Estimating the ratio between travelling and standing vibration waves under non-stationary conditions

This paper presents a new method of spatially decomposing vibration patterns in real time into their travelling and standing parts. This method creates a localized parametric model describing the nature of the developed vibrations leading to a scalar measure of the travelling to standing waves ratio. Despite being rather simple compared with spatial Fourier transform that yields space-averaged results, the current method seems superior for a localized description. Due to its superior localization, the decomposition conveys important insight and valuable information for vibrating structures that can be used for identification, diagnosis and for control purposes of ultrasonic motors and rotating discs. Several features make the proposed method advantageous over existing schemes, in particular when the number of deployed sensors is small and confined to a small region. It is shown that the presented approach differs from the commonly used Fourier-based methods in several ways: (a) the proposed method does not require equally distributed sensors and does not require a spatially complete coverage of the analyzed domain nor does it require equally spaced sensing elements; (b) the algorithm makes neither use of the spatial wavelength nor requires its estimate to curve-fit the instantaneous spatial deformation patterns; (c) the method is most suitable for cases where a localized pattern needs to be estimated and is therefore robust to imperfections in the vibrating structure; (d) the presented formulation has a recursive form that is suitable for real-time implementations by a digital signal processor.