Spiral bevel gears are widely used in aeroengine central and accessory drives because their curved teeth combine high load capacity, a large contact ratio, smooth motion, good reversing behavior, and relatively low noise. These advantages do not eliminate the internal excitations of a real transmission. Time-varying mesh stiffness, transmission error, bearing clearance, and tooth backlash make the mesh force piecewise and nonlinear. Under high-speed and heavy-load operation, the resulting vibration can change qualitatively, not merely in amplitude.
Backlash is often treated as a static assembly clearance, but in a dynamic model it defines the contact state. While the mesh-line displacement remains inside the clearance, the elastic contact force vanishes; when the displacement crosses the clearance boundary, contact is restored with an impact. The same gear pair can therefore move from periodic to multi-periodic and chaotic response as backlash, mesh frequency, or support stiffness changes. The relevant design question is not only how large the vibration is, but which contact regimes the system can enter during operation.
Model Structure and Nonlinear Mesh Force
The system is represented by an eight-degree-of-freedom lumped-parameter model. Each gear has three translational coordinates and one rotational coordinate, giving a bending–torsion–axial coupled formulation. Bearings are modeled as linear springs and dampers, while the mesh is represented by a time-varying stiffness \(k(t)\), a mesh damping \(c_h\), and a backlash function. Tooth friction and the spiral-contact effect are neglected in this formulation, so the model isolates the influence of the principal internal excitations.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 28 | 34 |
| Module / mm | 4.2 | 4.2 |
| Helix angle / (°) | 25 | 25 |
| Pressure angle / (°) | 20 | 20 |
| Shaft angle / (°) | 90 | |
| Hand | Right | Left |
| Rotational inertia / (kg·m²) | 1.7992 × 10−3 | 3.0367 × 10−3 |
| Equivalent mass / kg | 0.5024 | 0.6323 |
The mesh equivalent mass is 0.2855 kg. The generalized coordinate vector is
$$
\mathbf{q}=[X_p,\;Y_p,\;Z_p,\;\theta_p,\;X_g,\;Y_g,\;Z_g,\;\theta_g]^T ,
$$
where subscripts \(p\) and \(g\) denote pinion and gear, \(X_i,Y_i,Z_i\) are translational displacements, and \(\theta_i\) is the torsional displacement. The equations of motion can be written compactly as
$$
\begin{aligned}
m_i\ddot X_i+c_{ix}\dot X_i+k_{ix}X_i &= \sigma_i F_x,\\
m_i\ddot Y_i+c_{iy}\dot Y_i+k_{iy}Y_i &= \sigma_i F_y,\\
m_i\ddot Z_i+c_{iz}\dot Z_i+k_{iz}Z_i &= \sigma_i F_z,\\
J_i\ddot\theta_i &= \sigma_i\left(T_i-F_z r_i\right),
\end{aligned}
$$
with \(i=p,g\), \(\sigma_p=+1\), \(\sigma_g=-1\). Here \(m_i\) is the equivalent mass, \(J_i\) is the rotational inertia, \(c_{ij}\) and \(k_{ij}\) are support damping and stiffness, \(T_i\) is the driving or load torque, and \(r_i\) is the mid-face base radius. The normal mesh force \(F_n\) is resolved into three coordinate directions through the normal pressure angle \(\alpha_n\), the helix angle \(\beta_m\), and the pinion cone angle \(\delta_p\):
$$
\begin{aligned}
F_x &= F_n\left(\cos\alpha_n\sin\beta_m\cos\delta_p+\sin\alpha_n\sin\delta_p\right),\\
F_y &= F_n\left(\cos\alpha_n\sin\beta_m\sin\delta_p-\sin\alpha_n\cos\delta_p\right),\\
F_z &= -F_n\cos\alpha_n\cos\beta_m.
