Abstract: The Newmark method is a cornerstone of the development of most modern time integration methods. Although the classical Newmark method has numerical damping, it cannot be precisely controlled, which limits its applicability and makes it difficult to compare with other methods with controllable numerical damping, such as generalized – α method. According to the spectral analysis method, this article utilizes \rho _\infty , which is the spectral radius corresponding to the infinite frequency, to represent the two free parameters in the Newmark method, generating the one-parameter Newmark- \rho _\infty with precisely controllable numerical damping. On this basis, the analog system corresponding to the Newmark- \rho _\infty method is constructed, and its analytical solutions agree with those of the Newmark- \rho _\infty method. In addition, according to the analytical solutions of the analog system, the amplitude decay rate and the period elongation rate can be defined in theory. This helps clarify the mechanism of amplitude and phase errors of time integration methods, and also provide a theoretical basis for designing structure dependency time integration methods.