Multiple analytical mode decompositions (M-AMD) for high accuracy parameter identification of nonlinear oscillators from free vibration

In this study, multiple analytical mode decompositions (M-AMD) are proposed to identify the parameters of nonlinear oscillators from free vibration. The time-varying damping (stiffness) coefficient of an oscillator is divided into slow- and fast-varying components by a bisecting frequency in reference to velocity (displacement). The slow-varying damping and stiffness components are estimated from the oscillator's responses and their Hilbert transforms, and filtered with the adaptive low-pass filter AMD. Each fast-varying component is estimated from the oscillator's responses and the determined slow-varying components, and corrected with AMD using an approximate bisecting frequency of the two fast-varying components. The computational efficiency and accuracy of the proposed M-AMD are illustrated with three characteristic oscillators described by Duffing, Bouc-Wen, and spherical bearing models. The errors in estimation of all model parameters of the representative nonlinear oscillators are less than 3% from uncontaminated displacement responses and 9% when the root-mean-square value of displacement noises correspond to 0.05% of the peak displacement. In the case of Duffing oscillator, the error of the proposed method is less than 1.2% unless the ratio between the nonlinear and linear terms of the restoring force (degree of nonlinearity) exceeds 25. In the case of spherical bearing, the error of the proposed method is less than 1.2% when the coefficient of friction is less than 0.017. The instantaneous stiffness determined from Hilbert spectral analysis differs from the system stiffness by amount that rapidly increases with the degree of nonlinearity.