My research focuses on the vibration and noise behavior of a two-stage helical gear reducer specifically designed for an electric vehicle. With the rapid growth of the electric vehicle market, the powertrain NVH (Noise, Vibration, and Harshness) performance has become one of the most important attributes for customer satisfaction. Unlike internal combustion engine vehicles, an electric vehicle does not have an engine to mask gear noise. Therefore, the electric vehicle reducer, which operates at high speed and with high torque, often becomes a dominant noise source. In the present work, I integrate finite element analysis, multi-body dynamics, acoustic boundary element simulation, and experimental validation to systematically analyze and optimize the NVH performance of an electric vehicle reducer.
The work is carried out following a complete technical route: modeling, simulation, experiment, and optimization. First I build a high-fidelity multi-degree-of-freedom dynamic model of the complete reducer system. Then I perform static strength analysis and modal analysis. After that, I evaluate the transmission error under different input torques and use the worst case as the excitation to calculate the housing vibration response. Next, I predict the radiated noise by using boundary element acoustics and validate it with semi-anechoic chamber bench tests. Finally, I apply the NSGA-II multi-objective genetic algorithm to optimize the gear micro-geometry parameters so that both the excitation source and the structural response are improved.
1. Research Background and Significance
The global energy crisis and severe environmental pollution have forced the transportation sector to turn to electrification. Electric vehicles have gained a rapidly growing market share, and many countries have announced progressive plans to phase out traditional internal combustion engine vehicles. China is among the largest electric vehicle markets, and the domestic industry has invested heavily in improving the driving range, safety, and comfort of electric vehicles. The passenger perception of the vehicle quality depends strongly on acoustic comfort. Because an electric vehicle lacks the masking effect of an engine, the noise generated by auxiliary components such as the reducer, motor, inverter, and pumps becomes clearly audible. Among these, the electric vehicle reducer is especially critical because it transmits the full motor torque to the wheels through meshing gear pairs, which inevitably produce vibrations.
The reducer analyzed in this study belongs to a development-stage electric vehicle and is configured as a coaxial two-stage reduction gearbox with an integrated differential. Its main working parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Rated torque | 140 Nm |
| Peak torque | 320 Nm |
| Rated power | 60 kW |
| Peak power | 130 kW |
| Maximum speed | 10000 rpm |
During the actual operation of an electric vehicle, the reducer input speed may exceed 10,000 rpm, which is much higher than the speed levels normally encountered in traditional transmissions. Gear meshing frequencies thus sweep over a wide band, and if they coincide with the structural natural frequencies of the housing or the shafts, severe resonances occur. The so-called “electric vehicle whine” is a tonal noise at gear mesh harmonics, and it is recognized as one of the most annoying issues in electric vehicles. Therefore, improving the NVH performance of the electric vehicle reducer is not only a technical problem, but also an economic and market-oriented challenge.
A large number of previous studies have investigated gearbox NVH using both analytical and numerical methods. Some researches focused on dynamic tooth loads and bearing forces, while others studied the sound radiation efficiency of gearbox housings. With the emergence of electric vehicle drivetrains, researchers have attempted to optimize gear macro-geometry and micro-geometry to reduce transmission error and dynamic excitation. Nevertheless, most early approaches rely on single-objective parameter tuning or experience-based rules. These methods often fail when multiple conflicting objectives, nonlinear gear-housing interactions, and wide speed ranges are considered. To overcome these limitations, my work establishes a multi-physics framework that couples the structural dynamics of the electric vehicle reducer with acoustic radiation and uses an intelligent optimization algorithm to search for the best gear micro-modification parameters under realistic operating conditions.
2. Dynamic Modeling of the Reducer System
2.1 General system dynamic equation
The physical structure of the reducer is rather complex: it consists of the input shaft assembly, the intermediate shaft assembly, the differential output shaft assembly, two helical gear pairs, six rolling element bearings, the differential housing, and the outer monolithic housing. To preserve all important dynamic characteristics while retaining computational efficiency, I adopt a finite-element-based substructure technique. The complete drive train is discretized into a set of one-dimensional beam elements, lumped mass nodes, gear mesh springs, and bearings. The governing equation for the entire reducer system can be represented in the standard second-order matrix form:
$$
[\mathbf{M}]\{\ddot{\mathbf{q}}\} + [\mathbf{C}]\{\dot{\mathbf{q}}\} + [\mathbf{K}]\{\mathbf{q}\} = \{\mathbf{F}(t)\}
$$
where \([\mathbf{M}]\), \([\mathbf{C}]\), and \([\mathbf{K}]\) are respectively the global mass, damping, and stiffness matrices assembled from each component; \(\{\mathbf{q}\}\) is the vector of generalized displacements, and \(\{\mathbf{F}(t)\}\) is the external and internal excitation vector. To obtain the global matrices, the reducer housing and differential housing are first modeled in HyperMesh with three-dimensional tetrahedral elements, then reduced by dynamic condensation in the frequency range below 4000 Hz. The condensation is performed at the centers of the bearing housings connected to the internal surfaces by rigid elements. By doing so, the large finite element models are transformed into small equivalent stiffness and mass matrices while preserving their modal behavior in the frequency band of interest.
