Spiral bevel gears carry power between intersecting shafts in a single compact mesh, which is why they dominate the final stage of aero-engine accessory trains and helicopter tail drives. The same curved tooth flank and conical pitch surface that make this layout compact also make its vibration behaviour difficult to control: the contact ratio changes continuously along the tooth, the mesh point sweeps across the face width, and the resulting force carries simultaneous tangential, radial and axial components. Under high speed and heavy load, the axial component in particular is transmitted straight into the shaft, the tapered roller bearings and the casing.
Active remedies — profile modification, lead crowning, tighter manufacturing tolerance, redistribution of the macro-geometry — attack the excitation at its source but require the gear to be redesigned and remanufactured. A passive alternative is to insert a split or solid metal ring into a groove machined in the gear rim. The ring is lightly pressed against the groove wall; when the gear vibrates, relative slip at the ring–rim interface converts part of the vibration energy into heat. The ring adds little mass, tolerates high temperature, and can be retrofitted to an existing rim. Its performance, however, is entirely governed by two quantities that are awkward to estimate in practice: the normal contact load generated by the interference fit at operating speed, and the friction coefficient of that interface when the gear is spray-lubricated.
Two ring topologies and where the damping comes from
Three ring constructions are used in practice. A solid (integral) ring is shrink-fitted by heating the gear or chilling the ring; a split (C-type) ring is elastically compressed at its free ends and springs outward once seated; a helical ring is threaded into the groove and expands radially under centrifugal load. The helical form gives the most adaptive contact pressure, but its assembly tolerance and manufacturing cost are the highest. The integral ring is the most mature to manufacture and is the configuration examined here.
Because the ring is constrained in a groove, its motion relative to the gear reduces to two degrees of freedom: translation along the gear axis and rotation about it. Both are resisted by Coulomb-type friction. A one-degree-of-freedom equivalent of the gear–ring pair can be written as
$$m\ddot{x}+c\dot{x}+kx+f\,\mathrm{sgn}(\dot{x})=0$$
where $f$ is the sliding friction force at the interface. The important structural point is that the damping is proportional to the normal load, and the normal load is not a design constant — it is the interference fit minus the centrifugal growth of the ring, filtered through the elastic compliance of both bodies. Getting this quantity right is the prerequisite for any quantitative prediction of ring performance.
Contact load at speed: from curved beam to reduced-order model
Split ring
A split ring behaves as a thin curved beam in bending. Treating one half of the ring as a cantilever with a distributed contact load $q$, the opening displacement follows from the unit-load method, combining bending, transverse shear and hoop terms:
$$\Delta=\int_{0}^{\pi}\frac{M M_1}{E J}R_0\,\mathrm{d}\varphi+\int_{0}^{\pi}\frac{K_y Q Q_1}{G A}R_0\,\mathrm{d}\varphi+\int_{0}^{\pi}\frac{N N_1}{E A}R_0\,\mathrm{d}\varphi$$
with $J=bh^{3}/12$ for a rectangular section ($b$ width, $h$ thickness) and $K_y=1.2$. At standstill the internal moment, shear and hoop force vary with the angular position $\varphi$ measured from the free end as $qR_0^{2}\sin(\varphi/2)$-type functions. Rotation adds a centrifugal body-force term proportional to $\rho A\omega^{2}R_0$, so that the moment, shear and hoop force each acquire a component scaling with $\rho\omega^{2}R_0^{3}$. Inverting the resulting relation gives the contact load as a closed-form function of the opening $\Delta$, the section geometry and $\omega^{2}$: the centrifugal term progressively relieves the interference, which is why a ring that is adequately tight at standstill can lose contact pressure at high speed.
Integral ring on a conical rim
For a solid ring the standard treatment is a thick-walled cylinder pair, with the gear rim as the outer body and the ring as the inner body. Solving the plane-stress Lamé problem for both, the radial displacements at the common radius $a$ give the interference
$$\delta=u_g(a)-u_d(a)$$
This is acceptable for a cylindrical hub but not for a spiral bevel gear, whose rim is a cone. Two corrections are needed. First, the cone is sliced into thin discs normal to the axis, and the plane-stress solution is applied slice by slice, with the local radii taken from the cone geometry. Second, because the true three-dimensional compliance of the rim and the ring — including the tooth root region — differs from the idealised cylinder, a finite-element basis is used instead of the closed-form Lamé field, and the model is condensed before solving.
