-
Research ArticleOn Dynamic Mesh Force Evaluation of Spiral Bevel
Gears
Xiaoyu Sun , Yongqiang Zhao , Ming Liu, and Yanping Liu
School of Mechatronics Engineering, Harbin Institute of
Technology, No. 92, Xidazhi Street, Nangang District,
Harbin,Heilongjiang, China
Correspondence should be addressed to Yongqiang Zhao;
[email protected]
Received 22 June 2019; Revised 16 September 2019; Accepted 25
September 2019; Published 20 October 2019
Academic Editor: Miguel Neves
Copyright © 2019 Xiaoyu Sun et al. ,is is an open access article
distributed under the Creative Commons Attribution License,which
permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
,e mesh model and mesh stiffness representation are the two main
factors affecting the calculation method and the results of
thedynamic mesh force. Comparative studies considering the two
factors are performed to explore appropriate approaches toestimate
the dynamic meshing load on each contacting tooth flank of spiral
bevel gears. First, a tooth pair mesh model is proposedto better
describe the mesh characteristics of individual tooth pairs in
contact. ,e mesh parameters including the mesh vector,transmission
error, and mesh stiffness are compared with those of the
extensively applied single-point mesh model of a gear pair.Dynamic
results from the proposed model indicate that it can reveal a more
realistic and pronounced dynamic behavior of eachengaged tooth
pair. Second, dynamic mesh force calculations from three different
approaches are compared to further investigatethe effect of mesh
stiffness representations. One method uses the mesh stiffness
estimated by the commonly used average slopeapproach, the second
method applies the mesh stiffness evaluated by the local slope
approach, and the third approach utilizes aquasistatically defined
interpolation function indexed by mesh deflection and mesh
position.
1. Introduction
Spiral bevel gears are commonly used in power
transmissionbetween intersecting shafts but often suffer from
fatiguefailures caused by excess dynamic loads. ,e estimation
ofdynamic loads carried by each pair of teeth in contact laysthe
root for analyses of bending and contact stresses of gearteeth,
gear tooth surface wear (pitting and scoring), andlubrication
performance between the mating tooth flanks,which are all effective
ways to investigate the mechanism ofgear failures. Consequently, a
reasonably accurate predictionof dynamic mesh force (DMF) is the
cornerstone for failureanalysis.
Two factors have a dominant influence on the calcula-tion
approaches and results of DMF: firstly the mesh modelwhich
determines the calculation of mesh parameters in-cluding effective
mesh point coordinates, line of actionvectors, mesh stiffness, and
transmission errors and secondlythe mesh stiffness
representations.
,e determination of mesh parameters for spiral bevelgears is
complicated not only because of the complex toothflank geometries
but also because the mesh properties are
strongly influenced by load conditions. Nowadays, loadedtooth
contact analysis (LTCA) based on the conventionalfinite element
method [1–4] or finite element/contactmechanism method [5] is
widely applied to obtain the meshparameters of spiral bevel gears
under the loaded condition.However, in the absence of the
sophisticated commercialfinite element software (e.g., ABAQUS and
ANSYS) and aspecialized LTCA tool, [6] the loaded mesh parameters
werenot available, and hence, experimental or simple
analyticalmodels which preclude exact mesh geometries were used
inmost early dynamic studies [7, 8]. Until the beginning of
thiscentury, Cheng and Lim first attempted to integrate
thetime-variant mesh parameters obtained from tooth contactanalysis
(TCA) [9] and LTCA [10] of gears with exact toothflank geometry
into a mesh model. ,e model was exten-sively applied in the
subsequent studies on dynamics of thehypoid and bevel gear pair or
geared system [10–17]. Inaddition to the time-dependent mesh
characteristics, someother special factors affecting the dynamic
responses wereinvestigated, such as backlash nonlinearity [10–13],
friction[13–15], mesh stiffness asymmetry [12], gyroscopic
effect[13], assembly errors [13], and geometric eccentricity [13]
as
HindawiShock and VibrationVolume 2019, Article ID 5614574, 26
pageshttps://doi.org/10.1155/2019/5614574
mailto:[email protected]://orcid.org/0000-0001-8371-1171https://orcid.org/0000-0002-8317-5409https://orcid.org/0000-0003-2438-2587https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/5614574
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well as other driveline components such as the bearing [16]and
elastic housing [17].
In all contributions mentioned above, the single-pointcoupling
mesh model where the gear mesh coupling ismodeled by a single
spring-damper unit located at the ef-fective mesh point and acting
along the instantaneous line ofaction was applied. Although the
model is believed to be ableto catch the overall effect of the mesh
characteristics of a gearpair, it cannot reflect the mesh
properties of each meshinterface when more than one tooth pair is
engaged inmeshing, and hence, the DMF for each couple of teeth
incontact cannot be predicted directly. Karagiannis et al.
[15]utilized the load distribution factor defined and calculated
inquasistatic LTCA to estimate the dynamic contact load pertooth
when load sharing occurs. ,is method may be fea-sible, but the
results cannot accurately reflect the real fea-tures of the contact
force under the dynamic condition. Toovercome the deficiency of the
single-point coupling meshmodel, Wang [18] proposed a multipoint
coupling meshmodel in which each mesh interface is represented by
therespective effective mesh point, line of action, and
meshstiffness. ,e mesh properties and dynamic responses fromthe
model were then compared with those from a meshmodel whose
parameters are obtained from pitch conedesign [19].,e differences
between them stem from the factthat the mesh positions and line of
action vectors which aretime-variant for the multipoint coupling
mesh model areconstant for the mesh model based on the pitch cone
designconcept. ,e authors also believed that the former is
capableof describing the exact mesh properties of loaded
gears,while the latter is only valid for no-load and
light-loadedconditions, as the mesh parameters of them are
obtainedunder loaded and unloaded conditions, respectively. So
far,the multipoint coupling mesh model has not been used aswidely
as the single-point coupling mesh model for dynamicanalysis
probably because most studies focus on the overalleffect of mesh
characteristics on dynamics of a gear pairrather than the detailed
dynamic responses of each meshingtooth pair. Among recent studies,
maybe the multipointcoupling mesh model was only applied by Wang et
al. [20].,eir research proved that the model has the capability
ofpredicting the DMF-time history for each tooth pair within afull
tooth engagement cycle. Even so, because the model stilluses the
transmission error of a gear pair to determine thetooth contact and
contact deformation of each engagedtooth pair, the results cannot
properly reflect the real contactstate of each pair of teeth when
multiple pairs of teeth are inthe zone of contact.
,e accuracy of DMF calculation results is, to a largeextent,
dependent on the accuracy of mesh stiffness. ,edeformation of an
engaged gear pair or an individualcontacting tooth pair is
nonlinear with the applied loadbecause of the nonlinearity of the
local contact deformationwith the contact force. It means that mesh
stiffness shouldvary with mesh deformation to accurately describe
thenonlinear force-deflection relationship. However, in pre-vious
studies, although the variation of mesh stiffness withthe mesh
position was considered, the change in meshstiffness with
deformation at a given mesh position was
rarely taken into consideration in DMF calculation. ,ereare
generally two methods to calculate mesh stiffness: thecommonly used
one is called the average slope approach andthe other, which is
seldom applied but may be more ap-propriate for dynamic analysis,
is called the local slopeapproach [21]. ,e mesh stiffness evaluated
by the twoapproaches can both be applied in DMF estimation, but,
toaccurately calculate the DMF, an error-prone point whichshould be
paid enough attention is that the calculationformulas of DMF are
entirely different for different types ofmesh stiffness applied. It
is worth mentioning that not alldynamic models utilize prespecified
mesh stiffness for thecalculation of DMF. In researches on the
dynamics ofparallel axis gears, [22, 23], for example, the mesh
stiffness isevaluated according to instantaneous mesh deflection
ateach time step of the dynamic simulation.,is kind of modelis
referred to as the coupled dynamics and contact analysismodel, as
the dynamic and contact behaviors are mutuallyinfluential. ,e
variation of mesh stiffness with deformationis realized in this
kind of model. However, the model iscurrently not computationally
feasible for spiral bevel gearsbecause of the more complex and
time-consuming TCAcomputation.
,is study aims to explore appropriate methods for moreaccurate
prediction of the DMF of each meshing tooth pairof spiral bevel
gears through comparative analyses consid-ering the two factors
mentioned above. First, an improvedmesh model of a tooth pair
(MMTP) which represents thevariation of the mesh properties of a
tooth pair throughout afull engagement cycle was proposed. ,e model
applies thetransmission error curve of a tooth pair instead of that
of agear pair to estimate mesh stiffness and identify the
contactfor each couple of teeth. ,is improvement enables themodel
to capture more realistic mesh characteristics of eachcontacting
tooth pair than the multipoint coupling meshmodel. Second, we
compared the dynamic responses fromthe proposed mesh model and the
widely used single-pointcoupling mesh model for a generic spiral
bevel gear pair onthe one hand to validate the proposed one and on
the otherhand to demonstrate the superiority of the proposed
modelin the evaluation of DMF per tooth pair. It is shown that
thetwo models predict the similar variation tendencies of
DMFresponse with the mesh frequency. Although some differ-ences
exist in the response amplitudes particularly in ahigher mesh
frequency range, the resonance frequenciesestimated by the two
models are almost the same. Besides,the time histories of DMF per
tooth pair predicted by the twomodels are significantly different,
but the results from theproposed model can provide much better
insight into dy-namic contact behavior of each contacting tooth
pair. ,ird,we performed a comparative study on three
calculationapproaches of DMF to further investigate the effect of
meshstiffness representations on DMF calculations. One methoduses
the mesh stiffness estimated by the average slope ap-proach,
another one applies the mesh stiffness evaluated bythe local slope
approach, and the third method utilizes aquasistatically defined
interpolation function indexed by themesh deflection and the pinion
roll angle. Weighing theaccuracy and cost of the computation, we
found that the
2 Shock and Vibration
-
local slope approach is preferred for DMF evaluation ofspiral
bevel gears.
