In modern aero-engine development, the push for higher power-to-weight ratios has necessitated the design of transmission systems that are lightweight yet capable of handling significant loads at extreme rotational speeds. Within this context, the accessory drive system, particularly the bevel gears responsible for power off-take, plays a critical role. The failure of such a component can lead to catastrophic outcomes. Historically, vibration and noise issues stemming from gear dynamics have emerged as a significant bottleneck. This research focuses on the experimental characterization of traveling wave resonance in a high-speed driven bevel gear from a turboshaft engine’s accessory drive train. Our aim was to directly measure its vibrational behavior under operational conditions to provide essential data for validating design models and ensuring structural integrity.
The core challenge with high-speed rotating disk-like structures, such as the web of a bevel gear, is the excitation of nodal diameter vibration modes. Unlike stationary structures, these modes manifest as traveling waves when the gear’s rotational speed couples with its natural frequencies. A standing vibration pattern with *m* nodal diameters can be decomposed into two counter-rotating traveling waves: a forward traveling wave (FTW) moving in the direction of rotation and a backward traveling wave (BTW) moving against it. The resonance conditions are defined by the coincidence of the excitation frequency (often a multiple of the gear mesh frequency) with these traveling wave frequencies. The relationship is given by:
$$f_F = f_D + m n_F / 60$$
$$f_B = f_D – m n_B / 60$$
where $f_D$ is the natural frequency of the *m*-nodal diameter mode in a stationary frame (the dynamic frequency), $n_F$ and $n_B$ are the rotational speeds (in rpm) at which forward and backward traveling wave resonance occur, and $f_F$, $f_B$ are the corresponding resonant frequencies. For bevel gears operating at speeds often exceeding 30,000 rpm, avoiding all such resonances within the operational envelope is nearly impossible, making their identification and stress measurement paramount.
Prior to any physical testing, a comprehensive finite element analysis (FEA) was conducted to predict the modal characteristics of the driven bevel gear. The 3D model was constrained using grounded springs at bearing locations to simulate support stiffness. The Campbell diagram, plotting natural frequencies against rotational speed, revealed numerous intersections between the gear’s natural frequency lines (for modes with 2 to 5 nodal diameters) and potential excitation lines (like the 1X and 43X engine order lines) within the operating range. This simulation clearly indicated a high risk of exciting traveling wave vibrations in these bevel gears. The stress distribution for these modes showed that high dynamic stresses for the 3rd to 5th nodal diameter modes were concentrated at the fillet region on the convex side of the gear teeth at the small end of the bevel gear.
To capture the vibrational response experimentally, two complementary techniques were employed: acoustic noise measurement and dynamic strain measurement. The accessory gearbox presented a highly complex acoustic environment with multiple noise sources. To isolate the sound emanating specifically from the target bevel gear, an acoustic waveguide technique was adopted. A straight copper tube with an internal diameter of 8 mm and a length of 200 mm was installed in the gearbox casing, with its open end positioned close to the gear of interest. A microphone was fitted inside this tube. This setup exploits the cutoff frequency principle of a circular duct, which for our tube was approximately 25 kHz. Frequencies below this cutoff propagate predominantly as plane waves, effectively filtering out incoherent high-frequency noise from other sources and drastically improving the signal-to-noise ratio for monitoring the target bevel gear’s resonant responses. This method is highly effective for identifying resonant speeds and the precise frequencies of traveling wave vibrations.
While noise measurement identifies resonant conditions, it does not directly quantify the associated structural stress. For this, a dynamic strain gauge technique was implemented on a separate engine test. This was a considerably more complex endeavor due to the need for signal transmission from the rotating bevel gear. Miniature strain gauges were installed at three locations on the gear web, positioned radially at the same distance near the tooth root fillet on the convex side—a region identified by FEA as a high-stress area for higher nodal diameter modes. The wiring from these bevel gear strain gauges was carefully routed through the hollow gear shaft and connected to a slip-ring assembly mounted on a modified stationary shaft. This allowed continuous transmission of the dynamic strain signals to the data acquisition system during engine operation. It is crucial to note that a strain gauge rotating with the bevel gear measures the dynamic frequency $f_D$. When a traveling wave resonance occurs, the gauge, being in the rotating frame, does not directly measure $f_F$ or $f_B$, but rather captures the peak dynamic response at the resonant speed, from which $f_D$ can be derived.
