Research On Dynamic Characteristics And Calculation Method Of Multi-span Functionally Graded Material Pipes Conveying Fluid

Fluid-conveying pipes have wide applications in engineering fields,and functionally graded materials(FGMs)also have attracted many attentions of researchers.In this paper,dynamic characteristics and calculation method of multi-span FGM pipes conveying fluid are investigated.The governing equation of FGM pipes conveying fluid is derived by Newton's method and Hamilton's principle.And then,natural frequencies and mode shapes of multi-span viscoelastic FGM pipes conveying fluid are determined by finite element method(FEM),dynamic stiffness method(DSM)and a hybrid method which combined by reverberation-ray matrix method and wave propagation method.The results determined by present methods are compared with FEM results,experimental results and those in existing literature.Then,the effects of volume fraction laws,fluid velocity,fluid pressure,internal damping,material length scale parameter,nonlocal parameter and location and number of supports on free vibration and stability of multi-span FGM pipes conveying fluid are discussed.The main research contents and the main conclusions are presented as follows:1.The material properties of the FGM pipe are considered as graded distribution along the thickness direction according to a power-law.The governing equation is derived by Newton's method and Hamilton's principle.The DSM and FEM are adopted to calculate the dynamic characteristics and responses of a two-span FGM pipes conveying fluid.The effects of fluid velocity and volume fraction exponent on free vibration and stability of FGM pipes conveying fluid are discussed.The results show that the natural frequencies obtained by DSM are in good agreement with those in the existing literature and FEM results,which validates the correctness of the method in this paper.It is found that both fluid velocity and volume fraction exponent have significant effects on free vibration and stability of FGM pipes conveying fluid.The natural frequencies decrease with the increase of fluid velocity,when the natural frequency of first mode becomes zero,the pipeline loses its stability,and the corresponding velocity is critical velocity.The natural frequencies and critical velocities increase markedly with the increase of exponent n when it is less than 10.As the exponent n increases further,the effect on the stability becomes less pronounced.2.A hybrid method which combines reverberation-ray matrix method and wave propagation method is developed to determine the natural frequencies and mode shapes of multi-span FGM pipes conveying fluid.The continuity conditions of displacements and equilibrium conditions of forces at each node as well as wave propagation relations in each sub-span are utilized to build the characteristic equation.The results determined by hybrid method are compared with FEM results,experimental results and those in existing literature,and the efficiency of the hybrid method is also discussed.Finally the effects of fluid pressure,volume fraction laws and internal damping on free vibration and stability of multi-span FGM pipes conveying fluid are discussed.The results demonstrate that the present hybrid method has high precision in vibrational analysis of multi-span pipes conveying fluid by using only a small number of elements.It is also found that internal pressure could induce the divergence.The internal damping does not affect the critical velocities,but it can change the dynamic behavior after divergence significantly.3.The power law,sigmoid law and exponential law are adopted to describe the variation of material properties in FGM pipes.The hybrid method is adopted to calculate the natural frequencies.Then,a comparative study is performed to investigate the effects of these volume fraction laws on free vibration and stability of the FGM pipes conveying fluid.The results demonstrate that the volume fraction laws can easily modulate the stiffness of FGM piping systems.If the dominant frequency contents of external loads are known,a proper design for FGM pipes to reduce the vibration is possible.4.Based on modified couple stress theory,the governing equation and boundary conditions of FGM micropipes conveying fluid are derived by Hamilton's principle.Subsequently,the hybrid method is adopted to determine the natural frequencies.Then,the effects of material length scale parameter,volume fraction exponent,location and number of supports on stability of multi-span FGM micropipes conveying fluid are discussed.The results show that the size effect is significant when the diameter of micropipe is comparable to the length scale parameter,and the natural frequencies determined by modified couple stress theory are larger than those obtained by classical beam theory.For higher values of diameter of the micropipe(D_o/l>10),the results predicted by modified couple stress theory converge to the classical ones.In general,the intermediate supports can significantly increase the stiffness and expand the stability region of pipe conveying fluid.5.Based on nonlocal elasticity theory,the governing equation of FGM nanopipes conveying fluid is derived by Newton's method.The hybrid method is applied to calculate the natural frequencies.Then,the effects of nonlocal parameter and volume fraction exponent on stability of multi-span FGM nanopipes conveying fluid are discussed.The results show that natural frequencies and critical velocities of FGM nanopipes conveying fluid decrease with the increase of nonlocal parameter.It is also found that the effect of volume fraction exponent on dynamic behavior is obvious when it is less than 10,and the natural frequencies and critical velocities increase with increasing volume fraction exponent,which is same as the FGM macroscopic pipe conveying fluid.Generally,the dynamic characteristics and calculation method of multi-span FGM pipes conveying fluid are investigated.This study might be helpful for the design and applications of FGM pipes conveying fluid.