Spiral bevel gear pairs transmit torque between intersecting shafts in aero-engine and helicopter drivetrains. Their gradual tooth engagement gives high load capacity, smooth transmission and low noise, but higher shaft speeds and power densities push mesh excitation and bearing loads into speed ranges where support damping controls vibration. A squeeze film damper (SFD) is a common way to add external damping: a thin oil film in an annular clearance produces a hydrodynamic force as the journal orbits. The damping is useful near critical speeds, yet the force is strongly nonlinear in eccentricity. At high eccentricity, a conventional finite-length SFD can generate large nonsynchronous forces that offset its attenuation capacity.
An elastic ring squeeze film damper (ERSFD) modifies the geometry. A slotted elastic ring with inner and outer bosses sits in the oil chamber. The inner bosses contact the bearing outer ring, which acts as the journal; the outer bosses contact the housing. The bosses divide the inner and outer films into separate chambers linked by oil holes. When the journal translates, the inner bosses deform the ring, changing both the inner and outer film thicknesses. The oil film pressure and ring deformation therefore form a two-way fluid–structure interaction, not a rigid-film damper with a fixed clearance.
For a spiral bevel gear drive (SBGD), the relevant question is whether this compliance reduces the nonlinear oil-film force without weakening damping. A model with eight degrees of freedom couples the lateral and axial translations and rotations of the pinion and gear to the ERSFD forces. The film pressure is obtained from a generalized Reynolds equation, the ring deformation from a semi-analytical circular-arc model, and the time-domain response from a fourth-order Runge–Kutta integration with a block iteration between fluid and structure. The results identify the pressure distribution, the force–eccentricity behavior, and the influence of boss thickness on bifurcation and vibration amplitude.
ERSFD Architecture and Lubrication Model
The ERSFD in the model has eight bosses and eight oil holes. The ring length is 20 mm, the nominal inner and outer clearances are 0.2 mm, and the lubricant viscosity is 0.01844 Pa·s. The bearing inner ring is press-fitted to the gear shaft and rotates with it; the bearing outer ring is the journal and translates with the shaft but does not rotate about the shaft axis. The inner bosses and the bearing outer ring remain in contact without relative sliding in the circumferential direction. The outer bosses are treated as rigidly connected to the housing because their contact state changes little under journal translation. This leaves the inner film as the main pressure-generating gap and the outer film as a compliant backing connected through the ring and oil holes.
The film model assumes an incompressible, Newtonian, laminar, continuous film with constant viscosity; fluid inertia and body forces are neglected. For a chamber of the ERSFD, the reduced Reynolds equation in polar coordinates is
\[
\frac{1}{R^2}\frac{\partial}{\partial\theta}\left(h^3\frac{\partial p}{\partial\theta}\right)
+\frac{\partial}{\partial z}\left(h^3\frac{\partial p}{\partial z}\right)
=\frac{6\mu}{R}(u_0-u_h)\frac{\partial h}{\partial\theta}
+12\mu(v_h-v_0)
\]
where \(p\) is gauge pressure, \(h\) is the local film thickness, \(\mu\) is viscosity, \(R\) is the radius of the lower surface, and \(z\) is the axial coordinate. The subscripts 0 and h denote the lower and upper surfaces. For the inner film, the journal surface moves with velocity components \(u_0=-\dot{x}\sin\theta+\dot{y}\cos\theta\) and \(v_0=\dot{x}\sin\theta+\dot{y}\cos\theta\), while the ring surface has \(u_h=0\) and \(v_h=\partial r/\partial t\). For the outer film, only the ring radial velocity enters, giving a right-hand side of \(-12\mu\,\partial r/\partial t\).
The inner and outer film thicknesses are
\[
h_1=c_1\left\{1+\varepsilon\cos\left[\pi-(\beta-\theta)\right]\right\}+r,
\qquad
h_2=c_2-r
\]
where \(c_1=R_2-R_1\) and \(c_2=R_4-R_3\) are the nominal clearances, \(\beta\) is the journal attitude angle, \(\varepsilon=e/c_1\) is the eccentricity ratio, and \(r\) is the radial deformation of the elastic ring. Because \(r\) and \(\partial r/\partial t\) appear in the film equation, the pressure is transient and must be updated as the journal moves.
At the axial ends and at the circumferential edges of each chamber, gauge pressure is set to zero. The bosses are assumed to prevent circumferential leakage between chambers. The pressure equation is discretized by central finite differences on a grid of \(m\) circumferential and \(n\) axial intervals per chamber. The resulting linear system is solved by a QR algorithm, which is faster than successive over-relaxation for the repeated transient solutions.
Elastic Ring Compliance and Oil Film Force
The ring is not discretized with finite elements at every time step. Its deformation is approximated by circular arc segments between fixed contact points. The displacement of the \(i\)-th inner boss is
\[
\delta_i=\max(0,e\cos\phi_i)
\]
where \(e\) is the journal eccentricity and \(\phi_i\) is the angle between the boss and the line of eccentricity. The deformed arc gives the radial ring displacement \(r(\alpha)\) at any circumferential position \(\alpha\); a forward difference gives \(\partial r/\partial t\). This semi-analytical treatment keeps the ring compliance in the film equation at low computational cost.
