# Research on Semi-Active Vibration Control of Pipeline Based on Magneto-Rheological Damper
## 3 Research on Semi-Active Pipeline Vibration Control System  
### State Equation of Pipeline System  
The pipeline vibration control system is composed of pipeline, fluid medium in pipeline and MR damper, as shown in Figure 6. In the working process of pipeline system, fluid pulsation inside pipeline and vibration excitation outside pipeline act on the pipeline together, which will cause pipeline vibration. The pipeline vibration control system based on the MR damper can dynamically adjust the damping force of the MR damper according to the vibration characteristics of the pipeline, thereby controlling the vibration of the pipeline.  
Figure 5: Dynamic characteristics of MR damper under control current of 0, 2, 4 A: (**a**) Force-displacement curve (0 A); (**b**) Force-velocity curve (0 A); (**c**) Force-displacement curve (2 A); (**d**) Force-velocity curve (2 A); (**e**) Force-displacement curve (4 A); (**f**) Force-velocity curve (4 A).  
Figure 6: Schematic diagram of pipeline system.  
The motion equation of pipeline under environmental disturbance can be expressed as:  
\[\left\{\begin{array}{l}M\ddot{\mathbf{X}}(t)+\dot{\mathbf{X}}(t)+\mathbf{K}\mathbf{X}(t)=\mathbf{D}_ {s}\mathbf{F}(t)\\ \mathbf{X}(t_{0})=\mathbf{X}_{0}\quad\dot{\mathbf{X}}(t_{0})=\dot{\mathbf{X}}_{0}\end{array}\right. \tag{3}\]  
where, \(\mathbf{X}\) is the pipeline displacement vector, \(M\), \(\mathbf{C}\) and \(\mathbf{K}\) are the mass, damping and stiffness matrix of pipeline respectively, \(\mathbf{D}_{s}\) is the environmental disturbance location matrix, \(\mathbf{X}(t_{0})\) and \(\dot{\mathbf{X}}(t_{0})\) are the initial displacement and velocity of pipeline.  
In order to control the vibration response of pipeline system, MR dampers are installed in the pipeline system. The control force provided by MR damper is \(\mathbf{U}(t)\), and the corresponding position matrix of MR damper is \(\mathbf{B}_{s}\). Then the motion equation of the pipeline under the control of the MR damper can be expressed as:  
\[M\ddot{\mathbf{X}}(t)+\dot{\mathbf{C}}\dot{\mathbf{X}}(t)+\mathbf{K}\mathbf{X}(t)=\mathbf{D}_{s}\mathbf{F}( t)+\mathbf{B}_{s}\mathbf{U}(t) \tag{4}\]  
Since the displacement \(\mathbf{X}(t)\) and velocity \(\dot{\mathbf{X}}(t)\) of the pipeline system are independent variables, the system state vector can be expressed as:  
\[\mathbf{Z}(t)=\left[\begin{array}{c}\mathbf{X}(t)\\ \dot{\mathbf{X}}(t)\end{array}\right]_{2n\times 1} \tag{5}\]  
In the state space, the controlled piping system described by Equation (4) can be expressed as:  
\[\left\{\begin{array}{l}\dot{\mathbf{Z}}(t)=\mathbf{A}\mathbf{Z}(t)+\mathbf{B}\mathbf{U}(t)+\mathbf{ D}\mathbf{F}(t)\\ \mathbf{Z}(t_{0})=\mathbf{Z}_{0}\end{array}\right. \tag{6}\]  
In Equation (6):  
\[\mathbf{A}=\left[\begin{array}{cc}\mathbf{0}_{n}&\mathbf{I}_{n}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array}\right]_{2n\times 2n} \tag{7}\]  
\[\mathbf{B}=\left[\begin{array}{c}\mathbf{0}_{n\times p}\\ \mathbf{M}^{-1}\mathbf{B}_{s}\end{array}\right]_{2n\times p} \tag{8}\]  
\[\mathbf{D}=\left[\begin{array}{c}\mathbf{0}_{n\times r}\\ \mathbf{M}^{-1}\mathbf{D}_{s}\end{array}\right]_{2n\times r} \tag{9}\]  
where \(\mathbf{I}_{n}\) is the unit matrix.  
The output state of pipeline system controlled by MR damper can be expressed as:  
\[\mathbf{Y}(t)=\mathbf{C}_{0}\mathbf{Z}(t)+\mathbf{D}_{0}\mathbf{F}(t)+\mathbf{B}_{0}\mathbf{U}(t) \tag{10}\]  
where \(\mathbf{Y}(t)\) is the output vector, \(\mathbf{C}_{0}\) is the output matrix, \(\mathbf{D}_{0}\) and \(\mathbf{B}_{0}\) are the direct transfer matrix.  
Synthesizing (6) and (10), the controlled pipeline system under environmental disturbances can be described as:  
\[\left\{\begin{array}{l}\dot{\mathbf{Z}}(t)=\mathbf{A}\mathbf{Z}(t)+\mathbf{B}\mathbf{U}(t)+\mathbf{ D}\mathbf{F}(t)\quad\quad\mathbf{Z}(t_{0})=\mathbf{Z}_{0}\\ \mathbf{Y}(t)=\mathbf{C}_{0}\mathbf{Z}(t)+\mathbf{B}_{0}\mathbf{U}(t)+\mathbf{D}_{0}\mathbf{F}(t)\end{array}\right. \tag{11}\]