Analysis of Spur Gears with Error Tooth Surfaces: Internal Excitation and Dynamic Characteristics

In modern power transmission systems, particularly with the rise of electric vehicles, the demand for low vibration and noise performance in gear drives has become increasingly stringent. Gear transmission systems are inherently parametric self-excited systems. Even in the absence of external excitations, the internal excitations generated within the system can induce self-excited vibrations. Among these, the time-varying mesh stiffness (TVMS) and the composite meshing error (CME) are the two most significant internal excitation sources. Therefore, accurate calculation of these parameters for spur gears and understanding the influence of various manufacturing and assembly errors on the dynamic characteristics are crucial for predicting and controlling vibration and noise levels. Existing research often simplifies these internal excitations, using harmonic functions for CME and ideal tooth profiles for TVMS calculation, neglecting the complex interactions and non-ideal contact conditions induced by real-world errors. This study aims to bridge this gap by developing a comprehensive model that directly incorporates error distributions into the tooth surface geometry.

The primary objective of this work is to establish a refined model for spur gears that explicitly accounts for the statistical distribution of manufacturing and assembly errors. Based on this “error tooth surface” model, a novel Loaded Tooth Contact Analysis (LTCA) algorithm is proposed to accurately compute the TVMS and CME. Subsequently, a bending-torsion coupled dynamic model of a spur gear pair is established to analyze the impact of different error types on the system’s dynamic response. This integrated approach allows for a more realistic simulation of gear pair operation under non-ideal conditions.

Error Tooth Surface Model for Spur Gears

To construct a realistic model, the tooth flanks of a standard involute spur gear are first discretized into a grid of $n_1 \times n_2$ small elements. Each element is approximated as a cylindrical surface with a specific curvature radius and is represented by a control point at its center, indexed by $i$ and $j$. The coordinates $(X^{1ij}_{L/R}, Y^{1ij}_{L/R}, Z^{1ij}_{L/R})$, the normal vectors $\mathbf{n}^{1ij}_{L/R}$, and the curvature radii $\rho^{1ij}_{L/R}$ for the left (L) and right (R) flanks of a reference tooth are calculated from the involute equations.

Manufacturing errors, including single pitch deviation $f_{pt}$, cumulative pitch error $F_p$, and profile form error $F_{\alpha}$, are then introduced according to the ISO 1328-1 standard. The generation of an error tooth surface is treated as an equidistant offset from the theoretical profile. Therefore, the coordinates of a control point on the $n$-th tooth, considering manufacturing errors, are obtained by rotating the coordinates of the corresponding point on the reference tooth:

$$
\begin{bmatrix}
X_{MEL/R}^{nij} \\
Y_{MEL/R}^{nij} \\
Z_{MEL/R}^{nij}
\end{bmatrix}
=
\begin{bmatrix}
\cos(\Delta\theta_{MEL/R}^{nij}) & \sin(\Delta\theta_{MEL/R}^{nij}) & 0 \\
-\sin(\Delta\theta_{MEL/R}^{nij}) & \cos(\Delta\theta_{MEL/R}^{nij}) & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
X_{L/R}^{1ij} \\
Y_{L/R}^{1ij} \\
Z_{L/R}^{1ij}
\end{bmatrix}
$$

Where the rotation angles $\Delta\theta_{MEL}^{nij}$ and $\Delta\theta_{ME\R}^{nij}$ are defined as:

$$
\Delta\theta_{MEL}^{nij} = \left[ \frac{2\pi(n-1)}{z} + \frac{\sum_{z=1}^{n} f_{pt}^z + (F_{\alpha}^{nij})_L}{r} \right]
$$
$$
\Delta\theta_{ME\R}^{nij} = \left[ \frac{2\pi(n-1)}{z} + \frac{\sum_{z=1}^{n} f_{pt}^z + E_s^n + (F_{\alpha}^{nij})_\R}{r} \right]
$$

Here, $z$ is the number of teeth, $r$ is the pitch radius, $E_s^n$ is the tooth thickness error for the $n$-th tooth, and $(F_{\alpha}^{nij})_{L/R}$ is the profile error for the specific discrete element. These error values are generated as random variables within their respective tolerance bands, following prescribed distributions (e.g., uniform for $f_{pt}$ and $E_s$, and a modified random for $F_{\alpha}$). The generated gear is validated by checking if its cumulative pitch error $F_p$ and span measurement $W_k$ fall within the specified tolerance limits.

