Table 2.6 to compute the natural frequencies of uniform flexural beams with different supporting conditions. Methods are also available for dynamic analysis of continuous beams (Clough and Penzien 1993). Shear Vibration of Beams

Beams can deform by flexure or shear. Flexural deformation normally dominates the deformation of slender beams. Shear deformation is important for short beams or in higher modes of slender beams. Table 2.7 gives the natural frequencies of uniform beams in shear, neglecting flexural deformation. The natural frequencies of these beams are inversely proportional to the beam length L rather than L2, and the frequencies increase linearly with the mode number.

TABLE 2.7 Frequencies and Mode Shapes of Beams in Shear Vibration

L = length

K = shear coefficient (Cowper 1966) G = shear modulus = E/[2(1+v)j r = mass density

Boundary condition

Mode shape, fni-^


Fixed-fixed nn, n = 1, 2, B, nnx sin-, n = 1,2, B, i

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