Emergence of traveling waves in linear arrays of electromechanical oscillators*

Traveling waves of mechanical actuation provide a versatile strategy for locomotion and transport in both natural and engineered systems across many scales. These rhythmic motor patterns are often orchestrated by systems of coupled oscillators such as beating cilia or firing neurons. Here, we show that similar motions can be realized in linear arrays of electromechanical oscillators that move and interact via electrostatic forces. Conductive spheres oscillate between biased electrodes through cycles of contact charging and electrostatic actuation. The combination of repulsive interactions among the particles and spatial gradients in their natural frequencies lead to phase locked states characterized by gradients in the oscillation phase. The frequency and wavelength of these traveling waves can be specified independently by varying the applied voltage and the electrode separation. We demonstrate how traveling wave synchronization can enable the directed transport of material cargo. Our results suggest that simple energy inputs can power complex patterns of mechanical actuation with potential opportunities for soft robotics and colloidal machines.