<p>We investigate the interaction between vibration modes coupled through nonlinear friction effects. Specifically, we consider the case where the mode shapes differ significantly, resulting in distinct friction cycles at contact interfaces. This scenario is common in mechanical systems such as bladed disks in turbomachinery, where friction, although localized to a few contact nodes, plays a key role in energy dissipation and induces strong nonlinearities in the system response. Standard time integration methods are often inefficient due to the numerical stiffness introduced by the small effect of friction. Harmonic Balance Methods offer a more efficient alternative and have recently been extended to handle nonlinear friction and two-frequency forcing. In this work, we present an alternative asymptotic approach based on a multiple scales expansion, yielding a reduced-order model that captures frictional effects on the slow timescale where they are developed. For the case of two simultaneously excited modes, the method leads to two coupled amplitude equations, where the influence of nonlinear friction is encoded through complex-valued functions that describe contact transitions and energy dissipation. The resulting model enables efficient parametric studies and analytical computation of nonlinear resonance curves. Validation on a lumped parameter system demonstrates good agreement with time-domain simulations and reveals that linear superposition significantly overestimates the system response.</p>