THE SUPERPOSITION OF DAMPING CHARACTERISTICS OF A TAUT STRING WITH CONCENTRATED VISCOUS DAMPING SYSTEMS OF A QUARTILE

The taut string system with concentrated damping belongs to hybrid dynamic system in mechanical model. In order to understand the damping characteristics of the system to meet the needs of engineering application, the approximate method is usually used to solve its eigenvalue problem. In order to understand the dynamic characteristics of the system more deeply, this paper takes the taut string system with three concentrated viscous damping as the research object, analyzes the variation law of the damping characteristics of the system from an analytical point of view, and focuses on the superposition of the damping characteristics. The complex frequency equation in the form of transcendental function is derived when the damping is arranged at the quartering point, and the general algebraic form of the equation after substitution is given. On this basis, the algebraic complex frequency equation is simplified to the specific equations of three kinds of degenerate systems, namely, single damping system sequence, double damping system sequence and triple damping system. The complex eigenvalues of the three kinds of systems are solved analytically at the algebraic level and expressed as explicit analytical expressions with damping coefficients as parameters. The influence of damping coefficient on attenuation characteristics of various systems is analyzed, and the superposition of attenuation characteristics of various systems is discussed by using symmetric polynomials, and the proportional relationship of damping characteristics among various systems considering finite-order vibration is derived. The results show that the sum of the real parts of complex eigenvalues between systems with the same number of concentrated damping is equal and does not change with the position coordinates of concentrated damping. The sum of the real parts of complex eigenvalues between systems with different numbers of concentrated damping is proportional and does not change with damping coefficient. Finally, taking the twenty equal points damping string system as an example, it is shown that superposition is an inherent characteristic of the system itself and does not depend on the solution method of complex eigenvalues.