Assembly errors, including center distance errors and shaft misalignments, are modeled by transforming the pinion’s errors onto the gear’s axis. Four parameters $(\Delta x, \Delta y, \phi, \gamma)$ define the assembly errors: $\Delta x$ and $\Delta y$ represent center distance offsets, while $\phi$ and $\gamma$ represent angular misalignments in two orthogonal planes. The final coordinates of a control point on the assembled error tooth surface are given by:

$$
\begin{bmatrix}
X_{AEL/R}^{nij} \\
Y_{AEL/R}^{nij} \\
Z_{AEL/R}^{nij} \\
1
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & M_{13} & M_{14} \\
0 & 1 & M_{23} & M_{24} \\
0 & 0 & M_{33} & M_{34} \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
X_{MEL/R}^{nij} \\
Y_{MEL/R}^{nij} \\
Z_{MEL/R}^{nij} \\
1
\end{bmatrix}
$$

The coefficients $M_{13}$ to $M_{34}$ are functions of the assembly error parameters $(\Delta x, \Delta y, \phi, \gamma)$ and the nominal center distance $a$.

Loaded Tooth Contact Analysis for Error Tooth Surfaces

Traditional LTCA methods can be complex and numerically unstable for highly discontinuous contact patterns arising from errors. This study proposes a novel algorithm tailored for the discrete error tooth surface model. The core idea is to determine tooth pair contact by detecting interference states between the discrete surface elements of the pinion and gear.

The analysis is an iterative process. For each angular position of the gear, the pinion is rotated in steps. In each iteration, potential contact element pairs are identified, and the total resisting torque is calculated. Using a bisection method, the pinion rotation step is adjusted until the calculated torque balances the applied load. The outputs for each gear position are the TVMS and the CME.

To calculate mesh stiffness, a modified “nominal slice” method is introduced. The tooth width is divided into $n_1$ slices, each comprising $n_2$ discrete elements. Under load, some slices may lose contact due to errors. Adjacent contacting and non-contacting slices with similar global deformation are grouped into a “nominal slice” $S_N^t$. The slice with the maximum deformation within this group is considered the contact center. The mesh stiffness for a contacting tooth pair $ic$ and its nominal slice $S_N^t$ is:

$$
K_{ic}^t = \frac{1}{\frac{1}{\sum_{i=i_g-n_{gl}}^{i_g+n_{gn}} K_{G_g}^i} + \frac{1}{\sum_{i=i_p-n_{pl}}^{i_p+n_{pn}} K_{G_p}^i} + \frac{1}{K_{h}^{i_g i_p}}}
$$

Where $i_g, i_p$ are the indices of the contact center slices; $n_{gl}, n_{pl}, n_{gn}, n_{pn}$ account for adjacent non-contacting slices; $K_{h}^{i_g i_p}$ is the Hertzian contact stiffness between the central elements; and $K_{G_g}^i, K_{G_p}^i$ are the global deformation stiffnesses (combining bending, shear, axial compression, and foundation deflection) for the slices, calculated using formulas from established literature. The total TVMS is the sum over all contacting tooth pairs and their nominal slices:

$$
K_{mesh} = \sum_{ic=1}^{n_3} \sum_{t=1}^{n_4} K_{ic}^t
$$

Contact between two discrete cylindrical elements is determined by calculating the distance $d_c$ between their axes. If the axes are parallel (no misalignment), the total deformation is $\delta_{ic}^t = \max(\rho_g + \rho_p – d_c, 0)$. If misalignment exists, the average of the maximum and minimum distances is used: $\delta_{ic}^t = \max(\rho_g + \rho_p – (d_{c}^{max}+d_{c}^{min})/2, 0)$. A positive value indicates contact.

The CME $e$ is calculated based on the kinematic relationship between the pinion and gear rotations under no-load and loaded conditions. If $\theta_g$ is the gear angle at initial no-load contact and $\theta_p$ is the corresponding pinion rotation angle under load, then:

$$
e = (\theta_g – \theta_p \cdot z_p / z_g) \cdot r_{bg}
$$

Where $z_p, z_g$ are the numbers of teeth and $r_{bg}$ is the gear base radius.