\end{aligned}
$$
The mesh-line relative displacement includes the translational and torsional contributions of both gears and the normal transmission error \(e_n(t)\):
$$
\lambda_n=-a_1(X_p-X_g)-a_2(Y_p-Y_g)-a_3(Z_p-Z_g)-a_3(\theta_p r_p-\theta_g r_g)-e_n(t),
$$
where the geometric coefficients are
$$
\begin{aligned}
a_1 &= \cos\alpha_n\sin\beta_m\cos\delta_p+\sin\alpha_n\sin\delta_p,\\
a_2 &= \cos\alpha_n\sin\beta_m\sin\delta_p-\sin\alpha_n\cos\delta_p,\\
a_3 &= \cos\alpha_n\cos\beta_m.
\end{aligned}
$$
The dynamic mesh force is then
$$
F_n=k(t)f(\lambda_n)+c_h\dot\lambda_n,
$$
and the backlash function is
$$
f(\lambda_n)=
\begin{cases}
\lambda_n-b, & \lambda_n\gt b,\\
0, & |\lambda_n|\le b,\\
\lambda_n+b, & \lambda_n\lt -b,
\end{cases}
$$
where \(b\) is half the normal average mesh clearance. This piecewise function is the main source of contact switching. Inside the clearance band, the elastic mesh force is zero; outside it, the restoring force changes linearly with the penetration. The equations are non-dimensionalized using the half-clearance \(b_c\) and the time scale \(\tau=\omega_n t\), with \(\omega_n=\sqrt{k_m/m_e}\), where \(k_m\) is the mean mesh stiffness and \(m_e\) is the mesh equivalent mass. The resulting second-order system is reduced to 14 first-order state equations and integrated with a Runge–Kutta method. The baseline dimensionless support stiffness and damping are 1.25 and 0.01, and the mesh damping ratio is 0.1, within the typical 0.03–0.17 range.
Response Classification and Baseline at Cruise
Response regimes are identified from time histories, phase-plane portraits, and Poincaré maps. A closed phase trajectory with one point in the Poincaré section corresponds to period-1 motion; two or three isolated points correspond to period-2 or period-3 motion; a filled, non-closing set is treated as chaotic. The standard deviation of the steady mesh-line displacement is also used as a scalar indicator of fluctuation. In the cases examined, the period count and the standard deviation increase together: stronger nonlinearity produces larger and less regular vibration.
For the representative aeroengine central drive, cruise operation is taken as an input speed of 13,175 r/min, corresponding to a dimensionless mesh frequency \(\omega_H=0.2071\). Under this condition, the backlash parameter changes the response as summarized below.
| Backlash parameter \(b\) / µm | Poincaré signature | Response regime |
|---|---|---|
| 80 | One isolated point | Period-1 |
| 100 | Two isolated points | Period-2 |
| 120 | Three isolated points | Period-3 |
| 160 | Filled, non-repeating set | Chaotic |
Between 80 and 140 µm, the period count and standard deviation grow relatively slowly. Between 140 and 160 µm, they rise sharply; at \(b=160\) µm the phase trajectory becomes complex and non-closing, and the Poincaré map is filled with scattered points. Under cruise conditions, a 160 µm backlash parameter therefore places the system in a chaotic regime and should be avoided in the design of this transmission.
Backlash: Contact Loss and Impact
The physical role of backlash is to enlarge the interval in which the teeth can separate. A larger clearance allows a longer free-flight segment before re-engagement, so the relative velocity at impact and the resulting force discontinuity become larger. The mesh stiffness path is repeatedly switched on and off, and the system no longer behaves as a weakly perturbed linear oscillator. Instead, the contact state itself becomes part of the dynamics.
For the studied pair, increasing \(b\) from 80 to 160 µm changes the response from period-1 to period-2, then period-3, and finally to chaos. The largest sensitivity appears in the 140–160 µm interval. This means that backlash should not be selected solely from a static clearance or assembly tolerance perspective. The relevant quantity is the dynamic mesh-line displacement relative to the clearance boundary: if the displacement repeatedly crosses that boundary, impact and contact loss dominate even when the nominal clearance appears modest.
