# Developing a Simplified Method to Investigate the Dynamic Behavior of Fluid Conveying Pipes under Mean Internal Pressure

Authors: Mahdi Bayrami Atashgah; Mehdi Iranmanesh; Alireza Mojtahedi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Natural frequencies and critical velocities are the basic dynamic characteristics of fluid conveying pipes. Investigation of vibrations of fluid conveying pipes to control their dynamic behavior is one of the main topics of these structures. This paper presents a simplified approach to determine the natural frequencies and critical velocities of fluid conveying pipes with different typical boundary conditions under the mean internal pressure due to the fluid flow. The approach is suggested by reducing the order of mass, damping, and stiffness based on the concepts of the Galerkin method. The results of the method are validated by experimental results and compared with various numerical methods reported in the literature. The effects of mean internal pressure on the natural frequencies of fluid conveying pipes are assessed, and critical pressure values are calculated for the different mode shapes of the pipes. The method presents significantly better results in all the velocities in a range less than the critical flow velocity. Also, the natural frequencies of fluid conveying pipes in the presence and absence of mean internal pressure are calculated by the explicit formulas obtained by the simplified approach. The method has a lower computational cost than other more complex techniques, and its relative errors are less than 5%.


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## Body

## 1. Introduction

Fluid conveying pipes are used most widely in different fields of engineering, such as marine engineering, oil exploration, refining, and chemical transport. Flow-induced vibrations in these pipes require the study of the interaction between the fluid and the pipe [1–3]. The precise prediction of the natural frequencies and critical flow velocities are two important significant factors in studying the vibrations of the fluid conveying pipes. These factors are strongly dependent on the velocity of the internal fluid. In this regard, predicting the critical fluid velocity to determine the onset of instability is a challenge in the fluid conveying pipes. Critical velocity relates to the value at which the imaginary part of the natural frequency becomes zero.The vibration of fluid conveying pipes is a profound challenge in many areas. The hot topic started when Paidoussis [2] realized the vibration in pipes caused by the fluid flow and introduced two types of instabilities for them. The first type is the divergence instability that occurs at lower velocity, and the second type is the instability of the flutter that occurs at higher velocity. Benjamin [4] discussed the flutter instability of a cantilevered fluid conveying pipe. Gregory and Paidoussis [5] concluded that the flutter instability occurs earlier than divergence instability for a cantilevered fluid conveying pipe. Paidoussis and Li [6] described in detail the instability of cylindrical pipes due to the fluid by the linear vibration equations. Buckling instability and flutter instability were two issues addressed by them. Paidoussis and Issid [7] proposed a variety of methods to study the dynamic behavior of fluid conveying pipes. So far, different methods have been presented to solve the vibration problem of fluid conveying pipes both in linear and nonlinear dynamics, such as the Galerkin method [7–12] and finite element method (FEM) [13–16]. Housner [17] analytically determined the flow-induced vibration of pipes with pinned-pinned boundary conditions. However, the analytical solutions are problematic for other kinds of boundary conditions. Ibrahim [18, 19] and Li et al. [1] have systematically summarized research on fluid conveying pipes. Also, Dai et al. [20] studied the vibrations of a flexible fluid conveying pipe for transmitting oscillating currents, and the main parameters were analyzed and discussed.In this regard, some complicated methods have been developed by other researchers. A homotopy perturbation method was proposed to investigate the dynamic behavior of fluid conveying pipes. Xu et al. [21] evaluated a pinned-pinned fluid conveying pipe and validated the results using the finite element method and experimental data. Yun-dong and Yi-ren [22] proposed He’s variational iteration method for calculating the natural frequencies and critical velocities of fluid conveying pipes under different boundary conditions. The dynamic response of the Timoshenko beam under random stimulation using the superposition method was analyzed by Zhai et al. [23]. Kuiper and Metrikine [24] studied the instability of a free-hanging riser using power series expansion and the D-decomposition method. Lin and Qiao [25] used the differential quadrature method (DQM) under the harmonic excitation to obtain the vibrational response of fluid conveying pipes. Ni et al. [26] used the differential transformation method (DTM) to analyze the vibration of the fluid conveying pipes with several scenarios. A semianalytical method was proposed