An erection drive on a missile launch vehicle must raise a heavy missile from its transport position to a launch angle and hold it with minimal vibration. The final right-angle stage commonly uses a spiral bevel gear because its curved teeth engage gradually, spread contact across several teeth, and transmit high torque with relatively smooth motion. The same properties make the stage sensitive to load and alignment. A change in missile mass or a small deviation in erection angle alters the contact pattern, mesh stiffness, backlash, and friction, and the dominant vibration frequency of the gearbox drifts. The control problem is therefore not only speed regulation. It is regulation of an operating vibration signature under coupled mechanical disturbances.
A practical way to handle that problem is to divide it into two layers. Offline, the gear geometry and clearances are optimized to reduce vibration caused by manufacturing and assembly error. Online, the measured dominant vibration frequency is fed back to a fractional-order internal-model PID controller, which corrects the command as load and angle change. This division of labor matters because the two error sources have different time scales: manufacturing error is fixed in the hardware, while payload and angle disturbances vary during the erection cycle.
The Control Problem in an Erection Drive
During erection, two disturbances are coupled. A heavier missile increases the gravitational moment required from the drive. An angle error changes both the moment arm and the position at which the gear teeth carry the load. These effects are not independent: together they shift the mesh force, the contact ellipse, the effective backlash, and the friction at the tooth surfaces. The resulting vibration spectrum moves, and a controller tuned for one load case can be detuned for another.
The controlled variable in this architecture is the operating vibration frequency of the spiral bevel gear stage. It is an empirical condition indicator extracted from the gearbox signal, not the theoretical gear mesh frequency and not the undamped natural frequency of the gear pair. The distinction is important. Mesh frequency is fixed by tooth number and shaft speed; the measured dynamic frequency reflects the whole transmission state, including load, clearance, alignment, and contact conditions.
Measuring the Dynamic Frequency
A piezoelectric accelerometer is mounted on the gearbox housing close to the meshing region. When the gear pair runs, the seismic mass inside the sensor loads the piezoelectric element, which produces a charge proportional to acceleration. The charge is converted to a voltage by the sensor circuit, and a calibrated sensitivity converts that voltage back to acceleration:
$$q=\phi m a,\qquad U=\frac{q}{C_e},\qquad a=\delta_s U$$
Here \(q\) is charge, \(\phi\) is the piezoelectric coefficient, \(m\) is the seismic mass, \(a\) is vibration acceleration, \(C_e\) is the equivalent capacitance, \(U\) is output voltage, and \(\delta_s\) is the calibrated sensitivity. The measurement is simple in principle, but the signal is not clean. Bearing rotation, friction, impacts between components, and structural resonances all contribute. If the raw signal is used directly, the gear-related frequency component can be buried under high-frequency noise.
Independent component analysis is used as a blind source separation step. The measured signal is treated as a mixture of gear vibration and other sources. After centering and whitening, a separation matrix is estimated by maximizing negentropy with a Newton-type iteration. The separated component \(\hat a(t)\) is intended to approximate the gear-related vibration rather than the full sensor signal. A fast Fourier transform then converts the time-domain record into a spectrum:
$$G(\omega)=\int_{-\infty}^{\infty}\hat a(t)e^{-j\omega t}\,dt$$
The dominant peak in \(G(\omega)\) provides the dynamic frequency used by the controller. In the reported tests, vibration data were sampled at 20 kHz using a multi-point excitation, single-point response modal arrangement. The accelerometer was calibrated before testing, and the sensor-to-data-acquisition chain was checked for stability.