2.2 Housing and differential housing finite element modeling
The housing of the electric vehicle reducer is assembled from three parts: the main reducer housing, the motor housing, and the motor end cover. This assembly not only protects the internal gears from contaminants but also acts as the main radiator of noise. A detailed three-dimensional geometric model was first cleaned and simplified. Small fillets, chamfers, oil holes, and thread holes that have negligible influence on NVH were deleted to avoid low-quality elements. Then the model was meshed with second-order tetrahedral elements with a base size of 2 mm. The resulting mesh statistics are shown in Table 2.
| Component | Element count | Node count |
|---|---|---|
| Reducer housing assembly | 5,475,153 | 1,203,156 |
| Differential housing | 553,567 | 123,502 |
The finite element model of the differential housing is a non-axisymmetric rotating structure that cannot be defined directly by the one-dimensional beam elements in Romax. Hence, its reduced mass and stiffness matrices are generated in the same manner as the housing, and then imported into the Romax environment as an FE shaft. Table 3 lists the material properties used in the simulation.
| Material | Elastic modulus (MPa) | Density (kg/m³) | Poisson’s ratio |
|---|---|---|---|
| A380 aluminum alloy | 70,000 | 2700 | 0.33 |
| QT500 cast iron | 169,000 | 7100 | 0.275 |
| 45 steel | 206,000 | 7850 | 0.29 |
| 20CrMnTi | 210,000 | 7870 | 0.30 |
For the reducer housing, the six bearing bore centers are selected as master nodes. Each bearing bore surface is connected to the corresponding center point through an RBE2 element, and the mass of the actual bolt group is nodalized at the bolt locations. The dynamic condensation involves computing the frequency-dependent Guyan reduction over the range 0–4000 Hz. After condensation, the reduced matrices are expressed by submatrices as
$$
[\mathbf{M}_{h}] =
\begin{bmatrix}
M_{11} & M_{12} & \cdots & M_{16} \\
M_{21} & M_{22} & \cdots & M_{26} \\
\vdots & \vdots & \ddots & \vdots \\
M_{61} & M_{62} & \cdots & M_{66}
\end{bmatrix}, \quad
[\mathbf{K}_{h}] =
\begin{bmatrix}
K_{11} & K_{12} & \cdots & K_{16} \\
K_{21} & K_{22} & \cdots & K_{26} \\
\vdots & \vdots & \ddots & \vdots \\
K_{61} & K_{62} & \cdots & K_{66}
\end{bmatrix}
$$
where the subscripts refer to the six bearing center nodes retained in the condensation. These reduced matrices contain the full information of the housing elastic behavior in the 0-4000 Hz band, including all local modes that influence the NVH response of the electric vehicle reducer.
2.3 Elastic shaft element model
Every shaft in the reducer is divided into Timoshenko beam elements. Each element has two nodes and six degrees of freedom per node: three translational displacements and three rotations, so the element displacement vector is
$$
\{\mathbf{q}_s\} = [x_A, y_A, z_A, \theta_{xA}, \theta_{yA}, \theta_{zA}, x_B, y_B, z_B, \theta_{xB}, \theta_{yB}, \theta_{zB}]^T
$$
By considering the shear deformation, rotatory inertia, and gyroscopic effects, the consistent mass matrix \([\mathbf{M}_s]\), stiffness matrix \([\mathbf{K}_s]\), and damping matrix \([\mathbf{C}_s]\) can be derived from the elastic energy and kinetic energy. The entries of the mass matrix depend on the element length \(L\), cross-sectional area \(A\), density \(\rho\), and the transverse shear parameter \(\Phi\), defined as
$$
\Phi = \frac{12EI}{\kappa A G L^2}
$$
where \(E\) is Young’s modulus, \(G\) is the shear modulus, \(I\) is the second moment of area, and \(\kappa\) is the shear correction factor which is \(6(1+\nu)/(7+6\nu)\) for a solid circular section and \(2(1+\nu)/(4+3\nu)\) for a hollow circular section. The damping matrix is assumed to be proportional damping:
$$
[\mathbf{C}_s] = \alpha [\mathbf{M}_s] + \beta [\mathbf{K}_s]
$$
where the coefficients \(\alpha\) and \(\beta\) are obtained from the damping ratios at two selected frequencies. These shaft models enable the exact representation of the torsional and bending flexibilities of the input, intermediate, and output shafts.
2.4 Gear mesh element model
Each gear pair is represented as a pair of rigid disks connected by a spring-damper element along the line of action. The gear mesh element has 12 degrees of freedom. The displacement vector for pinion and gear is
$$
\{\mathbf{q}_g\} = [x_1, y_1, z_1, \theta_{x1}, \theta_{y1}, \theta_{z1}, x_2, y_2, z_2, \theta_{x2}, \theta_{y2}, \theta_{z2}]^T
$$
I denote the mass matrix of the gear mesh element as a diagonal matrix containing the mass and moments of inertia of the two gears:
$$
[\mathbf{M}_g] = \mathrm{diag}[m_1, m_1, m_1, I_{x1}, I_{y1}, I_{z1}, m_2, m_2, m_2, I_{x2}, I_{y2}, I_{z2}]
$$
The mesh stiffness matrix is obtained from the average mesh stiffness \(k_g\) and the displacement projection vector along the line of action:
$$
[\mathbf{K}_g] = k_g \{\mathbf{a}\}\{\mathbf{a}\}^T
$$
with
$$
\{\mathbf{a}\} =
\begin{bmatrix}
\sin\phi\cos\beta_b, & \cos\phi\cos\beta_b, & \sin\beta_b, & \pm r_1\sin\beta_b, & \cos\beta_b, & r_1\cos\phi\cos\beta_b \\
-\sin\phi\cos\beta_b, & -\cos\phi\cos\beta_b, & -\sin\beta_b, & \mp r_2\sin\beta_b, & -\cos\beta_b, & -r_2\cos\phi\cos\beta_b
\end{bmatrix}^T
$$
where \(\phi\) is the angle between the mating plane and the positive \(Y\)-axis, and \(\beta_b\) is the base helix angle. In the same way, the mesh damping matrix is written as
$$
[\mathbf{C}_g] = c_g \{\mathbf{a}\}\{\mathbf{a}\}^T
$$
where \(c_g\) is the mesh damping coefficient. Since a pair of helical gears may have multiple pairs of teeth in contact, the time-varying mesh stiffness is actually a periodic function of the mesh cycle. In the NVH analysis, I use the mean mesh stiffness for eigenvalue problems, while the first-order excitation is introduced through the transmission error. This yields an effective and accurate model for the electric vehicle reducer system.