by Liang et al. [27] to evaluate the dynamic behavior of a fluid conveying pipe under transverse external fluid flow, employing the differential quadrature method and the Laplace transform. The method was validated by an exact solution. Also, a mathematical model for the free vibration of the fluid-conveying cantilevered pipe-in-pipe system considering the thermal effect and two-phase flow was proposed by Guo et al. [28].Some researchers were focused on sensitivity analysis of fluid conveying pipe. Gregory and Paidoussis [5] examined the effects of the internal damping on stability of fluid conveying pipes. Chen [29] discussed and evaluated the forced vibrations of a fluid conveying pipe. Linear governing equations of the fluid conveying pipe and the critical velocities were discussed by Paidoussis and Issid [7] for different boundary conditions. Long Jr [30] observed the vibration characteristics of fluid conveying pipes experimentally and measured natural frequencies. He realized that as the internal fluid velocity of the fluid conveying pipes increased, the natural frequencies decreased. A mathematical model for the lateral motion of a marine riser was developed to examine the effect of the internal flow and bending rigidity of the pipe on the dynamic behavior of the riser by Wu and Lou [31]. Meng and Chen [32] investigated nonlinear-free vibrations and vortex-induced vibrations of a fluid-conveying steel catenary riser. The dynamic behavior of catenary pipelines subjected to internal slug-flow was investigated by Chatjigeorgiou [33]. He et al. [34] reported a theoretical investigation of an elastic and slender fluid-conveying pipe with a top-end excitation subjected to uniform cross flows. Oke and Khulief [35] discussed how the internal surface damage is reflected in the vibration behavior of a composite pipe conveying fluid. Gu et al. [36] proposed a stochastic dynamic model for the dynamic characteristics analysis of pipe-conveying fluid. Askarian et al. [37] investigated the gravity effects in vertical and horizontal fluid conveying pipes. An analytical solution for nonlinear vibration and postbuckling of functionally graded pipes conveying fluid considering the rotary inertia and shear deformation effects was presented by Khodabakhsh et al. [38]. Shahali et al. [39] investigated the nonlinear dynamic response of a viscoelastic pipe conveying fluid subjected to a uniform external cross flow based on the Euler–Bernoulli theory. The nonplanar vibrations and multimodal responses of pinned-pinned risers in shear cross flow were numerically studied by Jiang et al. [40]. Ma et al. [41] presented an experimental study of large-amplitude intermittent vibrations of an inclined bendable riser transporting an air-water flow with an unsteady slug-flow pattern. Liang et al. [42, 43] developed the dynamical modeling for the spinning pipe conveying fluid and analyzed its vibrations.The effects of mean internal pressure on the natural frequencies of fluid conveying pipes with different typical boundary conditions have not been studied in detail. The main aim of this study is to introduce a simplified approach to determine the natural frequencies and critical velocities of fluid conveying pipes. For this purpose, the elements of the diagonal matrices of mass, damping, and stiffness are considered to calculate the natural frequencies of pipe before occurring the instability (flow velocity in a range higher than the critical velocity), instead of all ones that gained by the Galerkin discretization. At the first step, a linear equation of motion for an Euler–Bernoulli beam is considered to simulate the vibration of fluid conveying pipes with mean internal pressure. For the second step, the Galerkin method is used to solve the partial differential equation for the condition ofN = 4 (N is the number of mode shapes). At the next step, the element in the row n and column n of the mass, damping, and stiffness matrices is used to calculate the nth natural frequency by which the matrix form is converted to an algebraic form. Then, the results are validated by experimental data and four numerical methods besides two analytical solutions. Finally, the effects of mean internal pressure variations on the natural frequencies are discussed, and critical values related to internal pressures are investigated. In each vibration mode of the pipe and fluid system, there is a critical value of internal pressure for which the natural frequency of the system becomes zero. This value is named Mean Critical Pressure. This study is the first stage of our work that aims is to justify the applicability of an efficient and simplified method to determine and evaluate the natural frequency of fluid conveying pipes. The advantage of the presented method is the simplicity of equations, and the disadvantage of the method is that the real part of the natural frequency is not calculated.


## 2. Methodology

In this section, the dynamic equation of the fluid conveying pipe is considered, and the Galerkin discretization method is formulated using first four modes to calculate the natural frequencies. The details of the proposed simplified method are also explained.


