The relentless pursuit of higher power-to-weight ratios in modern aero-engines imposes severe demands on ancillary drive systems, necessitating transmissions that are lightweight, compact, and capable of operating under high speeds and heavy loads. Within this context, the vibration and dynamic integrity of gear systems emerge as critical factors limiting performance and reliability. Among these, the high-speed bevel gear, often employed in accessory drive trains, presents a significant design challenge. Its complex geometry and operating conditions make it particularly susceptible to resonant vibrations, with travelling wave resonance of nodal diameters being a predominant and dangerous failure mode. This phenomenon occurs when the mesh frequency or its harmonics coincide with the natural frequencies of the bevel gear’s disc-like structure, potentially leading to catastrophic fatigue failure.
Traditional methods for assessing gear vibration, such as strain gauge measurements, offer direct stress data but are limited in scope due to the finite number of measurement points that can be practically instrumented on a rotating bevel gear. Consequently, critical modal responses may be missed. Conversely, acoustic noise measurement offers a global, non-contact monitoring technique but is often plagued by low signal-to-noise ratios in complex gearbox environments where multiple acoustic sources coexist. This study presents a comprehensive experimental investigation combining a novel acoustic measurement technique based on an acoustic waveguide with conventional rotating strain gauge telemetry. The objective is to fully characterize the travelling wave resonance behavior of a high-speed accessory drive bevel gear under operational conditions, providing validated data to inform and improve future design practices for these critical components.
Theoretical Foundation of Travelling Wave Vibration
For a rotating, centrally-clamped structure like a bevel gear disc, the natural modes of vibration can be categorized into nodal circle types, nodal diameter types, and compound types. For a high-speed bevel gear, the excitation from gear meshing—specifically the tooth passing frequency and its harmonics—most readily excites the nodal diameter modes. The vibration displacement at any point on the web of the bevel gear for a mode with \(m\) nodal diameters can be mathematically described. Consider a point on the bevel gear web. After the gear rotates through an angle \(\theta\), the corresponding point on the modal deflection curve for the \(m\)-nodal diameter mode will have experienced a phase shift of \(m\theta\). The transverse vibration displacement \(y\) at a radial location \(r\) can be expressed as:
$$y(r, \theta, t) = B(r) \cos(m\theta) \cos(\omega_m t)$$
Where \(B(r)\) is the radial amplitude function, \(\omega_m = 2\pi f_{D_m}\) is the natural angular frequency (the dynamic frequency) of the \(m\)-nodal diameter mode in the rotating frame, and \(t\) is time. Using a trigonometric identity, this equation can be decomposed into two travelling waves:
$$y(r, \theta, t) = \frac{1}{2}B(r)\left[\cos(\omega_m t + m\theta) + \cos(\omega_m t – m\theta)\right]$$
This decomposition reveals that the standing wave pattern of an \(m\)-nodal diameter mode is equivalent to the superposition of two waves travelling in opposite directions relative to the bevel gear. One travels in the direction of rotation (Forward Travelling Wave, FTW) and the other against it (Backward Travelling Wave, BTW). Their respective resonance frequencies, as observed in a stationary (non-rotating) reference frame, are given by:
$$f_{F_m} = f_{D_m} + \frac{m \cdot n_F}{60}$$
$$f_{B_m} = f_{D_m} – \frac{m \cdot n_B}{60}$$
Here, \(f_{F_m}\) and \(f_{B_m}\) are the forward and backward travelling wave resonance frequencies, \(f_{D_m}\) is the dynamic frequency of the bevel gear mode, and \(n_F\) and \(n_B\) (in revolutions per minute) are the rotational speeds of the bevel gear at which resonance occurs for the forward and backward waves, respectively. Resonance is excited when the gear mesh frequency \(f_{mesh} = (N \cdot n)/60\) (where \(N\) is the number of teeth) or its integer harmonic coincides with \(f_{F_m}\) or \(f_{B_m}\).