2.5 Bearing element model
Rolling element bearings are high-stiffness components that directly couple the shafts to the housing. Since their mass is small compared to the shaft and housing, only their stiffness and damping matrices are retained. The bearing is modeled as a 6-degree-of-freedom elastic support including cross-coupling stiffness terms:
$$
[\mathbf{K}_b] =
\begin{bmatrix}
k_{xx} & & & & & \\
k_{yx} & k_{yy} & & & & \\
k_{zx} & k_{zy} & k_{zz} & & & \\
k_{\theta_xx} & k_{\theta_xy} & k_{\theta_xz} & k_{\theta_x\theta_x} & & \\
k_{\theta_yx} & k_{\theta_yy} & k_{\theta_yz} & k_{\theta_y\theta_x} & k_{\theta_y\theta_y} & \\
k_{\theta_zx} & k_{\theta_zy} & k_{\theta_zz} & k_{\theta_z\theta_x} & k_{\theta_z\theta_y} & k_{\theta_z\theta_z}
\end{bmatrix}
$$
A similar full 6×6 matrix is used for the damping matrix. These bearing matrices are calculated from the contact deformation theory of rolling elements and are subsequently imported into the system model of the electric vehicle reducer.
3. Static Strength Analysis of the Reducer Housing
Before performing the dynamic NVH analysis, I investigate the structural strength of the housing under the most severe driving conditions. The extremes are the full positive torque condition and the full regenerative braking (negative torque) condition, both at peak torque \(320\,\text{Nm}\). The load on each bearing is obtained from the Romax system model, and the corresponding bearing force and moment components are listed in Table 4.
| Condition | Bearing | Force (N) | Moment (Nm) | ||||
|---|---|---|---|---|---|---|---|
| X | Y | Z | X | Y | Z | ||
| Positive torque 320 Nm | Input left | 1252.1 | -1772.8 | 1935.4 | -21.0 | 0 | 14.1 |
| Input right | -5413.4 | -1941.4 | -12718.4 | 51.7 | 0 | -20.8 | |
| Intermediate left | 6616.4 | 1745.6 | 1726.6 | 5.0 | 0 | -36.5 | |
| Intermediate right | 11365.1 | -12005.2 | -26440.3 | 178.1 | 0 | 60.1 | |
| Output left | -18411.0 | 18638.7 | 20095.2 | 182.4 | 0 | 138.1 | |
| Output right | 4590.8 | -4665.0 | 15401.4 | -131.8 | 0 | 36.7 | |
| Negative torque -320 Nm | Input left | 12.7 | 1256.5 | -1414.9 | -13.3 | 0 | -0.1 |
| Input right | -4174.2 | 2457.7 | 12198.0 | 70.0 | 0 | 22.6 | |
| Intermediate left | 7502.7 | 17402.1 | -923.5 | 88.2 | 0 | -69.0 | |
| Intermediate right | 10479.1 | -7142.5 | 25637.0 | -162.6 | 0 | 71.2 | |
| Output left | 2592.9 | 5907.9 | -20051.4 | -162.1 | 0 | -16.9 | |
| Output right | -16413.1 | -19881.7 | -15445.2 | 165.5 | 0 | -113.0 | |
In the finite element strength model, the bearing outer rings are inserted into the bearing seats. The outer ring surfaces are rigidly connected to the seat holes, and the inner raceway surface of each outer ring is connected to its center point by an RBE2 element. The bearing load is then applied at that center point. This method avoids artificially increasing the local stiffness while preserving the correct load transfer path. The housing mounting conditions have been modeled by constraining all bolt holes and the end faces that are bolted to the vehicle axle housing.
Table 5 lists the maximum von Mises stress and the maximum displacement computed for both extreme torque conditions.
| Condition | Maximum stress location | Maximum stress (MPa) | Maximum displacement (mm) |
|---|---|---|---|
| Positive torque | Output left bearing inner wall | 110.889 | 0.082 |
| Negative torque | Output left bearing inner wall | 98.861 | 0.080 |
Since the allowable stress of the A380 aluminum alloy is 160 MPa, the housing strength is adequate. According to the national standard QC/T 534-1999, the maximum allowable deformation per meter of wheel track is 1.5 mm/m. The wheel track of the target electric vehicle is 1630 mm, which gives a maximum allowable housing deformation of 2.445 mm. Both computed values are far below this threshold. Therefore, the static structural behavior of the electric vehicle reducer housing is considered safe, and the same model can be used for subsequent modal and NVH simulations.
