Resonance risk and global sensitivity analysis of a straight–curved combination pipe based on active learning Kriging model

Abstract

The resonance failure of straight–curved combination pipes conveying fluid which are widely used in engineering is becoming a serious issue. But there are only few studies available on the resonance failure of combination pipes. The resonance failure probability and global sensitivity analysis of straight–curved combination pipes conveying fluid are studied by the active learning Kriging method proposed in this article. Based on the Euler–Bernoulli beam theory, the dynamic stiffness matrices of straight and curved pipes are derived in the local coordinate system, respectively. Then the dynamic stiffness matrix and characteristic equation of a straight–curved combination pipe conveying fluid are assembled under a global coordinate system. The natural frequency is calculated based on the characteristic equation. A resonance failure performance function is established based on the resonance failure mechanism and relative criterions. The active learning Kriging model based on expected risk function is introduced for calculating the resonance failure probability and moment-independent global sensitivity analysis index. The importance rankings of input variables are obtained with different velocities. According to the results, it is shown that the method proposed in this article provides a lot of guidance for resonance reliability analysis and anti-resonance design in combination pipes conveying fluid.

Keywords

Straight–curved combination pipe conveying fluid anti-resonance moment-independent global sensitivity analysis active learning Kriging model

Introduction

Pipes conveying fluid are widely used in aerospace, aircraft, nuclear, ocean, and petrification engineering. Because of the restriction on the construction room, the layout of combination pipes is always complicated such as L-shaped, U-shaped, and Z-shaped combination layouts rather than single straight pipes or curved ones. The L-shaped combination pipe is most especially used in engineering. The uncertainty of pipe and fluid will have an obvious effect on the natural frequency of the combination pipes. When the natural frequency is close to the excitation frequency, this will lead to the resonance failure. The straight–curved combination pipes have more complicated resonance failure relative to a single straight or curved pipe. It is essential and valuable to study the resonance reliability of a straight–curved combination pipe conveying fluid.

There has been a lot of research on the natural frequency of single straight or curved pipes considering fluid–structure interaction (FSI).^1–6 MP Paidoussis¹ and MP Paidoussis and NT Issid² made a detailed study in his monographs on linear and nonlinear dynamics of the fluid-conveying pipes. HB Wen et al.³ proposed a kinetic model of the straight–curved pipe conveying fluid. YJ Hu and WD Zhu⁴ established a dynamic model of an extensible fluid-conveying curved pipe with an arbitrary undeformed configuration. N Egidi et al.⁵ proposed a mathematical model for U-shaped geothermal heat exchangers based on the unsteady Navier–Stokes problems. Many researchers studied the in-plane and out-of-plane vibration modes of the curved pipe based on the total extensible and in-extensible assumption.^6–9 The straight–curved combination pipes are always used in engineering. For the combination pipes, it is hard to establish the analytical solution, so the semi-analytical and numerical solutions such as finite element method,¹⁰ dynamic stiffness matrix (DSM),¹¹ transfer matrix method (TMM),¹² substructure method,¹³ and He’s vibration iteration method are employed.¹⁴ The DSM method is independent of the element number. For a large-size element, the DSM method is also accurate, so the DSM is suitable for the vibration analysis of combination pipes. Koo used the Euler–Bernoulli beam theory to establish the DSM of a straight pipe, deduced the DSM of a curved pipe based on TMM, and then completed the combination of a straight and curved pipe. His method is directive for computing the straight–curved combination pipes. HL Dai et al.¹⁵ used the TMM to establish the DSM of combination pipes by adding the axial combination force into the control equations of a straight pipe. Up to now, the methods for establishing the DSM of straight–curved combination pipes are always based on the straight pipe element, also called straight–straight combination. The straight–curved pipe element combination is not considered. Finite straight pipe elements are used for simulating the curved pipe combined with TMM. This will lead to the increase of dimensionality of DSM and the procedure is complex. Therefore, the accurate curved pipe elements are used for simulating the curved pipe in this study. In the global coordinate system, the whole DSM of straight–curved combination pipes is obtained, and then the natural frequency of the combination pipes can be computed.

