In an aircraft engine’s central and accessory drive, a spiral bevel gear pair transmits power through a right angle while carrying high torque. Its curved teeth provide a high contact ratio, gradual engagement, and relatively smooth motion, which explains the layout’s wide use in engine gearboxes. The same mesh is exposed to time-varying stiffness, transmission error, and clearance. Backlash creates a dead band in which teeth can separate and re-enter contact with impact. The response can therefore be periodic at one operating point and chaotic at another. The design issue is not simply the nominal clearance, but how clearance, mesh frequency, and bearing support stiffness combine to keep the mesh in a bounded periodic state.
An eight-degree-of-freedom mesh model
The gear pair is represented by two lumped masses. Each gear has three translational coordinates and one rotation, giving eight physical degrees of freedom. Bearings are modeled as linear spring–damper elements in the X, Y, and Z directions. The mesh carries a time-varying stiffness \(k(t)\), a viscous mesh damping \(c_h\), and a clearance function. Tooth friction and the detailed spiral contact pattern are neglected, so the model isolates the consequences of stiffness variation, transmission error, and clearance.
The equations of motion for the pinion and gear follow Newton’s law:
$$
\begin{aligned}
m_p\ddot X_p+c_{px}\dot X_p+k_{px}X_p&=F_x, &
m_g\ddot X_g+c_{gx}\dot X_g+k_{gx}X_g&=-F_x,\\
m_p\ddot Y_p+c_{py}\dot Y_p+k_{py}Y_p&=F_y, &
m_g\ddot Y_g+c_{gy}\dot Y_g+k_{gy}Y_g&=-F_y,\\
m_p\ddot Z_p+c_{pz}\dot Z_p+k_{pz}Z_p&=F_z, &
m_g\ddot Z_g+c_{gz}\dot Z_g+k_{gz}Z_g&=-F_z,\\
J_p\ddot\theta_p&=T_p-F_zr_p, &
J_g\ddot\theta_g&=-T_g+F_zr_g.
\end{aligned}
$$
Here \(X_i,Y_i,Z_i\) are translational displacements, \(\theta_i\) is rotation, \(m_i\) is lumped mass, \(J_i\) is rotary inertia, \(c_{ij}\) and \(k_{ij}\) are support damping and stiffness, \(F_j\) are mesh-force components, and \(T_i\) are torques. The normal mesh force \(F_n\) is projected onto the coordinate axes through the normal pressure angle \(\alpha_n\), spiral angle \(\beta_m\), and pinion cone angle \(\delta_p\):
$$
\begin{aligned}
F_x&=F_n\left(\cos\alpha_n\sin\beta_m\cos\delta_p+\sin\alpha_n\sin\delta_p\right),\\
F_y&=F_n\left(\cos\alpha_n\sin\beta_m\sin\delta_p-\sin\alpha_n\cos\delta_p\right),\\
F_z&=-F_n\cos\alpha_n\cos\beta_m.
\end{aligned}
$$
For compactness, write the three projection terms as \(a_1,a_2,a_3\). The mesh-line displacement then combines the relative translation of the two gear centers with the relative rotation and the normal transmission error \(e_n(t)\):
$$
\lambda_n=-a_1(X_p-X_g)-a_2(Y_p-Y_g)-a_3(Z_p-Z_g)-a_3(\theta_pr_p-\theta_gr_g)-e_n(t),
$$
$$
a_1=\cos\alpha_n\sin\beta_m\cos\delta_p+\sin\alpha_n\sin\delta_p,\quad
a_2=\cos\alpha_n\sin\beta_m\sin\delta_p-\sin\alpha_n\cos\delta_p,\quad
a_3=\cos\alpha_n\cos\beta_m.
$$
The normal dynamic force and the clearance law close the model:
$$
F_n=k(t)f(\lambda_n)+c_h\dot\lambda_n,
$$
$$
f(\lambda_n)=
\begin{cases}
\lambda_n-b, & \lambda_n \gt b,\\
0, & |\lambda_n|\le b,\\
\lambda_n+b, & \lambda_n \lt -b.
\end{cases}
$$
Here \(b\) is the half-width of the clearance band. The piecewise law is the essential source of nonlinearity: the slope of the force changes discontinuously whenever the mesh crosses the clearance boundary, and the recontact event injects impact energy.
For numerical treatment, the two rotational coordinates are replaced by the relative mesh-line displacement, reducing the dimensionless description to seven coordinates. Displacements are normalized by the half-clearance \(b_c\), and time by \(\omega_n^{-1}\), where \(\omega_n=\sqrt{k_m/m_e}\), \(k_m\) is the mean mesh stiffness, and \(m_e=m_pm_g/(m_p+m_g)\) is the equivalent mesh mass. The resulting first-order state vector is
$$
\mathbf u=\left[x_p,\dot x_p,y_p,\dot y_p,z_p,\dot z_p,x_g,\dot x_g,y_g,\dot y_g,z_g,\dot z_g,\lambda,\dot\lambda\right]^T.
$$
The second-order system is recast as \(\dot{\mathbf u}=\mathbf H\mathbf u+f(\lambda)\mathbf A+\mathbf B\) and integrated with a Runge–Kutta method. The initial state is zero and the transient is discarded. The dimensionless support stiffness and damping are 1.25 and 0.01, respectively, and the mesh damping ratio is 0.1.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 28 | 34 |
| Module (mm) | 4.2 | 4.2 |
| Spiral angle (°) | 25 | 25 |
| Pressure angle (°) | 20 | 20 |
| Shaft angle (°) | 90 | |
| Hand | Right | Left |
| Rotary inertia (kg·m²) | 1.7992 × 10−3 | 3.0367 × 10−3 |
The equivalent masses of the pinion, gear, and mesh are 0.5024 kg, 0.6323 kg, and 0.2855 kg. The cruise condition used for the baseline parametric study is 13,175 r/min, which corresponds to a dimensionless mesh frequency \(\Omega_h=\omega_h/\omega_n=0.2071\).
