Wilson-ρ_∞ method and analytical solution of analog system

The Wilson-ρ_∞ technique is formed by establishing a relationship between ρ_∞ and θ, as the high-frequency dissipation of the Wilson-θ method cannot be properly controlled by ρ_∞ (spectral radius at infinite frequency). The numerical performances of this approach are compared with those of the Generalized-α method. In the Wilson-ρ_∞ method, there are two different θ for a given ρ_∞. The characteristic roots of the Jacobi matrix corresponding to both θ are different, and the corresponding Wilson-ρ_∞ method has different numerical performances. A better θ is recommended according to the properties of the spectral radius. In addition, an analog system of a single degree-of-freedom forced vibration system is constructed with the dissipation and frequency of the Wilson-ρ_∞ method, and the initial conditions of which and the forces acting on the analog system are the same with those of the original system. It is evident that the steady state responses have no cumulative amplitude errors and phase errors, and the results of the Wilson-ρ_∞ method match the analytical solutions of the analog system.