in reference. Most models were use a discrete (lumped parameter) representation involving
rigid gear components and combinations of discrete elastic and dissipative elements to
represent the meshing teeth and support/bearing stiffness. Such models have varying
complexity in their treatment of the tooth mesh, shaft, bearing, and housing modeling. In
essence, the required analytical modelling to capture the complex gear dynamic response
has not been established.
Even when attention was restricted to modelling the tooth mesh, a variety of
possible representations exist, and the optimal treatment of time-varying mesh stiffness,
contact loss, use of static transmission error as a dynamic input, frictional effects, etc.,
remains unsettled. Their study analytically investigated the dynamics of a spur gear pair
for which comprehensive experimental data exist.
Their tooth mesh was the most complex aspect in gear dynamics, and the gear
system in that work was selected to isolate tooth mesh effects. Their primary analytical tool
was infinite elements/contact mechanics (FE/CM) formulation that offers significant
advantages in its representation of the crucial tooth contact. Their purpose was to further
expose the basic non-linear and time-varying phenomena at play in the tooth mesh,
demonstrate the modelling fidelity and advantages of the FEM method used, and compare
the ability of two s.d.o.f models to represent the experimentally observed phenomena.
They studied the gear pair that used in a series of experiments by Kahraman and
Blankenship. Tests on their system were initially reported in reference, where the details of
the system were given. Their test stand was designed to isolate the impact of tooth mesh
interactions on the dynamic response and exclude complications from the shafts, bearings,
and housing. In particular, the bearing and shaft configuration was such that the support
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structure was nearly rigid and the response was purely gear rotation. The test gears were
dynamically isolated from the slave gears in the back-to-back configuration. Despite their
reduction to the simplest case of s.d.o.f response, measurements of dynamic transmission
error show distinct, repeatable, non-linear, time varying system response in the form of
classical jump phenomena, sub- and super harmonic resonances, parametric instabilities,
and even apparently chaotic response. The non-linear tooth mesh forces causing these
complex behaviors were what they seek to model in their study.
A primary motivation was to establish the ability of the unique FE/CM formulation
to capture complex gear mesh forces in dynamics simulations. Similar analysis tools with
their advantages presented in what follows were not known to the authors. Conventional
infinite element analysis, and even the currently available commercial software, require
prohibitively redefined meshes to represent the tooth contact and precise tooth surface
description needed for gear mechanics, particularly when one seeks to go beyond static
analyses to dynamic response analyses. Their subject gear system was selected to validate
the FE/CM approach as a research tool because the complex, non-linear behavior was
suitably demanding benchmark, and carefully conducted, high-quality experiments exist.
The finite element formulation was unique in its combination of detailed contact
modelling between the elastic teeth with a combined surface integral finite element solution
especially capture tooth deformations and loads with a relatively coarse mesh. Details were
available in the references and a short description of the surface integral finite element
solution was given in reference. The contact analysis was briefly described there. They
mesh for the gear pair in their study was shown. Each of the gears undergoes large rotation
according to a prescribed, kinematic trajectory. In that two-gear case, there trajectory was
that of conjugate action of the gears at specified operating speed.
The elastic gear motions that superpose on their prescribed trajectory are small. If
the infinite element displacement vector x fit for a particular gear i was measured with
respect to a reference frame that follows that known trajectory, then it was possible to
represent its behavior by a linear system of equations. The contact analysis was briefly
described there. The mesh for the gear pair in their study was shown. Each of the gears
undergoes large rotation according to a prescribed, kinematic trajectory.
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In that two-gear case, the trajectory was that of conjugate action of the gears at
specified operating speed. (13)
Figure 4 shows the single degree of freedom system having equal in radius and
moment of inertia of two different gears which was connected by spring and damper in
between two gears which was shown in figure 4.
Figure 4: Single- degree-of-freedom modelling of the two gear system
R.G.Parker, X. Wu (2002) they use a planetary gear at particular strategy to reduce
vibration by using planet phasing, where the planet configuration and tooth numbers were
chosen such that self-equilibration of the mesh forces reduces the net forces and torques on
the sun, ring, and carrier, thereby reducing vibration. Their idea was proposed by Schlegel
and Mard where experimental measurements on a spur gear system demonstrated a noise
reduction of 11 dB Seager gave a more detailed analysis using a static transmission error
model of the dynamic excitation. Palmer and Fuehrer also demonstrate the effectiveness of
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Planet phasing and support their arguments with limited experiments. Kahraman and
Kahraman and Blankenship studied the use of planet phasing in the context of helical
planetary systems.
