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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.

Long-term global analysis of lower-dimensional systems can be performed by in a computationally effi cient manner by using cell mapping methods. Scaling limitations associated with higher order systems can be addressed by using the multiple degree-of-freedom cell mapping (MDCM) method. However, the MDCM algorithm consists of a number of sequential operations which limits its ability to utilize the parallel processing capabilities of modern computers. A parallelized multiple degree-of-freedom cell mapping (PMDCM) method is introduced here to address this limitation. By restructuring the MDCM algorithm to enable parallel operations, the effi ciency of the algorithm is signifi cantly enhanced. The details of the PMDCM algorithm are presented and performance is compared to the MDCM method and baseline obtained by using the grid-of-starts method. By running the PMDCM method on a workstation with a quad-core processor and by parallelizing 100 numerical integration operations, the total computation time is decreased by 93% from the time required of the MDCM method. Parallelizing the integration operations also reduces the error which can accumulate when by using the sequential operations of the MDCM method.