# Research on Semi-Active Vibration Control of Pipeline Based on Magneto-Rheological Damper
## 2 Model Description of MR Damper for Pipeline Vibration Control  
According to the authors' previous research [17; 18], the double-ended shear-valve mode MR damper prototypes are fabricated in accordance with the characteristics of pipeline vibration. The testing system (as shown in Figure 2) includes the INSTRON-8801 test system, MR damper prototype and current controller. When the control current is 0-4 A, the dynamic characteristics of the MR damper are shown in Figure 3.  
It can be seen from Figure 3 that, the damping force of the developed MR damper will enhance with the increasing current. As the current increases from 0 A to 1.5 A, the damping force is significantly proportional to the increment of the current. When the current exceeds 1.5 A, the damping force will slightly increase as the current increases. It is obvious that as the displacement  
Figure 1: Schematic diagram of pipeline vibration control system based on a magneto-rheological (MR) damper: (**a**) Structural diagram; (**b**) Control block diagram. the damping force is slightly influenced under the same current, which implies that the damping force of the MR damper is not critically dependent on the displacement and the velocity. In order to simulate the damping effect of MR damper under different vibration control algorithms, it is necessary to establish a dynamic mathematical model of MR damper and define the model parameters. [PERSON] et al. (2019) proposed an double-sigmoid model, which took into account the effects of control current and excitation properties. The model had concise function form and high recognition accuracy, which could accurately and simply describe the output characteristics of MR dampers. The damping force of MR damper based on the improved double-sigmoid model can be expressed as:  
\[F=f_{d}\frac{1-e^{-a(\dot{x}-x_{0}\text{sgn}(x))}}{1+e^{-a(\dot{x}-x_{0}\text{ sgn}(x))}}+c_{d}\dot{x} \tag{1}\]  
where, \(f_{d}\) is the adjustable coulomb damping force, \(c_{d}\) is the viscous damping coefficient, \(x\) and \(\dot{x}\) are the relative displacement and speed of MR damper piston and cylinder, \(x_{0}\) is the displacement value of the MR damper force-velocity hysteresis curve when the damping force is zero, \(a\) is the velocity adjustment coefficient of coulomb damping force. Therefore, there are four undetermined parameters \(f_{d},c_{d},x_{0}\) and \(a\). Among them, the Coulomb damping force \(f_{d}\) can be calculated, \(c_{d}\) is the MRF zero field dynamic viscosity, \(x_{0}\) can be obtained through the MR damper test data, and so only \(a\) needs to be identified in the model. The parameters of the double-sigmoid model are identified by genetic algorithm, Figure 4a shows the basic flow chart of genetic algorithm. The MATLAB genetic algorithm toolbox is used as shown in Figure 4b. Using the experimental data of the MR damper, the undetermined parameters in the double-sigmoid model can be identified. Figure 3: Dynamic characteristics of the MR damper: (a) Force-displacement curve; (b) Force-velocity curve. Figure 2: MR damper prototype and damping force testing platform. The function type selected through parameter identification is the minimum square value of the difference between simulation force and experimental force, to ensure that the final result of parameter identification can approximate the real value as possible. The fitness function can be expressed as:  
\[\mathit{fitness}=\frac{1}{m}\sum_{i=1}^{m}\left(\frac{\mathbf{F}_{if}-\mathbf{F}_{is}}{ \mathbf{F}_{\max}-\mathbf{F}_{\min}}\right) \tag{2}\]  
where, \(\mathbf{F}_{if}\) is the simulation value, \(\mathbf{F}_{is}\) is the experimental value, \(m\) is the number of test sites, \(\mathbf{F}_{\max}\) is the the maximum value of the initial population, \(\mathbf{F}_{\min}\) is the minimum value of the initial population. The recognition results are shown in Figure 5.  
Figure 4: Genetic algorithm of parameter identification for the double-sigmoid model: (**a**) Flow chart of genetic algorithm; (**b**) Genetic algorithm toolbox. Figure 5: _Cont._