Abstract: This paper analyzes the mechanism of limit cycle structural resonance in electromechanical servo systems under low-frequency modal load conditions, identifies the load effect model, conducts targeted experiments to avoid limit cycle structural resonance, and analyzes the experimental results. Simulation and experimental results show that the proposed method can increase the system load resonant frequency, eliminate the limit cycle structural resonance phenomenon, and significantly improve the dynamic performance of the servo system.
1 Overview
In recent years, with the rapid development of power electronics technologies such as digital integrated circuits, power devices, and rare-earth permanent magnet motors, electromechanical servo and its control technologies have been widely applied.
In the application of electromechanical servo systems for thrust vector control of aircraft engines, the ideal performance is rapid response, stable dynamic performance, and high precision. However, in reality, the three indicators of speed, stability, and precision in a servo system are a contradictory combination.
To meet the requirements of high dynamic performance, electromechanical servo systems generally adopt the method of increasing gain. However, high gain is prone to system resonance when facing nonlinear low-frequency modal load conditions. Moreover, due to the influence of nonlinearity, the occurrence of this resonance phenomenon is not regular and it is difficult to analyze it using the method of linear system stability criteria.
The coupling between structural resonance and servo electrical system resonance can easily lead to a combined resonance with low frequency and small damping coefficient. This combined resonance severely affects the stability of the servo system and limits its bandwidth, which is a major factor limiting tracking error and transient response quality. Servo bandwidth is limited by both the combined system resonance and structural resonance characteristics. A typical resonance curve of the electromechanical servo system described in this paper is shown in Figure 1.
Figure 1 Resonance curve of the limit ring structure
Previous studies have analyzed the limiting cycle oscillation problem of systems with gaps driving large inertia loads by using gap description functions and momentum theorem. However, this method requires solving the problem of how to determine the equivalent mass of two transmission bodies. In systems with multiple transmission components, it is difficult to determine the equivalent mass.
As the load of the thrust vector electromechanical servo system, the aircraft engine is also an important factor determining the performance of the servo system. The load model of the engine includes a variety of nonlinear characteristics such as clearance and damping, which are difficult to obtain through analytical methods. In order to meet the needs of servo system development and production, the identification of the load model is crucial. It is used to simulate the load of the servo system during system operation, mainly including inertial load, elastic load, frictional load and constant torque [6-11].
This paper analyzes the mechanism and preventive measures of limit loop structural resonance in electromechanical servo systems under low-frequency modal load conditions by experimentally identifying the load and load effect.
2 Model Identification of Electromechanical Servo System under Load Condition
The principles of electromechanical servo systems and hydraulic servo systems are basically the same. Electromechanical servo systems control the servo motor by controlling the magnitude and direction of the current, and convert the high-speed, low-torque motion of the servo motor into high-torque, low-speed motion through a certain reduction mechanism. According to the system's integrated design, the sensor that measures the rotation angle and participates in the closed-loop feedback is not installed at the last stage of the entire motion chain, but rather at the motion output interface of the servo system. Therefore, the servo's thrust vector control of the engine is a semi-closed-loop control loop at the global level, and its basic structure is shown in the figure below.
Figure 2 Basic control structure of electromechanical servo system under load conditions
Taking a certain electromechanical servo system with a small liquid engine load as an example, the mechanical model of the servo system output shaft and the engine load is shown in Figure 3 below.
Figure 3 Load dynamics model of a certain type of servo system
The load force balance equation is
Where J<sub> L </sub> is the equivalent load moment of inertia; K <sub>sr</sub> is the equivalent load stiffness; TL is the load torque. According to the characteristics of liquid engines, the load torque is complex and includes major factors such as friction, Coriolis force, and additional torque caused by thrust deflection. It is not significantly related to the sway angle, so it is considered as a constant value to simplify the model; B<sub> p</sub> is the equivalent load damping.
Applying the Laplace transform to the above equation, we get
As shown in the above equation, the load of the servo system is a typical second-order element with a resonant frequency of and a damping ratio of . Therefore, the load model is mainly determined by three physical quantities: the load equivalent stiffness, the load moment of inertia, and the load equivalent damping. The load moment of inertia can be calculated. The load equivalent stiffness depends on the transmission stiffness, clearance, and installation stiffness of the structural body. The load equivalent damping is closely related to lubrication and friction, and is nonlinear. The load equivalent stiffness and damping are difficult to obtain analytically in engineering and require experimental identification. The limit cycle structural oscillation and instability phenomenon that occurs in the servo system under low-frequency modal load conditions is a typical closed-loop control system in a critical stable state. Through modal analysis experiments, the characteristics of the controlled object or each component in the system can be determined.
Therefore, the system was fixedly mounted on a fixed foundation, and a sinusoidal scanning signal with an amplitude of 1° and a frequency ranging from 2Hz to 30Hz was applied to the servo system. The responses of each transmission link from the motion output interface of the servo system to the engine nozzle terminal were measured, and the frequency characteristics under command input were calculated. The frequency characteristics of the response at measurement points 1 to 6 in the figure below were tested using the displacement feedback signal of the servo system itself (i.e., linear displacement in the traditional sense) as input.
Measurement point locations: 1. Front end of pin; 2. Front end of end face gear crank; 3. Connection between constant level frame and actuator; 4. Connection between squirrel cage and shaft; 5. Upper end of squirrel cage; 6. Rear end of engine nozzle.
