Exponential sine sweeps for the autonomous estimation of nonlinearities and errors assessment by bootstrap Application to thin vibrating structures

Abstract

Vibrating structures are generally assumed to behave linearly and in a noise-free environment. This is in practice not perfectly the case. First, nonlinear phenomena such as jump phenomenon, hysteresis or internal resonance appear when the transverse vibration of a bi-dimensional structure exceeds amplitudes in the order of magnitude of its thickness. Secondly, the presence of plant noise is a natural phenomenon that is unavoidable for all experimental measurements. In order to perform reliable measurements of vibrating mechanical structures one should thus keep in mind these two issues and care about them. However, it turns out that they are actually coupled. Indeed, all the noise that is not correctly removed from the measurements could be misinterpreted as nonlinearities, thus polluting measurements. And if nonlinearities are not accurately estimated, they will end up within the noise signal and information about the structure under study will be lost. We thus try here to solve simultaneously both issues. The underlying idea consists in extracting the maximum of available linear and nonlinear deterministic information from measurements without misinterpreting noise. The aim of this talk is thus to provide a methodology that allows for the autonomous estimation of nonlinearities and errors assessment by bootstrap on a given vibrating structure. Nonlinearities are estimated by means of a block-oriented nonlinear model approach based on parallel Hammerstein models and on exponential sine sweeps. Estimation errors are simultaneously assessed using repetitions of the input signal (multi exponential sine sweeps) as the input of a bootstrap procedure. Mathematical foundations and practical implementation of the method are discussed on an experimental example. The experiment chosen here consists in exciting a steel plate under various boundary conditions with exponential sine sweeps and at different levels, in order to assess the evolutions of nonlinearities and of signal to noise ratio over a wide range of frequencies and input amplitudes.