1. Introduction and Research Background
Spur gear transmission is one of the most widely employed mechanical power transmission mechanisms in modern industry. From precision machine tools and aerospace equipment to automotive drivetrains and micro-mechanical devices, the demand for higher transmission accuracy has grown consistently. As mechanical transmission equipment moves toward higher precision and greater load capacity, the transmission error and its associated characteristics have become critical performance indicators. The transmission error of a spur gear system directly influences the dynamic behavior, noise level, vibration characteristics, and service life of the entire machinery.
A spur gear transmission system typically comprises gears, shafts, bearings, and the housing or gearbox structure. Under real operating conditions, every component undergoes elastic deformation due to external loads, and each manufactured component inevitably contains geometrical errors. These errors and deformations are coupled through the meshing interface and collectively contribute to the overall transmission error. It is therefore essential to establish an equivalent precision characteristic model that accounts for the form errors, elastic deformations, and applied loads of all constituent parts.
The research presented in this thesis addresses this challenge through the lens of kinematic geometry. The fundamental concept is to treat the gear transmission system as a multi-body system where each component’s error and elasticity are represented through equivalent kinematic models. The bearing assemblies are modeled using equivalent cam mechanisms with spring-constrained followers; the gear meshing is described through conjugate contact equations that incorporate both geometric errors and elastic deformations. This approach enables a systematic formulation of the transmission error problem, transforming it into a solvable system of equations that links component-level imperfections to system-level performance.
The specific objectives of this research are as follows:
- To establish a kinematic geometry model for a spur gear transmission system supported by floating shafts, accounting for the errors and elasticity of bearing components.
- To develop a transmission error model that incorporates gear tooth profile errors, center distance variations, and shaft axis misalignments.
- To formulate an elastic conjugate meshing model that considers time-varying mesh stiffness and alternating single/double tooth engagement.
- To construct a comprehensive transmission precision model that integrates both errors and elasticity, and to analyze the influence of individual factors on transmission accuracy.
This research was supported by the National Natural Science Foundation of China under Grant No. 51375065, focusing on the multi-conjugate composite model for precision characteristics of gear transmission systems.
2. Kinematic Geometry Model of Floating Shaft Support
2.1 System Composition and Equivalent Model
A spur gear transmission system consists of two meshing gears mounted on shafts, which are in turn supported by bearings connected to the housing. In practice, the supporting shafts must be considered as floating supports, meaning they can undergo small translational and rotational displacements under load. The bearing components introduce both geometric errors and elastic compliance at the support points.
To model this system effectively, each bearing assembly is represented as a plate conjugate cam mechanism. The geometric errors of the bearing inner raceway are mapped onto the cam profile, while the elasticity of the bearing and housing is represented by spring-constrained cam followers. Each cam follower has a single translational degree of freedom. The springs can only sustain compressive forces, which accurately simulates the physical behavior of rolling element bearings. For each shaft, two cam mechanisms (corresponding to two bearings) provide support, and each cam mechanism consists of four cam followers arranged along mutually perpendicular directions.
2.2 Coordinate System Definition
To facilitate analysis, a fixed global coordinate system $\{O; X, Y, Z\}$ is established. Two moving coordinate systems $\{O_i; X_i, Y_i, Z_i\}$ ($i=1,2$) are attached to the centers of the driving gear ($i=1$) and driven gear ($i=2$). The distance between $O_1$ and $O_2$ represents the instantaneous center distance of the gear pair.
For the cam mechanisms, local coordinate systems are defined at each cam center. The cam profile is expressed in the form of a Fourier series:
$$
\begin{equation}
\boldsymbol{r}_{OC_{ij}} = \left[ r_{0ij} + \sum_{n=0}^{\infty} E_{nij} \sin(n\phi + \psi_{ij})\right] \boldsymbol{e}(\phi)
\end{equation}
$$
where $\boldsymbol{e}(\phi)$ represents the circular vector equation, $r_{0ij}$ is the base circle radius of the cam, $E_{nij}$ are the Fourier coefficients representing profile errors, and $\psi_{ij}$ is the initial phase angle. For practical purposes, the Fourier series is truncated to the first four terms, where $n=1$ corresponds to eccentric circle, $n=2$ to elliptical, $n=3$ to trilobal, and $n=4$ to quadrilobal profiles.
