15 Lessons Learned from Operational Modal Analysis Courses at the University of Molise 163 Even if the input is immeasurable, it is important to remark that some assumptions about it are needed. For instance, the assumption of broadband excitation ensures that all the structural modes in the frequency range of interest are excited. Assuming that the combined system is excited by a random input, the second order statistics of the response carry all the physical information about the system and play a fundamental role in output-only modal identification. The focus on second order statistics has to be justified by referring to the central limit theorem, pointing out that the structural response is approximately Gaussian in most cases, no matter of the distributions of the (independent) input loads, which are often not Gaussian. The spatial distribution of the input also affects the performance of OMA methods, in particular in the presence of closely spaced modes. A distribution of random in time and space inputs provides better modal identification results [4]. In order to illustrate this concepts and the role of sensor layout, it is possible to present the problem of modal identification as one of matrix rank computation. In fact, taking into account that the output power spectral density (PSD) matrix [SYY(¨)] can be expressed in terms of the Frequency Response Function (FRF) matrix [H(¨)] of the structure and the input PSD matrix [SFF(¨)] as follows: ŒSYY .!/ DŒH.!/ ŒSFF .!/ ŒH.!/ T (15.1) its rank cannot be larger than the rank of the individual matrices appearing in the product. This implies that closely spaced modes cannot be estimated if the rank of the input PSD matrix is equal to one. Based on this simple mathematical concept, it becomes easy to show that closely spaced modes cannot be identified if only one input is present, or the inputs are fully correlated. Rank deficiency over a limited frequency band in the proximity of the considered closely spaced modes can partially hide the actual physical properties of the structure (for instance, by revealing only one of the modes, or a combination of the two modes). Thus, a proper design of sensor layout and a preliminary evaluation of the sources of excitation acting on the structure in its operational conditions play a primary role in ensuring the possibility to obtain high quality information from modal testing. The following general rule is mentioned: a large number of sensors allow maximizing the rank of the FRF matrix, while several uncorrelated inputs ensure the maximization of the rank of the input PSD matrix. On the contrary, correlated inputs or input applied in a single point limit the rank of the input PSD matrix; sensors placed in nodes of the mode shapes or multiple sensors measuring the same DOF (thus, adding no new independent information) limit the rank of the FRF matrix. This rule gives valuable suggestions for proper test execution. Having shown that all the physical information can be found in second order statistics, correlation functions and power spectral density functions are introduced. The basic ideas behind Fourier analysis and relevant theory behind correlation functions and power spectral densities are reviewed. Then, their practical estimation based on measured data is illustrated. The proper computation of correlation functions and power spectral densities is critical in view of the application of OMA methods. Thus, specific applications have been designed for the students to learn how to compute these functions from measured random vibrations with the support of their laptop (Fig. 15.1). High quality measurements represent another fundamental step for a successful modal identification. Any OMA method is ineffective if measurements are totally corrupted by noise. Poor measurements can be the result of an incorrect choice of sensors or measurement hardware, but they can be due also to incorrect wiring. In fact, for a given choice of the measurement hardware and sensors, different measurement schemes can often be adopted. The choice of the most appropriate cabling scheme and the adoption of the related specifications for the entire analog signal path play a primary role in the collection of high quality data. The sensing technologies available on the market are presented, pointing out advantages and limitations of different measurement solutions. In particular, the difference between grounded and floating sensors, and between single ended and differential measurement systems are illustrated in detail. In order to encourage the participation of the students and simplify the presentation of concepts that are usually far from the typical background of civil engineers, datasheets of different accelerometers and measurement devices are delivered to the students in order to set up an appropriate measurement chain for output-only modal identification. To this aim it is also important to summarize the main characteristics of different sensing solutions. In this perspective, a comparative analysis of the sensors usually used for modal testing of civil structures (piezoelectric and force balance accelerometers) is proposed. The presentation of the measurement principle of the sensors is fundamental to understand why different sensors show different advantages and limitations. For instance, the comparative analysis makes easy to understand why force balance accelerometers can measure up to the DC component while piezoelectric accelerometers cannot, or why different accelerometers, even of the same type (i.e. piezoelectric accelerometers), are characterized by a different bandwidth.
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