Modal decomposition method for response spectrum based analysis of nonlinear and non-classically damped systems

Modal decomposition methods have been widely used in the analysis of dynamically excited linear systems. However, they have been found to be unsuitable for the analysis of nonlinear vibratory systems. The application of nonlinear modal analysis in structural dynamics is limited due to various shortcomings including the inability to apply the principle of superposition in expressing the general motion in physical coordinates even with a nonlinear version of superposition. Thus, the development of alternative routes for the analysis of nonlinear systems is highly desirable in the field of structural engineering. In this setting, there are different forms of nonlinearities that fairly arise in real structural systems and should be carefully considered by the modal decomposition method. In the majority of systems of engineering interest, non-classically damping is evident considering the general non-diagonal nature of the modal damping matrix and the coupled modal equations.

To address the above concerns and challenges, Assist. Professor Ioannis P. Mitseas at University of Leeds and Professor Michael Beer from Leibniz University of Hannover proposed a novel inelastic modal decomposition method for random vibration analysis. This innovative method took into account nonlinear and non-classically damped multi-degree-of-freedom systems in alignment with contemporary aseismic code provisions under a coherent stochastic perspective. The research work is currently published in the Journal, Mechanical Systems and Signal Processing.

In brief, the structural systems were excited by elastic response uniform hazard spectra to eliminate the need for complex nonlinear response time-history analysis. The proposed methodology was presented in terms of response-spectrum variant rather than the time-history version primarily to unleash its practical abilities among engineers of practice. The practicability of the approach in the analysis of the seismic design of yielding structures was evaluated. The induced seismic processes were characterized by power spectra compatible in a stochastic sense with an assigned elastic response spectrum of specified modal damping ratio.

The framework exhibited the ability to solve a series of inverse stochastic dynamic problems. The statistical linearization and state-variable formulation of the complex eigenvalue problem was addressed by considering inelastic multi-degree-of-freedom systems subjected to a vector of stochastic seismic processes. The equivalent modal properties of the linearized system: pseudo-undamped natural frequencies and modal damping ratios were efficiently provided. The determination of forced vibrational modal properties was repeated iteratively for each monitored mode of vibration upon updating the damping ratio of the corresponding excitation response spectrum with the equivalent modal damping ratio. Upon convergence, the peak modal nonlinear response estimates were obtained. Lastly, the real-valued modal participation factors were determined for the complex-valued mode shapes and generalized square-root-of-sums-squared was employed as the modal combination rule for determining the peak total responses of the system in physical coordinates.

The modal nature of the proposed methodology provided physical insights that cannot be found in the classic time-history methods. Results showed that the displacements of the non-classically damped nonlinear multi-degree-of-freedom system can be expressed as a linear combination of the displacements of the various excited equivalent modal oscillators by response spectra adjusted appropriately to the forced vibrational modal characteristics. The feasibility of the method was successfully illustrated using a three-storey bilinear hysteretic frame structure exposed to Eurocode 8 elastic response spectrum. Based on the findings, that study was identified by Advances in Engineering as a key scientific article with significant contribution to the dynamic structural engineering field.