Abstract: The BN-stable method is a novel time integration method for solving dynamic problems, offering accurate and efficient solutions to complex nonlinear vibration systems. Since its introduction in 2021, this method has garnered significant attention from the academic communities. This paper presents the method's design philosophy, solution framework, and parameter optimization principles. Through benchmark examples—a higher-order nonlinear single-degree-of-freedom system and a rigid satellite model—the BN-stable method is compared with two classical approaches: the Generalized-α method and the ρ_∞-Bathe method. Results demonstrate that the BN-stable method exhibits superior accuracy and numerical stability. Additionally, this work serves as an innovative educational resource for courses in vibration mechanics, enhancing students' ability to analyze and solve nonlinear vibration problems while fostering their innovative thinking.