Condensation follows the Guyan static reduction. Partitioning the degrees of freedom into a retained set $b$ (interface nodes) and a set to be eliminated $a$, the static equilibrium $\mathbf{K}\mathbf{X}=\mathbf{F}$ is reduced by the transformation $\mathbf{T}=\left[-\mathbf{K}_{aa}^{-1}\mathbf{K}_{ab};\ \mathbf{I}\right]$, giving
$$\mathbf{K}^{*}=\mathbf{T}^{\mathsf{T}}\mathbf{K}\mathbf{T},\qquad \mathbf{F}^{*}=\mathbf{T}^{\mathsf{T}}\mathbf{F}$$
Only the interface nodes survive, so the contact problem becomes small. The residual interference under rotation is obtained from the reduced stiffness of gear and ring and the centrifugal displacement difference $\mathbf{C}=\left[\mathbf{K}^{*}_{bb,g}\right]^{-1}\mathbf{F}^{*}_{b,g}-\left[\mathbf{K}^{*}_{bb,r}\right]^{-1}\mathbf{F}^{*}_{b,r}$, solved as a linear program subject to the geometric non-penetration condition at every interface node. The nodal contact forces follow as $\mathbf{F}_{b}=[\mathbf{K}^{*}_{bb,r}]\mathbf{X}_{b}$, and the residual interference is obtained by superposing the nodal radial displacements. The corresponding distributed load then follows from the Lamé relation for the sliced section, using the ring width $L$ as the contact length and the elastic constants of the two materials.
The approach is static in nature, which is legitimate here because the quantity of interest is the steady contact state at a constant rotational speed; inertia effects on the eliminated degrees of freedom are negligible for that problem. Compared with a full nonlinear contact simulation of the assembled gear, ring and rim, the cost drops by orders of magnitude, which matters when the contact load must be re-evaluated for every candidate ring geometry inside an optimization loop.
Validation against a full finite-element contact model of the assembled pair used the gear data in Table 1, with interface interferences of 2, 4, 6 and 8 µm and speeds from 1000 to 10 000 rpm. Both approaches produced the same monotonic trend of contact load with speed, and the maximum deviation stayed below 5% over the whole range.
| Parameter | Value |
|---|---|
| Number of teeth (pinion / gear) | 34 / 34 |
| Face width | 30 mm |
| Big-end transverse module | 3 mm |
| Spiral angle at mid-face | 35° |
| Normal pressure angle | 20° |
| Shaft angle | 90° |
Friction under oil-jet lubrication
Dry-friction estimates are optimistic for an aero-engine gearbox, where oil is sprayed directly onto the mesh and reaches the rim groove. The interface then operates in mixed elastohydrodynamic lubrication, with part of the normal load carried by asperities and part by a lubricating film. The split is described by the film-load ratio $\gamma$, a monotonic function of the film ratio
$$\lambda=\frac{h_{\min}}{\sqrt{\sigma_1^{2}+\sigma_2^{2}}}$$
where $\sigma_1,\sigma_2$ are the RMS roughness of the two surfaces. The minimum film thickness follows the Dowson–Higgins line-contact relation
$$h_{\min}=\frac{2.65\,\alpha^{0.54}\left(\eta_0 v_e\right)^{0.7}R_E^{0.43}}{E^{*\,0.03}\,w^{0.13}}$$
with $\alpha$ the viscosity–pressure coefficient, $\eta_0$ the dynamic viscosity, $v_e$ the entrainment speed, $R_E$ the equivalent radius of curvature, $w$ the load per unit contact length and $E^{*}$ the equivalent elastic modulus,
$$E^{*}=\left[\frac{1}{2}\left(\frac{1-\nu_1^{2}}{E_1}+\frac{1-\nu_2^{2}}{E_2}\right)\right]^{-1}$$
The film contribution to friction is evaluated from a regression fit that depends on the maximum Hertz pressure $P_h=\sqrt{QE^{*}/(\pi R_E L)}$, the sliding speed and the roll–slide ratio, with surface roughness entering as a correction. The composite coefficient is then obtained by linear superposition,
$$\mu=\gamma\mu_e+(1-\gamma)\mu_b$$
with a boundary (dry) value $\mu_b=0.12$. The friction force and the friction moment on the ring are $F_d=\mu Q$ and $M_d=F_d R_c$, where $R_c$ is the radius of the ring outer surface.
