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Research Papers

An Alternative Formulation of the Dynamic Transmission Error to Study the Oscillations of Automotive Hypoid Gears

[+] Author and Article Information
I. Karagiannis

e-mail: I.Karagiannis@lboro.ac.uk

S. Theodossiades

e-mail: S.Theodossiades@lboro.ac.uk
Wolfson School of Mechanical and
Manufacturing Engineering,
Loughborough University,
Loughborough LE11 3TU, UK

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 22, 2013; final manuscript received July 23, 2013; published online September 12, 2013. Assoc. Editor: Prof. Philippe Velex.

J. Vib. Acoust 136(1), 011001 (Sep 12, 2013) (12 pages) Paper No: VIB-13-1060; doi: 10.1115/1.4025206 History: Received February 22, 2013; Revised July 23, 2013

A new modeling approach on the torsional dynamics of hypoid gear pairs is presented in this work. The current formulation is characterized by an alternative expression of the dynamic transmission error (DTE), accounting for the variation of the effective mesh position. Speed dependent resistive torque is introduced on the gear wheel, enabling the system to reach dynamic equilibrium based on realistic vehicle operating conditions. The above are supplementing past research studies, where simplifications were introduced in the calculation of DTE, while the operating angular velocity was defined a priori. The analysis is accompanied by numerical results, indicating the rich dynamic behavior captured by the new formulation. The dynamic complexity of the system necessitates the identification of the various response regimes. A solution continuation method (software AUTO) is employed to follow the stable/unstable periodic response branches over the operating range of the differential under examination.

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References

Figures

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Fig. 1

Free body diagram of the hypoid gear pair model

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Fig. 2

Resistance forces acting on vehicle

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Fig. 3

Response characteristics of the gear pair: (a) pinion angular velocity, (b) gear angular velocity, (c) dynamic mesh force, and (d) dynamic transmission error

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Fig. 4

Illustration of stability issues: (a) pinion angular velocity, (b) dynamic mesh force, (c) phase plots and (d) energy balance; integral form, equation (5), simplified form, Eq. (9)

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Fig. 5

Angular velocities of the gear wheels between 1100–1120 meshing periods: (a) pinion, (b) gear; velocity dependent torque, velocity independent torque

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Fig. 6

Angular velocities of the gear wheels between 100,100–100,120 meshing periods: (a) pinion, (b) gear; velocity dependent torque, velocity independent torque

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Fig. 7

(a) Mesh and natural frequencies, (b) frequency ratio

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Fig. 8

Comparison of the response characteristics: (a) maximum amplitude, (b) minimum amplitude; current methodology, · methodology presented in Ref. [12]

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Fig. 9

(a) Time histories of the dynamic transmission error and (b) phase portraits at cruising speed of 78 kph and input torque of 121.3 Nm; upper branch (current methodology) and lower branch (current methodology), · methodology presented in Ref. [12]

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Fig. 10

Effect of the mesh damping coefficient on the maximum amplitude of the dynamic response: (a) c = 1.9130e + 003 Ns/m, (b) c = 2.8695e + 003 Ns/m, (c) 5.7390e + 003 Ns/m and (d) 1.1478e + 004 Ns/m; stable branch, unstable branch

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Fig. 11

Effect of the mesh damping coefficient on the minimum amplitude of the dynamic response: (a) c = 1.9130e + 003 Ns/m, (b) c = 2.8695e + 003 Ns/m, (c) 5.7390e + 003 Ns/m and (d) 1.1478e + 004 Ns/m;    stable branch, unstable branch

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Fig. 12

Effect of the out of phase stiffness variation on the dynamic response: (a) maximum and (b) minimum amplitude; stable branch, unstable branch

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Fig. 13

Effect of the in phase stiffness variation on the dynamic response: (a) maximum and (b) minimum amplitude; stable branch, unstable branch

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Fig. 14

Effect of contact radii variation: (a) maximum (b) minimum amplitude;  stable branch,   unstable branch

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Fig. 15

Effect of the kinematic transmission error, in phase case: (a) maximum (b) minimum amplitude;   stable branch,   unstable branch

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Fig. 16

Effect of the kinematic transmission error on the dynamic response (π/2 phase difference): (a) maximum and (b) minimum amplitude; stable branch,  unstable branch

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Fig. 17

Effect of the second harmonic of the static transmission error on the dynamic response: (a) maximum and (b) minimum amplitude;   stable branch,  unstable branch

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