A principle of similarity for nonlinear vibration absorbers

• Adopting a principle of similarity, a nonlinear vibration absorber is defined.
• The vibration absorber should have the same functional form of the primary system.
• The absorber generalizes Den Hartog’s equal-peak method for nonlinear systems.
• A compact analytical formula for optimal tuning is defined.
• If more nonlinearities are present the formula exploiting a principle of additivity.

This paper develops a principle of similarity for the design of a nonlinear absorber, the nonlinear tuned vibration absorber (NLTVA), attached to a nonlinear primary system. Specifically, for effective vibration mitigation, we show that the NLTVA should feature a nonlinearity possessing the same mathematical form as that of the primary system. A compact analytical formula for the nonlinear coefficient of the absorber is then derived. The formula, valid for any polynomial nonlinearity in the primary system, is found to depend only on the mass ratio and on the nonlinear coefficient of the primary system. When the primary system comprises several polynomial nonlinearities, we demonstrate that the NLTVA obeys a principle of additivity, i.e., each nonlinear coefficient can be calculated independently of the other nonlinear coefficients using the proposed formula.