Dynamic Modeling of the Spur Gear Pair

A bending-torsion coupled dynamic model of the spur gear pair is established, neglecting friction. The equations of motion are derived using Newton’s second law:

$$
\begin{aligned}
m_p \ddot{y}_p + c_{py} \dot{y}_p + k_{py} y_p – [c_m \dot{\delta} + k_m(t) \delta] &= 0 \\
I_p \ddot{\theta}_p – [c_m \dot{\delta} + k_m(t) \delta] r_{bp} + T_p &= 0 \\
m_g \ddot{y}_g + c_{gy} \dot{y}_g + k_{gy} y_g + [c_m \dot{\delta} + k_m(t) \delta] &= 0 \\
I_g \ddot{\theta}_g + [c_m \dot{\delta} + k_m(t) \delta] r_{bg} – T_g &= 0
\end{aligned}
$$

Here, subscripts $p$ and $g$ denote pinion and gear; $m$ and $I$ are mass and moment of inertia; $k_y$ and $c_y$ are bearing support stiffness and damping in the lateral direction $y$; $T$ is the applied torque; $r_b$ is the base circle radius. The mesh damping $c_m$ is estimated using an empirical formula: $c_m = 2 \xi \sqrt{\bar{k}_m m_e}$, where $\xi$ is the damping ratio (taken as 0.1), $\bar{k}_m$ is the average TVMS, and $m_e$ is the equivalent mass. The dynamic transmission error (DTE) along the line of action $\delta$ is defined as:

$$
\delta = -y_p – r_{bp}\theta_p + y_g – r_{bg}\theta_g – e(t)
$$

The time-varying functions $k_m(t)$ and $e(t)$ are the TVMS and CME, respectively, obtained from the LTCA for the error tooth surfaces.

Results and Discussion

Model Validation

The proposed LTCA algorithm was first validated against published results for a perfect spur gear pair. The calculated TVMS showed excellent agreement, with differences in mean and peak-to-peak values of only 2.05% and 0.90%, respectively, confirming the model’s accuracy.

Influence of Manufacturing Errors on Internal Excitation

A spur gear pair with the parameters listed in Table 1 was analyzed. Both gears were assigned ISO Grade 5 precision, with error parameters as per Table 2.

Parameter	Pinion/Gear
Number of Teeth, $z$	25
Module, $m_n$ (mm)	2
Pressure Angle, $\alpha$ (°)	20
Face Width, $B$ (mm)	20
Young’s Modulus, $E$ (N/mm²)	2.0 × 10⁵
Poisson’s Ratio, $\nu$	0.3
Load Torque, $T_g$ (Nm)	28

Precision Parameter	Value (μm)
Single Pitch Deviation, $\pm f_{pt}$	±5.0
Cumulative Pitch Deviation, $F_p$	14.0
Total Profile Deviation, $F_{\alpha}$	5.0
Tooth Thickness Tolerance, $E_s$	-52 / -104

The effects of individual error types were isolated. The results, as shown in comparative plots of TVMS and CME, reveal distinct mechanisms:

Pitch Error ($f_{pt}$, $F_p$): This is the most influential manufacturing error. It causes the CME to exhibit step-like changes during the mesh cycle. When the step height (initial gap between tooth pairs) exceeds the compensation capacity of tooth deflection, tooth pair disengagement occurs. This leads to a sudden drop in TVMS, reducing the contact ratio from two pairs to one pair prematurely. The TVMS突变 (mutation) is directly linked to these CME steps.
Tooth Thickness Error ($E_s$): This error primarily thins the teeth, increasing tooth compliance and slightly reducing the mean TVMS. However, it does not affect the CME for the working flank in this model and thus induces minimal variation in the TVMS waveform.
Profile Error ($F_{\alpha}$): This error causes “off-line-of-action” contact, where the instantaneous contact line shifts from its theoretical position. This results in small, random fluctuations in both TVMS and CME around their ideal values but does not alter the overall contact pattern significantly.

These findings demonstrate that manufacturing errors, especially pitch errors, deeply couple the TVMS and CME. Modeling CME as a simple harmonic function while ignoring its impact on TVMS is insufficient for accurate dynamic analysis of spur gears.

