Interval sensitivity analysis of dynamic response envelopes for uncertain mechanical structures

Non-deterministic approaches are gaining momentum in the field of finite element analysis. The ability to include non-deterministic properties is of great value for a design engineer. It enables realistic reliability assessment that incorporates the uncertain aspects of the design. Furthermore, the design can be optimised for robust behaviour under varying external influences. Recently, criticism arises on the general application of the probabilistic concept in this context. Especially when objective information on the uncertainties is limited, the subjective probabilistic analysis result proves to be of little value, and does not justify its high computational cost. Consequently, alternative non-probabilistic concepts are used for non-deterministic finite element analysis, as e. g. the fuzzy and interval concept.

Recently, a fuzzy finite element methodology to calculate a fuzzy frequency response function (FRF) of uncertain undamped structures was developed by the authors. The procedure consists of the solution of a sequence of interval problems. The goal of each interval analysis is to calculate the envelope of the FRF taking into account that the input uncertainties can vary within the bounded space defined by their combined intervals. The resulting envelope response function gives a clear view on the possible variation of the response in the frequency domain. The applicability of this interval response analysis was proven on realistic case studies. In this result, all uncertain parameters are considered to act simultaneously. While this is often the most realistic representation of the physical condition of the actual product, for design purposes, it can be of great value to know the contribution of the individual uncertainties to the response range obtained from the interval analysis. This enables a designer to distinct between the non-deterministic influences that have an important contribution to the fuzziness on the dynamic behaviour of the design, and those that have little or no influence. This distinction can be very valuable in the definition of e. g. tight design tolerances or realistic allowable working conditions, and as such, could lead to less conservative designs.

This paper introduces an interval sensitivity procedure that calculates the sensitivity of the envelope response function in the outcome of the interval FRF analysis to each individual interval model uncertainty. The procedure focusses on the calculation of the sensitivity of the bounds defining the FRF response range to the width of each individual uncertain input parameter. The approach differs from the classical sensitivity analysis in the fact that it does not calculate local changes on the output resulting from local changes in the input. The interval sensitivity result describes the change of the response interval width, taking into account a change in the parameter interval width. The paper first describes the theoretical background of the fuzzy and underlying interval FRF procedure. Next, it introduces the methodology for interval sensitivity analysis. Finally, the method is illustrated on a numerical example.