# Study of Lead Rubber Bearings for Vibration Reduction in High-Tech Factories
## 2 Finite Element Modeling of Lead Rubber Bearings  
The LRB, as shown in Figure 1, is a single or multiple lead core built into laminated rubber to reduce structural horizontal vibration during earthquakes. Because the laminated rubber has high vertical stiffness, low horizontal stiffness, and high recovery and lead metal has low yield stress, combining the characteristics of the two makes the LRB a good vibration isolation device. [PERSON] et al. [12] proposed a two-way LRB model as below:  
\[\{\text{P}\}=\left\{ \begin{array}{c}P_{x}\\ P_{y}\end{array} \right\}=\alpha\frac{F_{y}}{Y}\{\text{U}\}+(1-\alpha)F_{y}\{ \text{Z}\}=k_{d}\{\text{U}\}+(1-\alpha)F_{y}\{\text{Z}\} \tag{1}\]  
Figure 1: Illustration of the lead rubber bearing (LRB) containing a lead core and laminated rubber. \[Y\left\{ \begin{array}{c}\dot{Z}_{x}\\ \dot{Z}_{y}\end{array} \right\}=\left(A[\mathrm{I}]\ -\ \left[ \begin{array}{cc}z_{x}^{2} \left(\gamma Sign\left(\dot{U}_{x}Z_{x}\right)+\beta\right)&Z_{x}Z_{y}\left( \gamma Sign\left(\dot{U}_{y}Z_{y}\right)+\beta\right)\\ Z_{x}Z_{y}\left(\gamma Sign\left(\dot{U}_{x}Z_{x}\right)+\beta\right)&z_{y}^{2 }\left(\gamma Sign\left(\dot{U}_{y}Z_{y}\right)+\beta\right)\end{array} \right] \right)\left\{ \begin{array}{c}\dot{U}_{x}\\ \dot{U}_{y}\end{array} \right\} \tag{2}\]  
where \(\{P\}=[P_{x},\ P_{y}]^{\mathrm{T}}\) is the LRB force vector, \(\{\mathrm{U}\}=[U_{x},\ U_{y}]^{\mathrm{T}}\) is the in-plane displacement vector between the LRB two sides, \(\alpha\) is the ratio of the final LRB stiffness over the initial LRB stiffness (\(\alpha=k_{d}/k_{e},\ k_{e}=F_{y}/\gamma=\) initial LRB stiffness), \(F_{y}\) is the LRB yielding force, \(Y\) is the LRB deflection at the yielding force, \(k_{d}\) is the yielding LRB stiffness, \(\{Z\}=[Z_{x},\ Z_{y}]^{\mathrm{T}}\) is the LRB nonlinear variable, \(\gamma,\ \beta\), and \(A\) are dimensionless parameters to control the shape of the hysteresis loop used in the two-way theory, where \(A/((\gamma+\beta)\)=1, and [I] is a unit vector. Equation (2) is nonlinear, and the Newton-Raphson method can be used to find \(\{Z\}\) using \(\left\{\dot{U}\right\}\) obtained from the finite element analysis. The details can be found in [16]. The finite element stiffness matrix of the shear force contribution is:  
\[K_{shar}=\left[ \begin{array}{cc}K_{11}&K_{12}\\ K_{21}&K_{22}\end{array} \right] \tag{3}\]  
\[ \begin{array}{c}\mathrm{Where},\ K_{11}=\frac{\partial P_{x}}{\partial U_{x} }=\alpha\left(\frac{F_{y}}{\gamma}\right)+(1-\alpha)F_{y}\frac{\partial Z_{x} }{\partial U_{x}},\ K_{12}=\frac{\partial P_{y}}{\partial U_{y}}=(1-\alpha)F _{y}\frac{\partial Z_{x}}{\partial U_{y}}\\ K_{21}=\frac{\partial P_{y}}{\partial U_{x}}=(1-\alpha)F_{y}\frac{\partial Z_ {y}}{\partial U_{x}},\ \mathrm{and}\ K_{22}=\frac{\partial P_{y}}{\partial U_{y}}= \alpha\left(\frac{F_{y}}{\gamma}\right)+(1-\alpha)F_{y}\frac{\partial Z_{y}} {\partial U_{y}}\end{array}  \tag{4}\]  
where \(\frac{\partial Z_{x}}{\partial U_{x}},\frac{\partial Z_{x}}{\partial U_{y}}, \frac{\partial Z_{y}}{\partial U_{x}}\), and \(\frac{\partial Z_{y}}{\partial U_{y}}\), can be found in [16]. Equations (3) and (4) produce an unsymmetrical global stiffness matrix, which may cause the double requirement of computer memory and time. Thus, one can empirically set \(K_{21}=K_{12}=(K_{12}+K_{21})/2\), and use the Newton-Raphson method to obtain the solution with small unbalance forces. The solution is still accurate, since equation (1) is used to find the LRB internal force vector without errors, but the Newton-Raphson iterations may increase when one direction is loading and the other is unloading. For the vertical direction of the LRB stiffness (\(K_{e}\)), a linear spring is used. The original LRB hysteretic curve under low speed loads can be modified as the functions of the wave frequency, wave speed, and axial load [17; 18]. For simplicity, we used the original LRB model for finite element analyses.