Multi-Point Mesh Modeling and Nonlinear Multi-Body

INTRODUCTIONHypoid and spiral bevel gears are widely used in the rear

axle power-train systems of vehicles. The dynamic behaviorof hypoid and spiral bevel gear pair is important mainly intwo reasons: (1) the vibration and noise problems are relatedto the dynamic mesh force, which is believed to be excited bygear transmission error (TE) and parametric variation of meshcharacteristics, (2) the prediction of dynamic mesh force iscritical to the study of gear fatigue and gear life. Moreover,the vibration of the meshing gear pair is transmitted throughshaft-bearing assembly into gear housing, which affects theoverall system dynamics of driveline system.

An accurate mesh model is critical for the prediction ofdynamic mesh force. The spring-damper model is usuallyused to represent the mesh characteristics of gear pairs. Overthe past 20 years, there are a number of studies assuming atime and spatial-invariant spring damper model. With thedevelopment of tooth surface contact program, there are morestudies recently based on the 3-dimensinal gear toothgeometry results [1, 2, 3, 4, 5]. In their models, the time-varying mesh characteristics, including mesh stiffness,transmission error, mesh point and line-of-action (LOA), areobtained from the contact results of quasi-static loaded toothcontact analysis (LTCA), and the contact results aresynthesized to a single spring-damper coupling, which

represents the gear mesh. Besides, Wang [3] also investigatedthe multiple spring-damper coupling and compared the meshparameters and dynamic responses with those by assumingsingle spring-damper coupling.

The basic frame work of hypoid and spiral bevel geardynamics is established by Cheng and Lim [1], a nonlineartime-varying (NLTV) model is applied, both backlashnonlinearity and time-varying mesh parameters areconsidered. Later, Peng [4] derived the coupled multi-bodydynamic and vibration model by including the large shaftrotation. Yang [6-7] further investigated the nonlineardynamic response of hypoid gearbox with time-varyingbearing stiffness and elastic housing.

Previous hypoid gear mesh model based on single-pointcoupling may lose accuracy when gear teeth are large andcontact ratio is high. The multi-point coupling mesh model isbelieved to be able to catch the true mesh characteristics withless assumption. In addition, the lack of the detailed analysisof the time history of dynamic mesh force on each engagingtooth pair, limited the further study on gear surface wear andfatigue model. In this paper, a multi-point hypoid gear meshmodel based on 3-dimensional loaded tooth contact analysisis incorporated into a coupled multi-body dynamic andvibration hypoid gear model to predict more detailed dynamicbehavior of each tooth pair. Linear dynamic responses are

2013-01-1895Published 05/13/2013

Copyright © 2013 SAE Internationaldoi:10.4271/2013-01-1895saepcmech.saejournals.org

Multi-Point Mesh Modeling and Nonlinear Multi-BodyDynamics of Hypoid Geared System

Yawen Wang, Teik C. Lim and Junyi YangUniversity of Cincinnati

ABSTRACTA multi-point hypoid gear mesh model based on 3-dimensional loaded tooth contact analysis is incorporated into a

coupled multi-body dynamic and vibration hypoid gear model to predict more detailed dynamic behavior of each toothpair. To validate the accuracy of the proposed model, the time-averaged mesh parameters are applied to linear time-invariant (LTI) analysis and the dynamic responses, such as dynamic mesh force, dynamic transmission error, arecomputed, which demonstrates good agreement with that predicted by single-point mesh model. Furthermore, a nonlineartime-varying (NLTV) dynamic analysis is performed considering the effect of backlash nonlinearity and time-varyingmesh parameters, such as mesh stiffness, transmission error, mesh point and line-of-action. Simulation results show thatthe time history of the mesh parameters and dynamic mesh force for each pair of teeth within a full engagement cycle canbe simulated. This capability enables the analysis of durability of the gear pair and more accurate prediction of the systemresponse.

CITATION: Wang, Y., Lim, T. and Yang, J., "Multi-Point Mesh Modeling and Nonlinear Multi-Body Dynamics of HypoidGeared System," SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):2013, doi:10.4271/2013-01-1895.

