A Surrogate Reduced Order Free Vibration Model of Linear and Non-Linear Beams using Modified Modal Coefficients and HOSVD Approaches

In the present work, two low-dimensional models are presented and used for vibration simulation of the linear and non-linear beam models. These models help to compute the dynamical responses of the beam with fast computation speed and under the effects of different conditions. Also the obtained results can be used in the conceptual and detailed design stages of an engineering system overall design. First, a finite element analysis based on Euler-Bernoulli beam elements with two primary variables (deflection and slope) at each node is used to find static and dynamic responses of the considered linear and non-linear beams. Responses to three different static load cases are obtained and applying them as initial conditions, the time responses of the beam are calculated by the Newmark's time approximation scheme. A low-dimensional POD model which was extracted from the ensemble under the effect of an arbitrary loading is reconstructed. To apply the model to simulate the response of beam under the effect of other loads, POD modal coefficients are updated due to change of initial condition. This modification is performed based on the recalculation of the eigenvalues due to a new initial condition. Also, another low-dimensional model is constructed which is developed based on an ensemble under the effect of several parameters. To apply the model to simulate the response of the beam under the effect of other loads and variations of beam thickness, POD-HOSVD modal coefficients are updated due to the change of desired parameters. The results obtained from the low-dimensional model are showing good agreement to the benchmark data and proving high level accuracy of the model.