## 4 Kinematic Conceptual Design of a Six-Link VCR Engine Mechanism
### Dimensional Synthesis Using Displacement Equations  
For the dimensional syntheses of the Stephenson III and Watt II mechanisms, the displacement equations are derived. For this, the four-bar linkage in both mechanisms with links \(\mathbf{r}_{0}\), \(\mathbf{r}_{1}\), \(\mathbf{r}_{2}\), and \(\mathbf{r}_{3}\) shown in the skeleton diagram of Table 1 is analyzed as follows [24].  
The vector loop-closure equation of the four-bar linkage can be written as  
\[\mathbf{r}_{1}+\mathbf{r}_{2}=\mathbf{r}_{0}+\mathbf{r}_{3}, \tag{8}\]  
or, equivalently as  
\[\mathbf{r}_{2}=\mathbf{r}_{0}+\mathbf{r}_{3}-\mathbf{r}_{1}. \tag{9}\]The \(x\) and \(y\) components of Equation (9) can be written respectively as  
\[r_{2}\cos\theta_{2}=r_{0}\cos\theta_{0}+r_{3}\cos\theta_{3}-r_{1}\cos\theta_{1}, \tag{10}\]  
\[r_{2}\sin\theta_{2}=r_{0}\sin\theta_{0}+r_{3}\sin\theta_{3}-r_{1}\sin\theta_{1}, \tag{11}\]  
where \(r_{i}\) is the length of \(\mathbf{r}_{i}\) and \(\theta_{i}\) is the angle of \(\mathbf{r}_{i}\) measured from the positive \(x\) axis.  
Square Equations (10) and (11), and sum to obtain  
\[\begin{array}{ll}r_{2}^{2}=&r_{0}^{2}+r_{1}^{2}+r_{3}^{2}+2r_{0}r_{3}(\cos \theta_{0}\cos\theta_{3}+\sin\theta_{0}\sin\theta_{3})\\ &-2r_{0}r_{1}(\cos\theta_{0}\cos\theta_{1}+\sin\theta_{0}\sin\theta_{1})-2r_{ 1}r_{3}(\cos\theta_{1}\cos\theta_{3}+\sin\theta_{1}\sin\theta_{3}).\end{array} \tag{12}\]  
Equation (12) can be written in terms of \(cos\theta_{3}\) and \(sin\theta_{3}\) as  
\[A\cos\theta_{3}+B\sin\theta_{3}+C=0, \tag{13}\]  
where  
\[\begin{array}{ll}A=2r_{0}r_{3}\cos\theta_{0}-2r_{1}r_{3}\cos\theta_{1},\\ B=2r_{0}r_{3}\sin\theta_{0}-2r_{1}r_{3}\sin\theta_{1},\\ C=r_{0}^{2}+r_{1}^{2}+r_{3}^{2}-r_{2}^{2}-2r_{0}r_{1}(\cos\theta_{0}\cos\theta _{1}+\sin\theta_{0}\sin\theta_{1}).\end{array} \tag{14}\]  
Substitute the tangent half-angle identities into Equation (13) and solve the resulting equation to obtain  
\[\theta_{3}=2\tan^{-1}\left(\frac{-B\pm\sqrt{B^{2}-C^{2}+A^{2}}}{C-A}\right), -\pi\leq\theta_{3}\leq\pi. \tag{15}\]  
Equation (15) states that there are two \(\theta_{3}\) values for a given angle \(\theta_{1}\), which correspond to two assembly modes. Dividing Equation (11) by Equation (10) gives the coupler angle \(\theta_{2}\) as  
\[\theta_{2}=\tan^{-1}\left(\frac{r_{0}\sin\theta_{0}+r_{3}\sin\theta_{3}-r_{1} \sin\theta_{1}}{r_{0}\cos\theta_{0}+r_{3}\cos\theta_{3}-r_{1}\cos\theta_{1}} \right). \tag{16}\]  
Now, the position of joint \(P_{2}\) can be written as  
\[\mathbf{P}_{2}=\left[\begin{array}{c}P_{2x}\\ P_{2y}\end{array}\right]=\left[\begin{array}{c}r_{0x}\\ r_{0y}\end{array}\right]+\left[\begin{array}{c}r_{3}\cos\theta_{3}\\ r_{3}\sin\theta_{3}\end{array}\right]. \tag{17}\]  
For the candidate mechanisms in Table 1 to be used in a VCR engine, there must be a crank connected to the ground link by an \(R\) joint in the four-bar linkage. In this research, it is assumed that the four bar is a crank and rocker. By Grashof criteria [25], the crank must be the shortest link and the following relation must hold for a crank-rocker four-bar linkage.  
\[L+S<I2, \tag{18}\]  
where \(L\) is the length of the longest link, \(S\) is the length of the shortest link, and \(I2\) is the sum of the remaining two link lengths.  
The positions of the other joints in the each mechanism in Table 1 can be determined as follows.