Structure And Properties Of A Taut String With A Lumped Damping At A Unit-Fraction-Span

As a key force transmission component of cable-stayed bridges,the vibration control of cable-stayed cables has received widespread attention in bridge engineering practice.The vast majority of completed cable-stayed bridges have taken necessary measures to suppress various vibrations of their cables.At the same time,the vibration control problem of cables represents a large class of engineering dynamics problems,characterized by involving a hybrid system composed of "continuous+discrete" subsystems,which has unique research value in theory and has always been valued by the academic community.Therefore,studying the fundamental issues of vibration control for cable-stayed cables is not only a universal demand in engineering practice,but also an inevitable requirement for the development of dynamic theory.This article studies the simplest model in cable vibration control-the concentrated damping string model,and explores the characteristics of the derivation,solution,and solution of its dynamic equations.When a cable-damper system uses only one damper and is laid across the position of the unit fraction(1/n)of the cable(referred to as the unit fraction concentrated damping chord),its dynamic system is naturally linked to algebra.For such cases,this thesis studies the functional conversion method of the frequency equation of the system,and discusses the analytical solution method of the original equation by using the characteristics of the converted equation(new equation)to obtain a possible closed solution.For the situation where the solution cannot be solved analytically,the basic properties of the new equation are used to discuss the structure of the original equation solution(including the principal value branch of the original equation solution,the relationship between each principal value branch,the characteristics of each branch solution with the same principal value branch,etc.),so as to overcome the inherent difficulties of non-analytic methods(approximate analysis,etc.)in discussing the structure of the solution(such as the interrelationship between solutions,etc.).The specific research work of this thesis is as follows:1.Formal transformation of the frequency equation of the cable-single damper system.In view of the situation where a single damper is installed in the 1/n span position of the cable,the transcendental function form(original equation)of the frequency equation of the single-set damping chord dynamics system is formally converted,and its algebraic form(new equation)is derived by commutation method,and the relationship between the original equation and the new equation,as well as the relevant properties of the new equation,are elaborated as the basis for further analysis later.2.The structure of the frequency equation solution of the damping string system concentrated by unit fraction.Using the basic properties of the new equation(an algebraic equation with a logarithmic function as an unknown element),the structural analysis of the original equation solution is carried out.According to the fundamental theorem of algebra,the transformation relationship between the roots of the new equation and the eigensolutions of the original equations,the countable infinite solutions of the system are classified as finite several solutions,and its countable infinite single-valued branches are discussed under each solution.On this basis,the circulation phenomenon of solution branch is discussed through the expression of solution(logarithmic function form)of algebraic equation.Using the conjugation of the roots of algebraic equations,the conjugate relationship between the unbranches was analyzed.According to the multivalued characteristics of the logarithmic function,the relationship between the singlevalued branches under the same debranch is analyzed.3.Several closed solutions of the concentrated damping chord system and the corresponding frequency and attenuation characteristics.When the damper is located in the 1/2,1/3,1/4 and 1/5 span positions of the cable(string),the derived algebraic equation is solved analytically,and on this basis,the total closed solution of the original system is further obtained.Using the analytic expression of the eigenvalue closed solution,the influence of damper position and damping coefficient on the self-resonance frequency and damping attenuation characteristics(logarithmic attenuation rate per unit time)of the system was analyzed.According to the different characteristics of damping attenuation,the demobilization is classified and discussed.4.Characteristics and classification of eigenfunctions(modes)of concentrated damping string systems.The variation characteristics of eigenfunctions with damping coefficients in different damping coefficients range were analyzed.Firstly,the eigenequation of the system is derived,and its segmented expression is obtained.The characteristics of eigenfunctions under three limit cases(i.e.,non-oscillating attenuation motion,non-attenuated self-oscillation,and infinitely close to critical damping)were analyzed;Finally,the characteristics of various eigenfunctions with the damping coefficient are analyzed.