Symmetric and asymmetric vehicle models: coincidence of natural frequencies and the role of modal and interwheel correlations under random excitation

Abstract This study addresses the random vibration analysis of a seven-degree-of-freedom full vehicle model with repeated or closely spaced natural frequencies. It first revisits existing results: the literature has established that an ideal symmetric vehicle without anti-roll bars, with the origin of the reference frame at the center of the sprung mass, possesses four identical natural frequencies associated with the vertical displacements of the wheels. The study then generalizes these results by accounting for both the anti-roll bars and the height of the center of gravity. Closed-form expressions are derived for the critical heights at which two body frequencies coincide, in addition to the four identical wheel frequencies. For the symmetric vehicle, the heave-pitch frequency coincidence occurs at a critical height of 0.761 m, whatever the stiffness of the anti-roll bars. The heave-roll coincidence occurs at a critical height of 0.408 m without anti-roll bars and 1.280 m with them. For the asymmetric vehicle, by contrast, no coincidence is observed for the heave–pitch coupling; the roll frequency, however, coincides with a body mode at critical heights of 0.356 and 0.522 m without anti-roll bars, and 1.089 m with them. For the vibration analysis, two excitations are considered: a stationary white noise and a sinusoidal speed-bump profile, both applied to the four wheels with time delays determined by the wheelbase, the track and the vehicle speed. Two types of correlation are identified in the autocorrelation matrix of the generalized velocities: the modal cross-correlations and the correlations between the wheel excitations. Omitting the former, which produces the 75% error reported by Long et al. when a single wheel is excited, no longer matters when all four wheels are excited: the error remains below 0.16% under white noise and 0.8% under the speed bump, including at the critical heights. Omitting the latter, which amounts to assuming that the wheels are excited independently, leads instead to errors of 1,453 to 6,064% in roll and of several hundred percent in heave and pitch at low speeds. Such an omission removes both the left–right compensation and the interference between axles that govern the body response. The critical speed range of the vehicle lies between 5 and 30 km/h, where the delays between axles and between sides are equal to half a period of the pitch or roll vibration. Since the speed bump is a deterministic transient excitation, the RMS indicator is replaced by the vibrational energy. For an aspect ratio of 1/10, the body energy is maximal between 5 and 15 km/h, well below the 50 km/h threshold reported in the literature, whereas the wheel energy increases linearly with speed: maximum loading and maximum error therefore do not occur at the same speeds.