Frequency response design of uncertain systems using performance indices and meta-models

Systems that are operated near their resonance frequencies experience vibrations that can lead to impaired performance, overstressing, fatigue fracture and adverse human reactions. Frequency response (FR) analysis can be invoked to mitigate the effects. When components of a system are described by random variables, modal frequencies and modal shapes, or, amplitudes and phases, are also random variables and the frequency response (FR) design of the system becomes more complex since it requires the solution of a frequency-variant probability problem. This paper presents a methodology to provide the frequency response design of uncertain systems using a transfer function approach. The methodology is found to be robust, expandable and flexible and can be applied to multi-disciplinary systems with n-dof and multiple design constraints. The novelty of the approach is the creation of a frequency-invariant probability problem through: (a) the discretization of the frequency band of interest into multiple contiguous point frequencies, (b) the introduction of new performance indices that measure the probability of success over the entire frequency band, and (c) the introduction of explicit meta-models to provide sufficiently fast probability evaluations through Monte Carlo simulation. The key to the performance indices are limit-state functions formed at all discrete, contiguous, frequencies. Each limit-state function establishes a conformance region in terms of the random design variables. The probabilities of the conformance regions are correctly combined to provide a single series-system index to be maximized by adjusting distribution parameters. The simple explicit meta-model is based on Kriging and performance measures at arbitrary design sets are efficiently calculated. Error analysis suggests ways to predict and control the errors with regards to meta-model fitting and probability calculations and so the method appears sufficiently accurate for engineering applications. The proposed methodology has applications in numerous areas such as electrical filters and structural mechanics - all with n-dof and multiple responses. The Performance indices can be evaluated at any frequency over any number of frequency ranges. A case study of a vibration absorber mechanism shows how the new methodology provides an improved and timely design with controllable accuracy when compared with previous proposals that employed modal frequencies.