2.3 Displacement, Equilibrium, and Compatibility Equations
The displacement relationship for the bearing assembly, derived from the closed-loop vector diagram of the cam-follower system, is expressed as:
$$
\begin{equation}
\boldsymbol{r}_{OC_{ij}} = \boldsymbol{r}_{O_{ij},J}^V – \boldsymbol{L}_{ij,J}^V – \boldsymbol{r}_{ij,J}^V, \qquad V = X, Y
\end{equation}
$$
where $\boldsymbol{r}_{O_{ij},J}^V$ is the position vector from theoretical support point to each cam follower, $\boldsymbol{L}_{ij,J}^V$ represents the vector from cam-follower contact point to the fixed support, and $\boldsymbol{r}_{ij,J}^V$ is the vector from cam center to contact point.
The deformation relationship of the cam mechanism under load can be written as:
$$
\begin{equation}
\boldsymbol{r}_{O_{ij},J}^V – \boldsymbol{\delta}_{ij,J}^V – \boldsymbol{r}_D^V = 0
\end{equation}
$$
where $\boldsymbol{\delta}_{ij,J}^V$ represents the elastic deformation vector of each cam follower and $\boldsymbol{r}_D^V$ is the vector without follower deformation.
The force equilibrium of shaft $i$ is governed by:
$$
\begin{equation}
\sum_{j=1}^{2} \sum_{J=1}^{2} (\boldsymbol{F}_{ij,J}^X + \boldsymbol{F}_{ij,J}^Y) + \boldsymbol{F}_{gi} = 0, \qquad i=1,2
\end{equation}
$$
The moment equilibrium equation about the shaft axis is:
$$
\begin{equation}
\sum_{j=1}^{2} \sum_{J=1}^{2} (\boldsymbol{r}_{ij,J}^{fX} \times \boldsymbol{F}_{ij,J}^X + \boldsymbol{r}_{ij,J}^{fY} \times \boldsymbol{F}_{ij,J}^Y) + \boldsymbol{r}_{gi} \times \boldsymbol{F}_{gi} + \boldsymbol{T}_i = 0
\end{equation}
$$
The elastic recovery force and deformation relationship at each cam follower is:
$$
\begin{equation}
\boldsymbol{F}_{ij,J}^{VK} = \boldsymbol{K}_{ij,J}^V \boldsymbol{\delta}_{ij,J}^V
\end{equation}
$$
The shaft deformation under load is characterized by the stiffness matrix formulation:
$$
\begin{equation}
\boldsymbol{K}_S [\Delta_{ij,i}^X, \Delta_{ij,i}^Y, \Delta_{ij,i}^Z, \theta_{ij,i}^X, \theta_{ij,i}^Y, \theta_{ij,i}^Z]^T = \boldsymbol{P}
\end{equation}
$$
2.4 Time-Varying Center Distance and Axis Alignment
The center distance between the two gears is obtained through coordinate transformation:
$$
\begin{equation}
\boldsymbol{r}_{O_1O_2} = \boldsymbol{r}_{O_1} – \boldsymbol{r}_{O_2} = \boldsymbol{M}_{O_{11}c_{11}} \boldsymbol{M}_{c_{11},g_1} \boldsymbol{r}_{g_1,O_1} – \boldsymbol{M}_{O_{22}c_{22}} \boldsymbol{M}_{c_{22},g_2} \boldsymbol{r}_{g_2,O_2}
\end{equation}
$$
The rotation matrix from the moving coordinate system to the fixed system is:
$$
\begin{equation}
\boldsymbol{R}_{cij} =
\begin{bmatrix}
\cos\theta_i & -\sin\theta_i & 0 \\
\sin\theta_i & \cos\theta_i & 0 \\
0 & 0 & 1
\end{bmatrix}
\end{equation}
$$
The shaft axis misalignment of the driven gear relative to the driving gear can be determined from the cam center positions:
$$
\begin{equation}
\beta_j = \frac{(\boldsymbol{r}_{21,j} – \boldsymbol{r}_{11,j}) – (\boldsymbol{r}_{22,j} – \boldsymbol{r}_{12,j})}{L}, \qquad j = x, y
\end{equation}
$$
2.5 Numerical Example for Center Distance Variation