Once the pressure field is known, the radial and tangential film forces follow from integration:
\[
\begin{bmatrix}F_r\\F_t\end{bmatrix}
=-\int_{\theta_1}^{\theta_2}
\left[\int_{-L/2}^{L/2}p(\theta,z)\,dz\right]
\begin{bmatrix}\cos\theta\\\sin\theta\end{bmatrix}
R\,d\theta
\]
The Cartesian components are
\[
F_x=-F_r\cos\beta+F_t\sin\beta,
\qquad
F_y=-F_r\sin\beta-F_t\cos\beta
\]
Even with the semi-analytical ring, solving the Reynolds equation at every Runge–Kutta stage is costly. A three-layer radial basis function (RBF) network therefore approximates the map from journal attitude \((\beta,\varepsilon)\) to \((F_x,F_y)\). The network uses 50 Gaussian hidden nodes and was trained on 10,000 samples generated by the numerical film model, split 98:1:1 for training, validation and testing. The fitted model reproduces the force components with sufficient accuracy for time integration and reduces the evaluation time from 2.280 s to 0.00764 s, roughly a 300-fold reduction.
| Item | Value |
|---|---|
| Inputs | Attitude angle \(\beta\), eccentricity ratio \(\varepsilon\) |
| Hidden layer | 50 Gaussian nodes |
| Outputs | Film force components \(F_x\), \(F_y\) |
| Samples | 10,000; split 98:1:1 (training/validation/test) |
| Numerical pressure-to-force evaluation | 2.280 s |
| RBF evaluation | 0.00764 s |
Eight-Degree-of-Freedom SBGD Model
The gear pair is represented by two rigid disks with translational coordinates \(x_l,y_l,z_l\) and rotation \(\theta_l\), where \(l=p,g\) denotes pinion and gear. The eight generalized coordinates are \(x_p,y_p,z_p,\theta_p,x_g,y_g,z_g,\theta_g\). The mesh is a time-varying stiffness and damping element along the line of action, with backlash and static transmission error. The normal dynamic mesh force is
\[
F_n=k_h(t)f(\lambda_n)+\xi_h\dot{\lambda}_n
\]
where \(\lambda_n\) is the relative displacement along the line of action and \(f(\lambda_n)\) is the backlash function
\[
f(\lambda_n)=
\begin{cases}
\lambda_n-b_c, & \lambda_n>b_c,\\
0, & |\lambda_n|\le b_c,\\
\lambda_n+b_c, & \lambda_n<-b_c.
\end{cases}
\]
The mesh stiffness and static transmission error are periodic at the mesh frequency:
\[
k_h(t)=k_m+\sum_{i=1}^{N_k}A_{ki}\cos(i\omega_e t+\phi_{ki}),
\qquad
e_n(t)=\sum_{i=1}^{N_e}A_{ei}\cos(i\omega_e t+\phi_{ei})
\]
The ERSFD forces enter the lateral equations of the pinion and gear. The ring deformation is not an independent dynamic coordinate; it is solved from the instantaneous journal position and fed back into the film force. This is the bidirectional coupling: gear motion sets \(\beta\) and \(\varepsilon\), the film model returns \(F_x\) and \(F_y\), and the updated forces drive the next integration step.
The fluid and solid equations are solved alternately in a time-splitting scheme. At each time step, the film pressure and ring deformation are updated from the current journal attitude; the gear equations are integrated with fourth-order Runge–Kutta; the new attitude is returned to the film model. The exchange repeats until the coupled step converges. The steady-state response is recorded after transients decay.
| Subsystem | Parameter | Value |
|---|---|---|
| ERSFD | Bosses / oil holes | 8 / 8 |
| ERSFD | Angular pitch per boss | 45° (\(\pi/4\) rad) |
| ERSFD | Ring length | 20 mm |
| ERSFD | Fluid viscosity | 0.01844 Pa·s |
| ERSFD | Inner / outer nominal clearance | 0.2 / 0.2 mm |
| ERSFD | Boss thickness cases | 0.15, 0.20, 0.25 mm |
| SBGD | Pinion / gear teeth | 47 / 53 |
| SBGD | Pinion / gear mass | 0.8 / 1.5 kg |
| SBGD | Pinion / gear inertia | 1040 / 3120 kg·mm² |
| SBGD | Hand of teeth | left / right |
| SBGD | Gear mass eccentricity | 0.5 µm |
| SBGD | Pressure angle | 20° |
| SBGD | Mid-spiral angle | 20° |
| SBGD | Shaft angle | 108° |
| SBGD | Face width | 30 mm |
| SBGD | Mean transmission error | 0.02 mm |
| SBGD | Mean mesh stiffness | 3 × 108 N/m |
| SBGD | Mesh damping | 1000 N·s/m |
| SBGD | Radial support stiffness / damping | 3 × 108 N/m / 500 N·s/m |
| SBGD | Axial support stiffness / damping | 5 × 108 N/m / 500 N·s/m |
| SBGD | Driving torque | 500 N·m |
Pressure Distribution and Nonlinear Oil Film Force
The inner-film pressure is segmented around the circumference. The bosses divide the film into chambers, and the pressure is discontinuous between chambers. Near an inner boss the film is very thin, and the model predicts a sharp local drop in pressure rather than a peak. The pressure maximum moves with the boss pattern. If no inner boss lies on the line of eccentricity, the journal at that attitude produces a relatively large pressure. The field is also asymmetric along the line of eccentricity because the ring deformation and chamber boundaries break circumferential symmetry.
