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Active vibration isolation technology is the key technique to solve the vibration isolation problems related to the multisources complex excitations vibration isolation system. The electromagnetic actuators-based multisources complex excitations active vibration isolation system is built. Additionally, in view of the complex structure and strong coupling of the system, the least-squares method to identify and obtain the mathematical model of the vibration isolation system is adopted. Furthermore, this paper also sets up the acceleration feedback-based PID control model for multisources complex excitations active vibration isolation system, proposes an improved particle swarm optimization (PSO) algorithm of dynamic inertia weight factors used to optimize parameters of the built PID control model, and conducts simulation analysis. The simulation results show that, compared with the passive system before the control, the multisources complex excitations active vibration isolation system under the PID control has the far less peak-to-peak amplitude of acceleration which is transmitted to the foundation and has the much better vibration isolation effect. Finally, the paper conducts experimental verification, which demonstrates that active vibration control effect is identical to the simulation results and the vibration control effect is significantly improved.

Along with the development of ships, the mechanical structure inside the ships becomes more and more complex, and so do the excitation mode and the components of excitation signals. Due to the limited load capacity and inner space of the ships, a majority of power devices are installed together rigidly or flexibly to form the isolation object with multiple excitation sources. As the vibration excitations generated by each equipment are different in amplitude, phase, frequency, and direction, those excitation signals are mixed and overlapped, resulting in complicated and variable frequency and phase of excitation signals which happen on the entire vibration isolation object [

Active vibration isolation technology can be carried out from three aspects: structural optimization of the active vibration isolation system, control of active vibration isolation system, and active vibration isolator (actuator) [

Among the abovementioned research papers related to active vibration isolation, a majority focus on single-layer and double-layer active vibration isolation system with single excitation source, studying dynamics model and control strategy and algorithm, while some propose active control policies and guidelines of multifreedom degree vibration isolation system. However, there have yet been no deep and comprehensive studies about whether resultant effect can be achieved by jointly using multiple active vibration isolators. Multisources complex excitations active vibration isolation system is the system which achieves vibration isolation by giving play to combined effect of active and passive vibration isolation components. It is the typical mechatronic system. Due to the complicated and changeable excitation, strong nonlinear relationship, and interconnection among different excitation sources and different isolators via structure, the system is very complicated. Thus, it is difficult to build a mathematical model which meets the requirements with the analysis method. Experimental data include all information of the model and the use of model identification method in the experimental data is an effective way to solve such issues [

The response amplitude of the acceleration delivered to the foundation or the attenuation ratio in the course of acceleration passing is an important indicator to measure the vibration isolation effect of mechanical equipment. Therefore, taking the strong coupling, nonlinear characteristics of complex multisources excitation active vibration isolation system, and the difficulty in establishing an accurate model using traditional kinetic methods into account, the least-squares model identification method is adopted to establish the active vibration isolation system model under the multiple complex excitation sources and build the intelligent PID control model which is based on acceleration feedback. In addition, this paper proposes the improved PSO algorithm of dynamic inertia weight factor, optimizes the built PID control parameters, and conducts simulation analysis and finally experimental verification.

The model of multisources complex excitation active vibration isolation system is shown in Figure

Multisources complex excitations active vibration isolation system.

