This paper focuses on predicting the shape, duration, and peak magnitude of the displacement, velocity, and acceleration curves while dropping weight over a viscoelastic fractal cushioning packaging system. Furthermore, to capture high-frequency harmonic components observed during the impact time span, the approximate frequency–amplitude expression of the governing equation of motion will be obtained via an ancient Chinese algorithm and He’s formulation by assuming initial trial solutions based on Jacobi elliptic functions. Numerical simulations confirmed the ability of the Jacobi elliptic functions to capture high-frequency harmonics observed during the cushioning packaging system dynamic response with a root mean square error (RMSE) value that does not exceed 0.0218, which is an indication of the great accuracy attained from our derived solution when compared to the exact numerical one. Furthermore, our derived approximate solution predicts that when a weight is dropped over the cushioning packaging, the cushion material with smaller porosity will absorb the produced kinetic energy faster.