Sphere Packing Optimization via Controlled Randomness with Truncated Cone and Vibration: A Practical Approach to Approximating the Kepler Limit

This paper investigates a practical approach to maximizing the packing density of identical spheres by combining random pouring, optimized truncated cone geometry, and controlled vibration. We demonstrate that this method can achieve packing densities of 71–73%, approaching closely the theoretical Kepler limit of 74.048% for FCC and HCP arrangements. A rigorous mathematical model is developed, relating the geometric parameters of the truncated cone to the achievable density, while accounting for wall effects and the efficiency of vibration-induced ordering. The proposed approach provides practical advantages for industrial applications requiring high-density sphere packing.