Influence of Assembly Errors on Internal Excitation

Assembly errors were introduced via parameters $\Delta y$ (center distance), $\phi$, and $\gamma$ (angular misalignments). Their effects are summarized below:

Center Distance Error ($\Delta y$): A positive $\Delta y$ (increased center distance) decreases TVMS and shifts the contact path toward the tooth tip, shortening the double-tooth contact region. A negative $\Delta y$ increases TVMS and shifts contact toward the root. The CME becomes a constant, non-zero value (positive for decreased center distance, negative for increased).
Angular Misalignment ($\phi$, $\gamma$): These errors cause load distribution across the face width, effectively shortening the contact line. This significantly reduces the TVMS. Misalignment $\phi$ has a more pronounced effect than $\gamma$ for the same angular magnitude. The CME also becomes a constant value, which can be zero or non-zero depending on the error direction.

A key observation is that under assembly errors, the CME is essentially constant over a mesh cycle, unlike the step-like variation from pitch errors. The reduction in TVMS is primarily due to the loss of effective contact length.

Dynamic Characteristics Under Different Errors

The computed $k_m(t)$ and $e(t)$ were fed into the dynamic model. The Dynamic Transmission Error (DTE) was analyzed in time and frequency domains for various error conditions. The pinion speed was 2400 rpm, corresponding to a mesh frequency $f_m = 1$ kHz.

The analysis of spur gears dynamics yielded the following insights:

Theoretical Surfaces: DTE peak-to-peak ($DTE_{pp}$) was 4.16 μm. The spectrum was dominated by the mesh frequency $f_m$ and its harmonics.
Combined Manufacturing Errors: $DTE_{pp}$ increased to 7.44 μm. The spectrum showed significant high-frequency content between 1-2 kHz, alongside reduced harmonic amplitudes.
Individual Error Contributions:
- Pitch Error Alone: Caused the largest $DTE_{pp}$ increase (8.39 μm) and was the primary source of high-frequency spectral content.
- Profile Error Alone: Led to a moderate $DTE_{pp}$ increase (5.40 μm) and some high-frequency components.
- Tooth Thickness Error Alone: Resulted in a slight $DTE_{pp}$ decrease (4.04 μm) and a spectrum similar to the theoretical case, with only mesh harmonics.
Assembly Errors:
- Center Distance: Negative $\Delta y$ had a smaller impact on $DTE_{pp}$ than positive $\Delta y$.
- Angular Misalignment: Small misalignments could reduce $DTE_{pp}$ by lowering TVMS. However, excessive misalignment (e.g., $\phi = -0.1^\circ$ causing ~50% contact loss) increased $DTE_{pp}$ despite lower TVMS, likely due to highly nonlinear contact conditions. The spectrum for assembly errors remained dominated by mesh harmonics.

Conclusions

This study presented a comprehensive framework for analyzing the internal excitation and dynamic characteristics of spur gears considering realistic manufacturing and assembly errors. The key conclusions are:

Error Coupling: Manufacturing and assembly errors simultaneously and deeply couple the Time-Varying Mesh Stiffness (TVMS) and Composite Meshing Error (CME). Accurate dynamic analysis must account for this coupling rather than treating CME as an independent harmonic input.
Dominant Manufacturing Error: Among manufacturing errors, pitch deviation is the most critical factor affecting the internal excitation of spur gears. It causes step-like changes in CME, and when the step exceeds the load compensation capacity, it triggers sudden tooth pair disengagement and a突变 (abrupt change) in TVMS.
Assembly Error Effects: Assembly errors generally render the CME constant over a mesh cycle. Center distance errors alter TVMS magnitude and contact ratio by shifting the contact path. Angular misalignments reduce TVMS significantly by causing偏载 (biased load distribution) and shortening the effective contact line.
Dynamic Response Implications:
- To minimize Dynamic Transmission Error (DTE) peak-to-peak, pitch deviations should be allocated considering the operating load to avoid disengagement-induced TVMS突变.
- Excessive angular misalignment should be avoided. While minor misalignment might reduce DTE slightly by lowering stiffness, severe misalignment increases DTE and leads to high contact stress, jeopardizing fatigue life.
- Although a negative center distance error may have a smaller impact on DTE_pp than a positive error, it can increase meshing impact incentives at high speeds. Its application should be combined with profile modification.

The proposed error tooth surface model and LTCA algorithm provide a powerful tool for high-fidelity simulation of spur gears under non-ideal conditions, offering valuable insights for gear design, tolerance allocation, and vibration/noise prediction in advanced transmission systems.