The tests were executed following carefully designed speed ramps. The noise test was performed on a dedicated gearbox test rig, while the dynamic stress test was conducted on a full engine. The results from both methods were strikingly consistent and validated the simulation predictions.
The time-frequency spectrogram from the acoustic test revealed distinct resonant tracks corresponding to the 2nd through 5th nodal diameter traveling waves. The resonant frequencies and speeds were extracted from these tracks. The dynamic strain data, presented as waterfall plots and order-tracked spectra, showed clear peaks in stress response at specific speeds. The following table summarizes the peak dynamic stress measured at one of the strain gauge locations during resonant events:
| Resonant Speed (kr/min) | Dynamic Freq. $f_D$ (kHz) | Dynamic Stress (MPa) | Harmonic (Engine Order) | Mode Identified |
|---|---|---|---|---|
| 30.75 | 20.00 | 38 | 39 | 4th ND, FTW |
| 18.44 | 12.27 | 47 | 40 | 3rd ND, FTW |
| 25.70 | 20.07 | 20 | 47 | 4th ND, BTW |
| 35.84 | 28.65 | 112 | 48 | 5th ND, BTW |
For the 5th nodal diameter backward traveling wave resonance, the measured dynamic stress was approximately 115 MPa across the gauges, representing the most severe vibratory stress condition identified for these bevel gears within the operating range.
A critical step was correlating the findings from the two independent test methods and with the simulation. Using the dynamic frequency ($f_D$) and resonant speed ($n$) from the strain gauge test, the traveling wave frequencies were calculated using the formulas above. These calculated frequencies showed excellent agreement with the mesh-frequency harmonics (43X of the driven bevel gear speed) that excited the resonance, and also with the frequencies directly measured by the acoustic method. The comparison is consolidated below:
| Vibration Mode | Simulation $f_D$ / $f_{F,B}$ (kHz) | Acoustic Test $f_{F,B}$ (kHz) | Strain Test $f_D$ / Calc. $f_{F,B}$ (kHz) | Freq. Error (Sim. vs. Noise Test) | Freq. Error (Sim. vs. Strain Test) |
|---|---|---|---|---|---|
| 2nd ND FTW | 6.680 | 6.710 | 6.660 | 0.4% | -0.3% |
| 2nd ND BTW | 6.101 | 6.016 | 6.071 | -1.4% | -0.5% |
| 3rd ND FTW | 12.846 | 13.210 | 13.195 | 2.8% | 2.7% |
| 3rd ND BTW | 11.171 | 11.520 | 11.471 | 3.0% | 2.6% |
| 4th ND FTW | 21.454 | 22.110 | 22.054 | 3.0% | 2.7% |
| 4th ND BTW | 17.802 | 18.400 | 18.291 | 3.3% | 2.7% |
| 5th ND BTW | 24.981 | 25.920 | 25.666 | 3.6% | 2.7% |
The analysis confirms that the tested bevel gears exhibit multiple traveling wave resonance modes within their operational speed range. The acoustic waveguide method proved highly effective in non-intrusively identifying the resonant speeds and the absolute traveling wave frequencies with high clarity. The dynamic strain measurement, though more challenging to implement, provided the essential quantitative data on vibratory stress levels. The close agreement between the two experimental methods, and their consistency with the finite element simulation (with frequency errors generally below 3.6%), validates the overall approach. The results underscore that for this class of high-speed bevel gears, the 5th nodal diameter backward traveling wave resonance presents a critical design consideration, as it occurs within the normal operating envelope and generates the highest measured dynamic stress. This integrated test strategy provides a robust framework for the experimental validation and refinement of future aero-engine bevel gear designs.