A pair of spur gears with module $m=4$ mm and tooth numbers $z_1 = z_2 = 25$ is used as an example. The input torque is $T_{in} = 4.7 \times 10^4$ N·mm. The deep groove ball bearing type 6312 is selected with the following structural parameters:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Bearing inner diameter d (mm) | 30 | Inner raceway curvature coefficient | 0.515 |
| Bearing outer diameter D (mm) | 55 | Outer raceway curvature coefficient | 0.525 |
| Number of rolling elements N | 11 | Initial radial clearance (mm) | 0.018 |
| Rolling element diameter (mm) | 7.144 | Bearing span L (mm) | 200 |
| Stiffness | K_sr1^X | K_sr2^X | K_sr1^Y | K_sr2^Y |
|---|---|---|---|---|
| Left bearing housing | 8.5 | 8.5 | 10.1 | 10.1 |
| Right bearing housing | 8.5 | 8.5 | 10.1 | 10.1 |
The bearing housing and shaft deformation are coupled through the bearing support stiffness, which is calculated as $K_{ij,J}^V = 871208 \delta^{1/2}$. Three types of cam profile errors are considered with amplitudes: $E_1 = 10\ \mu m$, $E_2 = 5\ \mu m$, and $E_3 = 2.5\ \mu m$.
The analysis reveals important insights. The center displacement of the cam exhibits periodic variation with the rotation angle. The nature of this variation depends significantly on both the cam profile shape and the initial phase angle. For the same profile geometry, different phase angles shift the displacement curve along the rotation axis. Similarly, for a fixed phase angle, different profile shapes yield different displacement amplitudes and patterns. The results confirm that the cam follower mechanism model successfully captures the floating support behavior of the bearing system.
When examining the gear center displacement, it becomes evident that the center distance error exhibits a periodic pattern with a fundamental frequency corresponding to one rotation of the gear. The amplitude of this center distance error is strongly correlated with both the profile error magnitude and the applied torque. Larger bearing raceway errors produce proportionally larger center distance variations. Increasing the input torque from $T_1 = 9.4$ N·m to $T_3 = 47$ N·m increases the center distance error amplitude accordingly, confirming the elastic nature of the bearing support.
The analysis also demonstrates an important practical finding: the center distance error can be minimized by appropriately selecting the initial phase angles of the bearing components. For the eccentric circle error case, the minimum center distance error of 18.58 μm was achieved with the specific phase arrangement denoted as parameter set 4.
3. Transmission Error Model for Gear Drives
3.1 Conjugate Meshing Equation with Errors
To establish the relationship between gear errors and transmission error, a three-dimensional gear transmission model is formulated. The model incorporates the following assumptions: (1) the gears are rigid with no elastic deformation; (2) no torsional or bending deformation of the shafts under load; (3) the shafts are supported by floating bearings, allowing spatial misalignment; (4) tooth profile errors and center distance errors are considered.
