Thin-web spiral bevel gears are used in helicopter transmissions where low mass and high power density are required at the same time. The web behaves like an annular disk with families of nodal-diameter modes. Because the teeth are curved and the pitch cone is steep, the mesh force has a substantial axial component. As the gear rotates, this axial force repeats at the tooth mesh frequency and its harmonics. When a harmonic approaches a nodal-diameter natural frequency, the disk can support a forward or backward traveling wave. The wave alternately strains the tooth root and the web, and a sustained response can lead to web cracking or fracture. The design question is therefore not only whether a natural frequency lies close to a mesh harmonic, but also how large the resonant strain becomes and where it occurs.
Resonance condition for nodal-diameter waves
For a gear with \(z\) teeth rotating at \(N\) r/min, the tooth mesh frequency is
$$f_m=\frac{zN}{60}.$$
A nodal-diameter mode with \(n\) nodal diameters has a forward wave that rotates in the direction of gear rotation and a backward wave that rotates against it. Resonance occurs when a mesh harmonic coincides with the traveling-wave frequency:
$$K_1\frac{zN}{60}=f_d\pm \frac{nN}{60},$$
where \(K_1\) is the harmonic order, \(f_d\) is the dynamic natural frequency of the gear, and \(n\) is the nodal diameter. Solving for the forward and backward resonant speeds gives
$$N_f=\frac{60f_d}{K_1z-n},\qquad N_b=\frac{60f_d}{K_1z+n}.$$
The dynamic frequency is slightly above the static frequency because of centrifugal stiffening. In the operating speed range considered here, the increase is below about 3%, so the static modal frequency is a practical substitute for initial screening. A resonance margin can be defined as
$$\phi=\left|\frac{f_g-f_j}{f_g}\right|\times100\%,$$
where \(f_g\) is the mesh frequency at resonance and \(f_j\) is the mesh frequency at the operating point. A small margin means that the operating speed is close to a nodal-diameter resonance. Plotting the mesh-order lines against the nodal-diameter frequencies in a Campbell diagram identifies the critical intersections.
Reconstructing the mesh load as a traveling-wave excitation
At the mid-face section, the tangential force transmitted by a torque \(T_0\) is
$$F_t=\frac{2T_0}{d_m},$$
where \(d_m\) is the mid-face pitch diameter. For a right-hand driven gear with right-hand rotation, the axial force has the form
$$F_x=\frac{2T_0}{d_m\cos\beta_m}\left(\tan\alpha\sin\delta-\sin\beta_m\cos\delta\right),$$
where \(\beta_m\) is the helix angle, \(\alpha\) is the pressure angle, and \(\delta\) is the pitch cone angle. The sign indicates direction; the magnitude is used as the static amplitude of the axial mesh load.
The load is shared between teeth according to the contact ratio. For a 20° pressure angle, the profile contact ratio can be written as
$$\varepsilon_\alpha=K\left(\varepsilon_{\alpha1}+\varepsilon_{\alpha2}\right),$$
where \(\varepsilon_{\alpha1}\) and \(\varepsilon_{\alpha2}\) are the profile contact ratios of the pinion and gear, and \(K\) is a coefficient obtained from the midpoint helix angle by cubic-spline interpolation. For pressure angles other than 20°, the profile contact ratios can be interpolated over equivalent tooth number. The overlap contact ratio is approximated by
$$\varepsilon_\beta\approx\frac{1}{1-0.5\phi_R}\frac{b\tan\beta_m}{\pi m},$$
with face width \(b\), face-width coefficient \(\phi_R\), and module \(m\). The total contact ratio is
$$\varepsilon_\gamma=\sqrt{\varepsilon_\alpha^2+\varepsilon_\beta^2}.$$
For the validation case, \(\varepsilon_\gamma=1.9214\). In single contact the axial tooth load is \(F_x\); in double contact the two teeth are assumed to share the load equally:
$$F=\begin{cases}
F_x, & \text{single contact},\\[4pt]
\frac{1}{2}F_x, & \text{double contact}.
\end{cases}$$
The tooth-pass period at the operating point is \(T=60/(Nz)=0.4976\) ms. The elementary load history is represented by a half-sine pulse,
$$F(t)=F_x\sin\left(\frac{\pi t}{T}\right),$$
applied over the active contact interval. The contact ratio determines how successive tooth pulses overlap and whether one or two teeth carry the load. This pulse train preserves the periodic, pulsating character of the mesh force without resolving the full contact process.
Expanding the pulse train in a Fourier series gives
$$F(t)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k\cos(k\omega t)+b_k\sin(k\omega t)\right].$$
Near a nodal-diameter resonance, the response is dominated by the mode whose nodal pattern matches the excitation. For the forced-response step, only the Fourier component at the resonant nodal-diameter frequency is retained. This is the component that can do net work on the selected traveling wave.