2. Mesh Model
,is study applies the mesh model with prespecified
meshparameters obtained through quasistatic LTCA which isperformed
with the aid of the commercial finite elementanalysis software
ABAQUS [24]. For the development offinite element models, the
technique without the applicationof the CAD software proposed by
Litvin et al. [1] is adopted.In this way, not only are the nodes of
the finite element meshlying on tooth surfaces guaranteed to be the
points of theexact tooth surfaces (which are calculated through
themathematical model of tooth surface geometry [25]) but alsothe
inaccuracy of solid models due to the geometric ap-proximation by
the spline functions of the CAD software canbe avoided. ,e
procedure of using ABAQUS for LTCA hadbeen introduced in detail by
Zhou et al. [4] and Hou andDeng [26] and thus will not be described
here. ,e cor-rectness of the finite element model is verified by
comparingthe obtained kinematic transmission error (KTE) curve
withthat obtained by traditional TCA, as shown in Figure 1,where
the two curves are approximately consistent. A pair ofmismatched
face-milled Gleason spiral bevel gears is se-lected to be the
example for analysis. ,e contact pattern islocated in the center of
the tooth surface, and there is nocontact in the extreme part
including the tip, toe, and heel ofthe teeth. Table 1 lists the
design parameters, with which themachine tool setting parameters
used to derive tooth ge-ometry are obtained by Gleason
CAGE4Win.
For the convenience of the following discussions, thecoordinate
systems (Figure 2) used in this study are in-troduced first. ,e
LTCA is performed in the fixed referenceframe Oxyz, whose origin is
at the intersection point of thepinion and gear rotation axes.
Other coordinate systems areused for dynamic analysis. ,e
coordinate systemsOpx
0py
0pz
0p and Ogx
0gy
0gz
0g are the local inertial reference
frames of the pinion and gear, respectively, whose originsand
z-axes coincide with the mass centroids and rolling axesof each
body under the initial static equilibrium condition.,e body-fixed
coordinate systems Cpx4py4pz4p andCgx
4gy
4gz
4g for the pinion and gear are also located with their
origins at the mass centroids, and all the coordinate axes
areassumed to coincide with the principal axes of the inertia
ofeach body.
2.1. MeshModel of a Gear Pair (MMGP). In the
single-pointcoupling mesh model (Figure 3), the two mating gears
arecoupled by a single spring-damper element whose stiffnesskm,
damping cm, acting point, and direction vary with re-spect to the
angular position of the pinion θp. ,e springstiffness km, which is
the function of bending, shear, andcontact deformations of all
meshing tooth pairs in the zoneof contact and the base rotation
compliance, represents theeffective mesh stiffness of a gear pair.
,e KTE ek due totooth surface modification, machining error, and
assemblymisalignment also changes with the pinion roll angle
θp,
while the backlash b is often regarded as a constant
forsimplicity.
,e synthesis of the single-point coupling mesh modelrequires
reducing the distributed contact forces obtained byLTCA to an
equivalent net mesh force Fm(Fmx, Fmy, Fmz)(Figure 4(a)). ,e
direction of the net mesh force is definedas the line of action
vector Lm which points from the pinionto the gear, and the point of
action is the effective mesh pointrepresenting the mean contact
position. ,e calculationmethod of contact parameters applied in
previous studies[10–14] can be used here, while if LTCA is
undertaken usingABAQUS, another similar but more convenient
approachcan be applied. As long as the history outputs of CFN,
CMN,and XN for the contact pair are requested [24], the
effectivemesh force Fm,i(Fmx,i, Fmy,i, Fmz,i) and the mesh
momentMm,i(Mmx,i, Mmy,i, Mmz,i) due to contact pressure on
eachmeshing tooth pair i and the mesh force acting point
vectorrm,i(xm,i, ym,i, zm,i) can be obtained directly at each
specifiedtime step. ,us, the magnitude of equivalent net mesh
forceFnet is calculated by
Fnet ���������������F2mx + F
2my + F
2mz
,
Fml �
N
i�1Fml,i, l � x, y, z.
(1)
,e line of action vector Lm(nx, ny, nz) of the net meshforce is
obtained by
nl �FmlFnet
, l � x, y, z. (2)
,e total contact-induced moments about each co-ordinate axis are
as follows:
Mml �
N
i�1Mml,i, l � x, y, z. (3)
,e effective mesh point position rm(xm, ym, zm) can becalculated
from the following equation:
xm �
Ni�1xm,iFm,i
Ni�1Fm,i
,
zm �Mmy + Fmzxm
Fmx,
ym �Mmx + Fmyzm
Fmz.
(4)
In above-mentioned equations, N is the number ofengaged tooth
pairs at each pinion roll angle. In this work,the single-point
coupling mesh model is named the meshmodel of a gear pair (MMGP),
as it reflects the overall meshproperties of a gear pair.
2.2. Mesh Model of a Tooth Pair (MMTP). In the
multipointcoupling mesh model (Figure 5(b)) [18], there are a
variablenumber of spring-damper units coupling the pinion andgear
in each mesh position. ,e spring-damper element
Shock and Vibration 3
-
represents the effective mesh point, line of action, and
meshstiffness of each mesh interface, and the number of
elementswhich depends on the gear rotational position and
loadtorque indicates the number of tooth pairs in the zone
ofcontact. ,is model describes the variation of the
meshcharacteristics of each tooth pair in the contact zone over
ameshing period, and thereby, the mesh characteristics of
theindividual tooth pair can be considered the basic unit.
Based
on this point of view, the mesh model of a tooth pair(MMTP)
(Figure 5(a)) which describes the mesh charac-teristic variation of
a tooth pair over one tooth engagementcycle is proposed. ,e MMTP
facilitates the development ofmodels handlingmore complex
situations, such as themodelwith nonidentical mesh characteristics
of the tooth pair (e.g.,one or several teeth have cracks, surface
wear, or any otherfeatures influencing mesh properties).
,e synthesis of the effective mesh point and line ofaction for
the MMTP is the same as that for themultipoint coupling mesh model
(Figure 4(b)) [18]. How-ever, as mentioned before, the effective
mesh force Fm,i(Fmx,i, Fmy,i, Fmz,i) due to contact pressure on
each meshingtooth pair i and the corresponding acting point
vectorrm,i(xm,i, ym,i, zm,i) can be obtained directly from
LTCAperformed with ABAQUS. ,e direction of the effectivemesh force
is defined as the line of action for each contactingtooth pair, and
the acting point vector is just the effectivemesh point position
vector. ,e calculation approach ofmesh stiffness and DMF for the
MMTP is modified to makethe model describe the meshing state of
each pair of teethmore accurately than the multipoint coupling mesh
model.,ese will be elaborated in the following
correspondingsections.
0 0.2 0.4 0.6 0.8 1 1.2Pinion roll angle (rad)
–16–14–12–10
–8–6–4–2
024
Ang
ular
TE
(rad
)
×10–5
DA B C
Contact zone 3Contact zone 1
Profileseparationof toothpair 2
Contact zone 2
Profileseparationof toothpair 2
KTE-tooth pair 3KTE-tooth pair 2
LTE-gear pairKTE-gear pairKTE-tooth pair 1
Figure 1: Angular transmission error curves.
O
y
x
z
Og
Cg
Cp
x0gx0p
z0p
y0p
x4p
x4g
z4p
z4gz0g y
4py
4g
Op
y0g
Figure 2: Coordinate systems of a spiral bevel gear pair.
Table 1: Design parameters of the example spiral bevel gear
pair.
Parameters Pinion (left hand) Gear (right hand)Number of teeth
11 34Helical angle (rad) 0.646Pressure angle (rad) 0.349Pitch angle
(rad) 0.313 1.257Pitch diameter (mm) 71.5 221Pitch cone distance
(mm) 116.139Width of the tooth (mm) 35Outer transverse module
6.5Virtual contact ratio 1.897
Instantaneous line of action at angular position i
Instantaneous line of action at angular position i + 1
Gear
Pinion
km (θp)
ek (θp)
Lm (θp)
b
rpm (θp)
cm (θp) rgm (θp)
Figure 3: Single-point coupling mesh model.
4 Shock and Vibration
-
2.3. Comparison of Mesh Parameters. In addition to com-paring
the mesh properties embodied on the individualtooth pair and the
gear pair, the effect of the applied load ontheir variations was
investigated. To this end, we synthesizedthe mesh parameters for
the MMTP and the MMGP at50Nm and 300Nm torque loads. ,e mesh
vectors, whichrepresent the magnitude, direction, and point of
action ofthe equivalent mesh force, at each mesh position
throughoutone tooth engagement cycle for the MMTP and over onemesh
cycle for the MMGP are presented in Figures 6(a) and6(b),
respectively. As can be seen, the magnitude of theequivalent mesh
force for the MMTP (i.e., the effective meshforce on a single pair
of meshing teeth) varies obviously withthe mesh position, while
only the tiny change can be ob-served in the size of the equivalent
net mesh force for theMMGP (i.e., the combined mesh force of all
engaged toothpairs). ,e former is mainly attributed to the load
sharingamong the tooth pairs in contact simultaneously, whereasthe
latter is only caused by small changes in the directionalrotation
radius (which can be regarded as the arm of force) at
different mesh positions. Figure 6 also shows that the
di-rection of the equivalent mesh force changes little for boththe
MMTP and the MMGP.
Furthermore, the line of action directional cosines andeffective
mesh point coordinates of the two models over afull tooth (tooth
pair 2) engagement cycle are compared inFigure 7. It shows clearly
that all mesh parameters of thetooth pair are altered significantly
with the gear rotationrelative to the corresponding changes in the
mesh param-eters of the gear pair. In addition, from both Figures 6
and 7,we can observe that the range of change in mesh parametersof
the gear pair becomes small under the heavy-loadedcondition
(300Nm), but no obvious change for the meshparameters of the tooth
pair was observed. A reasonableexplanation is as follows: with the
increase of load, theextended tooth contact appears because of the
deformationof gear teeth. It is manifested in the advance
engagement ofthe incoming tooth pair 3 and the deferred
disengagement ofthe receding tooth pair 1 in Figure 7. ,is change
increasesthe contact ratio and meanwhile decreases the amplitude
of
Lm,i (θp)
bi ek,i (θp)
km,i (θp)
cm,i (θp)
(a) (b)
Figure 5: Illustrations of the (a) mesh model of a tooth pair
and (b) multipoint coupling mesh model.
x
yz
Mm
Effective mesh point of a gear pair
Fm (Fmx, Fmy, Fmz)
rm (xm, ym, zm)O
(a)
O
x
y zMm,i
Effective mesh point of a tooth pair
Fm,i (Fmx,i, Fmy,i, Fmz,i)
Fm,i+1 (Fmx,i+1, Fmy,i+1, Fmz,i+1)
rm,i (xm,i, ym,i, zm,i)
rm,i+1 (xm,i+1, ym,i+1, zm,i+1)
(b)
Figure 4: Synthesis of the effective mesh parameters of the (a)
single-point coupling mesh model and (b) multipoint coupling mesh
model.
