Chapter 18 Developing a Correlation Criterion (SpaceMAC) for Repeated and Pseudo-repeated Modes Pranjal M. Vinze, Randall J. Allemang, and Allyn W. Phillips Abstract One of the most important factors in the validation of a finite element model is whether the modal vectors obtained from the finite element solution match sufficiently with the modal analysis results of the part or a test prototype. Modal Assurance Criteria (MAC) (Allemang, Brown, A correlation coefficient for modal vector analysis. In: Proceedings of the international modal analysis conference, pp 110–116, 1982) is usually a very effective way to check this condition. However, in case of repeated modal frequencies and vectors, MAC can give misleading results. Hence, there is a need for a method that could indicate how well the finite element method-based estimates for the repeated modes correlate with the modal analysis modes. This work is an attempt to develop a correlation criterion between a set of repeated roots from a finite element method solution and an experimental modal analysis solution. Building on a low dimensional modal vector example, a vector subspace-based approach was identified to help properly define the solution to a characteristic equation with repeated roots. This analogy was extended to higher dimension modal vector cases and vector subspace or hyper planes were identified as a way to model a repeated mode case. Similar to MAC consistency of the solution was considered as ideal way to establish correlation. But in this case the consistency of the solution subspace was found to be more important than that of normalized modal vectors. The smallest principal angle between the two solution subspaces was identified as a way of measuring the consistency. The criterion, referred to as spaceMAC, was developed as a function of this angle such that the range of the criterion is 0–1, similar to MAC was defined as 1- sin(θ) where θ is the principal angle between the two solution subspaces. This criterion was tested with two datasets 2001 Circular Plate Dataset and 2014 Circular Plate Dataset. Keywords Modal Assurance Criterion (MAC) · Repeated modes · Correlation · Modal vector · Vector subspaces 18.1 Introduction Modal Assurance Criterion (MAC) is the most common way of measuring the consistency of the estimates of modal vectors when the lengths of the modal vectors are same. It can be used in a ‘black box’ manner where the user can input the modal vectors and get a direct indicator of correlation between various modal vectors. However, there is a drawback in this approach. MAC can give misleading results in case of repeated and pseudo-repeated vectors. A usual workaround that is used in these cases is to observe a wireframe animation or to observe the mode shape for the modes in question. Then it can be checked if any shape can be rotated in space to get the shape for some other mode. Apart from using the wireframe animation, some methods exist to detect and correlate repeated roots. For example, it has been shown that by taking the singular value decomposition (SVD) of unity scale factor (USF), which is defined similarly to modal scale factor (MSF), the number of repeated roots can be found [1]. A method developed for correlating repeated modes is shown in [2], where the test mode is expressed as a linear combination of all possible combinations of analytic modes. The cross-MAC values for all these combinations and the test mode shape are calculated. From these cross-MAC values it is found which combination is a repeated mode combination. A new method has been developed based on vector subspace theory to address some limitations of existing methods and make the criteria more ‘black-box’ approach friendly. P. M. Vinze ( ) · R. J. Allemang · A. W. Phillips Department of Mechanical Engineering, College of Engineering and Sciences, University of Cincinnati, Cincinnati, OH, USA Structural Dynamics Research Laboratory, College of Engineering and Sciences, University of Cincinnati, Cincinnati, OH, USA e-mail: vinzepm@mail.uc.edu; allemarj@ucmail.uc.edu; philliaw@ucmail.uc.edu © The Society for Experimental Mechanics, Inc. 2021 B. Dilworth (ed.), Topics in Modal Analysis & Testing, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47717-2_18 189
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