effect of the tooth surface waviness on the dynamics and

INTRODUCTIONThe static transmission error is a well-recognized source,

and thus tooth modifications are often undertaken tominimize the excitation at specific loads to reduce gear whinenoise [1]. Yet, at high torque loads, noise levels are stillrelatively high [2]. This suggests that tooth surface wavinessand sliding friction could manifest itself as an alternate noisesource. For instance, Mark [3] found that the surfacewaviness is related to the machining process kinematics.Furthermore, the experiments conducted by Mitchell [4] andAmini et al. [5] on gears and by Othman et al. [6] on rotatingdisks show an increase in the sound level with an increase inthe surface waviness amplitude. The role of tooth surfacewaviness on gear noise is not well understood, especially forstructure-borne noise source(s) or path(s), due to thecomplexity in modeling micro-surface characteristics. Thisarticle employs a six degree of freedom (6DOF) linear time-varying (LTV) model of a spur gear pair to quantify thestructure-borne noise source and to illustrate a relationshipbetween waviness amplitude and wave number to geardynamics source and resulting sound radiation. The overallprocedure for predicting the sound pressure level (L) ispreviewed in Fig. 1, where θ is the angular displacement ofthe pinion or gear, while x and y are the translational motionsalong the line-of-action (LOA) and the off-line-of-action(OLOA) directions, respectively. Subscripts p and g represent

pinion and gear, respectively. (Also, refer to the list ofsymbols for the identification of variable and parameters.)

LINEAR TIME-VARYING SPURGEAR MODEL

The proposed 6DOF LTV model is schematically shownin Fig. 2. The gear and pinion are considered rigid discs ofpolar moments of inertia Jp and Jg with external torques Tpand Tg. Here, hp(t) and hg(t) represent tooth surface wavinesswith respect to perfect involute profiles. The governingequations are described by torsional and translationalmotions. The effective shaft-bearing stiffness elements aregiven by kpSx and kgSx in the X direction (LOA) and kpSy andkgSy in the Y direction (OLOA). The time-varying meshstiffness (k(t)) is calculated for a range of torques by using awell-known gear contact mechanics code (Load DistributionProgram or LDP) [7].

The parameters of the unity gear pair example used in thisstudy are as follows: Number of teeth = 28, outside diameter= 94.95 mm; root diameter = 79.73 mm; diametral pitch =0.315 m−1; center distance = 88.9 mm; pressure angle = 20°;face width = 6.35mm; tooth thickness = 4.851 mm; andelastic modulus = 206.9 kN/mm2. Further, it is assumed thatthe bearings of the gear and pinion are frictionless, and thewaviness amplitude is independent of the load.

2013-01-1877Published 05/13/2013

Copyright © 2013 SAE Internationaldoi:10.4271/2013-01-1877saepcmech.saejournals.org

Effect of the Tooth Surface Waviness on the Dynamics andStructure-Borne Noise of a Spur Gear Pair

Sriram Sundar, Rajendra Singh, Karthik Jayasankaran and Seungbo KimThe Ohio State University

ABSTRACTThis article studies the effects of tooth surface waviness and sliding friction on the dynamics and radiated structure-

borne noise of a spur gear pair. This study is conducted using an improved gear dynamics model while taking into accountthe sliding frictional contact between meshing teeth. An analytical six-degree-of-freedom (6DOF) linear time varying(LTV) model is developed to predict system responses and bearing forces. The time varying mesh stiffness is calculatedusing a gear contact mechanics code. A Coulomb friction model is used to calculate the sliding frictional forces.Experimental measurements of partial pressure to acceleration transfer functions are used to calculate the radiatedstructure-borne noise level. The roles of various time-varying parameters on gear dynamics are analyzed (for a specificexample case), and the predictions from the analytical model are compared with prior literature.

CITATION: Sundar, S., Singh, R., Jayasankaran, K. and Kim, S., "Effect of the Tooth Surface Waviness on the Dynamicsand Structure-Borne Noise of a Spur Gear Pair," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):2013, doi:10.4271/2013-01-1877.

