In modern mechanical transmission systems, involute spur gears are ubiquitous due to their well-known advantages in design simplicity, manufacturing, and power transmission efficiency. However, the operational performance and reliability of these spur gear pairs are profoundly influenced by their dynamic behavior, with the time-varying mesh stiffness (TVMS) acting as a primary internal excitation source for vibration and noise. Under harsh operating conditions involving heavy loads and high speeds, gear teeth are susceptible to surface failures. Among these, spalling—a form of surface pitting that leads to material detachment—is a critical failure mode. The presence of a spall defect locally alters the contact conditions and the load distribution along the tooth flank, thereby modifying the TVMS and consequently the system’s vibration signature, which can escalate to catastrophic transmission failure if left undetected.
Extensive research has been dedicated to modeling spall defects and analyzing their impact on gear dynamics. Previous studies have effectively employed the potential energy method and finite element analysis to derive analytical expressions for TVMS in the presence of rectangular or shaped spalls. Dynamic models incorporating spalling have been developed, revealing characteristic modulation sidebands in the vibration spectrum centered around the gear mesh frequency (GMF). A significant gap, however, remains in most of these models: they often neglect the thermomechanical effects arising from sliding friction during meshing. As two spur gear teeth engage, high contact pressures and relative sliding velocities generate frictional heat at the interface. This leads to a rapid, localized temperature rise known as “flash temperature.” This flash temperature induces thermal expansion and subtle yet non-negligible distortions in the tooth profile, effectively introducing an additional “thermo-elastic stiffness” component. This paper bridges this gap by developing a comprehensive dynamic model for a spur gear pair that integrates both a spall fault model and a flash temperature model. We derive the combined meshing stiffness and investigate its influence on the vibration characteristics of the spur gear transmission system.
Modeling Spur Gear Mesh Stiffness with a Spall Defect
We consider a standard involute spur gear pair. A rectangular spall defect is assumed to exist on the tooth flank of the driving gear. The geometry of the spall is defined by its length \(a_s\), width \(b_s\), and depth \(h_s\). The TVMS of the healthy and faulty gear pair is calculated using the potential energy method, which considers the energy stored in the gear teeth during deformation. The total mesh stiffness \(K_M(t)\) for a single tooth pair in contact is the series combination of several stiffness components:
$$ \frac{1}{K_M(t)} = \frac{1}{k_h} + \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{f1}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}} + \frac{1}{k_{f2}} $$
where \(k_h\) is the Hertzian contact stiffness, and for each gear (\(i=1,2\)), \(k_{bi}\) is the bending stiffness, \(k_{si}\) is the shear stiffness, \(k_{ai}\) is the axial compressive stiffness, and \(k_{fi}\) is the fillet foundation stiffness. The spall defect primarily affects the area moment of inertia \(I_i\) and the cross-sectional area \(A_i\) of the tooth slice within the spalled region, which in turn modifies the bending, shear, and axial stiffnesses of the driving gear.
For a healthy tooth section at a distance \(x\) from the tooth base, the area and moment of inertia are:
$$ A_{i,healthy}(x) = 2 h_x L $$
$$ I_{i,healthy}(x) = \frac{(2 h_x)^3 L}{12} $$
where \(h_x\) is the variable tooth thickness and \(L\) is the face width.
For a section within the spall region on the driving gear, these become:
$$ A_{1,spall}(x) = 2 h_x L – a_s h_s $$
$$ I_{1,spall}(x) = \frac{(2 h_x)^3 L – a_s h_s^3}{12} $$
The individual stiffness components are derived by integrating the strain energy over the effective tooth height. The bending stiffness \(k_b\) is derived from the bending energy \(U_b\):
$$ U_b = \int_0^d \frac{[F_b (d – x) – F_a h]^2}{2 E I(x)} dx = \frac{F^2}{2 k_b} $$
where \(F\) is the total normal load, \(F_b\) and \(F_a\) are its bending and axial components, \(d\) is the distance from the load application point to the tooth base, \(h\) is the height from the tooth centerline, and \(E\) is Young’s modulus.
The shear stiffness \(k_s\) and axial compressive stiffness \(k_a\) are derived similarly from their respective energy expressions:
$$ U_s = \int_0^d \frac{1.2 F_b^2}{2 G A(x)} dx = \frac{F^2}{2 k_s} $$
$$ U_a = \int_0^d \frac{F_a^2}{2 E A(x)} dx = \frac{F^2}{2 k_a} $$
where \(G\) is the shear modulus. The fillet foundation stiffness \(k_f\) is calculated using empirical formulas from literature. The Hertzian contact stiffness \(k_h\) for two cylinders in contact is given by:
$$ k_h = \frac{\pi E L}{4(1-\nu^2)} $$
where \(\nu\) is Poisson’s ratio. The overall mesh stiffness \(K_M(t)\) for the spur gear pair is the sum of the stiffnesses of all tooth pairs in simultaneous contact, considering the double-tooth and single-tooth engagement regions.
