Abstract: Based on a differential constitutive relationship of linear viscoelastic materials，Hamilton’s principle for viscoelastic pipe conveying fluid is established，and the solid-liquid coupling vibration equation of viscoelastic pipe conveying fluid is derived by using the Hamilton’s principle．The non-dimensional critical flutter flow velocities and complex frequencies of the Maxwell viscoelastic cantilever pipe conveying fluid，which is a gyroscopic non-conservative system，are calculated by the normalized power series method，and the complex frequencies curves of the first mode，second mode and third mode (Argand plot) are plotted．Then，the dynamic behaviors and stability of the pipe are discussed．The above numerical calculation results show that relaxation time of viscoelastic materials with Maxwell model has remarkable effect on dynamic behaviors and stability of the cantilevered pipe conveying fluid．

Key words: Solid-liquid coupling vibration, Stability Relaxation time, Viscoelastic pipe conveying fluid