Theoretical treatment of crowd–structure interaction dynamics

The equations governing crowd–structure interaction dynamics are derived from first principles. No assumptions are made about structural support conditions, crowd or beam displacements or crowd-induced forces in this derivation. The general equations of motion are shown to exist in pairs of two-degrees-of-freedom systems, for every ith beam–crowd mode. Numerical optimisation of the system response functions is preferred to both classical undamped eigenvalue analysis and a damped complex eigenvalue analysis using a state space formulation. Single parameter continuation of optima of frequency response functions is employed. In addition, two parameter continuations of folds of these optima are used. As a result, a fold loci plot in parameter space defines the region where multiple beam optima exist. This results in the possibility of complicated response behaviour. Increases in predominant frequency of the structure with increasing crowd mass are observed. In other cases, sudden drops in predominant frequency with increasing crowd mass are observed.