Chapter 1
2
low and high frequencies, changing with the age and the physical conditions of each
individual.
The basic reason why any system vibrates, producing noise, is that it is excited and, in
the case of a gear system, the most important excitation is transmission error.
Transmission error (TE) is defined as difference between the actual position of the
driven gear and its theoretical one in case of prefect transmission and it is equal to:
1
21
2
z
TE
z
=θ− θ (1.1)
where θ
1
and θ
2
are respectively the angular position of the driving and driven gear,
while z
1
and z
2
are the number of teeth of the two wheels. Usually, TE is expressed in
seconds of degree or, multiplying by the base radius of the driven gear, in microns.
More in detail, transmission error can be classified as static transmission error (STE)
and dynamic transmission error (DTE).
Figure 1.1 Gear noise generation process from transmission error.
Pitch errors
Transmission Error
(TE)
Thermal
distortions
Gears bodies
deformations
Teeth
deformations
Shafts
misalignments
Profile errors
Helix and
lead errors
Gears
runouts
Supports vibrations
Gearbox vibrations
Noise
Introduction
3
The former mainly depends on teeth deformations under load, misalignments and
profile errors, the latter is also influenced by the dynamics of the whole mechanical
system. It can be reasonably said that static transmission error is a little vibration,
originating at the gear mesh, pushing the driven gear backwards and forwards. This
little imposed displacement, responsible of impacts at the beginning of the gear mesh
and vibrations of the gear blanks, can be amplified by the dynamics of the system and
it can turn into a greater vibration called dynamic transmission error. This vibration,
originating at the gear mesh and amplified by the dynamics of the gear transmission,
then propagates through shafts and supports, reaching the gearbox walls that radiate
them outwards as loudspeakers, generating noise.
Figure 1.2 Transmission path of gear noise and vibrations.
On one hand, tooth modifications, such as profile or lead modifications, represent the
commonest solution to the reduction of static transmission error; on the other hand, a
static analysis is not always sufficient to reduce gear noise and it should be
Ceiling
Wall
Floor
Human
ear
Airborne noise
Structure borne noise
Chapter 1
4
accompanied by an accurate study of the dynamics of the whole gear transmission: all
efforts done to reduce static transmission error must not be nullified by its possible
dynamic amplifications occurring in the mechanical system.
In this study, the influence of profile modifications on transmission error and noise of
spur gears is investigated. To this aim, a test facility is setup in order to measure
transmission error and noise of parallel axis gear sets. Both static models, to predict
STE in function of gears macro-geometry and profile modifications, and dynamic
models, to predict mesh forces and DTE, are developed; the experimental validation
of these models is performed through the comparison between theoretical and
experimental results, obtained testing different gear sets on the same test facility
previously considered. Finally, further experimental tests are conducted to find an
experimental correlation between gear noise and dynamic mesh forces.
1.2 Literary Review
1.2.1 Review of the Test Facilities for Gear Noise Studies
The test facilities developed to study gear noise and vibrations can be grouped into
two main classes. The first class includes the power recirculation test rigs. This
design solution consists of a test gearbox connected to a slave one having the same
gear ratio: the isolation of the tested gear set is performed by long and slender shafts.
Typically a small motor is employed, since only power losses of bearings and gears
have to be reintroduced in the system; the resistant torque is generated by adjustable
torsion bars. The main advantage of this mechanical configuration is the possibility to
apply torque independently from the motor characteristic curve: the maximum load
depends only on the strength of shafts and gears. The system configuration is quite
"rigid": only gear sets with fixed centre distance and the same gear ratio can be tested.
Some examples of test rigs designed on the basis of the power recirculation principle
are given in [2] and [3].
Introduction
5
Figure 1.3 Power recirculation and power absorption principle.
The second group includes the power absorption test rigs: the input torque is applied
by a driving dynamometer, while the resistant torque is generated by a power
absorbing dynamometer; a speed increaser drive, positioned between the test gearbox
and the brake, can also be designed in order to bring the output rotational speed into
the favourable operating range of the dynamometer. This solution is more "flexible"
than the first one, since the gears supports can be designed in order to test gear sets
with different macro-geometry and misalignments. Moreover, motor and brake can be
re-used to test complete gear transmissions, in place of the original gears supports.