The excitation of the pipes conveying fluid, especially aircraft pipes, is intense and wideband frequency, which will lead to resonance failure easily. Although there are a lot of works in dynamic research of pipes conveying fluid, there are only few studies researching the resonance failure of pipes; especially, there is only much less research on combination pipes. Because the combination pipes are widely used in engineering, it is valuable and essential to study the resonance failure of combination pipes conveying fluid. HB Zhai et al.¹⁶ studied the dynamic reliability and local sensitivity analysis (LSA) of a straight pipe conveying fluid based on a refined response surface method, and the fluid profile in the pipe is uniform. AA Alizadeh et al.¹⁷ used the Monte Carlo method to study the self-vibration and stability analysis of a fluid-conveying pipe. YL Li et al.¹⁸ computed the reliability of subsea pipelines under spatially varying ground motions using subset simulation. The sensitivity analysis is used in many researches to identify the key parameters influencing the function of structures or systems.^19–21 The sensitivity analysis has been applied to identify the key input parameters influencing the energy conversion factor of flexoelectric materials.¹⁹ Hamdia et al.²⁰ presented a methodology for stochastic modeling of the fracture in polymer nanocomposites. Vu-Bac et al.²¹ provided a sensitivity analysis toolbox consisting of a set of MATLAB functions that offer utilities for quantifying the influence of uncertainty input parameters on uncertain model outputs. LSA indices are defined as the partial derivate of the failure probability with respect to the distribution parameters of input variables. LSA indices can only reflect the local effect of the distribution parameters on the failure probability but cannot tell us how the uncertainty of the input variables affects the failure probability globally. The global sensitivity analysis (GSA) indices can help measure the effect of uncertainty of each input variable on the variance or the distribution of the model outputs of interest.²² The GSA techniques include variance-based GSA, moment-independent GSA, and the screening method.^23–27 In this article, the moment-independent GSA will be adopted to research the resonance failure. There are two main methods for calculating the moment-independent indices, the one is based on probability density function (PDF) proposed by Borgonovo et al.,^23,24 Wei et al.,²⁵ Cui et al.,²⁶ and Wei et al.,²⁷ and the other one is based on cumulative distribution function (CDF) of the model output such as the PAWN method.²⁸ KM Hamdia et al.²⁸ have presented an improvement to the PAWN method that reduces the computational cost. Comparably, the moment-independent GSA indices, that is, $δ$ indices, are not only global, quantitative, and model free, but also make no assumption on the independence among the input variables. GSA aims at measuring the contribution of input uncertainty to the model’s output uncertainty, and the importance ranking obtained from GSA helps the designer carry out anti-resonance designing and optimization.

This article is organized as follows: section “Motion equations of combination pipes” introduces the DSM method and how to calculate the natural frequency of a combination pipe conveying fluid. Section “Resonance performance function” introduces the anti-resonance performance function of the combination pipe conveying fluid. Section “Moment-independent GSA” introduces the moment-independent GSA of resonance reliability on pipes conveying fluid. Section “The ALK model based on ERF” introduces the active learning Kriging (ALK) method based on expected risk function (ERF). Section “Example” gives an example of straight–curved combination pipes, and the resonance failure probabilities and moment-independent GSA indices with different velocities. Section “Conclusion” gives the conclusions.

Motion equations of combination pipes

The single straight pipe is shown in Figure 1. The Cartesian coordinates are built as shown in Figure 1.

Figure 1.

The straight pipe conveying fluid.

Suppose that the damping of pipe materials and gravity are ignored, the fluid is in-viscous and incompressible. The pipe is based on Euler–Bernoulli beam theory. The FSI vibration equation of the straight pipe conveying fluid is^1,29

E A_{p} \frac{\partial^{2} w_{x}}{\partial x^{2}} - m_{f} U \frac{\partial^{2} w_{x}}{\partial x \partial t} - (m_{p} + m_{f}) \frac{\partial^{2} w_{x}}{\partial t^{2}} = 0