Their work use static transmission error to represent the dynamic excitation in a
lumped parameter dynamic model. Their work examines the analytical basis for planet
phasing in spur planetary systems. The results were developed in terms of the physical
mesh forces and were not tied to any lumped parameter model. In fact, they make no
attempt to characterize the factors affecting the mesh forces or quantify their magnitudes.
The fundamental issue was that the inherent symmetries of planetary gears imply distinct
relationships between the dynamic forces at the individual meshes. Their approach was
more appealing to physical intuition and makes no supposition about the use of static
transmission error to model the dynamic excitation.
The symmetries lead naturally to specific conclusions for the suppression of
particular harmonics of mesh frequency in the net forces and torques on the sun, ring, and
carrier. Simple rules with clear design application emerge to suppress expected resonances
that occur when the mesh frequency M or one of its harmonics coincides with a system
natural frequency. Systems with equal planet spacing were examined in detail, although
the methods adapt easily to systems with unequal spaced, diametrically opposed planets,
which was the basis for the work presented there. The unique structure of planetary gear
vibration modes was essential in what follows. Planetary gears with equal sun planet mesh
stiffness at each mesh, equal ring planet mesh stiffness at each mesh, and equal planet
inertia properties have exactly three types of modes for systems with equally spaced or
diametrically opposed planets: rotational, translational, and planet modes. When the elastic
deformation of the ring gear was included that same mode types persist and a fourth
category of purely ring modes was added. In order to maximize the power density and
improve load sharing among the planets, planetary gears in numerous industries were
designed to have thin ring gears, and this leads to elastic deflection of the ring gear.
Most works model the ring gear as a rigid body. Wu and Parker established an elastic
discrete model that includes planetary gear discrete degrees of freedom (rotational and
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translational) and ring gear elastic deflection. That model was adopted in, which was the
basis for the material in that paper.
Gear vibration was driven by changing mesh stiffness as the number of teeth in
contact changes. That was often modelled as time varying parametric excitation, which was
closely related to modelling based on static transmission error as an external "right hand
side" driving force. Parametric instabilities were demonstrated clearly in experiments on a
spur gear pair Measurements showed large resonant vibration when the mesh frequency
was twice the natural frequency, which was a classical signature of parametric instability.
The amplitudes of vibration became sufficiently large as a result of parametric instability
that nonlinear phenomena such as tooth contact loss, period- doubling, and chaos were also
observed. Mathematical modelling with parametric excitation from varying mesh stiffness
as the driving excitation source agreed well with the experiments, including for speed
ranges where mesh frequency nearly equals a natural frequency, which was the most widely
studied gear resonance condition Parametric instability in single - pair gears has been
investigated in . Only a few studies exist on parametric instabilities of multiple mesh gear
systems. Tordion and Gauvin and Benton and Seireg analyzed the instabilities of two stage
gear systems but with contradictory conclusions. That was clarified by Lin and Parker, who
derived formulae that allow designers to suppress particular instabilities by choice of
contact ratios and mesh phasing. Liu and Parker analytically investigated the nonlinear
resonant vibration of idler gears parametrically excited by mesh stiffness variation.
The impact of mesh stiffness variation on tooth loads and load sharing in planetary
gears was studied by August and Kasuba and Velex and Flamand. They numerically
computed the dynamic response of planetary gears with three sequentially phased meshes
and found the impact of mesh stiffness variations on dynamic response was significant. Lin
and Parker analytically investigated the parametric instability of planetary gears using a
purely rotational model, and Bahk and Parker extend that to examine the nonlinear
dynamics. All of their works adopt a rigid ring model. Their work which was from,
examines planetary gear parametric instability using a model that includes the translational
vibration of all components and the elastic deformation of the ring gear. With the modal
expressions of the elastic-discrete model from, the instability boundaries were
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obtained as simple expressions. They show that many modes cannot interact to create
combination instabilities, and general instability existence rules were obtained for
equally spaced planets. By adjusting the tooth numbers, contact ratios, and mesh phases,
one can minimize or completely suppress much potential instability. (11)
Kahraman C. Yuksel (2004) studied on Planetary gear sets, also known as epicyclic
gear drives, were commonly used in a large number of automotive, aerospace and industrial
applications. They possess numerous advantages over parallel-axis gear trains including
compactness of design, availability of multiple speed reduction ratios, and less demanding
bearing requirements. Most common examples of planetary gear sets can be found in
automatic transmissions, gas turbines, jet engines, and helicopter drive trains. A typical
simple planetary gear set consists of a sun gear, a ring gear and a number of identical planet
gears (typically 36) meshing both with the sun and ring gears.
