TY - CHAP

T1 - Three dimensional vibration of a ballooning string

AU - Hall, Kevin J.

AU - Zhu, Fang

AU - Rahn, Christopher D.

PY - 1995/12/1

Y1 - 1995/12/1

N2 - 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 yarn stiffness. The effect of yarn elasticity on the first three vibration modes of a single loop balloon is analyzed. The steady state and stability analyses are experimentally verified.

AB - 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 yarn stiffness. The effect of yarn elasticity on the first three vibration modes of a single loop balloon is analyzed. The steady state and stability analyses are experimentally verified.

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M3 - Chapter

AN - SCOPUS:0029428639

T3 - American Society of Mechanical Engineers, Design Engineering Division (Publication) DE

SP - 1411

EP - 1418

BT - 15th Biennial Conference on Mechanical Vibration and Noise

A2 - Wang, K.W.

A2 - Yang, B.

A2 - Sun, J.Q.

A2 - Seto, K.

A2 - Yoshida, K.

A2 - et al, al

T2 - Proceedings of the 1995 ASME Design Engineering Technical Conference. Part C

Y2 - 17 September 1995 through 20 September 1995

ER -