Shock and Vibration 5
-
40
30
20
10
0
–10–25
–30–35
–40 50 6070 80
90 100110
z m (m
m)
ym (mm) xm (mm)
300Nm50Nm
(a)
300Nm50Nm
40
30
20
10
0
–10–25
–30–35
–40 50 6070 80
90 100110
z m (m
m)
ym (mm) xm (mm)
(b)
Figure 6: Mesh vectors of the MMTP (a) and the MMGP (b).
Tooth pair 1 at 300NmTooth pair 2 at 300NmTooth pair 3 at
300NmGear pair at 300Nm
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1 1.11Pinion roll angle
(rad)
Tooth pair 1 at 50NmTooth pair 2 at 50NmTooth pair 3 at 50NmGear
pair at 50Nm
–0.67
–0.66
–0.65
–0.64
–0.63
–0.62
LOA
dire
ctio
nal c
osin
e nx
(a)
Tooth pair 1 at 300NmTooth pair 2 at 300NmTooth pair 3 at
300NmGear pair at 300Nm
Tooth pair 1 at 50NmTooth pair 2 at 50NmTooth pair 3 at 50NmGear
pair at 50Nm
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1 1.11Pinion roll angle
(rad)
–0.17
–0.165
–0.16
–0.155
–0.15
–0.145
–0.14
LOA
dire
ctio
nal c
osin
e ny
(b)
Figure 7: Continued.
6 Shock and Vibration
-
Tooth pair 1 at 300NmTooth pair 2 at 300NmTooth pair 3 at
300NmGear pair at 300Nm
Tooth pair 1 at 50NmTooth pair 2 at 50NmTooth pair 3 at 50NmGear
pair at 50Nm
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.1Pinion roll angle
(rad)
0.735
0.74
0.745
0.75
0.755
0.76
0.765
LOA
dire
ctio
nal c
osin
e nz
(c)
Tooth pair 1 at 300NmTooth pair 2 at 300NmTooth pair 3 at
300NmGear pair at 300Nm
Tooth pair 1 at 50NmTooth pair 2 at 50NmTooth pair 3 at 50NmGear
pair at 50Nm
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1 1.11Pinion roll angle
(rad)
80
85
90
95
100
105
EMP
coor
dina
te x
m (m
m)
(d)
Tooth pair 1 at 300NmTooth pair 2 at 300NmTooth pair 3 at
300NmGear pair at 300Nm
Tooth pair 1 at 50NmTooth pair 2 at 50NmTooth pair 3 at 50NmGear
pair at 50Nm
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.1Pinion roll angle
(rad)
–32
–31
–30
–29
–28
–27
EMP
coor
dina
te y m
(mm
)
(e)
Figure 7: Continued.
Shock and Vibration 7
-
the variation of the mesh parameters for the gear pair. Notethat
it does not influence the mesh frequency of the gear pairbecause
according to the compatibility condition, the toothrotation due to
deformation for each couple of teeth shouldbe the same to maintain
tooth contact and continuity inpower transmission, and hence, the
tooth pitch does notchange. Overall, the results indicate that
although the time-varying mesh characteristics exhibited on each
pair of teethare prominent, their overall effect on the gear pair
isweakened, and this weakening will become more pro-nounced as the
contact ratio increases.
2.4. Remark on the Mesh Model Based on Quasistatic LTCA.From the
above analysis, it is substantiated again that themesh parameters
of spiral bevel gears are load-dependent.,erefore, the mesh model
based on quasistatic LTCA issuperior to the mesh model relying on
the geometric re-lationship in terms of describing the mesh
properties ofloaded gear pairs. However, as the mesh parameters
areestimated at a constant applied torque, the mesh model
istheoretically only suitable for the dynamic analysis of a
gearpair subjected to this torque load, and it seems to be
in-competent in the study of a gear pair subjected to a
time-variant torque load. Nevertheless, it has been discovered
thatthe mesh stiffness shows more prominent time-varyingparametric
excitation effects than the mesh vector [11, 13].,erefore, a
compromising way to calculate the DMF isapplying the mesh vector
predicted at the mean appliedtorque and the calculation approach
using the mesh forceinterpolation function (as introduced in DMF
Calculation)which takes the load effects on mesh stiffness
intoconsideration.
3. Mesh Stiffness Evaluation
,e force-deflection curve (a typical one is illustrated inFigure
8) describes the nonlinear relationship between the
force and the deformation precisely. ,erefore, using thiscurve
to predict the DMF for a known mesh deformationcould be the most
accurate approach. However, it is by nomeans the most efficient
method since the establishment ofthe curve requires a series of
time-consuming LTCAs atdifferent load levels. Usually, the DMF for
a given torqueload is predicted by using the mesh stiffness
evaluated at thesame load. Note that although at each rolling
position themesh stiffness is a constant, it varies with respect to
meshpositions. Hence, the mesh stiffness has a spatially
varyingfeature.
In general, there are two calculation approaches for
meshstiffness, namely, the average slope approach and the
localslope approach [21]. ,e corresponding calculated meshstiffness
is named the average secant mesh stiffness (ASMS)
Tooth pair 1 at 300NmTooth pair 2 at 300NmTooth pair 3 at
300NmGear pair at 300Nm
Tooth pair 1 at 50NmTooth pair 2 at 50NmTooth pair 3 at 50NmGear
pair at 50Nm
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1 1.11Pinion roll angle
(rad)
–6
–4
–2
0
2
4
6
EMP
coor
dina
te z m
(mm
)
(f )
Figure 7: Comparison of time-varying line of action directional
cosines (a–c) and effective mesh point coordinates (d–f) between
theMMTP and the MMGP at 50Nm and 300Nm torque loads.
Fm,e
Fm,a
Fm,l
Fm
FmT
LTMS
ASMS
Local tangent
Average secant
0
2000
4000
6000
8000
10000
12000
14000
16000
18000M
esh
forc
e (N
)
0.005 0.01 0.015 0.02 0.0250Mesh deflection (mm)
δd
δT
kTma = FmT /δT
kTml = dFm/dδ|δ=δT
Figure 8: A typical fitted force-deflection curve and
illustration ofthe local tangent mesh stiffness (LTMS) and the
average secantmesh stiffness (ASMS).
8 Shock and Vibration
-
and local tangent mesh stiffness (LTMS) in this paper. Asshown
in Figure 8, they are the slope of the secant (bluedashed line) and
the slope of the tangent (red dash-dottedline) of the
force-deflection curve, respectively.
3.1. Average Secant Mesh Stiffness. ,e average secant
meshstiffness has been extensively used to calculate DMF
fordifferent types of gears, such as spur gears [27–29],
helicalgears [29, 30], and spiral bevel or hypoid gears [9,
10,13–18, 20]. It is calculated through dividing the magnitudeof
mesh force Fm by the mesh deflection along the line ofaction δ:
kma �Fm
δ. (5)
,e difference between the translational KTE ε0 and theloaded
transmission error (LTE) ε can be attributed to themesh deflection
δ:
δ � ε0 − ε. (6)
In practice, the KTE always exists for spiral bevel gears,even
if there are no machining and assembly errors, becausegear tooth
flank profiles are often modified to be mis-matched to avoid
contact of extreme parts (top, heel, or toe)of the teeth and to
reduce the sensitivity of transmissionperformance to installation
and machining errors. Becauseof the mismatch, the basic mating
equation of the contactingtooth flanks is satisfied only at one
point of the path ofcontact. It has been proved that although the
KTE is alwayssmall compared to the LTE, the KTE cannot be omitted
inthe calculation of mesh stiffness, especially for the
light-loaded condition [13].
,e translational KTE and LTE, ε0 and ε, are the pro-jections of
the corresponding angular transmission errors, e0and e, along the
line of action. As y-axis of the coordinatesystem Oxyz is the
gear-shaft rotation axis (Figure 2), atwhich the angular
transmission error is expressed about, thetranslational KTE and LTE
are calculated by
ε0 � e0λy,
ε � eλy,(7)
where λy � nxzm − nzxm is the directional rotation radiusfor the
angular transmission error. Note that equations(5)–(7) are derived
based on the assumption that the meshvectors for unloaded and
loaded conditions are identical.However, the results shown in
Figure 7 indicate that thehypothesis deviates from the fact and
thereby inevitablybrings about calculation errors.
,e angular KTE and LTE, e0 and e, can be determined as
e0 � Δθg0 −Np
NgΔθp0,
e � Δθg −Np
NgΔθp,
(8)
where Δθp0 and Δθg0 (obtained by TCA) are the unloadedrotational
displacements of the pinion and gear relative to
the initial conjugated contact position where the trans-mission
error equals zero, Δθp and Δθg (obtained by LTCA)are the
corresponding loaded rotational displacements, andNp and Ng are the
tooth numbers of the pinion and gear,respectively.
Equations (5)–(8) can be directly applied to calculate theASMS
of a gear pair. In this case, Fm is the magnitude of thenet mesh
force of the gear pair and δ is the total deformationof all meshing
tooth pairs in the zone of contact.
