Keywords: dependability, parameter diagram, dependable regions, harmonic balance method, proper orthogonal decomposition, nonlinear dynamics.

The extensive applications of nonlinear elastic structures have drawn attention to the need for understanding their global behaviour and dependability and integrity from the view point of nonlinear dynamics. However, current nonlinear dynamics theory is limited to handle systems with a few degrees-of-freedom and with known mathematical equations. Meanwhile, current numerical techniques, including the finite element method, can provide reliable numerical results for nonlinear structures with huge large number of degrees of freedom (DOFs), given a specified set of system parameters. The aim of this paper is to bridge the gap by proposing an approach to build up a reduced model from MDOFs data. Then one can take advantage of the the power of these two different techniques to investigate the dependability and integrity of nonlinear dynamic structures.

The proposed approach to obtain a reduced model has three stages. The first stage collects displacement time series at different spatial locations from numerical simulations using a multiple degrees of freedom (MDOFs) model. The second stage conducts a dimension reduction using proper orthogonal decomposition (POD). POD is a powerful multivariate statistical method and has been widely applied in performing model reduction in structural dynamics. The final stage deduces a single DOF system with known governing equation using the harmonic balance method (HBM). HBM is known to identify system parameters of highly nonlinear systems. The combination of POD and HBM makes it possible to transform a nonlinear dynamical structure with huge DOFs and unknown governing equation, into a model with a few DOFs with known governing equations. This then is used to investigate the dependability and integrity of the nonlinear dynamical structure.

In this paper, as an example, firstly, ANSYS was used to perform dynamical analyses of a cantilever with a one-side stop, subjected to a sinusoidal base excitation with the the same amplitude but different frequencies. The frequencies of excitation were swept from 3Hz to 10Hz with each step 1Hz and displacement time series from 10s to 40s were recorded in ten uniformly distributed, spatial locations of the cantilever sampled using a constant sampling period. Secondly, for each excitation frequency, ten displacements corresponding to ten DOFs were simplified to one abstract displacement time series corresponding to a SDOF, using POD. A polynomial was assumed to represent unknown stiffness nonlinearity in the governing equation of the reduced model. Long segments with length of at least 60T (T is the period of excitation) from these time series, were used to identify all parametric coefficients of the governing equation. Finally, by varying the frequency and amplitude of external excitation, parameter diagrams of the system were obtained which show a global picture of its complicated structural behaviour.

The parameter diagrams can help identify the dependable region in the parameter space of the model. The approach reported here can help the structural engineer to better understand vulnerability and integrity of nonlinear elastic structures.

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