4. Modal Analysis and Experimental Verification
4.1 Theoretical background
The modal analysis of the reducer housing is carried out to determine its natural frequencies and mode shapes. For an undamped free-vibration system, the dynamic equation reduces to
$$
[\mathbf{M}]\{\ddot{\mathbf{x}}\} + [\mathbf{K}]\{\mathbf{x}\} = 0
$$
Assuming a harmonic solution \(\{\mathbf{x}\} = \{\mathbf{x}_0\}\sin(\omega_0 t + \varphi_0)\), the eigenvalue problem is
$$
([\mathbf{K}] – \omega_0^2 [\mathbf{M}])\{\mathbf{x}_0\} = 0
$$
Nontrivial solutions exist when
$$
\det ([\mathbf{K}] – \omega_0^2 [\mathbf{M}]) = 0
$$
Solving this characteristic equation yields the natural frequencies and mode shapes of the housing.
4.2 Free modal simulation
In the free modal simulation, the housing is unconstrained in space, and the calculation is limited to 0-4000 Hz, because frequencies above 4000 Hz are less important for global NVH perception in most electric vehicle applications. Table 6 lists the first six computed natural frequencies.
| Mode | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Frequency (Hz) | 1236.7 | 1301.4 | 1539.7 | 1846.3 | 1891.6 | 1941.5 |
The corresponding mode shapes show that the first mode mainly involves deformation at the joint between the reducer housing and the motor housing and at the motor end cover; the second mode is primarily a stretching deformation along the Z axis; the third mode is a motor end cover bending mode; and the higher modes also tend to concentrate at the motor end cover and the side walls of the reducer housing.
4.3 Free modal test
To verify the finite element model, I performed an experimental free modal test on the actual electric vehicle reducer housing. The housing was suspended by soft elastic cords. A force hammer was used as the exciting source, and tri-axial accelerometers were used as response transducers. The multiple-input single-output? Actually I used the single-input multi-output (SIMO) method for data acquisition. The LMS Test.lab acquisition system recorded the frequency response functions and computed the modal parameters using the polyreference least-squares complex frequency method.
The modal test was divided into seven groups of measurement points to keep the accelerometer mass loading small. Altogether 28 response points were distributed over the housing surface. The coordinates of these test points were imported into the LMS geometry module, where they were connected to create a wireframe model of the housing. The measured frequency response functions show clear peaks at several resonance frequencies. Table 7 compares the measured natural frequencies with the simulated values.
| Mode | Simulation (Hz) | Test (Hz) | Relative error (%) |
|---|---|---|---|
| 1 | 1236.7 | 1267.8 | 2.45 |
| 2 | 1301.4 | 1291.9 | 0.74 |
| 3 | 1539.7 | 1559.8 | 1.29 |
| 4 | 1846.3 | 1798.3 | 2.67 |
| 5 | 1891.6 | 1884.2 | 0.39 |
| 6 | 1941.5 | 1965.0 | 1.20 |
All errors are below 5%, demonstrating that the finite element model has sufficient accuracy for NVH prediction. The Modal Assurance Criterion (MAC) matrix is also calculated to evaluate the quality of the extracted mode shapes. The diagonal entries are close to unity, while the off-diagonal entries are relatively low. Thus, the experimental setup and the modal parameter estimation are reliable.
4.4 Frequency separation check
For a rotating machine, resonances are likely to occur when the gear mesh frequency or the shaft rotation frequency coincides with the structural natural frequencies. The mesh frequencies are calculated by
$$
\omega = \frac{n z}{60}
$$
and the shaft rotation frequency is
$$
s = \frac{n}{60}
$$
where \(n\) is the motor speed in rpm and \(z\) is the number of teeth. In the worst case, when the motor reaches 10,000 rpm, the first stage gear pair has 33 teeth, and the second stage gear pair has 22 teeth. The excitation frequencies are listed in Table 8.
| Frequency | Value (Hz) |
|---|---|
| 1st stage gear mesh frequency | 5500 |
| 2nd stage gear mesh frequency | 1142 |
| Input shaft rotation frequency | 166.7 |
| Intermediate shaft rotation frequency | 51.9 |
| Output shaft rotation frequency | 16.2 |
Comparing Table 8 with the housing natural frequencies of Table 7, the first six natural frequencies do not coincide with the second-stage gear mesh frequency or with any major shaft rotating harmonics. Although the first-stage gear mesh frequency range is broad because of the widely varying speed, at the nominal test point used in this work the mesh frequency at 5800 rpm is around 3190 Hz, which is close to the 32nd system mode at 3237 Hz. This proximity is later studied in detail through the vibration response analysis.
5. Gear Transmission Error and Dynamic Excitation
The static transmission error (STE) of a gear pair is defined as the difference between the actual angular position of the driven gear and its theoretical position when the driving gear rotates at a constant angular velocity. Transmission error is widely considered as the most important internal excitation source for gear whine. In the electric vehicle reducer, the first stage gear pair operates at higher speed and therefore often dominates whine generation. Thus I focus on the first-stage gear pair.
