Modelling hysteresis of energy dissipation at vibration of mechanical systems

The paper discusses a phenomenological approach to describe the energy dissipation hysteresis at nonstationary vibrations of mechanical systems. The mechanical system is considered as a “black box” with the known relationships between external factors and the observed reactions. A mathematical model is constructed for which the identification of parameters is carried out using the experimental data for steady-state vibrations of a system. To describe the hysteresis the authors have suggested the differential approach, called kinematic one, according to which the force and kinematic parameters are associated by the special differential equation of the first order, the right part of which is selected from a class of functions that provides the asymptotic approximation of the solution to the curves of the limit hysteresis cycle. The limit cycle is formed by curves of direct and reverse processes (“loading-unloading” processes), which are based on experimental data for the maximum possible or permissible intervals of parameter changes during the steady vibrations. Coefficients in the right part are determined from experimental data for the limit hysteresis cycle. Approximation curves of the limit cycle are constructed using the methods of minimizing the discrepancy of analytical representations to the number of experimental points. The proposed approach allows (based on one differential equation) describing the trajectory of hysteresis with a random starting point within the area of the limit cycle. As an example, the problem of forced vibrations of the pendulum-type damper for damping low-frequency vibrations is considered. The authors have considered the dampers, which are used for damping low-frequency vibrations of conductors of overhead transmission lines. The algorithm for energy dissipation analysis is suggested, allowing optimizing the damper constructions at the design stage. The proposed approach has the generality and can be used in the modelling of nonlinear behavior of different structures, mechanisms and systems where it is necessary to take into account the energy dissipation hysteresis in conditions of nonstationary vibrations.