Abstract This article represents Part I of a two‐part paper which provides a rigorous mathematical foundation of the modeanalysis method for analysing the periodic and quasi‐periodic oscillations observed in various types of coupled oscillators. Although the results predicted by this method had been confirmed by experiments to some extent, the crucial assumptions used to derive the averaged equations are based on engineering intuition. Moreover, while it is intuitively reasonable to associate an equilibrium solution of the averaged equations whose eigenvalues have negative real parts with that of a stable periodic or quasi‐periodic solution of the original equation, the relationship between the solution of the averaged equations and that of the original equations as t tends to infinity, is not clear in a mathematical sense. This paper resolves the theoretical ambiguities of the mode analysis method by using the theory of integral manifolds. In particular, we recalculate the averaged equations in a rigorous way, and show that they coincide with those obtained before. Therefore, the theory of integral manifolds guarantees the existence of an integral manifold in the original system which corresponds to a steady‐state periodic or quasi‐periodic solution, provided the equilibrium point of the averaged equation has no eigenvalues with a zero real part (i.e. hyberbolic). This rigorous analysis proved that all our previous results obtained from the mode analysis method, i.e. averaged equations and stability analysis were correct.
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