## Appendix A Dynamical Results When Using Only Three Counterweights  
This appendix provides a more detailed justification for disregarding counterweights whose thickness is close to zero. In addition, it reports the balancing results obtained when the mechanism operates with only three counterweights instead of five, illustrating that the dynamic response remains practically unchanged when the removed counterweights are assumed to have negligible thickness. For direct comparison, exactly the three solution vectors examined in Section 5 are reconsidered.  
A detailed examination of the constituent terms of the mass matrix **M** (Equation (4)) shows that every entry involving counterweight parameters is directly proportional to the corresponding thickness. Consequently, when a counterweight's thickness approaches zero, the associated terms vanish, and the counterweight ceases to influence either the **ShF** or the \(ShM\). The explicit expressions demonstrating this proportionality are provided in [22], thereby validating the cancellation criterion.  
The results for the representative cases selected in Section 5 are presented below. They confirm that removing counterweights 4 and 5--whose computed thicknesses are close to zero--yields a balancing performance virtually indistinguishable from that obtained with the complete five-counterweight configuration.  
The first solution corresponds to the vector given in Equation (19). The associated performance indices are \(\beta_{\text{ShF}}(X)=0.088122439\) and \(\beta_{\text{ShM}}(X)=0.986658705\), representing reductions of 91.19% in the shaking force (ShF) and 1.33% in the shaking moment (ShM), respectively.  
After removing counterweights four and five, the reduced solution vector becomes:  
\[X =[-0.030831847,-0.039661495,0.035057972,-0.075997350,\] \[-0.007025791,0.038659613,-0.053661036,-0.011334326, \tag{1}\] \[0.039975214,0,0,0,0,0,0].\]  
Substituting this vector into Equations (12) and (13) yield \(\beta_{\text{ShF}}(X)=0.0878687871\) and \(\beta_{\text{ShM}}(X)=0.9866074648\), corresponding to the reductions of 91.21 % in ShF and 1.34 % in ShM.  
Figure 11 displays the shaking force (ShF) and shaking moment (ShM) obtained with this solution. Compared with the balanced-mechanism response shown in Figure 9, the two curves are virtually indistinguishable.  
Alternatively, consider the solution obtained by SMS-EMOA (Equation (20)), corresponding to \(\beta_{ShM}(X)=0.223958809\) and \(\beta_{ShF}(X)=0.904631455\). This solution yields a 77.60% reduction in ShM, while ShF is reduced by 9.54%.  
By discarding counterweights four and five, the solution vector can be represented according to (Equation  
Finally, consider the third solution obtained by SMS-EMOA (Equation (21)), corresponding to \(\beta_{\text{Shf}}(X)=0.495578094\) and \(\beta_{\text{Shf}}(X)=0.495728512\). This solution yields a 50.44% reduction in ShF.  
When discarding counterweights four and five, the solution vector can be represented with Equation (A3)  
\[X= \left[\right.-0.026785451,-0.033672311,0.038997486,-0.074123172,\] \[\left.-0.006617802,0.037072145,-0.057625145,0.006778145,\] (A3) \[\left.0.039998738,0,0,0,0,0,0\right]\]  
Substituting this vector into Equations (12) and (13) yield \(\beta_{\text{Shf}}(X)=0.4955516213\) and \(\beta_{\text{Shf}}(X)=0.4956823663\), which corresponds to a reduction of 50.44 % in ShM, and a reduction of 50.43 % in ShF.  
Figure A3 shows the balanced **Shf** and \(ShM\) obtained when using only three counterweights. When comparing these results with the ones obtained using five counterweights (Figure 11), it can be seen that they are almost identical.