Dynamic load identification under unknown initial conditions for long-term response prediction in nonlinear dynamical systems

Dynamic load identification in nonlinear dynamical systems is a challenging task due to its high sensitivity to initial conditions, which are often unknown or difficult to be measured accurately. To address this limitation, we propose the conjugate gradient method combined with the improved Powell (CGM-IP) method. Firstly, an objective function involving dynamic loads and unknown initial conditions is established in a nonlinear dynamical system with the displacement-dependent parameters. The conjugate gradient method, based on the perturbation theory, iteratively estimates dynamic loads and unknown initial conditions by minimizing the objective function. To enhance the accuracy and efficiency of dynamic load identification, an improved Powell method is introduced to optimize the initial conditions during iteration. This method adopts a multi-stage progressive optimization strategy and incorporates a hybrid line search method, which combines the golden section search with parabolic interpolation. For identifying loads over long durations, the moving time window method is applied. The identified loads enable reliable prediction of the long-term system response from arbitrary initial conditions and system parameters. This encompasses the prediction of the time series and the global properties that delineate its steady-state behavior, whether periodic or chaotic. Numerical simulations validate the CGM-IP method under varying system parameters, dynamic loads, and measurement errors, while the consistency in predicting long-term responses further confirms the identification accuracy of the proposed method.