Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations

Area of Science:

• Nonlinear Dynamics
• Vibrational Analysis
• System Identification

Background:

• Spectral submanifolds (SSMs) are crucial for understanding nonlinear oscillatory systems.
• Backbone curves derived from SSMs are essential for experimental nonlinear model identification.
• Existing methods often lack analytical precision in determining SSM shapes and backbone curves.

Purpose of the Study:

• To develop an analytical methodology for computing spectral submanifolds (SSMs) and their backbone curves.
• To integrate data-assimilating models with experimental vibration signals for accurate nonlinear system analysis.
• To validate the proposed approach using both synthetic and real-world experimental data.

Main Methods:

• Utilized Taken's delay-embedding theorem for model identification.
• Applied a least-squares fit to the Taylor expansion of the sampling map.
• Employed the parametrization method for invariant manifolds to construct SSMs.
• Integrated data assimilation with experimental vibration signals.

Main Results:

• Successfully computed the analytical shape of SSMs.
• Accurately reproduced backbone curves from experimental data.
• Demonstrated high accuracy in reproducing dynamics for both synthetic and real experimental datasets.
• Validated the effectiveness of the data-assimilating model.

Conclusions:

• The developed methodology provides an accurate analytical approach to identify nonlinear oscillatory systems.
• This method enhances the precision of nonlinear model identification using experimental vibration data.
• The findings offer a robust tool for analyzing and understanding complex dynamical systems.