Dynamic response of freestanding objects under horizontal and vertical eccentricity

The dynamic response of freestanding objects has always been a central topic in mechanics and civil engineering, and such objects are commonly found in various infrastructure systems. Since the pioneering work of Housner, numerous studies have focused on symmetric freestanding objects, whose dynamic responses are relatively simple. However, in most practical engineering scenarios, freestanding objects exhibit eccentricity, i.e., a shift between their center of gravity and geometric center. Moreover, this eccentricity often occurs in both the horizontal and vertical directions. To investigate the motion characteristics of freestanding objects with combined horizontal and vertical eccentricities, this paper presents a theoretical derivation of the rocking motion of bi-directionally eccentric freestanding objects. An aluminum alloy apparatus capable of adjusting both the horizontal and vertical positions of the center of gravity was developed. Based on this apparatus, free-rocking tests and sinusoidal excitation tests were conducted for various center-of-gravity positions, and the experimental results were compared with the analytical solutions. In free-rocking tests, dynamic characteristics such as the energy reduction factor, damping ratio, and half period were obtained. Under the non-eccentric condition, the experimental half periods agreed well with the analytical values. As the horizontal and vertical eccentricities increased, discrepancies emerged, indicating that both horizontal and vertical eccentricities are key parameters affecting the rocking period. Due to the asymmetry induced by eccentricity, the rocking period splits into two distinct curves. Similar splitting phenomena were observed for the energy reduction factor and damping ratio. The experimentally obtained energy reduction factors were generally slightly higher than Housner's analytical values, but their mean values agreed well with the analytical solutions derived in this paper; similar results were found for the damping ratio. Under sinusoidal pulse excitation, the Housner formulation was extended to account for left-right asymmetry due to eccentricity. Overturning stability tests under multiple sinusoidal pulses were conducted, and the corresponding overturning spectral maps were developed. During the experiments, phenomena differing from the analytical predictions were observed, and the analytical solutions tended to underestimate the overturning risk of the objects. This work is meaningful for understanding the complex dynamic response of freestanding objects.