A Frenet frame $\{\boldsymbol{r}_i; \boldsymbol{e}_1^{(i)}, \boldsymbol{e}_2^{(i)}, \boldsymbol{e}_3^{(i)}\}$ is established at each point on the gear tooth flank, where $\boldsymbol{e}_1^{(i)}$ represents the tangent direction of the tooth profile, $\boldsymbol{e}_2^{(i)}$ the tooth width direction, and $\boldsymbol{e}_3^{(i)}$ the normal direction. For the ideal contacting case without errors, the conjugate condition requires:
$$
\begin{equation}
\boldsymbol{r}_1 – \boldsymbol{r}_2 = \boldsymbol{a}, \qquad \boldsymbol{e}_3^{(1)} = \boldsymbol{e}_3^{(2)} = \boldsymbol{e}_{03}
\end{equation}
$$
When errors are present, the actual meshing equation becomes:
$$
\begin{equation}
\boldsymbol{r}_1^* – \boldsymbol{r}_2^* = \boldsymbol{a} + \Delta\boldsymbol{a}, \qquad \boldsymbol{e}_3^{(1)*} = \boldsymbol{e}_3^{(2)*} = \boldsymbol{e}_3^*
\end{equation}
$$
where $\Delta\boldsymbol{a}$ represents the time-varying center distance error vector, and the asterisk indicates quantities evaluated at the actual (erroneous) contact point.
3.2 Gear Tooth Profile Error Representation
The tooth profile error is measured along the normal direction of the theoretical involute profile. For gear $i$, the actual tooth flank can be expressed as:
$$
\begin{equation}
\boldsymbol{R}_i'(u_i, v_i) = \boldsymbol{R}_i(u_i, v_i) + h_i(u_i, v_i)\boldsymbol{e}_{03}^{(i)}
\end{equation}
$$
where $h_i(u_i, v_i)$ represents the profile error magnitude at parameter coordinates $(u_i, v_i)$. Through differential geometry analysis, the infinitesimal variation of the actual flank is derived as:
$$
\begin{equation}
\delta\boldsymbol{R}_i’ = \boldsymbol{R}_{i,u}’ du_i + \boldsymbol{R}_{i,v}’ dv_i
\end{equation}
$$
with:
$$
\begin{equation}
\boldsymbol{R}_{i,u}’ = \boldsymbol{R}_{i,u} + (\sigma_1^{(i)} + h_{1,u}^{(i)} c_{11}^{(i)} + h_{2,u}^{(i)} c_{21}^{(i)})\boldsymbol{e}_1^{(i)} + h_{2,u}^{(i)}\boldsymbol{e}_3^{(i)}
\end{equation}
$$
$$
\begin{equation}
\boldsymbol{R}_{i,v}’ = \boldsymbol{R}_{i,v} + (\sigma_2^{(i)} + h_{1,v}^{(i)} c_{12}^{(i)} + h_{2,v}^{(i)} c_{22}^{(i)})\boldsymbol{e}_2^{(i)} + h_{1,v}^{(i)}\boldsymbol{e}_3^{(i)}
\end{equation}
$$
The unit normal vector to the erroneous tooth profile is:
$$
\begin{equation}
\boldsymbol{e}_3^{(i)’} = \boldsymbol{e}_3^{(i)} – h_{1,u}^{(i)} \boldsymbol{e}_1^{(i)} – h_{2,u}^{(i)} \boldsymbol{e}_2^{(i)}
\end{equation}
$$
3.3 Meshing Error Equations
By substituting the erroneous tooth profile equations into the conjugate meshing condition and applying the rotation transformation, the meshing error equation system is obtained:
$$
\begin{equation}
\begin{cases}
\sigma_2^{(1)} – \sigma_2^{(2)} – \delta\phi_2 (\boldsymbol{k} \times \boldsymbol{r}_2) \cdot \boldsymbol{e}_2 = (\Delta\boldsymbol{a} + \boldsymbol{\beta} \times \boldsymbol{r}_2) \cdot \boldsymbol{e}_2 \\
h_1^{(1)} – h_1^{(2)} + c_{11}^{(1)}\sigma_1^{(1)} + c_{12}^{(1)}\sigma_2^{(1)} – c_{11}^{(2)}\sigma_1^{(2)} – c_{12}^{(2)}\sigma_2^{(2)} + \delta\phi_2(\boldsymbol{k} \times \boldsymbol{r}_2)\cdot\boldsymbol{e}_1 = 0 \\
c_{21}^{(1)}\sigma_1^{(1)} + c_{22}^{(1)}\sigma_2^{(1)} – c_{21}^{(2)}\sigma_1^{(2)} – c_{22}^{(2)}\sigma_2^{(2)} – \delta\phi_2(\boldsymbol{k} \times \boldsymbol{e}_2)\cdot\boldsymbol{e}_1 + 0 = 0
\end{cases}
\end{equation}
$$
Considering the driving gear rotation angle as the reference ($\delta\phi_1 = 0$), the system has five unknown variables: $\sigma_1^{(1)}, \sigma_2^{(1)}, \sigma_1^{(2)}, \sigma_2^{(2)}, \delta\phi_2$. The equations are linear and can be solved directly. Here, $\delta\phi_2$ represents the transmission error of the driven gear.
