# chap92_add - Example Given a uniform cantilever beam Model...

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1 Example Given a uniform cantilever beam. Model the beam with a single finite element . Calculate the natural (modal) frequencies and mode shapes. Solution: Only two unrestrained dof in the system. Only the 2x2 lower right portion of the element stiffness and mass matrices will be used. From Eqs. 2.3-2 and 9.3-5 (given in the next pages):

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2 Plane Beam Element: Stiffness matrix = 2 3 6 1 3 2 3 6 3 6 2 2 2 2 L L L L L L L L EI k (SYMMETRIC)
3 Consistent Mass Matrix of a Plane Beam

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4 Example: The modal frequencies Eigenvalues are found by setting determinant to zero Expanding the determinant gives the characteristic equation , a quadratic equation in a. Solving we obtain the following two frequencies The exact numerical coefficients are 3.516 and 22.03. Errors are then 0.5 % and 58 %, respectively.
5 Example (cont.) The mode shape corresponding to each frequency is obtained by

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