Upper and Lower Frequencies Tapered Beams

Two equations for obtaining lower bounds of the natural frequencies of lateral vibrations of beams are presented. The effects of shear and rotatory inertia are included. One of the equations is employed in conjunction with the Rayleigh-Ritz method to obtain upper and lower bounds for the three lowest natural frequencies of vibration of linearly tapered beams with free-free end conditions. For cases in which the effects of shear and rotatory inertia are small, the upper and lower bounds are in good agreement for all three modes of vibration. In cases in which the effects of shear and rotatory inertia are significant the upper and lower bounds are in good agreement for the first two modes; for the third mode, however, the percent deviation is, in some cases, about 20% for beams with small length to depth ratios.