NVH Optimization for Electric Axles: A Case Study

In the simulation of electric axles using a Multi-Body System (MBS) environment, the vibrational characteristics of components—such as gears and bearings —are inherently accounted for as long as these components are properly modeled. The system automatically considers their interaction and the frequencies they generate.

However, while the simulation provides detailed results, it is essential for engineers to understand and identify the fundamental orders of these frequencies. Knowing the expected frequency ranges allows engineers to:

Validate simulation outputs.
Anticipate potential resonances or harmonics.
Design countermeasures effectively during the early stages of development.

This understanding bridges the gap between numerical simulation and practical engineering insight. Below, we outline the critical frequency sources in electric axles and provide analytical formulas to calculate them. These calculations serve as a guide for evaluating the dynamic behavior of the system and improving NVH performance.

Section 1: Analytical Formulas for NVH Analysis

Objective

To improve the NVH performance of a prototype electric axle by identifying and mitigating sources of vibration and noise, primarily caused by gear interactions, bearing dynamics, and motor harmonics.

Key Formulas: Gear Excitation Orders

1. Fundamental Gear Mesh Frequency (GMF)

f_m = z × f_s

Explanation: The gear mesh frequency is calculated as the product of the number of teeth (z) and the shaft frequency (f_s).

2. Epicyclic Gearbox GMF

f_m = (z_r × z_s) / (z_r + z_s) × n_s / 60

Explanation: For an epicyclic gearbox, the GMF depends on the teeth count of the ring (z_r) and sun gears (z_s), and the rotational speed of the sun gear (n_s).

Key Formulas: Bearings Excitation Orders

Key Parameters: D₁: Outer diameter D₂: Bore diameter PD: Pitch diameter, calculated as (D₁ + D₂) / 2 D_b: Rolling element diameter β: Contact angle n: Number of rolling elements rps: Revolutions per second

1. Fundamental Train Frequency (FTF)

f_FTF = (rps / 2) × (1 - (D_b / PD) × cos(β))

Explanation: The fundamental train frequency is related to cage rotation defects. It depends on the relative speed (rps), rolling element diameter (D_b), pitch diameter (PD), and contact angle (β).

2. Ball Pass Frequency of the Outer Race (BPFO)

f_BPFO = (n × rps) / 2 × (1 - (D_b / PD) × cos(β))

Explanation: The BPFO represents the frequency at which rolling elements pass by a defect on the outer raceway. It is a function of the number of rolling elements (n), relative speed, and geometric parameters.

3. Ball Pass Frequency of the Inner Race (BPFI)

f_BPFI = (n × rps) / 2 × (1 + (D_b / PD) × cos(β))

Explanation: The BPFI calculates the frequency at which rolling elements pass by a defect on the inner raceway, using similar parameters as BPFO but with an additive component.

4. Ball-Spin Frequency (BSF)

f_BSF = (PD × rps) / (2 × D_b) × [1 - ((D_b / PD) × cos(β))²]

Explanation: The BSF represents the frequency of impacts between rolling elements and the raceway. It accounts for the interaction of pitch diameter, rolling element diameter, and contact angle.

Key Formulas: Electric Machine Excitation Orders

Electric motor noise is primarily caused by electromagnetic forces in the air gap. These forces are categorized as:

Tangential forces: Generate motor torque.
Radial forces: Cause motor noise without affecting operation.

Key Parameters: p: Number of pole pairs s: Number of stator slots f: Motor rotational frequency (Hz) k: Integer sequence (0, 1, 2, ...) k₁: Integer multiplier for harmonic calculations n: Integer values (e.g., 6k ± 1)

Main Electromagnetic Frequencies

1. Product of Stator Spatial Harmonics

f₁ = 2p × n × f

Explanation: This frequency results from the stator spatial harmonics, where p is the number of pole pairs, n is an integer, and f is the motor rotational frequency (Hz).

2. Product of Rotor Spatial Harmonics

f₂ = 2p × n × (1 ± 2k₁) × f

Explanation: This frequency is derived from rotor spatial harmonics, with k₁ being an integer (0, 1, 2, ...).

3. Stator Winding and Rotor Spatial Harmonics

f₃ = 2p × n × (1 + k₁) × f

f₃ = 2p × n × k₁ × f

Explanation: This accounts for interactions between stator windings and rotor spatial harmonics.

4. Rotor Magnetic Field and Stator Slots Interaction

f₄ = p × μ_λ × f

Where:

μ_λ: Integer value defined as μ_λ = int[k × s / p], where s is the number of stator slots.

Explanation: This frequency is caused by the interaction of the rotor's magnetic field with the slotted stator core.

Section 2: Evaluating Gear Design Changes Using MBS Simulation

In this section, we demonstrate the application of MBS simulations to evaluate the impact of gear design changes on NVH performance. Specifically, we compare the noise and vibration levels of a system with Low Contact Ratio (LCR) gears versus High Contact Ratio (HCR) gears.

Methodology:

Model Setup:
- An MBS model of the electric axle is created, including detailed representations of the gear train, bearings, and motor components.
- Gear meshing parameters, such as stiffness and damping, are calibrated to reflect realistic conditions.
Analysis:
- The simulation is run for both LCR and HCR gear designs.
- Noise levels are assessed using sound pressure and vibration amplitudes at the housing.
- Dominant excitation frequencies are correlated with analytical calculations to ensure consistency.