Figure 4 Schematic diagram of test points for fixed installation state
Using the above methods, the response characteristics of each transmission link from the servo system output to the engine motion terminal were obtained with linear displacement as input. The load model parameters were matched to obtain the servo system load model as shown in Figure 5.
Figure 5 Load Fitting
The load effect is the reaction force of the load on the closed-loop internal structure of a servo system under actual load conditions. To test the load effect of this servo system, the open-loop transfer function is derived using the formula G(S) = Φ(S)/（1-Φ(S)）, based on the closed-loop transfer function. The open-loop dynamic characteristics are then calculated from the measured closed-loop dynamic characteristics. Under completely identical external conditions, the open-loop characteristics are measured under both no-load and loaded conditions. The load effect is then obtained by subtracting the no-load open-loop characteristic from the loaded open-loop characteristic. Finally, the load effect results are calculated and fitted to derive the load effect model, as shown in Figure 6.
Figure 6 Fitting of Load Effect
The load model and load effect model of the servo system can be obtained from the analysis as shown in Table 1 below. Among them, the mathematical model of the electromechanical servo system can be easily obtained through analytical methods when the servo motor, driver, reducer and control algorithm are known, and will not be elaborated on in this paper.
Table 1 Fitting of Load and Load Effect Transfer Function
3 Limit Ring Structure Resonance Mechanism
To analyze the mechanism of limit-loop structural resonance, we first calculate its open-loop stability margin from the perspective of linear system stability margin. Using the above mathematical model, we conduct frequency response tests and simulations in the open-loop case, and obtain the open-loop Bode plot as shown in Figure 7.
Figure 7 Open-loop Bode plot of the theory
After calculating the open-loop stability margin, it was found that the servo system satisfies the stability criterion of linear systems and has sufficient stability. In order to further analyze the limit loop structural oscillation problem, the dynamic characteristic analysis point was set on the engine nozzle angular displacement in the closed-loop state, and a 1° dynamic characteristic simulation was performed. The closed-loop angular displacement dynamic characteristic curve is shown in Figure 8 below.
Figure 8. Closed-loop angular displacement dynamic characteristic curve
It can be concluded that the closed-loop dynamic characteristics show an oscillation peak of about +8dB, while the phase at this time is about -180°, and the system is prone to oscillation.
Based on the above theoretical analysis, it was determined that the structural oscillation of the limiting loop after the servo system was installed on the engine was caused by the positive oscillation peak of the closed-loop system at a -180° phase shift. Although the system is stable under small signal conditions, it is prone to nonlinear limiting loops under the combined influence of nonlinear elements such as transmission clearance and power saturation. The occurrence of the limiting loop is related to the high gain of the servo system itself and the low load resonant frequency.
4 Load Mode and Structural Optimization Measures
The load resonant frequency and load damping are closely related to the limit cycle structure resonance. While these can be optimized through servo system control strategies, such as appropriately reducing gain or adding notch filtering algorithms, the latter comes at the cost of reduced dynamic performance. When the load resonant frequency is close to the servo system's own bandwidth, the loss in dynamic performance is significant. Increasing load damping weakens the resonance peak, but at low frequencies, large load damping introduces significant phase lag, requiring an increase in control gain to compensate. Therefore, the most effective method is to increase the load resonant frequency, placing it in the frequency band where the servo system experiences significant attenuation.
There are generally three ways to increase the load resonant frequency: increase the stiffness of the load and transmission links; reduce the clearance of the transmission links; and reduce the load rotational inertia.
5. Analysis of Experimental Results
This paper improves the stiffness of the transmission link to increase the load resonant frequency from 81 rad/s to 100 rad/s, and obtains the load closed-loop angular displacement dynamic characteristic curve, as shown in Figure 9.
Figure 9 Closed-loop dynamic characteristics of angular displacement after load stiffness improvement.
It can be concluded that after the stiffness of the transmission link is increased, the closed-loop amplitude-frequency characteristic of the angular displacement is below 0dB, thus demonstrating system stability. Subsequently, low-stiffness and high-stiffness servo systems were installed on the engine respectively, and external excitation was introduced into the engine nozzle for comparative testing. The comparative tests were identical except for the difference in stiffness. The feedback curves of the two states during the test are shown in Figures 10 and 11 below. (The vertical axis in the figures below is in V, and 0.1V corresponds to a swing angle of 0.2°.)
Figure 10 Low-stiffness external excitation resonance curve (vertical axis V, horizontal axis s)
Figure 11 Stability curves of high-stiffness external excitation (twice per channel) (vertical axis V, horizontal axis s)
It can be concluded that under the condition of introducing external excitation, the low transmission stiffness servo system exhibits continuous resonance, while the high transmission stiffness servo system does not exhibit oscillation (the fluctuation in the figure is a normal yaw caused by external excitation).
6. Conclusion
This paper takes a thrust vectoring electromechanical servo system as an application example and studies the mechanism of limit cycle structural resonance in the electromechanical servo system under low-frequency modal load conditions. A load model and load effect model of the thrust vectoring electromechanical servo system are established through identification methods. Based on simulation and experiments, the fundamental cause of limit cycle structural resonance in the electromechanical servo system under low-frequency modal load conditions is explored, and preventive measures to avoid structural resonance are proposed. The results show that the identification method can effectively and accurately derive the load model and load effect model of the electromechanical servo system; by increasing the load resonance frequency of the system, the limit cycle structural resonance phenomenon can be effectively eliminated without sacrificing the dynamic performance of the servo system, ensuring the stable operation of the system.