3.4 Force Analysis of the Gear Transmission System
For the gear transmission system under load, the force equilibrium of shaft $i$ is:
$$
\begin{equation}
\sum_{j=1}^2 \sum_{J=1}^2 (\boldsymbol{F}_{ij,J}^X + \boldsymbol{F}_{ij,J}^Y) + \boldsymbol{F}_{gi} = 0, \qquad i=1,2
\end{equation}
$$
The moment equilibrium about the three orthogonal axes is expressed through the system of equations:
$$
\begin{equation}
\begin{cases}
\sum_{j,J} [F_{ij,J}^X \cdot r_{ij,J}^{Y} – F_{ij,J}^Y \cdot r_{ij,J}^{X}] + F_{gi} \cdot (r_{gi,z} – r_{O_i,z}) + T_{ix} = 0 \\
\sum_{j,J} [F_{ij,J}^X \cdot r_{ij,J}^{Z} – F_{ij,J}^Z \cdot r_{ij,J}^{X}] + F_{gi} \cdot (r_{gi,x} – r_{O_i,x}) + T_{iy} = 0 \\
\sum_{j,J} [F_{ij,J}^X \cdot r_{ij,J}^{fY} – F_{ij,J}^Y \cdot r_{ij,J}^{fX}] + F_{gi} \cdot (r_{O_i} + \boldsymbol{r}_{gi}) + T_i = 0
\end{cases}
\end{equation}
$$
3.5 Example Analysis of Error Effects
A numerical example is presented for a gear pair with parameters listed in Table 3.
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Number of teeth | 25 | 25 |
| Module (mm) | 4 | 4 |
| Pressure angle (°) | 20 | 20 |
| Addendum coefficient | 1 | 1 |
| Input torque (N·m) | 260 | — |
| Face width (mm) | 30 | 25 |
| Modification coefficient | 0 | 0 |
| Elastic modulus (MPa) | 2.05×10⁵ | 2.05×10⁵ |
| Poisson’s ratio | 0.3 | 0.3 |
| Total profile error, grade 6 (μm) | 13 | 0 |
| Total profile error, grade 7 (μm) | 19 | 0 |
For the eccentric circle center distance error, the transmission error follows the same periodic pattern as the center distance variation. The meshing point position also exhibits synchronization with the transmission error. Under the influence of the tooth profile error represented by $h_k(u_k) = F_\alpha \sin(\pi \theta_k/\theta_{AB} + \phi_0)$, the transmission error shows amplification proportional to the profile error magnitude. A 7th-grade precision gear produces larger transmission error amplitude than a 6th-grade gear, confirming the direct proportionality between profile error magnitude and transmission error.
An important finding is the possibility of error compensation through phase arrangement. When both the driving and driven gears have 6th-grade profile errors but their initial phases are offset by 180°, the combined transmission error is significantly reduced compared to the case where only one gear has errors. Specifically, the transmission error decreases from $15 \times 10^{-3}$ degrees to $3 \times 10^{-3}$ degrees, and the meshing point position variation decreases from 0.6 mm to 0.05 mm. This demonstrates the practical design principle that careful phase arrangement of gear errors can substantially improve transmission accuracy.
