A Comparison of Gear Mesh Stiffness Modeling Strategies

Jim Meagher1, Xi Wu1, Dewen Kong2 , and Chun Hung Lee1

1-Mechanical Engineering Department, California Polytechnic State University, San Luis

Obispo, CA93407, USA

2-College of Mechanical Science and Engineering, Jilin University, Changchun 130025, China

jmeagher@calpoly.edu

Nomenclature:

A Cross Sectional Area W Face Width E Elastic Modulus r Gear Body Radius F Normal Contact Force Poisson’s Ratio

G Shear Modulus L Tooth Length I Area Moment of Inertia Ng Number of teeth in gear C Compliance Contact Ratio

V Shear Force LCR,HCR Low, High Contact Ratio

Abstract：

Gearboxes are prone to numerous faults that require vibration health monitoring to ensure proper operation. Monitoring methods typically employ casing mounted accelerometers. This data is difficult to interpret because of the many gear mesh frequencies present and is further complicated by the presence of significant amounts of noise. Several different dynamic system modeling strategies are currently being used by researchers to identify diagnostic indicators of gear health: a strength of materials based lumped parameter model, non-linear quasi-static finite element modeling, and rigid multi-body kinematic modeling with nonlinear contact stiffness. This study contrasts these methods of modeling gear dynamics by comparing their predicted stiffness cycle and its effect on dynamic response. A pair of ideal high contact ratio spur gears and a pair with a low contact ratio are considered. Data from experiments are shown for the high contact ratio pair. Results show that stiffness may be poorly matched either by exclusion of significant compliance, incomplete sensitivity and convergence studies, or by high frequency variations due to discretization.

Keywords: gear stiffness, vibration health monitoring, finite element, multi-body kinematic model

Proceedings of the IMAC-XXVIIIFebruary 1–4, 2010, Jacksonville, Florida USA

©2010 Society for Experimental Mechanics Inc.

Background:

The approaches to modeling gear system dynamics are as varied as the goals of the study; to predict operational loads and life, to design system performance, or to study diagnostic indicators of gear defects. Three major approaches used are finite element modeling, lumped parameter modeling, and rigid body kinematics with nonlinear contact stiffness. This study is primarily concerned with the modeling of the nonlinear meshing stiffness for the purpose of condition monitoring using vibration signals.

Two notable review papers that discuss the numerical modeling of gear dynamics are by Ozguven and Houser in 1988 [1] and by Parey and Tandon in 2003 [2]. Ozyguven categorized the models as dynamic factor models, models with tooth compliance, models for gear dynamics, those for rotor dynamics, and those for torsional vibration. The listed goals for the studies included reliability, life, stress, loading, noise, and vibratory motion. Curiously, condition monitoring was not included. Early work modeled the meshing stiffness as either an average or piecewise linear variation. Parey and Tandon’s review concentrated mostly on the modeling of defects but includes an extensive compilation of various lumped parameter models that have been used.

Early work to model the meshing stiffness was based upon a mechanics of materials approach. Lin [3] has published extensive details regarding the estimation of the time varying stiffness using beam approximations. The contribution of bending, axial loading, shear deformation, and local contact deformation of the teeth are considered and delineated for points along the line of action of the teeth.

Finite element approaches use a quasi-static method, whereby, after the stiffness is calculated in one position, the solid bodies are rotated to a new position and the FEA is performed again. Variations of the FEA method include partial gear body models [4], single teeth gear models [5], and whole tooth bodies [6]. Recent models include variations of the tooth profile [7]. Kuang and Yang [8] developed a closed form curve fit equation for stiffness based on modeling low contact ratio gears that included addendum modification.

Multi-body dynamics software can also be used to model gear mesh stiffness using a rigid-elastic model [9]. Hertzian contact at the gear interface is used to represent gear elasticity as a compromise over fully elastic models; thereby reducing computational effort over FEA approaches.

Lumped Parameter Model:

A lumped parameter model for the gear pairs defined in table 1 below was created based upon the work of Lin [3] with modifications for torsion of the gear body and Hertzian contact. The compliance calculated is defined as the deflection along the line of action for a unit load of contact force. For the gear body compliance in torsion, the effective stiffness in the first torsional mode of vibration is used [10]. For the Hertzian contact stiffness, a linearization from Yang and Sun [11] is used. The compliance of a single tooth due to bending, axial loading, and shear, can be found using Castigliano’s Method [12]. The stiffness of a single tooth is the reciprocal of the sum of the compliances. The resulting tooth compliance is evaluated numerically from:

(1)

The final two terms are from torsion of the gear body and Hertzian contact. For a tooth pair, the

stiffnesses add in series. For multiple pairs in engagement, the stiffness adds in parallel.

Number of

Teeth

Pressure

Angle

Face

Width mm.

Diameter of

Pitch Circle

mm.

Diameter of

Addendum

Circle mm.

