The stable and unstable oscillations of a rod attached to many springs

Abstract

The oscillation equations of a multi-spring system connected to a rod are developed. Oscillations can be stable or unstable depending on the distribution of the spring constant and the number of springs. The symmetrical spring distribution with respect to the rod center tends to stabilize. The system becomes more stable as the number of springs increases. The equations of motion are identified to satisfy the Mathieu's equation. The system could be thought of as a rough imitation of the crazy bamboo attraction, a traditional attraction by old civilization that is frequently associated with the mystical phenomenon. One spring represents one performer. The behavior of the performer's force (opposite to the direction of the bamboo movement and providing a stronger force as the displacement of the bamboo increases) is similar to that of the spring force.