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Cantilever grandstands are susceptible to vertical vibrations due to jumping or bobbing (vertical motion without loss of contact) of human occupants. Several major sports stadia have had vibration problems as a result. For bobbing, a simple mass-spring-damper-actuator model of the human (or group of humans) has been proposed and has been incorporated in the Institution of Structural Engineers guidelines on the dynamic performance of grandstands. For jumping, laboratory experiments have shown that in certain conditions regular periodic jumping close to the natural frequency of the supporting structure is not possible. However, there has previously been no model of the human-structure interaction in this situation. A fundamental difference from the bobbing case is that there are discontinuities between contact and non-contact phases of the motion. The behaviour is therefore highly nonlinear. This paper proposes a model similar to the human bobbing model but allowing for the loss of contact during jumping and the condition for landing, which is clearly influenced by the motion of the structure during the no-contact phase. The behaviour of the model is explored numerically. It is shown that in certain parameter ranges there are stable periodic solutions, but in others the periodic motion is unstable and instead the system exhibits chaotic motion. This could possibly explain the physical difficulty of regular periodic jumping close to the natural frequency of the supporting structure in the previous laboratory experiments.

Numerical investigation of a simple model of human jumping on an oscillating structure