Heavy-lift rotorcraft UAVs depend on a compact right-angle reduction stage to transfer torque from the powerplant to the rotor. The spiral bevel gear is the critical element in that stage: its curved teeth engage gradually and provide a high contact ratio, allowing high torque to be carried in a small gearbox. The same tooth geometry also makes the mesh sensitive to transient speed and load. At tooth entry, elastic deformation and a normal velocity difference produce a contact impulse; a sudden torque change adds a second impulse; and if a structural natural frequency is excited, the resulting dynamic load can exceed the steady design value. The engineering question is therefore not only whether the gear can carry nominal torque, but which speed and load transients keep mesh shock and resonance margin acceptable.
Contact Model for Mesh Impact
In a multibody contact model, the gear pair is treated as two rigid bodies whose tooth flanks exchange a penalty force when they penetrate. The UG motion model with an Adams solver uses a Hertzian elastic contact force together with a step damping term. For two curved bodies, the contact stiffness is
$$k=\frac{4}{3}\frac{\left[R_1R_2/(R_1+R_2)\right]^{1/2}}{(1-\nu_1^2)/E_1+(1-\nu_2^2)/E_2},$$
where \(R_1\) and \(R_2\) are the equivalent radii of the pinion and wheel at the mesh point, \(E_1\) and \(E_2\) are their elastic moduli, and \(\nu_1\) and \(\nu_2\) are their Poisson ratios. The contact force is represented as
$$F_c=\begin{cases}
0, & \delta\le 0,\\
\max\left[0,\;k\delta^{e}-C(\delta)\dot{\delta}\right], & \delta>0,
\end{cases}$$
where \(\delta=l_1-l\) is the penetration, \(l_1\) is the pre-contact distance, \(l\) is the current distance, \(e\) is the contact-force exponent, and \(C(\delta)\) is a step damping coefficient that rises toward its maximum value \(C_{\max}\) over the penetration depth \(d\). The maximum force is limited by the penetration and damping parameters, so the model reproduces both the elastic contact stiffness and the energy loss associated with impact.
The model parameters are summarized in Table 1. The gear material was carburized steel. The pinion was driven through a speed schedule and the wheel carried a constant resisting load. The simulation covered 0.3 s with 5,000 solution steps.
| Parameter | Value |
|---|---|
| Gear material | Carburized steel |
| Elastic modulus, \(E_1=E_2\) | \(2.0675\times10^5\) MPa |
| Poisson ratio, \(\nu_1=\nu_2\) | 0.30 |
| Contact stiffness, \(k\) | \(6.73\times10^5\) N/mm3/2 |
| Contact-force exponent, \(e\) | 1.5 |
| Maximum damping, \(C_{\max}\) | 45 N·s/mm |
| Full-damping penetration, \(d\) | 0.1 mm |
| Static friction coefficient | 0.08 |
| Dynamic friction coefficient | 0.06 |
| Constant resisting load | −100 N·mm |
| Speed stages | 1000, 2000, 3000 r/min at 0.05, 0.15, 0.25 s |
Speed Transients and Mesh Ripple
The pinion speed was increased in three plateaus: 1000, 2000 and 3000 r/min, corresponding to 6000, 12,000 and 18,000 deg/s. The command was built from step functions rather than a sudden start, so the gear train passed through each plateau before reaching the next. At the first two plateaus, the pinion had no external load; the wheel carried the constant resisting load.
The pinion followed the command and settled at 6000, 12,000 and 18,000 deg/s. The passive wheel reached average speeds of 5999.97, 12,000.06 and 17,999.97 deg/s, respectively. The deviations are below 0.06 deg/s, which is negligible for the transmission ratio. The instantaneous wheel speed, however, oscillated around the theoretical value. The cause is the periodic change in mesh stiffness as single and double tooth pairs alternate. Each transition from one contact state to the other produces a small impulse, so the output speed carries a periodic ripple even when the mean speed is correct.
| Stage | Pinion command (deg/s) | Equivalent speed (r/min) | Wheel average (deg/s) | Deviation (deg/s) |
|---|---|---|---|---|
| 1 | 6000 | 1000 | 5999.97 | −0.03 |
| 2 | 12,000 | 2000 | 12,000.06 | +0.06 |
| 3 | 18,000 | 3000 | 17,999.97 | −0.03 |
The ripple is more important than the mean-speed error. As the speed rises, both the amplitude and the frequency of the speed fluctuation increase, and the contact-force spikes at tooth entry become larger. The 3000 r/min plateau is therefore the most demanding operating point: it combines the highest mesh velocity with the strongest impact excitation. A continuous high-speed dwell should be avoided unless the tooth root and flank strength have been verified for the resulting dynamic load. If the mission requires high rotor speed, the acceleration phase and the steady high-speed phase should be treated as separate load cases.
