Abstract: The treatment of nonlinear oscillations is explored as a part of the fundamental course of vibration mechanics. The teaching goal is to understand a basic approximate analytical method in nonlinear oscillations and typical nonlinearly oscillating phenomena. The main contents include harmonic balance method, jump phenomenon in nonlinear forced oscillations, self-excited vibrations, saturation and double jumping in internal resonance, nonlinear energy sink for scaled whole-spacecraft vibration reduction and circular nonlinear vibration isolation via ring. The case studies are adopted to highlight simplifications in modeling and analysis, applicability of mechanics, and a close affinity between mathematics and mechanics.