Determine the smallest natural frequency of a beam with clamped ends, and of constant cross-sectional area A, moment of inertia I, and length L. Use the symmetry and two Euler—Bernoulli beam elements in the half beam


Note that the beam problem is a hyperbolic equation, hence the eigenvalue is the square of the natural frequency of flexural vibration, ω. For a mesh of two Euler—Bernoulli elements in a half beam (i.e., h = L/4), the assembled equations are given by

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The boundary conditions are: U1 = U2 = U6 = 0 and Q2/ 3 = 0. The condensed equations are given by

The determinant of the coefficient matrix yields a cubic polynomial in ω2. Note that by considering the half beam we restricted the natural frequencies to those of symmetric modes. The antisymmetric modes (only) can be obtained by using U5 = 0 instead of U6 = 0.