### 2.1. Dynamic Equation of the Fluid Conveying Pipes

For a pipe, considered as an Euler–Bernoulli beam, the linear equation of motion for the vibration of the fluid conveying pipe with an external excitation is given as follows [44]:(1)EI∂4wx,t∂x4+MfV2+p¯A1−2μ∂2wx,t∂x2+2MfV∂2wx,t∂x∂t+Mf+Mp∂2wx,t∂t2=f,where EI is the flexural rigidity, A stands for the flow area, Mf and Mp are the mass-per-unit-length of fluid and pipe, V is the fluid flow velocity, p¯ is the mean internal pressure, μ is the Poisson ratio which only considered in the clamped-clamped pipe, wx,t is the transverse deflection of the pipe, f is external excitation load, x is the horizontal coordinate along the centerline of the pipe, and t stands for the time. In the case of f = 0, the fluid conveying pipe vibrates due to the motion of the internal fluid, and the dynamic equation is(2)EI∂4w∂x4+MfV2+p¯A1−2μ∂2w∂x2+2MfV∂2w∂x∂t+Mf+Mp∂2w∂t2=0.The related nondimensionless parameters are defined as defined as follows:(3)η=wL,τ=EIMf+MptL2,u=MfEILV,ξ=xL,P=p¯AEIL2,β=MfMf+Mp,ω=Mf+MpEIω0L2,where ω is the dimensionless natural frequency and ω0 is the natural frequency rad/s. Substituting equation (3) into (2), the dimensionless form of equation (2) can be expressed as follows:(4)∂4η∂ξ4+u2+P1−2μ∂2η∂ξ2+2βu∂2η∂ξ∂τ+∂2η∂τ2=0.The framework of this study is presented in Figure1. The framework is performed in two main directions. In the first direction, by using the Galerkin method and considering 4 to 8 mode shapes of the pipes, the dynamic matrices of the system are extracted, and the natural frequencies in different boundary conditions are calculated. In the second direction, instead of using four mode shapes of the pipes, only one mode shape of the pipes is used, and the relationships related to natural frequencies and critical velocities are explicitly obtained. The main idea of this research is to use one mode shape of the pipe and convert the matrix form of dynamic equations to algebraic form. Consequently, only the diagonal components of dynamic matrices are used to calculate the natural frequencies.Figure 1
The framework of this study.


### 2.2. Galerkin Discretization

The main purpose of this study is to provide explicit formulas to determine the natural frequencies of the fluid conveying pipe under different boundary conditions presented in Figure2. The Galerkin method is used to obtain the explicit formulas and reduce complexities compared to the other methods such as the finite element method and finite difference method.Figure 2
Fluid conveying pipes with different boundary conditions.Equation (4) can be discretized using the Galerkin method to solve the related partial differential equation. The dimensionless lateral deflection is represented as follows:(5)ηξ,τ=∑i=1Nφiξqiτ+ε,where φiξ is the shape function that satisfied all considered boundary conditions for the pipe (see Table 1), and qiτ is the generalized coordinate of the discretized pipe structure, and ε is the order of the Galerkin truncation error (N is the number of mode shapes).Table 1
Mode shape functions for a Bernoulli beam in the different boundary conditions [45].
Boundary conditionsExpressionPinned-pinnedϕnξ=2sinλnξλn=nπClamped-pinnedϕnξ=e−λnξ−cosλnξ+1+e−λn/coshλnsinλnξ−e−λn/coshλnsinhλnξλ1=3.9266,λ2=7.0685,λ3=10.2102,λ4=13.3518Cantileveredϕnξ=e−λnξ−cosλnξ+1+e−λn−cosλn+sinλn/sinhλn−sinλnsinλnξ−−e−λn−cosλn−sinλn/coshλn+cosλnsinhλnξλ1=1.8751,λ2=4.6941,λ3=7.8547,λ4=10.9955Clamped-clampedϕnξ=e−λnξ−cosλnξ+1+e−λn−cosλn+sinλn/sinhλn−sinλnsinλnξ−e−λn−cosλn+sinλn/sinhλn−sinλnsinhλnξλ1=4.7300,λ2=7.8532,λ3=10.9995,λ4=14.1372The convergence can be obtained when the condition ofN ≥ 4 is satisfied [9]. In this study, it is considered N = 4. Therefore, we have(6)ηξ,τ≅∑i=14φiξqiτ,≅φ1ξq1τ+φ2ξq2τ+φ3ξq3τ+φ4ξq4τ.By considering theϕ=φ1φ2φ3φ4 and Q=q1q2q3q4 and introducing the parameters presented in (7), (8), and (5) is changed into the matrix form as shown in (9),(7)ηξ,τ=ϕTQ=QTϕ∂4η∂ξ4=ϕ4TQ∂2η∂ξ2=ϕ″TQ∂2η∂ξ∂τ=ϕ′TQ˙∂2η∂τ2=ϕTQ¨,(8)2βu=Λ,u2+P1−2μ=Π,(9)ϕTQ¨+Λ,ϕ′TQ˙+ϕ4T+Πϕ″TQ=0.By multiplyingϕ=φ1φ2φ3φ4 in both sides of (9) and integration respect to ξ on the interval [0, 1], and considering the orthogonality of the mode shape functions, the discrete form of dynamic equation is obtained as follows:(10)∫01ϕϕTdξQ¨+∫01Λ,ϕϕ′TdξQ˙+∫01ϕϕ4T+Πϕϕ″TdξQ=0.Equation (10) can be written as(11)MQ¨+CQ˙+KQ=0,where M, C, and K are the matrices of mass, damping, and stiffness, respectively. Q¨,Q˙, and Q are the generalized acceleration, velocity, and displacement. The elements of above matrices are presented in Appendix, equations (A1) to (A3). The dimensionless natural frequency of the pipe (ω) depends on Λ and Π parameters, as implied in equations (A2) and (A3).Then, (11) is converted to the problem of solving the eigenvalues of the matrix A, as follows:(12)A=04×4I4×4−M−1K−M−1C8×8=04×4I4×4−K−C8×8.The complex eigenvalues ofA can be calculated by using software, such as the MATLAB, where the coefficient of imaginary part represents the dimensionless natural frequency.