5.1 Simulation setup
Using the Romax model, I vary the input torque from 10 Nm to 140 Nm in increments of 10 Nm, while keeping the input speed at 5800 rpm. This input speed represents a common highway cruising condition for the target electric vehicle. The load cases are summarized in Table 9.
| Case number | Input torque (Nm) | Input speed (rpm) |
|---|---|---|
| 1 | 10 | 5800 |
| 2 | 20 | 5800 |
| 3 | 30 | 5800 |
| 4 | 40 | 5800 |
| 5 | 50 | 5800 |
| 6 | 60 | 5800 |
| 7 | 70 | 5800 |
| 8 | 80 | 5800 |
| 9 | 90 | 5800 |
| 10 | 100 | 5800 |
| 11 | 110 | 5800 |
| 12 | 120 | 5800 |
| 13 | 130 | 5800 |
| 14 | 140 | 5800 |
5.2 Results and discussion
Figure 1? Actually I cannot refer to figures by numbers in this text. The maximum transmission error first decreases as the torque increases from 10 to about 80 Nm, then increases. This is caused by the fact that the tooth deflection under load tends to compensate the manufacturing and assembly errors at low torques. When the torque further increases, the load-induced deformation dominates and causes the peak-to-peak transmission error to rise again. The largest peak-to-peak transmission error is found at 70 Nm. The mean transmission error at this torque is still relatively small.
By applying a fast Fourier transform to the transmission error time history at 70 Nm, I obtain the harmonic components. The first harmonic of the mesh frequency exhibits the largest amplitude. Therefore, the 70 Nm input torque, 5800 rpm input speed, and first-order gear mesh excitation are selected as the worst-case NVH load condition for the subsequent vibration and noise simulations. This condition is representative of a real-world acceleration phase of an electric vehicle.
6. Vibration Response Analysis of the Reducer System
6.1 System natural frequencies
The complete electric vehicle reducer system model includes the housing, shafts, gears, bearings, and the differential. After dynamic condensation, the eigenvalue analysis in Romax over 0-4000 Hz yields 41 system natural frequencies. Table 10 lists all the computed natural frequencies.
| Mode | Frequency (Hz) | Mode | Frequency (Hz) | Mode | Frequency (Hz) |
|---|---|---|---|---|---|
| 1 | 324.8 | 15 | 1638.9 | 29 | 2987.8 |
| 2 | 579.1 | 16 | 1684.1 | 30 | 3046.8 |
| 3 | 617.2 | 17 | 1728.6 | 31 | 3150.5 |
| 4 | 657.0 | 18 | 1928.0 | 32 | 3237.4 |
| 5 | 665.4 | 19 | 2000.2 | 33 | 3405.6 |
| 6 | 746.9 | 20 | 2211.5 | 34 | 3467.5 |
| 7 | 808.4 | 21 | 2299.3 | 35 | 3513.3 |
| 8 | 937.5 | 22 | 2451.4 | 36 | 3521.6 |
| 9 | 1144.7 | 23 | 2594.9 | 37 | 3595.9 |
| 10 | 1219.8 | 24 | 2723.6 | 38 | 3675.0 |
| 11 | 1348.0 | 25 | 2838.5 | 39 | 3734.6 |
| 12 | 1451.3 | 26 | 2878.2 | 40 | 3757.6 |
| 13 | 1525.1 | 27 | 2881.0 | 41 | 3899.3 |
| 14 | 1629.2 | 28 | 2959.0 |
6.2 Housing surface velocity RMS
With the first-stage gear pair transmission error applied as a displacement excitation, the forced response of the reducer system is calculated. The root mean square (RMS) velocity of the housing surface is a widely used indicator for structure-borne noise. Figure? I do not mention figure numbers. The RMS velocity spectrum exhibits three dominant peaks near 816 Hz, 2640 Hz, and 3200 Hz. By comparing these frequencies with the system natural frequencies, the peak near 816 Hz is associated with the 7th system mode at 808.4 Hz, while the peaks near 2640 Hz and 3200 Hz are associated with the 23rd, 24th and 32nd system modes.
6.3 Bearing vibration response
Further, I analyze the vibration response at each bearing location to trace the energy-transmission paths from the gears to the housing. The input shaft right bearing and the output shaft left bearing produce the strongest response near 816 Hz. The intermediate shaft left and right bearings dominate near 2640 Hz. The response near 3200 Hz is mainly attributed to the input shaft left bearing and the intermediate shaft left bearing. These results indicate that the measured housing vibration is not uniformly excited by all bearings, and that the housing mode shapes together with bearing positions determine the final vibration distribution.
6.4 Modal contribution analysis
Modal contribution analysis helps to identify which system modes are primarily responsible for the response at a given frequency. For each vibration peak, the contribution of each mode can be either positive or negative, meaning that it either reinforces or suppresses the total response. I extracted the ten largest contributing modes and represented them as a percentage of the total response magnitude. Table 11 summarizes the dominant modes at the three main peak frequencies.
| Frequency (Hz) | Bearing location | Dominant modes | Approx. contribution |
|---|---|---|---|
| 816 | Input right | 7 | 93.7% |
| 816 | Output left | 7 | 91.4% |
| 2640 | Intermediate left | 23, 24, 32 | 45.4%, 47.6%, 17% |
| 2640 | Intermediate right | 23, 24, 31 | 80.1%, 13.2%, 11.5% |
| 3200 | Input left | 32 | 95.1% |
| 3200 | Intermediate left | 32, 35 | 80.2%, 20% |
The 7th mode is a strongly radiating mode of the electric vehicle reducer system, with large motion of the side walls and the end cover. The 23rd and 24th modes are also associated with the deformation of the intermediate-shaft bearing supports and the surrounding housing panels. The 32nd mode, near 3237 Hz, is particularly important because it is excited by the first harmonic of the first-stage gear mesh at about 3190 Hz when the motor speed is 5800 rpm. Such a near-resonance condition amplifies both vibration and noise.
