Active vibration control of loop antenna structure

Abstract

The loop antenna is a kind of large and flexible space truss structure commonly with weak damping and low modal frequencies. It is recognized the active vibration control performance of loop antenna structure is limited with use of piezoelectric stack actuator due to small actuating displacement. Besides, the introduction of piezoelectric stack actuator may obstruct the unfolding of antenna. A new active control way is thus put forward in the work where the voice coil motor is used as actuator and connected to the antenna structure by Kevlar fiber. This kind of actuation mode has the advantages of large output displacement, fast response, large output power and convenient installation. The governing equation of vibration control system of loop antenna employing the voice coil motor - Kevlar fiber actuator is firstly established according to finite element theory. In practical engineering, the first modal frequency of antenna is such low that it might be very close to the operational frequency of attitude adjusting equipment. As a result, the first modal vibration of antenna is easy to be stimulated and thus is often of most concerned in vibration control of antenna structure. Accordingly, aiming to suppress effectively the first modal vibration of loop antenna, the active vibration control strategies with PD and Fuzzy control algorithm are studied respectively. Moreover, PD-Fuzzy composite controller is also designed to improve the control effects near equilibrium position. The numerical and experimental results show that the voice coil motor - Kevlar fiber actuator can suppress the first modal vibration response of the antenna significantly and the performance of Fuzzy control is better than that of PD control. The control performance near equilibrium position can be improved obviously by use of PD-Fuzzy composite control algorithm.

Keywords

Loop antenna vibration control voice coil motor Fuzzy control

1. Introduction

In the past several decades, vibration control of large space structures (e.g. loop antenna) has received extensive and in-depth researches. The various control techniques have been employed in the relevant work, for example, passive control [1], semi-active control [2,3] and active control [4–6] and so on. Among these vibration control techniques, active control technique is most promising one for controlling low frequency vibration of large space structures due to its adaptivity and programmability. To acquire the desired control performance, diverse smart material including PZT [4,5], ER [3] and SMA [6] etc. have been utilized in active vibration control. PZT has become increasingly popular in the subject because of the prominent advantages such as large actuating force, fast response, light weight and low power consume. Luo [4] proposed a novel piezoelectric stack bending actuator to actively control the vibration of hoop truss with PD algorithm. A new distributed PZT control strategy based on characteristic model was also presented for space frame structure by Xu et al. [5]. With the rapid development of space technology, the size of antenna structures becomes more and more larger, and the modal frequencies of antenna structures become more and more lower. Consequently, it becomes difficult to control effectively such low frequency vibrations for antenna structures with piezoelectric stack actuator due to its small actuating displacement though good control performances were obtained in the previous work [4,5]. In contrary, the voice coil motor [7] can generate large displacement and force simultaneously compared with PZT actuator and may provide a promising candidate for actuator applied to active vibration control of loop antenna.

Regarding this, this paper proposes the voice coil motor-Kevlar fiber actuator to carry out the active vibration control of a loop antenna. The system modeling, active control simulations and experiments are performed respectively to demonstrate the feasibility and validity of this new control way.

2. Modelling of loop antenna vibration control system

2.1. Dynamic model of loop antenna

Figures 1 and 2 show the loop antenna studied in the paper. The diameter of the loop antenna is 5 m and it contains the upper and lower rings, each of which consists of sixteen connecting rods and a oblique supporting rod connecting the fixed end with the antenna torus part.

With the finite element method, the motion of equation of the antenna is firstly obtained according to the matrix assembly rule of the finite element theory as $\begin{eqnarray}\displaystyle M\ddot{x}+Kx=0 & & \displaystyle\end{eqnarray}$ (1) where M and K is mass and stiffness matrix. As for the influence of damping, Rayleigh damping is chosen. Because the space damping is small, the damping ratio is set to be 0.005 according to the empirical value, thus the damping matrix C is expressed as $\begin{eqnarray}\displaystyle C={\alpha}M+{\beta}K. & & \displaystyle\end{eqnarray}$ (2)

Considering the external excitation, the dynamic equation of loop antenna control system can be obtained further $\begin{eqnarray}\displaystyle M\ddot{x}+C{\dot{x}}+Kx=P_{1}f(t) & & \displaystyle\end{eqnarray}$ (3) where P ₁ is the position matrix of disturbance excitation, and f(t) is the disturbance excitation vector.