| Parameter | Pinion (Driver) | Gear (Driven) |
|---|---|---|
| Number of Teeth, \(z\) | 19 | 48 |
| Module, \(m\) (mm) | 2 | 2 |
| Pressure Angle, \(\alpha\) (deg) | 20 | 20 |
| Face Width, \(L\) (mm) | 16 | 16 |
| Young’s Modulus, \(E\) (GPa) | 206 | 206 |
| Poisson’s Ratio, \(\nu\) | 0.3 | 0.3 |
| Density, \(\rho\) (kg/m³) | 7833 | 7833 |
| Spall Length, \(a_s\) (mm) | 1.5 | – |
| Spall Width, \(b_s\) (mm) | 1.0 | – |
| Spall Depth, \(h_s\) (mm) | 0.5 | – |
Modeling Tooth Surface Flash Temperature and Thermo-Elastic Deformation
The total contact temperature \(\Theta_T\) at the meshing interface of the spur gears is the sum of the bulk (or bulk) temperature \(\Theta_M\) and the instantaneous flash temperature rise \(\Theta_f\):
$$ \Theta_T = \Theta_M + \Theta_f $$
The bulk temperature is assumed constant and uniform for both gears. The flash temperature rise \(\Theta_f\) is calculated using the classical Block theory, which models the contact as a moving heat source on a semi-infinite body. The formula for two contacting gears is:
$$ \Theta_f = \frac{\mu_{th} f_m f_e |v_{s1} – v_{s2}|}{(\sqrt{\lambda_1 \rho_1 c_1 v_{s1}} + \sqrt{\lambda_2 \rho_2 c_2 v_{s2}}) B} $$
where:
\(\mu_{th}\) is a thermal partition coefficient (~0.8-0.9),
\(f_m\) is the coefficient of friction,
\(f_e\) is the normal load per unit face width (N/m),
\(v_{s1}, v_{s2}\) are the sliding velocities of the pinion and gear at the contact point,
\(\lambda_i\) is the thermal conductivity,
\(\rho_i\) is the density,
\(c_i\) is the specific heat capacity,
\(B\) is the semi-width of the Hertzian contact band, given by \(B = \sqrt{\frac{4 f_e R_{eff}}{\pi L E_{eff}}}\), with \(R_{eff}\) and \(E_{eff}\) being the effective radius and modulus.
The normal load per unit face width \(f_e\) changes when the spall enters the contact zone, as the effective load-bearing width is reduced:
$$ f_e = \begin{cases}
\frac{K_a F_n}{L} & \text{(Healthy Contact)} \\
\frac{K_a F_n}{L – a_s} & \text{(Spall in Contact)}
\end{cases} $$
where \(K_a\) is an application factor and \(F_n\) is the total normal force.
The localized flash temperature causes thermal expansion. The resulting thermo-elastic deformation \(\delta_i(t)\) of the involute profile for gear \(i\) at the contact point can be approximated by considering the thermal strain along the line of action. A simplified expression accounting for expansion at the base circle radius \(r_{bi}\) is:
$$ \delta_i(t) \approx -\Theta_f \alpha_{th} \left[ r_{bi} + u_{bi} – \frac{s_i}{2} \right] $$
where \(\alpha_{th}\) is the coefficient of linear thermal expansion, \(u_{bi}\) is the thermal displacement of the base circle, and \(s_i\) is the tooth thickness. This deformation acts as a displacement in the direction of the line of action, effectively creating a compliance. We define a “flash temperature stiffness” \(K_i(T)\) for each gear as the ratio of load to this thermal deformation:
$$ K_i(T) = \frac{f_e}{\delta_i(t)}, \quad (i=1,2) $$
The combined flash temperature stiffness \(K_T(t)\) for the spur gear pair in contact is then:
$$ K_T(t) = \frac{K_1(T) \cdot K_2(T)}{K_1(T) + K_2(T)} $$
| Parameter | Symbol | Value |
|---|---|---|
| Thermal Partition Coefficient | \(\mu_{th}\) | 0.83 |
| Coefficient of Friction | \(f_m\) | 0.01 |
| Thermal Conductivity | \(\lambda_i\) (W/m·K) | 46.47 |
| Specific Heat Capacity | \(c_i\) (J/kg·K) | 481.48 |
| Coefficient of Thermal Expansion | \(\alpha_{th}\) (1/K) | 1.16e-5 |
| Bulk Temperature | \(\Theta_M\) (°C) | 100 |
Integrated Time-Varying Mesh Stiffness for Spur Gears
The total effective mesh stiffness \(K_{total}(t)\) for the spur gear pair, accounting for both the mechanical deformation (including spall effects) and the thermo-elastic deformation due to flash temperature, is obtained by considering these two stiffnesses acting in series. The mechanical path (gear body deflection) and the thermal compliance path are coupled at the contact interface. Therefore, the combined stiffness is:
$$ \frac{1}{K_{total}(t)} = \frac{1}{K_M(t)} + \frac{1}{K_T(t)} $$
This formulation implies that the flash temperature effect generally reduces the overall mesh stiffness of the spur gears, as \(K_T(t)\) introduces an additional compliance. The reduction is most significant away from the pitch point where sliding velocities and hence \(\Theta_f\) are high, and less significant near the pitch point where pure rolling occurs.