A common carrier holds the planets in place. Dynamic analysis of planetary gears
was essential for eliminating noise and vibration problems of the products they are used
in. The dynamic forces at the sun- planet and ring-planet meshes were the main sources of
such problems. Although planetary gear sets have generally more favorable noise and
vibration characteristics compared to parallel-axis gear systems, planetary gear set noise
still remains to be a major problem. The dynamic gear mesh loads that were much larger
than the static loads were transmitted to the supporting structures, in most cases,
increasing gear noise. Larger dynamic loads also shorten the fatigue life of the
components of the planetary gear set including gears and bearings.
Surface wear was considered one of the major failure modes in gear systems. In
case of planetary gear sets, experimental data has shown that especially the sun gear meshes
might experience significant surface wear when run under typical operating conditions.
While wear was a function of a large number of parameters, sliding distance and contact
pressure were shown to be most significant parameters influencing gear wear. Wear of
tooth profiles results in a unique surface geometry that alters the gear mesh excitations in
the form of kinematic motion errors, enhancing the dynamic effects. Modelling of planetary
gear set dynamics received significant attention for the last 30 years. A number of studies
proposed lumped-parameter models to predict free and forced vibration characteristics of
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planetary gear sets. Their models assumed rigid gear wheels, connected to each other by
springs representing the flexibility of the meshing teeth. In their studies, nonlinear effects
due to gear backlash and time-varying parameters due to gear mesh stiffness fluctuations
were neglected. The corresponding Eigen value solution of the linear equations of motion
resulted in natural modes. Modal summation technique was typically used to find the forced
response due to external gear mesh displacement excitations defined to represent motion
transmission errors. That lumped-parameter model varies in degrees of freedom included,
from purely torsional models to two or three-dimensional transverse-torsional models.
While that model served well in describing the dynamic behavior of planetary gear
sets qualitatively, they lacked certain critical features. First, the gear mesh models were
quite simplistic with a critical assumption that complex gear mesh contact interaction can
be represented by a simple model formed by a linear spring and a damper. That models
demand that the values of the gear mesh stiffness and damping, as well as the kinematic
motion transmission error excitation, must be known in advance.
It was also assumed that these parameter values determined quasi-statically remain
unchanged under dynamic conditions. In addition, gear rim deflections and Hertzian
contact deformations were also neglected. Another group of recent models used more
sophisticated finite element-based gear contact mechanics models. There computational
models address all of the shortcomings of the lumped-parameter models since the gear
mesh conditions were modelled as individual nonlinear contact problems. The need for
externally defined gear mesh parameters was eliminated with their models. In addition,
rim deflection and spline support conditions were modelled accurately.
Their models were also capable of including the influence of the tooth profile
variations in the form of intentional profile modifications, manufacturing errors or wear on
the dynamic behaviour of the system. 696 C. Yuksel, A. Kahraman the study of wear of
gear contact was becoming one of the emerging areas in gear technology. A number of
recent gear wear modelling efforts form a solid foundation for more accurate, larger
system analyses. All of these models use Archard's wear model in conjunction with a
gear contact model and relative sliding calculations.
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Their studies focused on prediction of wear of either spur or helical gear
pairs in a parallel-axis configuration. The tooth contact pressures were computed in
that models using either simplified Hertzian contact or boundary element formulations
under quasi-static conditions. Sliding distance calculations were carried out
kinematically by using the involutes geometry and Archard's wear model was used with
an empirical wear coefficient to compute the surface wear depth distribution.
A number of studies investigated the influence of wear on gear dynamics response.
Among them, Kuang and Lin simulated the tooth profile wear process, and predicted the
variations of the dynamic loads and the corresponding frequency spectra as a function of
wear for a single spur gear pair. Wojnarowski and Onishchenko performed analytical and
experimental investigations of the influence of the tooth deformation and wear on spur gear
dynamics. They stated that the change in the profiles of the teeth due to wear must be taken
into account when dealing with the durability of the gear transmissions as well.