(1)

\begin{matrix} EI \frac{\partial^{4} w_{k}}{\partial x^{4}} + (m_{f} U^{2} + P_{i} A_{i}) \frac{\partial^{2} w_{k}}{\partial x^{2}} + 2 m_{f} U \frac{\partial^{2} w_{k}}{\partial x \partial t} \\ + (m_{p} + m_{f}) \frac{\partial^{2} w_{k}}{\partial t^{2}} = 0, k = y, z \end{matrix}

(2)

GJ \frac{\partial^{2} φ_{x}}{\partial x^{2}} - I_{x} \frac{\partial^{2} φ_{x}}{\partial t^{2}} = 0

(3)

where $w_{x}$ is the axial displacement of the pipe, $w_{k}$ is the radial displacement, $φ_{x}$ is the twist angle along the pipe axis, $t$ denotes the time, $P_{i}$ is the inner pressure, $A_{i}$ is the cross-sectional area of the pipe, $m_{p}$ is the pipe mass per length, $m_{f}$ is the fluid mass per length, $E$ is Young’s modulus, $G$ is the shear modulus, $U$ is the fluid velocity, $I$ is the moment of inertia of the cross section, $J$ is the polar moment of inertia, and $E A_{p}, EI, and GJ$ represent the traction rigidity, bending rigidity, and twist rigidity, respectively.

Equations (1)–(3) have a solution in the frequency domain as follows

{\begin{matrix} w_{x} (x) = A \exp (k_{a} x) \\ w_{k} (x) = B_{k} \exp (k_{b} x) \\ φ_{x} (x) = C \exp (k_{c} x) \end{matrix}

(4)

where $k_{a}, k_{b}, and k_{c}$ , respectively, represent the axial wave number, radial wave number, and twist wave number, which are determined by the frequency dispersion equation, and $A, B_{k}, and C$ are the displacement items. There are six roots in solving the frequency dispersion equation. According to the Euler beam theory, the discrete modes and DSM of the straight pipe can be obtained.

Define the displacement state vector $W_{s}$ , force state vector $F_{s}$ , and coefficient array $C_{s}$ , the superscript $s$ represents the straight pipe. The relationship among them can be expressed as follows

W_{s} = D_{1 s} C_{s}

(5)

F_{s} = D_{2 s} C_{s}

(6)

According to equations (5) and (6), the dynamic relationship is

F_{s} = D_{s} W_{s}

(7)

where $D_{s} = D_{2 s} D_{1 s}^{- 1}$ is the DSM of the straight pipe.

The curved pipe conveying fluid is shown in Figure 2. The curvilinear coordinates are built as shown in the figure.

Figure 2.

The diagram of the curved pipe conveying fluid.

According to the research of Paidoussis,¹ for the curved pipe conveying fluid, ignoring pipe damping, the fluid is in-viscous and incompressible, ignoring the gravity; the three-dimensional (3D) dynamic equation of the curved pipe based on Euler–Bernoulli beam theory is

\begin{matrix} EI (\frac{\partial^{4} w_{y}}{\partial s^{4}} + \frac{1}{R} \frac{\partial^{3} w_{x}}{\partial s^{3}}) + \frac{\partial}{\partial s} [(A_{i} P_{i} - N_{x}) (\frac{\partial w_{y}}{\partial s} + \frac{w_{x}}{R})] \\ + \frac{1}{R} (A_{i} P_{i} - N_{x}) \\ + m_{f} U^{2} (\frac{\partial^{2} w_{y}}{\partial s^{2}} + \frac{1}{R} \frac{\partial w_{x}}{\partial s} + \frac{1}{R}) \\ + 2 m_{f} U (\frac{\partial^{2} w_{y}}{\partial t \partial s} + \frac{1}{R} \frac{\partial w_{x}}{\partial t}) + (m_{p} + m_{f}) \frac{\partial^{2} w_{y}}{\partial t^{2}} = 0 \end{matrix}

(8)

\begin{matrix} EI (\frac{\partial^{4} w_{z}}{\partial s^{4}} - \frac{1}{R} \frac{\partial^{2} φ_{x}}{\partial s^{2}}) - \frac{GJ}{R} (\frac{\partial^{2} φ_{x}}{\partial s^{2}} + \frac{1}{R} \frac{\partial^{2} w_{z}}{\partial s^{2}}) \\ + \frac{\partial}{\partial s} [(A_{i} P_{i} - N_{x}) \frac{\partial w_{z}}{\partial s}] + m_{f} U^{2} \frac{\partial^{2} w_{z}}{\partial s^{2}} \\ + 2 m_{f} U \frac{\partial^{2} w_{z}}{\partial t \partial s} + (m_{p} + m_{f}) \frac{\partial^{2} w_{z}}{\partial t^{2}} = 0 \end{matrix}

