Nonlinear Model Updating Based On Dynamic Substructure

Dynamic behaviors of the engineering structures are substantially affected by different nonlinearities,which increase the difficulty for dynamic characteristic analyses,dynamic response prediction,vibration control,and structural optimization.The equivalent linearization modeling and analytical methods for nonlinear structures have been implemented to structures;however,the problems of incorrect and inaccurate analyses still exist.When the nonlinearities are considered in engineering structure,some phenomena,e.g.,internal resonance,limit cycle vibration,and quasi-periodic vibration,can be explained well,and a more accurate numerical model can also be constructed.The application of the nonlinear vibration theory to real complex structures is challenging to time-consuming computation and low accuracy.Therefore,the finite element model updating(FEMU)method based on nonlinear dynamics is essential for improving the framework of the model updating method to nonlinear structures.Firstly,a method for analyzing the dynamic characteristics of structures with local nonlinearity based on the nonlinear normal modes is proposed,the influence of structural parameters on the frequency-energy curve is also investigated.The method is used for analyzing and selecting the targeted responses in nonlinear FEMU procedure,e.g.,the steady-state time-domain response,which is calculated using the time integration method by applying the initial conditions at the frequency-energy curve.To improve the computational efficiency in nonlinear dynamic analysis,a high-order residual mode-based substructuring method for calculating the nonlinear time-domain responses is proposed.The effects of retained modes in model order reduction method and different condensation method on analytical accuracy are investigated.Results show that the nonlinear normal mode is useful for determining the dynamic characteristics of nonlinear structures and the targeted responses in nonlinear FEMU procedure.The computational efficiency is increased by using the dynamic substructuring method,and the computational accuracy can be improved by increasing the number of retained modes or considering the high order residual flexibility.Secondly,to quantify the influence of the structural parameters on targeted responses,a sensitivity analysis method based on directly differentiating the equations of motion is proposed.The accuracy of nonlinear response sensitivity analysis for different vibration behaviors,e.g.,quasi-periodic vibration,chaotic,and nonlinear normal modes,is investigated.Based on the perturbation theory,a convenient sensitivity analysis method based on the real-imaginary parameter perturbation is proposed.The effects of parameter perturbation steps on calculation accuracy are investigated.Results reveal that the direct dynamic sensitivity analysis method has high computational efficiency for complex structures and determines the consistent solutions for chaotic response sensitivity analysis.The real-imaginary perturbation-based sensitivity analysis method improves the accuracy compared to the traditional finite difference method and decreases the computational time compared to the complex variable derivatives method.The stability of the transient response sensitivity is significantly affected by the parameter perturbations in the real and imaginary parts.The imaginary sensitivity is feasible even for a tiny perturbation in the imaginary part,and the sensitivity trends to a stable minimal error that will not be affected by the condition error.The real sensitivity is stable only within a limited real perturbation range;the error of real sensitivity goes significant with the decrease of the real perturbation steps.Thirdly,a dynamic sensitivity-based finite element model updating method is proposed to update the nonlinear structures effectively.The effect of the vibration behaviors,the initial parameters,and the noise in the time-domain responses on the updating accuracy and the selection scheme of the targeted response points in nonlinear FEMU procedures are investigated.Results illustrate that the accurate finite element model can be updated using the proposed method for different initial parameters,free and forced vibrations,and low-level measurement noise.The selection of targeted responses is related to the initial relative errors of dynamic responses.Large numbers of response points are selected to the case that initial relative error is small;on the contrary,a small number of response points are selected when the initial relative error is high.The smallest number of the response points is greater than that of updating parameters.Finally,a dynamic substructuring method based nonlinear FEMU method is proposed to improve the iteration efficiency for complex nonlinear finite element models.The formulations of sensitivity analysis for dynamic substructural targeted responses,the framework of sensitivity-based dynamic substructure finite element model updating method is constructed,and the influence of different substructuring methods on updating accuracy is investigated.Results reveal that the proposed dynamic substructure based finite element model updating for the linear and nonlinear structure is more efficient compared to the full-order model.The initial modeling errors caused by model reduction can be eliminated using the proposed FEMU method.The ECB(Enhanced Craig-Bampton)method is more accurate to update the nonlinear dynamic substructure as considering the high order residual flexibility in model reduction.