Assessment of the applicability and error of eigendecomposition in the problems of nonlinear multidimensional system dynamics

Real structural elements exhibit various nonlinear properties, which are most significantly manifested during intensive dynamic processes. The deviation of mechanical characteristics from the linear ones adopted in idealised models, for which the problem of eigenvalues is solved, can lead to unacceptable calculation errors or completely false results. For this reason, the accuracy and applicability of linear models and resulting methods based on solving problems of eigenvalues remain an open question. The study is aimed at the analysis of errors associated with the application of spectral methods in “best” approximations of nonlinear characteristics by linear dependences, obtained on the basis of root-mean-square approximations, which prevent superfluous doubts about the result formation. A dynamic model of an unsupported beam with two concentrated masses oscillating in directions perpendicular to the stiffness axis was considered. For the linear model, the accuracy of approximating the initial nonlinear stiffness was assessed by comparing the amplitude displacement values and velocities of the system at the linearized and initial stiffness. In addition, a comparison of the above linearization method with the linearization by a zero first derivative of the nonlinear stiffness function is considered. The discrepancies in the results represent the functions of initial conditions. The limit values of deviations at the maximum point of the function, describing the nonlinearity of stiffness, comprised 2.02 and 10.55% for the methods of standard deviation and zero first derivative, respectively. The obtained results require clarification with regard to structural systems used in construction practice.

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