| Lubricant property | Value | Reference temperature |
|---|---|---|
| Density | 971.2 kg·m⁻³ | 25 °C |
| Kinematic viscosity | 1.205×10⁻⁵ m²·s⁻¹ | 25 °C |
| Dynamic viscosity | 1.170×10⁻² Pa·s | 25 °C |
Excitations feeding the gear–ring model
Mesh stiffness from loaded tooth contact analysis
The tooth flank of a spiral bevel gear cannot be described accurately by simple analytical profiles, so the solid model was generated in a dedicated gear design package and exported as a precise NURBS body. Hexahedral meshing produced 315 882 C3D8R elements with density 7850 kg·m⁻³, elastic modulus 210 GPa and Poisson ratio 0.3. The static analysis constrains each gear to rotate about its own axis through a kinematic coupling at the bore, and is split into three steps: a small reverse rotation of 3.5×10⁻³ rad applied to the driven gear to close the contact, a release step that allows the contact deformation to spring back while contact is maintained, and the loaded solve. The first step exists purely to suppress the rigid-body displacement that would otherwise contaminate the unloaded transmission error.
Mesh stiffness is then extracted from the loaded transmission error, defined as the difference between the actual angular position of the driven gear and its ideal kinematic position, projected onto the line of action:
$$k_m=\frac{P}{TE_L-TE_{\text{unloaded}}}$$
After discarding the transient part of the loading history, one stable period is retained and replicated to build the full time history, which is then represented by a Fourier series
$$k_m(t)=k_{m0}+\sum_{i=1}^{N}A_i\cos\left(i\Omega_h t+\Phi_i\right)$$
where $\Omega_h$ is the mesh frequency. Retaining a small number of harmonics keeps the dynamic integration tractable without smearing the stiffness peaks that drive the parametric excitation.
Time-varying support stiffness of the tapered roller bearings
Bearings are commonly modelled as constant springs, which removes an excitation that is present at every revolution: as the cage rotates, the number of rollers in the loaded zone changes from nine to ten and back, and the load on the bottom roller migrates to its neighbour. For the bearing considered here the radial and axial loads are distributed over the roller set with the standard load-zone factor $J_r\approx 1/4.08$, giving the maximum roller loads
$$Q_r=\frac{4.08F_r}{Z\cos\alpha},\qquad Q_a=\frac{4.08F_a}{Z\sin\alpha}$$
Roller–raceway compliance is treated as a Hertzian line contact, with convex–convex geometry at the inner raceway and convex–concave at the outer. Because the roller is a truncated cone, its curvature varies along the generatrix; the roller is therefore sliced, and the local equivalent radius at slice $i$ is obtained from the local section diameter and the semi-cone angle. The lubricant film, evaluated with the same line-contact film-thickness relation used for the gear–ring interface, separates the surfaces, so the effective deformation entering the stiffness is $\delta^{*}=\delta-h$. The stiffness $K=\mathrm{d}F/\mathrm{d}\delta$ is then decomposed along the $x$ and $y$ axes, and the radial component couples with the axial load through the contact angle,
$$K_R=\frac{\mathrm{d}F_r}{\mathrm{d}\delta_r}+\frac{\mathrm{d}F_a}{\mathrm{d}\delta_r}\tan\left(\alpha+2\beta\right)$$
with $\alpha$ the contact angle and $\beta$ the roller half-cone angle. Table 2 lists the bearing geometry used.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Outer diameter | 90 mm | Roller small-end diameter | 9.8 mm |
| Bore diameter | 50 mm | Roller large-end diameter | 10.8 mm |
| Inner-ring width | 23 mm | Contact angle | 12.24° |
| Outer-ring width | 19 mm | Roller half-cone angle | 1.7° |
| Roller contact length | 16.8 mm | Number of rollers | 20 |
Ring–rim contact stiffness and damping
The ring–rim pair is idealised as two cylinders in contact. The Hertzian normal stiffness scales with the cube root of the load, and the tangential stiffness follows from the same contact patch:
$$k_n=\left[\frac{2}{3}Q R_E\left(\frac{2}{E}\right)^{2}\left(1-\nu^{2}\right)\right]^{1/3},\qquad k_d=\left[\frac{3}{2}\frac{Q R_E\left(2\right)^{2}}{E\left(1-\nu\right)\left(1-2\nu\right)}\right]^{1/3}$$
Contact damping is related to the tangential stiffness and the ring mass through the material loss factor $\eta_d$:
$$c_d=0.5\,\eta_d\sqrt{k_d m_d}$$
Coupled ten-degree-of-freedom formulation
The two gears and the ring are represented as rigid bodies on elastic supports, giving the generalized coordinate vector
$$\mathbf{q}=\left\{X_p,Y_p,Z_p,\theta_p,X_g,Y_g,Z_g,\theta_g,Z_d,\theta_d\right\}^{\mathsf{T}}$$
where the $x$ and $z$ axes coincide with the pinion and gear axes respectively; the ring translates along, and rotates about, the gear axis. The $z$-direction response of the gear is therefore its axial response, and the only dissipation channel available to the ring is sliding along that direction — a point that will explain the anisotropy of the results below.