Their previous models considered surface wear effects for only a single spur gear
pair, voiding multi-mesh gear systems such as the planetary gear sets. They focused on
only external gears and used lumped-parameter dynamic models excluding nonlinear and
time-varying effects. (4)
Yichao Guo, (2011) discussed on planetary gears were widely used in all kinds of
transmission systems, such as wind turbines, aircraft engines, automobiles, and machine
tools and they were classified into two categories: simple and compound planetary gear.
Simple planetary gears have one sun, one ring, one carrier, and one planet set (i.e., single
stage). There was only one planet in each planet train
Compound planetary gears involve one or more of the following three types of
structures: meshed-planet (there are at least two more planets in mesh with each other in
each planet train), stepped-planet (there exists a shaft connection between two planets in
each planet train), and multi-stage structures (that system contains two or more planet
sets). Compared to simple planetary gears, compound planetary gears have the advantages
of larger reduction ratio, higher torque-to-weight ratio, and more flexible configurations.
In spite of their advantages, vibration remains a major concern in planetary gear
applications. Vibration creates undesirable noise, reduces fatigue life of the whole system,
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and decreases durability and reliability. Vibration reduction was key to the applications of
compound planetary gears. They require analytical study on compound planetary gear
dynamics to provide fundamental understanding of the dynamics and guide vibration
reduction.
Most research on gear dynamics focuses on single gear pairs or multi-mesh gear
systems. Recently, considerable progress has been made in the modeling and analysis of
simple planetary gear. Studies on compound planetary gears, however, were limited. Many
fundamental analyses that were proved to be essential in other systems and studies have
not been performed, including the purely rotational system modeling and the associated
modal properties, the impact of system parameter changes on natural frequencies and
vibration modes (Eigen sensitivity analysis), the natural frequency veering and crossing
patterns, the classification of mesh phase relations, the suppression of selected dynamic
responses through mesh phasing, and the parametric instability caused by mesh stiffness
variations were not performed. (11) Their study aims at their research gaps and the main
objectives were:
-
To develop a purely rotational model for general compound planetary gears that
can clarify the confusion in previous rotational planetary gear models and
analytically prove the associated modal properties.
-
To perform an Eigen sensitivity analysis based on Kiracofe and Parker's rotational-
translational model and derives the Eigen sensitivities in compact formulae.
-
To inspect the natural frequency veering/crossing phenomena and identify any
patterns or general rules.
-
To find a way to analytically describe and calculate all the relative mesh phases in a
compound planetary gear.
-
To investigate the existence of mesh phasing rules for deferent compound planetary
Gear models that can suppress vibration.
-
To study the parametric instability caused by mesh stiffness variations and to
Analytically determine the boundaries for instability regions.
-
To examine the back-side mesh stiffness and to quantify the impact of backlash on
the back-side mesh stiffness.
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A. Palermo et al. (2010) studied on gears which were extensively employed in
mechanical systems since they allow the transfer of motion in a wide range of working
conditions, with a variety of gear ratios, and at reasonable production costs. The gear
meshing was a complex process because it involves moving and multiple contact points,
variable load sharing on the meshing teeth, contact mechanics (which is nonlinear), and all
of them from a dynamic standpoint. Furthermore tooth micro geometry, manufacturing
imperfections and assembly errors have relevant effects on the behavior of gear systems
And cannot be ignored. That complexity has to be faced in the design phase,
which must address both endurance and noise requirements. For that reasons, a
considerable amount of research works on gear dynamics was available in literature, but
still many aspects remain unresolved. Moreover, today's markets were highly
competitive and therefore reaching valid solutions in shorter timeframes represents a
clear advantage. In that context, numerical models and simulations allow to achieve
solutions to improve the dynamic behavior of gear systems, and to limit the testing
phase saving time and money. That explains why efforts continue to be spent in the
gear dynamics research field, with applications especially in helicopter, wind turbine
and automotive industries.
