Dynamic Factor Formulae

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Dynamic tooth loads and quasi-static transmission errorsin helical gears Approximate dynamic factor formulae

P. Velex *, M. Ajmi

Laboratoire de Mecanique des Contacts et des Solides, UMR CNRS 5514, Batiment Jean dAlembert, INSA de Lyon 20,

Avenue Albert Einstein, 69621 VILLEURBANNE Cedex, France

Received 17 July 2006; accepted 14 December 2006Available online 8 February 2007

Abstract

Based on both single and multi degree of freedom models, some original results are presented which establish a formallink between dynamic tooth loads and quasi-static transmission errors in helical gear sets. It is demonstrated that, as longas linear behaviour is considered, a perturbation technique leads to approximate formulae which make it possible to com-pute effectively dynamic tooth loads. Comparisons with the results obtained by a time-step integration method are favour-able with considerable gains in computational time when multi degree of freedom models are employed. The proposedmethodology looks promising in the context of statistical approaches to dynamic tooth loads when parameter variabil-ity/uncertainties cannot be ignored.

2007 Elsevier Ltd. All rights reserved.

Keywords: Helical gear; Dynamics; Tooth loads; Dynamic factor; Simulations; Analytical formulae

1. Introduction

Although geared systems have a long history and are widely used in industry, the scientific study of theirdynamic behaviour has taken place mainly in the last 20 years. The motivation to analyse gear dynamics stemsfrom both noise and dynamic loading. It has been shown that noise comes from the vibrations established asthe gear teeth deform under load and transmission error (TE) [14] has been recognised as a relevant indicator

of the noise generated by a gear set. Since then, the interest of the concept has been experimentally confirmedfor narrow-faced gears (automotive applications mostly), and transmission error is now widely used in geardynamics models and for tooth modification design [59]. On the other hand, dynamic loading is critical intooth failure by either bending or contact fatigue and, for instance, its influence is incorporated in design stan-dards by a dynamic rating factor [1012].

Typical gear dynamic models are based on mass-spring representations in which the springs account for thetime-varying, possibly non-linear, tooth stiffness whereas tooth shape deviations and geometrical errors are

0094-114X/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechmachtheory.2006.12.009

* Corresponding author. Tel.: +33 4 72 43 84 51; fax: +33 4 78 89 09 80.E-mail address: [email protected] (P. Velex).

Mechanism and Machine Theory 42 (2007) 15121526

www.elsevier.com/locate/mechmt

MechanismandMachine Theory
mailto:[email protected]:[email protected]


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often considered as initial separations between the mating flanks [1318]. Some more sophisticated deform-able-body models have recently been proposed which include not only tooth deflections but also blank con-tributions [1921]. A comparatively small number of experimental investigations have been conducted mostlyon single-stage spur and helical gear sets [2,13,15,2224]. Transmission errors are measured by using encodersor accelerometers and tooth loads are evaluated by strain gauges placed at tooth fillets. Recently, Tamminanaet al. [25] have found by experimental means that, for linear behaviour, tooth loads in spur gears seemed tocorrelate to dynamic TE variations. If confirmed, this finding would allow simpler dynamic tooth force mea-surements by encoders and/or accelerometers as opposed to the direct measurement of dynamic contact forceswhich necessitates strain gauges at the root of the teeth. In a recent paper [26], the authors have investigatedfrom a theoretical perspective the contributions of TE as an excitation source in three-dimensional models.

The conclusion reached was that, rather than the time variations of quasi-static TE under load, the difference

Nomenclature

e(Mi) shape deviation with respect to ideal flanks at a potential point of contact MiFS;FD static, dynamic mesh forceki individual mesh stiffness at a potential point of contact Mikm average mesh stiffnesskUk modal stiffness associated with the kth mode shapemeq J1J2J1Rb22J2Rb21 equivalent massmUk modal mass associated with the kth mode shapeMi generic potential point of contactNLTE no-load transmission errorR FD

FSRmax maximum of R at a given speedTES quasi-static transmission error under loadvx0 VTX0 FSkm static mesh deflectionvxD V