Fig. 1.

The loop antenna structure.

Fig. 2.

Experimental model.

2.2. Dynamic model of voice coil motor-Kevlar fiber actuator

Here, assuming n voice coil motors are used as actuators to control structural vibration and connected to the antenna structure by Kevlar fibers. Based on ampere theory, the voice coil motor can produce electromagnetic actuating force in the magnetic field. The force produced is proportional to the length of the coil, the current in the wire and the intensity of magnetic field. The exact expression for i-th (i = 1, …, n) actuator is as follows $\begin{eqnarray}\displaystyle F_{i}=BIL & & \displaystyle\end{eqnarray}$ (4) where B is magnetic field strength of voice coil motor, I is the current of voice coil motor and L is the coil length.

Fig. 3.

Dynamic model of actuator.

According to the configuration of voice coil motor-Kevlar fiber actuator, its dynamic model can be illustrated in Fig. 3, where m _e, c _e and k _e are the mass, damping and stiffness of voice coil motor, k _f is the stiffness of Kevlar fiber, y _i is the displacement of inertial body of i-th voice coil motor, F _i is the output force of i-th voice coil motor, x _i1 and x _i2 are the displacement of the connecting nodes at both ends of the i-th actuator. This dynamic model takes the dynamic of i-th voice coil motor and property of Kevlar fiber into account and can be formulated as $\begin{eqnarray}\displaystyle m_{e}\ddot{y_{i}}-c_{e}\dot{y_{i}}-(k_{e}+k_{f})y_{i}+c_{e}\dot{x_{i1}}+k_{e}x_{i1}+k_{f}x_{i2}=F_{i}. & & \displaystyle\end{eqnarray}$ (5)

2.3. Governing equation of the active control system

Combining Eq. (5) with Eq. (3) and rearranging the terms associated with actuators, the governing equation of loop antenna control system can be expressed as $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cc@{}}M & 0\\ 0 & M_{a}\end{array}\right]\left\{\begin{array}{@{}c@{}}\ddot{x}\\ \ddot{y} \end{array}\right\}+\left[\begin{array}{@{}cc@{}}C & 0\\ C_{e} & C_{a}\end{array}\right]\left\{\begin{array}{@{}c@{}}{\dot{x}}\\ {\dot{y}}\end{array}\right\}+\left[\begin{array}{@{}cc@{}}K & 0\\ K_{e} & K_{a}\end{array}\right]\left\{\begin{array}{@{}c@{}}x\\ y\end{array}\right\}=P_{1}F(t)+P_{2}F_{a} & & \displaystyle\end{eqnarray}$ (6) where M _a, C _a, K _a are the mass, damping and stiffness matrixes of actuators, C _e and K _e are the coupling damping matrix and coupling stiffness matrix introduced by actuators, F is the external excitation vector of loop antenna, F _a is the driving force vector of voice coil motors and F _a = [F ₁, F ₂, …, F _n]^T, P ₁ and P ₂ are the position matrixes of excitation applied on the loop antenna and the driving forces of voice coil motors.

2.4. Positions placement of actuators

Imposing fixed constraints in six directions at the fixed end of the loop antenna, the characteristic equation of the loop antenna can be obtained as $\begin{eqnarray}\displaystyle (K-{\omega}^{2}M){\phi}=0. & & \displaystyle\end{eqnarray}$ (7)

From Eq. (7), the modal frequencies and modal shapes of the loop antenna can be computed. The first modal frequency of antenna is 0.33 Hz and the first mode shape of the antenna is the rotation around the fixed end in the XOY plane as shown in Fig. 4 which is referred to as “shaking-head” mode.

Fig. 4.

The first modal shape of loop antenna.

Fig. 5.

Positions of actuators.

In order to get good vibration control effect for the first mode of loop antenna, two voice coil motors (denoted by the actuator A and B) are used. Each actuator is installed on the straight bar of loop antenna and connected to the stand bar by Kevlar fiber as shown in Fig. 5. With this placement, when the loop antenna rotates anticlockwise around the fixed end, the voice coil motor A can be used to reject the motion of antenna. By contrary, the voice coil motor B can be used to reject the clockwise rotation of antenna. As a result, the first modal vibration of loop antenna may be suppressed by driving the voice coil motors A and B alternately with appropriate control algorithm.