Establishing the model of multisources complex excitations active vibration isolation system is the base and premise for the active vibration isolation. Modeling methods for the active vibration isolation system can be divided into two types: the first is analytical method, including multibody dynamics method, finite element method, and four terminal parameters’ method, which make certain reasonable assumptions and provide description with accurate mathematical expression. For a more complex system, however, there are general cases where certain parameters cannot be determined and then meet the demand for modeling. The second is system identification method, which is suitable for the system whose mechanism is unclear or overly complex. For a system with unclear or overly complicated mechanism, the system model structure and parameters are determined by input/output data collected from the system under a certain condition, with no need to have a deep understanding of its internal mechanism, which is especially a complex one. The multisources complex excitations active vibration isolation system is a system that achieves the vibration isolation by the combined action of active and passive vibration-isolating elements, which is the typical mechatronics system. There is the strong nonlinearity due to the complex and changeable excitation signals and excitation sources intercouple via structures. Excitation sources and active vibration isolators intercouple with each other as well. Active and passive vibration isolators are also of features of coupling and nonlinearity. Besides, as the system is complex, it is hard to establish the mathematical model meeting requirements with the analytical method adopted. As the experiment data measured includes all the information related to the model, adopting the model identification method is an effective way to solve such issues. The least-squares system identification method is to determine the system model parameters with squares and functions of generalized error minimized by adopting the least-squares principle. The method can be used both for the linear and for the nonlinear systems, which is of features of concise principle, rapid convergence, being easy to implement, and so forth and is most widely used in the system identification. In addition, the recursive least-squares method considering both the amount of calculation and accuracy for the model identification of the multisources complex excitation active vibration isolation system is adopted. The experimental system of experimental data required by the identification model is shown in Figure

Data acquisition system of model identification experiment.

The amplitude frequency curves and phase frequency curves of identification model and experimental data model for the input signal voltage of active vibration isolator 3-1 and output acceleration signal of accelerometer 6-2 are shown in Figure

Frequency response curves of VA12 experiment model and identification system model.

The amplitude frequency curves and phase frequency curves of identification model and experimental data model for the input signal voltage of active vibration isolator 3-1 and output acceleration signal of accelerometer 6-5 are shown in Figure

Frequency response curves of VA15 experiment model and identification system model.

The amplitude frequency curves and phase frequency curves of identification model and experimental data model for the input signal voltage of active vibration isolator 3-2 and output acceleration signal of accelerometer 6-2 are shown in Figure

Frequency response curves of VA22 experiment model and identification system model.

The amplitude frequency curves and phase frequency curves of identification model and experimental data model for the input signal voltage of active vibration isolator 3-2 and output acceleration signal of accelerometer 6-5 are shown in Figure

Frequency response curves of VA25 experiment model and identification system model.

In addition, the models between the signal of accelerometer 6-2 and the signals of 6-1, 6-3, 6-4, and 6-6 accelerometers are, respectively, AA21, AA23, AA24, and AA26; the models between the 6-5 signal of accelerometer and the signals of 6-1, 6-3, 6-4, and 6-6 accelerometers are, respectively, AA51, AA53, AA54, and AA56. The approach to obtain identification model is consistent with the foregoing.

Control is the core part of active vibration isolation system and can be divided into feedback control and feedforward control. Feedback control is applied to the situation in which there are many disturbance factors and they are not detectable. It is particularly suitable for the control of complex systems and systems with uncertain parameters. In terms of multisources excitation active vibration isolation system, the system is complicated and has complex disturbance signals. Therefore, feedback control is suitable for the system. Furthermore, PID controller is the most widely used feedback controller. It has the features of simple structure, good stability, reliable working performance, and strong robustness. Particularly, when the structure and parameters of the controlled object cannot be fully grasped or when precise mathematical model cannot be obtained, feedback controller also has very good control effect. The control effect of PID controller mainly depends on the three parameters of P, I, and D. Determining the control parameters is the main content of designing PID control system. Conventional PID controller obtains the PID control parameters by means of manual adjustment which is time-consuming and does not guarantee the best performance. The PID controller based on intelligent and improved algorithm obtains PID control parameters through the improvement of intelligent algorithm by means of integrating the PID controller with modern intelligent algorithms such as genetic algorithm, ant colony algorithm, and PSO. Compared with conventional PID control, it has the abilities of self-learning and self-adaption. Such PID controller can automatically identify the parameters of controlled process and adapt to the changes of the parameters of controlled process. Besides, like the conventional PID controller, it also has the features of simple structure, good stability, reliable working performance, and strong robustness. Compared with other intelligent algorithms, PSO has the advantages of simple structure, fewer parameters, fast convergence, easy implementation, and so forth. For this reason, the PID controller of multisources complex excitations active vibration isolation system based on PSO is established. The purpose of active vibration isolation is to reduce the transmission of vibration acceleration, so the acceleration signals of 6-2 and 6-5 are as the feedback signals of PID controller. The controller carries out calculation and feedback output based on the control parameters obtained through the improved PSO algorithm. The amplified output signal regulates the current in active vibration isolation electromagnetic coil to generate a controllable electromagnetic force, and then active control can be conducted for multisources vibration. The control schematic diagram is shown in Figure

Control schematic diagram of multisources complex excitations active vibration isolation system.