TXDvUk VTUk UTkVx Rb1h1 Rb2h2xS;xD quasi-static, dynamic value of xK;M constant stiffness matrix, total mass matrixK averaged total stiffness matrixn1 outward unit normal vector with respect to pinion flanksVMi;V structure vector, averaged structure vector (14) and (15)X;XS;XD respectively total displacement vector, quasi-static displacement vector and dynamic displace-

ment vectorX_

0 K1FSVX0 K1F0 static solution with averaged mesh stiffnessbb base helix angled vector of the modal unknownsde(Mi) difference between the maximum ofe(Mi) and e(Mi) at time t, e(Mi) normal composite deviation

(pinion + gear) at MiU a truncated modal basisUk kth mode shapeX1, X2 angular velocity of the pinion, of the gearqk percentage of modal strain energy stored in the mesh for mode kA derivative of A with respect to the angular variable # X1tB B divided by the static mesh deflection vx0A/k kth component of vector A

P. Velex, M. Ajmi / Mechanism and Machine Theory 42 (2007) 15121526 1513


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between quasi-static TE under load and no-load TE should be considered as the relevant excitation indicator.In the present article, the relation between dynamic tooth loads (i.e., dynamic factors), and transmission errorsis examined from a theoretical point of view. Based on the formulation proposed in [26], some new results arepresented which relate quasi-static TE to dynamic forces. A general formula for dynamic tooth loads in helicalgears is then derived which, provided the variations of quasi-static TE are known (either experimentally or

numerically), enables almost instantaneous calculations of dynamic tooth loads over a range of speeds.

2. Transmission errors

Transmission error (TE) is defined as a measure of departure from perfect motion transfer between the pin-ion and gear where perfect refers to rigid bodies with ideal geometry and positioning [13]. The definition oftransmission error under load is somewhat ambiguous as its value can vary depending on the chosen measure-ment points (positions of encoders or accelerometers) or cross-sections of reference for calculating deviationsbetween actual and perfect rotation transfers from the pinion to the gear [27]. In the proposed formulation,transmission errors are defined by extrapolating the usual experimental practice, i.e., from the actual totalangles of rotation at one section of reference on the pinion shaft (subscript I) and on the gear shaft (subscriptII). The projection of transmission error on the theoretical base plane leads to the following generic definitionwhich is used throughout the text:

TE Rb1Zt

0

X1 df hI !

Rb2Zt

0

X2 df hII !

Rb1hI Rb2hII NLTE 1

with X1 X2, angular velocity of the pinion, of the gear; f, a dummy integration variable; hI; hII; the angularperturbations with respect to rigid-body rotations (degrees of freedom) at node I on the pinion shaft andat node II on the gear shaft (hI h1 and hII h2 in the case of the single d.o.f torsional model).

The normal practice is to separate, (i) no-load transmission error (NLTE) which accounts for geometricaldeviations, (ii) quasi-static transmission error under load (TES) obtained at low speeds when torques are trans-mitted and, (iii) dynamic transmission error (DTE) corresponding to loaded high-speed behaviour.

3. Single degree torsional model

3.1. Theory

For the sake of clarity, the theoretical developments are first presented based on the classical torsionalmodel in Fig. 1. It can be shown that the original semi-definite system associated with the two angles h1 (pin-ion) and h2 (gear) can be transformed into the differential equation:

X21meqx

Xi

ki

cos2 bbx Ft

Xi

kideMi

cos bb X21J2

Rb22NLTE 2

with x Rb1h1 Rb2h2; meq J

1J

2J1Rb22J2Rb21, equivalent mass; Ft

Cm0Rb1

CrRb2, tangential static mesh force; ki, indi-

vidual mesh stiffness at a potential point of contact Mi; e(Mi), shape deviation with respect to ideal flanks at apotential point of contact Mi; deMi is the difference between the maximum of e(Mi) and e(Mi) at time t, e(Mi)is the normal composite deviation (pinion + gear) at Mi; NLTE, no-load transmission error; bb, base helixangle; A , derivative of A with respect to the angular variable # X1t.