The forward and backward wave displacements can be written as
$$w(r,\theta,t)=\frac{1}{2}A(r)\cos\left(n\theta\pm\omega t\right),$$
where \(A(r)\) is the vibration amplitude at radius \(r\), \(\theta\) is the circumferential position, and the sign distinguishes the two wave directions. An energy check compares the work performed by the pressure pulse with the kinetic energy of the wave:
$$W_1=\int_{\theta_0}^{\theta_t}\int_0^t\int_{r_1}^{r_2}f(t)w(r,\theta,t)\,dr\,dt\,d\theta,$$
$$W_2=\int_{\theta_0}^{\theta_t}\int_{r_1}^{r_2}dW_2,\qquad dW_2=\frac{1}{2}dm’\left(\frac{\partial w}{\partial t}\right)^2,$$
$$dm’\approx\rho h r\,dr\,d\theta.$$
Here \(\rho\) is the material density and \(h\) is the web thickness. Because \(f(t)>0\), the work input over a cycle is much larger than the stored vibration energy at the considered amplitude, so the prescribed pulse train is not energy-limited.
Finite-element forced-response procedure
The gear is held stationary and the nodal-diameter pressure pattern is advanced around the circumference with the phase progression of the selected wave. This is a rotating-frame equivalent that avoids a full transient simulation of rotating teeth in contact. The gear and hollow shaft are assembled with a bonded interface, and the shaft boundary conditions are applied. The mesh uses tetrahedral elements; the validation model contained 288,203 elements.
The pulse train is applied to the loaded tooth surfaces, and the dynamic response is solved in the time domain. Strain is extracted at the rim and web, giving a full-field strain distribution rather than a single contact stress. Time-frequency analysis of the strain history identifies the nodal-diameter pattern, the apparent response frequency, and the migration of the strain wave. Compared with an explicit transient contact solution, this forced-response route removes contact convergence difficulties and long run times. It is an approximation intended for cases in which one nodal-diameter mode dominates the resonant response.
Case study: four-nodal-diameter forward wave
The validation case is a twin-input helicopter transmission spiral bevel gear. The input speed is 1,193.8 r/min and the total input power is 2,000 kW. The basic gear parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Number of teeth, \(z\) | 101 |
| Module, \(m\) | 3.85 mm |
| Helix angle, \(\beta_m\) | 25° |
| Pitch cone angle, \(\delta\) | 81.852° |
| Pressure angle, \(\alpha\) | 22.5° |
| Face width, \(b\) | 72 mm |
| Addendum coefficient | 1 |
| Hand and rotation | Right-hand, right rotation |
Modal analysis gave nodal-diameter frequencies of 530.28 Hz for 1ND, 555.28 Hz for 2ND, 1,122.1 Hz for 3ND, 1,891.7 Hz for 4ND, and 2,694.3 Hz for 5ND. The predicted resonances and margins are listed in Table 2. At 1,193.8 r/min the mesh frequency is 2,009.9 Hz. The 4ND forward wave has a resonance frequency of 1,969.7 Hz at 1,170 r/min, leaving a margin of only 1.98%. The 4ND backward wave has a 9.45% margin, while the 3ND margins exceed 42%. The 4ND forward wave is therefore the critical case.
| Nodal diameter | Wave direction | Resonance frequency \(f_g\) (Hz) | Resonant speed (r/min) | Margin \(\phi\) (%) |
|---|---|---|---|---|
| 3 | Backward | 1,089.7 | 647 | 45.7727 |
| 3 | Forward | 1,156.4 | 687 | 42.4527 |
| 4 | Backward | 1,819.6 | 1,081 | 9.4512 |
| 4 | Forward | 1,969.7 | 1,170 | 1.9833 |
The static axial load obtained from the mesh-force expression is 132.17 kN. With a total contact ratio of 1.9214, the tooth-pass period is 0.4976 ms, and the corresponding half-sine excitation is constructed. The forced-response solution produces a strain field with a clear four-lobe pattern. The maximum strain appears at the small-end tooth root, a critical region for bending fatigue in thin-web aviation spiral bevel gears. At the location corresponding to the later strain-gauge comparison point, the simulated strain is 226.16 × \(10^{-6}\).
Time-frequency analysis of the simulated strain gives a peak interval of 0.00023 s and an apparent strain-wave migration speed of 123.62 rad/s. The rotational frequency is 19.9 Hz and the mesh frequency is 2,009.9 Hz. The apparent 4ND response frequency is 1,929.98 Hz, close to \(f_m-4f_z=2,009.9-4(19.9)=1,929.9\) Hz and consistent with the modal 4ND value of 1,891.7 Hz.