(9)

\begin{matrix} \frac{EI}{R} (\frac{\partial^{3} w_{y}}{\partial s^{3}} + \frac{1}{R} \frac{\partial^{2} w_{x}}{\partial s^{2}}) - \frac{\partial}{\partial s} (A_{i} P_{i} - N_{x}) \\ + \frac{1}{R} (A_{i} P_{i} - N_{x}) (\frac{\partial w_{y}}{\partial s} + \frac{w_{x}}{R}) \\ + \frac{m_{f} U^{2}}{R} (\frac{\partial w_{y}}{\partial s} + \frac{w_{x}}{R}) - m_{f} U (\frac{\partial^{2} w_{x}}{\partial t \partial s} - \frac{1}{R} \frac{\partial w_{y}}{\partial t}) \\ - (m_{p} + m_{f}) \frac{\partial^{2} w_{x}}{\partial t^{2}} = 0 \end{matrix}

(10)

- GJ (\frac{\partial^{2} φ_{x}}{\partial s^{2}} + \frac{1}{R} \frac{\partial^{2} w_{z}}{\partial s^{2}}) + \frac{EI}{R} (\frac{φ_{x}}{R} - \frac{\partial^{2} w_{z}}{\partial s^{2}}) + I_{x} \frac{\partial^{2} φ_{x}}{\partial t^{2}} = 0

(11)

where $s$ represents the natural coordinate and $R$ is the radius of curvature. The definitions of the other symbols are described already. The axial force is $N_{x} = E A_{p} ε$ , while under the total extensible assumption it is $ε = \partial w_{x} / \partial s - w_{y} / R$ .

The general solution of equations (8)–(11) in the frequency domain is

{\begin{matrix} w_{x} (s) = A \exp (k_{a} s) \\ w_{y} (s) = A' \exp (k_{a} s) \\ w_{z} (s) = B \exp (k_{b} s) \\ φ_{x} (s) = B' \exp (k_{b} s) \end{matrix}

(12)

where $k_{a}$ represents the in-plane vibration wave number including axial wave number and in-plane bending wave number and $k_{b}$ represents the out-of-plane vibration wave number including the twist wave number and out-of-plane vibration bending wave number. $A$ and $B$ are the displacement items. Six in-plane and out-of-plane wave numbers can be obtained by solving the frequency dispersion equation. Combined with Euler beam theory, the discrete model of the curved pipe is obtained.

Define the displacement state vector $W_{c}$ , force state vector $F_{c}$ , and coefficient array $C_{c}$ , and the superscript $c$ represents the curved pipe. The relationship among them can be expressed as

W_{c} = D_{1 c} C_{c}

(13)

F_{c} = D_{2 c} C_{c}

(14)

Combining equations (13) and (14), the relationship can be rewritten as

F_{c} = D_{c} W_{c}

(15)

where $D_{c} = D_{2 c} D_{1 c}^{- 1}$ is the DSM of the curved pipe.

The steps for calculating the natural frequency of straight–curved combination pipes are shown as follows:

Step 1. Discretize the straight–curved combination pipes into a single straight pipe element and a curved one, based on the Euler–Bernoulli beam theory, and build 3D vibration governing equations of the straight pipe and curved elements.

Step 2. Deduce discrete modes and DSM of the pipe elements based on the frequency dispersion relation.

Step 3. Assemble all straight pipe elements and the curved ones under a global coordinate system and establish the straight–curved combination pipe’s DSM.

Step 4. Considering the boundary conditions, build the characteristic equations of the straight–curved combination pipes. The natural frequency can be obtained by solving the characteristic equation.

Resonance performance function

To avoid resonance failure, the excitation frequency must be far away from the natural frequency of the straight–curved combination pipes conveying fluid. The resonance reliability analysis aims at estimating the resonance failure probability of the pipes conveying fluid.