Numerical Simulation for Bevel Gear Dynamics
Prior to experimental validation, a finite element analysis (FEA) was conducted to predict the dynamic characteristics of the subject bevel gear. A detailed three-dimensional model of the bevel gear was constructed and discretized using tetrahedral elements. The bearing supports were simulated using grounded spring elements at the appropriate locations, with stiffness values representative of the actual bearings. Constraints were applied to the internal spline to reflect the mounting conditions. A modal analysis was performed to extract the natural frequencies and mode shapes, followed by the construction of a Campbell diagram to identify potential resonance crossings within the engine’s operational speed range.
The Campbell diagram analysis revealed several intersections between the excitation lines (primarily the 1st order and the 43rd mesh order, corresponding to the bevel gear’s tooth count) and the predicted modal frequency lines within the operating envelope. This confirmed the high susceptibility of this bevel gear design to travelling wave resonance. The mode shapes for the 2nd through 5th nodal diameter modes were examined. The analysis indicated that for the 3rd, 4th, and 5th nodal diameter modes, areas of high dynamic stress were concentrated in the fillet region on the convex side of the teeth at the smaller end of the bevel gear. This stress concentration informed the subsequent placement of strain gauges during experimental testing.
Experimental Methodology for Bevel Gear Vibration Characterization
Two complementary experimental techniques were employed to capture the dynamic behavior of the bevel gear: an acoustic noise measurement method and a rotating dynamic stress measurement technique. The tests were conducted on separate, nominally identical bevel gears. The noise test was performed on a dedicated gearbox test rig, while the dynamic stress test was integrated into a full engine test to capture the effects of the real operational environment.
Acoustic Noise Measurement via Acoustic Waveguide
The accessory gearbox environment is acoustically complex, with multiple gear pairs generating significant broadband noise. To isolate the sound radiation from the specific bevel gear of interest and improve the signal-to-noise ratio for its resonant frequencies, an acoustic waveguide was employed. A straight tube made of copper, with an internal diameter of 8 mm and a length of 200 mm, was installed in the gearbox casing, with its inlet positioned close to the bevel gear. A high-frequency pressure microphone was mounted at the outer end of the tube.
The waveguide acts as a high-pass filter for acoustic modes. For a circular duct, the cutoff frequency \(f_c\) below which only plane waves can propagate is given by:
$$f_c = \frac{1.84 \cdot c_0}{2\pi a}$$
where \(c_0\) is the speed of sound in air (approximately 340 m/s) and \(a\) is the radius of the duct. For the 8 mm diameter tube, this yields a cutoff frequency of approximately 25 kHz. Acoustic energy from the bevel gear at frequencies below \(f_c\) propagates as a plane wave with minimal attenuation, while energy from other, more distant sources is attenuated. Frequencies above \(f_c\) can propagate but with a more complex modal structure. The attenuation introduced by the waveguide was quantified and found not to obscure the primary spectral features of the bevel gear’s resonance noise.
Rotating Dynamic Stress Measurement via Telemetry
To obtain direct quantitative data on dynamic stresses, strain gauges were installed on the rotating bevel gear. Based on the FEA results, three strain gauges were mounted at identical radial positions on the bevel gear web, close to the tooth root fillet on the convex side, and spaced circumferentially. This arrangement was designed to capture the bending stress associated with the radial-dominated nodal diameter vibrations. The signals from the rotating gauges were transmitted via precision slip-ring units mounted on a dedicated shaft extension. The signals were then conditioned and recorded using high-speed dynamic strain amplifiers and data acquisition systems. The bevel gear was run through a carefully controlled speed ramp, and data was acquired across the entire operational range.
Analysis of Experimental Results for Bevel Gear Vibration
The data from both experimental methods were processed and analyzed to identify resonant conditions and characterize the dynamic response of the bevel gear.
Acoustic Noise Test Results
The time-frequency spectrogram (rpm vs. frequency) from the acoustic test clearly revealed distinct hyperbolic tracks characteristic of travelling wave resonance. The excitation source, being the 38th order of the driving pinion speed, was evident. Multiple resonance crossings were identified, corresponding to the forward and backward travelling waves of the 2nd, 3rd, 4th, and 5th nodal diameter modes of the bevel gear. The resonance frequencies at specific speeds were extracted from the spectral data. For example, the spectrum at a resonance condition showed a dominant peak corresponding precisely to the calculated mesh frequency, confirming the excitation mechanism.
