The central drive of an aero-engine uses a spiral bevel gear to connect the engine rotor system with the accessory gearbox. Its service conditions combine high speed, heavy tooth load, and elevated temperature. Because the driven gear is a disk-like body with a relatively thin web, its nodal-diameter modes can be excited by the meshing force. Once a nodal-diameter mode splits into forward and backward traveling waves and one of those waves matches the tooth-mesh excitation, the gear enters traveling-wave resonance. The practical failure mode is not a localized tooth-surface defect but a circumferential fracture at the web root; in a fault-reproduction test, the driven gear fractured through twelve whole teeth together with the web. This makes the problem twofold: the resonance speed must be moved away from the operating range where possible, and the resonant stress at the web and tooth root must be kept within acceptable limits when full avoidance is impossible.
Mechanism of Nodal-Diameter Traveling-Wave Resonance
A disk-like structure can vibrate in nodal-circle, nodal-diameter, or combined modes. For a gear body, the nodal-diameter family is the relevant one because the web and rim form a disk-like elastic system. When the gear is stationary relative to the coordinate frame, the transverse displacement of a point on the web can be written as
$$ z(r,\theta,t)=A(r)\cos(m\theta)\cos(\omega t) $$
where \(z\) is the transverse displacement, \(A(r)\) is the radial amplitude, \(m\) is the number of nodal diameters, \(\theta\) is the circumferential angle, and \(\omega\) is the angular frequency. In a rotating gear, this standing pattern decomposes into two counter-rotating waves:
$$ z=\frac{1}{2}A(r)\left[\cos(m\theta-\omega t)+\cos(m\theta+\omega t)\right] $$
The two terms are the forward and backward traveling waves. Their frequencies in the fixed frame are
$$ f_f=f_d+\frac{i n m}{60},\qquad f_b=f_d-\frac{i n m}{60},\qquad i=\frac{Z_1}{Z_2} $$
where \(f_f\) and \(f_b\) are the forward- and backward-wave frequencies, \(f_d\) is the nodal-diameter dynamic frequency, \(n\) is the driving-gear speed in r/min, \(Z_1\) is the number of driving-gear teeth, and \(Z_2\) is the number of driven-gear teeth. The mesh excitation frequency is \(f_e=nZ_1/60\). Resonance occurs when \(f_e=f_f\) or \(f_e=f_b\). Solving for the driving speed gives
$$ n=\frac{60f_d}{i(Z_2\mp m)} $$
with the minus sign for the forward wave and the plus sign for the backward wave. For the gear considered here, \(Z_1=47\), \(Z_2=35\), and \(i=47/35\approx1.343\). The third and fourth nodal-diameter modes are the critical ones because their resonance speeds fall within the practical operating range.
The spiral bevel gear is not symmetric about the mid-plane of its web. Its front and back web surfaces therefore do not carry equal vibration stress. This asymmetry matters when stress is used to judge fatigue risk: a single “web stress” value is insufficient, and the front surface, back surface, and tooth root must be evaluated separately.
Finite Element Model and Bench Validation
The driven gear has 35 teeth and the driving gear has 47 teeth; the large-end module is 3.875 mm, the normal pressure angle is 20°, and the helix angle is 35°. The gears run in opposite hand, with normal backlash of 0.10–0.15 mm and normal mesh clearance of 0.15–0.35 mm. The gear material has a density of \(7.86\times10^3\ \mathrm{kg/m^3}\). Its elastic modulus is 215, 213, and 210 GPa at 20, 100, and 200 °C, respectively; Poisson’s ratio is 0.38, 0.39, and 0.41 over the same temperature range.
Modal analysis of the driven gear was performed without applied load. Fixed constraints were imposed on the bearing and support surfaces, and the mesh used hexahedron-dominated elements: 13,120 SOLID187 elements and 65,034 SOLID186 elements. The first 12 modes were extracted. The computed static nodal-diameter frequencies are listed in Table 1.
| Nodal-diameter mode | 1-ND | 2-ND | 3-ND | 4-ND |
|---|---|---|---|---|
| Frequency (Hz) | 3474.5 | 4510.9 | 7996.9 | 12768 |
Bench modal tests on three driven gears provided an independent check of the finite element model. The comparison is given in Table 2. The three-nodal-diameter error was below 2%, and the four-nodal-diameter error was below 5%. Because the static and dynamic frequencies differ only slightly, the static values were subsequently used in place of dynamic values to calculate forward- and backward-wave resonance frequencies and speeds.