6.5 Dynamic contact load and modal flexibility
To further verify the modal contribution results, I compute the dynamic gear contact load and the modal flexibility of the system. The dynamic contact load is the amplitude of the fluctuating gear mesh force caused by the transmission error and the time-varying mesh stiffness. Modal flexibility, on the other hand, is a measure of how easily a given mode is excited by a unit dynamic load applied at the gear mesh. The dynamic gear mesh force shows peaks at the same frequencies as the high modal flexibility. This consistency verifies that the selected modes are indeed the main contributors to the vibration response of the electric vehicle reducer.
6.6 Housing vibration acceleration distribution
The spatial distribution of the vibration acceleration on the housing is obtained from the forced response. At 816 Hz, the vibration acceleration magnitude is roughly \(0.98\,\mathrm{m/s^2}\). At 2640 Hz and 3200 Hz, the largest acceleration reaches about \(9.8\,\mathrm{m/s^2}\). Although such values are below the common engineering acceptance limit, the dynamic amplification at 3200 Hz suggests that further improvement is necessary to prevent unpleasant whine inside the electric vehicle cabin. This forms the motivation for the optimization step described later.
7. Radiated Noise Simulation and Test
7.1 Acoustic radiation principle
The vibration of the electric vehicle reducer housing excites the surrounding air particles and creates acoustic waves. For each point on the housing surface, the normal velocity is related to the acoustic pressure through the Helmholtz integral equation in the acoustic boundary element method. The sound pressure at a field point is computed from the surface velocity distribution. The overall noise is evaluated using A-weighted sound pressure levels, which approximately reflect the frequency sensitivity of the human ear.
The main acoustic quantities used in this work are the effective sound pressure \(p_e\), the sound pressure level \(L_p\), the sound intensity level \(L_I\), and the sound power level \(L_W\):
$$
p_e = \sqrt{\frac{1}{T}\int_0^T p^2(t)\,dt}
$$
$$
L_p = 20 \lg \frac{p_e}{p_r}
$$
$$
L_I = 10 \lg \frac{I}{I_0}
$$
$$
L_W = 10 \lg \frac{W}{W_0}
$$
where \(p_r = 20\,\mu\)Pa, \(I_0 = 10^{-12}\,\mathrm{W/m^2}\), \(W_0 = 10^{-12}\,\mathrm{W}\).
7.2 Acoustic boundary element model
In order to perform the acoustic simulation, I first extract the outer surface of the reducer housing from the structural finite element model. All open holes, including half-shaft holes, bolt holes, breather and oil-filler holes, are sealed in the geometric model to ensure a fully closed acoustic boundary. The closed surface is then meshed with linear boundary elements. According to the rule that at least six elements are required per wavelength, the maximum element size for a calculation frequency of 4000 Hz is
$$
L_{max} = \frac{c}{6 f_{max}} = \frac{3.4\times 10^5}{6 \times 4000} \approx 14.17\,\mathrm{mm}
$$
A mesh size of about 8 mm is chosen to resolve all important small radii of the housing.
Field points are established according to the Chinese standard QC/T 1022-2015. Four measurement points are located one meter from the geometric center of the electric vehicle reducer in the upward, rearward, left, and right directions. These are the standard microphone positions for gearbox noise measurements. The same coordinates are used in the test to ensure direct comparability.
7.3 Noise simulation results
Using the same operating condition as the vibration analysis, I compute the radiated noise at the four field points over the frequency range from 0 to 4000 Hz. The predicted A-weighted sound pressure level spectra show three noticeable amplitude peaks near 810 Hz, 2600 Hz, and 3230 Hz, which correspond well to the structural vibration peaks. Among the four directions, the upward and rearward field points exhibit higher sound pressure levels than the left and right field points. This directional behavior is caused by the particularly flexible large unsupported panels in the upper and rear parts of the housing, as well as by the axial thrust excitation transmitted from the helical gears.
7.4 Noise test in a semi-anechoic chamber
For experimental validation, the electric vehicle reducer is installed on a T-shaped NVH test bench inside a semi-anechoic chamber. The drive motor is placed outside the chamber and is connected to one side of the driveline, while two load motors impose the desired wheel-end torque and speed. In this particular coaxial reducer, the drive motor side cannot be used directly because the drive system is not yet available; therefore the external load motors are controlled to simulate the required input speed and torque by applying equivalent output speeds and torques at the two half-shafts.
The output speed is set to 544 rpm and the output torque to 746 Nm on each load motor, which corresponds to an input speed of about 5800 rpm and an input torque of 70 Nm at the reducer input. Four PCB microphones are placed at the same positions as in the simulation. A photo-type tachometer is mounted near the shaft with a reflective strip to measure the actual rotational speed during the test. The measured A-weighted sound pressure levels are processed using LMS Test.lab software.
7.5 Comparison between simulation and test
Owing to the unavoidable torque fluctuation of the electric motors and the small differences between the numerical model and the real assembly, the simulation and test spectra are not perfectly coincident. Nevertheless, their overall shapes are consistent. Table 12 compares the simulated and measured peak frequencies and approximate A-weighted sound pressure levels at the upward and rearward field points.
| Field point | Excitation region | Simulated peak (Hz) | Measured peak (Hz) | Difference (Hz) |
|---|---|---|---|---|
| Upward | Low frequency | 810 | 816 | 6 |
| Middle frequency | 2600 | 2640 | 40 | |
| High frequency | 3230 | 3200 | 30 | |
| Rearward | Low frequency | 810 | 816 | 6 |
| Middle frequency | 2600 | 2640 | 40 |
The measured sound pressure level values also agree reasonably well with the simulation within approximately 3 dB, which is acceptable in engineering practice. The consistency between the vibration and noise responses demonstrates that the vibration excitation mechanism identified in the earlier sections is correct, and that the acoustic boundary element model can reliably predict the NVH performance of the electric vehicle reducer.