Gear dynamics were mentioned because, at the occurrence, the proposed
methodology enables to evaluate the dynamic meshing loads, which in transients can be
several times higher than the static ones. From that perspective, the proposed technique will
also allow the simulation of load sharing in planetary gear trains, which was currently a
major issue for that kind of transmissions. Coming back to gear noise purposes, the
proposed methodology takes into account the dynamic transmission error (DTE), which
was defined for a gear pair as the dynamic relative displacement between meshing teeth.
The transmission error was widely regarded as one of the main causes of gear
noise. According to Munro, the transmission error was defined first by Harris in 1958,
which started its analytical investigation. Nevertheless references prove that their
concept was already applied before, but using an empirical approach. The two main
factors affecting the TE were the mesh stiffness, which accounts for tooth flexibility and
number of meshing tooth pairs, and the tooth micro geometry, in terms of intentional
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modifications and manufacturing errors. Variations in the TE, during the gear meshing,
trigger vibrations and then airborne noise. Their analysis will be focused on the case of
involutes parallel spur and helical gears, which were the most common ones.
Several methodologies were available in literature to simulate and estimate
meshing vibrations, using analytical lumped parameter, Finite Element (FE), and multi
body approaches. The analytical model proposed by Umezawa, later corrected by Cai
to consider the influence on tooth stiffness of the gear tooth number, describes the
gear meshing with a single degree of freedom (SDOF) system aligned along the line of
action. Assuming a time-varying function for the mesh stiffness, defined within one mesh
period (or, a dimensionally, within one mesh cycle), and the damping, the equation of
motion can be solved. Their models allow considering the effects of tooth micro
geometry, assembly and manufacturing errors, lumping them on the line of action with a
displacement-driven excitation for the SDOF system.
With that assumptions it was impossible to consider the three-dimensionality of
the contact problem, and the quality of the results that can be obtained depends on how
realistic were the mesh stiffness function for the given gear pair and the displacement
excitation. Different analytical models improved the accuracy of the results using a FE
model to take into account shaft deflections and three-dimensional geometry, but the
lumped parameters description was still suitable to analyze simple cases. Full FE models
allow more accurate representations of gear systems and avoid a-priori assumptions on
the TE, but since tooth contact happens in a very small area, and it spans the teeth from
root to tip, highly refined mesh or contact detection followed by re meshing was needed
along the whole tooth face. That causes high computational costs to run a simulation.
Moreover, it was also an issue to correctly describe the tooth three-dimensional micro
geometry. In the FE field, an interesting technique was proposed by Parker et al.
They use a semi-analytical finite element formulation specifically devised for
contact problems. The tooth was divided in a contact zone (extending beneath the tooth
surface) and a FE zone, separated by a matching interface. The contact zone was
analytically solved by means of the Boussinesq's solution. That solution was evaluated at
the matching FE nodes and the obtained nodal parameters were used to solve the remaining
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FE part. However, FEM techniques were not able to consider with good computational
efficiencies nonlinear entities such as gear lashes, assembly clearances, bearings, clutches,
and other nonlinear effects which arise from large angle rotations. Such nonlinear effects
require time-domain integration, which was typical of the multi body environment.
Moreover, to assess the noise and vibration performances of a geared system it was usually
desirable to test a wide range of working conditions (e.g., torques, regime run-up,). With
such demands, the use of large scale finite element models in time domain becomes
computationally expensive and maybe impractical.
The technique proposed in that paper was an improvement of the Static Transmission
Error method described by Morgan et al. The basic idea of the method was to let
specialized, thus highly efficient, software for gear contact analysis (abbreviated GCAS
from now on) execute the calculation of the static mesh stiffness, which can then be used
in the dynamic multi body simulation. The gear meshing, in the multi body
simulation, was in fact governed by the dynamic equilibrium of the contact forces applied
to the gears, rather than the ideal kinematic contact ratio. Considering a single spur or
helical gear pair described by the standard gear parameters, the static mesh stiffness can
be calculated using GCAS which enables to take into account three-dimensional teeth
micro geometric modifications and manufacturing errors, teeth global and contact stiffness,
shaft deflections and assembly misalignments. That mesh stiffness was obtained, for one
static working condition, as a function of the position along the mesh cycle.
