Comments on “Free vibration of functionally graded beams and frameworks using the dynamic stiffness method [J. Sound Vib. 422(2018) 34–47]”

Popov comments on free vibration of functionally graded beams and frameworks using the dynamic stiffness method. He has demonstrated that it is not necessary in the application of the transfer matrix method to derive beforehand explicit analytical expressions for achieving the exact solutions in Ref. [1]. An effective and efficient numerical procedure has been implemented. The approach relies on computed matrix exponentials based on the system matrix originating in the system of differential equations of the mechanical system. The author was able to replicate results in Refs. [1,7] for functionally graded beams by employing the matrix exponential computation. It is thought that the potential of using functions of matrices goes beyond the discussed specific application.