Finite Linear Representation of Nonlinear Structural Dynamics Using Phase Space Embedding Coordinate

Modeling of structural nonlinear dynamic behavior is a central challenge in civil and mechanical engineering communities. The phase space embedding of response time series has been demonstrated to be an efficient coordinate basis for data-driven approximation of the modern Koopman operator, which can fully capture the global evolution of nonlinear dynamics by a linear representation. This study demonstrates that linear and nonlinear structural dynamic vibrations can be represented by a universal forced linear model in a finite dimension space projected by time-delay coordinates. Compared with the existing methods, the proposed approach improves the performance of finite linear representation of nonlinear structural dynamics on two essential issues including the robustness to measurement noise and applicability to multidegree-of-freedom (MDOF) systems. For linear structures, the dynamic mode shapes and the corresponding natural frequencies can be accurately identified by using the time-delay dynamic mode decomposition (DMD) algorithm with acceleration response data experimentally measured from an 8-story shear-type linear steel frame. Modal parameters extracted from the time-delay DMD matched well with those identified from traditional modal identification methods, such as frequency domain decomposition (FDD) and complex mode indicator function (CMIF). In addition, numerical and experimental studies on nonlinear structures are conducted to demonstrate that the finite-dimensional DMD based on the discrete Hankel singular value decomposition (SVD) coordinate is highly symmetrically structured, and is able to accurately obtain a linear representation of structural nonlinear vibration. The resulting linearized data-driven equation-free model can be used to accurately predict the responses of nonlinear systems with limited training data sets.