For the MMTP, the ASMS of a single tooth pair needs tobe
estimated. ,e effective mesh force on each meshingtooth pair Fm,i
can be obtained directly through LTCA, whilethe troublesome issue
is how to obtain the mesh deflection ofa single tooth pair when
multiple sets of teeth are in contact.In this work, the
compatibility condition is applied to ad-dress this problem. It is
based on the fact that the total toothrotation Δθ (which is the sum
of the composite deformationof the tooth and base ΔB caused by
bending moment andshearing load, contact deformation ΔH, and
profile sepa-ration ΔS divided by the directional rotation radius
aboutthe rolling axis λ) must be the same for all
simultaneouslyengaged tooth pairs to maintain the tooth contact
andcontinuity in power transmission [2]. If three pairs of teethare
in contact, the compatibility condition can be expressedbyΔSi− 1 +
ΔBi− 1 + ΔHi− 1
λi− 1�ΔSi + ΔBi + ΔHi
λi
�ΔSi+1 + ΔBi+1 + ΔHi+1
λi+1� Δθ.
(9)
If the ASMS of a tooth pair is defined as the mesh forcerequired
to produce a unit total deformation (which con-tains the composite
deformation of the tooth and base ΔBand the contact deformation
ΔH), equation (9) can be re-written in a form with the ASMS of each
meshing tooth pairkma,i as follows:
ΔSi− 1 + Fm,i− 1/kma,i− 1 λi− 1
�ΔSi + Fm,i/kma,i
λi
�ΔSi+1 + Fm,i+1/kma,i+1
λi+1� Δθ.
(10)
3.2. Local Tangent Mesh Stiffness. ,e local tangent
meshstiffness can be approximately calculated by the
centraldifferencemethod which requires the LTCA to be performedat
three different torque loads: one at the nominal torqueload,
another above it, and the third below it:
kml �dFmdδ δ�δ
T
≈12
FTm − FT− ΔTm
δT − δT− ΔT+
FT+ΔTm − FTm
δT+ΔT − δT , (11)
where ΔT is a specified small change in torque load. Asδ � ε0 −
ε, equation (11) can be transformed to
Shock and Vibration 9
-
kml ≈12
FTm − FT− ΔTm
εT− ΔT − εT+
FT+ΔTm − FTm
εT − εT+ΔT . (12)
Equation (12) can be directly used to calculate the LTMSof both
a gear pair and a tooth pair, as the profile separationof each
tooth pair ΔSi which influences the calculation of theASMS of a
tooth pair does not function in the calculation ofthe LTMS of a
tooth pair. ,e reason is that the KTE ε0which contains the profile
separation does not appear inequation (12). In contrast with the
calculation of the ASMS,for the evaluation of the LTMS, assuming
themesh vectors atthe torque loads of T, T + ΔT, and T − ΔT are
identical isreasonable when ΔT is small.
3.3. Example. Taking the evaluation of the mesh stiffness ata
torque load of 300Nm as an example, we demonstrate thegeneral
calculation approaches of the ASMS and LTMS.Figure 9 shows the
effective mesh force of three consec-utively engaged tooth pairs
and the net mesh force of thegear pair over one tooth engagement
cycle. Note that theactual length of the period of one tooth
engagement islonger than the one shown in Figure 9 since tooth pair
2comes into contact at some pinion roll angle less than
0.171radians and leaves the contact at some pinion roll
anglegreater than 1.107 radians. By assuming that, at the
be-ginning and end of a tooth engagement cycle, the meshforce of
each meshing tooth pair increases and decreaseslinearly, the pinion
roll angles where tooth contact beginsand ends can be approximately
predicted through linearextrapolation. Using the data given in
Figure 9, we canpredict that tooth pair 2 engages at approximately
0.159radians and disengages at nearly 1.110 radians, and toothpair
3 engages at about 0.730 radians.,erefore, if the pitcherror is not
considered, the tooth meshing is repeated every0.571 radians, and
the actual length of one tooth engage-ment cycle is 0.951 radians.
With the known roll angles atthe beginning of engagement and
disengagement, meshparameters at the two points can be predicted.
In this work,the LS-SVMlab toolbox [31] is utilized to estimate
thefunctions of mesh parameters with respect to the rotationangle
of the pinion tooth, and then the parameters at thestarting
positions of engagement and disengagement areevaluated by the
estimated functions.
Figure 1 shows the angular KTE and LTE of the examplegear pair.
,e KTE of the gear pair is obtained by LTCAperformed at a small
torque load. ,is method is named theFEM-based TCA.,e torque load
should be carefully chosen(3Nm in this work) to keep the teeth in
contact but not tocause appreciable deformation. ,e KTEs of tooth
pairs 1, 2,and 3, which are necessary for the calculation of the
meshdeflection of the tooth pair, are acquired by traditional
TCA[32] (to the authors’ knowledge, the complete KTE curve of
atooth pair cannot be obtained by the FEM-based TCA).
,eappropriateness of the selected torque for FEM-based TCAcan be
validated if the obtained KTE is approximatelyconsistent with KTEs
acquired by traditional TCA. It can beseen from Figure 1 that, in
contact zone 1, the KTE (absolutevalue) of tooth pair 1 is less
than that of tooth pair 2, which
means in this region tooth pair 1 is in contact, whereas
toothpair 2 is not in contact in the no-loading condition. It
isbecause the mating tooth surfaces of tooth pair 2 are sep-arated
from each other by the profile separation which canbe measured by
the difference in KTEs between tooth pairs 1and 2. Likewise, for
contact zones 2 and 3, only tooth pairs 2and 3 are in contact,
respectively, in the unloaded condition.,at is why the KTEs of the
gear pair in contact zones 1, 2,and 3 correspond to the KTEs of
tooth pairs 1, 2, and 3,respectively.
,e total deformation of a gear pair is usually consideredthe
difference between the KTE and the LTE of the gear pair.,is
difference, however, is not always themesh deflection ofa tooth
pair caused by the load it bears. ,e schematicdiagram for
illustrating the deflection of the meshing toothpair is shown in
Figure 10. As previously mentioned, at anyrolling position in
contact zone 1 (shown in Figure 1), onlytooth pair 1 is in contact
in the no-loading condition and themating tooth surfaces of tooth
pair 2 are separated by theprofile separation, as shown in Figure
10. With the augmentof torque load, the LTE of the gear pair
increases graduallybecause of the mesh deflection of tooth pair 1.
If the LTE ofthe gear pair at any rolling position in contact zone
1 isgreater than the KTE of tooth pair 2, which means the
profileseparation of tooth pair 2 at this position has been
closedbecause of tooth rotation, tooth pair 2 will be in contact
andshare the whole load with tooth pair 1. ,erefore, as
illus-trated in Figure 10, part of the rotation of tooth pair
2compensates the profile separation between the matingflanks and
the remaining part is the result of mesh deflection.At any rolling
position in contact zone 2, as tooth pair 2 is incontact initially,
the tooth rotation of tooth pair 2 is whollyattributed to its
deflection. In contact zone 3, however, theprofile separation of
tooth pair 2 must be closed beforecontact, and hence, the remaining
part of the tooth rotationwhich excludes the angular profile
separation is the rota-tional deformation of tooth pair 2. Based on
above analysis,it can be speculated that, at the torque load of
300Nm, toothpairs 1 and 2 are in contact in region AB, only tooth
pair 2 isin contact in region BC, and tooth pairs 2 and 3 are
incontact in region CD. ,is speculation is proved to be trueby the
time histories of mesh force on each engaged toothpair shown in
Figure 9.
Above discussions elaborate the meaning of equation(10). Namely,
the tooth rotations Δθ for all engaged toothpairs are the same and
equal the rotation of the pinion andgear blanks which is measured
by the difference between theangular KTE and LTE of the gear pair,
while the meshdeflection along the line of action δi for the tooth
pair i dueto the load it shares should be calculated by subtracting
theprofile separation ΔSi from the projection of the tooth
ro-tation Δθ along the line of action. Actually, the mesh
de-flection for a tooth pair equals the difference between theKTE
of the tooth pair and the LTE of the gear pair, as shownin Figure
1.,erefore, in the MMTP, the KTE of a tooth pairis applied instead
of the KTE of a gear pair in the estimationof mesh stiffness and
the identification of the contact foreach pair of teeth. It is the
main difference from the mul-tipoint coupling mesh model.
10 Shock and Vibration
-
Figure 11 shows the mesh deflections along the line ofaction for
the gear pair and each tooth pair. ,e toothengagement cycle is
divided into regions where two meshingtooth pairs are in contact
and one meshing tooth pair is incontact (denoted by DTC and STC,
respectively, inFigures 11–13). In the double-tooth contact zone,
althoughthe rotational deformation of the gear pair equals the
ro-tational deformation of the tooth pair which is initially
incontact in the unloaded condition, their translational
de-flections along the line of action are somewhat different.,isis
because in the double-tooth contact zone, the effectivemesh point
of the gear pair is different from that of the toothpair, as shown
in Figure 4, and thus, the corresponding linesof action of the net
mesh force acting on the effective meshpoint are different as
well.
Figure 12 compares the ASMS calculations of the gearpair and the
tooth pair. It can be seen that, in the double-tooth contact zone,
the ASMS of the gear pair experiences agradual increase and then a
steady decrease. ,e variationsignificantly differs from that of
unmodified spur gearswhose mesh stiffness presents a sharp increase
and decreaseat the instants when the approaching tooth pair comes
intocontact and the receding tooth pair leaves the mesh,
re-spectively [21]. Apart from that, with the contact ratio
increase, the variation of the mesh stiffness of the gear
pairtends to become smaller. ,is conclusion can be proved bythe
comparison of the ASMS calculations between this studyand studies
of Peng [13] and Wang et al. [20].
An important fact shown in Figure 12 is that, in thedouble-tooth
contact zone, the ASMS of a gear pair does notequal the sum of the
ASMS of each meshing tooth pair, if theKTE of a tooth pair is
applied to estimate the mesh stiffnessof a tooth pair, except for
the position where the twomeshing tooth pairs are in contact in the
unloaded condi-tion, such as point A in the graph. ,e reason is
explained indetail in Appendix A. In many previous studies on
meshstiffness calculation of spur gears [33–36], the total
meshstiffness of the gear pair in the multiple-tooth
engagementregion is calculated by summation of the mesh stiffness
of alltooth pairs in contact. ,is estimation method is
probablyinapplicable for spiral bevel or hypoid gears in terms of
theASMS evaluation, whereas it is proved to be feasible for theLTMS
calculation.