3. The simulated studies of active vibration control

3.1. Active vibration control system

Figure 6 shows the principle of the active vibration control system of loop antenna. The displacement response of feedback point is fed into the controller and then drive two voice coil motors alternately through the selector switch. Here, the motion direction of loop antenna is judged according to the positive and negative of displacement response of the feedback point, and then the corresponding actuator is driven. The displacement response of the monitoring point is used to evaluate the vibration control effect.

Fig. 6.

Principle of active vibration control system.

3.2. The control simulation of the first modal vibration

Firstly, the vibration control simulation under sinusoidal excitation is carried out in this paper. Figure 7 shows the control effect under the sine excitation. It is seen that the displacement response amplitude of node 1 (shown in Fig. 5) is 3.7 mm without control. Under the PD control, the displacement response amplitude is 0.9 mm with 76% attenuation compared to the uncontrolled one. Moreover, the displacement response amplitude of system is decreased to 0.3 mm (92% attenuation) under Fuzzy control as shown in Fig. 7(b).

Fig. 7.

The control effect under the sine excitation.

The results shown in Fig. 8 are the vibration control effect under impulse excitation with PD control. In this paper, the decay time of the system is defined as the time it takes to reduce system maximum displacement response by 95%. The maximum displacement amplitude of node 1 is 2.2 mm and the decay time is 250 s without control as shown in Fig. 8(a). With PD control the system decay time is shortened to 45 s and reduced by 82%. In order to improve the pointing accuracy of loop antenna further, the Fuzzy PD control (F-PD control) strategy is investigated as well. It is observed from Fig. 8(b) that the system decay time is shortened to 39 s and reduced by 85% under F-PD control.

Fig. 8.

The control effect under the impulse excitation.

4. The experimental studies of active vibration control

4.1. The experimental system

Figure 9 shows the experimental setup of control system. It mainly includes a dSPACE system and a driving system, which converts the feedback signals collected by the sensors into the control signals by controller, and then sends the control signals to the current amplifier to drive the voice coil motors. The voice coil motors are installed according to the position layout of actuators adopted in the simulated studies so as to restrain the “shaking-head” modal vibration as shown in Fig. 10. The displacement response of feedback point along Y axis is measured using a laser displacement meter. In the active vibration control experiments, two kinds of external excitations, i.e. sine excitations and half sine impulse are considered.

Fig. 9.

The experimental setup.

Fig. 10.

Position of actuator.

4.2. The active control experiment

Figure 11 shows the control effect under the sine excitation. It is seen that the displacement response amplitude of feedback point is 1.2 mm without control. The displacement response amplitude is attenuated to 0.25 mm with 79% reduction compared to the uncontrolled one under PD control. While, the displacement response amplitude is decreased to 0.05 mm with 96% reduction under Fuzzy control.

The results shown in Fig. 12 are the vibration control effect under impulse excitation with PD control. The decay time is 310 s without control. The system decay time is shortened to 21 s with 93% reduction under PD control. Moreover, the system decay time is shortened to 13 s under F-PD control as seen in Fig. 12(b).

5. Conclusions

The paper proposed the voice coil motor-Kevlar fiber actuator to control the low frequency vibration of loop antenna. The governing equation of active control system was established with finite element method taking the characteristics of voice coil motor into consideration. The position placement of actuators was determined to obtain good control effect. The different control strategies, i.e. PD control and Fuzzy control were investigated. The simulated and experimental results show that for the first modal vibration of loop antenna, the displacement response amplitude can be attenuated by more than 76% under the sine excitation and the decay time can be reduced by more than 79% under the impulse excitation with the present control method. Besides, Fuzzy control exhibits better control performance than PD control. It should be noted that control experiments exhibit better control performance than control simulations. These may attribute to underestimation on structural damping ratio in simulations and introduction of addition damping by structure suspension system in experiments. As a matter of fact, the work demonstrates the feasibility and effectiveness of voice coil motor-Kevlar fiber actuator applied to vibration control of loop antenna. The work provides a promising control way for active control of low frequency and large displacement vibration in space engineering.

Fig. 11.

The control effect under the sine excitation.

Fig. 12.

The control effect under the impulse excitation.

Footnotes

Acknowledgements

This research was supported by the National Natural Science Foundations of China (No. 11872290 and No. U1430129).

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