Abstract: Time integration algorithms are the principal numerical tools for computing the dynamic response of large structural systems. Textbooks on vibration typically introduce classical schemes such as the Newmark method and the generalized-α method. When these algorithms are applied to structures under harmonic excitation, the numerical solution contains both the steady state response and the accompanying free vibration response, which raises the question of how to extract the steady state component from the integrated results. Departing from the conventional practice of discarding the transient stage or introducing artificial damping, this study proposes an equivalent initial condition method. The method exploits the strong similarity between the accompanying free vibration response and the response induced by suitable initial conditions, enabling accurate extraction of the steady state response. Using the steady state response of a cantilever beam under harmonic loading as a case study, the effectiveness of the equivalent initial condition method is demonstrated with the generalized-α method. Assigning this class of problems as a major coursework project in vibration courses can provide comprehensive training on key concepts and foster the development of students’ integrative skills.