Three dimensional vibration of a ballooning string

In many textile manufacturing processes, yarn is rotated at high speed forming a balloon. In this paper, Hamilton's principle is used to derive the nonlinear partial differential equations of a ballooning string. Jacobian elliptical sine functions satisfy the nonlinear steady state equations. The steady state eyelet tension is related to the string length for a constant balloon height. For high tension and low string length cases, single loop balloons occur. As the string length increases, tension decreases and multiple loop solutions are obtained. The nonlinear partial differential equations are linearized about the steady state solutions, resulting in three coupled equations with spatially varying coefficients. The equations involve a positive definite mass matrix operator, skew symmetric gyroscopic matrix operator, and symmetric stiffness matrix operator. It is shown using a Galerkin approach that only single loop balloons are stable for practical yarn elasticity. The natural frequencies of the single loop balloon increase with decreasing balloon size and increasing yam stiffness. The effect of yam elasticity on the first three vibration modes of a single loop balloon is analyzed. The steady state and stability analyses are experimentally verified.