Nevertheless, the main disadvantage of this configuration is due to the enormous
waste of electrical energy: all energy introduced by the motor is lost and, in case of
high mechanical powers, the cost of electricity can be very incisive for the research.
Finally, contrary to the power recirculation principle, the applied torque depends on
the motor characteristic curve: at low speeds the motor could not supply the requested
load. Some test rigs designed according to the power absorption principle are given in
[4] and [5].
The measurements of static transmission error is a well consolidate practice: two
optical encoders are employed and each one measures the position of the driving and
Motor
Slave
gearbox
Test
gearbox
Encoder
Adjustable
torsion bar
Motor
Brake
Encoder
Encoder
POWER RECIRCULATION
TEST RIG
POWER ABSORPTION
TEST RIG
Chapter 1
6
the driven gear; the time history of transmission error is easily calculated as the
difference between the angular position of the driving gear and the angular position of
the driven gear, multiplied by the inverse of the gear ratio. The most recent encoders
have more than 90000 lines and thanks to electronic interpolation, the original
resolution can be further increased. The commonest errors associated to this technique
are related to mechanical couplings alignments responsible for twice-per-revolution
errors.
Nevertheless, the measurement of dynamic transmission error is not a consolidate
procedure as the previous one. According to Rosinski [6], the commonest methods
used for measuring TE under high speed are:
1. Angular acceleration: two accelerometers are mounted tangentially on each
gear body [7]; the two signals of each shaft are added and the resulting signal,
proportional to the rotational acceleration, is transmitted via slip-rings or by
telemetry. The output signals of the two shafts are double integrated and then
subtracted in order to calculate transmission error. Two main problems are
associated to this technique: the first one is that, if there is a very large
torsional vibration, the subtraction of two large numbers can give a very small
value of transmission error; the second one is that the constant component of
TE cannot be measured.
2. Pulse series generated by gratings: gratings are attached to input and output
shaft via mechanical couplings. In most cases, due to the mechanical
resonances of the couplings, the maximum measuring speed is limited to 2500
rpm [8].
3. Optical encoders: for high measuring speeds, a high number of lines can cause
serious problems related to signals acquisition. Solutions to these problems are
obtained by measuring the phase difference between the two encoder pulses
[9], or by comparing the phase between the encoders pulses and a string of
pulses generated by a digital oscillator [10]. A further method is based on
Introduction
7
phase demodulation [11]. As the previous methodology, measuring problems
can be given by mechanical couplings between transducers and shafts.
4. Laser interferometry: it is a non-contact measurement. It would be very useful
in TE measurement of a gear transmission, since no modifications of the
system layout are necessary, as it occurs when encoders are used.
Nevertheless, the signal resolution is not sufficient to discern the small
amplitude variations of transmission error.
5. Inertial transducers: strain gauges are used to detect seismic mass movement
in order to measure torsional vibrations of the shafts. The main limits are due
to the limited bandwidth and to the insufficient resolution.
6. Munro's system: it is based on an optical system and it is very accurate for high
speed. Nevertheless, the method is applicable only to gear sets having a gear
ratio equal to one [12].
7. Newcastle system: it is based on the Moiré's fringe method and it allows to
calculate TE for unlimited speed of rotation in gear drives of fixed gear ratio
[6].
Comparisons of experimental results, obtained using the different techniques, are also
dealt in [8].
1.2.2 Review of Static Transmission Error Studies
According to Munro [16], the idea of transmission error has been introduced first by
Harris [17] in 1958, starting from a previous work of Walker [18] and, some years
later, by Niemann and Baethge [19]. Nevertheless, before the analytic definition of
transmission error, it was universally recognized that teeth deformation under load
affects gears kinematics, causing impacts between the meshing teeth and then noise.
Chapter 1
8
Profile modifications, such as K profiles, were born just from these ideas, but
sometimes they have been based on empirical rules or selected according to similar
successful experiences. Therefore, the practice of applying profile modifications, in
order to reduce gear noise and vibrations, has been introduced before the definition of
transmission error, as confirmed by papers of D.W. Dudley [20] and H. Walker [21],
[22], [23] and [23] in the 40's.