Projecting the relative motion onto the line of action through the directional cosines $c_1$, $c_2$, $c_3$ (functions of the normal pressure angle, the mid-face spiral angle and the pitch cone angle) gives the mesh deflection
$$\lambda_n=c_1\left(X_p-X_g\right)+c_2\left(Y_p-Y_g\right)+c_3\left(Z_p-Z_g\right)+r_p\theta_p-r_g\theta_g-e(t)$$
with $e(t)=e_0+e_r\sin\left(\omega t+\varphi\right)$ the equivalent sinusoidal transmission error representing manufacturing and assembly deviations. The normal mesh force and its components are
$$F_n=k_m(t)\lambda_n+c_m\dot{\lambda}_n,\qquad F_x=c_1F_n,\quad F_y=c_2F_n,\quad F_z=c_3F_n$$
Assembly of the ten equations yields a coupled bending–torsional–axial system of the form $\mathbf{M}\ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}}+\mathbf{K}(t)\mathbf{q}=\mathbf{F}(t)$, in which the time-varying mesh stiffness and the decomposed bearing stiffnesses occupy the corresponding diagonal and coupling blocks. The ring contributes a linear spring–damper path along $z$ and a friction moment about the gear axis,
$$M_d=\mu N R_c\,\mathrm{sgn}\left(\dot{\theta}_g-\dot{\theta}_d\right)$$
where $N$ is the normal force in the groove. The system was integrated in the time domain with an explicit Runge–Kutta solver, and the response was evaluated in both time and frequency domains, with the mesh frequency $f_m$ and its second harmonic $2f_m$ used as the reference components.
What the ring removes from the response
With a ring of 30 mm outer diameter, 6 mm width, 6 mm thickness and 2 µm interference, running at 4000 rpm and 45 kW, the comparison against the ring-free system is consistent across all three indicators. The dynamic mesh force decays faster after start-up and settles to a slightly lower mean with a narrower fluctuation band; in the spectrum, the second harmonic is attenuated most strongly, which is the signature of an added dissipation path acting on the periodic mesh excitation rather than on a single resonance.
The acceleration and velocity results are direction-dependent, as Table 3 shows. The $z$ (axial) direction dominates, with the acceleration RMS falling by 25.4% and the velocity by 30.43%, while the two transverse directions improve by only a few percent. The asymmetry follows directly from the kinematics of the ring: energy is dissipated by axial sliding between the ring and the groove wall, so the axial channel is the one that is damped. The $z$-direction acceleration also shows short sawtooth excursions, most likely because the axial support stiffness is lower than the transverse stiffness and the ring contact therefore modulates the axial force more visibly.
