# Development of Vibration Control Structure on Suspended Ceiling Using Pulley Mechanism
## 2 Pulley-Damper Ceiling System
### Derivation of the Constitutive Equation  
Constitutive equations were established for the proposed pulley-damper system considering wire-cable deformation under the static condition. Hence, the friction force, thermal changes, and the wire mass are not included in the equation. Figure 3 illustrates a schematic diagram for the PDCS. A continuous wire-cable is reciprocated \(n\) times between the movable pulley, \(P_{L1}\), and the fixed pulley, \(P_{L2}\), and similarly between the movable pulley, \(P_{R1}\), and the fixed pulley, \(P_{R2}\). When the SCS deformed displacement \(D\) in the horizontal direction and force, \(f_{L}\) and \(f_{R}\) act on the wire-cable each side (use plus for tension); the force-deformation relationship of the wire-cable in the diagonal part and horizontal part can be expressed both in the left-hand side of Equation (1) and the right-hand side of Equation (2), as follows:  
\[f_{L}=\frac{K_{L1}}{n}\delta_{L1}\cdots\text{Diagonal part} \tag{1a}\] \[f_{L}=K_{L2}\delta_{L2}\cdots\text{Horizontal part}\] (1b) \[f_{R}=\frac{K_{R1}}{n}\delta_{R1}\cdots\text{Diagonal part}\] (2a) \[f_{R}=K_{R2}\delta_{R2}\cdots\text{Horizontal part} \tag{2b}\]  
where \(L_{1}\) and \(\delta_{1}\) are the wire-cable length and deformation in the diagonal part, respectively; \(L_{2}\) and \(\delta_{2}\) are the wire-cable length and deformation in the horizontal part, respectively; \(E\) is the [PERSON]'s module of wire-cable; \(A\) is the cross-sectional area of wire-cable; \(K\) is the axial stiffness of the wire-cable (\(=EA/L\)).  
In this derivation, the wire-cable is modeled as equivalent truss-elements. Thus, the force-deformation relationship can be converted into the axial force, \(N\), and the deformation, \(\delta\), as follows for each side:  
\[N_{L}=nf_{L} \tag{3a}\] \[\delta_{L}=\frac{(\delta_{R1}+\delta_{R2}+x)}{n}=\frac{(\alpha_{L}f_{L}+x)}{n}\] (3b) \[N_{R}=nf_{R}\] (4a) \[\delta_{R}=\frac{(\delta_{R1}+\delta_{R2}-x)}{n}=\frac{(\alpha_{R}f_{R}-x)}{n}\] (4b) where \[\alpha=n/K_{1}+1/K_{2}\]  
Figure 2: Concept schematic for the proposed pulley–damper ceiling system (PDCS).  
As the PDCS consists of one continuous wire, if one side gets shorter, the other side must grow longer. Thus, the following relation should hold:  
\[(\delta_{L1}+\delta_{L2}+x)+(\delta_{R1}+\delta_{R2}-x)=\alpha_{L}f_{L}+\alpha_{ R}f_{R}=0 \tag{5}\]  
In addition, the damper force can be obtained from the difference in the tensile force of both wire sides. Therefore, by substituting Equation (5) into Equation (6), the damper force and the displacement can be established as Equation (7):  
\[Q(x,\dot{x})=f_{L}-f_{R} \tag{6}\]  
\[Q(x,\dot{x})=\Omega\big{(}f-f^{\prime}\big{)}=\bigg{(}\frac{\alpha_{L}+\alpha_ {R}}{\alpha_{R}}\bigg{)}f_{L}=-\bigg{(}\frac{\alpha_{L}+\alpha_{R}}{\alpha_{L} }\bigg{)}f_{R} \tag{7}\]  
Utilizing Equations (3a), (4a), and (7), the axial force of the truss elements can be finally written, as follows:  
\[N_{L}=n\bigg{(}\frac{\alpha_{R}}{\alpha_{L}+\alpha_{R}}\bigg{)}Q(x,\dot{x}),\ N_{R}=n\bigg{(}\frac{\alpha_{L}}{\alpha_{L}+ \alpha_{R}}\bigg{)}Q(x,\dot{x}) \tag{8}\]  
The deformation of the damper is given by:  
\[x=-n\delta_{L}+\delta_{L1}+\delta_{L2}=-n\delta_{1}+\frac{\alpha_{L}N_{L}}{n} \tag{9}\]  
Therefore, the simulation analysis can be implemented by the following procedures:  
* Step 1: Calculate the damper displacement using truss force and displacement from Equation (9);
* Step 2: Determine the damper force \(Q(x,\dot{x})\);
* Step 3: Update the truss axial forces using Equation (8).