# Influence of tip mass on dynamic behavior of cracked cantilever pipe conveying fluid with moving mass

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## Abstract

In this paper, we studied about the effect of the open crack and a tip mass on the dynamic behavior of a cantilever pipe conveying fluid with a moving mass. The equation of motion is derived by using Lagrange’s equation and analyzed by numerical method. The cantilever pipe is modelled by the Euler-Bernoulli beam theory. The crack section is represented by a local flexibility matrix connecting two undamaged pipe segments. The influences of the crack, the moving mass, the tip mass and its moment of inertia, the velocity of fluid, and the coupling of these factors on the vibration mode, the frequency, and the tip-displacement of the cantilever pipe are analytically clarified.

## Key Words

Dynamic Behavior Open Crack Cantilever Pipe Conveying Fluid Moving Mass Tip Mass Euler-Bernoulli Beam Flexibility Matrix## Nomenclature

*ac*Maximum depth of crack

*A*Cross-sectional area

*b*Half-length of crack

*C*Flexibility matrix

*d*Tip-displacement of pipe, dimensionless

*F*_{f}Tangential follower force

*J*_{o}^{*}Moment of inertia of tip mass, dimensionless

*K*_{I}Stress intensity factor (fracture mode I)

*K*_{R}Rotating spring coefficient

*k*Number of segment

*m*Mass per unit length of pipe

*mf*Fluid mass per unit length of pipe

*m*_{m}Moving mass

*m*_{p}Tip mass

*q*Deflection of pipe

*u*Velocity of fluid

*U*Velocity of fluid, dimensionless

*v*Velocity of moving mass

*V*Velocity of moving mass, dimensionless

*μ*_{m}Ratio of tip mass to its moment of inertia

*θ*^{*}Half-angle of crack, dimensionless

- f
Distance measured along pipe, dimensionless

*ξ*_{c}Crack position, dimensionless

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