| Gear | Mode | Simulation (Hz) | Test (Hz) | Error (%) |
|---|---|---|---|---|
| A | 3-ND | 7997 | 7964 | 0.41 |
| A | 4-ND | 12768 | 13402 | 4.73 |
| B | 3-ND | 7997 | 7928 | 0.87 |
| B | 4-ND | 12768 | 13360 | 4.43 |
| C | 3-ND | 7997 | 7908 | 1.13 |
| C | 4-ND | 12768 | 13320 | 4.14 |
Transient dynamics of the meshing pair were modeled with Hertzian contact. The contact was frictional with a coefficient of 0.05, the augmented Lagrangian method was used, and the contact stiffness was set to 1 after trial calculations. The driving gear carried the speed boundary condition and the driven gear carried a resisting torque, calculated from
$$ T_1=9.55\times10^6\frac{P}{iN} $$
where \(P\) is the load power in kW, \(N\) is the speed in r/min, and \(T_1\) is the torque in N·mm. The solver time was \(9\times10^{-5}\) s with an initial substep of 24. For a web-thickness ratio of 60% at 200 °C, the four-nodal-diameter backward-wave speed was 14,402 r/min and the corresponding torque was 98,759 N·mm. The simulated front-web stress was within a factor of 1.7 of the measured value, and the predicted fracture pattern agreed with the fault-reproduction result. This is an engineering-level stress estimate rather than a high-accuracy absolute stress measurement, but it is adequate for comparing parameter trends.
The Hertzian contact pressure used in the contact model is
$$ \sigma_{H,\max}=\sqrt{\frac{F\left(\frac{1}{\rho_1}+\frac{1}{\rho_2}\right)}{\pi w\left(\frac{1-\mu_1^2}{E_1}+\frac{1-\mu_2^2}{E_2}\right)}} $$
where \(F\) is the contact force, \(w\) is the face width, \(\rho_1\) and \(\rho_2\) are the local curvature radii, \(\mu_1\) and \(\mu_2\) are Poisson’s ratios, and \(E_1\) and \(E_2\) are elastic moduli.
Web Thickness and Temperature Sensitivity of Resonance
The full-web thickness of the driven gear is \(h=17.28\) mm. For a reduced web thickness \(h’\), the thickness ratio is defined as \(\xi=h’/h\). Seven models were evaluated: \(\xi=48\%,54\%,57\%,60\%,62\%,65\%\), and \(71\%\), corresponding to \(h’=8.28,9.28,9.78,10.28,10.78,11.28\), and \(12.28\) mm. The operating temperatures were 20, 60, 100, 150, and 200 °C. Table 3 summarizes the 20 °C results for the third and fourth nodal-diameter resonance speeds, together with the static frequencies.
| \(\xi\) (%) | 3-ND \(f_3\) (Hz) | 3-ND forward speed (r/min) | 3-ND backward speed (r/min) | 4-ND \(f_4\) (Hz) | 4-ND forward speed (r/min) | 4-ND backward speed (r/min) |
|---|---|---|---|---|---|---|
| 48 | 7496 | 10,466 | 8,814 | 12,147 | 17,508 | 13,916 |
| 54 | 7736 | 10,801 | 9,096 | 12,439 | 17,929 | 14,251 |
| 57 | 7864 | 10,980 | 9,247 | 12,600 | 18,161 | 14,435 |
| 60 | 7997 | 11,166 | 9,403 | 12,768 | 18,403 | 14,628 |
| 62 | 8135 | 11,359 | 9,565 | 12,946 | 18,659 | 14,832 |
| 65 | 8280 | 11,561 | 9,736 | 13,136 | 18,933 | 15,049 |
| 71 | 8589 | 11,993 | 10,099 | 13,548 | 19,527 | 15,521 |
At 20 °C, increasing \(\xi\) from 48% to 71% raised the three-nodal-diameter forward-wave frequency from 8198 to 9395 Hz and the backward-wave frequency from 6904 to 7911 Hz. The corresponding forward speed increased by 1527 r/min and the backward speed by 1285 r/min. For the four-nodal-diameter mode, the forward-wave frequency rose from 13,714 to 15,296 Hz and the backward-wave frequency from 10,901 to 12,158 Hz; the forward speed increased by 2019 r/min and the backward speed by 1605 r/min. The upward shift indicates that the stiffness increase from a thicker web outweighs the accompanying mass increase.
Temperature acts in the opposite direction. At \(\xi=60\%\), raising the temperature from 20 to 200 °C reduced the three-nodal-diameter static frequency from 7997 to 7874 Hz, the forward-wave frequency from 8747 to 8612 Hz, and the backward-wave frequency from 7366 to 7252 Hz. The forward speed fell by 172 r/min and the backward speed by 145 r/min. For the four-nodal-diameter mode, the static frequency fell from 12,768 to 12,571 Hz, the forward-wave frequency from 14,415 to 14,193 Hz, and the backward-wave frequency from 11,458 to 11,282 Hz; the forward speed fell by 284 r/min and the backward speed by 226 r/min. The comparison in Table 4 shows the same trend over the full temperature range at \(\xi=60\%\).
