# Design and Experiment of a Moving Magnet Actuator Based Jetting Dispenser
## 3 MMA Dispenser Design
### Calculation of Driving Force
\(C_{\rm c}\) = 0.64 is the contraction coefficient at the contraction section depending on the ratio of the cross-section area of the nozzle chamber and nozzle seat. \(Q_{B}\) is the backflow rate in the chamber. \(A_{\rm Con}\) is the contraction area of the nozzle chamber depending on the existing position of the needle tip. \(A_{\rm AN}=2.19\) mm\({}^{2}\) is the cross-section area limited by the ball-seat and needle tip in the annular duct. \(r_{\rm con}\) is the contraction-section radius at the existing needle-position. \(z_{\rm m}=2.5\) mm is the maximum distance from the ball-seat to the needle tip. \(z_{\rm n}\) are the needle displacement at the considered position. \(\varphi\) is the inclined angle of the ball-seat. \(l_{\rm s}\) = 1.25 mm is the ball-seat length. \(r_{\rm s}=1.24\) mm is the ball-seat radius. \(r_{\rm n}\) = 1 mm is the needle radius. \(n\) = 0.5 is the flow behavior index [29]. \(\sigma\) = 0.81 is the needle's dimensionless radius \(\sigma=r_{\rm n}/r_{\rm s}\). \(\lambda\) = 0.72 is the function of \(\sigma\) and n. \(K\) = 0.365 Pa.s\({}^{\rm n}\) is the consistency index. \(Q_{\rm N}\) is the flow rate created by the downward motion of the needle. \(Q_{\rm Noz}\) is the flow rate through the needle. \(a\)(t) is the flow velocity at the nozzle center. The reference value of \(a\)(t) is 4 m/s. The friction force between the O-ring and the needle (\(F_{\rm F}\)) can be calculated by [30]:  
\[F_{\rm G}=F_{\rm D}+F_{\rm P}. \tag{2}\]3  
where \(D_{\rm g}\) = 6 mm and \(d_{\rm np}\) = 3 mm are the cylinder and needle-piston diameters, respectively. \(r_{\rm or}\) = 0.8 mm is the cross-section radius of O-ring, and \(E_{\rm or}\) = 4 N/m\({}^{2}\) is the elastic modulus of the nitrile rubber material. In Equation (13), the precondition is that the value of \(D\)g-\(d\)np must be less than these of 4\(r_{\text{or}}\) so that it is not a complex expression. The acceleration force of load (\(F_{\text{M}}\)) occurs at the beginning of the filling and dispensing processes and is calculated by equation \(F_{\text{M}}=m_{L}.a\), in which, \(m_{L}\) is the mass total of the needle (\(m_{\text{n}}\)), connector nut (\(m_{\text{n}}\)), MMA core (\(m_{\text{s}}\)), permanent magnets (\(m_{\text{pm}}\)), and ferromagnetic discs (\(m_{\text{fd}}\)). \(a\) is the acceleration \(a=dv_{n}/dt\). The spring force is calculated using [PERSON]'s law \(F_{\text{sp}}=kz_{\text{c}}\). k is the spring stiffness \(k=2\) N/mm. The RMS force is expressed by the equation as below [31]:  
\[F_{\rm G}=F_{\rm D}+F_{\rm P}. \tag{2}\]4  
\[F_{\rm G}=F_{\rm D}+F_{\rm P}. \tag{2}\]5  
where \(t_{1},t_{2},t_{3}\), and \(t_{4}\) are the acceleration time, run time, deceleration time, and dwell time, respectively. The results obtained from the theoretical calculation of the dispensing cluster include the damping force (\(F_{\text{D}}=57\times 10^{-3}\) N), upward force (\(F_{\text{P}}=0.81\) N), resistance force of the glue (\(F_{\text{G}}=0.867\) N), friction force of the O-ring (\(F_{\text{F}}=1.84\) N), acceleration force of the mass (\(F_{\text{M}}=6.15\) N), and spring force (\(F_{\text{Sp}}=5\) N), peak force (\(F_{\text{Pe}}=13.65\) N), and the root-mean-square force (\(F_{\text{RMS}}=8.08\) N).