PSO algorithm has the advantages of simple structure, fewer parameters, fast convergence, easy implementation, and so forth. Its space complexity and time complexity are relatively low, and it has been proved that good optimal solutions can be obtained with relatively small calculation cost [

In PSO algorithm, each particle represents a potential solution to the problem, and each particle corresponds to a fitness value determined by the fitness function. The speed of the particles determines the direction and distance of the movement of those particles. In

Inertia weight factor

The optimization of PID controller is to determine a suitable set of parameters

The bridge between improved PSO algorithm and PID control model is the particle and the corresponding fitness value of the particle. The particle is the parameters of PID controllers, and the format of particle is

The optimization process of PID parameters based on improved PSO is as follows.

Set up the optimum fitness and maximum number of evolution generations to determine the termination condition: the optimum fitness is achieved, or the maximum evolution generation number is reached.

Initial particle swarm is generated randomly. Determine the number of initial populations, the maximum speed and minimum speed of particle swarm, and the positions and speeds of all randomly generated particles. Set up the upper and lower limits of particle parameters to determine the initial inertia weights at the beginning and at the end of the improved PSO algorithm.

Assign the value of the particles to PID control parameters

Transmit the fitness values to PSO as the adaptation value of the particle. Find the best fitness value to see whether the termination condition is satisfied. If so, exit the program.

If the termination condition is not satisfied, use the algorithm of (

Detailed optimization process of PID parameters based on improved PSO can be found in Figure

Optimization process for PID using improved PSO.

Optimization for the parameters of PID controllers based on improved PSO is carried out. Initial particle swarms are generated randomly, and the number of initial particle swarms is 100. In terms of the positions and speeds of all particles generated randomly, the largest and smallest speeds of particle swarms are 1 and −1, and the best fitness value setting is

Improved PSO iterative process and the optimum fitness value.

Improved PSO iterative process and the parameter values of PID controller 1.

Improved PSO iterative process and the parameter values of PID controller 2.

There are multiple excitation sources and complex interference signals of active isolation system. In order to determine whether the established PID controllers and control parameters obtained through optimization are valid, simulation analysis for active isolation system is carried out under the double sine interference signals and double linear swept sine interference signals based on actual conditions.

Taking the actual operating conditions into account, when both motor vibration sources are sinusoidal signals, the 1-1 source signal is 50 Hz sinusoidal signal, and the 1-2 source signal is 100 Hz sinusoidal signal. By using the control parameters obtained through the optimization of improved PSO algorithm, the isolation effects before and after control of the isolation system are analyzed, and the simulation time is 1 s. Figures

Comparison time-frequency domain curves of acceleration response of position 6-1 on the isolation system before and after being controlled under double sinusoidal interference.

Comparison time-frequency domain curves of acceleration response of position 6-2 on the isolation system before and after being controlled under double sinusoidal interference.

Comparison time-frequency domain curves of acceleration response of position 6-3 on the isolation system before and after being controlled under double sinusoidal interference.

Comparison time-frequency domain curves of acceleration response of position 6-4 on the isolation system before and after being controlled under double sinusoidal interference.

