The operational integrity of gear transmission systems, particularly those subjected to high speeds and heavy loads, is paramount for mechanical reliability. Among the most critical failure modes in such systems is the initiation and propagation of fatigue cracks. The pitch circle region of a gear tooth is especially vulnerable due to the significant alternating stresses it endures during meshing. A crack originating at the pitch circle can progressively deepen, eventually leading to tooth fracture, catastrophic failure, and unplanned downtime. Therefore, a profound understanding of the dynamic behavior of a spur and pinion system with such a defect is crucial for developing effective condition monitoring and predictive maintenance strategies. This study delves into the fault mechanism by establishing a comprehensive nonlinear dynamic model that incorporates the crack’s influence and analyzing the resulting vibration signatures.
The dynamic analysis of a geared rotor system requires a model that accurately captures the interactions between its key components. A lumped-parameter model is developed here, representing a single-stage spur gear transmission. The system comprises six disks: the driving and driven gears (pinion and wheel) and four disks representing the supporting shafts and bearings at both ends. The model accounts for twelve degrees of freedom: transverse vibrations and torsional rotations for each disk. The gear mesh is modeled as a spring-damper element acting along the line of action, with its stiffness being the critical time-varying parameter influenced by the crack. The supporting shafts provide bending and torsional stiffness, while the bearings are modeled as linear spring-damper elements in the transverse direction.
The fundamental equations of motion for this 12-DOF gear-rotor-bearing system are derived using Newton’s second law. The relative displacement along the line of action, \( u \), governs the mesh force. It is defined as the combination of the transverse displacements and the torsional rotations of the gear masses:
$$ u = y_p – y_g + r_p \theta_p – r_g \theta_g $$
where \( y_p \) and \( y_g \) are the transverse displacements, \( \theta_p \) and \( \theta_g \) are the angular displacements, and \( r_p \) and \( r_g \) are the base circle radii of the pinion and gear, respectively. The resulting mesh force \( F_m \) is:
$$ F_m = k_m(t) f(u) + c_m \dot{u} $$
Here, \( k_m(t) \) is the time-varying mesh stiffness, \( c_m \) is the constant mesh damping, and \( f(u) \) is a nonlinear displacement function accounting for gear backlash \( b_n \):
$$
f(u) =
\begin{cases}
u – b_n/2, & u > b_n/2 \\
0, & |u| \le b_n/2 \\
u + b_n/2, & u < -b_n/2
\end{cases}
$$
The complete set of dimensionless differential equations, after introducing the characteristic displacement \( b_c = b_n/2 \) and characteristic frequency \( \omega_n = \sqrt{k_{m0} (1/m_p + 1/m_g)} \), where \( k_{m0} \) is the average mesh stiffness, is presented below. The dimensionless parameters for stiffness and damping are denoted with bars.
For the Pinion Rotor (Disks 1, 2, 3):
$$ \ddot{Y}_1 + \bar{K}_{s11}(Y_1 – Y_2) + \bar{C}_{s11}(\dot{Y}_1 – \dot{Y}_2) + \bar{K}_{bp}Y_1 + \bar{C}_{bp}\dot{Y}_1 = 0 $$
