In this thesis, I focus on the dynamic behavior of spur gear transmission systems under the influence of tooth profile thermal deformation. I first developed a mathematical model to compute the tooth profile thermal deformation based on the Blok flash temperature theory and the Hertz contact theory. Then, I incorporated this thermal deformation into a nonlinear dynamic model of a spur gear pair and systematically analyzed the resulting dynamic characteristics. Finally, I validated the proposed model through experimental tests on a spur gear transmission test rig. The key contributions and findings are summarized below.
1. Introduction and Background
Spur gears are among the most widely used mechanical components in industrial applications. Their dynamic characteristics directly affect the stability, reliability, and noise level of the entire machine. With the increasing demand for high-speed operations, the thermal deformation of gear tooth profiles caused by friction heat has become an important factor that cannot be ignored. However, most classical dynamic models of spur gear systems simplify the problem by neglecting thermal effects. To improve the accuracy of dynamic prediction, I have developed a comprehensive modeling framework that includes the thermal deformation of the tooth profile and its interaction with gear backlash and time-varying mesh stiffness.
The main research topics in this thesis include:
- Construction of a tooth profile thermal deformation calculation model considering different lubrication conditions.
- Incorporation of thermal deformation into the tooth backlash and time-varying mesh stiffness models.
- Establishment of a nonlinear torsional dynamic model of a spur gear pair considering thermal deformation.
- Numerical simulation of the system’s dynamic responses using the Runge-Kutta method.
- Experimental validation of the proposed model on a gear vibration test platform.
2. Calculation Model of Tooth Profile Thermal Deformation
2.1 Flash Temperature Model of Gear Tooth Surfaces
Based on the Blok flash temperature theory, the instantaneous flash temperature rise of a spur gear tooth surface can be expressed as:
$$ \Delta T_f = \frac{ u f_m f_e (v_1 – v_2) \sqrt{B} }{ \sqrt{\pi} \left( \sqrt{g_1 \rho_1 c_1 v_1} + \sqrt{g_2 \rho_2 c_2 v_2} \right) } $$
where \( u \) is the temperature rise coefficient, \( f_m \) is the friction coefficient, \( f_e \) is the normal load per unit tooth width, \( v_1 \) and \( v_2 \) are the tangential velocities of the driving and driven gear tooth surfaces, \( g_1, g_2 \) are the thermal conductivities, \( \rho_1, \rho_2 \) are the material densities, \( c_1, c_2 \) are the specific heats, and \( B \) is the half-width of the contact band.
The tangential velocities of the tooth surfaces are obtained from the gear geometry:
$$ v_i(t) = \omega_i r_{ki}(t) \sin \alpha_i(t), \quad i = 1, 2 $$
where \( \omega_i \) is the angular velocity, \( r_{ki}(t) \) is the distance from the mesh point to the gear center, and \( \alpha_i(t) \) is the pressure angle at the mesh point. The contact half-width is calculated using the Hertz contact theory:
$$ B(t) = \sqrt{ \frac{2 \eta F_n (1 – \mu^2) R_1(t) R_2(t) }{ \pi E b (R_1(t) + R_2(t)) } } $$
where \( \eta \) is a coefficient, \( \mu \) is Poisson’s ratio, \( E \) is the elastic modulus, \( F_n \) is the normal force, \( b \) is the face width, and \( R_i(t) \) is the curvature radius of the tooth profile at the mesh point.
2.2 Friction Coefficient under Different Lubrication Conditions
The friction coefficient plays a critical role in the flash temperature calculation. I considered two lubrication states: dry friction and mixed lubrication. For mixed lubrication, the friction coefficient is given by:
$$ \mu_m(t) = 1.13 \times 10^{-2} \left( \frac{ \eta_0 \, p_{ei}(t) }{ S_{av} } \right)^{0.127} \text{sign}\left( v_s(t) \right) \log_{10} \left( \frac{ 29700 \, \eta_0 }{ \sqrt{ \left( \frac{ v_e(t)^2 }{ v_s(t) } \right) } } \right) $$
where \( \eta_0 \) is the dynamic viscosity of the lubricant, \( S_{av} \) is the average surface roughness, \( p_{ei}(t) \) is the normal load per unit width, \( v_e(t) \) is the entrainment velocity, and \( v_s(t) \) is the sliding velocity.
