Modelling effects of passive crowds on grandstand vibration

Abstract

The possibility of excessive vibration on cantilever grandstands caused by lively human-induced loading is causing concern. One problem to be faced is the creation of a model of the crowd-loaded structure when the mass of the passive (seated and standing) crowd is significant compared with the mass of the structure. This paper presents the derivation of a crowd model and investigates the effect the passive crowd has on the response of the structure. A two-degree-of-freedom (2DOF) crowd model is obtained from published research on the apparent mass frequency response of people either seated or standing on a vibrating platform. Using the MATLAB® Control Toolbox, the 2DOF crowd model is combined with a single-degree-of-freedom (SDOF) model of the structure using a feedback system. Analysis is carried out for SDOF structures with natural frequency from 1 to 10 Hz, occupied with crowds of various sizes, quantified by the ratio of the mass of the crowd to the mass of the structure, ranging from 5 to 40%. The results show that the passive crowd adds significant mass and damping to the SDOF structure. The combined system shows a reduction in the natural frequency and dynamic response compared with the empty structure. These effects, which vary depending on the natural frequency of the structure, are quantified and presented in charts, which allow designers to estimate the natural frequency and response of an occupied structure quickly and easily.