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compared between single-point and multi-point meshcoupling to validate the model. The nonlinear dynamicanalysis is also performed considering the effect of backlashnonlinearity and time-varying mesh parameters. Moredetailed dynamic responses on each engaging gear tooth pairare examined, thus it can be applied to investigate theinteraction between gear dynamics and surface wear [8].

MESH AND DYNAMIC MODELGear Mesh Model

The hypoid gear mesh model, including mesh point, line-of-action, mesh stiffness and kinematic TE can be derivedfrom the load distribution results calculated by a 3-dimensional quasi-static loaded tooth contact analysisprogram for hypoid and spiral bevel gears [9]. This programcombines the semi-analytical theory with finite elementmethod [10], which can solve the gear tooth contact problemvery efficiently.

A hypoid gear pair with multiple contact interfaces isshown in Figure 1(a), and Figure 1(b) shows contact cells oneach tooth pair. The position vector of the contact cell in themesh coordinate system is ri(rix, riy, riz), the contact force isfi, and the normal vector is ni(nix, niy, niz).

Figure 1. Illustrations of: (a) multi-point coupling ofhypoid gear pair, and (b) contact cells on engaging tooth

surface.

The total contact force is calculated by summing thecontact forces on each contact cell (assuming there are Ncontact cells)

(1)

The line-of-action vector L(nx, ny, nz) can be obtainedfrom

(2)

The total contact moment is given by

(3)

The mesh point R(xm, ym, zm) can be obtained from

(4a)

(4b)

(4c)

The translational loaded and unloaded transmission errorseL and e0 are the projections of corresponding angulartransmission error along the line of action.

Finally, the mesh stiffness is defined by

(5)

Above formulation is applied to each contact interface,thus the multi-point mesh parameters in one mesh cycle canbe obtain and applied to the dynamic model. The number ofcontact tooth pairs at any given time is a function of load,pinion roll angle and tooth geometry.

Dynamic FormulationThe coupled multi-body dynamic and vibration model,

proposed by Peng [4], is employed in this paper. The pinion,gear, engine and load component are considered as rigidrotors. The engine and load component each has only onerotational coordinate. The shaft-bearing assembly stiffness isused to represent the flexibility at both pinion and gearlumped support. Multi-point mesh coupling is appliedbetween the engaging gear teeth.

The equations of motion in matrix form can be written as

Wang et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013)1128


(1)

where

(2)

(3)

The stiffness matrixis [K] the lumped shaft-bearingassembly support stiffness. The damping matrix [C] isassumed to be viscous type and derived from damping ratio.The gyroscopic matrix [G] and [Ga] are associated with theabsolute rolling velocities and accelerations of pinion andgear respectively. The force vector can be written as

(5)

The dynamic mesh force Fm and directionaltransformation vectors hl are defined as

(6)

(7)

Where bc is the gear backlash, ε0 is unloaded transmissionerror and rotational radius are defined as

(8)

(9)

(10)

where Ll and Rl is the line of action and mesh point in pinioncoordinate system (l = p) or gear coordinate system (l = g),and Xl, Yl, Zl are the unit vectors in the pinion or gearcoordinate system.

Figure 2. 14-DOF hypoid geared rotor system model

Finally, the dynamic transmission error δd is given by

(11)

The mesh parameters km, cm and ε0 are obtained frommulti-point mesh model, thus the dynamic mesh force foreach contact gear pair can be calculated. Numericalintegration is applied to solve the nonlinear equations in thetime domain.

SIMULATION RESULTSMesh Analysis

Mesh parameters, such as kinematic TE, mesh stiffness,mesh point and line-of-action vector, that vary with pinionroll angle are obtained from quasi-static tooth contactanalysis. At any instant of time, there are variable number ofmesh interfaces linking the pinion and gear. Each meshinterface is represented by its mesh point, LOA and stiffness.