In the equation above, it is assumed that the input angular velocity X1 is kept constant and imposedwhereas the resisting torque Cr on the gear is constant. As long as the dynamic contact conditions on the teethare close to quasi-static ones and following the procedure defined in [26], the excitation functions can beexpressed via transmission errors and (2) re-written in terms of the dynamic displacement xD x xS as

X21meqxD

km cos

2 bb

1

ag

#

xD

meqX

21 TE

S

J2

J1

Rb21

Rb

2

2

NLTE" # 3

1514 P. Velex, M. Ajmi / Mechanism and Machine Theory 42 (2007) 15121526


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where km is the average mesh stiffness such thatP

iki km1 ag# and a 0 1, TES is the quasi-static

transmission error under load.After neglecting the viscous forces in comparison with the elastic forces, the dynamic tooth loading is given

by

FD FS km1 ag# cosbbxD 4with FS Ftcos bb, the static normal mesh force, from which, the dynamic factor R, defined as the ratio of thetotal dynamic to the total static mesh force, is expressed as

R FDFS

1 xD ag#xD 5

where

xD xDx0S

x0S

FS

km cosbb; static normal deflection:

There is no exact solution to the parametrically excited differential Eq. (3) but approximate solutions can befound by using perturbation methods [15,26]. Assuming that a 00 1 (this assumption being more suited tohelical gears), particular stable solutions can be sought as straightforward asymptotic expansions of the form

xD xD0 axD1 a2xD2 6The identification of the main order terms leads to

xD0 2n-xD0 -2xD0 TES kNLTE 7with - 1

X1

ffiffiffiffiffikmmeq

qcosbb, normalised eigenfrequency of the undamped system with averaged mesh stiffness; n,

damping factor; k

J2

J1

Rb21

Rb22 :

T

Z

O 21

O

Gear

Pinion

S

1

2

iM

1n

X

Z

1T

'

1T

2T

'

2T

Fig. 1. One degree of freedom torsional model mesh stiffness model on base plane.



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Assuming that the transmission error function TES kNLTE can be developed in Fourier series of theform

TES kNLTE B0 XnP1

An sin n# Bn cos n# 8-a

hence

TES kNLTE XnP1

n2An sin n# Bn cos n# 8-b

a particular solution of (7) can be sought as

xD0 XnP1

an sin n# bn cos n# 9

which, when re-injected in (7) and after identification of like terms in cosine and sine, leads to

an An -n 2 1 ! 2Bnn -n

-

n

2 1

!2 4n2 -

n

2 10-a

bn Bn

-

n

2 1

! 2Ann -

n

-

n

2 1

!2 4n2 -

n

2 10-bFrom (5), (9) and (10), the main order approximation of the dynamic factor is derived as

R ffi 1 xD0 ffi 1 XnP1

An-

n

2 1

! 2Bnn -

n

-

n

2 1

!2 4n2 -

n

2 sin n# Bn

-

n

2 1

! 2Ann -

n

-

n

2 1

!2 4n2 -

n

2 cos n# 11

with An Anx0S and Bn Bnx0S

, normalised coefficients of the Fourier series decomposition of quasi-static transmis-sion errors.

To the authors knowledge, (11) is original as it establishes a formal link between quasi-static transmissionerror variations (via An and Bn and dynamic tooth load factor R for helical gears (the latter being a conse-quence of the assumed small variation in mesh stiffness, i.e., a

001). Further analytical results can be derived

in the case of perfect helical gears, i.e., high precision unmodified gears for which the excitations termsPikideMi

and X21

J2Rb2

2

NLTE vanish. The quasi-static transmission error is determined by solving (2) whenX1 shrinks to zero as

TES Ftkm cos2 bb1 ag#

ffi x0S1 ag# 12

from which, it can be observed that the normalised coefficients An and Bn of TES in the dynamic factor Eq. (11)are totally independent of the transmitted load. Furthermore, if one assumes that mesh stiffness is propor-tional to instantaneous contact length, the time-varying function in (12) can be decomposed as follows [28]:

ag

#

Xk kk cos kZ1# gk sin kZ1# 13



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with

kk 12eaebp2k

2cos2pkeb cos2pkea cos2pkea eb 1

gk 1

2eaebp2k2sin2pkeb sin2pkea sin2pkea eb

which shows that the excitations are mostly controlled by profile and overlap contact ratios.