The shaft eccentricity analysis reveals that spatial misalignment transforms the line contact into point contact at the boundary of the tooth flank. This contact condition is less favorable for load distribution and indicates the need for tooth modification or profile optimization when misalignment is present.
4. Elastic Transmission Model for Spur Gears
4.1 Elastic Conjugate Meshing Model
The elastic deformation of gear teeth under load is incorporated into the conjugate meshing model. The assumption is that the gears have compliance characterized by time-varying mesh stiffness, without manufacturing errors. The model includes torsional deformation of the input and output shafts.
The elastic deformation of the gear tooth can be decomposed into three components: contact deformation $h_{ci}$, bending deformation $h_{bi}$, and gear body torsional deformation $h_{ti}$. Each contributes to the total transmission error through corresponding rotation angles $\delta\phi_{bi}$, $\delta\phi_{ti}$, and $\delta\phi_{Ti}$.
The actual flank equation including elastic deformation becomes:
$$
\begin{equation}
\boldsymbol{R}_i'(u_i,v_i) = \boldsymbol{R}_i(u_i,v_i) + h_{ci}(u_i,v_i)\boldsymbol{e}_3^{(i)}
\end{equation}
$$
where $h_{ci}(u_i,v_i)$ represents the contact deformation at point $(u_i,v_i)$ on the tooth flank. The meshing equation for the elastic contact case is:
$$
\begin{equation}
\begin{cases}
\delta\boldsymbol{r}_1 – \delta\boldsymbol{r}_2 + (\delta\phi_1 \boldsymbol{k} \times \boldsymbol{r}_1 + \delta\phi_{E1}\boldsymbol{k} \times \boldsymbol{r}_1 + \delta\phi_{b1}\boldsymbol{k} \times \boldsymbol{l}_{c1} + \delta\phi_{t1}\boldsymbol{k} \times \boldsymbol{r}_1) \\
– (\delta\phi_2 \boldsymbol{k} \times \boldsymbol{r}_2 + \delta\phi_{E2}\boldsymbol{k} \times \boldsymbol{r}_2 + \delta\phi_{b2}\boldsymbol{k} \times \boldsymbol{l}_{c2} + \delta\phi_{t2}\boldsymbol{k} \times \boldsymbol{r}_2) – \boldsymbol{\beta} \times \boldsymbol{r}_2 = \Delta\boldsymbol{a} \\
\boldsymbol{e}_3^{(1)*} – \boldsymbol{e}_3^{(2)*} = 0
\end{cases}
\end{equation}
$$
4.2 Gear Elastic Deformation Equations
The elastic deformations are calculated from the applied load and the corresponding stiffness components using:
$$
\begin{equation}
h_{ci} = \frac{F_{Hi}}{K_{ci}}, \qquad \delta\phi_{bi} = \frac{F_{Hi}}{K_{bi}}, \qquad \delta\phi_{ti} = \frac{F_{Hi}}{K_{ti}}
\end{equation}
$$
where $F_{Hi}$ is the mesh force acting on gear $i$, and the stiffness values are computed from the following formulas:
$$
\begin{equation}
K_{ci} = \frac{\pi E B}{4(1-\nu^2)}
\end{equation}
$$
$$
\begin{equation}
K_{bi} = \frac{F_{Hi}}{h_{bi}}
\end{equation}
$$
The bending and contact stiffness calculation is performed through a finite-element-based formulation using the Weber-Banashek method. The tooth is modeled as a cantilever beam on an elastic foundation. The stiffness expressions involve the gear geometry, material properties, and contact point position.
The shaft torsional stiffness contribution is:
$$
\begin{equation}
\delta\phi_{Ti} = \frac{T_i l_i}{G I_p}
\end{equation}
$$
where $T_i$ is the applied torque, $l_i$ is the distance from the torque application point to the gear center, $G$ is the shear modulus, and $I_p$ is the polar moment of inertia of the shaft cross-section.