For dry friction, the friction coefficient is assumed to be in the range 0.1–0.5. The friction coefficient curves under both conditions were numerically simulated for a spur gear pair with the parameters listed below.
2.3 Tooth Profile Thermal Deformation Model
When the tooth surface temperature increases, the tooth profile undergoes both radial and circumferential thermal expansion. Based on the non-uniform steady-state temperature field theory, the temperature distribution in the gear body along the radial direction is:
$$ T(r_i) = \frac{ \ln r_x – \ln r_i }{ \ln r_x – \ln r_a } \left[ T(r_a) – T(r_x) \right] + T(r_x) $$
where \( r_i \) is the radial distance, \( r_x \) is the shaft radius, \( r_a \) is the addendum radius, and \( T(r_a) \), \( T(r_x) \) are the temperatures at the addendum and shaft, respectively.
The actual tooth profile after thermal deformation can be expressed by the following parametric equations:
$$ \begin{cases} r\’_k = r_k + \lambda \int_{r_b}^{r_k} \left[ T(r_i) – T_0 \right] dr_i + u_b \\ \theta\’_k = \theta_k – \frac{ \lambda \left[ inv(\alpha_k) – inv(\alpha) \right] \left[ r_k – r \right] }{ r_k \cos \alpha_k } \end{cases} $$
where \( \lambda \) is the linear expansion coefficient, \( u_b \) is the base circle thermal deformation, and \( T_0 \) is the initial temperature. The geometric relationship between the actual deformed profile and the theoretical involute profile gives the tooth profile thermal deformation \( \Delta f \) as:
$$ \Delta f = \frac{ \lambda \left[ T(r_k) – T_0 \right] }{ r_k \cos \alpha_k } \left[ s_k – 2 r_k \left( inv(\alpha_k) – inv(\alpha) \right) \right] + \lambda \int_{r_b}^{r_k} \frac{ T(r_i) – T_0 }{ \cos \alpha_k } dr_i $$
I performed numerical simulations for different operating conditions. The results show that both the friction coefficient, the flash temperature, and the tooth profile thermal deformation increase with increasing rotational speed and torque. The strongest positive correlation exists among these three quantities.
3. Dynamic Model of a Spur Gear System Considering Thermal Deformation
3.1 Effect of Thermal Deformation on Backlash
The tooth profile thermal deformation reduces the backlash of the spur gear pair. The modified backlash half-gap \( b\’ \) is defined as:
$$ b\’ = \frac{ 2b – \Delta f_1 – \Delta f_2 }{ 2 } $$
where \( b \) is the original half-backlash, and \( \Delta f_1 \), \( \Delta f_2 \) are the thermal deformations of the driving and driven gear tooth profiles, respectively. The backlash function is therefore:
$$ f(x) = \begin{cases} x + b\’, & x < -b\’ \\ 0, & |x| \leq b\’ \\ x – b\’, & x > b\’ \end{cases} $$
3.2 Effect of Thermal Deformation on Time-Varying Mesh Stiffness
I used the potential energy method to calculate the time-varying mesh stiffness of spur gear pairs. The total mesh stiffness includes Hertzian contact stiffness, bending stiffness, shear stiffness, axial compressive stiffness, and gear body stiffness. The thermal deformation adds an equivalent stiffness change:
$$ k_{Ti}(t) = \frac{ F_n }{ \Delta f_i \, b_i } $$
The equivalent thermal stiffness of the gear pair is:
$$ k_T(t) = \frac{ k_{T1}(t) k_{T2}(t) }{ k_{T1}(t) + k_{T2}(t) } $$
Then the total mesh stiffness considering thermal deformation is:
$$ k_e(t) = k_e^{0}(t) + k_T(t) $$
where \( k_e^{0}(t) \) is the stiffness without thermal deformation. The numerical results show that the thermal deformation slightly increases the mesh stiffness and changes the stiffness fluctuation amplitude. For the studied gear pair, the average mesh stiffness is approximately \( 4.875 \times 10^8 \, \text{N/m} \), and the fluctuation amplitude is approximately \( 2.75 \times 10^8 \, \text{N/m} \).
