By Farzad Ebrahimi

ISBN-10: 9533072091

ISBN-13: 9789533072098

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**Extra resources for Advances in Vibration Analysis Research**

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Finite elements method (FEM) derivations Two different FEM models were derived for the curved beam model. The first one has three degrees of freedom (DOFs) per node and uses a linear approximation for the axial displacement and a Hermite type polynomial approximation for the bending displacement. v( y ) =< N ( y )v > { v j v j + 1 } = N 1 v ( y )v j + N 2 v ( y )v j + 1 w( y ) =< N ( y )w > { w j w ' j w j + 1 w ' j + 1 } = N 1 w ( y )w j + N 2 w ( y )w ' j + N 3 w ( y )w j + 1 + N 4 w ( y )w ' j + 1 (8) (9) In the equations above, vj, vj+1, wj and wj+1 are the nodal values at j and j+1 corresponding to the circumferential and radial displacements, respectively (these can be likened to the axial and flexural displacements for a straight beam).

40-Elem. 61 FEM; 4-DOF/node 20-Elem. 40-Elem. 02 Table 1. 88rad/s Fig. 3. 53% for the fifth natural frequency for 20-element models 46 Advances in Vibration Analysis Research when comparing the DFE and the 4-DOF FEM. 51% again for the fifth mode when comparing the DFE and 4-DOF FEM. Also, the first four normalized modes were computed using DFE model for the cantilevered curved sandwich beam and are shown in the Figure 3 below, generated using a 40-element DFE model. The curved beam has a large radius of curvature compared to its span, so the mode shapes of a straight beam can be used as a rough guideline to gauge the acceptability of the current modes.

Each element is defined by nodes j and j+1 with the corresponding co-ordinates (l=xj+1–xj). The admissibility condition for finite element approximation is controlled by the undiscretized forms of equations (4) and (5). 3. Finite elements method (FEM) derivations Two different FEM models were derived for the curved beam model. The first one has three degrees of freedom (DOFs) per node and uses a linear approximation for the axial displacement and a Hermite type polynomial approximation for the bending displacement.

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