For the calculation of the LTMS expressed in equation(12), LTCA
has to be performed at the torque loads of T,T + ΔT, and T − ΔT to
obtain the corresponding LTEs εT,εT+ΔT, and εT− ΔT. Since the mesh
parameters are all load-dependent, ΔT should be properly chosen to
ensure that theeffective mesh points and the directions of the line
of actionare almost the same for the three torque loads at each
rollingposition. However, it does not mean that the smaller the
ΔTis, the better the result will be since the effect of
numericalcalculation errors on the calculation results will
becomemore pronounced as ΔT decreases. In the present study,ΔT �
10Nm can weigh the two factors mentioned above.Figure 13 compares
the LTMS calculations of the gear pairand the tooth pair. It can be
seen that, in the double-toothcontact zone, the LTMS of the gear
pair approximatelyequals the sum of the LTMS of simultaneously
contactingtooth pairs. ,e reason is mathematically clarified in
Ap-pendix A. Comparing calculations in Figures 12 and 13, wecan
observe that, for both the mesh stiffness of a gear pairand a tooth
pair, the LTMS is larger than the ASMS over theentire tooth
engagement cycle. It is consistent with the
5500500045004000350030002500200015001000
5000
0 0.2 0.4 0.6 0.8 1 1.2
Mes
h fo
rce (
N)
X: 0.207Y: 638.7
X: 0.171Y: 161.8
X: 0.171Y: 4436
X: 0.711Y: 4416
X: 0.783Y: 720.5
X: 0.747Y: 230.4 X: 1.107
Y: 25.56
X: 1.071Y: 358.4
Pinion roll angle (rad)
One tooth engagement
Tooth pair 1Tooth pair 2
Tooth pair 3Gear pair
Figure 9: Finite element calculation of the effective mesh force
of the tooth pair and the net mesh force of the gear pair over one
toothengagement cycle.
Tooth pair 2
Tooth pair 1
Profile separation
Deformed gear
Undeformedgear
Pinion
Gear
Figure 10: Deflection of the meshing tooth pair at the
meshingposition in contact zone 1.
Shock and Vibration 11
-
theoretical prediction from the concave force-deflectioncurve
shown in Figure 8. It has been known that thenonlinear contact
deformation due to the growth of the
contact area as applied load increases results in the
concaveshape of the force-deflection curve. Note that the
differencebetween the computed results of the ASMS and LTMS is
the
DTC STC DTC
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Mes
h de
flect
ion
alon
gth
e LO
A (m
m)
0.2 0.4 0.6 0.8 1 1.20Pinion roll angle (rad)
Tooth pair 1Tooth pair 2
Tooth pair 3Gear pair
Figure 11: Mesh deflection along the line of action of the gear
pair and the tooth pair over one tooth engagement cycle at 300Nm
appliedtorque.
DTC STC DTC
Tooth pair 1Tooth pair 2Tooth pair 3
Gear pair
×105
0.2 0.4 0.6 0.80 1.21Pinion roll angle (rad)
1
2
3
4
5
6
ASM
S (N
/mm
)
A
A
A
X: 0.351Y: 5.783e + 05
X: 0.351Y: 3.227e + 05
X: 0.351Y: 2.67e + 05
Figure 12: ,e ASMS of the gear pair and the tooth pair over one
tooth engagement cycle at 300Nm applied torque.
0.2 0.4 0.6 0.8 1 1.20Pinion roll angle (rad)
Tooth pair 1Tooth pair 2
Tooth pair 3Gear pair
×105
STC DTCDTC
1
2
3
4
5
6
7
LTM
S (N
/mm
)
Figure 13: ,e LTMS of the gear pair and the tooth pair over one
tooth engagement cycle at 300Nm applied torque.
12 Shock and Vibration
-
result of their definitions and calculation approaches, and
wecannot make a conclusion about which prediction is moreaccurate.
Both the ASMS and LTMS can be used to calculateDMF, but the
corresponding calculation formulas are en-tirely different.
4. DMF Calculation
4.1. Calculation Approach Using ASMS. A common repre-sentation
of DMF Fm d utilizing the ASMS is [13, 14, 18, 20]
Fmd �
kma εd − ε0( + cm _εd − _ε0( , εd − ε0 > 0( ,
0, − b≤ εd − ε0 ≤ 0( ,
kma εd − ε0 + b( + cm _εd − _ε0( , εd − ε0 < − b( ,
⎧⎪⎪⎨
⎪⎪⎩
(13)
where kma and cm are the ASMS and proportional meshdamping and
εd and ε0 are the translation form of the dy-namic transmission
error and KTE along the line of action.,e gear backlash b is
expressed on the coast side here. It isnoted that here the
translational KTE ε0 should be de-termined by ε0 � ((Np/Ng)Δθp0 −
Δθg0)λ which is differentfrom the traditional definition of the KTE
(equation (8)).
,e DMF is composed of the elastic contact force Fm,a �kma(εd −
ε0) and the viscous damping force Fc � cm(_εd − _ε0).,e elastic
contact force Fm,a is illustrated by the blue dashedline in Figure
8. It can be observed that as long as the dynamicmesh deflection δd
is not equal to the static mesh deflectionδT caused by the nominal
load, the calculated contact forcewill differ from the actual one
Fm. ,e deviation can beacceptable if the dynamic mesh deflection
fluctuates aroundthe nominal static mesh deflection within a small
range.However, as for moderate and large amplitude vibrationswhere
a substantial difference exists between δd and δ
T, thevalidity of Fm,a may be questioned.
4.2. Calculation Approach Using LTMS. As shown in Fig-ure 8, for
small mesh deflections relative to the nominalstatic deflection δT,
the elastic contact force Fm,l (demon-strated by the red
dash-dotted line) evaluated by using theLTMS is closer to the true
value Fm than the contact forceFm,a. ,erefore, in this case, it
could be more appropriate toapply the LTMS to calculate DMF. ,e DMF
Fm d consid-ering backlash nonlinearity can be defined as
Fmd �
Fm0 + kml εd − εl( + cm _εd − _εl( , Fr > 0( ,
0, Fr ≤ 0 and εd − ε0 ≥ − b( ,
kma εd − ε0 + b( + cm _εd − _ε0( , Fr ≤ 0 and εd − ε0 < − b(
,
⎧⎪⎪⎨
⎪⎪⎩
(14)
where kml is the LTMS and εl is the translational LTE alongthe
line of action. ,e nominal contact force Fm0 can beacquired from
LTCA.,e reference force Fr which is used todetermine the
operational condition is given by
Fr � Fm0 + kml εd − εl( . (15)
Since the calculation approach using the LTMS is onlysuitable
for small mesh deflections about the nominal static
deflection, this approach is only applied to calculate theDMF
for the drive side, while the DMF for the coast side isstill
estimated by the approach using the ASMS. It has beenfound that
mesh properties for the coast side have less effecton dynamic
responses than those of the drive side, and theirinfluences appear
only in double-sided impact regions forlight-loaded cases [12].
,erefore, if the evaluation of DMFfor the drive side is of primary
interest, it is reasonable toignore the mesh characteristic
asymmetry. In this study, themesh parameters of the drive side are
used to approximatelyevaluate the DMF of the coast side for
simplicity.
It is noteworthy that the representations of the elasticcontact
force in equations (13) and (14) are fundamentallydifferent, and
hence, it is incorrect to apply the ASMS inequation (14) and apply
the LTMS in equation (13). Forexample, the use of the LTMS in
equation (13) will give afalse result Fm,e (demonstrated by the
green dash-dotted linein Figure 8) which differs from the actual
one Fmdramatically.
4.3. Calculation Approach Using Mesh Force
InterpolationFunction. As illustrated in Figure 8, both the
calculationapproach using the ASMS and the one applying the
LTMSwill lose accuracy when the dynamic mesh deflection
sub-stantially differs from the nominal static mesh
deflectionbecause at any mesh position, the elastic contact is
modeledas a linear spring, and therefore, the nonlinear
force-de-flection relationship is ignored. ,e effect of such
simpli-fication will be examined later by comparing the
DMFcalculated by the two approaches with the DMF interpolatedfrom
the force-deflection function. Since the mesh force alsovaries with
the contact position of meshing tooth pairs, theinterpolation
function indexed by the nominal mesh posi-tion and the mesh
deflection at this position is required.,ismethod is also applied
by Peng [13] and recently by Dai et al.[37] to the prediction of
the total DMF of hypoid gear pairsand spur gear pairs,
respectively. Here, the interpolationfunction for the mesh force of
individual tooth pairs at eachmesh position over one tooth
engagement cycle is developedthrough a series of quasistatic LTCAs
with the appliedtorque load ranging from 10Nm to 3000Nm. ,is range
farexceeds the nominal load (50Nm and 300Nm in the presentwork)
that the gear bears because we need to obtain therelationship
between the mesh force and the mesh deflectionat the beginning and
end of the tooth meshing cycle underthe nominal load. Looking back
at Figure 6(a), we canobserve that the mesh force at these two
positions will bezero if the applied load is below the nominal
load, or be verysmall if the applied load is just over the nominal
load.,erefore, in order to obtain sufficient data for the
devel-opment of the force-deflection relationship at these
twopositions, the range of the applied torque load should bewide
enough. ,e force-deflection function at each meshposition over one
tooth engagement cycle is graphicallyshown in Figure 14. Note that
even when the load applied forLTCA is 3000Nm, the data for the mesh
deformation ex-ceeding 0.004mm at the position of tooth engagement
anddisengagement are not available. Likewise, because of load
Shock and Vibration 13
-
sharing, the data points marked in Figure 14 cannot beobtained
by LTCA with the maximum applied load of3000Nm. ,ese marked data
are estimated by the LS-SVMlab toolbox [31] based on the data
obtained from LTCA.Although the predicted data may not be very
precise, theyhave a marginal effect on the steady-state dynamic
responsesince the dynamic mesh deformation of a tooth pair in
theregion of tooth engagement and disengagement is also small,and
thereby, the precise data from LTCA are used.