Table 1Gear data

Tooth number 16 59Pressure angle () 20Helix angle () 11.00Profile shift coefficient 0.4925 0.42375

Module (mm) 8.00Face width (mm) 171.40 158.75Addendum coefficient 1.0 1.0Dedendum coefficient 1.4 1.4Tool radius/module 0.4 0.4Centre distance (mm) 312.43Pitch diameter (mm) 133.30 491.56Base diameter 122.26 450.84External diameter (mm) 154.28 503.62Root diameter (mm) 115.88 465.22Theoretical profile contact ratio ea 1.448Total contact ratio ea eb (theoretical) 2.653

P

T

c

1'

T1''

T2''

T2'

Fig. 2. Definition of tooth modifications. (a) Definition of profile modifications Pc is the tip relief amplitude, Lc is the extent ofmodification measured on the base plane. (b) Definition of parabolic lead modifications; (a) is the lead modification amplitude, (b) is the

face width.



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3.2. Comparisons with time step integrations

The validity of the approximate expression of the dynamic factor (11) has been assessed via comparisonswith the results obtained by the time-step integration of (2) using a Newmarks scheme combined with a nor-mal contact algorithm [18]. The gear data are given in Table 1 and the profile and lead modifications are

defined in Fig. 2. The system is loaded by a constant torque of 5976 N m applied on the pinion shaft. The

1.0

1.4

1.8

2.2

2.6

3.0

0 200 400 600 800 1000 1200

1 (rad/s)

Rmax

Time-step integrationApproximate formula (11)

Fig. 3. Maximum of dynamic factor versus pinion speed comparisons between time-step integrations and formula (11). Single degree offreedom torsional model (linear profile relief on pinion and gear tips with Pc 2 lm and Lc 0:2T1T2 + parabolic crowning ofamplitude 10 lm on the pinion).

1.0

1.8

2.6

3.4

4.2

5.0

0 200 400 600 800 1000 1200

1 (rad/s)

R

max

Time-step integration Approximate formula (11)

Fig. 4. Maximum of dynamic factor versus pinion speed comparisons between time-step integrations and formula (11). Single degree of

freedom torsional model (unmodified pinion and gear).



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tooth mesh stiffness is determined by using the ISO formulae [10] and a unique damping factor of 0.01 is used.Fig. 3 shows the evolution of the maximum of RRmax versus the pinion speed and a good agreement isobserved between the analytical results from (11) and the dynamic factor obtained by numerical integration.One limitation of the proposed approach is illustrated in Fig. 4 which represents the same curve but forunmodified tooth flanks. The dynamic effects are more marked and, near the major tooth critical speed, con-

tacts between the teeth are momentarily lost and shocks occur which cannot be reproduced by (11).

4. Generalisation to three-dimensional models

4.1. Dynamic factor formula for helical gears

The objective of this paragraph is to show that the above approach is not limited to the one-degree-of-free-dom model but can be generalised to three-dimensional models. When eccentricities are ignored, the equationsof motion for the undamped system can be written as [26]:

X

2

1MX

K Xi kiVMiVMiT" #X F0 Xi kideMiVMi X21NLTEE 14

with M: total mass matrix; K: constant stiffness matrix; V(Mi): structure vector [15,18,26] which accounts forthe particular gear geometry and whose restriction to the pinion-gear degrees of freedom is (the other compo-nents being zero) eVMiT n1;O1Mi n1;n1;O2Mi n1 (see Fig. 1 for the parameter definition); Mi: ageneric potential point of contact; n1: the outward unit normal vector with respect to pinion flanks; E: vectorcontaining the polar moments of inertia associated with the nodes on the output member.

Using a similar approach to that employed for the single degree of freedom model, (14) can be re-written interms of the dynamic displacements and with the excitations expressed as functions of transmission errors:

X

2

1MXD K akmg#VVT

XD cosbbX

21

vx0 TESMX

_

0 X2

1 NLTE E cosbbvx0 MX

_

0 ! ! 15with K: averaged total stiffness matrix; XD X XS, the dynamic displacement vector; VT n1;O1M0n1;n1;O2M0 n1 is an averaged structure vector proportional to the co-ordinates of the mesh forcewrenches on the pinion in O1 and on the gear in O2 assuming that the mesh force distribution is reduced

to a single sliding vector applied along a line M0; n1 [26,27]. X_

0 K1FSV.The total dynamic mesh force FD can be written in a simple form analogous to the single degree-of-freedom

Eq. (4) as

FD FS km1 ag#vxD 16

where vxD VT

XD and the dynamic factor is deduced as

R FDFS

1 vxD ag#vxD 17

with vxD vxDvx0 ; vx0 VTX0 FSkm: static mesh deflection; X0 K1F0: static solution with averaged mesh

stiffness.A particular stable solution for dynamic displacements is sought under the form of an asymptotic

development:

