Parametric exploration of a simple model of human jumping on an oscillating structure

Lightweight and long-span civil structures, such as grandstand tiers, are susceptible to vertical vibrations due to jumping or bobbing of human spectators. After a careful review of experimental evidence, including new data, a simple mathematical model is investigated in order to characterise human-structure interactions observed during human rhythmic jumping on a perceptibly moving surface. A passive mass-spring-damper is used to model a human jumper whilst subjected to an input structural oscillation. The coupled system is modelled as a piecewise-smooth contact dynamics problem allowing for the loss of contact during rhythmic jumping. Parametric sweeps are interpreted using methods from non-linear dynamical systems theory, including bifurcation analysis and the use of Poincaré maps. Hysteresis and coexistence of qualitatively different stable periodic motions over a broad range of parameter values of the system are observed. The presence of such coexisting non-intuitive motions is verified by preliminary experimental results that indicate period-doubled jumping behaviour (a repeating sequence of large-jump-small-jump) of test subjects both above and below a structure’s natural frequency. At large amplitudes of structural motion, the model shows that the behaviour can become chaotic. This provides an explanation for experimental findings that it is difficult to jump periodically around the natural frequency of a supporting structure. The simple model provides a detailed map of where different motions occur in parameter space, which should be amenable to further experimental validation.