The time-frequency domain curves of the output acceleration of the 6-1 position in the isolation system before and after being controlled are shown in Figure ^{2} without the control to active system’s 0.17 m/s^{2} under the control. The peak of active system’s maximum acceleration response is only 1/4 of passive system. In the frequency domain curves, the vertical axis represents the magnitude of the acceleration response, and the horizontal axis is frequency. Likewise, the dotted line represents the response curve of acceleration before being controlled (passive), and the solid line is the response curve of acceleration after being controlled (active). It can be discovered that, at 100 Hz, acceleration amplitude is reduced from 0.22 m/s^{2} without the control to 0.066 m/s^{2} under the control; the amplitude is reduced by 2/3. After being controlled at 50 Hz, there is also corresponding decrease in terms of the amplitude.

The time-frequency domain curves of the output acceleration of the 6-2 position in the isolation system before and after being controlled are shown in Figure ^{2} without the control to active system’s 0.073 m/s^{2} under the control. The peak of active system’s maximum acceleration response is less than 1/3 of passive system. In the frequency domain curves, the vertical axis represents the magnitude of the acceleration response, and the horizontal axis is frequency. Likewise, the dotted line represents the response curve of acceleration before being controlled (passive), and the solid line is the response curve of acceleration after being controlled (active). It can be discovered that acceleration amplitude is reduced from 0.085 m/s^{2} before being controlled to 0.026 m/s^{2} after being controlled; the amplitude is reduced by around 70%. At 50 Hz, the acceleration response amplitude reduces from 0.038 m/s^{2} without the control to 0.0076 m/s^{2} under the control; the amplitude is reduced by around 80%.

The time-frequency domain curves of the output acceleration of the 6-3 position in the isolation system before and after being controlled are shown in Figure ^{2} without the control to active system’s 0.107 m/s^{2} under the control. The peak of active system’s maximum acceleration response is only about 1/4 of passive system. In the frequency domain curves, the vertical axis represents the magnitude of the acceleration response, and the horizontal axis is frequency. Likewise, the dotted line represents the response curve of acceleration before being controlled (passive), and the solid line is the response curve of acceleration after being controlled (active). It can be discovered that, at 100 Hz, acceleration amplitude is reduced from 0.124 m/s^{2} before being controlled to 0.038 m/s^{2} after being controlled; the amplitude is reduced by around 60%. At 50 Hz, the acceleration response amplitude reduces from 0.040 m/s^{2} without the control to 0.019 m/s^{2} under the control; the amplitude is reduced by more than 50%.

The time-frequency domain curves of the output acceleration of the 6-4 position in the isolation system before and after being controlled are shown in Figure ^{2} without the control to active system’s 0.097 m/s^{2} under the control. The peak of active system’s maximum acceleration response is less than 1/3 of passive system. In the frequency domain curves, the vertical axis represents the magnitude of the acceleration response, and the horizontal axis is frequency. Likewise, the dotted line represents the response curve of acceleration before being controlled (passive), and the solid line is the response curve of acceleration after being controlled (active). It can be discovered that, at 50 Hz, acceleration amplitude is reduced from 0.106 m/s^{2} before being controlled to 0.0446 m/s^{2} after being controlled; the amplitude is reduced by around 60%. At 100 Hz, the acceleration response amplitude reduces from 0.061 m/s^{2} without the control to 0.016 m/s^{2} under the control; the amplitude is reduced by more than 75%.

Given the complexity of the actual interference, in order to further validate the effectiveness of the PID control model of multiple excitation active vibration isolation system and PID control parameters obtained through the optimization of improved PSO algorithm, and considering the actual operating conditions, simulation analysis is carried out against the linear swept sine signals when the 1-1 and 1-2 vibration source signals are both 10–200 Hz under the situation that both motor vibration sources are linear swept sine signals. The simulation time is 1 s. Figures

Comparison time-frequency domain curves of acceleration response of position 6-1 on the isolation system before and after being controlled under double linear swept sine interference.

Comparison time-frequency domain curves of acceleration response of position 6-2 on the isolation system before and after being controlled under double linear swept sine interference.

Comparison time-frequency domain curves of acceleration response of position 6-3 on the isolation system before and after being controlled under double linear swept sine interference.

Comparison time-frequency domain curves of acceleration response of position 6-4 on the isolation system before and after being controlled under double linear swept sine interference.