XD XD0 aXD1 a2XD2 18which, after projection of the main order solution in a modal basis by the transformation:

XD0 Ud 19



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leads to a series of uncoupled differential equations of the form

d=k 2nk-kd=k -2kd=k CkTES DkNLTE 20

where U is a truncated modal basis associated with the constant coefficient equations (e.g. the modes of the

differential system (15) with g# 0; d is the vector of the modal unknowns; A/k is the kth component ofvector A; -k 1X1

ffiffiffiffiffiffikUkmUk

q; kUk is the modal stiffness associated with the kth mode shape; mUk is the modal mass

associated with the kth mode shape; Ck cosbbvx0/TMX

_

0=kmUk

; Dk UTE=kmUk

Ck.Assuming that the excitation functions can be decomposed into a Fourier series of the same form (8) as for

the single degree of freedom model, one obtains

CkTES DkNLTE

XnP1

n2An sin n# Bn cos n# 21

and a solution to (20) can be sought as

d=k XnP1

ak

n sin n# bk

n cos n# 22

in which coefficients akn and akn are determined by identification of like terms in sine and cosine.

17

814

4

11

S

Z

90 100

120130

80

120

120

120

120

80

Fig. 5. Gear set three-dimensional finite element model.



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Introducing the asymptotic development (18) in the expression of the dynamic factor R, (17) gives a firstorder approximation of the form

R ffi 1 Xk

vUkNk 23

with

vUk VTUk UTkV;

1.0

1.2

1.4

1.6

1.8

2.0

0 200 400 600 800 1000 1200

1 (rad/s)

R m

ax

Time-step integration

Approximate formula (25)

Fig. 6. Maximum of dynamic factor versus pinion speed comparisons between time-step integrations and formula (25). Three-dimensional multi-degree of freedom model (unmodified pinion and gear).

1

1.2

1.4

1.6

1.8

2

0 200 400 600 800 1000 1200

1 (rad/s)

Rma

x

20 modes 10 modes 5 modes

Fig. 7. Influence of the number of modes on the dynamic factor computed by (25).



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Uk; kth mode shape;

Nk

XnP1An

-k

n

2 1

! 2Bnnk

-k

n

-k

n 2

1 !

2

4n2k

-k

n 2

sin n#Bn

-k

n

2 1

! 2Annk

-k

n

-k

n 2

1 !

2

4n2k

-k

n 2

cos n#;

An Anvx0

; Bn Bnvx0

:

The coefficients vUk can be related to the percentage of modal strain energy qk stored in the mesh for mode k(considering the system with averaged stiffness matrix) which is defined as

qk kmU

TkVV

TUk

UTkKUk

kmkUk

v2Uk 24

1

1.2

1.4

1.6

1.8

2

0 200 400 600 800 1000 1200

1 (rad/s)

Rmax

20 harmonics 10 harmonics 5 harmonics

Fig. 8. Influence of the number of transmission error harmonics on the dynamic factor computed by (25).

Table 2Frequencies with qk above 5%

Eigenfrequency (Hz) Percentage of modal strain energy

2390 34.94239 22.93585 14.12695 8.71290 7.25605 6.9

600 6.48574 6.36455 5.6

2580 5.6



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and the main order approximation of the dynamic factor is finally written under the form

R ffi 1 Xk

ffiffiffiffiffiffiffikUk

km

r ffiffiffiffiffiqk

pNk 25

Table 3Pitch error distribution on the pinion

-15

-10

-5

0

5

10

15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

tooth number

(m)

1.0

1.1

1.2

1.3

1.4

0 200 400 600 800 1000 1200

1 (rad/s)

Rmax

Time-step integrationApproximate formula (25)

Fig. 9. Maximum of dynamic factor versus pinion speed comparisons between time-step integrations and formula (25). Three-dimensional multi-degree of freedom model (linear profile relief on pinion and gear tips with Pc 12 lm and Lc 0:2T1T2 + paraboliccrowning of amplitude 10 lm on the pinion).