During the years, numerous theoretical models have been developed in order to
predict static transmission error, anyway they can be all classified into two main
groups: the first one includes the models based on an analytical formulation of the
problem, while the second one comprises all models based on a finite elements
approach. The most important contribution to the development of analytical models
has been due to Munro [24]: by means of his approach, each pair of meshing teeth is
modeled as a spring and when double contact occurs on the line of action, the load is
shared by the two pairs according to two springs in parallel. On the other hand, finite
elements models consist in a discretization of the meshing teeth and some references
are given in [25] and [26].
All these studies have set the basis for the growth of different software tools for the
gears design, with particular attention to the prediction of static transmission error.
Among these software, it is worth mentioning KISSsoft [27], based on an analytical
formulation of the problem, and LDP [28], based on a boundary elements method. The
main advantage of analytical approaches in comparison to a FEM/BEM is, with no
doubts, the greater speed of calculus; nevertheless, the last numerical techniques are
necessary when an accurate analysis of the complete gear transmission is required, in
particular when the contribution of compliances of bearings, gear bodies or shafts to
the total transmission error becomes significant.
1.2.3 Review of Gears Dynamics Studies
The importance of an accurate study of the dynamics of a gear transmission is dual: it
can reduce the sources of excitations responsible for gear noise and it can prevent
loads amplifications affecting the fatigue life of the mechanical system. For this
Introduction
9
reason, in the course of time, different mathematical models have been developed in
order to study gears dynamics. A good review of these models has been given by
Ozguven and Houser [33], Blankeship and Singh [34] and Velex [35]. Next studies
have been focused on the influence of nonlinearities and time varying parameters on
the dynamics of a spur gear pair. In [36] Houser has considered the effect of gear
backlash in a spur gear pair model excited by loaded static transmission error at the
gear mesh; in the same study he has compared the results of the numerical simulation
to the experimental results presented by Kubo in [37] and [38]. Kahraman has gone
deeper in this study, considering a time varying mesh stiffness: its influence on
nonlinearities due to backlash has been investigated for both spur gear pairs [39] and
geared rotors [40], [41]. Nevertheless, in most of these researches, geared rotor
systems have been studied using lumped parameters models: they match very well
with the study of nonlinearities and time-varying parameters, but they are not suitable
to deeply investigate couplings between torsional and transverse vibrations of geared
systems due to shafts compliances. One of the first contributions to the study of
geared rotors using finite elements has been due to Kahraman in [42]: he has
developed a finite elements model of a geared rotor system, considering a constant
mesh stiffness and investigating the effects of bearing compliances on system
dynamics. A similar study has been performed by the same author for a multi-shaft
helical gear transmission [43]. Finally, analogous researches have been presented by
Ozguven [44] and Velex [45].
1.2.4 Review of Gear Noise Studies
The majority of the technical papers published on gear noise considers transmission
error as the main responsible for gears acoustic emission, nevertheless no rigorous
correlation between the two phenomena has been still found. The most part of these
papers are experimental and deal with the influence of gear design parameters on
noise. The first researches have been focused on the experimental relationship
between static transmission error and noise. In a recent study, Houser [48] has
compared the measured noise levels of eight different spur and double helical gear
Chapter 1
10
sets to the predicted values of static transmission error; an analogous study has been
performed by Smith [49]. Nevertheless, in a similar work [50] the same Houser has
stated that, the understanding of the relationship between transmission error and noise
cannot prescind from an accurate dynamic analysis of the gear system, since system
resonances are responsible for dynamic amplifications of transmission error
harmonics. On the basis of that idea, low-contact-ratio spur gears have been tested by
Oswald [51] and a correlation between predicted dynamic tooth overloads and
measured sound level has been found.
Later studies have been focused on the characterization of gear noise sources. To this
scope, Umezawa [52] has employed the acoustical holography technique to locate the
sound sources of a gear machine and to study the mechanism of sound propagation
from the gear mesh. Automated acoustic measurements of spur gears, having different
tooth profiles, have been even dealt by Lewicki [53]. Anyway, the conclusions of both
authors are that the biggest sound emission is always located in correspondence of the
top of the gearbox.
