Stability and bifurcations investigation of an axially functionally graded beam coupled to a geometrically nonlinear absorber

In this paper, the forced vibration behavior of an axially functionally graded (AFG) cantilever-based beam is explored while a lightweight geometrically nonlinear absorber is connected to it. It is shown that the attachment can control the unwanted vibrations and prevent the structure from failure, especially in the resonance situation. The particular configuration of the absorber produces a stiffness with a negative linear and a cubic nonlinear term. The equations of motion of the system are solved both analytically (complexification-averaging method) and numerically. According to the results, in order to reach the best efficiency of the absorber, the best place for attaching it is the free-end of the cantilever beam. The saddle-node bifurcation occurs only for small amounts of the absorber damping. Besides, the interval of the saddle-node bifurcation would grow with reducing the elasticity modulus gradient and damping. Moreover, the strongly modulated response (SMR) region would be increased by rising the elasticity modulus gradient and reducing the density gradient. Hence, the performance of the absorber has the same relation as SMR region to the elasticity modulus gradient and density gradient. Furthermore, the frequency range for the existence of the SMR is apparent when considering the energy and entropy diagrams.