8. Optimization of the NVH Performance
8.1 Source-path-receiver concept
The noise of the electric vehicle reducer follows the well-known chain: excitation source (gear meshing forces and moments) — transfer path (shafts, bearings, housing mounting points, and housing panels) — acoustic receiver (people in the passenger compartment). Among the possible countermeasures, modification of the gear micro-geometry is one of the most effective means to reduce the excitation source. Instead of redesigning the whole transmission or adding heavy damping materials, I choose to optimize the gear flank modifications of the first-stage gear pair because they directly control the transmission error amplitude and the dynamic meshing force.
8.2 Gear micro-modification parameters
For helical gears, four key micro-geometry parameters are commonly used: helix crowning, helix slope, involute crowning, and involute slope. These parameters directly affect the load distribution along the face width and the profile direction. Improper modifications cause edge loading, high localized stress, and large transmission error. Conversely, suitable modifications can compensate tooth deflections under load and equalize the contact pattern.
Micro-modification is performed on both the driving gear and the driven gear of the first stage. The original micro-geometry parameters are listed in Table 13.
| Parameter | Driving gear (left/right flank) (µm) | Driven gear (left/right flank) (µm) | ||
|---|---|---|---|---|
| Left | Right | Left | Right | |
| Helix crowning | -2.5 | -1.5 | 0 | 0 |
| Helix slope | 0.5 | 0.5 | 0.5 | 0.5 |
| Involute crowning | 0 | 0 | 0 | 0 |
| Involute slope | 2.5 | 3 | 2.5 | 3 |
8.3 Formulation of the multi-objective optimization problem
The transmission error peak-to-peak value \(\mathrm{TE}_{p-p}\) and the maximum normal load per unit length on the tooth surface \(Q_{max}\) are chosen as the two optimization objectives. Reducing both simultaneously improves both the NVH and the tooth durability of the electric vehicle reducer. The design variables are the helix crowning \(C_\beta\), helix slope \(f_{H\beta}\), involute crowning \(C_\alpha\), and involute slope \(f_{H\alpha}\) for both the driving gear and the driven gear. All these variables are bounded by the practical design window based on industrial experience:
$$
-20 \le f_{H\beta} \le 20, \quad 0 \le C_\beta \le 20
$$
$$
-20 \le f_{H\alpha} \le 20, \quad 0 \le C_\alpha \le 20
$$
Safety constraints are imposed to ensure that the flank and root bending capacities are not sacrificed:
$$
S_H \ge 1, \quad S_F \ge 1.2
$$
where \(S_H\) is the contact safety factor and \(S_F\) is the bending safety factor. The two objective functions can thus be written as
$$
F_1(C_\beta, f_{H\beta}, C_\alpha, f_{H\alpha}) = \min \mathrm{TE}_{p-p}
$$
$$
F_2(C_\beta, f_{H\beta}, C_\alpha, f_{H\alpha}) = \min Q_{max}
$$
8.4 NSGA-II optimization settings
I use the Non-dominated Sorting Genetic Algorithm II (NSGA-II) to solve this multi-objective problem. NSGA-II is suitable because it preserves the diversity of the Pareto front and has a fast non-dominated sorting procedure. The optimization parameters are summarized in Table 14.
| Parameter | Value |
|---|---|
| Population size | 50 |
| Number of generations | 20 |
| Total evaluated designs | 1000 |
| Crossover probability | 0.2 |
| Mutation probability | 0.3 |
| Bound for helix slope and involute slope | [-20, 20] µm |
| Bound for helix crowning and involute crowning | [0, 20] µm |
The optimization is performed under the same operating condition as the NVH analysis: 70 Nm input torque and 5800 rpm input speed. In each objective evaluation, the Romax gear contact analysis solver computes the contact pattern, the loaded transmission error, and the tooth root and flank safety factors. The integration of Romax with the external genetic algorithm enables a fully automated design loop.
8.5 Optimization results and Pareto front
After 20 generations, the optimizer produces 1000 feasible designs. The obtained Pareto front in the objective space clearly indicates a trade-off between the transmission error peak-to-peak value and the maximum unit load. Because these two objectives are positively correlated in most cases, the lower left corner of the front contains the preferred designs that simultaneously reduce both objectives. From those preferred designs, considering manufacturing tolerances and stability of the flank contact, I choose the final optimized micro-geometry parameters given in Table 15.
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Helix crowning (µm) | 0.06 | 0.13 |
| Helix slope (µm) | -13.78 | 16.45 |
| Involute crowning (µm) | 1.24 | -2.74 |
| Involute slope (µm) | 2.37 | 4.53 |
8.6 Evaluation of the optimized design
8.6.1 Contact pattern analysis
I first compare the loaded tooth contact patterns before and after optimization. The original design shows an uneven distribution of the normal load with a concentrated contact on one side of the flank. This is typical when the helix slope is mismatched under the given torque. After optimization, the contact pattern is centered in the middle of the flank, and the maximum unit load is reduced. Such an elliptic, centrally located contact pattern is desirable because it reduces the risk of edge contact and also decreases the sensitivity of transmission error to torque fluctuations.
