## 4 Analysis Model
### One-Dimensional Vector Form Instinct Finite Element Method, VFIFE  
During the action of the ISAMD, the directional joint of the ISAMD is applied to lock or unlock the connection between the ISAMD and the structure. Thus, the Vector Form Instinct Finite Element Method (VFIFE) is applied to analyze the structural responses of the structure with the ISAMD. The steps of the analysis are as follows:  
Step 1: Discretization  
The VFIFE method is used to discretize the joints of the nodes and elements, where the nodes are the vertices of the element. Nodes are assigned to the mass of the element and become the contact point between two separate elements. The element provides internal force based on the overall coordinates of its nodes and is assigned to the node. The equation is as follows:  
\[\overset{\rightarrow}{F}_{in,j}^{k}(t_{i})=\overset{\rightarrow}{f}\left( \overset{\rightarrow}{u}(t_{i}),\overset{\rightarrow}{u}(t_{i}),\Omega^{k}\right) \tag{2}\]  
where  
\(\overset{\rightarrow}{F}_{in,j}^{k}(t_{i})\) is the internal force vectors of the \(k\) element at time point \(t_{i}\) to maintain this element at the time the node coordinate state is in the \(j_{th}\) node. This internal force is affected by the equation of motion, which includes mass matrix, damping matrix and stiffness matrix. \(\overset{\rightarrow}{u}(t_{i})\) and \(\overset{\rightarrow}{u}(t_{i})\) are the system displacement vector and velocity vector at time point \(t_{i}\), respectively;  
\(\Omega^{k}\) represents the properties of the \(k_{th}\) element, including the material properties and geometric definitions. Step 2: Force Equilibrium  
At each time point \(t_{i}\), the internal forces acting on each node associated with the element can be calculated by the node displacement and velocity vector at that time. The external and internal forces on the degrees of freedom of all nodes can be accumulated to achieve the unbalanced force vector on each node's degree of freedom. The equation can be expressed as follows:  
\[\Delta\overset{\rightarrow}{F}_{j}(t_{i})=\overset{\rightarrow}{F}_{extj}( t_{i})-\overset{n_{e}}{\underset{k=1}{\sum}}\overset{\rightarrow}{F}_{in,j}^{k}(t_{i}) \tag{3}\]  
where  
\(\Delta\overset{\rightarrow}{F}_{j}(t_{i})\) is the balance force vector of the \(j_{th}\) node at time point \(t_{i}\) and  
\(\overset{\rightarrow}{F}_{ext,j}(t_{i})\) is the external force vector of the \(j_{th}\) node at time point \(t_{i}\). Step 3: Predict displacement vector at the next time step \(t_{i+1}\)  
Figure 6: Control Low flowchart of the ISAMD. The centered difference scheme is applied to explicitly estimate the displacement reaction at time point \(t_{i+1}\) under the condition of acceleration responses as a known condition at time point \(t_{i}\):  
\[\ddot{\vec{u}}_{j}(t_{i})=\frac{\overrightarrow{u}_{j}(t_{i+1})-2\overrightarrow {u}_{j}(t_{i})+\overrightarrow{u}_{j}(t_{i-1})}{\Delta t^{2}} \tag{4}\]  
where  
\(\ddot{\overrightarrow{u}}_{j}(t_{i})\) is the acceleration vector of the \(j_{th}\) node at time point \(t_{i}\) and  
\(\Delta t\) is an analytical stride. Then, the displacement reaction at time point \(t_{i+1}\) can be expressed as follows:  
\[\dddot{u}_{j}(t_{i+1})={M_{j}}^{-1}\Delta\overrightarrow{F}_{j}(t_{i})\Delta t ^{2}+2\dddot{u}_{j}(t_{i})-\dddot{u}_{j}(t_{i-1}) \tag{5}\]  
where \(M_{j}\) is the mass matrix of the \(j_{th}\) node. It is a diagonal matrix. Step 4: Repeat the above steps to complete the analysis. Setting of the Analysis Model for the Shock Absorption Effect of a 10-Story Building with the ISAMD  
To explore the reaction to seismic forces of a structure with multiple degrees of freedom and the ISAMD and define the effective interval of the control parameters of the ISAMD for control system design, a 10-story shearing building under excitation of various earthquake records was investigated.