Abstract: Accidents involving shaft damage caused by variable frequency speed control systems occur frequently, and preliminary analysis suggests that they are caused by oscillations resulting from the elastic deformation of the shaft system. Based on the analysis of the torsional vibration mechanism of elastic shaft systems, this paper proposes to suppress torsional vibration of the system through virtual resistance based on its equivalent circuit model. Furthermore, the range of virtual resistance values is broadened by using virtual negative capacitance. The effectiveness of this method has been verified through simulation.
1 Introduction
Torsional vibration is widespread in electrical drive systems. In multi-mass systems, the elastic deformation of the long shaft is the main cause of torsional vibration. Torsional vibration can cause pulsations in the speed and torque of the motor, affecting the stability and dynamic performance of the system, and in severe cases, it can also damage the elastic shaft [1].
High-performance engineering frequency converters require research into methods for suppressing torsional vibration of the elastic shaft. This paper first analyzes the generation mechanism of torsional vibration of the elastic shaft; then, it proposes a method for suppressing torsional vibration of the elastic shaft based on equivalent circuits and virtual resistance and capacitance; finally, the effectiveness of the torsional vibration suppression method is verified through Matlab simulation.
2 Torsional Vibration Mechanism
When the motor drives the load through the long shaft, the whole system can be regarded as a two-mass system. The elastic shaft needs to be twisted by a certain angle ∆θ to transmit torque[2].
Figure 1 Two-mass transmission system
Shaft output torque:
The system's equations of motion:
In the formula: s is the differential operator; Ks is the elastic coefficient of the shaft; Ds, Dm, and DL are the mechanical damping coefficients, which are generally very small and can be ignored; Jm and JL are the moments of inertia of the motor and the load.
The system can be represented by the equivalent circuit model shown in Figure 2 (ignoring Dm and DL). Here, current sources Tm and TL represent the electromagnetic torque and load torque, respectively; current source Ts corresponds to the connecting shaft torque; voltages wm and wL correspond to the motor's rotational angular velocity and the load's rotational angular velocity, respectively; capacitors Jm and JL represent the motor's rotational inertia and the load's rotational inertia, respectively; and Ks and Ds represent the connecting shaft's elastic coefficient and damping coefficient, respectively, their reciprocals corresponding to the inductance and resistance in the circuit model. The relationships between the electrical quantities in Figure 2 correspond one-to-one with the relationships between the corresponding mechanical quantities in the transmission shaft.
Figure 2 Equivalent circuit model of the drive shaft system
From Figure 2, the transfer function between Ts and Tm can be obtained as follows:
(1)
As can be seen from equation (1), the transmission shaft system has an inherent oscillation frequency of .
(2)
From equation (1), the gain of Ts to Tm at the natural oscillation frequency can also be obtained as follows:
(3)
Therefore, it can be seen that if a frequency disturbance component of f0 exists in the system, the transmission shaft torque Ts will oscillate. The amplitude of this oscillation is related to the shaft elastic coefficient Ks and the shaft damping coefficient Ds; the larger Ks is and the smaller Ds is, the larger the oscillation amplitude. In actual systems, when the load changes abruptly, it is very likely to introduce a frequency disturbance component of f0 into the system, causing Ts to oscillate and producing torsional vibration.
3 Torsional vibration suppression
Let Ds = 0, then we can analyze the worst-case scenario. In Figure 2, the controllable quantities in the motor are Tm and wm. For a vector control system, since the speed loop is an outer loop with a slower response, while the torque loop is an inner loop with a faster response, we can choose to control Tm to suppress torsional vibration. The method chosen here to suppress torsional vibration is to create a virtual resistor for Tm, and use the damping effect of the resistor to suppress the torsional vibration oscillation. In Figure 2, Tm is a current source, and wm is the port voltage on Tm. For ease of small-signal analysis, let ∆Tm and ∆wm be the fluctuations of Tm and wm, respectively. Assuming the virtual resistor to be simulated is Rmv, then ∆TmR = -∆wm/Rmv. The damping effect of Rmv on the system is analyzed below.
As shown in Figure 3, when the load generates a disturbance signal ∆TL, the corresponding transmission shaft torque disturbance signal ∆Ts is:
(4)
Figure 3 shows the damping effect of Rmv on the shaft drive system.
From equation (4), we know that when Rmv approaches infinity, we have:
(5)
At this point, the oscillation frequency of the system dominated by JL, Jm and 1/Ks is given, the damping ratio of the system is given, and the DC gain is given by Jm/(JL+Jm).
When Rmv approaches 0, we have:
(6)
At this point, it's equivalent to short-circuiting JL. The system's oscillation frequency, dominated by Jm and 1/Ks, is [value missing]. The system's damping ratio is [value missing], and the DC gain is 1. Clearly, the oscillation frequency w1 [value missing].
To ensure good damping performance of the system, let (damping ratio can be taken as 0.6 , corresponding to 10% overshoot), then the range of values for Rmv is:
(7)
Where w0 > w1 and Jm < (Jm + JL)
For equation (7) to hold true, the following must be satisfied:
(8)
If we take it, then according to equation (8), it will take approximately Jm.
Figure 4 shows the amplitude-frequency curves of ∆Ts/∆TL as Rmv changes.
Figure 5 shows the vector composition diagram of ∆TmR and ∆TmC corresponding to Rmv and Cmv. In the figure, ∆TmR acts as a damper, and ∆TmC acts as an adjuster for the system oscillation frequency.
Figure 5 Vector diagrams of ∆Tm and ∆wm
Figure 6. Vector diagram of ∆Tm and ∆wm of the PI controller.