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One can notice that the individual contribution of a given mode is directly related to its percentage of strainenergy in the meshing teeth and to the ratio of its modal stiffness to the average mesh stiffness. These prop-erties can be used for identifying the usually limited number of critical mode shapes and frequencies withrespect to tooth contact loads. They may also serve to test the structural modifications aimed at avoiding crit-ical loading conditions over a range of speeds.

4.2. Elements of validation

The objective of this section is to critically assess the validity of the approximate theory presented in Section3.1 by comparing the dynamic responses obtained by using (25) with those given by time-step integrations ofthe equations of motion (14). The gear data are the same as in Section 2 and the geared train three-dimen-sional model under consideration is shown in Fig. 5. The first set of comparisons in Fig. 6 concerns an unmod-ified errorless pinion and gear which were simulated with a very low modal damping factor of 0.01 for allmodes. A good agreement is observed over the entire range of speeds both in terms of response peak positionsand associated amplitudes. It should be noted that, in this example, the proposed approach is at least 20 timesfaster than the conventional integration technique. A sensitivity analysis was performed on the effect of thenumber of modes and excitation harmonics retained in formula (23). The results in Figs. 7 and 8 indicate that,for the example treated, a high degree of agreement is achieved even for a very limited number of modes andharmonics (five modes and five harmonics of the mesh frequency in the example treated), thus emphasizing theinterest of the methodology in terms of computational time saving. The percentages of modal strain energy inthe mesh qk, listed in Table 2, clearly show that a limited number of modes have to be considered for dynamictooth load calculations. Comparisons are extended to more realistic gear geometries by introducing the toothmodifications already defined in Figs. 2 and 3 to which the pitch error distribution on the pinion defined inTable 3 is superimposed. The dynamic factor curves in Figs. 9 and 10 prove that the analytical formula com-pares very well with the results obtained by the Newmark scheme combined with a normal contact algorithm.It is observed that pitch errors broaden the response peaks because of the more complex excitation spectrathat they generate.

1

1.2

1.4

1.6

1.8

2

0 200 400 600 800 1000 1200

1 (rad/s)

Rmax

Time-step integration

Approximate formula (25)

Fig. 10. Maximum of dynamic factor versus pinion speed comparisons between time-step integrations and formula (25). Three-dimensional multi-degree of freedom model (unmodified pinion and gear, pitch errors defined in Table 3).



14/15

5. Conclusion

Based on straightforward expansions of the dynamic response, it is demonstrated that analytical expres-sions of dynamic tooth loads or tooth dynamic factors can be derived in terms of the harmonics of thequasi-static transmission error under load. The proposed formulae are essentially valid for helical gears since

mesh stiffness variations are supposed to be small compared with the average value of the stiffness function.Two gear models are successively examined: (i) a single degree-of-freedom one similar to that used for definingthe dynamic rating factor in the ISO 6336 standard [10], and (ii) a three-dimensional model condensed by apseudo-modal method. Two original formulae are presented which compare very well with the resultsobtained by numerical integration of the equations of motion (parametrically excited differential systems).From the simplest formula derived from the one degree of freedom model, it has been found that the dynamicfactors for perfect helical gears depend mostly on contact ratios and are largely independent of the transmittedload. The latter was verified by integrating the three-dimensional equations of motion (14) for three torques of2976, 5976 and 8976 N m on the pinion shaft and, as long as linear mesh stiffness was employed, the dynamicfactor curves were almost exactly superimposed.

From a theoretical point of view, the dynamic factor equations in this paper establish a formal link betweendynamic tooth loads and the fluctuations of quasi-static transmission errors under load for helical gears. This

correlation has been known and mathematically proven in the case of spur gears since the pioneering work ofHarris [1], Gregory et al. [2,3] but, to the authors knowledge, extension to helical gears had yet to be dem-onstrated. The proposed equations are also of practical importance as they lead to significant gains in com-putational time which makes it possible to conduct intensive parameter analyses and apply statisticalapproaches to the dynamic behaviour of actual gears, i.e., accounting for geometrical tolerances in relationwith quality grades for instance.

References

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Dynamic Factor Formulae

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