Static Control: Optimizing the Spiral Bevel Gear Set
Even a well-made spiral bevel gear pair contains tooth-profile error, pitch error, and small axis misalignment. These errors create a baseline frequency offset that feedback control alone would have to fight continuously. The static layer addresses that bias by selecting a single set of gear parameters that minimizes the RMS vibration level over a representative operating cycle:
$$a_{\mathrm{RMS}}=\sqrt{\frac{1}{T}\int_0^T a^2(t)\,dt}$$
Four parameters are adjusted: pressure angle, spiral angle, axial clearance, and radial clearance. Their search ranges and optimized values are summarized in Table 1. The pressure angle affects contact ratio and bending load; the spiral angle affects overlap, axial force, and smoothness; the axial and radial clearances affect backlash, mesh stiffness, and transmission error. Reducing clearance can improve precision, but the clearance cannot be made arbitrarily small because thermal growth, lubrication, and manufacturing tolerance still have to be accommodated.
| Parameter | Search range | Before optimization | After optimization | Primary influence |
|---|---|---|---|---|
| Pressure angle | 20°–25° | 22° | 23.5° | Contact ratio, tooth loading, mesh impact |
| Spiral angle | 25°–35° | 30° | 28.2° | Contact overlap, axial force, transmission smoothness |
| Axial clearance | 0.05–0.15 mm | 0.12 mm | 0.07 mm | Axial float, impact, transmission precision |
| Radial clearance | 0.8–2.4 mm | 1.05 mm | 0.92 mm | Mesh stiffness, backlash, transmission error |
The optimization uses a seagull optimization algorithm with 50 candidate parameter vectors and 200 iterations. Each candidate encodes the four parameters. The fitness value is the RMS vibration level computed from a time-domain record. The search alternates between a migration phase, which avoids collisions and moves toward the best position, and an attack phase, which performs a spiral search around that position. The initial exploration coefficient is 2.0, the decay factor is 0.9, the spiral coefficient and shape factor are both 1.0, and the maximum attack angle is \(2\pi\). Boundary handling resets any candidate that leaves the permitted range.
The optimized set was applied in manufacturing and assembly by adjusting the cutter mounting angle or generating train for pressure and spiral angle, and by using shims or housing dimensions to set axial and radial clearance. Transient finite-element analysis with nonlinear contact then estimated the vibration response of the optimized pair. The RMS vibration acceleration fell below 16 g. That result supports the static layer as a way to lower the baseline vibration level, but it does not make the gear immune to time-varying load.
Dynamic Control: Fractional-Order Internal-Model PID
Static optimization cannot follow a payload change or an angle deviation that develops during erection. The dynamic layer therefore uses the measured frequency error to adjust the transmission state. The controller combines internal-model control with fractional-order PID. Internal-model control provides an explicit plant model, a low-pass robustness filter, and a straightforward conversion to feedback form. The fractional PID adds two tuning degrees of freedom through the integral and derivative orders.
The plant model \(H(s)\) is split into an invertible part \(H_-(s)\) and a non-invertible part \(H_+(s)\). An internal-model controller is formed by inverting \(H_-(s)\) and adding a low-pass filter \(1/(\upsilon s+1)^r\), where \(\upsilon\) sets the filter bandwidth and \(r\) sets its order. Converting the resulting structure to a feedback controller and approximating the delay with \(e^{-\tau s}\approx 1-\tau s\) yields a PID-type law. In fractional form, the control command is
$$u(t)=K_p e(t)+K_i D_t^{-\alpha}e(t)+K_d D_t^{\beta}e(t),\qquad e(t)=\omega_{\mathrm{ref}}(t)-\omega(t)$$
where \(K_p\), \(K_i\), and \(K_d\) are the proportional, integral, and derivative gains; \(\alpha\) and \(\beta\) are the fractional integral and derivative orders; \(\omega_{\mathrm{ref}}\) is the target dynamic frequency; and \(\omega\) is the measured value from the accelerometer signal. The fractional orders change the shape of the loop response without adding a more complex controller structure. In the reported validation, the target dynamic-frequency variable was about 1.0 Hz.
The static and dynamic layers are complementary. Static optimization removes error built into the gear set. Dynamic feedback handles the part that cannot be predicted from a single parameter set: load steps, angle changes, friction variation, and nonlinear contact behavior. The dynamic loop does not replace accurate geometry; it reduces the residual error that remains after the hardware has been optimized.