In engineering, because of the uncertainty among materials, manufacturing, installation, and servicing, the excitation frequency, natural frequency, and vibration response are random variables. Based on the traditional design roles, if $1 - k_{1} < RR / S < 1 + k_{2}$ , there exists risk in resonance design, where $RR$ is the natural frequency, $S$ is the excitation frequency, and $k_{1}$ and $k_{2}$ are the constants based on the materials. For different materials, $k_{1}$ and $k_{2}$ are different. Now, it is assumed that $S_{1} = (1 - k_{1}) S$ and $S_{2} = (1 + k_{2}) S$ , and $S_{1}$ and $S_{2}$ are the minimum and maximum of excitation frequency, respectively.²¹ The performance function of pipe resonance failure is

{\begin{matrix} z_{j} (x) = (R_{j} - S_{1}) \cdot (R_{j} - S_{2}), j = 1, 2, \dots, m \\ z (x) = {z_{1} (x), z_{2} (x), \dots, z_{m} (x)} \end{matrix}

(16)

where $z_{j} (x)$ is the j-order performance function, $x$ is the input variable, $R_{j}$ is the j-order natural frequency, and $m$ is the number of orders and also the failure mode number.

Moment-independent GSA

The moment-independent GSA indices measure the relative importance of one input using the average shift between the unconditional and conditional PDFs of the model output when this input is fixed over its full distribution range.²² The delta indices can be expressed as the accumulation of the difference between unconditional PDF $f_{Y} (y)$ and conditional PDF $f_{Y | X_{i}} (y)$ over the full distribution range. Based on the idea of the moment-independent GSA, the moment-independent global sensitivity index $η_{X_{i}}$ of the basic variables on the failure probability is defined as

η_{X_{i}} = \frac{1}{2} E_{X_{i}} [| P_{f} - P_{f | X_{i}} |] = \frac{1}{2} \int_{- \infty}^{+ \infty} | P_{f} - P_{f | X_{i}} | f_{X_{i}} (x_{i}) d x_{i}

(17)

where $E (\cdot)$ is the operator of expectation, $F$ is the failure domain defined as $F = {X : g (X) < 0}$ , and $f_{X_{i}} (x_{i})$ is the joint PDF of the input variable $x_{i}$ .

The absolute value in the above equation is difficult to compute, and it is equally transformed into square operation.³⁰ The modified global sensitivity indices on the failure probability can be expressed as follows

δ_{i}^{p} = E_{X_{i}} {[P_{f} - P_{f | X_{i}}]}^{2} = \int_{- \infty}^{+ \infty} {[P_{f} - P_{f | X_{i}}]}^{2} f_{X_{i}} (x_{i}) d x_{i}

(18)

Because the resonance failure performance function is implicit, the response is hard to obtain. The Monte Carlo simulation method is used to calculate the failure probability $P_{f}$ and conditional failure probability $P_{f | X_{i}}$ . Substituting $P_{f}$ and $P_{f | X_{i}}$ into the moment-independent index $δ_{i}^{P}$ , the moment-independent index $δ_{X_{i}}^{NP}$ is defined as

δ_{X_{i}}^{NP} = E_{X_{i}} {[P_{f} - P_{f | X_{i}}]}^{2} = E_{X_{i}} {[E (I_{F}) - E (I_{F | X_{i}})]}^{2}

(19)

Compute the error of estimate by $Cov = \sqrt{(1 - {\hat{P}}_{f}^{MCS}) / N_{MCS} {\hat{P}}_{f}^{MCS}}$ , where ${\hat{P}}_{f}^{MCS}$ is the failure probability and $N_{MCS}$ is the total call count number of Monte Carlo simulation.

The GSA focuses on measuring the effects of the uncertainty of each input variable on the variance or distribution parameters of the failure probability and deciding how to reduce the failure probability by decreasing the uncertainty of the input variables. By decreasing the uncertainty of the input variables with high moment-independent indices, we can achieve the most reduction of failure probability. By ignoring the uncertainty of the input variables with small moment-independent indices, we can reduce the dimensionality and simplify the procedure of anti-resonance optimization design.