3.3 Nonlinear Torsional Dynamic Model
Using the lumped mass method, I established a two-degree-of-freedom torsional vibration model of a spur gear pair. The equations of motion are:
$$ \begin{cases} I_1 \ddot{\theta}_1 + c_m r_{b1} \dot{x} + k_e(t) f(x) r_{b1} = T_1 \\ I_2 \ddot{\theta}_2 – c_m r_{b2} \dot{x} – k_e(t) f(x) r_{b2} = -T_2 \end{cases} $$
where \( I_1, I_2 \) are the moments of inertia, \( r_{b1}, r_{b2} \) are the base circle radii, \( c_m \) is the mesh damping, \( k_e(t) \) is the time-varying mesh stiffness, \( f(x) \) is the backlash function, and \( T_1, T_2 \) are the input and output torques. The relative displacement along the line of action is:
$$ x = r_{b1} \theta_1 – r_{b2} \theta_2 – e(t) $$
where \( e(t) \) is the static transmission error. After non-dimensionalization, the dynamic equation reduces to:
$$ \ddot{x} + 2\zeta \dot{x} + \left( 1 + \varepsilon \cos(\omega_h t) + \varepsilon_T \right) f(x) = F_m + F_h \cos(\omega_h t) $$
where \( \zeta \) is the dimensionless damping ratio, \( \omega_h \) is the dimensionless excitation frequency, \( \varepsilon \) is the dimensionless mesh stiffness fluctuation amplitude, \( \varepsilon_T \) is the dimensionless thermal stiffness amplitude, \( F_m \) is the dimensionless mean excitation, and \( F_h \) is the dimensionless alternating excitation.
4. Simulation Results and Dynamic Characteristics Analysis
4.1 Comparison of Dynamics With and Without Thermal Deformation
I solved the non-dimensional dynamic model using the fourth-order Runge-Kutta algorithm. The time history, phase plane, Poincaré map, and frequency spectrum were obtained for both cases. Without considering thermal deformation, the spur gear system exhibits a stable single-periodic motion, as shown by the closed phase trajectory and the concentrated Poincaré points. When thermal deformation is included, the system becomes less stable. As the gear body temperature increased from 100 °C to 160 °C, the time history shows more fluctuations, the phase trajectory becomes thicker, and the Poincaré points spread over a wider region. The frequency spectrum reveals a growing half-frequency component, indicating a transition from stable single-periodic motion to a quasi-periodic motion.
4.2 Influence of Key Parameters on Dynamic Response
I further investigated the effect of the damping ratio \( \zeta \), excitation frequency \( \omega_h \), and mesh stiffness amplitude \( \varepsilon \) on the dynamic response with thermal deformation. The results are summarized in the following table:
| Parameter | Range | Observed dynamic behavior |
|---|---|---|
| \( \zeta \) | 0.18 → 0.10 | Single-periodic → quasi-periodic, stability decreases |
| \( \omega_h \) | 1.1 → 1.3 | Quasi-periodic → periodic with improved stability, complex bifurcations |
| \( \varepsilon \) | 0.1 → 0.9 | Single-periodic → double-periodic → quasi-periodic |
Specifically, with decreasing damping \( \zeta \), the system moves from a stable single-periodic motion to a multi-periodic motion and finally to chaos. Increasing the excitation frequency \( \omega_h \) causes the system to pass through a bifurcation from a two-periodic to a single-periodic motion, with an overall increase in vibration amplitude. Increasing the mesh stiffness amplitude \( \varepsilon \) induces a period-doubling bifurcation route to chaos.