4.3 Compatibility and Load Sharing in Double-Tooth Engagement
During double-tooth engagement, the deformation compatibility condition must be satisfied along both lines of action. The system is governed by:
$$
\begin{equation}
\begin{cases}
F_{H1} + F_{H2} = F_H \\
\delta_{m11} + \delta_{m21} = \delta_{m12} + \delta_{m22} \\
F_{Hi} = \sum_{i=1}^{2} \frac{K_i}{K_1 + K_2} F_H
\end{cases}
\end{equation}
$$
where $\delta_{mij}$ represents the elastic deformation at contact point $j$ of gear $i$. The load sharing between the two pairs of teeth is determined by their respective stiffness values. The meshing stiffness varies periodically due to the alternating single-double tooth engagement pattern.
4.4 Numerical Example and Discussion
The gear parameters are the same as in Table 3. The time-varying center distance function is obtained from the floating support model. The calculation reveals that the elastic deformation contributes a periodic transmission error component with abrupt changes at the transitions between single-tooth and double-tooth engagement zones. This is expected because the mesh stiffness changes discontinuously at these transition points.
The results show that the elastic deformation alone causes a transmission error of approximately 0.49 degrees, with a periodic fluctuation of about 0.01 degrees. When the center distance error is superimposed, the total transmission error reaches about 0.57 degrees. The center distance variation dominates the transmission error waveform, while the elastic deformation adds a higher-frequency modulation.
Similarly, the meshing point position variation is primarily driven by the center distance variation, with additional sudden jumps corresponding to changes in the contact point curvature radius when transitioning from single to double tooth engagement.
5. Comprehensive Transmission Precision Model Combining Errors and Elasticity
5.1 Comprehensive System Model
The comprehensive transmission precision model integrates all sources of error and elasticity within the spur gear transmission system. The model considers: (1) bearing component errors and elasticity represented through the cam follower mechanism; (2) shaft bending and torsional elasticity; (3) gear tooth profile errors; (4) gear tooth elastic deformation including contact, bending, and body torsion; (5) gear eccentricity.
For a planar gear transmission system, the actual center distance is determined from the floating shaft support model. The gear tooth flank including both error and elastic deformation is expressed as:
$$
\begin{equation}
\boldsymbol{R}_i'(u_i) = \boldsymbol{R}_i(u_i) + [h_i(u_i) + h_{ci}(u_i)]\boldsymbol{e}_3^{(i)}
\end{equation}
$$
The comprehensive transmission error equation combines all contributions:
$$
\begin{equation}
\delta\phi_2^* = \delta\phi_2^{center} + \delta\phi_2^{profile} + \delta\phi_2^{bending} + \delta\phi_2^{torsion} + \delta\phi_2^{contact} + \delta\phi_2^{shaft}
\end{equation}
$$
The load sharing and contact condition determination follows the modified compatibility equation that accounts for both elastic deformation and geometric errors:
$$
\begin{equation}
\delta_{m11} + \delta_{m21} + h_{11} + h_{21} = \delta_{m12} + \delta_{m22} + h_{12} + h_{22}
\end{equation}
$$
The contact condition is determined by:
$$
\begin{equation}
H_{contj} =
\begin{cases}
1 & \text{if } F_{Hj} \geq 0 \text{ and } j \text{ is engaged} \\
0 & \text{if } F_{Hj} < 0 \text{ or } j \text{ is not engaged}
\end{cases}
\end{equation}
$$
5.2 Numerical Results for the Comprehensive Model
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Number of teeth | 19 | 19 |
| Module (mm) | 2.5 | 2.5 |
| Pressure angle (°) | 20 | 20 |
| Input torque (N·m) | 260 | — |
| Face width (mm) | 20 | 20 |
| Elastic modulus (MPa) | 5.36×10⁵ | 5.36×10⁵ |
| Poisson’s ratio | 0.3 | 0.3 |
| Total profile error (μm) | 20 | 0 |
The comprehensive model is evaluated under three bearing error conditions (eccentric circle, ellipse, and trilobal). The results consistently demonstrate that the center distance variation, originating from bearing error and elasticity, constitutes the dominant source of transmission error. The shaft torsional deformation contributes a constant offset, while the gear tooth elasticity and profile error produce higher-frequency fluctuations superimposed on the base waveform.