The ALK model based on ERF

Because the resonance failure performance function is complex and implicit, the computational cost of the traditional Monte Carlo method is high and time consuming. To solve this problem, the approximation method such as support vector machines,³¹ neural networks,³² polynomial chaos expansions,³³ and Kriging model is introduced to approximate the implicit performance function which improves the computational efficiency greatly. The Kriging model method is widely adopted because of its high simulation accuracy. The ALK model means the Kriging model constructed by adding new training points probably located in the region of interest and updating the Kriging model. Through this iterative process, a large portion of the training points will be in the region of interest rather than throughout the design space.^34–36 The ALK model can further improve the accuracy.

The predicted value $μ_{G (x)}$ and the predicted variance $σ_{G}^{2} (x)$ of the Kriging model are as follows

μ_{G (x)} = f^{T} \hat{β} + r^{T} (x) R {(θ)}^{- 1} (Y - F \hat{β})

(20)

σ_{G}^{2} (x) = σ^{2} (1 + u^{T} {(F^{T} R {(θ)}^{- 1} F)}^{- 1} u - r^{T} (x) R {(θ)}^{- 1} r (x))

(21)

where $\hat{β}$ , $σ^{2}$ , and $θ$ are the parameters of the Kriging model. The parameter $θ$ has the largest effect on the accuracy of a Kriging model. It should be determined by solving the following optimization problem

θ^{*} = \arg min_{θ} ({| R |}^{\frac{1}{n}} {\hat{σ}}^{2})

(22)

The ALK model is established in the DACE toolbox of MATLAB.³⁷ The global optimization algorithm adopted in this article is the modified dividing rectangles algorithm for calculating the parameter $θ_{k}$ . The traditional Kriging method establishes the substituting model by random sampling, but the random sampling is aimless, and the accuracy of the Kriging model could not be guaranteed. ALK is updated by adding an optimal training point to the design of experiment (DoE) in iterations by learning function. In this article, the ERF is introduced as a learning function in the ALK model.

To improve the ability of predicting the objective function’s plus or minus sign, the point with the largest risk of being wrongly predicted would be picked out and added into the DoE. The ability of predicting the objective function’s sign would be improved mostly. The ERF is a learning function used to identify this point with a high risk.

The Kriging model provides a predicted value $\hat{G} (x)$ for the true value $G (x)$ at point $x$ , and there is a risk of predicting the wrong plus or minus sign of $G (x)$ . In the case of $\hat{G} (x) < 0$ , for measuring the risk of wrong prediction, the indicator is defined as

R (x) = max [(G (x) - 0), 0]

(23)

The indicator $R (x)$ can show the extent that $G (x)$ is larger than zero when $\hat{G} (x) < 0$ . Because $R (x)$ and $G (x)$ are random variables, $R (x)$ is averaged in the real space, and then the expectation of risk can be expressed as

E (R (x)) = s (x) ϕ (\frac{\hat{G} (x)}{s (x)}) + \hat{G} (x) Φ (\frac{\hat{G} (x)}{s (x)})

(24)

In the case of $\hat{G} (x) \geq 0$ , the indicator of risk is

R (x) = max [(0 - G (x)), 0]

(25)

And the expectation of risk can be expressed as

E (R (x)) = - G (x) Φ (- \frac{\hat{G} (x)}{s (x)}) + s (x) ϕ (\frac{\hat{G} (x)}{s (x)})

(26)

The ERF also called learning function of the predicted value $μ_{G (x)}$ can be expressed uniformly as

\begin{matrix} E (R (x)) = - sign (μ_{G} (x)) μ_{G} (x) Φ (- sign (μ_{G} (x)) \frac{μ_{G} (x)}{σ_{G} (x)}) \\ + σ_{G} (x) ϕ (\frac{μ_{G} (x)}{σ_{G} (x)}) \end{matrix}

(27)

where $ϕ (\cdot)$ and $Φ (\cdot)$ are the PDF and CDF of the standard normal distribution, respectively.