4.3 Bifurcation Characteristics
I plotted bifurcation diagrams for the system with and without thermal deformation as functions of \( \zeta \), \( \omega_h \), and \( \varepsilon \). The thermal deformation mainly affects the stability of the periodic motions, not the bifurcation structure. For example, in the \( \zeta \)-bifurcation diagram, the system undergoes period-1, period-2, period-4, quasi-periodic, and chaotic motions as \( \zeta \) decreases. The bifurcation points are nearly the same for the cases with and without thermal deformation, but the Poincaré points are more scattered when thermal deformation is included. This indicates that thermal deformation reduces the stability of the periodic motion.
5. Experimental Verification
5.1 Test Platform and Procedure
I built a spur gear vibration test platform consisting of a three-phase induction motor, a gearbox with a spur gear pair (module 5 mm, tooth numbers 25:75, pressure angle 20°), a magnetic powder brake, a DHDAS data acquisition system, accelerometers, and a temperature meter. Vibration acceleration signals were measured at rotational speeds of 1000 rpm, 2000 rpm, and 3000 rpm. The measured signals were filtered using the FIR filter in the DHDAS system and then integrated twice to obtain the displacement response.
5.2 Comparison Between Experimental and Simulation Results
The experimental displacement spectra show dominant frequencies close to the gear mesh frequency. For example, at 1000 rpm, the measured main frequency is 419.7 Hz, while the simulation with thermal deformation gives 413 Hz and that without thermal deformation gives 411 Hz. The corresponding displacement amplitudes are 10.4 μm (experiment), 8.7 μm (with thermal deformation), and 8.17 μm (without thermal deformation). Similar trends were observed at 2000 rpm and 3000 rpm. The following table summarizes the errors:
| Speed (rpm) | Item | With thermal deformation | Without thermal deformation | Experiment | Error with thermal (%) | Error without thermal (%) |
|---|---|---|---|---|---|---|
| 1000 | Frequency (Hz) | 413 | 411 | 419.7 | 1.60 | 2.07 |
| Displacement (μm) | 8.7 | 8.17 | 10.4 | 16.3 | 21.4 | |
| 2000 | Frequency (Hz) | 826 | 823 | 832.58 | 0.79 | 1.15 |
| Displacement (μm) | 15.9 | 15.15 | 19.3 | 17.6 | 18.5 | |
| 3000 | Frequency (Hz) | 1248 | 1245 | 1251.6 | 0.29 | 0.53 |
| Displacement (μm) | 41 | 39.6 | 48.5 | 15.4 | 18.4 |
The comparison clearly indicates that the dynamic model of spur gears accounting for tooth profile thermal deformation yields simulation results with much smaller errors and better agreement with the experimentally measured vibration responses. Therefore, the proposed model is more accurate for predicting the dynamic characteristics of spur gear transmission systems.
6. Conclusions and Outlook
In this thesis, I have systematically investigated the influence of tooth profile thermal deformation on the dynamic behavior of spur gear transmission systems. The main conclusions are:
- The friction coefficient, flash temperature, and tooth profile thermal deformation of spur gears are strongly correlated. They increase with increasing input speed and torque, and the mixed lubrication state produces lower values than the dry friction state.
- Thermal deformation reduces the backlash and slightly increases the time-varying mesh stiffness of spur gears. The developed non-dimensional dynamic model successfully captures these effects.
- Thermal deformation mainly affects the stability of the periodic motions, while the damping ratio, excitation frequency, and mesh stiffness amplitude control the periodicity and bifurcation behavior of the system.
- The experimental results confirm that the dynamic model of spur gears considering thermal deformation provides more accurate predictions of vibration frequencies and displacement amplitudes than the traditional model neglecting thermal effects.
Future work should consider other lubrication regimes (boundary and elastohydrodynamic lubrication), a multi-degree-of-freedom model including shaft flexibility and bearing clearance, and more precise temperature measurement inside the gearbox. The proposed modeling method provides a new approach for the dynamic design and fault diagnosis of spur gear transmission systems under realistic thermal conditions.