$$ \ddot{\Theta}_1 + \bar{K}_{t11}(\Theta_1 – \Theta_2) + \bar{C}_{t11}(\dot{\Theta}_1 – \dot{\Theta}_2) = \bar{T}_p $$
$$ \ddot{Y}_2 + \bar{K}_{s21}(Y_2 – Y_1) + \bar{K}_{s22}(Y_2 – Y_3) + \bar{C}_{s21}(\dot{Y}_2 – \dot{Y}_1) + \bar{C}_{s22}(\dot{Y}_2 – \dot{Y}_3) + \bar{F}_m = 0 $$
$$ \ddot{\Theta}_2 + \bar{K}_{t21}(\Theta_2 – \Theta_1) + \bar{K}_{t22}(\Theta_2 – \Theta_3) + \bar{C}_{t21}(\dot{\Theta}_2 – \dot{\Theta}_1) + \bar{C}_{t22}(\dot{\Theta}_2 – \dot{\Theta}_3) + \bar{F}_m \bar{r}_p = 0 $$
$$ \ddot{Y}_3 + \bar{K}_{s31}(Y_3 – Y_2) + \bar{C}_{s31}(\dot{Y}_3 – \dot{Y}_2) + \bar{K}_{bp}Y_3 + \bar{C}_{bp}\dot{Y}_3 = 0 $$
$$ \ddot{\Theta}_3 + \bar{K}_{t31}(\Theta_3 – \Theta_2) + \bar{C}_{t31}(\dot{\Theta}_3 – \dot{\Theta}_2) = 0 $$
For the Gear Rotor (Disks 4, 5, 6):
$$ \ddot{Y}_4 + \bar{K}_{s41}(Y_4 – Y_5) + \bar{C}_{s41}(\dot{Y}_4 – \dot{Y}_5) + \bar{K}_{bg}Y_4 + \bar{C}_{bg}\dot{Y}_4 = 0 $$
$$ \ddot{\Theta}_4 + \bar{K}_{t41}(\Theta_4 – \Theta_5) + \bar{C}_{t41}(\dot{\Theta}_4 – \dot{\Theta}_5) = 0 $$
$$ \ddot{Y}_5 + \bar{K}_{s51}(Y_5 – Y_4) + \bar{K}_{s52}(Y_5 – Y_6) + \bar{C}_{s51}(\dot{Y}_5 – \dot{Y}_4) + \bar{C}_{s52}(\dot{Y}_5 – \dot{Y}_6) – \bar{F}_m = 0 $$
$$ \ddot{\Theta}_5 + \bar{K}_{t51}(\Theta_5 – \Theta_4) + \bar{K}_{t52}(\Theta_5 – \Theta_6) + \bar{C}_{t51}(\dot{\Theta}_5 – \dot{\Theta}_4) + \bar{C}_{t52}(\dot{\Theta}_5 – \dot{\Theta}_6) – \bar{F}_m \bar{r}_g = 0 $$
$$ \ddot{Y}_6 + \bar{K}_{s61}(Y_6 – Y_5) + \bar{C}_{s61}(\dot{Y}_6 – \dot{Y}_5) + \bar{K}_{bg}Y_6 + \bar{C}_{bg}\dot{Y}_6 = 0 $$
$$ \ddot{\Theta}_6 + \bar{K}_{t61}(\Theta_6 – \Theta_5) + \bar{C}_{t61}(\dot{\Theta}_6 – \dot{\Theta}_5) = -\bar{T}_e $$
The heart of modeling the crack’s effect lies in accurately calculating the time-varying mesh stiffness \( k_m(t) \). The potential energy method is employed, considering Hertzian contact energy \( U_h \), bending energy \( U_b \), axial compressive energy \( U_a \), shear energy \( U_s \), and gear body foundation energy \( U_f \). The total potential energy for a single tooth pair is:
$$ U = \frac{F^2}{2k} = U_h + \sum_{i=1}^2 (U_{b,i} + U_{a,i} + U_{s,i} + U_{f,i}) $$
This leads to the expression for the mesh stiffness of a single tooth pair \( k_t \):
$$ k_t = \frac{1}{\frac{1}{k_h} + \sum_{i=1}^2 \left( \frac{1}{k_{b,i}} + \frac{1}{k_{a,i}} + \frac{1}{k_{s,i}} + \frac{1}{k_{f,i}} \right) } $$
For a spur gear pair in double-tooth contact, the total mesh stiffness is the sum of the stiffnesses of the two engaged pairs. A crack at the pitch circle primarily affects the bending and shear stiffness of the tooth by reducing its effective area \( A_x \) and moment of inertia \( I_x \) along the tooth height from the crack tip to the tip. For a crack of length \( q \) at an angle \( \alpha \), the modified effective dimensions are:
$$
A_x = \begin{cases}
(h_{c1} + h_x)L, & \phi_c \le x \le \phi_g \\
(2h_x)L, & \phi_b < x < \phi_c \text{ and } \phi_g < x < \phi_d
\end{cases}
$$
$$
I_x = \begin{cases}
\frac{1}{12}(h_{c1} + h_x)^3 L, & \phi_c \le x \le \phi_g \\
\frac{1}{12}(2h_x)^3 L, & \phi_b < x < \phi_c \text{ and } \phi_g < x < \phi_d
\end{cases}
$$
where \( h_{c1} = h_c – q \sin \alpha \), \( h_x \) is the distance from the tooth centerline at point \( x \), \( L \) is the face width, and \( \phi \) denotes angular positions along the tooth profile.
