Missile launch vehicles carry the missile horizontally and erect it to the launch position before firing. The erection device therefore has to hold and rotate a large offset mass through a controlled angle. As mission payload changes the missile mass, and as the erection angle sweeps through its range, the gravitational torque acting on the gearbox changes continuously. Spiral bevel gears are widely used at the right-angle stage of such drives because they provide high load capacity and relatively smooth motion, but their mesh state is sensitive to pressure angle, spiral angle, axial and radial clearance, housing deflection, and manufacturing error. The resulting vibration shifts the dominant operating frequency, increases tooth wear, and can feed back into positioning error.
The control problem is not simply speed regulation. A fixed gear geometry and fixed clearance set can be optimized for a nominal load, but it cannot follow a time-varying erection torque. A conventional feedback loop can reject disturbances, but it must then correct both the initial manufacturing error and the load-induced frequency shift, which often produces large overshoot. A more effective architecture separates the problem into two layers: a static layer that selects the best nominal gear parameters, and a dynamic layer that uses measured vibration frequency to adjust the drive command in real time.
Operating Mechanism and Control Problem
In a spiral bevel gear pair, the contact pattern moves diagonally across the tooth surface as the gears rotate. The mesh stiffness varies periodically with the contact ratio, and axial thrust is coupled to the spiral angle. When the erection device starts, stops, or changes angle, the load torque changes, the housing and shaft deflect, and the mesh force redistributes. These effects appear in the gearbox vibration signal as a shift in the dominant frequency and as changes in sideband energy.
In this context, “dynamic frequency” refers to the dominant vibration frequency observed during operation, not the undamped natural frequency of a stationary gear. It is an operating response shaped by mesh stiffness, transmission error, load, speed, and structural compliance. Monitoring that frequency provides an indirect but practical indicator of mesh state, especially when direct torque measurement at the tooth contact is impossible.
Static optimization addresses repeatable error: tooth form error, pitch error, shaft misalignment, and incorrect clearance. Dynamic control addresses repeatable and non-repeatable load variation: missile mass, erection angle, friction, and actuator lag. Neither layer is sufficient alone. Static optimization cannot track a changing load, and dynamic feedback alone can become aggressive when it has to compensate for a poor nominal mesh.
Vibration Measurement and Frequency Identification
A piezoelectric accelerometer mounted on the gearbox housing near the meshing zone converts gearbox vibration into charge, then into voltage. A simplified measurement chain is
$$q=S_q a,\qquad U=q/C_e,\qquad a=S_a U$$
where \(q\) is charge, \(a\) is acceleration, \(S_q\) is charge sensitivity, \(C_e\) is the equivalent capacitance of the sensor and cable, \(U\) is output voltage, and \(S_a\) is voltage sensitivity. The sensor must be calibrated before use because sensitivity drift directly biases the estimated frequency amplitude.
Raw gearbox signals also contain bearing rotation, friction, impact, and structural resonance. Independent component analysis can separate the gear-mesh component from these additional sources. After centering and whitening the mixed signal, a separation matrix is estimated by maximizing non-Gaussianity, commonly through negative entropy. The recovered gear signal is then transformed into the frequency domain:
$$F(f)=\int_{-\infty}^{\infty}\hat a(t)\,e^{-j2\pi ft}\,dt$$
The dominant peak within the expected mesh band is taken as the dynamic frequency. In the validation described below, acceleration was sampled at 20 kHz with a multi-point excitation and single-point response arrangement. This matters because a single accelerometer can miss a mode or an axial-thrust component that appears more strongly at another housing location.
