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.

Figure 1

Schematic drawing of different ways of modeling capturing the phase between the seat of the swing (blue) and the upper body (red). (a) Fixed frequency model. (b) Square-wave model. (c) Model proposed here. Top row: $A$ (amplitude of the swing) = ${10}^{\circ }$. Bottom row: $A={40}^{\circ }$. The unit for the upper body angle (ordinate axis on the right of each panel) is arbitrary. Note that in the bottom row a whole cycle of the swing to the phase of $2.0\pi$ is not shown because the period of the swing is extended with the gain of the amplitude.

Figure 2

Rigid body model with two degrees of freedom sitting on a swing. (a) Definitions of parameters. (b, c) Positive (b) and negative (c) range of motion of the hip and the knee joint.

Figure 3

Initial phase $\alpha$, amplitude of the swing $A$, and gain of the amplitude per cycle $\mathrm{\Delta }$ (left, short chain; middle, medium chain; right, long chain). The bright, medium, and dark colors indicate small $({\varphi }_0={25}^{\circ })$, medium $({\varphi }_0={35}^{\circ })$, and large $({\varphi }_0={45}^{\circ })$ amplitude of the upper body, respectively. Colored dots indicate the initial phase $\alpha$ to obtain the maximum $\mathrm{\Delta }$ for a given amplitude $A$. Triangles in dark colors are initial phases for zero $\mathrm{\Delta }$ values.

Figure 4

A participant pumping the swing in the lab.

Figure 5

Predicted and obtained gain of seat amplitude $\mathrm{\Delta }$ per cycle (from left to right: short, medium, and long chain).

Figure 6

Period of the (a) seat, (b) upper body, and (c) amplitude of maximum lean back as a function of the swing amplitude $A$. Red, green, and blue indicate short, middle, and long chains, respectively. The solid lines in panels (a) and (b) indicate the period of the swing predicted from Eq. (3).

Figure 7

Trajectories of upper body motion as a function of the phase of the seat. The colored thick curves are from the cycle of $A\approx {10}^{\circ }$ when the pump is required. The gray curves are from the cycle of maximum $A$ of the trial where the gain of the amplitude was no longer necessary. The curves in faint colors are from cycles in between. The initials of the corresponding participant, length of chain, and maximum $A$ of the trial are shown.

Figure 8

Initial phase $\alpha$ as a function of amplitude $A$. Colored dots indicate the initial phase and amplitude for each cycle in colors of red, green, and blue for chain lengths of S, M, and L, respectively. Lines with color show theoretical values of initial phase to achieve maximum gain $\mathrm{\Delta }$ with colors the same as dots. Gray lines indicate theoretical values of making $\mathrm{\Delta }=0$.