### 2.3. Simplified Method Based on Galerkin Discretization (SMGD)

In this method, the nth mode shape of the Galerkin discretization is used to determine the nth natural frequency of the fluid conveying pipe. The components of the diagonal matrices M, C, and K, which are calculated based on the Galerkin method, are used to calculate the natural frequencies, orders of the matrices of mass, damping, and stiffness being equal to 1. Hence, the matrix form is converted to an algebraic form. The characteristic equation for the nth dimensionless natural frequency can be expressed as follows:(13)Kn−Mnωn2+iCnωn=0,where Mn, Cn, and Kn are the nth components of the mass, damping, and stiffness matrices, respectively, as follows:(14)Mn=mcofn,(15)Cn=Λ,ccofn,(16)Kn=kcofn1+Πkcofn2,where mcofn, ccofn, kcofn1,kcofn2 are the components of the diagonal matrices of Mcof,Ccof, Kcof1 and Kcof2, respectively.By substituting (8) into equation (13) and using the components of the diagonal matrices of Mcof,Ccof, Kcof1, and Kcof2, the dimensionless form of nth natural frequency, as a function of u,P, and β, and the dimensionless critical velocities for the different types of the boundary conditions (pinned-pinned, clamped-pinned, cantilevered and clamped-clamped), can be obtained as shown in Table 2 and 3. We explicitly derive the dynamic matrices of the system, including the mass matrix, the damping matrix, and the stiffness matrix. Considering only one mode shape to determine the natural frequency is the main idea of the research. Next, by considering only one mode shape, the characteristic equation is obtained only to determine the natural frequency. The relevant relationships were listed in Table 2. According to the relationships of the mode shapes described in Table 1, the integral multiplication of the mode shape with its derivative, which is described in equation (12), becomes zero for pinned-pinned, clamped-pinned and clamped-clamped conditions. Therefore, the damping term does not appear in the dynamic equations. Furthermore, the relationships presented in Tables 2 and 3 show that increasing the mean internal pressure reduces the natural frequencies and critical flow velocities.Table 2
Formulas of SMGD for the first four dimensionless natural frequencies of the fluid conveying pipes with internal pressure for the different boundary conditions.
Boundary conditionsExpressionPinned-pinnedω1=97.4091−9.8696u2+Pω2=1558.5−39.4784u2+Pω3=7890.1−88.8264u2+Pω4=24937−157.9137u2+PClamped-pinnedω1=237.722−11.5125u2+Pω2=2496.5−42.8964u2+Pω3=10868−94.0377u2+Pω4=31780−164.9183u2+PCantileveredω1=12.3622+0.8583u2+P−4βuω2=485.5182−13.2943u2+P−4βuω3=3806.6−45.9043u2+P−4βuω4=14617−98.9169u2+P−4βuClamped-clampedω1=500.563−12.3026u2+P1−2μω2=3803.5−46.0501u2+P1−2μω3=14618−98.9047u2+P1−2μω4=39944−171.5861u2+P1−2μTable 3
Formulas of SMGD for the first four dimensionless critical velocities of the fluid conveying pipes with internal pressure for the different boundary conditions.
Boundary conditionsExpressionPinned-pinneduc1=9.8696−Puc2=39.4784−Puc3=88.8264−Puc4=157.9137−PClamped-pinneduc1=20.649−Puc2=58.1984−Puc3=115.571−Puc4=192.701−PCantilevereduc1=4β2−40.858312.3622+0.8583P/20.8583uc2=4β2−4−13.2943485.5182−13.2943P/2−13.2943uc3=4β2−4−45.90433806.6−45.9043P/2−45.9043uc4=4β2−4−98.916914617−98.9169P/2−98.9169Clamped-clampeduc1=20.649−P1−2μuc2=58.1984−P1−2μuc3=115.571−P1−2μuc4=192.701−P1−2μThe simplified method, as shown in Figure1, is fundamentally different from the Galerkin method. In the Galerkin method, the dynamic components of the system are matrices. But in the simplified method, the dynamic components of the system are algebraic.


