Dynamic Optimization of High-speed Herringbone Gear Transmission System

This study investigates the dynamic excitation characterization and robust optimization of high-speed herringbone gear transmission systems. Through systematic modeling and analysis, we propose methodologies to minimize vibration responses while considering manufacturing uncertainties. Key innovations include resonance point identification criteria and a 6σ robust design framework validated through experimental verification.

1. Dynamic Excitation Characterization

The time-varying meshing stiffness of herringbone gears is calculated using energy method and validated through FEM. The potential energy components include:

$$U_h = \frac{F^2}{2K_h},\ U_b = \int_0^d \frac{[F_b(d-x) – F_ah]^2}{2EI_x}dx$$

Comparative results show excellent agreement between analytical and numerical methods:

Method Average Stiffness (N/m) Computation Time
Analytical 2.68×107 15 min
FEM 2.72×107 4.5 hr

Transmission error and backlash are modeled as:

$$TE(t) = 0.5f_p \sin(\omega_p t + \phi_p) + 0.5f_m \sin(\omega_m t + \phi_m)$$
$$b(t) = \begin{cases}
b_{max} – \delta(t) & \delta(t) < -b \\
0 & |\delta(t)| \leq b \\
b_{max} + \delta(t) & \delta(t) > b
\end{cases}$$

2. Dynamic Modeling and Vibration Analysis

The flexible multibody dynamics model reveals critical modal characteristics:

Mode Frequency (Hz) Damping Ratio Energy Distribution
1 131.16 0.622 Y-axis swing
2 164.98 0.696 X-axis swing
3 202.49 0.645 Z-axis extension

Resonance identification criteria are established:

  1. Frequency principle: $f_{excite} \leq 6f_{shaft}$
  2. Damping principle: $0 < \zeta < 1$
  3. Energy principle: $\sum E_{rot} < 1$

3. Robust Optimization Design

The 6σ optimization model minimizes vibration acceleration variance:

$$Minimize\ F = w_1\left(\frac{\mu_y – M}{\Delta M}\right)^2 + w_2\left(\frac{\sigma_y}{\Delta S}\right)^2$$

Design Variable Initial (mm) Optimal (mm) Δ Mass (kg)
Cover thickness 14 12.5 -8.2
Base thickness 24 22.1 -12.7
Stiffener thickness 10 11.4 +3.6

Optimization results demonstrate significant vibration reduction:

$$a_{RMS}^{optim} = 0.55a_{RMS}^{initial}\ \pm 0.02σ$$

4. Experimental Validation

Prototype testing confirms theoretical predictions:

Parameter 6000 rpm Specification
Max shaft vibration 19.1 μm ≤25 μm
Noise level 106.5 dB(A) ≤110 dB(A)
Efficiency 98.86% ≥98.5%

The proposed methodology successfully addresses the dynamic challenges in herringbone gear systems through:

  1. Comprehensive excitation modeling
  2. Resonance avoidance strategy
  3. Uncertainty-aware optimization

Future research will focus on thermal-structural coupling effects and multi-objective reliability optimization for enhanced herringbone gear performance in extreme operating conditions.

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