Static Parameter Optimization
The static layer adjusts four assembly and geometry parameters within manufacturing limits: pressure angle, spiral angle, axial clearance, and radial clearance. The search is formulated as a constrained minimization of vibration acceleration over a steady operating window:
$$J=\sqrt{\frac{1}{T}\int_0^T a^2(t)\,dt}$$
A seagull optimization algorithm treats each candidate parameter set as a search agent. Migration and attack phases update the population, while boundary repair keeps every candidate inside the allowed range. The reported search used a population of 50, 200 iterations, an initial exploration coefficient of 2.0, a decay factor of 0.9, a spiral coefficient of 1.0, and a maximum attack angle of \(2\pi\). These settings balance exploration of the parameter space against convergence speed.
| Parameter | Allowed range | Before optimization | After optimization |
|---|---|---|---|
| Pressure angle | 20°–25° | 22° | 23.5° |
| Spiral angle | 25°–35° | 30° | 28.2° |
| Axial clearance | 0.05–0.15 mm | 0.12 mm | 0.07 mm |
| Radial clearance | 0.8–2.4 mm | 1.05 mm | 0.92 mm |
The optimized set moves the pressure angle toward the upper bound, reduces the spiral angle, and tightens both clearances. These changes alter contact ratio, axial thrust, and mesh stiffness. A larger pressure angle can improve load capacity but also raises normal tooth force; a smaller spiral angle can reduce axial thrust but may affect overlap and smoothness; reduced clearance improves positioning stiffness but leaves less margin for thermal growth and deflection. The result is therefore a compromise, not a universal optimum.
Finite-element simulation of the optimized gear pair gave an acceleration RMS value below 16g. This is a simulation result for the selected parameter set, not a direct bench measurement of the assembled gearbox. It indicates that static optimization can lower the nominal vibration level, but it does not establish long-term durability or field reliability.
Fractional-Order Internal-Model PID Control
The dynamic layer uses the difference between the measured dynamic frequency and the desired setpoint as the control input. Internal-model control provides a structured way to design a robust feedback loop from a process model. The model is factorized into invertible and non-invertible parts,
$$H(s)=H_+(s)H_-(s)$$
and the internal-model controller is chosen as an inverse of the invertible part with a low-pass filter:
$$Q(s)=H_-^{-1}(s)\frac{1}{(\lambda s+1)^r}$$
The equivalent feedback controller is
$$C(s)=\frac{Q(s)}{1-H(s)Q(s)}$$
After approximating the delay term and rearranging, the controller takes a PID-like form. Replacing the integer integral and derivative with fractional operators gives
$$e(t)=f_{\rm ref}-f_{\rm meas}(t)$$
$$u(t)=K_p e(t)+K_i D_t^{-\alpha}e(t)+K_d D_t^{\beta}e(t)$$
where \(f_{\rm ref}\) is the desired dynamic frequency, \(f_{\rm meas}\) is the measured value, \(u(t)\) is the control command, \(K_p\), \(K_i\), and \(K_d\) are gains, and \(\alpha\) and \(\beta\) are positive fractional orders. The fractional orders add two tuning degrees of freedom beyond an integer-order PID. They allow the controller to shape phase and gain over a wider frequency range, which is useful when the gearbox response changes with load and angle.
The internal-model structure contributes robustness to model mismatch; the fractional PID contributes flexibility in tracking and disturbance rejection. In the reported validation, the setpoint was approximately 1.0 Hz, and the controller was expected to restore the measured frequency to that value after a speed or load change.
Experimental Validation
Validation used a custom gear test bench configured as an erection-drive chain. The bench had a maximum speed of 5000 rpm, a torque capacity of 8000 N·m, and an axial load capacity of 50 kN. A MTS 244.21 electro-hydraulic loading system applied dynamic force up to ±50 kN with a 100 Hz frequency response. Torque was measured with an HBM T40B sensor rated from 0 to 5000 N·m at ±0.1% accuracy. Gearbox vibration was acquired with a Brüel & Kjær 4514B accelerometer, and transmission error was measured with a Keyence LK-G5000 laser displacement sensor. Control algorithms ran on a dSPACE DS1202 real-time system. Tests were performed at approximately 20°C and 40% relative humidity.