Dynamic Model of the Spur Gear Transmission System
We develop an 8-degree-of-freedom (8-DOF) lumped-parameter dynamic model of the spur gear transmission system, which includes rotational motions of the motor and load inertias, translational motions (x and y) of the gear centers due to bearing compliance, and rotational motions of the gears themselves. The model incorporates the integrated time-varying mesh stiffness \(K_{total}(t)\) and a corresponding mesh damping \(c_m\). The equations of motion are derived using Lagrange’s equations.
System Equations:
$$
\begin{aligned}
I_m \ddot{\theta}_m + c_{in}(\dot{\theta}_m – \dot{\theta}_1) + k_{in}(\theta_m – \theta_1) &= T_m \\
I_l \ddot{\theta}_l + c_{out}(\dot{\theta}_l – \dot{\theta}_2) + k_{out}(\theta_l – \theta_2) &= -T_l \\
I_1 \ddot{\theta}_1 – c_{in}(\dot{\theta}_m – \dot{\theta}_1) – k_{in}(\theta_m – \theta_1) &= -r_{b1} F_{mesh} \\
m_1 \ddot{x}_1 + c_{b1} \dot{x}_1 + k_{b1} x_1 &= F_f \\
m_1 \ddot{y}_1 + c_{b1} \dot{y}_1 + k_{b1} y_1 &= F_{mesh} \\
I_2 \ddot{\theta}_2 – c_{out}(\dot{\theta}_l – \dot{\theta}_2) – k_{out}(\theta_l – \theta_2) &= r_{b2} F_{mesh} \\
m_2 \ddot{x}_2 + c_{b2} \dot{x}_2 + k_{b2} x_2 &= -F_f \\
m_2 \ddot{y}_2 + c_{b2} \dot{y}_2 + k_{b2} y_2 &= -F_{mesh}
\end{aligned}
$$
where:
\(T_m, T_l\) are input motor and output load torques.
\(I_m, I_l, I_1, I_2\) are moments of inertia of motor, load, pinion, and gear.
\(m_1, m_2\) are masses of pinion and gear.
\(k_{in}, c_{in}\) and \(k_{out}, c_{out}\) are torsional stiffness and damping of input/output shafts.
\(k_{b1}, c_{b1}\) and \(k_{b2}, c_{b2}\) are bearing support stiffness and damping in x and y directions.
\(r_{b1}, r_{b2}\) are base circle radii.
\(\theta, x, y\) are rotational and translational displacements.
\(F_{mesh}\) is the dynamic mesh force along the line of action, and \(F_f\) is the friction force tangential to the tooth profile.
The dynamic mesh force \(F_{mesh}\) is given by:
$$ F_{mesh} = K_{total}(t) \cdot \delta_{mesh} + c_m \cdot \dot{\delta}_{mesh} $$
where \(\delta_{mesh}\) is the dynamic transmission error along the line of action, defined as:
$$ \delta_{mesh} = (r_{b1} \theta_1 – r_{b2} \theta_2) + (y_1 – y_2) \cos \alpha – (x_1 – x_2) \sin \alpha – e(t) $$
Here, \(e(t)\) represents static transmission error (unmodeled geometric errors).
| Parameter | Symbol | Value |
|---|---|---|
| Motor & Load Inertia | \(I_m, I_l\) (kg·m²) | 0.0021, 0.0105 |
| Gear Masses | \(m_1, m_2\) (kg) | 0.96, 2.88 |
| Gear Inertias | \(I_1, I_2\) (kg·m²) | 4.37e-4, 8.36e-4 |
| Input/Output Torque | \(T_m, T_l\) (N·m) | 11.9, 48.9 |
| Shaft Torsional Stiffness | \(k_{in}, k_{out}\) (N·m/rad) | 4.4e4, 4.4e4 |
| Bearing Support Stiffness | \(k_{b1}, k_{b2}\) (N/m) | 6.56e7, 6.56e7 |
| Mesh Damping Ratio | \(\zeta_m\) | 0.07 |
| Pinion Rotational Frequency | \(f_r\) (Hz) | 30 |
| Mesh Frequency | \(f_m\) (Hz) | 570 |
Results and Analysis of Vibration Characteristics
The system of ordinary differential equations is solved numerically using the 4th/5th order Runge-Kutta method with a fixed time step. The analysis focuses on the dynamic response in the vertical direction (\(y_1\)) of the pinion, as it is most directly excited by the mesh force.
