Properties of the Intrinsic Solution of a Four-point Centrally Damped String System
The tensioned string system with concentrated damping belongs to the hybrid dynamical system in the mechanical model,and the approximation method is usually used to solve the eigen values to meet the needs of engineering applications.In order to further clarify the vibration characteristics of this type of system,it is necessary to explore its eigen problems from the analytical point of view.For a four-point centrally damped string system,the algebraic form of its frequency equation was derived and simplified,the solution of the algebraic equation was obtained,and all the closed solutions of the original frequency equation were obtained after the commutative inverse process.The struc-ture of the closed solutions of the system was discussed according to the fundamental theorem of algebra.It is found that there exist three sets of closed solutions in total,two of which are conjugate to each other.The structure of the closed solutions was discussed according to the fun-damental theorem of algebra.The results show that the system has two identical motion characteristics,the logarithmic attenuation rate per unit time is the same,and the frequencies are opposite to each other.The logarithmic attenuation rate and frequency per unit time corre-sponding to the three de expenditures in the system always cycle periodically with the increase of order.Each single-valued branch under the same debranching also shows a periodic cycle,that is,with the increase of order,its corresponding frequency increases by an integer multiple of 4π,while the logarithmic attenuation rate per unit time remains unchanged.Based on the above,it can be concluded that the motion char-acteristics of the centralized damping chord system always change repeatedly with the increase of order,and the change period is related to the damping installation position.
taut stringeigenvalueconcentrated dampingeigenfunctionnonclassical linear system
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