The time-varying mesh point and line of action for eachtooth pair are synthesized from the load distribution results.Unlike most other gears, the normal force on each tooth pairacts in different directions for hypoid gears. As is shown inFigure 3, the mesh point and line of action in multi-pointmesh coupling presents stronger time-variant characteristicsthan that in the single-point mesh model. However, the meanvalues of multi-point mesh points and line of action areapproximately the same as those of single-point mesh model.

Figure 4 shows the time-varying mesh stiffness for single-point and multi-point mesh model. It can be seen that themesh stiffness of each tooth pair is changing greatly in onemesh cycle, which makes mesh force varies largely. Themesh stiffness of single-point has much smaller variation.

Wang et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 6, Issue 2(July 2013) 1129


Figure 3. Time-varying mesh point coordinates (a), (b),(c) and LOA directional cosines (d), (e), (f)

single-point, mesh point 1, mesh point2, mesh point 3.

Figure 4. Time-varying mesh stiffness in one mesh cycle single-point, mesh point 1, mesh point 2, mesh point 3.

Dynamic AnalysisThe time-averaged multi-point and single-point mesh

parameters are applied to linear time-invariant (LTI) analysis.Figures 5 and 6 show the comparison of dynamictransmission error and net mesh force, respectively, fromsingle-point and multi-point mesh model. The results indicate

that the dynamic responses by using multi-point mesh modelare in reasonably good agreement with single-point meshmodel results. The advantage of multi-point mesh model isthat it can predict the mesh force for each tooth pair. Figure 7shows the dynamic mesh force for each engaging tooth pair.The tooth pair which stays engaged in one mesh cyclecontributes the most to the total dynamic mesh force; theother two tooth pairs have nearly same contribution.

Then the time-varying mesh parameters and backlash aretaken into account, to examine the time history of dynamicresponses under nonlinear time-varying condition. Figure 8compares the angular DTE for single-point and multi-pointmesh model, by using torsional angular displacement. Thetranslational DTE then can be calculated and projected to theLOA of each mesh interface, as is shown in Figure 9. Thereare some variations of projected DTE at some certainfrequencies, but the basic trends are the same.

Figure 5. Dynamic transmission error single-point mesh model, multi-point mesh model

Figure 6. Dynamic mesh force single-point meshmodel, multi-point mesh model



Figure 7. Dynamic mesh force of each tooth pair mesh point 1, mesh point 2, mesh

point 3

Figure 8. Angular transmission error single-point mesh model, multi-point mesh model

Figure 9. Dynamic transmission error projected to theLOA of each mesh interface single point,

mesh point 1, mesh point 2, mesh point 3.

Figure 10 and 11 show the net mesh force predicted bysingle-point and multi-point mesh model, and the individualforce for each contact tooth pair. For single-point meshmodel, we define the dynamic mesh force as the variationaround the mean value in the time domain. However, for themulti-point mesh model, since the load sharing characteristics

are changing with time, the mean value of mesh force on eachtooth pair varies greatly. Thus the net mesh force in the timedomain is first obtained by summing the forcing vectors, andthen the individual force is obtained by the projection of netmesh force in the frequency domain. It is noticed that single-point mesh model tends to overpredict the mesh force at somelocal ranges, compared to multi-point mesh model. In theoverall frequency range, they are still in good agreement witheach other.

In Figure 12 and 13 the time history of net mesh force andindividual mesh forces are shown at 100Hz and 1700Hz,respectively, to compare the time history of dynamic meshforce at a non-resonant frequency and a resonant frequency. Itcan be observed that (1) the individual mesh forces alwayshave large variations, whereas the total mesh force only haslarge variation at frequencies that are near resonance; (2) theindividual forces vary in the same manner within a fullengagement cycle at a certain mesh frequency; (3) thevariation ranges of individual forces are almost the same, butin different manners at different frequencies.

Figure 10. Dynamic mesh force single-pointmesh model, multi-point mesh model

Figure 11. Dynamic mesh force mesh point 1, mesh point 2, mesh point 3.