Diameter of

Dedendum

Circle mm.

Contact

Ratio

High Contact Ratio - Two Stage Gearbox

Pinion 24 14.5⁰ 19.0 50.8 45.5 55.0 2.023

Stage 1

Stage 1

Gear 60 127 121.7 131.2

Pinion 36 14.5⁰ 19.0 76.2 80.4 2.065

Stage 2

Stage 2

Gear 48 101.6 105.8 96.3

Low Contact Ratio Pair

Pinion 23 20⁰ 25.4 58.4 63.5 57.0 1.626

Gear 31 78.7 83.8 77.4 Table 1: Gear Pair Parameters

Figure 1 represents all points along the line of

contact for the LCR pair of gears with the pinion

driving. The total gear roll angle corresponding to

a gear tooth traversing the load zone equals the

contact ratio, , times the angle a single gear

tooth rotates before reaching the same relative

position with respect to a pinion tooth: . Along

the load zone, the sequence of number of tooth

pairs in mesh is 2, 1, 2. This can be seen in figure

1 as higher relative stiffness over the first and last

portions where multiple pairs of teeth engage. As

the pair of gears rotates, the mesh stiffness

repeats the pattern; two pairs engaged, one pair

engaged, or the first degrees of the figure.

Also shown is the contribution of a single gear and pinion tooth along the line of action. As the gear tooth

enters the load zone it is loaded at the addendum so that its stiffness is low compared to being loaded

near the root as occurs as it exits the load zone. The pinion demonstrates the reverse effect, stiff then

compliant, since it is loaded near the root at the start of the load zone and exits the load zone with the

contact force at the addendum.

The individual contributions of bending, shear, gear body torsion, Hertzian contact, and axial compliance

are shown in figures 2 and 3. These components act in series for an individual tooth. The numerical

beam approximation shows that convergence of results occurs within 10 beam segments. These plots are

for contact as the pinion tooth driver enters the load zone. Since the pinion is loaded near its root the

torsional gear body compliance is the most dominant effect. The Hertzian Contact and tooth shear are the

next most significant for the pinion tooth at this point. For the gear tooth at the corresponding position,

the bending compliance dominates because the tooth is loaded at the tooth tip. This illustrates one

advantage to lumped parameter modeling; the relative contribution to the stiffness for each component is

available.

Figure 1: Low Contact Ratio Gear Mesh Stiffness for Contact throughout the Line of Action.

Figure 2: Results of numerical integration approximation for tooth stiffness at the beginning of the load zone for the gear tooth.

Figure 3: Results of numerical integration approximation for tooth stiffness at the beginning of the load zone for the pinion tooth.

Finite Element Model:

Finite element modeling has increasingly been

used as part of a dynamic analysis of meshing

gears. The meshing stiffness is typically obtained

by loading one gear and restraining the other to

calculate the stiffness for a given position and

operating load. The gear bodies must be

repeatedly rotated and the calculations repeated.

Figure 4 shows a gear pair with two different mesh

densities. The loading of two pairs of teeth is

apparent for this position. The FEM is flexible,

especially for non-standard tooth geometries, but

can be time intensive for the user to define the

geometry, and for the computer to solve the

loading. Besides convergence studies to

determine appropriate mesh densities, many other

factors must be considered to get accurate results.

Plane stress, plane strain, and three dimensional

modeling were performed using ABAQUS™. There were small differences in the results but

computational time for a full 3-D study was considerable. The results from linear or quadratic planar

elements are displayed in figure 5. Each data point on this plot is a separate FEA. The mesh density was

not sufficient for linear elements to give accurate results but quadratic elements worked well as shown by

the expected pattern and magnitude of the meshing stiffness.

Figure 4: Finite Element Gear Models with Different Mesh Densities.

For a nonlinear contact analysis a tolerance must

be set which defines whether contact occurs or

not. The results as shown in figure 6 show the

sensitivity of the results to this parameter. Over a

relatively small range the results change

dramatically. The middle three curves show the

correct pattern of stiffness variation between two

and one pair of teeth supporting the load.

Even with proper tolerance, enough elements of

high enough order, the results may not be

accurate in a very important characteristic,

frequency. This is displayed in figure 7. The beam

model matches the theoretical meshing period

exactly, it repeats every time a new pair of teeth

comes into mesh (360⁰/Nteeth). This is important when the meshing stiffness is used for dynamic analysis

where the vibration frequency is a primary consideration. Also, the point to point variation of stiffness for

the FEA shows additional frequency content that will obscure the results for ideal gears or gears with

defects.

Figure 5: Results for various element types.

