Simple formulas for the natural frequencies of non-uniform cables and beams

• The problem of free vibration of non-uniform cables and beam is addressed by asymptotic developments method.
• A solution exact up to the first order is obtained.
• The mechanical properties of the equivalent uniform beams and cable are obtained.
• Simple formulas for the frequencies are obtained for arbitrarily shaped beams and cables.
• An error estimation is proposed.

The asymptotic development method is used to obtain approximate analytical expressions for the natural frequencies of non-uniform cables and beams. By manipulating the first-order terms, we obtain the mechanical properties (mass, stiffness, etc.) of the equivalent uniform cables and beams having the same (up to the first order) frequencies of the non-uniform one. The second order terms provide an error estimation for the previous expressions. Some examples are reported to illustrate the effectiveness and simplicity of the proposed formulas.