Once the static mesh stiffness was imported in the multi body software, the contact
forces were calculated and applied to the gears by a user-defined force element which
reads the instantaneous value of the mesh stiffness based on the actual position along the
mesh cycle. In this way, the meshing complexity is captured avoiding the high
computational cost related to the full-scale model, since the multi body gears and shafts
models were rigid (while the bearings are compliant). The improvement brought by the
current work was the capability to import and use the static mesh stiffness as a function
of the instantaneous values of torque. In that paper, first the multi body model adopted
for the gear system is described, and then the Static Transmission Error (STE) and the
Dynamic Transmission Error (DTE) were defined. Subsequently, the static mesh stiffness
23
sensitivity to the main assembly errors was evaluated, in order to identify the most influent
ones. A description of the new technique to consider the variable torque follows. Finally, the
obtained results were discussed and compared to the ones obtained with the previous
technique and the static GCAS values. (15)
A Al-Shyyab et al. (2009) discussed on planetary gear sets which were widely used in
many applications including automotive transmission, rotorcraft, wind and gas turbine
gearboxes, as well as other marine and industrial power transmission systems. Planetary
gear trains have many advantages over fixed center counter-shaft gear systems. The flow
of power via multiple-gear meshes increases the power density, helping to reduce the
overall size of the gearbox. The axisymmetric orientation of the planet gears reduces the
radial bearings loads and in many cases allowing its central members (sun gear, ring gear,
or the planet carrier) to float radially. That was reduces the effect of gear and carrier
Manufacturing errors on planet load sharing. Finally, the ability of multi-stage
planetary sets in providing multiple speed reduction (gear) ratios has been the main reason
for their extensive use for automatic transmission applications. Compound planetary
trains were obtained from number of single-stage planetary gear sets whose central
members were connected according to a given power flow configuration. Input, output,
and fixed (stationary) member assignments were made to certain central members to
achieve a given gear ratio.
Most of the planetary gear train dynamic models were limited to single-stage
planetary gear sets. Early models were of linear time-invariant type (no backlash and
constant mesh stiffness) where the Eigen solutions and model summation techniques were
used to predict the natural modes and the forced response. That model were extended
to study the neutralization or cancellation of excitations at each gear mesh through
proper phasing of the gear meshing by specifying the planet position angles and numbers
of teeth of gears. A schematic of a three-stage segment of a stage compound gear train
was considered. A particular stage compresses three central elements, the sun gear, the
ring gear and carrier as well as number of planets. The planets of each stage were free to
rotate with respect to their common carrier, while the central elements of any stage were
candidates for being an input, output, or reaction member. Multiple stages can be
24
connected in different ways via a proper coupling central member of stage
n was
connected to a central member of stage- through a torsional spring) representing a
permanent or clutch connection.
An example set of permanent or clutch stage couplings were shown in solid lines,
while the model allows any user defined set of couplings. The torsional dynamic model
of the planetary gear set of stage-
n was shown. That model was similar to the one
proposed by Al-shyyab and Kahraman with the exception of torsional springs of
stiffnesses added there to represent coupling of the central member
j of stage-n with the
central member of stage-
m. That was discrete model employs a number of simplifying
assumptions as discussed in detail in reference.
The central elements were constrained by torsional linear springs of stiffnesses
respectively. A particular central member was held stationary by assigning a very large
constraint stiffness value to it. Likewise, a zero value for the constraint stiffness indicates
that was central member was not connected to the housing. The magnitudes of the stage
coupling stiffness's chosen to represent the torsional stiffness of the actual (24)
R. G. Parker et al. (2000) studied the dynamic response of a helicopter
planetary gear system was examined over a wide range of operating speeds and torques. The
analysis tool was a unique, semi analytical finite element formulation that admits
precise representation of the tooth geometry and contact forces that were crucial in gear
dynamics. Importantly, no a priori specification of static transmission error excitation
or mesh frequency variation was required; the dynamic contact forces were evaluated
internally at each time step. The calculated response shows classical resonances when a
harmonic of mesh frequency coincides with a natural frequency. However, particular
behavior occurs where resonances expected to be excited at a given speed were absent.
That absence of particular modes was explained by analytical relationships that depend
on the planetary configuration and mesh frequency harmonic.
The torque sensitivity of the dynamic response was examined and compared to
static analyses. Rotational mode response was shown to be more sensitive to input torque
than translational mode response.
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- , 2016, VIBRATION AND NOISE REDUCTION IN PLANETARY GEAR TRAIN BY PHASING, Hamburg, Bedey Media GmbH, https://www.anchor-publishing.com/document/317651
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