Figure 6: Sensitivity of FEA Results to Contact Tolerance Parameter

Figure 7: Comparison of FEA results to Lumped Parameter Model

Multi-body kinematic model:

A rigid-elastic model of a pair of meshing gears is shown in Figure 8. The gears

are rigid and the supports have lumped elasticity to model bearing support

compliance. The contact surfaces between the gears are modeled as deformable

flex-bodies. The nonlinear contact force is composed of an

elastic and damping portion [13]. The damping force, Cv, is proportional to impact

velocity, v. The stiffness coefficient, K, is taken to be the average value of stiffness

over one tooth mesh cycle as calculated from an FEA model. The force exponent,

e, was determined from trial simulations. The damping coefficient generally takes a

numeric value between 0.1 %-1% of K. The computational time of this type of

model is much less than a finite element model. The determination of force

exponents however is not obvious and must be based on experience. No direct

comparison of stiffness can made such as in figure 7 so the predicted responses

are compared. In figure 9 is shown the Gear support acceleration.

Figure 9: Theoretical Gear Vertical Acceleration

The gear mesh frequency and harmonics are clearly evident. These frequencies are modulated by a

40Hz signal of unknown origin. With slight modifications of support stiffness and damping they are not

prominent. A dynamics simulation using a lumped parameter approach predicts only the gear mesh

frequencies and the harmonics for an ideal set of gears.

Figure 8: Multi-body Kinematic Model

Experiment:

A parallel shaft gearbox was instrumented with accelerometers on the bearing housings.

Figure 10: Two Stage Parallel Shaft Gearbox

Figure 11: Accelerometer Mounting

The gearbox has high contact ratio pairs of spur gears as described in table 1. Since the gear shafts are

supported by rolling element bearings, vibration signals contain ball passing frequencies and gear

meshing frequencies from both stages, figure 12. These compared well to theoretical values [14], table 2.

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6x 10

-6 Power Spectrum(FFT): input rpm=1020(17Hz) Freq. Range:1-700Hz

Frequency (Hz)

Pow

er

Spectr

um

(V

olt

2)

f1

1/6Xfm1

1/4Xfm1

fORBP

1/3Xfm1

fIRBP

fIRBP

-2Xf3

fIRBP

+2Xf3

fm2

fm2

-f2

fm2

-2Xf2

fm2

+2Xf3 f

m2+4Xf

3

fm1

2Xfm2

5XfORBP

5XfORBP

-f1

5XfORBP

-2Xf2

4XfIRBP

Figure 12: FFT with Hanning Window.

Table 2: Identified Frequencies in the accelerometer response spectrum (0-700 Hz).

Conclusions:

Three distinct approaches to modeling gears for the purpose of dynamic system modeling are presented.

These methods each result in a model that predicts expected gear meshing frequencies for an ideal set of

involute spur gears but the effort of modeling, flexibility, and the sensitivity of input parameters that may

not be well known vary. The lumped parameter method calculates gear stiffness for involute gears with

only a few parameters to specify the gears and calculates the stiffness more quickly than the other

methods. The method also reveals the relative contribution of different parts of the compliance. However,

for other types of gears or with addendum modifications, additional effort would be required. FEA and

Multi-body kinematic approaches are flexible but take considerable user time to generate a model for

each specific set of gears. The FEA approach is sensitive to tolerances, mesh density, and element

choice. The calculated stiffness may cause errors in the response spectrum due to discretization errors. A

Multi-body kinematic approach has a contact stiffness that depends on parameters that are not well

known.

Item Symbol

Calc Value (Hz)

Exp. Value (Hz)

Shaft

Frequencies

Shaft 1 (Input Shaft) Rotating Frequency f1 17 16.85

Shaft 2 (N2 and N3 Shaft) Rotating Frequency f2 6.8 -

Shaft 3 (N4 Shaft) Rotating Frequency f3 5.1 -

Gear

Meshing

Frequencies

First Stage Meshing Frequency f_m1 408 408.7

Second Stage Meshing Frequency f_m2 244.8 245.4

2 times of Second Stage Meshing Frequency 2×f_m2 489.6 490

1/6 times of First Stage Meshing Frequency 1/6×f_m1 68 68.12

1/4 times of First Stage Meshing Frequency 1/4×f_m1 102 101.8

1/3 times of First Stage Meshing Frequency 1/3×f_m1 136 136.2

1/2 times of First Stage Meshing Frequency 1/2×f_m1 204 204.3

2/3 times of First Stage Meshing Frequency 2/3×f_m1 272 272.5

1/2 times of Second Stage Meshing Frequency

1/2×f_m2 122.4 122.3

1/3 times of First Stage Meshing Frequency 1/3×f_m1 136 136.2

Bearing

Elements

Frequencies

Outer Race Ball Pass Frequencies

f_ORBP 116.22 119.4

2× f_ORBP 232.44 238

3× f_ORBP 348.66 357.4

4× f_ORBP 464.88 476.8

5× f_ORBP 581.1 595.5

Inner Race Ball Pass Frequencies

f_IRBP 155.78 153.1

2× f_IRBP 311.56 306.2

4× f_IRBP 623.12 612.3

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