The time-frequency domain curves of the output acceleration of position 6-1 in the isolation system before and after being controlled under the double linear swept sine interference signals are shown in Figure ^{2} without the control to active system’s 0.34 m/s^{2} under the control. The peak of active system’s maximum acceleration response is only 1/10 of passive system. In the frequency domain curves, the vertical axis represents the magnitude of the acceleration response, and the horizontal axis is time. Likewise, the dotted line represents the response curve of acceleration before being controlled, and the solid line is the response curve of acceleration after being controlled. It can be discovered that the peak of maximum acceleration response is reduced from 0.094 m/s^{2} before being controlled to 0.011 m/s^{2} after being controlled. It is reduced by about 90%. The peaks of other frequency ranges are also significantly reduced, and the increase of isolation effect is obvious.

The time-frequency domain curves of the output acceleration of position 6-2 in the isolation system before and after being controlled under the double linear swept sine interference signals are shown in Figure ^{2} without the control to active system’s 0.378 m/s^{2} under the control. The peak of active system’s maximum acceleration response is only around 1/10 of passive system. In the frequency domain curves, the dotted line represents the response curve of acceleration before being controlled, and the solid line is the response curve of acceleration after being controlled. It can be discovered that the peak of maximum acceleration response is reduced from 0.114 m/s^{2} before being controlled to 0.0102 m/s^{2} after being controlled. It is reduced by more than 90%. The peaks of other frequency ranges are also significantly reduced, and the increase of isolation effect is obvious.

The time-frequency domain curves of the output acceleration of position 6-3 in the isolation system before and after being controlled under the double linear swept sine interference signals are shown in Figure ^{2} without the control to active system’s 0.324 m/s^{2} under the control. The peak of active system’s maximum acceleration response is less than 1/10 of passive system. In the frequency domain curves, the dotted line represents the response curve of acceleration before being controlled, and the solid line is the response curve of acceleration after being controlled. It can be discovered that the peak of maximum acceleration response is reduced from 0.0987 m/s^{2} before being controlled to 0.0126 m/s^{2} after being controlled. It is reduced by around 90%. The peaks of other frequency ranges are also significantly reduced, and the increase of isolation effect is obvious.

The time-frequency domain curves of the output acceleration of position 6-4 in the isolation system before and after being controlled under the double linear swept sine interference signals are shown in Figure ^{2} without the control to active system’s 0.302 m/s^{2} under the control. The peak of active system’s maximum acceleration response is around 1/5 of passive system. In the frequency domain curves, the dotted line represents the response curve of acceleration before being controlled, and the solid line is the response curve of acceleration after being controlled. It can be discovered that the peak of maximum acceleration response is reduced from 0.0423 m/s^{2} before being controlled to 0.00803 m/s^{2} after being controlled. It is reduced by more than 80%. The peaks of other frequency ranges are also significantly reduced, and the increase of isolation effect is obvious.

In order to further validate the effectiveness of the PID control model of multisources complex excitations active vibration isolation system and the PID control parameters obtained through the improved PSO algorithm, an experimental platform of multiple excitation active vibration isolation system is established, as shown in Figure

Multisources complex excitations active vibration isolation experimental platform.

Actual vibration acceleration signals of two excitation motors.

Time-frequency domain curves of the acceleration signal in 6-2 motor position

Time-frequency domain curves of the acceleration signal in 6-5 motor position

Control block diagram of multiexcitation active vibration isolation experiment platform.

By using the established PID control model and the improved PSO algorithm, the parameters of control system are obtained. The acceleration response amplitudes of positions 6-1 to 6-4 on the experiment platform before and after being controlled are shown in Figures

Comparison time-frequency domain curves of acceleration response of position 6-1 on the experimental platform before and after being controlled.

Comparison time-frequency domain curves of acceleration response of position 6-2 on the experimental platform before and after being controlled.

Comparison time-frequency domain curves of acceleration response of position 6-3 on the experimental platform before and after being controlled.

