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Research Papers

An Efficient Hybrid Analytical-Computational Method for Nonlinear Vibration of Spur Gear Pairs

[+] Author and Article Information
Xiang Dai

Department of Mechanical Engineering,
Virginia Tech,
Blacksburg, VA 24061

Christopher G. Cooley

Department of Mechanical Engineering
and Energy Processes,
Southern Illinois University Carbondale,
Carbondale, IL 62901

Robert G. Parker

Department of Mechanical Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: r.parker@vt.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 6, 2017; final manuscript received June 25, 2018; published online August 13, 2018. Assoc. Editor: Karsten Stahl.

J. Vib. Acoust 141(1), 011006 (Aug 13, 2018) (13 pages) Paper No: VIB-17-1405; doi: 10.1115/1.4040674 History: Received September 06, 2017; Revised June 25, 2018

This work develops a hybrid analytical-computational (HAC) method for nonlinear dynamic response in spur gear pairs. The formulation adopts a contact model developed in (Eritenel, T., and Parker, R. G., 2013, “Nonlinear Vibration of Gears With Tooth Surface Modifications,” ASME J. Vib. Acoust., 135(5), p. 051005) where the dynamic force at the mating gear teeth is determined from precalculated static results based on the instantaneous mesh deflection and position in the mesh cycle. The HAC method merges this calculation of the contact force based on an underlying finite element static analysis into a numerical integration of an analytical vibration model. The gear translational and rotational vibrations are calculated from a lumped-parameter analytical model where the crucial dynamic mesh force is calculated using a force-deflection function (FDF) that is generated from a series of static finite element analyses performed before the dynamic calculations. Incomplete tooth contact and partial contact loss are captured by the static finite element analyses and included in the FDF, as are tooth modifications. In contrast to typical lumped-parameter models elastic deformations of the gear teeth, including the tooth root strains and contact stresses, are calculated. Accelerating gears and transient situations can be analyzed. Comparisons with finite element calculations and available experiments validate the HAC model in predicting the dynamic response of spur gear pairs, including for resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC model is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy.

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Figures

Grahic Jump Location
Fig. 3

Finite element calculation of the FDF for the 28-tooth gear pair with the 25 deg pressure angle tooth side in contact

Grahic Jump Location
Fig. 2

Schematic of the elastic deformation in the gear teeth and blank. The black circles represent arbitrary material points on the elastic gear body.

Grahic Jump Location
Fig. 1

Schematic of the 28-tooth gear pair system. The vertical and horizontal dashed lines denote the line of action and off-line of action, respectively.

Grahic Jump Location
Fig. 4

Flowchart of the HAC method

Grahic Jump Location
Fig. 5

(a) Oscillating (RMS) and (b) mean components of the dynamic mesh deflections for the 50-tooth ICR 1.37 gear pair with unmodified teeth. The applied torque is 170 N·m. The black dots, blue circles, and green squares denote the HAC calculations, FE/CM calculations, and experiments from Ref. [41], respectively.

Grahic Jump Location
Fig. 11

(a) Dynamic mesh deflection, (b) dynamic mesh force, and (c) maximum dynamic tooth contact pressure for the 28-tooth spur gear pair at fm = 1600 Hz and 300 N·m torque with the 25 deg pressure angle tooth side in contact. The blue circles ∘ and black diamonds ◇ denote the FE/CM and HAC calculations, respectively. The plus signs +, crosses ×, and triangles △ denote the STE model, average slope model, and local slope model results, respectively.

Grahic Jump Location
Fig. 7

Hybrid analytical-computational calculation of dynamic tooth root strains compared with FE/CM results and experiments [52] for the 50-tooth spur gear pair. The gears have ICR 1.8 with tooth profile modifications. The strains are calculated at R = 70.7 mm where maximum strain occurs in the tooth root region. The gears are at (a) fm/fn = 0.545, (b) fm/fn = 0.702, and (c) fm/fn = 0.986 and 200 N·m applied torque. The solid black lines, dashed blue lines, and dotted green lines denote HAC, FE/CM, and experiments, respectively.

Grahic Jump Location
Fig. 8

Hybrid analytical-computational calculation of (a) dynamic mesh deflection, (b) dynamic mesh force, and (c) maximum dynamic contact pressure compared with FE/CM results for the 50-tooth spur gear pair at fm/fn = 0.986 and 200 N·m torque. The gears have ICR 1.8 with tooth profile modifications. The solid black lines and dashed blue lines denote HAC and FE/CM, respectively.

Grahic Jump Location
Fig. 10

(a) Root-mean-square of oscillating component and (b) mean value of the dynamic mesh deflection for the 28-tooth spur gear pair at 300 N·m torque with the 25 deg pressure angle tooth side in contact. The blue circles ∘ and black diamonds ◇ denote the FE/CM and HAC calculations, respectively. The plus signs +, crosses ×, and triangles △ denote the STE model, average slope model, and local slope model results, respectively.

Grahic Jump Location
Fig. 6

(a) Oscillating (RMS) and (b) mean components of the dynamic mesh deflections for the 50-tooth ICR 1.8 gear pair with tooth profile modifications. The applied torque is 200 N·m. The black dots, blue circles, and green squares denote the HAC calculations, FE/CM calculations, and experiments from Ref.[52], respectively.

Grahic Jump Location
Fig. 12

Hybrid analytical-computational calculation of dynamic mesh stiffness (solid red line) for the 28-tooth gear pair at 300 N·m torque with the 25 deg pressure angle tooth side in contact and fm/fn = 1.027. The dashed blue line denotes the static mesh stiffness calculated using the local slope method at the nominal load. The FE/CM calculation of instantaneous maximum tooth contact pressure on the gear teeth at mesh cycle 0.6 is shown in the subfigures.

Grahic Jump Location
Fig. 15

Hybrid analytical-computational calculation of (a) RMS of the oscillating component and (b) mean value of the dynamic mesh deflection compared with FE/CM results for the 28-tooth spur gear pair at 300 N·m torque with the 25 deg pressure angletooth side in contact. The gear translational bearing stiffnesses are ky1 = ky2 = 200 × 106 N/m. The black dots and blue circlesdenote the HAC calculations and FE/CM calculations, respectively.

Grahic Jump Location
Fig. 9

Hybrid analytical-computational calculation of dynamic tooth root strains compared with FE/CM results and experiments for the 28-tooth spur gear pair. The gears are at (a) fm/fn = 0.571 and (b) fm/fn = 0.893 and 300 N·m torque with the 25 deg pressure angle tooth side in contact. The solid blacklines, dashed blue lines, and dotted green lines denote the HAC calculations, FE/CM calculations, and experiments, respectively.

Grahic Jump Location
Fig. 14

Hybrid analytical-computational calculation of translational deformations compared with FE/CM results for the 28-tooth spur gear pair at fm/fn = 0.706 and 300 N·m applied torque with the 25 deg pressure angle tooth side in contact. The gear translational bearing stiffnesses are ky1 = ky2 = 200 × 106 N/m. The solid black lines and dashed blue lines denote the HAC calculations and FE/CM calculations, respectively.

Grahic Jump Location
Fig. 13

Hybrid analytical-computational calculation of RMS of oscillating component of the dynamic mesh deflection for the 28-tooth spur gear pair at 100 N·m torque with the 25 deg pressure angle tooth side in contact compared with FE/CM results and local slope model results. The blue circles ∘, black diamonds ◇, and black triangles △ denote the results from FE/CM model, HAC model, and local slope model, respectively.

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