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Research Papers: Power Transmissions and Gearing

# Dynamic Transmission Error Prediction of Helical Gear Pair Under Sliding Friction Using Floquet Theory

[+] Author and Article Information
Song He

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

Rajendra Singh

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

J. Mech. Des 130(5), 052603 (Apr 01, 2008) (9 pages) doi:10.1115/1.2890115 History: Received May 06, 2007; Revised August 17, 2007; Published April 01, 2008

## Abstract

An analytical solution to the dynamic transmission error of a helical gear pair is developed by using a single-degree-of-freedom model with piecewise stiffness functions that characterize the contact plane dynamics and capture the velocity reversal at the pitch line. By assuming a constant mesh stiffness density along the contact lines, a linear time-varying model (with parametric excitations) is obtained, where the effect of sliding friction is quantified by an effective mesh stiffness term. The Floquet theory is then used to obtain closed-form solutions to the dynamic transmission error, and responses are derived to both initial conditions and the forced periodic function under a nominal preload. Analytical models are validated by comparing predictions with numerical simulations, and the effect of viscous damping is examined. Stability analysis is also briefly conducted by using the state transition matrix. Overall, the sliding friction has a marginal effect on the dynamic transmission error of helical gears, as compared with spur gears, in the context of the torsional model.

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## Figures

Figure 1

Schematic of the helical gear pair system

Figure 2

Contact zones at the beginning of a mesh cycle within the contact plane. Key: PP′ is the pitch line; AA′ is the face width W; AD is the length of contact zone Z.

Figure 3

Individual effective stiffness Ke,i(t) along the contact zone, where Tmesh is one mesh cycle. Key: blue solid line, tooth pair 0 (μ=0); green solid line, tooth pair 1 (μ=0); red solid line, tooth pair 2 (μ=0); blue dotted line, tooth pair 0 (μ=0.4);green dashed line, tooth pair 1 (μ=0.4); and red dashed-dotted line, tooth pair 2 (μ=0.4).

Figure 4

Piecewise effective stiffness function defined in six regions within one mesh cycle withμ=0.4. Key: blue dotted line, tooth pair 0; green dashed line, tooth pair 1; red dashed-dotted line, tooth pair 2; and black solid line, combined stiffness function.

Figure 5

(a) Effective stiffness. (b) homogeneous response predictions within two mesh cycles, given x0=2×10−6in. and v0=20in.∕s at Ωp=1000rpm. Key: green dotted line, μ=0; red dashed-dotted line, μ=0.2; black dashed line, μ=0.4 (analytical solution by the Floquet theory); blue solid line, μ=0.2 (numerical solution).

Figure 6

Predictions of damped homogeneous responses within two mesh cycles, given x0=2×10−6in., v0=20in.∕s, and μ=0.2 at Ωp=1000rpm. Key: blue solid line, analytical solution by the Floquet theory with Ce0; red dashed-dotted line, numerical with Ce(t).

Figure 7

Predictions of (undamped) forced periodic responses within two mesh cycles, given x0=2×10−6in., v0=20in.∕s, Tp=2000lbin., and μ=0.2 at Ωp=1000rpm. Key: blue solid line, analytical solution by the Floquet theory; red dashed-dotted line, numerical solution.

Figure 8

Steady-state forced periodic responses given x0=2×10−6in., v0=20in.∕s, Tp=2000lbin., and μ=0.1 at Ωp=1000rpm: (a) DTE versus time; (b) DTE spectra. Key: blue solid line with ○, undamped analytical prediction; black dotted line with ◻, damped numerical simulation of the SDOF system with ζe=5%; red dashed-dotted line with ▵, damped numerical simulation of a 6DOF model with ζe=5% (with mean component compensated).

Figure 9

Predicted mesh harmonics of (undamped) forced periodic responses as a function of μ, given x0=2×10−6in., v0=20in.∕s, and Tp=2000lbin. at Ωp=1000rpm: (a) DTE; (b) slope of DTE. Key: blue line with ▵, n=1; green line with ○, n=2; red line with ×, n=3; and cyan line with ◻, n=4.

Figure 10

Mapping of eigenvalue κ (absolute value) maxima as a function of the ratio of time-varying mesh frequency fm(t) to the system natural frequency fn. Key: —, μ=0.01; -⋅-, μ=0.1; and ⋯, μ=0.2.

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