| Temperature (°C) | 3-ND \(f_3\) (Hz) | 3-ND forward speed (r/min) | 3-ND backward speed (r/min) | 4-ND \(f_4\) (Hz) | 4-ND forward speed (r/min) | 4-ND backward speed (r/min) |
|---|---|---|---|---|---|---|
| 20 | 7997 | 11,166 | 9,403 | 12,768 | 18,403 | 14,628 |
| 60 | 7973 | 11,133 | 9,375 | 12,730 | 18,348 | 14,584 |
| 100 | 7949 | 11,099 | 9,347 | 12,692 | 18,293 | 14,541 |
| 150 | 7904 | 11,036 | 9,293 | 12,619 | 18,188 | 14,457 |
| 200 | 7874 | 10,994 | 9,258 | 12,571 | 18,119 | 14,402 |
The two design variables therefore act on different scales. Web thickness changes the resonance speed by roughly 1,500–2,000 r/min over the studied range, whereas a 180 °C temperature rise changes it by only about 145–284 r/min. Temperature is still relevant because it lowers the resonance speed and because the elastic modulus decreases with temperature, but web thickness is the stronger geometric lever for moving a third- or fourth-nodal-diameter resonance out of the operating range.
Stress Distribution Under Resonant Operation
Stress was extracted at three locations on the driven gear: the web front surface, the web back surface, and the tooth root. Under both third- and fourth-nodal-diameter resonance, the fourth-nodal-diameter stresses were higher than the third-nodal-diameter stresses at the corresponding locations. At both modes, the stress magnitude decreased in the order tooth root, web back, web front. The tooth root is therefore the most heavily loaded region, while the web front is the least sensitive indicator of resonant severity.
Table 5 summarizes the stress changes produced by three parameter sweeps. The load sweep ran from 150 to 250 kW at 100 °C and \(\xi=60\%\). The temperature sweep ran from 20 to 200 °C at 210 kW and \(\xi=60\%\). The damping sweep ran from 0.03 to 0.17 at 100 °C, 210 kW, and \(\xi=60\%\).
| Parameter change | 3-ND front | 3-ND back | 3-ND root | 4-ND front | 4-ND back | 4-ND root |
|---|---|---|---|---|---|---|
| Load power 150 → 250 kW | +1.2 | +49.1 | +32.6 | +17.7 | +71.1 | +60.2 |
| Temperature 20 → 200 °C | +2.0 | +42.3 | +42.4 | +13.1 | +50.0 | +38.7 |
| Damping coefficient 0.03 → 0.17 | −4.0 | −11.4 | −33.8 | −6.4 | −63.4 | −32.7 |
Increasing load power raises all three stress measures, but not equally. In the three-nodal-diameter mode, the web front stress increased by only 1.2 MPa, while the web back increased by 49.1 MPa and the tooth root by 32.6 MPa. In the four-nodal-diameter mode, the corresponding increases were 17.7, 71.1, and 60.2 MPa. The web back is thus more load-sensitive than the web front, and the fourth-nodal-diameter condition amplifies the load effect.
Increasing temperature also raises stress. From 20 to 200 °C at 210 kW, the three-nodal-diameter web front stress increased by 2.0 MPa, the web back by 42.3 MPa, and the tooth root by 42.4 MPa. Under four-nodal-diameter resonance, the increases were 13.1, 50.0, and 38.7 MPa. The temperature effect is smaller than the load effect in the web front but remains substantial in the web back and tooth root.
Increasing damping reduces stress. From 0.03 to 0.17, the three-nodal-diameter web front stress fell by 4.0 MPa, the web back by 11.4 MPa, and the tooth root by 33.8 MPa. Under four-nodal-diameter resonance, the reductions were 6.4, 63.4, and 32.7 MPa. Among the three parameters, damping produced the strongest relative change in the web back and tooth root, which are also the locations with the highest stress. The web front remains relatively insensitive throughout.
Design Implications and Limits
The third- and fourth-nodal-diameter traveling-wave resonances are the governing dynamic cases for this driven spiral bevel gear. A thicker web shifts both the resonance frequency and the resonance speed upward; a higher temperature shifts them downward. Because the web-thickness effect is much larger, a design change that adjusts web thickness within the allowable strength and stiffness envelope can move the third- and fourth-nodal-diameter resonance speeds away from the normal operating range. Temperature must still be included in the final check, especially at the upper end of the thermal envelope, because it reduces the resonance speed and changes the material modulus.
When resonance cannot be fully avoided, stress control should focus on the tooth root and the web back rather than the web front. Load reduction and cooling both lower the resonant stress, but increased damping is the most effective of the three parameters examined, particularly for the web back and tooth root. The four-nodal-diameter condition produces higher stress than the three-nodal-diameter condition, so high-speed resonance carries a greater fatigue-fracture risk. The fault-reproduction result is consistent with this: sound gears had sufficient life within the four-nodal-diameter resonant speed range, whereas gears with initial defects could fracture through the teeth and web.
The quantitative conclusions apply to the tested geometry, material, and parameter ranges. The modal frequencies are static values used in place of dynamic values because the difference is small, and the transient stress prediction is an engineering estimate with an error within a factor of 1.7. These approximations are acceptable for ranking parameter effects and for guiding web-thickness selection, but absolute life predictions would require a more detailed dynamic stress and defect assessment.