Abstract
Effective mass for a particular mode in a particular direction is classically calculated using a combination of fixed base mode shapes, the mass matrix, and a rigid body mode shape from a finite element model. Recently, an experimental method was developed to calculate effective mass using free experimental mode shapes of a structure on a fixture (the base) along with the measured mass of the fixture and of the test article. The method required three steps. The first step involved constraining all the free modes of the fixture except one rigid body mode in the direction of interest. The second step involved calculating pseudo-modal participation factors for this case. The third step involved constraining the final fixture rigid body degree of freedom and utilizing the constraint matrices with pseudo-modal participation factors to obtain the estimate of the standard modal participation factors which can be converted to effective mass. This work provides a simpler formulation. After the constraint in step one above, the effective masses are calculated directly from the mass normalized mode shapes of the fixture. In most cases this method gives the same answer as the original approach, within experimental error. In some instances, it appears more robust with low signal to noise ratios. It also provides better physical insight as to which modes have significant effective mass in a particular direction. The new approach is illustrated by experimental example.
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy National Nuclear Security Administration under Contract DE-AC04-94AL85000.
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Abbreviations
- DoF:
-
Degree of freedom
- DP:
-
Drive point
- FRF:
-
Frequency response function
- PMPF:
-
Pseudo-modal participation factor
- H dp (ω) :
-
Drive point FRF at fixture after all fixture mode shapes except one are constrained
- m eff r :
-
Effective modal mass for mode r
- m :
-
Mass
- \( \ddot{x} \) :
-
Acceleration in one Cartesian axis direction
- q :
-
Generalized coordinate
- L:
-
Reduction matrix applying the constraint to equations of motion
- Φ:
-
Mode shapes from free modal test of test article mounted on fixture
- Θ:
-
Mode shapes of test article and fixture after all fixture mode shapes except one are constrained
- ζ :
-
Modal damping ratio
- η :
-
Generalized coordinate for partially constrained system
- ω :
-
Angular frequency (radians/second)
- Ψ:
-
Mass normalized real mode shape matrix of the fixture
- Γ:
-
Eigenvectors resulting from constraint equations
- b :
-
Subscript for the base or fixture sensor DoF
- fix :
-
Subscript for the fixture
- r :
-
Subscript for mode number
- res :
-
Subscript for residual effective mass of all higher modes not extracted
- RB :
-
Subscript for the single rigid body mode of a partially constrained system
- TA :
-
Subscript for the test article
References
Mayes, R.L., Schoenherr, T.F., Blecke, J., Rohe, D.P.: Efficient method of measuring effective mass of a system. Proceedings of the 31st International Modal Analysis Conference, Garden Grove, CA, February 2013, paper #194
Hensley, D.P., Mayes, R.L.: Extending SMAC to multiple references. Proceedings of the 24th International Modal Analysis Conference, Orlando, FL, pp. 220–230, January 2006
Acknowledgement
Notice: This manuscript has been authored by Sandia Corporation under Contract No. DE-AC04-94AL85000 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.
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Mayes, R.L., Hunter, P.S. (2017). A Simpler Formulation for Effective Mass Calculated from Experimental Free Mode Shapes of a Test Article on a Fixture. In: Allen, M., Mayes, R., Rixen, D. (eds) Dynamics of Coupled Structures, Volume 4. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-54930-9_24
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DOI: https://doi.org/10.1007/978-3-319-54930-9_24
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