The dynamics of pill millipedes’ flipping

Abstract

This study aims to find an equation of motion that can describe the flipping motion of the pill millipede (Zephronia siamensis Hirst). From observation, the flipping motion can be characterized into four types. Moreover, the pill millipede’s shell curve that contacts the flat surface while flipping can be described as a form of the hyperbolic cosine function. Therefore, the equation of motion of flipping can be derived. From the geometry of the cross-section of the pill millipede, an angle between the vertical axis and the symmetry axis, a critical angle that allows it to flip, was obtained. Moreover, the maximum angle, the angle between its symmetry axis to the last point that hyperbolic cosine is fitted the shell’s curve, could be defined. We have found that the critical angle must be smaller than the maximum angle to provide a successful flipping with a minimal initial velocity. The equation of motion of flipping has been solved and tested by plotting the flipping angle as a function of time compared to the flipping angle from observation.