Comparison time-frequency domain curves of acceleration response of position 6-4 on the experimental platform before and after being controlled.

The time-frequency domain curves of the output acceleration of position 6-1 in the isolation experimental platform before and after being controlled are shown in Figure ^{2} before being controlled to active system’s 0.1839 m/s^{2} after being controlled. The maximum acceleration response of the active isolation system is only 35% of the passive system. In the frequency domain curves, the vertical axis represents the magnitude of the acceleration response, and the horizontal axis is time. Likewise, the dotted line represents the response curve of acceleration before being controlled, and the solid line is the response curve of acceleration after being controlled. It can be discovered that the maximum peak amplitude (at 100 Hz) of acceleration response reduces from 0.1715 m/s^{2} before being controlled to 0.03744 m/s^{2} after being controlled, which means it is reduced by 78%. The decrease of other peak amplitudes is also quite significant; the increase of isolation effect is obvious.

The time-frequency domain curves of the output acceleration of position 6-2 in the isolation experimental platform before and after being controlled are shown in Figure ^{2} before being controlled to active system’s 0.1019 m/s^{2} after being controlled. The maximum acceleration response of the active isolation system is only around 1/3 of the passive system. In the frequency domain curves, the dotted line represents the response curve of acceleration before being controlled, and the solid line is the response curve of acceleration after being controlled. It can be discovered that the maximum peak amplitude (at 100 Hz) of acceleration response is reduced from 0.1021 m/s^{2} before being controlled to 0.02548 m/s^{2} after being controlled, which means it is reduced by 75%. The increase of isolation effect is obvious.

The time-frequency domain curves of the output acceleration of position 6-3 in the isolation experimental platform before and after being controlled are shown in Figure ^{2} before being controlled to active system’s 0.13078 m/s^{2} after being controlled. The maximum acceleration response of the active isolation system is only around 45% of the passive system. In the frequency domain curves, the dotted line represents the response curve of acceleration before being controlled, and the solid line is the response curve of acceleration after being controlled. It can be discovered that the maximum peak amplitude (at 200 Hz) of acceleration response is reduced from 0.04424 m/s^{2} before being controlled to 0.03103 m/s^{2} after being controlled, which means it is reduced by 25%. The decrease of other peak amplitudes is also quite significant.

The time-frequency domain curves of the output acceleration of position 6-4 in the isolation experimental platform before and after being controlled are shown in Figure ^{2} before being controlled to active system’s 0.09127 m/s^{2} after being controlled. The maximum acceleration response of the active isolation system is only around 28% of the passive system. In the frequency domain curves, the dotted line represents the response curve of acceleration before being controlled, and the solid line is the response curve of acceleration after being controlled. It can be discovered that the maximum peak amplitude (at 200 Hz) of acceleration response is reduced from 0.04336 m/s^{2} before being controlled to 0.01407 m/s^{2} after being controlled, which means it is reduced by more than 68%. The decrease of other peak amplitudes is also quite significant.

Multisources excitation active vibration isolation system has complex structure and strong coupling ability. For those features, the model identification method is adopted to establish active isolation system model, and the model identification accuracy is over 90%. PID control model based on acceleration feedback is established, and an improved PSO algorithm of a dynamic inertia weight factor is put forward. Optimization for established PID control parameters is conducted. In order to verify the effectiveness of control model and the control parameters obtained by improved PSO algorithm, simulation analysis is carried out under the functions of double sine interference signals and double sweep-frequency signals. The simulation results show that the vibration acceleration amplitudes of active multisources excitation isolation system after being controlled are much smaller than the passive system before being controlled, and the isolation effect of the active system is much better than those of the passive system. In the end, experimental verification is carried out. The vibration isolation effect of the active control of multisources excitation isolation system agrees with the simulation results. The maximum output acceleration amplitude of the active system is only 40% of the passive system, and the vibration isolation effect is significantly improved.

The authors declare that they have no competing interests.

The authors would like to acknowledge the National Natural Science Foundation of China (no. 51205296, no. 51275368, and no. U1537103).