A criterion using rigid-body modes to verify the conservation of mass inertias is presented. Conservation of rod element mass guarantees convergence to the exact eigensolutions of a rod. Conservation of beam element mass guarantees convergence to the exact eigensolutions of a Bernoulli-Euler beam without rotatory inertia. Conservation of element mass and rotatory inertia guarantees convergence to the exact solutions of a Bernoulli-Euler beam with rotatory inertia. Conservation of mass moment of inertia is not a requirement for convergence, but is important for a beam mass matrix with respect to their accuracy and consistency with various boundary conditions. Based on this criterion, a concept for the formulation of a non-consistent mass matrix is presented. The concept unifies the formulation of various kinds of rod and beam mass matrices, and facilitates the generation of new mass matrices for optimization. To gain more physical insight into the formulation, the shape functions for the non-consistent mass matrices are also introduced.
Four examples are considered. The first two examples are used to find the optimized mass matrices for rods and beams and to study their eigensolution errors. The optimized mass matrices minimize the root mean square errors of natural frequencies over a specified range of modes. The results of using a rod optimized mass matrix show that the root mean square error of natural frequencies for the first half of total extractable modes is reduced from 5%, obtained from using the consistent-mass and the lumped-mass matrices, to 1%. The results also show that if equally spaced elements are used for a rod, all the eigenvectors are exact. However, if unequal-length elements are used, both the frequency errors and eigenvector errors increase, and the upper half of total extractable modes are not reliable. The results of using a beam optimized mass matrix show that the root mean square error of natural frequencies is reduced from 0.16%, obtained from using a consistent-mass matrix, to 0.10%. The upper half of the total modes are not reliable. The remaining two examples are used to study the performances of all rod and beam mass matrices (consistent-mass, lumped-mass, and higher-order mass matrices) on a portal arch. According to the results, the higher-order mass matrix generates the most accurate eigensolutions. The use of the higher-order mass matrix in place of the consistent-mass matrix is recommended. The block-diagonal lumped-mass matrix performs better than the diagonal lumped-mass matrices at free ends of a structure. The eigensolution errors for all the mass matrices start to increase significantly after the first one third of the total modes.
Finally, a technique for finding the modal reduction mass matrices is proposed. Fully populated modal reduction mass matrices for a rod are successfully extracted. This type of models generate exact natural frequencies and mode shapes for all the extractable modes of a rod problem. Further investigation of this technique is recommended. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/38980 |
| Date | 28 July 2008 |
| Creators | Young, Kuao-John |
| Contributors | Mechanical Engineering, Mitchell, Larry D., Knight, Charles E., Wicks, Alfred L., Hallauer, William L., Hendricks, Scott L. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | xvii, 206 leaves, BTD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 23716241, LD5655.V856_1990.Y696.pdf |
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