Dynamic Analysis of a Passive Plate
The objective of using a high precision finite element is to carry out the dynamic analysis of laminated composite plates. Such analysis often requires repeated operations; each involves the computational effort equivalent to a single static solution for a single load vector. To reduce the amount of computation, it is sometimes helpful to reduce the size of matrices being manipulated. Various kinds of reduction can be identified. Each can be regarded as either a way of imposing an elastic constraint or as a way of providing a ‘reduced basis’. A basis is a set of linearly independent vectors that can be combined in various proportions to represent other vectors. In the context of structural vibration, ‘other vectors’ implies the complete set of eigen vectors of the FE model. A basis is called ‘reduced’ if it includes fewer vectors than the complete set.
The internal degrees of freedom do not have any physical significance; hence, they can be condensed from the system equations. In the element level for undamped free vibration, stiffness and mass matrices are partitioned as
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(19.6) |
Here D1 and D2 are the master and slave degrees of freedom respectively. Lower partition is solved for D2 and is substituted into the upper partition. Thus obtaining a smaller system that has only D1 as DOF. Matrices of such a reduced system would be frequency-dependent. To obtain a frequency-independent transformation, Guyan suggested that the relation between the slaves and the masters be dictated entirely by stiffness coefficients. So, the modified stiffness and mass matrices are
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(19.6) |
Modal analysis is a powerful tool to determine the free and forced vibration responses of multi-degree systems. The system vibrates with harmonic motion at a natural frequency when and only when the amplitudes of vibration of the masses in the system satisfy any one of its mode shape
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(19.7) |
Inbuilt solver of MATLAB and subspace iteration techniques is used to find the eigen frequencies. A standard FEM package ABAQUS is used for modal analysis to check the dynamic behavior for higher modes and also to verify the FEM code.
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