| Direction | Acceleration RMS without ring (m·s⁻²) | Acceleration RMS with ring (m·s⁻²) | Acceleration reduction | Velocity reduction |
|---|---|---|---|---|
| $x$ | 168.08 | 158.93 | 5.44% | 4.91% |
| $y$ | 191.16 | 182.29 | 4.63% | 3.38% |
| $z$ | 35.18 | 26.24 | 25.40% | 30.43% |
Speed and load sensitivity
Speed changes the interface twice over. As the sliding velocity rises, the entrainment speed rises with it, the film thickens and the film-load ratio climbs from 50.09% at 4000 rpm to 73.5% at 10 000 rpm, while the composite friction coefficient falls from 0.0597 to 0.0329. Both effects weaken the friction force available for dissipation. Superimposed on this is the frequency-matching effect: the ring suppresses the axial velocity response strongly at 4000 rpm and above 8000 rpm, but between 5000 and 7000 rpm the difference from the ring-free system narrows and at some speeds the response is marginally amplified. Ring damping is therefore not a monotonic benefit across the operating envelope.
Load behaves quite differently. Between 30 and 50 kW the interface film-load ratio stays at 50.7% and the friction coefficient at 0.0591 — the lubrication state is essentially unchanged over this range. The vibration level itself rises with power (axial velocity from 2.44 to 2.69 mm/s for the ring-free system), and the ring reduces it by a nearly constant margin at every power level. Load therefore sets the vibration amplitude while speed sets the damping effectiveness.
Geometry: two competing mechanisms
Sweeping the four ring dimensions one at a time reveals the same pattern in every case: an improvement range followed by a reversal. Table 4 collects the interface quantities; the vibration response is described in the text, since it is best read as a trend rather than a set of isolated values.
| Parameter swept | Range | Film-load ratio | Composite friction coefficient | Best vibration response |
|---|---|---|---|---|
| Rotational speed | 4000–10 000 rpm | 50.09% → 73.5% | 0.0597 → 0.0329 | 4000 rpm; ≥8000 rpm |
| Input power | 30–50 kW | 50.7% (constant) | 0.0591 (constant) | margin nearly constant |
| Ring width | 4–9 mm | 50.85% → 50.70% | 0.0589 → 0.0592 | 7 mm |
| Ring thickness | 4–9 mm | 51.99% → 49.70% | 0.0575 → 0.0604 | ≈7 mm |
| Ring outer diameter | 25–40 mm | 44.5% → 62.4% | 0.066 → 0.045 | 35 mm |
| Interference fit | 2–6 µm | 50.7% → 46.6% | 0.0591 → 0.064 | lowest value tested (2 µm) |
Width. Increasing the width lowers the residual interference and raises the load per unit contact length, so the film-load ratio falls slightly and the friction coefficient rises slightly. The vibration response improves up to about 7 mm, where the larger contact area provides more dissipative capability, and then worsens: beyond 8 mm the contact pressure is high and the ring itself becomes structurally stiffer, so relative slip — the very quantity that produces damping — is suppressed.
Thickness. Thickness acts mainly on ring stiffness. Between 4 and 7 mm the ring is compliant enough to slip while stiff enough to sustain a useful contact pressure, and the response decreases; beyond 8 mm the ring is effectively rigid in its own plane, slip amplitude collapses and the response climbs again despite the higher contact pressure.
Outer diameter. A larger diameter increases the contact area and reduces the unit pressure, which improves the lubrication state: the film-load ratio rises from 44.5% to 62.4% and the friction coefficient drops from 0.066 to 0.045. The response improves up to 35 mm because the additional contact area outweighs the loss of friction coefficient, then degrades sharply at 40 mm, where structural stiffness limits sliding. The frequency content is unchanged across the sweep — only amplitudes move, so the diameter does not alter the dominant excitation mechanism.
Interference. Interference is the parameter that sets the normal load directly. Raising it from 2 to 6 µm thickens nothing but thins the film through the higher contact pressure, so the film-load ratio falls from 50.7% to 46.6% and the friction coefficient rises from 0.0591 to 0.064. Normally that would be beneficial, yet the vibration response increases monotonically with interference over the tested range. The explanation is that the higher normal load constrains the slip displacement itself: with a larger normal force the interface approaches a stuck condition, the dissipated energy per cycle falls, and the ring becomes an increasingly rigid coupling rather than a damper. An optimum therefore exists at a low but non-zero interference, and the design problem is to find it without losing contact at speed.
