Dynamic response of thin strain gradient elastic beams

The governing equation of motion of thin flexural beams is studied in the context of strain gradient elasticity. Simple linear strain gradient elastic theory with surface energy is employed. The governing beam equations with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin beams. Those terms are due to couple stresses, and are missing from the existing strain gradient beam theories and increase highly the stiffness of the thin beam. With the full term equation of motion we will study the flexural wave propagation and free flexural vibrations of the beam. Moreover free vibrations of gradient elastic beams are also analyzed and natural frequencies and modal shapes are obtained. Furthermore, the problem of the dynamic stability of a cantilever beam compressed by a follower force (Beck’s problem) is considered. Finally there is a discussion for the results especially comparing existing theories with the present one.