# Effect of Tuned Spring on Vibration Control Performance of Modified Liquid Column Ball Damper
## 2 Mathematical Modelling
### Tuned Liquid Column Ball Spring Damper (TLCBSD)
#### 2.1.1 Tuned Liquid Column Ball Spring Sliding Damper (TLCBSSD)  
The TLCBSSD arrangement the total kinetic energy (\(T\)) of the system shown in Figure 1 is given as  
\[T=T_{f}+T_{s}+T_{b} \tag{1}\]  
where \(T_{f}\), \(T_{s}\), and \(T_{b}\) are the kinetic energy of the fluid, primary structure, and ball, respectively. \[T_{f}=\frac{1}{2}\int_{V_{1}}p\ (\dot{y}^{2}+\dot{x}_{s}^{2})dv+\frac{1}{2}\int_{V _{2}}p(\dot{y}+\dot{x}_{s})^{2}dv+\frac{1}{2}\int_{V_{3}}p\Big{(}\dot{x}_{s}^{2 }+\dot{y}^{2}\Big{)}dv \tag{2}\]  
where \(V_{1}\) and \(V_{3}\) are the fluid volume in the left and right vertical portion of the tube, respectively; \(V_{2}\) is the volume of liquid in the horizontal portion of the tube; \(x_{s}\) is the displacement of the primary structure; and \(y\) is the displacement of the fluid surface. The integration of Equation (2) results in  
\[T_{f}=phA(\dot{y}^{2}+\dot{x}_{s}^{2})+\frac{1}{2}pbA(\dot{y}+\dot{x}_{s})^{2} \tag{3}\]  
In Equation (3), \(p\) is the density of the fluid, \(A\) is the cross-sectional area of the tube, and \(h\) and \(b\) are the undisturbed length of fluid in the vertical section and horizontal section of the tube, respectively. The kinetic energy of the structure is given in Equation (4), where \(m_{s}\) is the mass of the structure attached to the TLCBSSD. \[T_{s}=\frac{1}{2}m_{s}\dot{x}_{s}^{2} \tag{4}\]  
The kinetic energy of the ball is given in Equation (5), where \(m_{b}\) is the mass of the ball. \[T_{b}=\frac{1}{2}m_{b}\dot{x}_{b}^{2} \tag{5}\]  
The total kinetic energy of the system is  
\[T=phA(\dot{y}^{2}+\dot{x}_{s}^{2})+\frac{1}{2}pbA(\dot{y}+\dot{x}_{s})^{2}+ \frac{1}{2}m_{s}\dot{x_{s}}^{2}+\frac{1}{2}m_{b}\dot{x}_{b}^{2} \tag{6}\]  
Figure 2: Structure with a tuned liquid column ball spring rolling damper (TLCBSRD). The total potential energy (\(U\)) of the system is as follows, where \(U_{f}\), \(U_{s}\), and \(U_{b}\) are the potential energy of the fluid, structure, and ball, respectively. \[U=U_{f}+U_{s}+U_{b} \tag{7}\]  
The potential energy associated with the fluid is  
\[U_{f}=\int_{V_{1}}pgzdv+\int_{V_{3}}pgzdv \tag{8}\]  
Integrating the above Equation (8) leads to  
\[U_{f}=\frac{1}{2}pgA(h-y)^{2}+\frac{1}{2}pgA(h+y)^{2} \tag{9}\]  
\[T_{f}=\frac{1}{2}\int_{V_{1}}p\ (\dot{y}^{2}+\dot{x}_{s}^{2})dv+\frac{1}{2}\int_{V _{2}}p(\dot{y}+\dot{x}_{s})^{2}dv+\frac{1}{2}\int_{V_{3}}p\Big{(}\dot{x}_{s}^{2 }+\dot{y}^{2}\Big{)}dv \tag{2}\]0  
where \(g\) is the gravitational acceleration. The potential energy associated with the structure can be expressed as  
\[T_{f}=\frac{1}{2}\int_{V_{1}}p\ (\dot{y}^{2}+\dot{x}_{s}^{2})dv+\frac{1}{2}\int_{V _{2}}p(\dot{y}+\dot{x}_{s})^{2}dv+\frac{1}{2}\int_{V_{3}}p\Big{(}\dot{x}_{s}^{2 }+\dot{y}^{2}\Big{)}dv \tag{2}\]1  
where \(k_{s}\) is the fundamental stiffness of the structure. The potential energy of the ball can be presented as  
\[T_{f}=\frac{1}{2}\int_{V_{1}}p\ (\dot{y}^{2}+\dot{x}_{s}^{2})dv+\frac{1}{2}\int_{V _{2}}p(\dot{y}+\dot{x}_{s})^{2}dv+\frac{1}{2}\int_{V_{3}}p\Big{(}\dot{x}_{s}^{2 }+\dot{y}^{2}\Big{)}dv \tag{2}\]2  
where \(k_{bs}\) is the stiffness of the spring attached to the ball. Combining Equations (10)-(12) results in  
\[T_{f}=\frac{1}{2}\int_{V_{1}}p\ (\dot{y}^{2}+\dot{x}_{s}^{2})dv+\frac{1}{2}\int_{V _{2}}p(\dot{y}+\dot{x}_{s})^{2}dv+\frac{1}{2}\int_{V_{3}}p\Big{(}\dot{x}_{s}^{2 }+\dot{y}^{2}\Big{)}dv \tag{2}\]3  
The external and damping forces associated with \(x_{s}\), \(y\), and \(x_{b}\) are \(Q_{1}\), \(Q_{2}\), and \(Q_{3}\), respectively.