# Design and Experiment of a Moving Magnet Actuator Based Jetting Dispenser
## 3 MMA Dispenser Design
### Calculation of Driving Force  
The driving/thrust force (\(F_{\rm th}\)) generated from the MMA must be larger than the root-mean-square force (\(F_{\rm RMS}\)). The \(F_{\rm RMS}\) is calculated based on the peak force (\(F_{\rm Pe}\)), including the resistance force of the glue (\(F_{\rm G}\)), friction force of the O-ring (\(F_{\rm F}\)), acceleration force of the mass (\(F_{\rm M}\)), and the spring force (\(F_{\rm Sp}\)) as:  
\[F_{\rm Pe}=F_{\rm G}+F_{\rm F}+F_{\rm M}+F_{\rm Sp}. \tag{1}\]  
The resistance force of the glue (\(F_{\rm G}\)) including the damping force (\(F_{\rm D}\)) and the upward force in the nozzle seat area (\(F_{\rm P}\)) when the needle moves downward, is expressed by:  
\[F_{\rm G}=F_{\rm D}+F_{\rm P}. \tag{2}\]  
The damping force can be calculated by [27]:  
\[F_{\rm D}=v_{\rm n}\cdot\sum C_{\rm d}=v_{\rm n}2\pi\mu d_{\rm ne}\left(\frac{l _{\rm nc}}{D_{\rm nc}-d_{\rm ne}}+\frac{l_{\rm c}}{D_{\rm c}-d_{\rm ne}}\right), \tag{3}\]  
where \(v_{\rm n}\) is the maximum speed of the needle (320 mm/s), \(C_{\rm d}\) is the damping coefficient, and \(\mu\) is the dynamic viscosity at room temperature (1.49 kg/m s). \(D_{\rm nc}\), \(D_{\rm c}\), and \(d_{\rm ne}\) are the diameters of the nozzle chamber, chamber, and the needle, respectively. \(l_{\rm nc}\) and \(l_{\rm c}\) are the lengths of the nozzle chamber and chamber, respectively. The upward force in the nozzle-seat area can be calculated by [11]:  
\[F_{\rm P}=P_{\rm s}\cdot A_{\rm en}+P_{\rm A}A_{\rm c}, \tag{4}\]  
Figure 4: Configuration of dispensing parts. (**a**) Dimensional specifications; and (**b**) dynamic analysis in the dispensing stage. where \(P_{\rm S}\) is the glue pressure between the nozzle-seat and the needle tip. (\(A_{\rm en}\) = 6.38 mm\({}^{2}\)) is the effective area at the needle end that is directly affected by the \(P_{\rm S}\). \(P_{\rm A}\) is the glue pressure in the chamber. \(A_{\rm C}\) = 4.71 mm\({}^{2}\) is the needle-cap area. In the chamber, the glue pressure \(P_{\rm A}\) is determined using the Bernoulli equation [28]:  
\[P_{\rm A}=P_{\rm sy}+\rho g\Big{(}h_{\rm sy}-h_{\rm ch}\Big{)}+\frac{1}{2} \rho\Big{(}v_{\rm sy}^{2}-v_{\rm ch}^{2}\Big{)}, \tag{5}\]  
where \(P_{\rm sy}\) is the maximum backpressure in the syringe (0.5 Bar); \(h_{\rm sy}=100\) mm and \(h_{\rm ch}=16\) mm are the glue height in the syringe and chamber, respectively; \(\rho\) is the Glycerin density (1261.3 kg/m\({}^{3}\)), and \(v_{\rm gs}=0.1\) mm/s and \(v_{\rm ch}=1.06\) mm/s are the glue speed at the syringe and chamber, respectively. The dot volume is assumed to be 0.5 mm\({}^{3}\). The glue pressure between the needle tip and nozzle seat \(P_{\rm S}\) is given by [20; 29]:  
\[P_{\rm S}=P_{\rm A}+\Delta P_{\rm ce,b}+\Delta P_{\rm ns,b}, \tag{6}\]  
\[\Delta P_{\rm ce,b}=\frac{1}{2}\rho C_{\rm c}\xi_{\rm B}\frac{Q_{\rm B}^{2}}{A _{\rm Con}^{2}}=\frac{1}{2}\rho C_{\rm c}\bigg{(}\frac{1}{2}\Big{(}1-\frac{A_ {\rm Con}}{A_{\rm en}}\Big{)}^{3/4}+\Big{(}1-\frac{A_{\rm Con}}{A_{\rm en}} \Big{)}^{2}\bigg{)}\frac{Q_{\rm B}^{2}}{A_{\rm Con}^{2}}, \tag{7}\]  
\[\Delta P_{\rm ns,b}=\Bigg{[}\frac{(3n+1)Q_{\rm B}+\frac{(3n+1)\pi r_{\rm S}^{2 }}{3n-1}\Big{[}\frac{(3n+1)\big{(}1-\sigma^{2}\big{)}-2n\big{(}1-\sigma^{2} \big{)}}{1-\sigma^{1/n}}\Big{]}|v_{\rm C}|}{n\pi r_{\rm S}^{3}\big{[}(1-\lambda ^{2})^{1+1/n}-\sigma^{1+1/n}(\lambda^{2}-\sigma^{2})^{1+1/n}\big{]}}\Bigg{]}^ {n}\frac{2l_{s}K}{r_{\rm s}}, \tag{8}\]  
\[A_{\rm Con}=2\pi r_{\rm con}(z_{\rm m}-z_{\rm n})cos\rho, \tag{9}\]  
\[F_{\rm G}=F_{\rm D}+F_{\rm P}. \tag{2}\]0  
\[F_{\rm G}=F_{\rm D}+F_{\rm P}. \tag{2}\]1  
\[F_{\rm G}=F_{\rm D}+F_{\rm P}. \tag{2}\]2  
where \(\Delta P_{\rm ce,b}\) is the drop in pressure because of the expansion and contraction of the flow at the needle tip. \(\Delta P_{\rm ce,b}\) is the backflow pressure drop due to the fluid passing through the annular channel between the ball-seat and the needle in the dispensing stage. \(\xi_{\rm B}\) is the local loss factor due to the contraction and expansion of the backflow.