,e interpolated DMF Fm d is expressed as follows:
Fm d �
fd εd − ε0, θt( + cm _εd − _ε0( , εd − ε0 > 0( ,
0, − b≤ εd − ε0 ≤ 0( ,
fc εd − ε0 + b, θt( + cm _εd − _ε0( , εd − ε0 < − b( ,
⎧⎪⎪⎨
⎪⎪⎩
(16)
where fd and fc represent the interpolation functions for
thedrive and coast sides, which are the same in this work; εd −
ε0and εd − ε0 + b express the dynamic mesh deformation along
the line of action for the drive and coast sides,
respectively;and θt is the mesh position of a tooth pair.
5. Gear Pair Dynamic Model
5.1. A 12-DOF Lumped-Parameter Model. ,e gear pairmodel is
illustrated in Figure 15(a). Each gear with itssupporting shaft is
modeled as a single rigid body. ,e gear-shaft bodies are each
supported by two compliant rollingelement bearings placed at
arbitrary axial locations. In thecurrent case, the pinion is
overhung supported, while thegear is simply supported.
As shown in Figure 15(b), the position of each gear-shaftbody in
the space is determined by its centroid position Cjand the attitude
of the body relative to the centroid, whichare defined by the
translational coordinates (xj, yj, zj) andthe Cardan angles (αj,
βj, cj), respectively. ,e rotationprocess of the body about the
centroid described by theCardan angles is as follows: the body
firstly rotates about thex1j-axis with the angle αj from the
original position indicated
18000
16000
14000
12000
10000
8000
6000
4000
2000
0.024 0.018Deformation along LOA (mm) Tooth pair m
esh position (rad)0.012 0.006 0
0
0.2 0.3 0.4 0.50.6 0.7 0.8 0.9
0 0.1
Elas
tic co
ntac
t for
ce (N
)
Figure 14: Relationship between the tooth contact force and its
corresponding deformation along the line of action at each mesh
position.
A
A
B
B
x
y
O
y0g
z0g
z0p
y0p
LAp
LAg
Lp
LBp
LBg
LgOg
Op
(a)
y1j
y0j
z0j
x0j
x3j
Cj
Oj
x1j (x2j )
z3j (z4j )
y2j (y3j )
z2j
z1j
γj
αj
αj
βj
βj
(b)
Figure 15: (a) Gear pair model and (b) coordinate systems of
each gear-shaft body.
14 Shock and Vibration
-
by the coordinate system Cjx1jy1jz
1j to the position repre-
sented by the coordinate system Cjx2jy2jz
2j , and then it ro-
tates about the y2j-axis with the angle βj reaching thelocation
shown by the coordinate system Cjx3jy
3jz
3j .,e last
rotation is about the z3j-axis with the angle cj, and the
bodyreaches the final instantaneous position represented by
thecoordinate system Cjx4jy4jz4j . Generally, αj and βj are
small
angles, while cj is the large rolling angle of the gear. In all
thecoordinates mentioned above and the following equations,the
subscript j � p, g indicates that the quantities belong tothe
pinion or gear component. ,e derivation of theequations of motion
is demonstrated in Appendix B. Here,the detailed equations of
motion of the gear pair with 12DOFs are given directly as
follows:
mp €xp � − xp l�A,B
klpxt − βp
l�A,B
klpxtZ
lp − _xp
l�A,B
clpxt −
_βp l�A,B
clpxtZ
lp +
N
i�1npx,iFm,i, (17a)
mp €yp � − yp l�A,B
klpxt + αp
l�A,B
klpxtZ
lp − _yp
l�A,B
clpxt + _αp
l�A,B
clpxtZ
lp +
N
i�1npy,iFm,i, (17b)
mp €zp � − zp l�A,B
klpzt − _zp
l�A,B
clpzt +
N
i�1npz,iFm,i, (17c)
Jxp€αp + J
zp
_βp _cp + Jzpβp€cp � − θpx
l�A,B
klpxr +
l�A,B
klpxtZ
lp − αpZ
lp + yp −
_θpx l�A,B
clpxr +
l�A,B
clpxtZ
lp − _αpZ
lp + _yp +
N
i�1λpx,iFm,i,
(17d)
Jxp€βp − J
zp _αp _cp − J
zpαp€cp � − θpy
l�A,B
klpxr −
l�A,B
klpxtZ
lp βpZ
lp + xp −
_θpy l�A,B
clpxr −
l�A,B
clpxtZ
lp
_βpZlp + _xp +
N
i�1λpy,iFm,i,
(17e)
Jzp€cp � Tp +
N
i�1λpz,iFm,i, (17f)
mg €xg � − xg l�A,B
klgxt − βg
l�A,B
klgxtZ
lg − _xg
l�A,B
clgxt −
_βg l�A,B
clgxtZ
lg +
N
i�1ngx,iFm,i, (17g)
mg €yg � − yg l�A,B
klgxt + αg
l�A,B
klgxtZ
lg − _yg
l�A,B
clgxt + _αg
l�A,B
clgxtZ
lg +
N
i�1ngy,iFm,i, (17h)
mg €zg � − zg l�A,B
klgzt − _zg
l�A,B
clgzt +
N
i�1ngz,iFm,i, (17i)
Jxg€αg + J
zg
_βg _cg + Jzgβg€cg � − θgx
l�A,B
klgxr +
l�A,B
klgxtZ
lg − αgZ
lg + yg −
_θgx l�A,B
clgxr +
l�A,B
clgxtZ
lg − _αgZ
lg + _yg +
N
i�1λgx,iFm,i,
(17j)
Jxg€βg − J
zg _αg _cg − J
zgαg€cg � − θgy
l�A,B
klgxr −
l�A,B
klgxtZ
lg βgZ
lg + xg −
_θgy l�A,B
clgxr −
l�A,B
clgxtZ
lg
_βgZlg + _xg +
N
i�1λgy,iFm,i,
(17k)
Jzg€cg � − Tg +
N
i�1λgz,iFm,i, (17l)
Shock and Vibration 15
-
where N is the maximum number of tooth pairs that
si-multaneously participate in the meshing; Tp and Tg are
thedriving torque acting on the pinion body and the torque
loadapplied to the gear body, respectively; Fm,i is the DMF ofeach
tooth pair and can be calculated by equations (13), (14),or (16)
according to the needs; and nj,i(njx,i, njy,i, njz,i) is thecontact
force directional cosine vector of eachmeshing toothpair i. ,e
calculation of the directional rotation radius λjx,i,λjy,i, and
λjz,i is given by
λjx,i, λjy,i, λjz,i � njz,iyjm,i − njy,izjm,i, njx,izjm,i
− njz,ixjm,i, njy,ixjm,i − njx,iyjm,i.
(18)
,e evaluation of sliding friction is beyond the scope ofthis
article, and thereby, the current dynamic model isformulated
without consideration of sliding friction.However, it should be
noted that friction is a significantexternal source of excitation
influencing the dynamic be-havior of gearing systems. Explanations
for other symbolsused above are given in Appendix B and
nomenclature. ,eequations of motion given above are expressed based
on theMMTP. If the MMGP is applied, the DMF and mesh pa-rameters of
the individual tooth pair should be replaced bythose of the gear
pair. ,e following description is also basedon the MMTP. ,e related
information about the MMGP isintroduced in the study by Peng
[13].
Geometric modeling of spiral bevel gears, TCA, andLTCA are
performed in the global fixed reference frameOxyz (as shown in
Figure 2), so it is necessary to transformthemesh point position
rm,i and the line of action vector Li tothe local inertial
reference frames of the pinion and gearOjx
0jy
0jz
0j in which the dynamic analysis is performed. ,e
relationship between rpm,i(xpm,i, ypm,i, zpm,i) and
rm,i(xm,i,ym,i, zm,i) is
xpm,i
ypm,i
zpm,i
1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
� MOpO′rm,i �
0 0 − 1 0
0 1 0 0
1 0 0 − OpO
0 0 0 1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
xm,i
ym,i
zm,i
1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
�
− zm,i
ym,i
xm,i − OpO
1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
(19)
,e directional cosine vector of the mesh force acting onthe
pinion np,i(npx,i, npy,i, npz,i) is
npx,i
npy,i
npz,i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦� MOpO − Li( �
0 0 − 1
0 1 0
1 0 0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
− nx,i
− ny,i
− nz,i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ �
nz,i
− ny,i
− nx,i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.
(20)
,e relationship between rgm,i(xgm,i, ygm,i, zgm,i) andrm,i(xm,i,
ym,i, zm,i) is
xgm,i
ygm,i
zgm,i
1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
� MOgO′rm,i �
0 0 − 1 0
1 0 0 0
0 − 1 0 − OgO
0 0 0 1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
xm,i
ym,i
zm,i
1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
�
− zm,i
xm,i
− ym,i − OgO
1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
(21)
,e directional cosine vector of the mesh force acting onthe gear
ng,i(ngx,i, ngy,i, ngz,i) is
ngx,i
ngy,i
ngz,i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦� MOgOLi �
0 0 − 1
1 0 0
0 − 1 0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
nx,i
ny,i
nz,i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ �
− nz,i
nx,i
− ny,i
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦. (22)
It is noted that the direction of the mesh force acting onthe
pinion is opposite to the direction of the line of actionvector
Li.
5.2. Dynamic Transmission Error Calculation. ,e trans-lational
dynamic transmission error vector is defined as thedifference
between the effective mesh point position vectorsof the pinion and
gear. ,e dynamic transmission erroralong the line of action εd can
be obtained by projecting thetranslational dynamic transmission
error vector along theline of action vector.
For the MMGP, the time-dependent mesh parametersare often
treated as the function of the position on the meshcycle (PMC)
[29], while for theMMTP, the mesh parametersof each tooth pair
should be treated as the function of theposition on the one tooth
engagement cycle (PTEC). Forcurrent formulation, all the teeth are
assumed to be equallyspaced and have the same surface topology. ,e
PMC andPTEC are calculated according to the following
equations:
PMC � θpz + θpz,0
− floorθpz + θpz,0
θpz,p⎛⎝ ⎞⎠θpz,p, (23)
PTEC � θpz + θt,0
− floorθpz + θt,0
Nθpz,p⎛⎝ ⎞⎠Nθpz,p, (24)
where θpz is the pinion roll angle which approximatelyequals cp
as shown in equation (B.11); θpz,0 is the initialangular position
of the pinion which can be arbitrarily se-lected; and θpz,p is the
angular pitch. If equation (24) isapplied to calculate the PTEC of
the reference tooth pair, θt,0is the initial angular position of
the reference tooth pairwhich can also be chosen discretionarily,
and N is themaximum number of tooth pairs that participate in
themeshing simultaneously. ,e PTEC of other simultaneouslyengaged
tooth pairs can be deduced by the PTEC of thereference tooth pair
and the number of angular pitchesbetween them. In both equations,
the “floor” functionrounds a number to the nearest smaller
integer.