Surrogate-based optimization of the ring
The single-parameter sweeps show the direction of each effect but not their interaction; the optimum is a four-variable problem. The design vector is $\mathbf{X}=[D,W,T,\delta]^{\mathsf{T}}$ over the box
$$\Omega=\left\{25\le D\le 40\ \text{mm},\ 2\le W\le 9\ \text{mm},\ 2\le T\le 9\ \text{mm},\ 2\le\delta\le 8\ \mu\text{m}\right\}$$
and the objective is the sum of the RMS vibration velocities of the driven gear in the three orthogonal directions at the rated condition (4000 rpm, 45 kW):
$$f(\mathbf{X})=\mathrm{RMS}(v_x)+\mathrm{RMS}(v_y)+\mathrm{RMS}(v_z)$$
Equal weights are used because no independent field calibration is available to justify otherwise; the sum of RMS values is chosen because it represents the overall energy level of the response over a long operating period rather than a single peak.
Each objective evaluation requires a full time-domain integration, so the optimization is run on a surrogate. A Latin hypercube design of 150 points was generated in the four-dimensional box and refined with a maximin criterion, which raised the minimum pairwise Euclidean distance in the normalized space from 0.071 to 0.188 and thus improved space filling. The points were then mapped to physical coordinates and evaluated with the dynamic model. A Kriging interpolator with a Gaussian correlation function was fitted, with the correlation lengths selected by logarithmic grid search and ten-fold cross-validation. The fit reached a coefficient of determination of 0.9971 on the training set; against 20 independently drawn validation points it reached 0.9839, with a relative RMSE of 0.99% and a maximum error of 2.12%, both inside a 5% acceptance threshold, and no visible overfitting between the training and validation scores.
Particle swarm optimization was used on the surrogate, with 50 particles, 100 generations, inertia weight 0.7 and cognitive and social coefficients of 1.49, terminating early if the global best changed by less than 10⁻⁶ over ten consecutive generations. The swarm converged in about 34 generations with a monotone decrease in the population mean, and the resulting design is listed in Table 5. All four variables settle in the upper part of their ranges, which is consistent with the single-parameter results: a large diameter for contact area, a large width and thickness for dissipative capacity without excessive stiffness, and an interference high enough to keep the interface loaded at speed.
| Quantity | Outer diameter $D$ | Width $W$ | Thickness $T$ | Interference $\delta$ | Total RMS velocity |
|---|---|---|---|---|---|
| Optimum design | 38.44 mm | 8.10 mm | 7.13 mm | 7.4 µm | 21.74 mm/s (surrogate) |
| Verification in dynamic model | — | — | — | — | 21.70 mm/s |
| Initial design | 30 mm | 6 mm | 6 mm | 2 µm | 24.55 mm/s |
The surrogate prediction of 21.74 mm/s differs from the direct dynamic simulation of 21.70 mm/s by less than 0.2%, so the surrogate introduced no meaningful bias into the optimum. Relative to the initial design the three-directional RMS total falls from 24.55 to 21.70 mm/s, a reduction of about 11.4%, with the largest contribution again coming from the axial direction. Time- and frequency-domain comparisons of the optimized and initial rings confirm that the improvement is spread across the whole record rather than confined to start-up, and that the spectral peaks at $f_m$ and $2f_m$ are both lowered.
Design implications and open questions
Three conclusions follow from the mechanics rather than from the numbers alone. First, the useful damping of a ring damper is set by the residual slip amplitude, not by the friction coefficient: every parameter that raises contact pressure eventually reduces dissipation, which produces the non-monotonic width, thickness and diameter trends and the monotonic penalty of excessive interference. Second, the interface lubrication state is an operational variable, not a material constant — over the 4000–10 000 rpm range the film-load ratio moves by more than twenty percentage points and the friction coefficient nearly halves, so a ring optimised at one speed will not be optimal at another. Third, the damping is intrinsically directional: axial sliding damps the axial response by roughly 25–30% while the transverse response improves by only 3–5%, so a ring should not be expected to solve a transverse resonance problem.
The analysis is numerical throughout; no bench measurements of ring damping were available, and the friction model relies on a regression fit for the film contribution with a fixed boundary coefficient. Wear of the contact interface over service life, and the resulting drift of interference and surface roughness, is not represented, nor is the possibility of adaptive contact pressure that would track speed and load. These are the natural next steps before the geometry reported here can be treated as a qualified design rather than an optimized candidate.