The noise radiated by a gearbox is constituted by two components: the structure and
the airborne noise. The airborne noise is produced by gear blank vibrations which
transmit outwards to the listener: it is relevant in the case of unhoused gears, but it is
negligible if a gearbox encloses the gears, since the housing walls are poor
transmitters and the airborne noise remains inside the gearbox, without reaching the
ear of the listener. For this reason, most of the time, structure noise is the most
important part of the total noise of a gear transmission. This kind of noise is due to the
interaction between bearing forces resulting from gear mesh excitation and housing
panels: gearbox panels act as loudspeakers, amplifying noise that is heard by the
listener. Gearbox design is then essential to solve problems related to gear noise: an
experimental investigation on the influence of gear design parameters on gearbox
radiated noise has been made by Oswald and Townsend [54]. Theoretical researches
on the influence of gearbox design on structure noise, using finite elements or
boundary elements method, are given in [55], [56], [57], [58], [59] and [60]. The
transmission path of the vibrations originating at gear mesh has been even investigated
by Singh in [61] and [62].
Introduction
11
Transmission error is not the only excitation at the gear mesh responsible for gear
noise. Recently, some authors have been focused on the contribution of tooth surface
roughness to gear noise. An experimental investigation on this topic has been given by
Ishida and Matsuda [63], while Singh has proposed the first theoretical model that
correlates sliding forces to gear noise [64]. Nevertheless, the most part of the previous
authors maintains that surface roughness has a negligible effect on gear noise, because
of the presence of film thickness between the mating teeth, and, moreover, they are
convinced that transmission error has still the major influence on gears acoustic
emission.
1.3 Scope and Objectives
It is universally recognized that transmission error is the main responsible for gear
noise, nevertheless no rigorous correlation is still found between the two phenomena,
that means it is not still possible to predict the acoustic emission of a gear
transmission since design phase. This study proposes to further investigate the
relationship between gear noise and transmission error, with particular attention to
those solutions commonly used to reduce both of them, i.e. profile modifications.
Accordingly, the main objectives of this study can be listed as follows:
1. Setup of a test rig for the study of transmission error and noise of parallel axis
gear sets and development of a measurement system to measure both static and
dynamic transmission error, using optical encoders.
2. Development of a theoretical model for the prediction of static transmission
error of spur gear sets, in function of the applied load and on the basis of gears
macro-geometry (module, number of teeth, centre distance, ...) and micro-
geometry (profile modifications, actual tooth profiles, pitch errors, ...).
Chapter 1
12
3. Experimental validation of the static model, by means of static transmission
error measurements, considering various gear sets with different profile
modifications.
4. Development of dynamic models of the test facility, in order to predict
dynamic transmission error, as well as dynamic mesh forces.
5. Experimental validation of the dynamic models, by means of dynamic
transmission error measurements and using the test facility previously
described.
6. Investigation on the influence of profile modifications on gear noise,
comparing measured sound levels to dynamic mesh forces predicted on the
basis of static transmission error measurements.
1.4 Dissertation Outline
Chapter 2 describes the setup of a test facility developed to study gear noise and
vibrations. In the same chapter, the development of a gear casing necessary to
guarantee the correct lubrication of the gear sets and to ensure the prescribed encoders
tolerances is presented. Then, a data acquisition system for the measurement of
transmission error up to 3000 rpm, using optical encoders, is disclosed and the
consequent data elaboration process is discussed.
In Chapter 3 a model for the prediction of static transmission error in function of the
applied torque, the gear macro and micro-geometry is presented. The model, based on
a previous existing theoretical formulation [24], is improved considering a real mesh
compliance due to the tooth bending compliance, the Hertzian compliance and the
tooth base rotation. The validation of the model is performed both numerically,
comparing the numerical results to those ones obtained using the commercial software
LDP [28], and experimentally, considering four gear sets, having the same macro-
Introduction
13
geometry, but different profile modifications, and measuring static transmission error
varying the applied load.
The dynamic analysis of the gear transmission is dealt in Chapter 4. Different dynamic
models of the test facility are considered: starting from a 2 DOF lumped parameters
linear model, scheme complexity is increased up to consider a finite elements model
of the test rig. The experimental validation of the models is performed measuring
dynamic transmission error up to 3000 rpm.
Gear noise measurements of the four gear sets are presented in Chapter 5. Sound
pressure levels are measured for different values of speed and torque and a correlation
between noise levels and predicted gear mesh dynamic forces is disclosed.
Finally, conclusions and recommendations for future works are listed in Chapter 6.
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