Load Steps, Contact Ratio and Unloading Shock
To isolate the load effect, the speed was held constant and the resisting load was stepped from −30 to −70 to −100 N at 0.03, 0.13 and 0.23 s. At each load level, a sharp contact force appeared at mesh entry, followed by a bounded fluctuation around a mean value. As the load magnitude increased, the contact ratio increased: more tooth pairs shared the load during the transition, the transmission became smoother, and the fluctuation amplitude decreased. The mean contact force still rose with load, so a smoother force trace does not imply lower stress.
The UAV flight cycle runs in the opposite direction from this load sweep. At takeoff, rotor speed rises while the rotor is relatively lightly loaded, so the speed transient dominates. After the UAV reaches full payload, the gear contact ratio is high and the transmission is comparatively stable. The critical event is payload release: the rotor load drops suddenly, the resisting torque on the wheel falls, and the gear train experiences a load-step impulse. That unloading shock can be more severe than a gradual load increase because it excites the torsional response of the drivetrain and can produce a tooth-root stress wave. Design should therefore reserve strength for the release transient and control the sequence of payload drop. Adjusting flight attitude or maintaining a controlled external load during release can reduce the sudden change seen by the bevel pair.
Modal Behavior and Resonance Margin
Mesh impact is a forced transient; resonance is a structural amplification problem. The gear modal properties were calculated in ANSYS Workbench using a tetrahedral mesh. Natural frequencies and mode shapes depend on the distribution of mass and stiffness, not on the external load, so the modal solution is a free-vibration eigenvalue problem. The equation of motion for the gear structure is
$$\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=\mathbf{F}(t),$$
where \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are the mass, damping and stiffness matrices, \(\mathbf{x}\) is the displacement vector and \(\mathbf{F}(t)\) is the external load vector. For free vibration with damping neglected, the eigenvalue equation is
$$(\mathbf{K}-\omega^2\mathbf{M})\boldsymbol{\phi}=0,$$
where \(\omega\) is the natural frequency and \(\boldsymbol{\phi}\) is the corresponding mode shape. The first six modes are listed in Table 3.
| Mode | Natural frequency (Hz) | Dominant motion | Maximum deformation region |
|---|---|---|---|
| 1 | \(3.5803\times10^{-3}\) | Near-zero free-free/circumferential mode | Increases from the bore toward the large end |
| 2 | 16,286.0 | Rocking about the gear Y axis | Large end near the tooth root |
| 3 | 16,445.0 | Rocking about the gear Y axis | Large end near the tooth root |
| 4 | 20,928.0 | Bending | Large-end trailing region at maximum bending |
| 5 | 20,977.0 | Torsion | Large-end root under axial twist |
| 6 | 23,233.0 | Bending | Large-end trailing region at maximum bending |
The first mode appears at essentially zero frequency because the modal model is unconstrained; it represents free-free motion rather than a practical resonance. The lowest non-zero natural frequency is the second mode at 16,286 Hz. The second and third modes are close, and both involve rocking of the gear about its Y axis. The fourth and sixth modes are bending-dominated, while the fifth is torsional. Low-order modes are more easily excited and less effectively damped than high-order modes, so the second mode sets the lowest practical separation margin.
In the rotor drive, excitation comes from gear mesh, shaft rotation, bearing defects and control transients. None of the corresponding harmonics should coincide with 16,286 Hz or the neighboring modes under the intended speed range. The modal calculation provides the separation criterion; it does not by itself predict fatigue life, but it identifies the frequencies at which a small mesh impulse can be amplified into a large dynamic displacement.
Operating Envelope and Design Consequences
Three operating rules follow from the contact and modal results. First, speed changes should be ramped, and high-speed dwell should be limited. The mesh-impact trend is monotonic with speed: higher pinion speed produces larger speed ripple, higher contact-force spikes and stronger excitation of the gear structure. Second, load changes should be controlled, with particular attention to unloading. Increasing load raises the contact ratio and smooths the force trace, but releasing load removes that stabilizing contact condition and introduces a step impulse. Third, the lowest non-zero natural frequency, 16,286 Hz, should be kept clear of the mesh and control excitation spectrum.
These conclusions apply to the modeled contact parameters and linear modal properties. They do not replace a full fatigue assessment, lubrication analysis or thermal study. Within that scope, the dominant mechanism is clear: mesh velocity sets impact severity, contact ratio sets ripple, and the lowest non-zero natural frequency sets the resonance threshold. A heavy-lift UAV rotor drive should therefore be qualified against speed transients and payload-release transients separately, rather than against a single nominal torque condition.