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Figure 12. Time history of dynamic mesh force at 100Hz net mesh force mesh point 1,

mesh point 2 mesh point 3.

Figure 13. Time history of dynamic mesh force at1700Hz net mesh force mesh point 1,

mesh point 2 mesh point 3.

SUMMARYThe accuracy of the conventional single-point mesh

model is challenged by heavy torque load and high contactratio. To obtain the detailed dynamic response of eachengaging tooth pair and to predict the system responses withless assumption, a multi-point mesh model is applied to thecoupled multi-body dynamic and vibration formulation ofgear rotor system. The mesh parameters and dynamicresponses are compared for single-point and multi-point meshmodel, and good agreement is obtained. Dynamic mesh forcefor each engaging tooth pair can be calculated by theprojection of net mesh force. The time history of net meshforce and individual forces within a full engagement cyclecan be simulated, and results show that multi-point meshmodel is able to give more insights on the dynamic behaviorof each pair of teeth. This result can be further applied toinvestigate the gear surface wear model. The effect of driveload and backlash nonlinearity on the individual dynamicmesh force can also be further studied.

REFERENCES1. Cheng, Y., Lim, T.C., “Vibration Analysis of Hypoid Transmissions

Applying an Exact Geometry-based Gear Mesh Theory,” Journal ofSound and Vibration 240, 519-543 (2001).

2. Cheng, Y., “Dynamics of High-Speed Hypoid and Bevel Geared RotorSystems,” PhD Dissertation, Ohio State University, 2000.

3. Wang, H., “Gear mesh characteristics and dynamics of hypoid gear rotorsystem,” PhD Dissertation, University of Alabama, 2002.

4. Peng, T., “Coupled multi-body dynamic and vibration analysis ofhypoid and bevel geared rotor system,”, PhD Dissertation University ofCincinnati, 2010.

5. Wang J., “Nonlinear Time-varying Gear Mesh and Dynamic Analysis ofHypoid and Bevel Geared Rotor System,” PhD Dissertation, Universityof Cincinnati, 2007

6. Yang, J. and Lim, T., “Dynamics of Coupled Nonlinear Hypoid GearMesh and Time-varying Bearing Stiffness Systems,” SAE Int. J.Passeng. Cars - Mech. Syst. 4(2):1039-1049, 2011, doi:10.4271/2011-01-1548.

7. Yang, J. and Lim T. C. (2011). “Nonlinear Dynamic Simulation ofHypoid Gearbox with Elastic Housing.” ASME Conference Proceedings2011(54853): 437-447.

8. Ding, H. and Kahraman A. (2007). “Interactions between NonlinearSpur Gear Dynamics and Surface Wear.” Journal of Sound andVibration 307(3-5): 662-679.

9. Vijayakar, S. (2003). “Hypoid Face Milled User's Manual,” (AdvancedNumerical Solutions LLC.).

10. Vijayakar, S., “A Combined surface integral and finite element solutionfor a three-dimensional contact problem,” International Journal forNumerical Methods in Engineering, 31, 525-545, 1991

CONTACT INFORMATIONName: Teik C. Lim, PhD, PE, Fellows (ASME, SAE)Position: Herman Schneider Professor of MechanicalEngineering, and Dean of College of Engineering andApplied ScienceDirector, Vibro-Acoustics and Sound Quality ResearchLaboratoryDirector, Hypoid Gear Mesh and Dynamic ModelingConsortiumDirector, UC Simulation CenterAddress: College of Engineering and Applied Science, Univ.of Cincinnati801 Engineering Research Center (ERC)P.O. Box 210018, Cincinnati, OH 45221, [email protected]

ACKNOWLEDGMENTSThe author wishes to express grateful thanks to Vibro-

Acoustics and Sound Quality Research Lab of the College ofEngineering and Applied Science at University of Cincinnatifor support of this effort.



http://dx.doi.org/10.4271/2011-01-1548

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Multi-Point Mesh Modeling and Nonlinear Multi-Body