____________________________________

http://dx.doi.org/10.4271/2013-01-1877

http://saepcmech.saejournals.org/

With reference to the geared system model of Fig. 2, thegoverning equations for θp(t) and θg(t) are:

(1)

(2)

The time-varying moment arms χpi(t) and χgi(t) for the ith

meshing pair with a contact ratio (σ) are the following with n= floor(σ):

(3 a, b)

Here, Ωp and Ωg are the nominal speeds (in rad/s); Λ isthe base pitch; and LAP, LXA, and LYC are the geometriclength constants as defined by He et al. [8]. The normal loads(Np and Ng) and friction forces (Fpf and Fgf) are defined asfollows, where μ(t) is the time-varying coefficient of frictionand Δh(t) = hp(t) - hg(t):

(4)

(5 a, b)

Equations of translational motion along X and Y are givenby the following, where ζ is modal damping ratio:

(6)

(7)

(8)

(9)

Fig. 1. Procedure for predicting gear accelerations and sound pressure levels.

Sundar et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)

Fig. 3. Normalized time-varying parameters along amesh cycle. (a) Mesh stiffness, ; (b) Coefficient of

friction, .

The normalized time-varying mesh stiffness andcoefficient of friction are shown in Fig. 3 for tooth pair #0

(solid line) and at tb for tooth pair # 1 (dotted line). Here, tarepresents the time from two teeth in contact to the first toothleaving contact, tb represents the time from two teeth incontact to the pitch point where the sliding velocity changesits direction, and tc represents the gear mesh period. Themesh stiffness parameters at a mean torque are calculated byusing LDP [7], given the kinematics.

Assuming that the gear tooth surface is one-dimensionalgiven in terms of mesh locations (s), the involute coordinatessp and sg are defined as follows in the involute coordinatesystem, where α is the roll angle, and j denotes the toothindex:

(10 a)

(10 b)

Expressions for known motion inputs to the gearedsystem are stated below in terms of tooth surface wavinesshp(s) and hg(s), as shown in Fig. 2:

(11 a)

Fig. 2. Proposed 6DOF linear time-varying gear dynamics model with prescribed tooth surface waviness given by hp(t) andhg(t).


(11 b)Here ϕ and λ represent the wave phase and wave length,

respectively, of the surface waviness. The wave number, κ =ω/vs = 2π/λ, where vs represents the sliding velocity,determines the spectral contents of surface-inducedexcitation, and the amplitudes Hp and Hg determine theamplitudes of excitation. Periodic waviness is generated witha constant ϕ, whereas the sinusoidal waviness is generated bychoosing a proper λ such that the surface waviness completesa full cycle at the end of tc. Random waviness is alsogenerated by choosing ϕj values with well-knowndistributions such as exponential, chi-square, Poisson, orRayleigh distribution.

The partial radiated pressure to gear acceleration transferfunctions along X and Y directions Γx and Γy, respectively,are measured using a microphone located 152 mm above thetop plate of the gearbox by Singh [9]; the microphone isplaced in the free field. Table 1 shows typical magnitudes ofΓx and Γy at certain frequencies corresponding to gear meshfrequencies of a high speed gearbox. Though themeasurements are made at a particular load, it is assumed thatthe measured transfer functions (Γx, Γy) are still valid with avariation in T.

Table 1. Measured magnitudes of partial radiatedpressure to gear acceleration transfer functions (Γx, Γy)

where the microphone is located at 152 mm above the topplate of a gearbox.

SOUND PRESSURE PREDICTIONSEffect of Surface Waviness

The sound pressure is predicted for a gear system withdifferent tooth surfaces at 22.6 N-m Where is the gearmesh harmonic. Fig. 4 shows the sound pressure levelprediction along the X direction (Lpx) and the Y direction(Lpy) with a smooth tooth surface, and random, periodic, andsinusoidal waviness at the first five gear mesh harmonics( ). The wave numbers for the periodic waviness excitationsare κp = 2π650 m−1 (λp = 1.5 mm) and κp = 2π800 m−1, andthe sinusoidal waviness excitation has κp = 2π857 m−1 (λp =

1.2 mm). In all cases, Hp = 1.0μm. Here, k(t) is a majorcontributor towards Lpx, and in the case of Lpy the majorcontribution is from μ(t) and h(t).