The bench reproduced the transmission chain of an erection device, but it was not a complete launcher. Structural completeness, component fit, accumulated wear, and aging differed from a service vehicle. The results therefore show control behavior under controlled laboratory conditions rather than field durability.
| Case | Excitation or environment | Key outcome |
|---|---|---|
| Static parameter optimization | Pressure angle, spiral angle, axial and radial clearance optimized within bounds | Finite-element acceleration RMS below 16g |
| Speed mutation | 1000 rpm steady, increased to 1500 rpm, then reduced to 1000 rpm | Overshoot below 1.0%; frequency settled near 1.0 Hz within about 2 s |
| Load mutation | 50% rated torque increased to 200% rated torque | Overshoot below 1.0% |
| Temperature cycling | 20°C to 25°C to 20°C, 2 h per stage | Stability index above 85% |
| Humidity cycling | 40% to 60% to 40% relative humidity, 2 h per stage | Stability index above 85% |
| Controller comparison | Same speed and load mutations with neural-network control, auto-tuning PID, and RBF neural-network control | Lower overshoot than the three comparison methods |
Before blind source separation, the acceleration signal was dominated by high-frequency noise and irregular amplitude fluctuation. After separation, gear-related frequency components became clearer, allowing the dominant dynamic frequency to be extracted by FFT. Static optimization reduced the simulated vibration RMS below 16g, but static control alone left a residual frequency drift, especially later in the run. This is consistent with the limitations of a fixed parameter set: as the gearbox heats, deflects, and wears, the nominal optimum no longer matches the operating state.
With the dynamic layer active, the frequency was restored to about 1.0 Hz within roughly 2 s. Under the speed mutation and load mutation cases, the fractional-order internal-model PID kept overshoot below 1.0%. Among the comparison methods, the RBF neural-network controller performed better than the neural-network and auto-tuning PID controllers under the speed step, but its overshoot remained higher than that of the fractional-order controller. Under the load step, the advantage of the fractional-order controller was more pronounced. This is plausible because the fractional orders provide additional phase compensation when the load-dependent mesh stiffness changes.
During separate temperature and humidity cycling, the test protocol’s stability index remained above 85%. This suggests that the controller retained useful authority as ambient conditions changed, but it does not by itself prove long-term reliability. The stability index is a test-specific measure, not a field availability guarantee.
Interpretation and Limits
The main engineering value of the scheme is the division of labor. Static optimization removes repeatable manufacturing and assembly error from the nominal mesh. Dynamic control then handles the time-varying load and angle that no fixed parameter set can follow. For spiral bevel gears in a missile erection device, this separation is more practical than trying to solve both problems with a single controller.
Several limits remain. The validation was performed on a simplified test bench rather than a complete launcher, so housing compliance, foundation flexibility, and service wear were not fully represented. The experimental evidence is short-term; long-term tooth wear, pitting, and fatigue were not evaluated. The signal chain also introduces delay: ICA and FFT require a finite window, and the resulting frequency estimate lags the actual mesh state. The 1.0 Hz setpoint is application-specific and would need revalidation for different gear ratios, erection speeds, and sensor locations. The actuation path and saturation limits are not detailed, yet they determine achievable settling time and overshoot in a full system. Finally, the environmental tests varied temperature and humidity separately; combined cycling and contamination were outside the reported scope.
Conclusion
Spiral bevel gears in missile launch-vehicle erection devices operate under time-varying gravitational torque, changing erection angle, and manufacturing error. A two-layer control approach addresses these conditions by combining accelerometer-based vibration monitoring, independent component analysis, and FFT for dynamic-frequency identification; seagull-algorithm optimization of pressure angle, spiral angle, axial clearance, and radial clearance for static compensation; and fractional-order internal-model PID control for real-time frequency regulation.
In validation, static optimization reduced the finite-element acceleration RMS below 16g. The dynamic controller settled near 1.0 Hz within about 2 s and kept overshoot below 1.0% under speed and load mutations, outperforming neural-network, auto-tuning PID, and RBF neural-network controllers. Temperature and humidity cycling maintained the test stability index above 85%. These results support the approach for high-load, time-varying right-angle drives, while long-term wear testing and full-launcher validation remain necessary before field deployment.