Hopf Bifurcation of a Standing Cantilever Pipe Conveying Fluid with Time Delay

Table 1

Summary of symbols.

Notation	Explanation
	Flexural rigidity of the pipe
	Coefficient of Kelvin–Voigt viscoelastic damping of the pipe material
	Pipe length
	Mass per unit length
	Mass of the conveying fluid per unit length
	Flow velocity
	Stiffness of the spring of the elastic support
	Stiffness of cubic spring which represents the effect of the motion constraints
	Location of the constraints
	Distance between the fixed end of the piezoelectric plate and the upper end of the pipeline
	Distance between the fixed end of the piezoelectric plate and the lower end of the pipeline
	Position coordinates of the sensor
	Envelope half angle of a piezoelectric plate
	Acceleration due to gravity
	Dirac delta function
	Lateral deflection of the pipe