An explicit expression of natural frequencies through crack parameters is derived for multiple cracked beams with simply supported boundaries using the Rayleigh quotient. The obtained provides not only a simple tool calculating beams, but also allows employing so-called scanning method detecting cracks in from measured frequencies. A numerical example demonstrates that method, combination quoti...

in the present work, study of the vibration of a functionally graded (fg) cylindrical shell made up of stainless steel, zirconia, and nickel is presented. free vibration analysis is presented for fg cylindrical shells with simply supported-simply supported and clamped–clamped boundary condition based on temperature independent material properties. the equations of motion are derived by hamilton...