# Seismic Response Analysis of Prestressed Concrete
## 3 Kinematics of the PCRF  
The geometrical relationship of the planar four-bar linkage mechanism for \(\theta_{1}<0\) is shown in Figure 3. Firstly, \(\theta_{1}\) is assumed as a known angle, and \(\varphi_{i}(i=1,2,3)\) are calculation auxiliary angles. From the geometry relationships, we have  
\[\theta_{2}=\varphi_{2}+\gamma-\varphi_{1}\qquad\theta_{3}=\varphi_{2}+\varphi_ {3}-\pi/2 \tag{1}\]  
The equations below can be obtained from sine and cosine law:  
\[\overline{\text{BD}}^{2}(\theta_{1})=(2L)^{2}+\left(2R_{\text{c}}^{\prime} \right)^{2}-8R_{\text{c}}^{\prime}L\cos(\beta-\theta_{1}) \tag{2}\]  
\[(2h)^{2}=\overline{\text{BD}}^{2}(\theta_{1})+\left(2R_{\text{b}}\right)^{2}- 4R_{\text{b}}\overline{\text{BD}}(\theta_{1})\cos\varphi_{1} \tag{3}\]  
\[\frac{\overline{\text{BD}}(\theta_{1})}{\sin(\beta-\theta_{1})}=\frac{2R_{ \text{c}}^{\prime}}{\sin\varphi_{2}} \tag{4}\]  
\[\frac{2h}{\sin\varphi_{1}}=\frac{2R_{\text{b}}}{\sin\varphi_{3}} \tag{5}\]  
The expressions for \(\theta_{2}\) and \(\theta_{3}\) can be derived from Equations (1)-(5). The expressions are written as  
\[\theta_{2}=\arcsin\left[\frac{2R_{\text{c}}^{\prime}}{\overline{\text{BD}}( \theta_{1})}\sin(\beta-\theta_{1})\right]+\gamma-\arccos\left[\frac{4R_{\text {b}}^{2}+\overline{\text{BD}}^{2}(\theta_{1})-4h^{2}}{4R_{\text{b}}\overline {\text{BD}}(\theta_{1})}\right] \tag{6}\]  
\[\theta_{3}=\arcsin\left[\frac{2R_{\text{c}}^{\prime}}{\overline{\text{BD}}( \theta_{1})}\sin(\beta-\theta_{1})\right]+\arcsin\left\{\frac{R_{\text{b}}}{ h}\sin\left\{\arccos\left[\frac{4R_{\text{b}}^{2}+\overline{\text{BD}}^{2}( \theta_{1})-4h^{2}}{4R_{\text{b}}\overline{\text{BD}}(\theta_{1})}\right] \right\}\right\}-\pi/2 \tag{7}\]  
Figure 4: Planar four-bar linkage mechanism of PCRF: (**a**) \(\theta_{1}<0\); (**b**) \(\theta_{1}>0\). Figure 3: Geometrical relationship of planar four-bar linkage mechanism for \(\theta_{1}<0\). Substituting Equation (2) into Equations (6) and (7) yields  
\[ \begin{array}{ll}\theta_{2}(\theta_{1})=&\arcsin\left[\frac{R_{\rm c}^{\prime} }{\sqrt{L^{2}+R_{\rm c}^{\prime 2}-2R_{\rm c}^{\prime}L\cos(\beta-\theta_{1})}}\sin(\beta- \theta_{1})\right]+\gamma\\ &-\arccos\left[\frac{R_{\rm b}^{2}+L^{2}+R_{\rm c}^{2}-2R_{\rm c}^{\prime}L\cos (\beta-\theta_{1})-h^{2}}{2R_{\rm b}\sqrt{L^{2}+R_{\rm c}^{\prime 2}-2R_{\rm c}^{ \prime}L\cos(\beta-\theta_{1})}}\right]\\ \\ \theta_{3}(\theta_{1})=&\arcsin\left[\frac{R_{\rm c}^{\prime}}{\sqrt{L^{2}+R_ {\rm c}^{\prime 2}-2R_{\rm c}^{\prime}L\cos(\beta-\theta_{1})}}\sin(\beta- \theta_{1})\right]\\ &+\arcsin\left\{\frac{R_{\rm b}}{h}\sin\left\{\arccos\left[\frac{R_{\rm b}^{2} +L^{2}+R_{\rm c}^{\prime 2}-2R_{\rm c}^{\prime}L\cos(\beta-\theta_{1})-h^{2}}{2R_{ \rm b}\sqrt{L^{2}+R_{\rm c}^{\prime 2}-2R_{\rm c}^{\prime}L\cos(\beta-\theta_{1})}} \right]\right\}\right\}-\pi/2\end{array}  \tag{8}\]  
Differentiating Equations (8) and (9) with respect to time \(t\), the corresponding angular velocities can be obtained and expressed for simplicity as  
\[\dot{\theta}_{2}\left(\theta_{1},\dot{\theta}_{1}\right)=\frac{\partial\theta_ {2}}{\partial\theta_{1}}\dot{\theta}_{1}=\partial_{\theta_{1}}\theta_{2}( \theta_{1})\cdot\dot{\theta}_{1}\qquad\dot{\theta}_{3}\left(\theta_{1},\dot{ \theta}_{1}\right)=\frac{\partial\theta_{3}}{\partial\theta_{1}}\dot{\theta}_ {1}=\partial_{\theta_{1}}\theta_{3}(\theta_{1})\cdot\dot{\theta}_{1} \tag{10}\]  
where \(\dot{\theta}_{1}\) is the angular velocity of bar AB and the upper dot denotes differentiation with respect to time \(t\). The symbol \(\partial_{\theta_{1}}\) denotes a partial derivative with respect to \(\theta_{1}\).