Effect of RandomnessThe sound pressure level is then predicted for a random

profile (with κp = 2π650 m−1 and κp = 2π800 m−1 and Hp =1.0 μm) with uniform distribution. Now non-mesh harmoniccomponents are also observed mainly along the X directionas shown in Fig. 5, where is the gear mesh harmonic.Along with non-mesh harmonics, the dominant harmonics aredisplayed. Along the X direction the random waviness withκp = 2π650 m−1 is more dominant until ; converselythe random waviness with κp = 2π800 m−1 is more dominantfrom to . Along the Y direction, there is adominant peak at 977 Hz, which is coincident with a naturalfrequency of the geared system.

Effect of TorqueTable 2 shows the predicted Lp range for three values of

Tp (22.6 N-m, 45.2 N-m, and 90.4 N-m) for a smooth toothsurface at the first five mesh harmonics. As Tp increases,there is a considerable increase in Lp (the slope isapproximately 5 dB per octave).

Table 2. Effect of mean load (Tp) on predicted soundpressure levels for smooth tooth surface using Δh(t) as

excitation

COMPARISON WITH PRIOREXPERIMENTS OR CALCULATIONS

The proposed model is used to examine experimental orcomputational results reported in the literature. First, theeffect of T on sound pressure is considered. Mitchell [4]documented a 5 dB increase in the sound pressure level withan octave increase in T. In the proposed model, an almost 6dB increase in sound pressure at all mesh harmonics (alongthe X and Y directions) is seen with a doubling of the Tvalue. Further, Mitchell [4] reported a 5 dB per octave slopewith speed variation; the current model is slightly off as itpredicts a slope of about 8.5 dB per octave for speedvariations. The proposed model (for the structure-bornesound) is, however, similar to the air-borne noise source


model suggested by Kim and Singh [10] that predicted aslope of about 9 dB per octave.

In order to compare the effect of surface wavinessamplitude H in the current model with the literature, asinusoidal waviness with κp = 2π857 m−1 is first assumed.Ishida et al. [11] documented a 10 dB decrease in sound whenΔH was reduced from 9 µm to 1 µm. The current model

predicts a 17 dB reduction in sound level for the same changein ΔH. Mitchell [4] reported an increase of about 1.5 dBwhen H was raised to 2.5 μm from 1 μm (and again by 1.5 dBwhen H was increased further to 5 μm). The currentprediction shows about a 6.5 dB increase for the samechanges in H. Hansen et al. [2] reported vibration levels at thefirst gear mesh frequency with a super-finished tooth surface.

Fig. 4. Sound pressure levels with Hp = 1.0μm and Tp = 22.6 N-m as excited by LOA and OLOA gear accelerations given

sinusoidal or periodic surface waviness. (a) Lpx, (b) Lpy. Key: , smooth surface; , Periodic waviness, κp = 2π650 m−1; ,Periodic waviness, κp = 2π800 m−1; , Sinusoidal waviness, κp = 2π857 m−1

Fig. 5. Sound pressures corresponding to the LOA gear acceleration when excited by random tooth surface with Hp = 1.0 μm.Key: , κp = 2π650 m−1; , κp = 2π800 m−1.


The vibration level for the third stage bull gear decreased by7 dB for a reduction in H from 0.38 μm to 0.07 μm; thecurrent model predicts a 6 dB reduction [2]. Furthermore,Hansen et al. [2] measured a 3 dB reduction when H waschanged from 0.38 μm to 0.09 μm for the second stage bevelpinion; in this case, the current model predicts a 5 dBdecrease in vibration (noise).

Table 3 shows the normalized accelerations (at the non-

gear mesh frequencies) along the X direction ( ) with an

increase in H. The values of at first five gear meshharmonics (for all the values of Hp) are given by thefollowing in dB re 10−2 (a normalized acceleration value):39, 34, 34, 25, and 10. Observe that the non-mesh harmonics

are dominant compared to the mesh harmonic values of ,which is attributed to the randomness in the surface waviness.With an octave increase in H there is a 6 dB increase in soundpressure along the X direction at non-mesh harmonics in thecase of random waviness. This trend explains some peaks inbetween gear mesh harmonics in the measured spectra. Ofcourse the “ghost frequencies” may also be generated as adirect consequence of a specific profile left on tooth surfacesby a honing machine [5].

