Numerical investigation of coexisting high and low amplitude responses and safe basin erosion for a coupled linear oscillator and nonlinear absorber system

Abstract

Over the last half century, numerous nonlinear variants of the tuned mass damper have been developed in order to improve attenuation characteristics. In the present study, the performance of a linear oscillator and an absorber with a strongly nonlinear cubic stiffness is evaluated by using numerical methods. This configuration has been of recent interest due to its capability of wide-band energy absorption. However, high amplitude solutions, which would amplify the response of the system, have been shown to often coexist with the low amplitude solutions. The present research is focused on numerically determining the relative strength of the coexisting solutions. Erosion profiles are presented, quantifying the integrity of the system, i.e. the likelihood of converging to a safe, low amplitude response, and providing an indication of the structural safety of a practical absorber system. The results indicate that the high amplitude solutions not only exist but significantly influence the response of the system within the range of expected operating conditions, particularly at excitation frequencies lower than the natural frequency of the linear oscillator. The erosion profiles indicate a 20-40% increase in system integrity for the case of zero damping compared to a small amount of damping, no significant integrity change when adding a small linear stiffness component to the nonlinear absorber, and no significant change in integrity between the midpoint and extreme of the bi-stable * Please address all correspondence to this author.Email address: andrew.j.dick@rice.edu (A. J. Dick)Preprint submitted to Journal of Sound and Vibration March 4, 2014 range. Additional higher-period solutions are also discovered and evidence of a chaotic response is presented.

Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber

“…The likelihood of converging to the safe, low-amplitude periodic solution in the region where the DRC exists should therefore be determined. Given a forcing amplitude and frequency, direct time integrations for a large set of random initial states provided the basins of attraction, similarly to what was achieved in [35] for a coupled linear oscillator and nonlinear absorber system. To limit the scope of the discussion, the NLTVA was considered at rest, as it would be the case, e.g., during an earthquake, but other configurations were also tested.…”

Analytical Investigation of Vibration Attenuation With a Nonlinear Tuned Mass Damper

“…Although this nonlinearity negatively affected the performance of the TMD at first, since it was a perturbation from an optimized linear state, the combination with the STMD was able to more than correct this issue. The results once again showed Eason et al also performed a numerical study on a linear oscillator with a strongly nonlinear Düffing NTMD to determine the relative strength of the high and low amplitude solutions that can coexist in Düffing systems [14]. They discovered that the high amplitude solution has a significant influence.…”

Bifurcation of Safe Basins and Chaos in Nonlinear Vibroimpact Oscillator under Harmonic and Bounded Noise Excitations

“…In some time the limitation of the vibration amplitude may be more important, since the structure of the system will be destroyed when the amplitude of the vibration passes through a critical value and thus leads to the researches of the safe basins [25,26]. There are some relations between erosion of safe basins and chaotic motions of the system.…”

“…When the safe basin is eroded, the boundary of the sage basin will have fractal structures, and the motions initial from some points within the safe basin will be chaotic. According to [25,26], the safe basins of the system may be defined using a bounded area in the space of phase trajectories. The trajectory starting from the safe basins will be stay in the area when the time tends to infinity.…”