The phenomenological equations of motion for the international roughness index (IRI) have been reviewed and discussed in terms of Lagrangian method in physics. The current paper proposes a practical, two-dimensional model for studying essentially three-dimensional, vibrating, and mechanical systems (vehicles). The purpose is to provide a new profiling method for IRI, which is practical in computations and compatible with traditional profiling for roughness of a road-surface. The modern technology employs elaborated sensors such as gyro sensor, Global Positioning System (GPS), magnetometer sensor, and accelerometer to measure high-speed longitudinal motions, resulting in time series of big-data expressed as compressed longitudinal spikes. The time series of longitudinal spikes obtained from high-speed longitudinal motions are traditionally considered as a background noise for constructing a profile. The conventional IRI is calculated from big-data of the road profile by employing statistical method, but the Lagrangian model dynamically determines the road profile. The useful concept and relation among the road-roughness function , associated roughness index (ARI), acceleration and position are introduced and examined in the present paper. The associated roughness index (ARI) defined by the current dynamical approach is examined by applying virtual simulations which represent roughness of a road-surface. The current theoretical model supports and compensates information of interpreting a profile of IRI and elucidates physical meanings for the roughness index of a road-surface.