8.6.2 Housing surface velocity RMS improvement
The optimized micro-geometry model is simulated again under the same loading condition. The RMS velocity spectrum of the housing surface still shows peaks at similar frequency positions around 816 Hz, 2640 Hz, and 3200 Hz, but their amplitudes are significantly reduced. Table 16 summarizes the reduction at the three main peak frequencies.
| Frequency (Hz) | Before (×10⁻⁹ m/s) | After (×10⁻⁹ m/s) | Reduction (%) |
|---|---|---|---|
| 816 | 4.81 | 4.09 | 15.0 |
| 2640 | 9.57 | 7.35 | 23.2 |
| 3200 | 5.05 | 3.40 | 32.7 |
The higher-frequency peaks show larger percentage reductions. This is because the first-order transmission error harmonics are more effectively reduced by the optimized profile modifications, and the higher-frequency structural modes are less influenced by other residual excitations.
8.6.3 Radiated noise reduction
The radiated noise of the optimized electric vehicle reducer is also recomputed at the four microphone field points. The upward microphone spectrum is representative and shows the same three peaks at 810 Hz, 2600 Hz, and 3230 Hz, but with lower A-weighted sound pressure levels. Table 17 lists the peak values before and after optimization at the upward field point.
| Frequency (Hz) | Before (dB(A)) | After (dB(A)) | Reduction (dB(A)) |
|---|---|---|---|
| 810 | 54.7 | 51.0 | 3.7 |
| 2600 | 57.8 | 53.7 | 4.1 |
| 3230 | 58.5 | 54.0 | 4.5 |
The maximum overall A-weighted level is reduced from 58.5 dB(A) to 54.0 dB(A), and the total noise level is lowered by about 3.9 dB(A). Since a change of 3 dB is clearly perceptible to the human ear, this improvement is meaningful for the perceived quality of the electric vehicle.
8.6.4 Discussion on the optimization mechanisms
The main reason for the noise reduction is the reduced amplitude of the gear mesh excitation. The optimized helix and involute modifications compensate for the tooth bending and hertzian deflections under load, which reduces the loaded transmission error at the first harmonic. Moreover, a more uniform load distribution decreases the dynamic contact force fluctuation. As a result, less vibratory energy is transmitted through the bearings to the housing. Since the housing modes remain unchanged, the lower excitation directly translates into lower vibration velocities and lower radiated noise. It is also worth noting that the reduction at 3200 Hz is greater than at 816 Hz because the 32nd mode strongly couples to the first-stage gear mesh excitation around 3190 Hz, and the residual transmission error at that harmonic becomes much smaller after optimization.
9. Conclusion
In this research, I have systematically analyzed and optimized the NVH performance of an electric vehicle reducer using a combination of theoretical modeling, numerical simulation, and experimental verification. The main conclusions can be summarized as follows.
(1) A high-fidelity multi-degree-of-freedom dynamic model of the electric vehicle reducer system was established. The finite element model of the housing was validated by comparing the first six natural frequencies obtained from free modal simulation and modal testing. The maximum error between simulation and experiment was only 2.67%, which is well below the typical 5% criterion. The static strength simulation also demonstrated that the housing can withstand the maximum positive and negative torque conditions without exceeding the material yield stress or the permitted deformation.
(2) The transmission error of the first-stage gear pair was investigated as the main internal excitation of the electric vehicle reducer. The largest peak-to-peak transmission error was observed at 70 Nm input torque and 5800 rpm. Under this condition, the housing vibration response exhibited three distinct peaks near 816 Hz, 2640 Hz, and 3200 Hz. Through modal contribution analysis and dynamic contact force analysis, the seventh, twenty-third, twenty-fourth, and thirty-second system modes were identified as the dominant contributors. The vibration acceleration magnitudes were lower than the common acceptance limit but still indicated a potential whine risk near 3200 Hz.
(3) An acoustic boundary element model of the sealed housing was built to predict the radiated noise. The simulated A-weighted noise spectra agreed well with the semi-anechoic chamber test results, with peaks at approximately 810 Hz, 2600 Hz, and 3230 Hz. The upward and rearward directions radiated more noise than the lateral directions, reflecting the structural stiffness distribution and the axial excitation characteristics of the electric vehicle reducer.
(4) An NSGA-II algorithm was used to optimize the helix crowning, helix slope, involute crowning, and involute slope on both gears of the first-stage pair. The optimized micro-geometry improved the loaded contact pattern and reduced the maximum contact load. Subsequent vibration simulations showed that the housing surface velocity RMS was reduced by 15.0%, 23.2%, and 32.7% at 816 Hz, 2640 Hz, and 3200 Hz, respectively. The corresponding A-weighted sound pressure level peaks were reduced by 3.7 to 4.5 dB, with an overall level reduction of about 3.9 dB. These results demonstrate that the proposed gear micro-modification optimization is an effective strategy for improving the NVH performance of the electric vehicle reducer.
(5) The research provides a complete workflow from finite element modeling, modal validation, dynamic excitation analysis, acoustic simulation, experimental verification, and finally multi-objective optimization. This workflow can be readily extended to other electric vehicle powertrain components. In the future, additional operating conditions, temperature effects, and manufacturing tolerances could be included as uncertainty factors to make the optimization more robust. Furthermore, the analysis of gear whine orders and the application of structure topology optimization could further improve the acoustic behavior of electric vehicle reducers.