<p>This paper develops a mathematically rigorous and computationally efficient decoupled reduced-order framework for the dynamic analysis of asymmetric piecewise linear oscillators under harmonic excitation. Such systems appear widely in mechanical and structural applications involving unilateral constraints, gaps, or asymmetric stiffness distributions, and they are governed by non-smooth second-order differential equations with switching restoring forces. The non-smoothness induces switching dynamics and nonzero mean offsets that challenge classical averaging and harmonic balance techniques. To address these difficulties, the system response is approximated using a decoupled reduced-order representation that explicitly includes a constant basis function to capture asymmetry-induced mean-offset dynamics. A Galerkin projection yields a low-dimensional nonlinear system of ordinary differential equations that preserves the mechanical structure and the piecewise linear character of the original model. Existence and uniqueness of reduced-order solutions are established under standard Lipschitz-type conditions, while nonlinear stability is studied using energy-based Lyapunov arguments, invariance principles, and linearized spectral analysis. We show that the reduced-order model inherits Lyapunov and asymptotic stability properties of the full non-smooth system, and that the stability conditions depend explicitly on the asymmetry parameters. For numerical realization, an implicit Newmark time-integration scheme combined with Newton-type iterations is employed to maintain stability across switching regimes. Numerical investigations, reported in tabular form, demonstrate excellent agreement with direct numerical integration and improved accuracy compared with classical harmonic balance approximations, and show that the proposed approach achieves high accuracy with substantially reduced computational cost, making it well suited for parametric studies and design-oriented analyses.</p>