Mesh Frequency: Resonance and Nonlinear Amplification
Mesh frequency changes the phasing of the internal excitation relative to the system natural frequency. With the backlash parameter fixed at 100 µm, the dimensionless mesh frequency was varied while all other parameters were held constant. The resulting regimes are listed below; the amplitude range refers to the dimensionless mesh-line displacement \(\lambda\).
| Dimensionless mesh frequency \(\omega_H\) | Amplitude range of \(\lambda\) | Response regime |
|---|---|---|
| 0.0308 | 1.2–1.7 | Period-1 |
| 0.0769 | 1.2–1.8 | Period-3 |
| 0.2437 | −3 to 4 | Chaotic |
| 0.3061 | 1–1.7 | Period-1 |
As mesh frequency increases, the response does not simply become more severe. It passes from periodic motion to chaotic motion and then returns to periodic motion. The largest nonlinear response occurs at \(\omega_H=0.2437\), which corresponds to an input speed of 15,500 r/min. This speed is close to the main resonance peak of the system, so the internal excitation is strongly amplified and the response occupies a much wider amplitude range. At \(\omega_H=0.3061\), the response is again period-1 with a narrower amplitude range.
The design implication is that the critical speed is set by the relationship between mesh frequency and natural frequency, not by the highest operating speed alone. For this configuration, sustained operation near 15,500 r/min should be avoided because it lies in the resonance-sensitive region where chaotic response is most likely.
Support Stiffness: Restoring Stiffness and Chaos Margin
Support stiffness controls the restoring force that keeps the gear axes aligned and limits the translational response. With \(b=100\) µm and \(\omega_H=0.2071\), the dimensionless support stiffness \(k_s\) was varied while the other parameters were kept unchanged.
| Dimensionless support stiffness \(k_s\) | Response regime |
|---|---|
| 1.25 | Period-2 |
| 0.70 | Period-7 |
| 0.20 | Period-8 |
| 0.10 | Chaotic |
Lower support stiffness increases the translational freedom of the gear bodies and allows larger excursions around the nominal mesh position. The contact-loss interval is encountered more often, impact velocities increase, and the system is more likely to leave periodic motion. At \(k_s=0.10\), the response becomes chaotic. Adequate bearing support stiffness therefore acts as a stability margin, while excessively soft support reduces the predictability of the transmission response. The related practical requirement is that bearing radial clearance should be controlled within a reasonable range; excessive clearance weakens the effective support and can promote the same nonlinear behavior.
Engineering Implications and Model Scope
The results show that the nonlinear behavior of spiral bevel gears is governed by the interaction of clearance, mesh frequency, and support flexibility. Backlash changes the contact topology; mesh frequency determines whether the internal excitation is amplified near resonance; support stiffness determines how far the gear bodies can move before the contact state changes. For the studied 28/34-tooth aeroengine central drive, three design guidelines follow:
- Keep the backlash parameter below the chaotic case observed at 160 µm under cruise conditions. The 140–160 µm interval is the most sensitive, where period count and vibration fluctuation rise rapidly.
- Avoid sustained operation near 15,500 r/min, where the dimensionless mesh frequency is close to the main resonance region and chaotic response appears for a 100 µm backlash parameter.
- Provide sufficient bearing support stiffness and control radial clearance. Low support stiffness increases the likelihood of multi-periodic and chaotic motion.
These trends are specific to the modeled gear pair, load level, and dimensionless parameter set. The formulation neglects tooth friction and the spiral-contact effect and assumes equal support stiffness and damping in all directions. Absolute regime boundaries may therefore shift in a more detailed three-dimensional contact model. The value of the present model lies in identifying the mechanisms and the parameter ranges that deserve attention in dynamic design, rather than in supplying universal clearance limits for all spiral bevel gear systems.