Backlash: period doubling and a chaotic window
At cruise, the clearance parameter controls both the duration of contact loss and the impact velocity at reengagement. The phase portraits and Poincaré maps show a sequence rather than a gradual amplitude change. At \(b=80\,\mu m\), the phase trajectory is a single closed loop and the Poincaré map is one point, which identifies single-period motion. At \(b=100\,\mu m\), two closed loops and two points indicate two-period motion. At \(b=120\,\mu m\), three intersecting closed curves and three points indicate three-period motion. At \(b=160\,\mu m\), the trajectory fills phase space and the Poincaré map becomes a diffuse cloud, the signature of chaos.
| Clearance parameter \(b\) (μm) | Phase-plane and Poincaré signature | Steady-state motion |
|---|---|---|
| 80 | One closed curve; one point | Single-period |
| 100 | Two closed loops; two points | Two-period |
| 120 | Three intersecting closed curves; three points | Three-period |
| 160 | Space-filling nonclosed trajectory; diffuse points | Chaotic |
The standard deviation of the mesh-line displacement rises with the steady-state period number. As the response becomes more strongly nonlinear, both the spread of the vibration and its energy increase. The relation is not linear. From 80 to 140 μm, the period number and standard deviation grow slowly; between 140 and 160 μm they rise steeply. For this engine model, a clearance parameter around 160 μm is therefore a high-risk design point, not a smooth extension of the low-clearance behavior.
Mesh frequency: periodic–chaotic–periodic
With \(b=100\,\mu m\), changing speed changes \(\Omega_h\) while the other parameters remain fixed. At \(\Omega_h=0.0308\), the dimensionless mesh-line displacement stays between 1.2 and 1.7 and the response is single-period. At 0.0769, it stays between 1.2 and 1.8 and becomes three-period. At 0.2437, it ranges from −3 to 4 and the response is chaotic; this case corresponds to 15,500 r/min and lies near the main resonance. At 0.3061, the displacement returns to 1–1.7 and the motion is again single-period.
| \(\Omega_h\) | Dimensionless displacement range | Motion state |
|---|---|---|
| 0.0308 | 1.2–1.7 | Single-period |
| 0.0769 | 1.2–1.8 | Three-period |
| 0.2437 | −3 to 4 | Chaotic |
| 0.3061 | 1–1.7 | Single-period |
As frequency increases, the response therefore passes from periodic to multiperiodic to chaotic and then back to periodic. The amplitude follows the same qualitative path: it first grows and then falls as nonlinearity weakens. The key variable is proximity to the natural frequency, not speed alone. A speed that is high in absolute terms is not necessarily chaotic; a speed near a resonance peak is. The 15,500 r/min case is a resonance-adjacent condition for the selected clearance and stiffness, and the model identifies it as a speed to avoid.
Support stiffness and bearing clearance
At cruise with \(b=100\,\mu m\), the dimensionless support stiffness \(\kappa\) determines how far the response moves from low periodicity. \(\kappa=1.25\) gives two-period motion. Reducing it to 0.7 gives seven-period motion, 0.2 gives eight-period motion, and 0.1 gives chaos. Lower stiffness reduces the restoring force that centers the gear, so larger excursions and more frequent contact loss become possible. The practical implication is that bearing radial clearance must be controlled: excessive clearance lowers the effective support stiffness and erodes the stability margin.
| \(\kappa\) | Steady-state motion |
|---|---|
| 1.25 | Two-period |
| 0.7 | Seven-period |
| 0.2 | Eight-period |
| 0.1 | Chaotic |
Even the highest stiffness case is not single-period at this clearance; it is two-period. The comparison is therefore about the distance from chaos, not about eliminating all nonlinearity. The model indicates a threshold-like loss of stability as support stiffness falls.
Design implications and limits
Three parameters emerge as qualitative design levers. First, backlash should be kept out of the steep transition band. For this model, 80–140 μm produces slow growth in period number and vibration spread, while 140–160 μm produces rapid growth and a chaotic state at 160 μm. Second, operating speed should maintain margin from the main resonance; 15,500 r/min is a high-risk speed for the selected clearance and stiffness. Third, support stiffness should be high enough, with bearing clearance controlled, to prevent large contact-loss excursions.
These are numerical predictions from a deterministic lumped-parameter model, not experimental measurements. The model omits tooth friction and detailed spiral tooth contact, uses constant clearance and isotropic bearing stiffness, and assumes fixed damping. Absolute thresholds therefore apply to the stated parameter set and should be validated against hardware. The broader result is that backlash, mesh frequency, and support stiffness do not merely scale vibration amplitude; they change the qualitative state of the mesh. Design choices that look acceptable in a linear amplitude calculation can move a spiral bevel gear system across a period-doubling or chaotic boundary.