## 5 Vibration Control of an Isolated Structure
### Basic Principle of Structural Control  
A scaled structural prototype model is built to verify and improve seismic mitigation performance. A base isolation device with a novel MRE (or with MWCNTs) is installed underneath the base to support the vertical upper loading, and at the same time to protect the building in case of lateral seismic shock. Laminated MRE can provide horizontal adaptive controllable flexibility as a smart base isolator while the steel sheets can provide adequate vertical loading capacity. Five viscoelastic models are incorporated to model the adjustable stiffness and damping properties of the \"smart\" isolator. When a seismic wave is approaching, online structural status is captured by the sensors installed on the structure, and the developed control algorithm is correspondingly applied to the isolation system, making it adaptively react to the changing structural status. At the same time, a changing magnetic field is generated across the isolation device which tunes the stiffness and damping properties of the smart material that reacts to external disturbances. Thus,  
Figure 16: The effect of suppressing acceleration between two materials.  
the structure can be well controlled with the developed control algorithms and the smart material. This process is clearly depicted in Figure 17.  
Assuming that a typical state-space form of a \((n+1)\) degree of freedom structure is expressed as the following structural dynamics expression:  
\[M\ddot{x}(t)+C\dot{x}(t)+Kx(t)=Pu(t)+Qw(t) \tag{16}\]  
where \(x(t)=[x_{n},\ x_{n-1},\cdots,x_{1},x_{p}]^{T}\) represents the \((n+1)\) order vector, and \(x_{i}\) is the displacement of the \(i\)th floor. \(M,K,C\) are \((n+1)\times(n+1)\) mass matrix, stiffness matrix and damping matrix, respectively, and \(m_{i},k_{i}\) and \(c_{i}\) are the corresponding elements of mass, stiffness and damping matrix. Specifically, \(k_{isl}(t)\) and \(c_{isl}(t)\) are stiffness and damping coefficients of the smart base isolator, respectively.  
\[M=\left[\begin{array}{cccc}m_{n}&&&\\ &m_{n-1}&&\\ &&\ddots&\\ &&&m_{1}\end{array}\right] \tag{17}\]  
\[K=\left[\begin{array}{cccc}k_{n}&-k_{n}&&\\ -k_{n}&k_{n}+k_{n-1}&-k_{n-1}&0\\ &\ddots&\ddots&\ddots&\\ 0&-k_{2}&k_{2}+k_{1}&-k_{1}\\ &&&-k_{1}&k_{1}+k_{isl}(t)\end{array}\right] \tag{18}\]  
\[C=\left[\begin{array}{cccc}c_{n}&-c_{n}&&\\ &-c_{n}&c_{n}+c_{n-1}&-c_{n-1}&0\\ &\ddots&\ddots&\ddots&\\ 0&-c_{2}&c_{2}+c_{1}&-c_{1}\\ &&&-c_{1}&c_{1}+c_{isl}(t)\end{array}\right] \tag{19}\]  
In Equation (16), \(P\) and \(Q\) respectively represent \((n+1)\times m\) location matrix of the control force and \((n+1)\times s\) location matrix of the external loads, \(u(t)=[u_{1},\ u_{2},\cdots,u_{m}]^{T}\) and \(u(t)=[u_{1},\ u_{2},\cdots,u_{m}]^{T}\) are respectively a vector of \(m\) control forces and \(s\) external environmental loads or disturbances. The state space equation of Equation (16) is expressed as follows:  
\[\dot{z}(t)=Az(t)+Bu(t)+Ew(t) \tag{20}\]  
Figure 17: Structural Vibration Control Principle.  
where \(z(t)=\left[z_{1},\,z_{2},\cdots,z_{2n+2}\right]^{T}\) is a \((2n+2)\) state vector, \(A\) is a \((2n+2)\times(2n+2)\) system matrix, \(B\) is a \((2n+2)\times m\) control matrix and \(D\) is a \((2n+2)\times s\) disturbance matrix. These have the following forms:  
\[z(t)=\left[\begin{array}{c}x(t)\\ \dot{x}(t)\end{array}\right] \tag{21}\]  
\[A=\left[\begin{array}{cc}0&I\\ -M^{-1}K\,-M^{-1}C\end{array}\right] \tag{22}\]  
\[B=\left[\begin{array}{c}0\\ M^{-1}P\end{array}\right] \tag{23}\]  
\[E=\left[\begin{array}{c}0\\ M^{-1}Q\end{array}\right] \tag{24}\]  
Herein, some parameters are assigned as follows: \(m_{3}=m_{2}=m_{1}=m_{b}=10\) kg, \(k_{3}=k_{2}=k_{1}=100\), and \(c_{3}=c_{2}=c_{1}=50\), \(P=Q=\left[0\,0\,0\,1\right]^{T}\) means the control force and the external excitation are applied at the bottom floor, \(k_{isl}(t)\) and \(c_{isl}(t)\) are stiffness and damping coefficients, which can be scaled and assigned according to the five MRE models and their modified models.