Chapter6 Fixing Degrees of Freedom of an Aluminum Beam by Using Accelerometers as References Kevin L. Napolitano Abstract Modal tests are performed to validate analysis models of structures, and it is important to support a test article use fixtures that allow an engineer to focus his time and effort on updating the analysis model instead of the supports. Oftentimes, however, inadequate boundary condition fixtures are used in modal surveys because the design and manufacture of a proper boundary condition may be too expensive for a program. An alternative approach of creating appropriate boundary conditions by using accelerations as references to fix degrees of freedom is presented in this paper and is demonstrated using test results from a tap test on an aluminum beam. Frequency response functions (FRF) are generated directly and indirectly using a partial inversion of the FRF matrix for several different boundary condition cases using the same set of test data. Modes are extracted from the resulting FRF and are compared to an analysis model. Keywords Modal testing · Vibrations · Structural modification · Fixed base · Constraint shapes 6.1 Introduction Modal tests are performed to validate analysis models of structures, and it is important to support a test article use fixtures that allow an engineer to focus his time and effort on updating the analysis model instead of the supports. Oftentimes, however, inadequate boundary condition fixtures are used in modal surveys because the design and manufacture of a proper boundary condition may be too expensive for a program. Great effort and expense is then spent during the model updating task accounting for the test fixture instead of the analysis model. Errors from the test fixture model then propagate to the updated analysis model, degrading the results of the model updating task. Both modal substructuring and frequency based substructuring methods have been used to help extract fixed base modes from structures mounted on uncertain boundary conditions. An excellent description of both methods are described in the following reference [1]. An approach of creating appropriate boundary conditions by using accelerations as references to fix degrees of freedom is presented in this paper and is demonstrated using test results from a tap test on an aluminum beam. This method is not new and follows frequency base substructuring techniques [2, 3]. Frequency response functions (FRF) are generated directly by using accelerations as references when calculating FRF [4] or indirectly using a partial inversion of the measured acceleration/force FRF matrix for several different boundary condition cases using the same set of test data. In doing so, modes extracted from the resulting FRF are associated with those same acceleration degrees of freedom fixed. 6.2 Theoretical Background The derivation of the partial inversion of the FRF matrix [H(ω)] is presented here. It is equivalent to the Structural Modification Using Frequency Response Functions (SMURF) method. An FRF matrix, [H(ω)] can be partitioned into two sets of degrees of freedom, those associated with internal DOF “i”, and those associated with boundary DOF “b”. A subset of the internal DOF can be further partitioned to DOF where forces are applied, specified by the subscript “f”. In this derivation, the boundary DOF are assumed to be impacted by a force. Thus, the FRF matrix can be partitioned as follows K. L. Napolitano ( ) ATA Engineering, Inc., San Diego, CA, USA e-mail: kevin.napolitano@ata-e.com © Society for Experimental Mechanics, Inc. 2020 M. L. Mains, B. J. Dilworth (eds.), Topics in Modal Analysis & Testing, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12684-1_6 53
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