16 Shock and Vibration
-
On the pinion side, it is convenient to express the
in-stantaneous position vector of the effective mesh point in
thecoordinate system Cpx3py
3pz
3p:
r3pm,i � xpm,i PTECi( i3p + ypm,i PTECi( j
3p + zpm,i PTECi( k
3p.
(25)
To consider the effect of large rotational angles θpz(orcp)and
θgz(orcg) on the dynamic transmission error, animaginary coordinate
system CgxIgyIgzIg considering theinstantaneous angular TE about
the z-axis on the gear side(θgz − (θpz/R)) is established. ,e
effective mesh pointposition vector of the gear expressed in
CgxIgy
Igz
Ig is
rIgm,i � xgm,i PTECi( iIg + ygm,i PTECi( j
Ig + zgm,i PTECi( k
Ig.
(26)
Transforming the coordinates to the global fixed refer-ence
frame Oxyz, we obtain
rOpm,i � MOOp′MOpCp′r3pm,i,
rOgm,i � MOOg′MOgCg′rIgm,i,
(27)
where the transformationmatrices forCpx3py3pz3p toOxyz are
MOOp′ �
0 0 1 OpO
0 1 0 0
− 1 0 0 0
0 0 0 1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
,
MOpCp′ �
1 0 βp xp0 1 − αp yp
− βp αp 1 zp0 0 0 1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
(28)
,e transformation matrices for CgxIgyIgzIg to Oxyz are
MOOg′ �
0 1 0 0
0 0 − 1 − OgO
− 1 0 0 0
0 0 0 1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
,
MOgCg′ �
1 − θgz − θpz/R βg xg
θgz − θpz/R 1 − αg yg− βg αg 1 zg0 0 0 1
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
(29)
,e dynamic transmission error along the line of actionfor each
pair of teeth in contact is calculated by
εd,i � rOpm,i − r
Ogm,i · Li. (30)
6. Numerical Simulation Analysis
,e proposed equations of motion are solved by MATLAB’sODE45 [38]
which implements the 4/5th order Runge–Kutta
integration routine with the adaptive step size. ,e second-order
differential form of equations (17a)–(17l) must berewritten in the
state space (first-order) form before they canbe solved by the
solver. Step speed sweep simulations areperformed to obtain the
dynamic response in a prescribedrange of mesh frequency. Numerical
integration at eachmesh frequency is performed up to the time when
thesteady-state response is reached. ,e variable values of thefinal
state at this mesh frequency are applied as the initialconditions
for dynamic analysis at the next one. Time-do-main DMF response at
a specified mesh frequency iscomputed directly from the solution of
numerical in-tegration. Frequency response is obtained by root
meansquare (RMS) values of the time-domain response over onetooth
engagement period at each frequency. Note that timehistories of DMF
on each pair of teeth over one tooth en-gagement cycle are not the
same at the frequencies wheresubharmonic or chaotic motion occurs,
which results indifferent RMS values for different tooth pairs. In
such cases,the average value is adopted for graphing.
,e system parameters are listed in Table 2. ,e ratedtorque load
of the gear pair is about 700Nm. Considering thedynamic
characteristics formedium- and heavy-load cases aresimilar to but
different from those for the light-load case [13],we performed
dynamic simulations at 50Nm and 300Nmtorque loads to examine
themodeling effects for the light-loadand medium-to-heavy-load
conditions, respectively.
6.1. Effects ofMeshModel. To preclude the influence of
meshstiffness on the calculation results of DMF, the more oftenused
ASMS is adopted for both the MMTP and the MMGP.
Figure 16 compares the DMF responses with respect tothe mesh
frequency from the two models. It can be seen thatthe simulation
results from the two models are overallconsistent, which partly
verifies the validity of the MMTPproposed in this work. Differences
exist in the responseamplitudes particularly at higher mesh
frequency, but theresonance frequencies estimated by the two models
areapproximately the same. In the lightly loaded condition(50Nm),
jump discontinuities in the vicinity of the reso-nance frequencies
are predicted by both models. ,e toothseparation and impact occur
in the frequency range betweenthe upward and downward jump
discontinuities. ,e jumpfrequencies near the resonance frequency of
about 3000Hzare different for speed sweep up and down
simulationsowing to the strong nonlinearity of the system. For
themedium-to-heavy-loaded case (300Nm), the nonlineartime-varying
responses from both models are continuouscurves, and the results
for speed sweep up and down casesare just the same. ,e
disappearance of nonlinear jumpsmeans the load has become large
enough to prevent the lossof contact and the backlash nonlinearity
is ineffectual.Generally, the increase of load tends to weaken the
effect ofbacklash nonlinearity not only because of the higher
meanload which plays the role in keeping tooth surfaces in
contact[11] but also because the variations of mesh stiffness,
meshvector, and LTE which all aggravate the degree of
backlashnonlinearity are alleviated with the increase of load.
Shock and Vibration 17
-
,e essential differences between the MMTP and theMMGP can be
explored further by comparing the timehistories of DMF per tooth
pair. Analysis results that candescribe some typical phenomena of
gear meshing are il-lustrated as follows.
Figure 17 compares the response histories at 50Hz meshfrequency
and 300Nm torque load. As can be seen, globally,the DMF predicted
by the MMGP waves smoothly aroundthe static MF obtained through
LTCA. By contrast, the DMFfrom the MMTP seems to fluctuate more
frequently withdramatic oscillation (due to the tooth engaging-in
impact)occurring at the start of tooth engagement which,
however,cannot be observed in the results from the MMGP. ,eoverall
difference in the predicted DMFs can be attributed tothe fact that
the MMGP uses the mesh parameters whichreflect the gentle overall
effect of time-varying mesh char-acteristics on the entire gear
pair, while the more stronglychanged mesh parameters of a tooth
pair are applied in theMMTP.
It is known that tooth impact always occurs at themoment of the
first contact of each tooth pair. ,is phe-nomenon can be explained
by the short infinite high ac-celeration appearing in the
acceleration graph (which is thesecond derivative of the motion or
KTE graph) at thechangeover point between two adjacent pairs of
teeth [39].,e results shown in Figure 17 indicate that the MMTP
iscapable of describing tooth impact at the instant of en-gagement
and disengagement, but the MMGP is in-competent in this respect. No
impact occurs in theestimation from the MMGP because the DMF on
each toothpair in simultaneous contact can only be roughly
estimatedthroughmultiplying the net mesh force by the
quasistaticallydefined load distribution factor which gradually
increasesfrom zero in the engaging-in stage and decreases to
zerowhen the tooth pair leaves the contact.
Another interesting phenomenon shown in the graph isthat the DMF
from the MMTP oscillates above the staticmesh force in the
engaging-in stage but below the staticmesh force in the
engaging-out stage. To find the reason, weapplied the MMTP to a
two-DOF dynamic model whichtakes into account only the torsional
motions of the pinionand gear. It is shown in Figure 17 that the
predicted DMFfluctuates around the static mesh force from LTCA.
,isseems to be reasonable because, in LTCA, the pinion and
gear are restricted to rotating about their rolling axes as
well.,e results thereby indicate that the deviation between theDMF
and the static mesh force is because of the change ofthe relative
position of the pinion and gear due to the releaseof additional
lateral degrees of freedom. For the MMGP, themesh force acting on
each pair of teeth obtained according tothe principle of load
distribution waves around the staticmesh force. ,e estimation,
however, may deviate from theactuality if the pose of the gear can
change in any degree offreedom. ,e simulation results also indicate
that the meshparameters obtained by the current LTCA method are
moresuitable for the two-DOF dynamic model.
At the light torque load of 50Nm (as shown in Fig-ure 18), the
deviation between the DMF and the static meshforce is reduced, and
the engaging-in impact is less signif-icant. ,is is because the
change of relative position of thepinion and gear caused by the
load is relatively small at thelightly loaded condition. ,erefore,
we can find that thetransmission performance of a spiral bevel gear
pair issusceptible to the relative position of the two gears, and
areasonable design of the external support structure plays
anessential role in improving the stability of the gear
trans-mission. Tooth flank modification is also necessary to
reducethe sensitivity of transmission performance to the
inevitablechange of relative position caused by the load or
installationerrors.
It is noteworthy that the prediction from the meshmodelwith
predefined mesh parameters has an unavoidable de-viation from the
reality as the mesh parameters under thedynamic condition differ
from those for the static case.,erefore, the impact force sometimes
may be overpredictedbecause of the inaccurate initial contact
position of a toothpair. ,e inaccuracy leads to overestimation of
the contactdeformation at the start of tooth engagement.
However,according to the contact conditions of gear teeth
discussedbefore, the tooth pair will get into contact ahead of
thequasistatically defined beginning of contact because the
largedeformation evaluated at the point of the predefined start
ofcontact means the profile separation between the matingflanks of
the extended contact area has been closed.
Figure 19 compares the DMF histories per tooth pair atthe mesh
frequency of 400Hz where loss of contact occurs.As displayed, near
resonant frequency, DMF varies drasti-cally with the peak value
much higher than static mesh force.,e two mesh models provide
almost identical results with aslight difference near the end of
tooth engagement wheretransient loss of contact only appears in the
results from theMMTP. It is worth noting that the differences
between thecalculations from the MMTP and MMGP usually appearnear
the beginning and end of tooth contact since in thesetwo regions,
the KTE curve of the tooth pair and the gearpair, which are
separately applied in the two models, isdistinct, as shown in
Figure 1.
6.2. Effects of Mesh Stiffness. Here, the MMTP is applied forall
analyses. ,e dynamic responses and time histories ofDMF per tooth
pair predicted by using the ASMS and LTMSare compared with the
results from the mesh force
Table 2: System parameters.