Experimental Validation
Tests were carried out in a laboratory at approximately 20 °C and 40% relative humidity. The custom gear test rig reproduced the transmission chain of an erection device, with a maximum speed of 5000 rpm, a torque capacity of 8000 N·m, and a 50 kN axial load capacity. Loading was applied with an MTS 244.21 servo-hydraulic system rated at ±50 kN dynamic force with a 100 Hz frequency response. Torque was measured with an HBM T40B transducer covering 0–5000 N·m at ±0.1% accuracy and 5 kHz bandwidth. Gearbox vibration was measured with a Brüel & Kjær 4514B accelerometer covering ±500 g and 0.5 Hz–10 kHz. A Keyence LK-G5000 laser displacement sensor provided 0.05 μm resolution at a 392 kHz sampling rate for transmission-error measurement. Control code ran on a dSPACE DS1202 real-time system with a minimum step of 10 μs, and a 24-bit NI PXIe-4499 system acquired data at 204.8 kS/s per channel.
The static-only result showed the expected limitation. The optimized gear set reduced the initial frequency deviation, but the trace drifted as the run progressed, and the deviation tended to grow in the later part of the record. That behavior is consistent with accumulated error under external and nonlinear effects. Adding the dynamic loop brought the measured frequency back toward the target and kept it close to 1.0 Hz. The combined controller suppressed the deviation within about 2 s.
Two step disturbances were used for comparison. In the speed step, the gear system ran at 1000 rpm, ramped to 1500 rpm, and returned to 1000 rpm. In the load step, it ran at 50% rated torque and then ramped to 200% rated torque. The proposed fractional-order internal-model PID controller kept overshoot below 1.0% in both cases and produced lower overshoot than a neural-network controller, a PID baseline, and an RBF neural-network controller. The reported tracking deviation remained below 0.1 mm; because the unit is millimetres, this metric is best read as a transmission or positional deviation rather than a frequency error.
| Test | Disturbance or condition | Reported result |
|---|---|---|
| Static optimization | Optimized pressure angle, spiral angle, axial and radial clearance | RMS vibration acceleration below 16 g |
| Speed step | 1000 → 1500 → 1000 rpm | Overshoot below 1.0%; frequency held near the 1.0 Hz target |
| Load step | 50% → 200% rated torque | Overshoot below 1.0% |
| Tracking deviation | Iterative dynamic control | Below 0.1 mm for the reported displacement-type error |
| Environmental cycling | 20–25 °C at fixed humidity; 40–60% relative humidity at fixed temperature; 2 h per stage | Stability index above 85% |
In the environmental tests, temperature was raised from 20 °C to 25 °C and returned to 20 °C, while humidity was held at 40%. In a second run, relative humidity was raised from 40% to 60% and returned to 40%, while temperature was held constant. Each stage lasted 2 h. The reported stability index stayed above 85% in both runs, which indicates that the control loop retained its regulatory effect under slow environmental drift.
What the Evidence Supports
The results support a division of labor between hardware optimization and feedback control. Static optimization reduced the baseline vibration level and lowered the RMS acceleration below 16 g. The dynamic loop then reduced overshoot during speed and load steps and kept the frequency near the target when the static set alone began to drift. This is the main engineering value of the two-layer approach: the gear set is designed for the best achievable baseline, and the controller manages the variability that remains.
Several limits qualify the conclusion. The validation used a custom test rig rather than a complete erection vehicle, so structural completeness, fit precision, wear, and aging were not fully represented. The tests were short-term; long-term drift, fatigue, and wear compensation remain open questions. The dynamic command path is described at the control-law level, but the available validation does not isolate whether the final actuator is a speed, torque, or alignment correction. The reported 85% stability index is a relative indicator without a stated formula, and the 0.1 mm error metric is a displacement-type measure rather than a frequency error. These points do not remove the observed improvements in RMS acceleration and overshoot, but they bound how far the results can be generalized.
For spiral bevel gear drives in missile launch vehicle erection systems, the practical lesson is to treat the gear pair and its controller as one system. The operating vibration signature should be measured, the hardware should be optimized for the baseline error, and closed-loop control should be reserved for the load and alignment variability that no fixed parameter set can cover.