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; zhyqhit@hit.edu.cn

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:zhyqhit@hit.edu.cnhttps://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

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

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valu

e of D

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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.

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h fo

rce (

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1500

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ASMSLTMSMF interpolation function

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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