The sloshing motion of a confined liquid inside a vertically moving tank is analyzed in the present series of paper. The main objective of the study is to understand the multiple resulting energy dissipation mechanisms, namely wall-liquid impacts and free surface phenomena, among others. This analysis is connected to the damping effects on the aircraft wings caused by the liquid action inside the fuel tanks. Due to the complexity and nonlinearity of the flow generated inside the tank, only experiments or efficient numerical solvers are suitable for studying the problem. In this paper, the tank-fluid system is periodically excited with a prescribed law of motion and the nonlinear features are observed, the force between the wall and the fluid and the global energy balance are computed. A weakly compressible smoothed particle hydrodynamics model has been reformulated and adapted to this kind of violent and turbulent flow. The evolution of the different terms that appear in the energy conservation law are computed, and the formulation is compared to other alternatives where the advantages of the present formulation are indicated. The comparison with the experimental results and the fluid-structure interaction case is carried out in Part II [Marrone, Numerical study on the dissipation mechanisms in sloshing flows induced by violent and high-frequency accelerations. II. Comparison against experimental data, Phys. Rev. Fluids 6, 114802 (2021)10.1103/PhysRevFluids.6.114802] of this work.