Vibrations of A Beam of Varying Cross Section with End Mass

The vibrations of beam with a mass attached to its end is discussed. This model applied to civil engineering towers. By using Bessel functions, the frequancy equation is obtained. The frequancy rallies of the first mode, 2nd mode and 3rd mode for many cases are calculated. The fundamental frequancy calculated by present paper are agreement with other approximate method, but present method can calculate the frequancy of any orders.