Keywords: reduced order models, piecewise-linear oscillators, unilateral constraints, proper orthogonal decomposition, nonlinear forced response, cracked structures.

A novel reduced order modeling method for vibration problems of elastic structures with localized piecewise-linearity is proposed. The modeling framework is based on a bi-linear modal representation of (localized) piecewise-linear oscillators. The focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity (such as cracked structures). First, the mathematical formulation for constructing reduced order models of piecewise-linear oscillators using linear projection is developed. Second, proper orthogonal decomposition of forced response results is examined using a vibration problem of a cracked plate, and a few key properties of proper orthogonal modes are discussed. In particular, it is observed that the most dominant proper orthogonal modes resemble the linear normal modes in the frequency range of interest, and the proper orthogonal modes can yield a compact reduced order model. More importantly, forced response analyses using the proper orthogonal modes reveal that capturing the local deformations near the discontinuous region is important for accurately predicting the nonlinear forced response. It is then shown that the subspace spanned by the set of dominant proper orthogonal modes can be accurately approximated by a slightly larger set of linear normal modes with special boundary conditions. These linear modes are referred to as bi-linear modes, and are selected by an elaborate methodology that utilizes certain similarities between the bi-linear modes and approximations of the dominant proper orthogonal modes. Several novel bi-linear modal representations of proper orthogonal modes are then proposed based on the observations on the proper orthogonal modes. Namely, it is shown that the most dominant proper orthogonal mode can be accurately approximated by the corresponding linear normal mode, and the proper orthogonal modes showing the local deformations can be represented by linear combinations of bi-linear modes. To circumvent the difficulty in choosing the most adequate set of bi-linear modes, bi-linear mode selection criteria using angle-based metrics are proposed. These metrics are based on interpolated proper orthogonal modes of smaller dimensional models. The applicability of the proposed method, including the bi-linear mode selection criteria, is investigated using a case study of a cracked plate. The proposed method is compared with traditional reduced order modeling methods such as component mode synthesis, and its advantages are discussed. It is demonstrated that the proposed bi-linear modal representation successfully produces low-dimensional reduced order models that accurately capture the vibration response of (localized) piecewise-linear oscillators.

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