Dynamic Stress Test Results
The dynamic strain data from the rotating bevel gear provided direct evidence of resonant stress amplification. A waterfall plot of strain versus speed and frequency showed clear ridges along the predicted dynamic frequency lines (\(f_{D_m}\)) for the 2nd through 5th nodal diameter modes. An order-tracked analysis focusing on the 48th engine order (closely aligned with the bevel gear’s mesh harmonics) revealed sharp strain peaks at specific speeds. The three strain gauges showed consistent results, with all recording significant dynamic stress peaks at the resonance speeds for the 3rd nodal diameter forward wave, the 4th nodal diameter forward and backward waves, and the 5th nodal diameter backward wave. The maximum oscillatory stress amplitude, approximately 115 MPa, was observed during the 5th nodal diameter backward travelling wave resonance.
Since the strain gauges rotate with the bevel gear, they measure the dynamic frequency \(f_{D_m}\) directly when resonance occurs. The corresponding travelling wave resonance frequencies can be calculated using the measured \(f_{D_m}\) and the resonant speed \(n\), via the equations \(f_{F/B} = f_{D_m} \pm (m \cdot n)/60\). These calculated frequencies showed excellent agreement with the mesh frequency at that specific speed, validating the identification of the resonance event.
Synthesis and Comparison of Results
The results from the acoustic and dynamic stress methods were cross-validated and compared with the initial FEA predictions. The table below summarizes the key resonance frequencies identified by the different methods for the bevel gear under investigation.
| Mode Description | Simulation (kHz) | Acoustic Test (kHz) | Dynamic Stress Test (kHz) | Relative Error (Sim vs. Noise) | Relative Error (Sim vs. Stress) |
|---|---|---|---|---|---|
| 2nd ND Backward Wave | 6.10 | 6.02 | 6.07 | 1.3% | 0.5% |
| 3rd ND Forward Wave | 12.85 | 13.21 | 13.20 | 2.8% | 2.7% |
| 4th ND Backward Wave | 17.80 | 18.40 | 18.29 | 3.4% | 2.8% |
| 5th ND Backward Wave | 24.98 | 25.92 | 25.67 | 3.8% | 2.8% |
The agreement between the two experimental techniques is remarkable, with resonance frequency differences generally below 1% and resonant speed differences below 2%. Both experimental sets show good correlation with the simulation, with frequency errors typically within 3-4%, which is considered acceptable given the complexities of modeling boundary conditions and material properties for a bevel gear. The data confirms that within the operational speed range, this bevel gear design is subject to multiple travelling wave resonances, from the 2nd to the 5th nodal diameter.
Conclusions
This integrated experimental study successfully characterized the travelling wave resonance behavior of a high-speed aerospace bevel gear. The acoustic waveguide method proved highly effective in isolating the target bevel gear’s noise signature within a complex gearbox, enabling clear identification of resonance speeds and the corresponding travelling wave frequencies (\(f_F\), \(f_B\)). The rotating strain gauge telemetry provided direct, quantitative measurement of dynamic stress amplitudes at resonance and confirmed the dynamic frequency (\(f_D\)) of the bevel gear modes.
The key findings are: The subject bevel gear exhibits several nodal diameter mode resonances within its operational speed envelope. Significant dynamic stress responses were measured for the 3rd nodal diameter forward wave, the 4th nodal diameter forward and backward waves, and the 5th nodal diameter backward wave. The most critical condition observed was the 5th nodal diameter backward travelling wave resonance, which generated the highest oscillatory stress amplitude of approximately 115 MPa. The close agreement between the independent acoustic and stress measurement techniques, and their reasonable correlation with numerical simulation, validates the methodologies employed. The results underscore the necessity for detailed dynamic analysis in the design of high-speed bevel gears and provide a valuable dataset for benchmarking future simulation models. The location of the 5th nodal diameter resonance firmly within the operating range highlights a specific design concern that requires mitigation, such as precise frequency tuning or damping strategies, to ensure the long-term structural integrity and reliability of the bevel gear and the broader accessory drive system.