1. Flash Temperature Distribution: The calculated flash temperature \(\Theta_f\) varies significantly throughout the mesh cycle of the spur gears. It is nearly zero at the pitch point due to pure rolling. The temperature rises in both the approach and recess regions due to increased sliding velocities. A distinct step increase occurs when the spall enters the contact zone because the unit load \(f_e\) increases abruptly (due to reduced contact width \(L-a_s\)), leading to a higher frictional heat flux. Another step change occurs when the spall exits.
2. Thermo-elastic Deformation: The profile deformation \(\delta_1(t)\) of the pinion tooth follows the trend of \(\Theta_f\). The deformation is minimal at the pitch point and increases away from it. The spall-induced load change causes a corresponding jump in the deformation magnitude, indicating a localized increase in thermo-elastic compliance.
3. Integrated Mesh Stiffness: The plot of \(K_{total}(t)\) over one mesh cycle reveals the combined effects. Compared to the mechanical-only stiffness \(K_M(t)\), the integrated stiffness \(K_{total}(t)\) is lower overall. The reduction is slight near the pitch point but more pronounced in the double-tooth engagement regions at the beginning and end of the cycle. The spall event manifests as a sharp drop in stiffness. Crucially, the depth of this drop is slightly greater when flash temperature is considered, because the spall also increases the local unit load and thus the flash temperature, further reducing \(K_T(t)\) and hence \(K_{total}(t)\) at that instant.
4. Vibration Signal Characteristics: The simulated vibration acceleration signal (derived from \(\ddot{y}_1\)) shows clear modulation.
- Time Domain: The signal exhibits periodic impacts corresponding to the spall’s entry into mesh once per revolution of the faulty pinion. The interval between these major impulse peaks is \(1/f_r = 33.3\) ms. Furthermore, these impulse responses are modulated at the gear mesh frequency \(f_m\), observable as a repetitive pattern within each shaft rotation period.
- Frequency Domain: The Fast Fourier Transform (FFT) of the signal reveals a rich spectrum. The dominant peak is at the mesh frequency \(f_m = 570\) Hz. Its higher harmonics (\(2f_m, 3f_m, …\)) are also present, indicating nonlinearities and periodic stiffness variations. Most importantly, families of sidebands appear around the mesh frequency and its harmonics. The spacing between these sidebands is equal to the pinion’s rotational frequency \(f_r = 30\) Hz. This is a classic signature of amplitude modulation caused by a localized fault like spalling. The presence of flash temperature subtly alters the amplitude distribution of these sidebands compared to a model without thermal effects, due to the modified stiffness excitation pattern.
Conclusion
This study developed a comprehensive dynamic model for a spur gear pair that integrates the effects of a surface spall defect and flash temperature-induced thermo-elastic deformation. The key conclusions are as follows:
- During the meshing of spur gears, sliding friction generates heat, leading to a localized flash temperature rise. This causes thermal expansion and distortion of the tooth profile, which introduces an additional, time-varying thermo-elastic compliance (flash temperature stiffness) into the system.
- The flash temperature and the resulting tooth deformation are minimal at the pitch point (pure rolling) and increase in the approach and recess regions. The presence of a spall defect causes an abrupt increase in unit load when it enters the contact zone, leading to a corresponding step increase in flash temperature and thermo-elastic deformation.
- The integrated mesh stiffness, combining mechanical and thermal components, is generally lower than the purely mechanical stiffness for spur gears. The reduction is most significant away from the pitch point. The spall-induced stiffness drop is slightly magnified when flash temperature is considered.
- The vibration response of the faulty spur gear pair exhibits clear periodic impulses at the shaft rotational frequency (\(f_r\)) due to the spall’s periodic entry into mesh. The frequency spectrum is characterized by the gear mesh frequency (\(f_m\)), its harmonics, and families of sidebands spaced at \(f_r\). This sideband structure is a direct consequence of the periodic modulation caused by the spall interacting with the time-varying mesh stiffness, an effect that is quantitatively influenced by the inclusion of flash temperature dynamics.
The integrated model provides a more physically accurate representation of spur gear dynamics under faulty conditions, considering a critical real-world phenomenon often neglected. The analysis of the resulting vibration signatures, particularly the sideband structure around the mesh frequency, offers valuable insights for condition monitoring and fault diagnosis of spur gear transmissions operating under significant load and speed, where thermal effects become non-negligible. Future work could explore the interaction between flash temperature, lubrication regimes, and the progression of spalling or other wear modes in spur gears.