The Vibrational Power Flow Analysis Of Functionally Graded Material Beams With Open Cracks

Functionally graded materials(FGM)are made from a continuous gradient mixture of many materials along its dimensions,which have more advantages than other composite materials and are used in a variety of engineering fields.Beams are one of the most important structures,while the existence of cracks in the beam affects its local stiffness and other properties and then changes its vibration and wave propagation characteristics.First,the free vibrational characteristics of functionally graded material(FGM)beams with open cracks are studied.It is assumed than the material properties varying continuously in the thickness direction according to the exponential function.Free vibration equations of beam are deduced based on Bernoulli-beam theory.The crack is simulated by using the rotational spring model.And a detailed parametric study is conducted to show the influences of the length,crack location,crack depth,material property gradient,and boundary conditions on the natural frequencies.The relusts indicate that the variation of parameters will affect the natural frequencies of the beams.Then the wave propagation and power flow of FGM beams with open cracks are studied.The governing differential equations of the infinite functionally graded beams are established and the vibration characteristics are obtained by using the wave propagation method with continuity conditions.The input power flow and transmitted power flow of intact and cracked beams are calculated.The influences of material gradient,excitation frequency,crack location and crack size on the input power flow and transmitted power flow are studied,which provides the basis for the future study on the functionally graded material structures' damage identification based on vibrational power flow method.Finally,a detailed parametric study is conducted to show the characteristics of modal power flow of the FGM beams with cracks.The FGM beam's mathematical formula of the modal power flow is derived,and a damage index based on the modal power flow is proposed.The damage identification of FGM beams with clamped-clamped,clamped-free and hinged-hinged end supports is analyzed by using the damage index based on modal power flow.Numerical experiments show that the index not only identify the crack lacation,but also predicate the severity of the crack.And material gradient can also influence the index.By compared with the damage indices based on the slope,bending strain and the strain energy,we can see that the index based on modal power flow also has its superiority.