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Nonlinear vibration effects on the fatigue life of fluid-conveying pipes composed of axially functionally graded materials

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Abstract

Fatigue is inevitable in pipes conveying fluid due to unwanted vibration. Internal resonance occurs in such pipes due to pre-pressure. For the first time, the effects of vibration on the fatigue of fluid-conveying pipes are investigated in this paper. The influences of the internal resonance and the axially functionally graded materials on the fatigue of the pipes are analyzed, aiming at improving mechanical properties and increasing fatigue life. The Galerkin method and the direct multi-scale method are used to construct the solvability condition for the primary resonance and 1:3 internal resonance. Approximate analytical solutions are derived for presenting the nonlinear dynamics of the pipes. The tensile, bending, and resultant stress distribution of the axially functionally graded pipe in internal resonance is determined. The results of the fatigue analysis demonstrate that internal resonance can shorten the fatigue life of axially functionally graded pipes. Reducing the distribution coefficient of functionally graded pipe is beneficial for reducing the resonance response and maximum stress of the pipe conveying fluid. The numerical integration results support the analytical results.

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Abbreviations

L :

Length of the pipe

A :

Cross-sectional area of the pipe

P :

Axial pre-pressure

E(x):

Elastic modulus of the pipe

\(\rho (x)\) :

Density of the pipe

\(\alpha _{E}\) :

Elastic modulus ratio between the two ends

\(\alpha _{\rho }\) :

Density ratio between the two ends

M :

Fluid mass per unit length

m :

Pipe mass per unit length

U :

Steady velocity of the fluid

w(x, t):

Transverse displacement of the pipe

x :

x-axial coordinate

y :

y-axial coordinate

\(\xi \) :

Dimensionless x-axial coordinate

\(\eta \) :

Dimensionless transverse displacement of the pipe

u :

Dimensionless steady velocity of the fluid

\(\beta \) :

Ratio between pipe mass and fluid mass per length at the initial end

\(\alpha (\xi )\) :

Dimensionless mass per unit length

p :

Dimensionless axial pre-pressure

\(u_{0}\) :

Dimensionless mean velocity of the fluid

\(\mu \) :

The perturbation amplitude of the fluid velocity

\(\varOmega _{1}\) :

The perturbation frequency of the fluid velocity

\(\varepsilon \) :

Book-keeping parameter

\(\varTheta _{n}(\xi )\) :

nth mode function

\(\omega _{\mathrm{n}}\) :

nth natural frequency

\(p_{kn}\) :

Harmonic coefficient

\(q_{kn}\) :

Harmonic coefficient

\(\theta _{1}\) :

Detuning factors

\(\theta _{2}\) :

Detuning factors

\(\Gamma \) :

Coefficients of nonlinear algebraic equations

\(a_{n}\) :

Amplitude of nth mode

\(A_{n}(T_{1},T_{2})\) :

The amplitude function of the nth mode

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11772181, 11872037 and 11572182) and the Innovation Program of Shanghai Municipal Education Commission (No. 2019-01-07-00-09-E00018).

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 149 Yan Chang Road, Shanghai, 200072, China

    Ze-Qi Lu, Kai-Kai Zhang, Hu Ding & Li-Qun Chen

  2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, China

    Ze-Qi Lu, Hu Ding & Li-Qun Chen

  3. School of Mechanics and Engineering Science, Shanghai University, Shanghai, China

    Ze-Qi Lu, Hu Ding & Li-Qun Chen

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  1. Ze-Qi Lu
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  2. Kai-Kai Zhang
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Correspondence to Hu Ding.

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Lu, ZQ., Zhang, KK., Ding, H. et al. Nonlinear vibration effects on the fatigue life of fluid-conveying pipes composed of axially functionally graded materials. Nonlinear Dyn 100, 1091–1104 (2020). https://doi.org/10.1007/s11071-020-05577-8

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