Vibration analysis using approximate methods for heavily damped systems with variability

Abstract

In many engineering fields, vibration analysis is required for a safe design of mechanical structures. At an early stage, the design process relies on computer models for assessing the performance of the structure. Computational methods, such as the Finite Element Method (FEM), are typically used in order to predict the vibration behaviour of the modelled structure. These methods, however, involve a high computational cost for obtaining the dynamic response of a system model, specially when the analyses are targeted at vibrations in the mid-frequency range. Furthermore, repeated vibration analyses are often required in structural design since the model parameters may be variable, e.g. in model updating from vibration measurements, in model optimisation algorithms, or in vibration analysis of systems subject to uncertainty. Therefore, in applications where repeated analyses need to be conducted intensively, the vibration analyses cost may add up, so that the overall computation time becomes unbearable for the engineering needs. For this reason, there already exist efficient methods for the approximate vibration reanalysis of systems with varying properties. However, these approximate methods most often neglect the energy dissipation mechanisms in the structure, and just address the system dynamics through undamped modal analysis. This produces sufficiently fair approximations for lightly damped structures, but it may not be the case for locally/heavily damped structures.

This thesis presents novel approximate vibration analysis methods for heavily damped structures, which are based on the state-space formulation for the modal analysis of generally damped systems. It is shown how, whereas the modal solution of the state-space equations of motion presents some added computational difficulties, linear approximations on state-space models present equivalent computational complexity to those on undamped models, while providing better estimates of the variation in the damped dynamics. In particular, the variations in (state-space) damped modes due to variations in the system model parameters are efficiently estimated through perturbation and interpolation methods. Moreover, a recently defined Rayleigh quotient for damped systems is proposed for improving the accuracy of the linear approximate methods. In order to enhance the efficiency of the approximate reanalyses, it is further proposed to reduce the state-space models through Component Mode Synthesis (CMS). Existing CMS methods based on state-space formulation are investigated for the reduction of large built-up systems which are locally and/or heavily damped. A perturbation propagation scheme on state-space CMS models is then introduced. By simplifying this general scheme for a given CMS method much efficient perturbation methods for damped systems are attained. Finally, these latter methods are used in a numerical case study in application to uncertainty analysis.

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Identifiers

ORCID for Neil Ferguson: orcid.org/0000-0001-5955-7477

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