The ERF is a measurement of the potential risk of a point at which the sign of the performance function can be expected to be changed form positive to negative. The point maximizing ERF should be added to the DoE. ERF indicates the probability that the predicted value is different from the real value, that is, the larger the value of ERF is, the more likely it is that the real value changes from positive to negative or from negative to positive. In addition, the points where the ERF values are relatively large are generally the points near the limit state surface and the points with large Kriging variances. Therefore, these ERF maximum points should be added to the DoE to improve the accuracy of the Kriging model. The ALK solution based on ERF for $δ_{X_{i}}^{NP}$ can be simply shown in Figure 3.

Figure 3.

The flow diagram of the ALK method proposed in this article.

Here N = 20 is the number of initial samples for establishing the initial Kriging model. And m = 10⁵ is the number of candidate points for selecting samples by ERF. Some candidate points (m*) will be selected to calculate the resonance failure and moment-independent GSA index through the Monte Carlo method based on the Kriging model. The total call count number of the performance function is N + m*. And X denotes the n-dimensional input vector which has the maximum ERF, which is not a number, but is an input vector, X = (x₁, x₂, …, x_n), where n is the number of total input variables.

Example

Combining two straight pipes and one curved pipe, the L combination pipe is obtained as shown in Figure 4. The information of geometry and materials is listed in Table 1. The boundary condition is clamped–clamped supported at both ends.

Figure 4.

The L combination pipe diagram.

Table 1.

Input variables of straight–curved combination pipes.

Input variables	Distribution	Mean value	Variance
Young’s modulus (Pa)	Gauss	200E9	10E9
Pipe density (kg/m³)	Gauss	7193	359.65
Fluid density (kg/m³)	Gauss	1000	50
Velocity (m/s)	Gauss	10	0.5
Pressure (MPa)	Gauss	14	0.7
Minimum excitation(rad/s)	Gauss	200	10
Maximum excitation(rad/s)	Gauss	400	20
Straight pipe1length (m)	Gauss	0.250	0.0125
Straight pipe2length (m)	Gauss	0.520	0.0260
Radius ofcurvature (m)	Gauss	0.5	0.0250
Angle (rad)	Gauss	$π / 2$	0.0785
Outerdiameter (m)	Gauss	0.010	5e–4
Thickness (m)	Gauss	0.0012	6e–5

Natural frequency

The natural frequency of L combination pipe is calculated, and the results are presented in Figure 5. According to Figure 5, it is shown that, in the natural frequency of L combination pipe, there exist two closed frequencies, which is because the pipe model in this article is 3D Euler–Bernoulli beam, not only the in-plane vibration but also the out-of-plane vibration.

Figure 5.

Natural frequency of straight–curved combination pipes.

Figure 6 shows the relationship of the natural frequency with the velocity. From Figure 6, it is shown that, with the increase of velocity, the natural frequency of the combination pipe decreases until the natural frequency falls to zero suddenly. This is because the centrifugal and Coriolis force items in the vibration equations increase along with the increase of velocity. When the velocity is $v = 53.67 m / s$ , the first natural frequency is close to zero and the pipe buckles, so the critical velocity is $v_{cr} = 53.67 m / s$ .

Figure 6.

First four natural frequencies with the increasing fluid velocity.

Resonance reliability

The resonance failure probability is calculated by the ALK method and compared with the value calculated by the Monte Carlo method as shown in Table 2. $P_{f}$ is the resonance failure probability, Total is the total call count number of the performance function, Cov is the coefficient of variance of $P_{f}$ , and Error is the difference of $P_{f}$ between those two methods. It is shown that the error between the ALK method proposed in this article and the Monte Carlo method is less than 5%. This indicates that the method proposed in this article is accurate for computing the resonance failure probability of assembled pipes conveying fluid. And the method proposed in this article is efficient and is feasible for dealing with the engineering problems.

Table 2.

The resonance failure probability with different methods of velocity v = 10 m/s.

	First mode		Second mode
	Monte Carlo	ALK	Monte Carlo	ALK
$P_{f}$	0.00189	0.00183	0.02136	0.02054
Total	100,000	20 + 111	100,000	20 + 122
Cov (%)	7.27	7.39	2.14	2.18
Error (%)	–	3.17	–	3.83

ALK: active learning Kriging.