Table 3. Predicted with increase in H for randomexcitation with κp = 2π800 m−1

CONCLUSIONAn improved 6DOF LTV analytical model has been

developed to predict structure-borne noise for spur gearsinduced by sliding friction and tooth surface waviness. Themodel utilizes time-varying k(t) over a range of T ascalculated with LDP [7]. Based on the experimental partialradiated pressure to gear acceleration transfer functions, freefield sound pressure levels are also predicted. The effect of anincrease in torque on Lp has been calculated and comparedwith results reported previously. In particular, non-meshharmonics are excited with random tooth surface. Noisetrends as found in the prior literature can be conceptuallyexplained using the proposed model though there is sufficientroom for improvement.

REFERENCES1. Houser, D. R., Singh, R., “Gear dynamics and gear noise” short course

notes, The Ohio State University, USA, 2012.

2. Hansen, B., Salerno, M., Winkelmann, L., “Isotropic superfinishing ofS-76C+ main Transmission gars”, AGMA Technical paper 06FTM02,2006.

3. Mark, W.D., “Contributions to the vibratory excitation of gear systemsfrom periodic undulations on tooth running surfaces”, The Journal of theAcoustical Society of America 91:166, 1992.

4. Mitchell, L.D., “Gear noise: the purchaser's and the manufacturer'sview”, Proceedings of the Purdue Noise Control Conference, 14-16,1971.

5. Amini, N., Rosen, B.G., “Surface topography and noise emission ingearboxes”, ASME Design Engineering Technical Conferences, Paper #DETC97/VIB-3790, 1997.

6. Othman, M., Elkholy, A., Seireg, A., “Experimental investigation offrictional noise and surface-roughness characteristics”, ExperimentalMechanics 30: 328-331, 1990.

7. Houser, D. R., Harianto, J., “Load Distribution Program manual”,GearLab, The Ohio State University, 2002.

8. He, S., Cho, S., Singh, R., “Prediction of dynamic friction forces in spurgears using alternate sliding friction formulations”, Journal of Soundand Vibration 309: 843-851, 2008.

9. Singh, R., “Dynamic analysis of sliding friction in rotorcraft gearedsystems”, Technical report submitted to the Army Research Office,Grant number DAAD19-02-1-0334, 2005.

10. Kim, S., Singh, R., “Gear surface roughness induced noise predictionbased on a linear time-varying model with sliding friction”, Journal ofVibration and Control 13:1045-1063, 2007.

11. Ishida, K., Matsuda, T., “Effect of tooth surface roughness on gear noiseand gear noise transmitting path”, American Society of MechanicalEngineers, Paper 80C2/DET-70, 1980.

CONTACT INFORMATIONProfessor Rajendra SinghDept. of Mechanical and Aerospace EngineeringAcoustics and Dynamics Laboratory, NSF SmartVehicle Concepts Center, The Ohio State UniversityColumbus, OH 43210 [email protected]: 614-292-9044.

LIST OF SYMBOLSa - gear accelerationc - viscous mesh dampingk - stiffnessF - friction forceh - surface wavinessH - waviness amplitudei - gear mesh indexj - tooth indexL - sound pressure levelJ - mass moment of inertiam - mass of the spur gearN - normal loadn - number of teethp - sound pressureR - base radius of the spur gears - involute coordinatet - timeT - torquev - velocity


mailto:[email protected]

χ - moment arm for friction forceX - line-of-action directionY - off-line-of-action directionx - translation along line-of-actiony - translation along off-line-of-actionα - roll angleζ - damping ratioθ - angular motion of spur gearΛ - base pitchΩ - speed of rotationκ - wave numberϕ - wave phaseλ - wave lengthχ - moment arm for friction forceΓ - partial pressure to acceleration transfer functionμ - coefficient of friction

Subscripts0 - mesh start point of pinion1,2 - gear pair in contactg - gearL - mesh start point of gearm - mesh pointp - pinions - slidingS - shaft - bearingx - line-of-action directiony - Off-line-of-action direction

Superscripts(·) - differentiation with respect to t

(−) - normalization

AbbreviationsDOF - degree-of-freedomLDP - load distribution programLOA - line-of-actionLTV - linear time-varyingOLOA - off-line-of-action


effect of the tooth surface waviness on the dynamics and

Documents