In vector control, when the shaft torque fluctuation is ∆Ts, a fluctuation of ∆wm will be generated in wm. The speed loop will detect this fluctuation, thus affecting its regulator output, i.e., the electromagnetic torque Tm. Therefore, the speed fluctuation ∆wm will cause the electromagnetic torque fluctuation ∆Tm to become...
(9)
As shown in equation (9), the speed loop is equivalent to introducing a resistor 1/KPw and an inductor 1/KIw. As shown in Figure 6, the synthesized ∆Tm of the PI regulator is beneficial for suppressing oscillations.
If the drive shaft system itself satisfies Jm
(10)
In the formula: fRmv needs to be much greater than the oscillation frequency f0.
If it is necessary to reduce Jm by introducing a virtual negative capacitor to broaden the selection range of Rmv, let the virtual capacitor be Cmv. Then, after Jm is reduced, Jm_d = Jm - Cmv. From the parameters of the PI controller, the virtual capacitor is Cmv_d = Cmv - KIw/w02. Based on the reduced Jm, the selection range of Rmv can be further determined, thus obtaining Rmv_d = Rmv/(1 - RmvKPw). When implementing Cmv, the form of equation (10) can still be used, and it can be rewritten as follows:
(11)
From equation (11), kd and Td can be approximately obtained based on Rmv_d and Cmv_d:
(12)
4. Simulation Verification
The simulation parameters of the asynchronous motor are shown in Table 1.
Table 1 Simulation parameters of asynchronous motor
The switching frequency of the inverter used for driving is fsw= 4.8kHz , the simulation control frequency is fcon= 4.8kHz , the cutoff frequency of the d and q axis current loops is approximately fc_i=270Hz, and the phase margin is approximately PMi=65°.
The load and drive shaft parameters are: JL = 0.1 kgm², Ks = 4000, Ds = 0. The calculated oscillation frequencies are f1 = w1/2p = 31.8 Hz and f0 = w0/2p = 45.5 Hz. Keeping Jm constant, we can calculate Rmv_max ≈ Rmv_min = 0.03 Ω.
Let the speed loop PI controller parameters KPw = 93 , 33.3, 6, and the PI controller cutoff frequency be 10Hz. Under these three sets of parameters, the cutoff frequency and phase margin of the speed loop are 230Hz (30°), 80Hz (65°), and 17Hz (50°), respectively. Under these three sets of parameters, the corresponding Rmv and damping ratios x0 and x1 can be calculated as shown in Table 2.
Table 2. Rmv and damping ratios x0 and x1 under three sets of PI controller parameters.
Based on these three sets of parameters, simulations were performed, and the simulation results are shown in Figure 7. Figure 7 shows that the higher the cutoff frequency of the speed loop, the smaller the speed fluctuation during load jumps. In Figure 7a, when KPw=93, due to x1= 0.211 , the damping is very weak, therefore Ts oscillates at a frequency of f1= 31.8Hz during load jumps; in Figure 7c, when KPw=6, due to x0= 0.113 , the damping is very weak, therefore Ts oscillates at a frequency of f0= 45.5Hz during load jumps; in Figure 7b, since x0≈x1= 0.6 , the system has good damping, and the Ts oscillation is quickly suppressed during load jumps. The simulation results agree well with the theoretical analysis.
KPw=93
KPw= 33.3
KPw=6
Figure 7 Simulation results with different speed ring parameters JL= 0.1 kgm2, Ks=4000, and Ds=0.
Because Jm and JL are relatively close under this set of transmission shaft parameters, the range of Rmv is very small when the condition x0≈x1=0.6 is met . Below, the scheme of virtual negative capacitance is used to realize the damping of Ts. From equation (7), the maximum and minimum value curves of Rmv corresponding to the change of Jm under the condition of x0=x1= 0.6 can be obtained, as shown in Rmvmax and Rmvmin in Figure 8. According to the limitation of the maximum oscillation frequency f0 of the system ( f0≤0.2fc_i ), a curve of the minimum value of Jm can also be obtained. The interval enclosed by these three curves is the range of Jm and Rmv that can be taken, as shown in Figure 8. In this paper, Jm= 0.05kgm2 and Rmv= 0.045Ω are selected to realize the damping. The PI regulator parameter KPw=6 is used. According to equation (12), kd=18 and Td= 0.0023 can be calculated. Figure 9 shows the simulation results with KPw=6 after adding the damping element. Comparing Figure 9 and Figure 7c, it can be seen that the oscillation of Ts is significantly damped when the load changes, and this damping effect is even better than that in Figure 7(b). This is because the range of Rmv values is widened after the virtual negative capacitance, making x0= 0.64 and x1= 0.68 under this set of parameters, thus resulting in a better damping effect.
Figure 8 shows the selection range of Jm and Rmv when x0=x1=0.6 after the virtual negative capacitor is applied .
The simulation results after adding a virtual negative capacitor under the simulation parameters shown in Figure 9 and Figure 7(c) are as follows.
Furthermore, as shown in Figure 8, the change in Rmvmin is not significant when Jm changes. Therefore, in cases where the transmission shaft parameters are uncertain, during actual debugging, Cmv can be set to 0 first, and then Rmv can be adjusted until the system has a good damping effect. Then, according to equation (12), Cmv can be gradually increased until the system has good damping.
5. Conclusion
The torsional vibration model of the elastic shaft in the transmission system can be equivalently represented by a circuit model, and its torsional vibration mechanism can be analyzed by analyzing the LC resonance mechanism in the equivalent circuit. In a vector control system, torsional vibration can be suppressed by controlling the characteristics of Tm. The analysis in this paper shows that the virtual resistance Rmv for Tm can suppress the torsional vibration of the shaft in the transmission system. In addition, the virtual negative capacitance Cmv for Tm can broaden the selection range of the virtual resistance Rmv, which can further improve the system's performance in suppressing torsional vibration.