Along with the increasing velocity, the first- and second-order resonance failure probabilities are listed in Table 3. It is shown that, along with the increasing velocity, the first-order resonance failure probability is getting smaller and the second-order resonance failure probability is getting larger. It is because the natural frequency is getting smaller along with the increasing velocity, the resonance failure area between the minimum excitation frequency and natural frequency is smaller, and the resonance failure area between the maximum excitation frequency and natural frequency is getting larger. The results calculated by the method proposed in this article coincide with the analytical results.

Table 3.

The resonance failure probability with different velocities.

Velocity (m/s)	First-order resonance failure probability	Second-order resonance failure probability
0	0.00324	0.00960
5	0.00202	0.01940
10	0.00183	0.02054
20	0.00049	0.02377

Moment-independent GSA

When the velocity is $v = 10 m / s$ , the moment-independent GSA indices of input variables are listed as follows. The total call count number of the performance function is 20 + 111.

Table 4 presents the importance measurement. It is shown that when the velocity $v$ is $10 m / s$ and the other parameters are the same as in Table 1, the importance measure ranking is $pd > hd > L 2 > R > ang > fd$ . The pipe density $pd$ has the largest effect on the failure probability, the next is thickness of the cross section $hd$ , the third one is the second pipe length $L 2$ , and the forth one is the radius of curvature $R$ . The other input variables, such as $s 2, E, wd, P, and v$ , have a little effect on the failure probability. Especially, the velocity $v$ has the smallest effect on the failure probability.

Table 4.

The moment-independent GSA indices of input variables.

Inputvariables	$E$	$pd$	$fd$	$v$	$P$	$s 1$	$s 2$	$L 1$	$L 2$	$R$	$ang$	$wd$	$hd$
GSAindices	0.03012	0.86218	0.30845	0.00188	0.00911	0.17351	0.05489	0.16635	0.48196	0.45816	0.31817	0.016944	0.50636

GSA: global sensitivity analysis.

Considering the input variables $pd, hd, and L 2$ , when their different variances are used, the failure probability is calculated as shown in Figure 7. It is easy to find that, along with the increase of variables’ variance, the input variable $pd$ has the largest effect on the failure probability, followed by $hd$ and the last is $L 2$ .

Figure 7.

The failure probability along with the increasing variance of $pd, hd, and L 2$ .

Conclusion

Based on the Euler–Bernoulli beam theory, the accurate curved pipe model is used for establishing the DSM. Then all straight pipe elements and the curved ones are assembled under a global coordinate system. The natural frequency can be calculated based on the characteristic equation. After establishing the performance function of resonance failure, the ALK model based on ERF is used to calculate the resonance failure probability and moment-independent GSA index. The method proposed in this article is guided for resonance reliability analysis and GSA.

First, the natural frequencies with different velocities are calculated. It is found that, along with the increasing velocity, the natural frequency is getting smaller. When the velocity reaches to the critical velocity, the first-order natural frequency of the combination pipe becomes zero and buckling instability occurred. Second, the ALK model based on ERF is used for calculating the resonance failure probability and moment-independent indices. Along with the increasing velocity, the first-order resonance failure probability is getting smaller, while the second-order resonance failure probability is getting larger. The result coincides with the theoretical analysis. Finally, the importance ranking is obtained from the moment-independent indices. The variance of input variables with high moment-independent index were changed from 0.03 to 0.06. The resonance failure probability is getting larger along with the increasing variance of the input variables with high moment-independent indices. The larger the moment-independent indices, the greater the resonance failure probability increase caused by changing the variance. If the input variable with larger indices is controlled strictly, the resonance failure probability can be reduced much more than the other variables. From the results, the method proposed in this article is meaningful for resonance failure analysis of straight–curved combination pipes and is referential for other combination pipes.

For the future research work, the resonance failure performance function will be improved, while the traditional performance function does not consider the multi-order resonance failure. The second point of future study is to improve the calculation efficiency. Because the performance function is implicit, the time cost for calculation is huge, especially for sensitivity analysis. Therefore, a more efficient solving method is needed.

Footnotes

Handling Editor: Mohammad Arefi

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Yuzhen Zhao

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