The stiffness of the shafts and bearings are also critical. The shaft’s bending stiffness \( k_s \) and torsional stiffness \( k_t \) are derived from Timoshenko beam theory:
$$ k_s = \frac{EI}{l + \frac{GAl}{4k_{xz}}} \quad , \quad k_t = \frac{GJ}{l} $$
where \( E \) and \( G \) are elastic and shear moduli, \( I \) and \( J \) are area and polar moments of inertia, \( l \) is length, and \( k_{xz} \) is a shear correction factor. The bearing radial stiffness \( k_b \) is approximated considering elastic deformation and contact fits:
$$ k_b = \frac{F}{\delta_1 + \delta_2 + \delta_3} $$
where \( \delta_1 \), \( \delta_2 \), and \( \delta_3 \) are deformations related to bearing internal clearance and housing fits.
| Component | Parameter | Symbol | Value | Parameter | Symbol | Value |
|---|---|---|---|---|---|---|
| Spur Gears | Module | \( m \) | 1.5 mm | Number of Teeth | \( z_p / z_g \) | 36 / 90 |
| Mass | \( m_p / m_g \) | 0.2 / 1.6 kg | Moment of Inertia | \( I_p / I_g \) | 9e-5 / 3.7e-3 kg·m² | |
| Input Speed | \( n_p \) | 1000 rpm | Avg. Mesh Stiffness | \( k_{m0} \) | 2.747e8 N/m | |
| Backlash | \( b_n \) | 1e-6 m | Mesh Damping | \( c_m \) | 6660 N·s/m | |
| Shafts | Bending Stiffness | \( k_s \) | 4.71e8 N/m | Bending Damping | \( c_{sp} / c_{sg} \) | 872 N·s/m |
| Torsional Stiffness | \( k_t \) | 3.45e4 N·m/rad | Torsional Damping | \( c_{tp} / c_{tg} \) | 75 N·s/m | |
| Bearings | Bearing Stiffness | \( k_b \) | 1.04e9 N/m | Bearing Damping | \( c_{bp} / c_{bg} \) | 144 / 408 N·s/m |
| Loads | Input Torque | \( T_p \) | 40 N·m | Resistive Torque | \( T_e \) | 100 N·m |
The primary effect of a pitch circle crack is a localized reduction in the mesh stiffness. The calculated single-tooth-pair stiffness for a healthy pinion and for pinions with cracks of depth \( q = 0.7 \) mm and \( q = 1.1 \) mm clearly demonstrates this. In a healthy spur and pinion mesh, the stiffness increases gradually from the root to a maximum near the pitch circle and then decreases towards the tip. When a crack is present, a distinct step-like drop in stiffness occurs precisely at the pitch circle engagement point. The depth of this drop is directly proportional to the crack depth \( q \), as the effective tooth section modulus is more severely compromised.
This local stiffness reduction manifests in the total mesh stiffness \( K_m(t) \) of the spur gear pair. The stiffness function remains unaffected during the initial double-pair contact and the first part of the single-pair contact (before the pitch point). However, during the second part of the single-pair contact and the following double-pair contact region, the stiffness profile shows a significant dip. The severity of this dip increases with crack depth, and the effect is more pronounced in the single-pair contact zone than in the double-pair contact zone, as the load is shared in the latter.
| Disk # | 1 | 2 (Pinion) | 3 | 4 | 5 (Gear) | 6 |
|---|---|---|---|---|---|---|
| Mass \( m \) (kg) | 0.41 | 0.20 | 0.41 | 0.41 | 1.60 | 0.41 |
| Moment of Inertia \( I \) (kg·m²) | 2.63e-4 | 9.00e-5 | 2.63e-4 | 2.63e-4 | 3.70e-3 | 2.63e-4 |
The dimensionless equations of motion are solved numerically using the fourth-order Runge-Kutta method for three distinct cases: healthy pinion (\( q = 0 \) mm), and pinions with pitch circle cracks of \( q = 0.7 \) mm and \( q = 1.1 \) mm. The dynamic response, specifically the vibration signal along the line of action, is analyzed in both the time and frequency domains.
In the time domain, the overall amplitude of vibration does not change dramatically with the introduction of a crack. However, a crucial feature emerges: periodic impulse responses are superposed on the signal. These impulses occur at an interval equal to the rotational period of the cracked pinion (0.06 s for a 1000 rpm pinion). The amplitude of these impulses increases with the depth of the crack. This is a direct consequence of the sudden change in mesh stiffness when the cracked tooth enters the damaged zone (post-pitch-point contact), creating a periodic impact excitation within the spur and pinion system.
