On the Inclusion of Time Derivatives of State Variables for Parametric Model Order Reduction for a Beam on a Nonlinear Foundation

The computational burden of parameter exploration of nonlinear dynamical systems can become a costly exercise. A computationally efficient lower dimensional representation of a higher dimensional dynamical system is achieved by developing a reduced order model (ROM). Proper orthogonal decomposition (POD) is usually the preferred method in projection-based nonlinear model reduction. POD seeks to find a set of projection modes that maximize the variance between the full-scale state variables and its reduced representation through a constrained optimization problem. Here, we investigate the benefits of an ROM, both qualitatively and quantitatively, by the inclusion of time derivatives of the state variables. In one formulation, time derivatives are introduced as a constraint in the optimization formulation—smooth orthogonal decomposition (SOD). In another formulation, time derivatives are concatenated with the state variables to increase the size of the state space in the optimization formulation—extended state proper orthogonal decomposition (ESPOD). The three methods (POD, SOD, and ESPOD) are compared using a periodically, periodically forced with measurement noise, and a randomly forced beam on a nonlinear foundation. For both the periodically and randomly forced cases, SOD yields a robust subspace for model reduction that is insensitive to changes in forcing amplitudes and input energy. In addition, SOD offers continual improvement as the size of the dimension of the subspace increases. In the periodically forced case where the ROM is developed with noisy data, ESPOD outperforms both SOD and POD and captures the dynamics of the desired system using a lower dimensional model.