Dynamics of Swing Pumping

This project is concerned with the study of the dynamics of a playground swing, primarily involving the examination of a variety of different standing pumping strategies. The case of pumping a swing from the standing position has been appraised in the literature by the development of two principal canonical models, namely the simple pendulum parametric oscillator and the double pendulum models that describe, respectively, variations in the length and angular orientation of the rider relative to the suspending rope. In an attempt to introduce originality to the project as well as incorporate an additional degree of complexity to the prevalent models, a third model is subsequently proposed as a refinement to the double pendulum model, by combining the effects of perturbations in both the length and angle of the rider. These models are then analysed and compared in their non-autonomous versions to grasp the feasibility of each model in capturing the actual motion of a child on a playground swing. It is found that the parametric oscillator model leads to a geometric increase in swing amplitude, as opposed to the double pendulum model, which leads to an non-oscillatory exponential increase in amplitude after each cycle. Interestingly enough, the third model is shown to produce pure oscillatory, sinusoidal variations in amplitude with time. These analytical results are then validated via numerical simulations. An autonomous description of the standing pumping strategy is also formulated, following the conduction of simple experiments, involving the testing and recording of various strategies with a video camera on a real playground swing. Finally, damping is introduced into the model through inclusion of a physical dissipation source such as friction in the swing support and is shown to have a limiting effect on the maximum amplitude of oscillation by causing it to gradually decay and approach equilibrium