Efficiency of the Pendulum Damper with a Mobile Suspension Point

The paper posed and solved a new dynamic problem, which is a generalization of the well-known classical problem of suppressing forced vibrations of high-rise flexible objects with the use of pendulum dampers. A mathematical model describing forced vibrations of a vibroprotection system equipped with a pendulum damper and a movable suspension point was built. The model is a system of nonlinear differential equations, after linearization and averaging of which for one period, the formula for the frequency response of the vibroprotection system was established. Within the framework of the obtained model, the frequency formula for small natural vibrations of the proposed absorber was also obtained and analyzed. The main regulating parameters of the damper was established, which determine its optimal tuning to the frequency of the carrier object. It is shown that the natural vibration frequency of the new damper coincides with the natural vibration frequency of a classical mathematical pendulum with an equivalent suspension length, which is equal to L_eq = L + mg/k. If the suspension point is fixed (k →∞), the frequency equation turns into a well-known formula for the frequency \( \upomega =\sqrt{g/L} \) of small natural oscillations of a mathematical pendulum. If the value of the stiffness coefficient of elastic elements tends to zero (k →0), the frequency of the absorber also tends to zero. An important design feature of the proposed pendulum is noted, which consists in the fact that due to the appropriate choice of three control parameters of the pendulum (k, L, and m), its frequency, if necessary, can be made any in the range from zero to \( \sqrt{g/L}. \) Numerical analysis of the dynamic behavior of the vibroprotection system showed a high efficiency of the proposed absorber. With the optimal adjustment of the damper parameters, the level of amplitudes of forced vibrations of the carrier body can be reduced by more than five times.