## 2 Mechanical Analysis
### Mass-Matrix Characterization of a Six-Bar Watt Linkage Using FCC  
Figure 1 depicts a six-bar Watt linkage consisting of five moving links and one degree of freedom. Each link is associated with a local coordinate system (\(x_{n}\) axis and \(y_{n}\) axis), where the origin is positioned at point \(i\) and the \(x\)-axis is oriented toward point \(j\). The physical properties of each link include its mass \(m_{bn}\) and a center of gravity defined by the local coordinates \((x_{bn},y_{bn})\) for \(1\leq n\leq 5\).  
Three important vectors must be defined to characterize the mass matrix of the mechanism: vector \(\mathbf{q}\) (Equation (1)), which represents the position of the basic points; vector \(\mathbf{\dot{q}}\) (Equation (2)), which represents the velocity of the basic points; and vector \(\mathbf{\ddot{q}}\) Equation (3)), which represents the acceleration of the basic points.  
\[\mathbf{q}=[A_{x}\ A_{y}\ B_{x}\ B_{y}\ C_{x}\ C_{y}\ D_{x}\ D_{y}\ E_{x}\ E_{y}\ F_{x}\ F_{y}\ G_{x}\ G_{y}]^{T} \tag{1}\]\[\dot{\textbf{q}}=[V_{Ax}\ V_{Ay}\ V_{Bx}\ V_{By}\ V_{Cx}\ V_{Cy}\ V_{Dx}\ V_{Dy}\ V_{Ex}\ V_{ Ey}\ V_{Fx}\ V_{Fy}\ V_{Gx}\ V_{Gy}]^{T} \tag{2}\]  
\[\ddot{\textbf{q}}=[A_{Ax}\ A_{Ay}\ A_{Bx}\ A_{By}\ A_{Cx}\ A_{Cy}\ A_{Dx}\ A_{Dy}\ A_{Ex}\ A_{Ey}\ A_{Fx}\ A_{Fy}\ A_{Gx}\ A_{Gy}]^{T} \tag{3}\]  
The mechanical method proposed to optimize dynamic balance in this research consists of adding fixed counterweights to each of the bars to redistribute the mass, thereby shifting the center of mass of the linkage to improve balancing. Figure 2 illustrates the mechanism with the added counterweights for each linkage. The physical properties of each counterweight include its mass (\(m_{cn}\)) and a center of gravity located at the local coordinates of its corresponding link, i.e., \((x_{cn},y_{cn})\) for \(1\leq n\leq 5\).  
Figure 1: Six-bar Watt linkage to be optimized.  
Figure 2: Six-bar Watt linkage with counterweights.  
After defining the mass matrix [22] for each linkage (considering the bar-counterweight system), it is possible to derive the mass matrix that characterizes the entire mechanism.  
Equation (4) presents the mass matrix (**M**) of the mechanism depicted in Figure 2, where \(a_{n}\), \(b_{n}\),..., \(j_{n}\) represent the elements of the individual mass matrices for each linkage, with \(n=1,\ldots,5\).  
\[\textbf{M}=\left[\begin{array}{cccccccccccc}a_{2}&0&0&0&b_{2}&c_{2}&0&0&0&0&0&0 &0&0\\ 0&a_{2}&0&0&-c_{2}&b_{2}&0&0&0&0&0&0&0\\ 0&0&c_{3}&0&0&0&0&0&s_{3}&0&f_{3}&0&0&0\\ 0&0&0&0&c_{3}&0&0&0&0&0&s_{3}&0&f_{3}&0&0\\ b_{2}&c_{2}&0&0&d_{2}+e_{1}&0&g_{1}&0&f_{1}&0&0&0&0\\ -c_{2}&b_{2}&0&0&0&d_{2}+e_{1}&0&g_{1}&0&f_{1}&0&0&0&0\\ 0&0&0&0&g_{1}&0&a_{4}+j_{1}&0&i_{1}&0&0&b_{4}&c_{4}\\ 0&0&0&0&0&g_{1}&0&a_{4}+j_{1}&0&i_{1}&0&0&-c_{4}&b_{4}\\ 0&0&g_{3}&0&f_{1}&0&i_{1}&0&h_{1}+b_{3}&0&i_{3}&0&0\\ 0&0&0&g_{3}&0&f_{1}&0&i_{1}&0&h_{1}+i_{3}&0&i_{3}&0&0\\ 0&0&f_{3}&0&0&0&0&i_{3}&0&4s+h_{3}&0&b_{5}&c_{5}\\ 0&0&0&f_{3}&0&0&0&0&0&i_{3}&0&a_{5}+h_{3}&-c_{5}&b_{5}\\ 0&0&0&0&0&0&h_{4}&c_{4}&0&0&b_{5}&c_{3}&d_{4}+d_{5}&0\\ 0&0&0&0&0&-c_{4}&b_{4}&0&0&-c_{5}&b_{5}&0&d_{4}+d_{5}\end{array}\right] \tag{4}\]  
The definitions of \(a_{n}\), \(b_{n}\), \(c_{n}\), \(d_{n}\), \(e_{n}\), \(f_{n}\), \(g_{n}\), \(h_{n}\), \(i_{n}\), and \(j_{n}\), where the \(n\) suffix denotes the corresponding linkage number, can be found in [22].