Electrohydrodynamic eigenfrequency and jetting frequency analysis via Lagrangian dynamical model

Electrohydrodynamic (EHD) printing has emerged as a promising technique for high-resolution additive manufacturing, but the frequency limits and optimal operating ranges for drop-on-demand EHD printing remain elusive. This lack of clarity has significant implications for printing efficiency and accuracy. Here, we present a theoretical model based on the Lagrange equation to calculate the eigenfrequency in EHD printing. Our study elucidates the dynamic relationships between the eigenfrequency and key process parameters (e.g., voltage, meniscus diameter, and center angle), and reveals the intrinsic relationship between the jetting frequency and the oscillation eigenfrequency, demonstrating that the jetting frequency is limited by the eigenfrequency. This conclusion has significant implications for the design of high frequency EHD printing systems; it highlights opportunities to increase the eigenfrequency by optimizing parameters such as meniscus radius/center angle. In addition, it is interesting to note that the high-order-mode jetting phenomenon can be utilized to break through the traditional frequency limit of the low-order mode. This work can serve as an instructive benchmark of the EHD printing frequency, which provides a basis for designing a high-frequency EHD printing system.