0
Research Papers: Power Transmissions and Gearing

Construction of Semianalytical Solutions to Spur Gear Dynamics Given Periodic Mesh Stiffness and Sliding Friction Functions

[+] Author and Article Information
Song He

Acoustics and Dynamics Laboratory, Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210he.81@osu.edu

Todd Rook

 Goodrich Aerospace, 101 Waco Street, Troy, OH 45373todd.rook@goodrich.com

Rajendra Singh

Fellow ASME, Acoustics and Dynamics Laboratory, Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210singh.3@osu.edu

J. Mech. Des 130(12), 122601 (Oct 07, 2008) (9 pages) doi:10.1115/1.2988478 History: Received October 16, 2007; Revised June 10, 2008; Published October 07, 2008

Gear dynamic models with time-varying mesh stiffness, viscous mesh damping, and sliding friction forces and moments lead to complex periodic differential equations. For example, the multiplicative effect generates higher mesh harmonics. In prior studies, time-domain integration and fast Fourier transform analysis have been utilized, but these methods are computationally sensitive. Therefore, semianalytical single- and multiterm harmonic balance methods are developed for an efficient construction of the frequency responses. First, an analytical single-degree-of-freedom, linear time-varying system model is developed for a spur gear pair in terms of the dynamic transmission error. Harmonic solutions are then derived and validated by comparing with numerical integration results. Next, harmonic solutions are extended to a six-degree-of-freedom system model for the prediction of (normal) mesh loads, friction forces, and pinion/gear displacements (in both line-of-action and off-line-of-action directions). Semianalytical predictions compare well with numerical simulations under nonresonant conditions and provide insights into the interaction between sliding friction and mesh stiffness.

FIGURES IN THIS ARTICLE
<>
Copyright © 2008 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

(a) Periodic mesh stiffness functions of the spur gear pair example (with tip relief) given nominal pinion torque T¯p=550lbin. Key: blue dashed line, k0(t); red solid line, k1(t). (b) Periodic frictional functions. Key: blue dashed line, f0(t); green dashed-dotted line, f1(t); red solid line, f2(t).

Grahic Jump Location
Figure 2

(a) Snap shot of the contact pattern (at t=0) for the sample spur gear pair. (b) Normal (mesh) and friction forces of the analytical spur gear system model.

Grahic Jump Location
Figure 3

Semianalytical versus numerical solutions for the SDOF model, expressed by Eq. 3, given T¯p=550lbin., Ω¯p=500rpm, and μ=0.04. (a) Time-domain responses; (b) mesh harmonics in frequency domain. Key: blue solid line and △, numerical simulations; black dashed line and ◻, semianalytical solutions using one-term HBM; red dashed-dotted line and ○, semianalytical solutions using five-term HBM.

Grahic Jump Location
Figure 4

Semianalytical versus numerical solutions for the SDOF model as a function of pinion speed with μ=0.04. (a) Mesh order n=1, (b) n=2, (c) n=3, and (d) n=4. Key: red ×, numerical simulations; blue solid line, semianalytical solutions using five-term HBM.

Grahic Jump Location
Figure 5

(a) 6DOF spur gear pair model and its subset of unity gear pair (3DOF model) used to study the natural frequency distribution. (b) Natural frequencies ΩN as a function of the stiffness ratio KB∕km. Key: black solid line, ΩSN of SDOF system (torsional only, in terms of DTE); blue dashed line, ΩN1 of 3DOF system; green dotted line, ΩN2 of 3DOF system; red dashed-dotted line, ΩN3 of 3DOF system.

Grahic Jump Location
Figure 6

Semianalytical versus numerical solutions for the 6DOF model as a function of Ω¯p with μ=0.04. (a) Mesh order n=1, (b) n=2, (c) n=3, and (d) n=4. Key: blue solid line, predictions using five-term HBM; green ○, numerical simulations (4) with nominal KB(KB∕km=0.37); red ×, numerical simulations with stiff KB(KB∕km=100).

Grahic Jump Location
Figure 7

Semianalytical versus numerical solutions of the LOA displacement xp for the 6DOF model as a function of Ω¯p with KB∕km=100, μ=0.04. (a) Mesh order n=1, (b) n=2, (c) n=3, and (d) n=4. Key: red ×, numerical simulations; blue solid line, predictions using five-term HBM.

Grahic Jump Location
Figure 8

Semianalytical versus numerical solutions of the OLOA displacement yp for the 6DOF model as function of Ω¯p with KB∕km=100, μ=0.04. (a) Mesh order n=1, (b) n=2, (c) n=3, and (d) n=4. Key: red ×, numerical simulations; blue solid line, predictions using five-term HBM.

Grahic Jump Location
Figure 9

Semianalytical versus numerical solutions of the LOA displacement xp for the 6DOF model as a function of Ω¯p with KB∕km=0.37, μ=0.04. (a) Mesh order n=1, (b) n=2, (c) n=3, and (d) n=4. Key: green ○, numerical simulations; blue solid line, predictions using five-term HBM.

Grahic Jump Location
Figure 10

Semianalytical versus numerical solutions of the OLOA displacement yp for the 6DOF model as a function of Ω¯p with KB∕km=0.37, μ=0.04. (a) Mesh order n=1, (b) n=2, (c) n=3, and (d) n=4. Key: green ○, numerical simulations; blue solid line, predictions using five-term HBM.

Grahic Jump Location
Figure 11

Semianalytical predictions for the ratio of OLOA displacement xp to the LOA displacement xp for the 6DOF model as a function of T¯p with KB∕km=0.37, μ=0.04. (a) Mesh order n=1, (b) n=2, (c) n=3, and (d) n=4. Key: red dashed line with ×, perfect involute gear; blue solid line with ○, gear with tip relief.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In