Fundamental insight on the performance of a nonlinear tuned mass damper

Abstract

To try and enhance the performance of a tuned mass damper, nonlinear stiffness can be introduced to good use. In this paper, an investigation is performed on the characteristics of a nonlinear tuned mass damper when attached to a linear oscillator. The whole system is modelled as a two degree-of-freedom system with cubic stiffness nonlinearity, and the effect of the nonlinearity and mass ratio of the attachment is investigated for both a softening and a hardening stiffness characteristics, using an analytical formulation. The corresponding linear case is presented as a benchmark, and the main critical issues are discussed. Depending on the frequency to which the vibration reduction is desired, recommendations are highlighted for proper tuning and parameters selection.

Appendix

Coefficients for the polynomial equation reported in Eq. (14)

$$\begin{aligned} a^{*} & = 1 \\ b^{*} & = 3W^{2} \gamma \left( {1 + \mu } \right) + 2\left( {1 + 2\varOmega^{2} + \left( {1 + \mu } \right)\varOmega_{0}^{2} } \right) \\ c^{*} & = 1 + \frac{27}{16}W^{4} \gamma^{2} \left( {1 + \mu } \right)^{2} + 2\varOmega^{2} + 6\varOmega^{4} + 2\left( {2 + \mu + \left( {1 + \mu } \right)\varOmega^{2} } \right)\varOmega_{0}^{2} \\ & \quad +\, \cdots \left( {1 + \mu } \right)^{2} \varOmega_{0}^{4} + 3W^{2} \gamma \left( {2 + \mu + \left( {1 + \mu } \right)\varOmega_{{}}^{2} + \left( {1 + \mu } \right)^{2} \varOmega_{0}^{2} } \right) \\ d^{*} & = \frac{27}{8}W^{4} \gamma^{2} \left( {1 + \mu + \left( {1 + \mu } \right)^{2} \varOmega_{{}}^{2} } \right) + 3W^{2} \gamma \left( {1 - \left( {1 + \mu } \right)\varOmega^{4} + 2\left( {1 + \mu } \right)\varOmega_{0}^{2} + 2\varOmega^{2} \left( { - 2 + \mu + \left( {1 + \mu } \right)^{2} \varOmega_{0}^{2} } \right)} \right) \\ & \quad +\, \cdots 2\left( {2\varOmega_{{}}^{6} + \varOmega_{0}^{2} + \left( {1 + \mu } \right)\varOmega_{0}^{4} - \varOmega^{4} \left( {1 + \left( {1 + \mu } \right)\varOmega_{0}^{2} } \right) + \varOmega^{2} \left( {1 + 2\left( { - 2 + \mu } \right)\varOmega_{0}^{2} + \left( {1 + \mu } \right)^{2} \varOmega_{0}^{4} } \right)} \right) \\ e^{*} & = \frac{27}{16}W^{4} \gamma^{2} \left( { - 1 + \left( {1 + \mu } \right)\varOmega_{{}}^{2} } \right)^{2} + \left( {\varOmega^{4} + \varOmega_{0}^{2} - \varOmega^{2} \left( {1 + \left( {1 + \mu } \right)\varOmega_{0}^{2} } \right)} \right)^{2} \\ & \quad +\, \cdots 3W^{2} \gamma \left( { - 1 + \left( {1 + \mu } \right)\varOmega_{{}}^{2} } \right) \times \left( { - \varOmega^{4} - \varOmega_{0}^{2} + \varOmega^{2} \left( {1 + \left( {1 + \mu } \right)\varOmega_{0}^{2} } \right)} \right) \\ \end{aligned}$$