Collisions can be classified as completely elastic or inelastic. Collision mechanics theory has gradually developed from elastic to inelastic collision theories. Based on the Hertz elastic collision contact theory and Zener inelastic collision theory model, we derive and explain the Hertz and Zener collision theory model equations in detail in this study and establish the Zener inelastic collision theory, which is a simple and fast calculation of the approximate solution to the nonlinear differential equations of motion. We propose an approximate formula to obtain the Zener nonlinear differential equation of motion in a simple manner. The approximate solution determines the relevant values of the collision force, material displacement, velocity, and contact time.

Needle-free injections are mainly used for administering human or mammalian vaccines or drugs. However, poultry vaccines, in ovo injections to embryos, subcutaneous injections to chickens, and intramuscular injections are administered using needle injections. This article presents a new needle-free in ovo injection device method that uses push-pull solenoids to eject liquid jets, mainly for embryonic eggs of chickens. Furthermore, our study investigated the suitable jet pressures for using this method and the post-injection hatching rates in 18-day-old embryonic eggs. Using this method, we could deliver the liquid to the allantoic and amniotic cavities or the muscle tissue through the egg membrane of the air chamber using a jet pressure of ~6–7 MPa or ~8 MPa. After injecting 0.25 mL of 0.9% saline into 18-day-old Lohmann breed layer embryonic eggs and specific pathogen-free (SPF) embryonic eggs at a jet pressure of ~7 MPa, we observed hatching rates of 98.3% and 85.7%, respectively. This study’s electromagnetic needle-free in ovo injection device can apply vaccine or nutrient solution injection for embryo eggs and serve as a reference for future studies on needle-free in ovo injection automation systems, jet pressure control, and injection pretreatment processes.