Initial phase and frequency modulations of pumping a playground swing

The playground swing is a dynamic, coupled oscillator system consisting of the swing as an object and a human as the swinger. Here, we propose a model for capturing the effect of the initial phase of natural upper body motion on the continuous pumping of a swing and validate this model from the motion data of ten participants pumping swings of three different swing chain lengths. Our model predicts that the swing pumps the most if the phase of maximum lean back, which we call the initial phase, occurs when the swing is at a vertical (midpoint) position and moving forward when the amplitude is small. As the amplitude grows, the optimal initial phase gradually shifts towards an earlier phase of the cycle, the back extreme of the swing's trajectory. As predicted by our model, all participants shifted the initial phase of their upper body movements earlier as swing amplitude increased. This indicated that swingers adjust both the frequency and initial phase of their upper body movements to successfully pump a playground swing.