The transmission error amplitude for the eccentric circle case varies from approximately -0.48° to -0.56°, reflecting the combined effect of all error sources. The elliptic and trilobal cases produce similar behavior with slightly different waveforms. The meshing point position variation tracks the transmission error closely, with additional abrupt changes at single-tooth/double-tooth transitions.
5.3 Influence of Gear Eccentricity on Transmission Error
Gear eccentricity, arising from manufacturing and assembly processes, creates an additional periodic center distance error. For a single gear with eccentricity $\Delta e$, the actual center distance is:
$$
\begin{equation}
a’ = \sqrt{(\Delta e \sin\theta)^2 + (a – \Delta e \cos\theta)^2}
\end{equation}
$$
The resulting center distance error is $\Delta a = a’ – a$, which varies with the rotation angle $\theta$.
Analysis for $\Delta e = 0.025$ mm and $\Delta e = 0.05$ mm shows that eccentricity amplitude increases the range of transmission error variation while preserving the fundamental error pattern. The shaft torsional deformation remains the largest single contribution, but the periodic component from eccentricity amplifies the fluctuation range.
An important design finding emerges from studying phase arrangements of two eccentrically mounted gears. When only the driving gear has eccentricity, the transmission error exhibits a particular waveform. When both gears have equal eccentricity with phase difference $\psi_1 = 0°$ and $\psi_2 = 180°$, the transmission error increases. However, when both gears are eccentric with $\psi_1 = 0°$ and $\psi_2 = 180°$ (out-of-phase mounting), the transmission error is reduced. This confirms that careful phase management of gear eccentricity can partially compensate for individual errors.
6. Conclusions and Future Work
This thesis presents a systematic investigation of transmission precision in spur gear systems using kinematic geometry methods. The key contributions and findings are summarized as follows:
(1) A kinematic geometry model of a spur gear transmission system with floating shaft support was established. Bearing component errors and elasticity were equivalently represented through plate conjugate cam mechanisms. The model successfully predicts time-varying center distance errors and shaft axis misalignment as functions of component errors, elasticity, and applied load. The results demonstrate that bearing raceway profile errors and bearing support stiffness significantly influence center distance variation, and that proper selection of bearing phase angles can minimize these effects.
(2) A transmission error model incorporating gear tooth profile errors, center distance variations, and shaft axis misalignments was developed. The model quantitatively describes how each error factor contributes to transmission error and meshing point position. The analysis reveals that shaft misalignment transforms line contact into point contact, adversely affecting load distribution. The phase arrangement of gear profile errors can be optimized to reduce combined transmission error.
(3) An elastic transmission model considering time-varying mesh stiffness and alternating single/double tooth engagement was formulated. The model accounts for contact deformation, tooth bending, and gear body torsion. The elastic deformation contributes periodic transmission error components with abrupt changes at engagement transitions.
(4) A comprehensive transmission precision model integrating both errors and elasticity was constructed. This model provides a framework for predicting overall transmission accuracy of spur gear systems from component-level design parameters. The eccentricity analysis demonstrates that appropriate phase arrangements can substantially reduce transmission error.
Future research directions include extending the model to spatial spur gear and helical gear transmissions with more complex tooth modifications, experimental verification of the theoretical model using the established test rig, and dynamic analysis of the transmission system incorporating error and elasticity excitations. The kinematic geometry approach presented in this thesis provides a theoretical foundation for precision design and error compensation in gear transmission systems, which is of significant practical importance for high-precision machinery and equipment.