In the frequency domain, the spectrum of the healthy system is dominated by the gear mesh frequency \( f_m \) (600 Hz) and its harmonics. With a pitch circle crack, these components persist, but sidebands appear around the mesh frequency and its harmonics. The spacing between these sidebands is equal to the rotational frequency of the pinion shaft \( f_p \) (16.7 Hz). This modulation phenomenon is classical in gear fault diagnosis and confirms the periodic nature of the disturbance. The amplitude of these sidebands grows as the crack deepens, reflecting the increased severity of the stiffness modulation.
While time-domain waveforms and frequency spectra provide visual evidence, dimensionless statistical indicators are often more robust and sensitive for automated fault detection. Several such indicators are calculated from the vibration signal \( x(n) \), where \( N \) is the total number of data points, \( X_{av} \) is the mean, \( \sigma \) is the standard deviation, \( X_{peak} \) is the peak amplitude, \( X_{rms} \) is the root mean square, and \( S_r \) is the square root amplitude.
Kurtosis: Measures the “peakedness” or impulsiveness of the signal. It is highly sensitive to impacts.
$$ K_q = \frac{1}{N} \sum_{n=1}^{N} \left( \frac{x(n) – X_{av}}{\sigma} \right)^4 $$
Margin Factor: Related to the ratio of peak value to a measure of the signal’s “root.”
$$ L = \frac{1}{N} \sum_{n=1}^{N} \left( \frac{X_{peak}}{S_r} \right) $$
Peak Factor (Crest Factor): Ratio of peak to RMS value.
$$ C = \frac{1}{N} \sum_{n=1}^{N} \left( \frac{X_{peak}}{X_{rms}} \right) $$
Impulse Factor: Ratio of peak to mean absolute value.
$$ I = \frac{1}{N} \sum_{n=1}^{N} \left( \frac{X_{peak}}{X_{av}} \right) $$
Shape Factor: Ratio of RMS to mean absolute value.
$$ S = \frac{1}{N} \sum_{n=1}^{N} \left( \frac{X_{rms}}{X_{av}} \right) $$
| Parameter | Healthy (\( q = 0 \) mm) | Cracked (\( q = 0.7 \) mm) | Cracked (\( q = 1.1 \) mm) |
|---|---|---|---|
| Kurtosis \( K_q \) | 2.867 | 3.065 | 3.116 |
| Margin Factor \( L \) | 1.369 | 1.467 | 1.477 |
| Peak Factor \( C \) | 1.399 | 1.499 | 1.500 |
| Shape Factor \( S \) | 1.014 | 1.015 | 1.015 |
| Impulse Factor \( I \) | 1.389 | 1.489 | 1.490 |
The analysis of the dynamic response from the 12-DOF model leads to several definitive conclusions regarding the fault mechanism of a spur gear with a pitch circle crack. Firstly, the presence of the crack induces a characteristic, abrupt reduction in the mesh stiffness precisely when the cracked tooth segment near the pitch circle comes into contact. This stiffness drop is proportional to the crack depth. Secondly, this periodic stiffness excitation generates a corresponding periodic impulse in the time-domain vibration response of the spur and pinion system, with an interval matching the pinion’s rotation period. The amplitude of this impulse increases with crack severity. Thirdly, in the frequency domain, this periodicity manifests as sidebands around the mesh frequency and its harmonics, with spacing equal to the pinion’s rotational frequency. The amplitude of these sidebands also escalates with crack depth.
Finally, the dimensionless parameter analysis provides quantifiable metrics for fault detection. Kurtosis, Margin Factor, Peak Factor, and Impulse Factor all show a clear and sensitive increase from the healthy state to the cracked states, reacting to the induced impulsiveness. Among these, Kurtosis demonstrates a consistent rising trend with increasing crack depth, making it a particularly useful indicator for gauging fault severity. The Shape Factor remains largely unchanged, as it is less sensitive to transient impacts. Therefore, monitoring these dimensionless indicators, especially Kurtosis, in conjunction with traditional spectral analysis for sidebands, offers a robust theoretical foundation for the early detection and severity assessment of pitch circle cracks in spur gear transmission systems, ultimately contributing to improved reliability and predictive maintenance protocols.