At fixed attitude, increasing eccentricity from 0.5 to 0.65 raises the pressure. At \(\beta=\pi/4\), the ERSFD force components are much lower than those of a finite-length squeeze film damper (FLSFD) over the same eccentricity range. For \(\varepsilon\) below about 0.7, both dampers show a nearly linear increase. Beyond 0.7, the FLSFD force grows sharply and tends toward infinity as \(\varepsilon\) approaches 1. The ERSFD force continues to grow in a nearly linear manner instead of diverging. The reason is that the ring deformation changes the film thickness in a way that offsets part of the eccentricity increment, so the pressure does not have to carry the full nonlinearity. The result is a weaker nonlinear relation between oil-film force and eccentricity.
Boss Thickness, Bifurcation, and Vibration
The boss thickness changes the ring’s bending compliance and therefore the effective film stiffness. Three values were compared at a driving torque of 500 N·m: 0.15, 0.20 and 0.25 mm. The baseline 0.20 mm case has its first two critical speeds at 2200 and 4200 r/min. Except near the first critical speed, the response is period-1 from 1000 to 5600 r/min. At 5600 r/min it bifurcates to quasi-period-2, returns to quasi-period-1 at 6300 r/min, and at 7100 r/min follows a quasi-periodic route into chaos. The chaotic interval is 7100–8200 r/min; above 8200 r/min the response returns to period-1.
Reducing the thickness to 0.15 mm widens the chaotic interval to 5200–8200 r/min, an increase of 1900 r/min, and produces a jump at 5000 r/min. The lateral vibration amplitude is also larger. Increasing the thickness to 0.25 mm suppresses chaos: a bifurcation still appears between 7300 and 8200 r/min, but the response remains quasi-periodic rather than chaotic, and the rest of the speed range is period-1 or quasi-period-1.
Frequency content supports the bifurcation classification. At 2200 r/min, the 0.20 mm case contains 0.4×, 0.8× and 1.2× components; the 0.25 mm case contains 0.4× and 0.8×; the 0.15 mm case has a continuous spectrum and an unstable phase portrait and Poincaré section. At 5800 r/min, the 0.20 mm case is dominated by 0.5× and 1×, the 0.25 mm case by 1×, and the 0.15 mm case is chaotic. At 7600 r/min, both 0.15 and 0.20 mm are chaotic, while 0.25 mm is quasi-periodic with 0.4×, 1× and 1.3× components plus sidebands.
Peak-to-peak lateral amplitudes at the three speeds are listed in Table 3. At every speed the amplitude falls monotonically as boss thickness increases. The model attributes this to a rapid decrease in oil-film stiffness coefficient with increasing boss thickness, which lowers the transmitted vibration. The same trend is consistent with the narrowing and disappearance of the chaotic interval.
| Pinion speed (r/min) | h = 0.15 mm (µm) | h = 0.20 mm (µm) | h = 0.25 mm (µm) |
|---|---|---|---|
| 2200 | 0.1174 | 0.0480 | 0.0378 |
| 5800 | 0.2313 | 0.1132 | 0.0866 |
| 7600 | 0.1951 | 0.0870 | 0.0603 |
Design Implications and Model Boundaries
Within the modeled parameter set, boss thickness is not a minor geometric detail. It shifts the bifurcation boundary and controls whether the SBGD enters chaos in the 5200–8200 r/min range. A 0.25 mm boss converts the baseline chaotic band into quasi-periodic motion and reduces the peak-to-peak amplitude at 2200 r/min from 0.0480 µm to 0.0378 µm, at 5800 r/min from 0.1132 µm to 0.0866 µm, and at 7600 r/min from 0.0870 µm to 0.0603 µm. A 0.15 mm boss has the opposite effect. The ERSFD therefore acts as both a damper and a nonlinearity-control element: the ring compliance lowers the oil-film force at high eccentricity, while the chambered film changes the bifurcation structure.
These conclusions are conditional on the model. The film is incompressible, Newtonian, laminar and constant-viscosity; inertia and body forces are neglected. The gear bodies are rigid disks, the support stiffness and damping are linear, and the ring is represented by circular arc segments with rigid outer-boss contact and no circumferential leakage at the bosses. The pressure is set to ambient at the axial ends and chamber edges. The analysis covers a 47/53 tooth pair, a 108° shaft angle, 0.2 mm nominal clearances and a 500 N·m driving torque. Only three boss thicknesses are compared, so the results identify a trend and a bifurcation-control mechanism rather than an optimized geometry. Experimental validation under rotating conditions would be required before the thickness trend is used for design.