This paper introduces a new explicit integration method for second-order ordinary differential equations (ODEs) commonly encountered in engineering applications. Traditionally, these problems are solved either by reformulating them as first-order systems to apply one-step methods such as Runge–Kutta schemes, or by using direct second-order approaches widely adopted in linear dynamics, including the generalized-α, central difference, and Newmark methods. The proposed method is derived from a Taylor series expansion truncated at the third derivative, resulting in a fully explicit algorithm that requires only one function evaluation per time step. Similar to Newmark’s formulation, it includes adjustable parameters that allow the user to balance accuracy and stability. For a specific parameter choice, the method exhibits convergence and stability properties comparable to those of the central difference scheme. An important advantage is that it remains explicit even when nonlinearities depend on first-derivative terms. The paper presents a theoretical analysis covering stability, local truncation error, spectral properties, numerical damping, and period elongation. The method is validated through four test cases from multibody dynamics, including linear and nonlinear problems. Results demonstrate that the Explicit Integration Grade 3 (EIG-3) method achieves accuracy comparable to existing explicit second-order integrators while significantly reducing computational cost, particularly in nonlinear applications.
Urkullu et al. (2026) studied this question.