Experimental validation
The response calculation was checked with a strain-gauge test on the gear assembly. Eight gauges were placed symmetrically around the circumference: two on the front rim, four on the front web near the rim, and two on the back rim near the large end. The gauges were oriented along the strain-gradient direction to maintain sensitivity. BAB250 series strain gauges rated for 250 °C were bonded with H610 high-temperature adhesive. The measured strain histories were processed by order analysis to separate the speed-related components.
At the comparison point, the measured mean strain was 240 × \(10^{-6}\), while the simulated value was 226.16 × \(10^{-6}\). The strain-response error is defined as
$$\zeta=\left|\frac{\varepsilon_s-\varepsilon_m}{\varepsilon_m}\right|\times100\%,$$
where \(\varepsilon_s\) is the simulated strain and \(\varepsilon_m\) is the measured strain. The largest discrepancy in the comparison is 5.77%. The measured 4ND frequency is 1,968 Hz, while the simulated modal value is 1,892 Hz, a difference of about 4%. The measured frequency is also close to the predicted 4ND forward-wave resonance of 1,969.7 Hz. The comparison is summarized in Table 3.
| Quantity | Simulation | Test | Difference |
|---|---|---|---|
| 4ND resonance frequency (Hz) | 1,892 | 1,968 | 4.0% |
| Strain at comparison point (× \(10^{-6}\)) | 226.16 | 240 (mean) | 5.77% |
The residual difference is consistent with gauge location and angle deviation, acquisition-system error, strain-gauge nonlinearity and sensitivity, and the assumptions of the forced-response model. Within these limits, the test supports the use of the half-sine excitation and finite-element forced-response procedure for engineering prediction of traveling-wave strain.
Effect of web holes
The validated procedure was then used to examine equally spaced web holes. The gear parameters and operating point were kept the same, and the hole count was varied from 0 to 6. The resulting 4ND natural frequencies, forward-wave resonance frequencies, and resonant speeds are listed in Table 4.
| Number of web holes | 4ND natural frequency (Hz) | 4ND forward-wave resonance frequency (Hz) | Resonant speed (r/min) |
|---|---|---|---|
| 0 | 1,891.7 | 1,969.7 | 1,170 |
| 2 | 1,879.8 | 1,957.3 | 1,162.8 |
| 4 | 1,879.2 | 1,956.7 | 1,162.4 |
| 6 | 1,878.2 | 1,955.7 | 1,161.8 |
The no-hole gear has the highest 4ND natural frequency. Adding web holes lowers both the natural frequency and the forward-wave resonance frequency. From 0 to 6 holes, the 4ND natural frequency falls by 13.5 Hz, or about 0.7%; the forward-wave resonance frequency falls by 14.0 Hz, also about 0.7%; and the resonant speed falls by 8.2 r/min. These changes are monotonic but small.
Because the frequency shift is small, the original operating point remains close to resonance. At 1,193.8 r/min the mesh frequency is 2,009.9 Hz, while the six-hole forward-wave resonance is 1,955.7 Hz, leaving a margin of about 2.8%. Hole count alone is therefore not an effective detuning measure for this geometry.
The location of the maximum strain is also stable. For all hole counts, the global maximum remains at the small-end tooth root. For gears with web holes, however, the local strain near the holes is elevated. The holes interrupt the hoop load path and introduce local compliance and strain concentration, but they do not move the global maximum of the resonant mode. If web holes are used for mass reduction, the local strain field around the holes should be checked in addition to the global resonance margin.
Limitations and practical use
The calculation is a forced-response approximation. It assumes equal load sharing between teeth in double contact and retains only the resonant Fourier component of the pulse train. It does not resolve friction, contact nonlinearity, mistuning, or interaction among several nodal-diameter modes. The stationary-gear formulation also treats centrifugal stiffening mainly through the dynamic-frequency correction rather than through a full rotating-frame solution. These assumptions are acceptable for screening and response ranking when a single nodal-diameter mode dominates, but transient contact analysis remains necessary when impact, nonlinear contact, or strongly coupled modes are important.
The validation is limited to one operating point and one gear geometry. The web-hole trend covers 0 to 6 equally spaced holes at the same load; other hole patterns, web thicknesses, or rim stiffnesses require a new forced-response calculation.
Summary
- The critical condition for the validation gear is the 4ND forward traveling wave, with a 1.98% resonance margin at 1,193.8 r/min.
- A half-sine pulse train combined with finite-element forced response predicts a strain of 226.16 × \(10^{-6}\) against a measured mean of 240 × \(10^{-6}\), a difference of 5.77%, and a frequency of 1,892 Hz against 1,968 Hz, a difference of about 4%.
- Increasing the number of equally spaced web holes from 0 to 6 lowers the 4ND natural frequency by about 0.7% and does not remove resonance risk. The global maximum strain remains at the small-end tooth root, while local strain near the holes increases.