Stability of delaminated composite beams subjected to retarded periodic follower force

Abstract

This work deals with the stability problem of elastic composite cantilever beams subjected to a delayed, periodically changing follower force. The equation of motion of the periodic system with time delay is deduced based on some previous works. Composite beams with and without delamination are considered, and the finite element method is applied to carry out the spatial discretization of the structures. Besides, for the delaminated case further two cases are involved. The first case is when the delamination is in the midplane of the beam, while the second case involves an asymmetrically placed delamination, respectively. The Floquet theory is applied to derive the transition matrix of the periodic system. An important aspect is that the time delay and the principal period of the dynamic force are equal to each other. The discretization over the time domain is performed by using the Chebyshev polynomials of the first kind. Basically, there are five parameters governing the dynamic problem including among others the time delay and the static and dynamic forces. The stability behavior is shown for the intact and delaminated beams on the parameter planes for large number of cases by using the unit circle criteria. The presence and absence of structural damping is also analyzed in each case. The results indicate that some planes are sensitive to the mesh resolution, others are not. Moreover, on some planes significant differences may take place between the intact and delaminated beams from the standpoint of stable zones.