Parameters Pinion GearMass m (kg) 1.82 6.71Torsional moment of
inertia Jz (kg·m2) 1.065E − 3 3.94E − 2Bending moment of inertia Jx
(kg·m2) 7.064E − 4 2E − 2Centroid position of the gear L (mm)
97.507 − 41.718Bearing A axial position LA (mm) 120 200Bearing B
axial position LB (mm) 80 80Translational bearing stiffness kxt
(N/m) 2.8E8 2.8E8Axial bearing stiffness kzt (N/m) 8E7 8E7Tilting
bearing stiffness kxr (Nm/rad) 6E6 8E6Mesh damping ratio
0.06Support component damping ratio 0.02Gear backlash (mm) 0.15
18 Shock and Vibration
-
NLTV analysis with MMGP(speed sweep up)NLTV analysis with
MMGP(speed sweep down)
NLTV analysis with MMTP(speed sweep up)NLTV analysis with
MMTP(speed sweep down)
1000 2000 3000 4000 5000 60000Mesh frequency (Hz)
400
600
800
1000
1200
1400
RMS
valu
e of D
MF
(N)
(a)
NLTV analysis with MMTPNLTV analysis with MMGP
1000 2000 3000 4000 5000 60000Mesh frequency (Hz)
2900
3000
3100
3200
3300
3400
3500
3600
RMS
valu
e of D
MF
(N)
(b)
Figure 16: RMS values of the DMF acting on the individual tooth
pair evaluated at (a) 50Nm and (b) 300Nm torque loads.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–0.1Pinion roll angle
(rad)
0
1000
2000
3000
4000
5000
Mes
h fo
rce (
N)
Static MF from LTCADMF from MMGP
DMF from MMTPDMF from MMTP/2-DOF
Figure 17: Time histories of DMF per tooth pair at 50Hz mesh
frequency and 300Nm torque load.
Shock and Vibration 19
-
interpolation function, which, in theory, are believed to bemore
accurate.
Figure 20 shows the comparison of the DMF responses at300Nm
torque load. Differences occur in both responseamplitudes and
resonant frequencies and become noticeablein the range of high mesh
frequency. Overall, both theapproaches using the ASMS and LTMS tend
to un-derestimate the effective values of DMF, and the
predictedresonance frequencies consistently shift to the lower
meshfrequency range in comparison with the results from themesh
force interpolation function. Generally, the resultsfrom the model
applying the LTMS seem to be closer to theinterpolated results. ,is
finding can also be observed fromthe time histories of the
individual DMF illustrated inFigure 21.
As shown in Figure 21, DMF predicted by the mesh
forceinterpolation function changes most smoothly, and thetooth
engaging-in impact is the minimum among the threeapproaches. It
manifests that if the nonlinear relationshipbetween the mesh force
and the mesh deflection (i.e., thevariation of mesh stiffness with
mesh deflection) is
considered, the fluctuation of DMF is actually moderate.From the
graph, we can also observe that the DMF estimatedwith the LTMS
varies smoothly relative to the DMF pre-dicted by the ASMS, and the
tooth engaging-in impact is lesspowerful. Although the two methods
do not give sub-stantially different results, the approach using
the LTMS ispreferred for vibration analysis of gears around the
nominalstatic deformation. ,is is not only because the
resultspredicted with the LTMS aremore similar to the results
fromthe mesh force interpolation function which precisely
de-scribes the force versus deformation behavior but also be-cause
the calculation approach using the LTMS shown inequation (14)
better represents the relationship between theDMF and the nominal
static mesh force when gears vibratearound the nominal static
deformation.
Similar to the results shown in Figure 21, for the examplegear
pair, in the predefined mesh frequency range (0–6000Hz), the impact
rarely occurs when the tooth pair leavesthe contact. It should be
noted that whether impact occurs atthe instant of tooth engagement
and disengagement and themagnitude of the impact largely depends on
the
0100200300400500600700800900
Mes
h fo
rce (
N)
–0.1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1Pinion roll angle
(rad)
Static MF from LTCADMF from MMGPDMF from MMTP
Figure 18: Time histories of DMF per tooth pair at the mesh
frequency of 50Hz and 50Nm torque load.
0 0.1 0.2
Loss ofcontact
–0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9–0.5
1.11Pinion roll angle (rad)
0
500
1000
1500
2000
2500
Mes
h fo
rce (
N)
0200400600800
Static MF from LTCADMF from MMGPDMF from MMTP
Figure 19: Time histories of DMF per tooth pair at the mesh
frequency of 400Hz and 50Nm torque load.
20 Shock and Vibration
-
transmission error which can be modified through toothsurface
modification.
7. Conclusion
In this paper, two critical factors influencing the
calculationof dynamic mesh force (DMF) for spiral bevel gears,
i.e., themeshmodel and themesh stiffness representation, have
beeninvestigated. An improved mesh model of a tooth pair(MMTP)
which better describes the mesh characteristics ofeach pair of
teeth is proposed. As the difference between thekinematic
transmission error (KTE) of a tooth pair and theloaded transmission
error (LTE) of a gear pair represents theactual deformation of the
tooth pair caused by the load itbears, instead of the KTE of a gear
pair which is commonlyused in the single-point and multipoint
coupling meshmodels, the KTE of a tooth pair is applied in the MMTP
forthe calculation of the mesh stiffness of individual tooth
pairs,
the identification of tooth contact, and the estimation
ofdynamic mesh deflection of each pair of teeth in contact.,emesh
parameters and dynamic responses are comparedbetween the proposed
model and the mesh model of a gearpair (MMGP) (i.e., the
single-point coupling mesh model).Overall, the twomodels predict
the similar variation trend ofdynamic responses with the mesh
frequency. Althoughnoticeable differences can be observed in the
responseamplitudes particularly in the range of high mesh
frequency,the resonance frequencies estimated by the two models
arealmost the same. ,e time histories of dynamic mesh force(DMF)
per tooth pair from the two models are significantlydifferent since
the variations of mesh parameters which actas the parametric
excitations in the dynamic model aredramatic for the MMTP but
moderate for the MMGP.Compared with previous mesh models, the MMTP
is su-perior in simulating the dynamic contact behavior of eachpair
of teeth especially at the instant of tooth engagement
2800
3000
3200
3400
3600
3800
RMS
valu
e of D
MF
(N)
2000 60000 1000 500040003000Mesh frequency (Hz)
ASMSLTMSMF interpolation function
Figure 20: RMS values of the DMF acting on each tooth pair
evaluated by the approaches using the ASMS, LTMS, and mesh
forceinterpolation function at 300Nm toque load.
0
1000
2000
3000
4000
5000
Mes
h fo
rce (
N)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–0.1Pinion roll angle
(rad)
1500
1000
5000.6
ASMSLTMSMF interpolation function
4400
4200
0.3 0.4 0.5
Figure 21: Time histories of DMF per tooth pair estimated by the
approaches using the ASMS, LTMS, andmesh force interpolation
functionat the mesh frequency of 50Hz and 300Nm torque load.
Shock and Vibration 21
-
and disengagement. ,e results from the MMTP also showthat the
DMF will deviate from the static mesh force if thepose of the gear
can change in any degree of freedom. ,isphenomenon, however, has
never been observed in previousstudies where the single-point or
multipoint coupling meshmodel is applied.
For the MMGP, the total mesh stiffness of all contactingtooth
pairs is applied, while for the MMTP, the stiffness ofthe
individual tooth pair is used. Both stiffnesses can becalculated by
the average slope and local slope approaches.,e average secant mesh
stiffness (ASMS) derived from theaverage slope approach
substantially differs from the localtangent mesh stiffness (LTMS)
estimated by the local slopeapproach owing to their definitions and
calculationmethods, and we cannot make a conclusion about
whichcalculation is more appropriate. Besides, because of
theparticularity of the tooth shape, the ASMS for the MMGPcannot be
directly computed by the summation of the ASMSof each tooth pair,
but the LTMS of a gear pair approxi-mately equals the sum of the
LTMS of all tooth pairs incontact. Both the ASMS and LTMS can be
used to calculateDMF, but the calculation formulas are rather
different. ,emisuse of mesh stiffness for a given method will lead
toerrors in the physical sense.
DMF calculations from the method applying the ASMSand LTMS are
compared with those from the mesh forceinterpolation function which
accurately represents thenonlinear force-deformation relationship.
Although theresults given by the methods using the LTMS and ASMS
arenot substantially different, the former approach is recom-mended
to calculate the DMF when simulating the vibrationof gears about
the nominal static deformation. One of thereasons is that the
results predicted with the LTMS are closerto the results from the
mesh force interpolation function.Another reason is that the model
using the LTMS betterrepresents the relationship between the DMF
and thenominal static mesh force when gears vibrate around
thenominal static deformation. ,e third reason is that
theassumption, which believes the position of the effective
meshpoint does not change before and after the tooth de-formation,
is more reasonable for the calculation of LTMS.,is is because the
change of the effective mesh point po-sition caused by the mean
tooth deflection relative to theundeformed state is noticeable, but
the change caused by thesmall deformation relative to the nominal
state is negligible.,e approach applying the mesh force
interpolation func-tion can provide the most accurate prediction.
However, thedevelopment of the mesh force interpolation function
re-quires multiple time-consuming loaded tooth contact ana-lyses at
different load levels, and the interpolation in solvingdynamic
problems needs more computational efforts thanthe other two
approaches.
Abbreviations
ASMS: Average secant mesh stiffnessDMF: Dynamic mesh forceFEM:
Finite element methodKTE: Kinematic transmission error
LTCA: Loaded tooth contact analysisLTE: Loaded transmission
errorLTMS: Local tangent mesh stiffnessMMGP: Mesh model of a gear
pairMMTP: Mesh model of a tooth pairRMS: Root mean squareTCA: Tooth
contact analysis.
Nomenclature
ΔB: Composite deformation of the toothand base
b: Gear backlashcm: Mesh dampingcxt: Translational motion
dampingczt: Axial motion dampingcxr: Tilting motion dampingek:
Angular kinematic transmission errorFm: Mesh force of the gear
pairFm,i: Mesh force of the tooth pair iFm d: Magnitude of dynamic
mesh forceFnet: Magnitude of net mesh forceFm,a: Magnitude of
contact force calculated
with the average secant mesh stiffness