## 2.1. Dynamic Equation of the Fluid Conveying Pipes

For a pipe, considered as an Euler–Bernoulli beam, the linear equation of motion for the vibration of the fluid conveying pipe with an external excitation is given as follows [44]:(1)EI∂4wx,t∂x4+MfV2+p¯A1−2μ∂2wx,t∂x2+2MfV∂2wx,t∂x∂t+Mf+Mp∂2wx,t∂t2=f,where EI is the flexural rigidity, A stands for the flow area, Mf and Mp are the mass-per-unit-length of fluid and pipe, V is the fluid flow velocity, p¯ is the mean internal pressure, μ is the Poisson ratio which only considered in the clamped-clamped pipe, wx,t is the transverse deflection of the pipe, f is external excitation load, x is the horizontal coordinate along the centerline of the pipe, and t stands for the time. In the case of f = 0, the fluid conveying pipe vibrates due to the motion of the internal fluid, and the dynamic equation is(2)EI∂4w∂x4+MfV2+p¯A1−2μ∂2w∂x2+2MfV∂2w∂x∂t+Mf+Mp∂2w∂t2=0.The related nondimensionless parameters are defined as defined as follows:(3)η=wL,τ=EIMf+MptL2,u=MfEILV,ξ=xL,P=p¯AEIL2,β=MfMf+Mp,ω=Mf+MpEIω0L2,where ω is the dimensionless natural frequency and ω0 is the natural frequency rad/s. Substituting equation (3) into (2), the dimensionless form of equation (2) can be expressed as follows:(4)∂4η∂ξ4+u2+P1−2μ∂2η∂ξ2+2βu∂2η∂ξ∂τ+∂2η∂τ2=0.The framework of this study is presented in Figure1. The framework is performed in two main directions. In the first direction, by using the Galerkin method and considering 4 to 8 mode shapes of the pipes, the dynamic matrices of the system are extracted, and the natural frequencies in different boundary conditions are calculated. In the second direction, instead of using four mode shapes of the pipes, only one mode shape of the pipes is used, and the relationships related to natural frequencies and critical velocities are explicitly obtained. The main idea of this research is to use one mode shape of the pipe and convert the matrix form of dynamic equations to algebraic form. Consequently, only the diagonal components of dynamic matrices are used to calculate the natural frequencies.Figure 1
The framework of this study.


## 2.2. Galerkin Discretization

The main purpose of this study is to provide explicit formulas to determine the natural frequencies of the fluid conveying pipe under different boundary conditions presented in Figure2. The Galerkin method is used to obtain the explicit formulas and reduce complexities compared to the other methods such as the finite element method and finite difference method.Figure 2
Fluid conveying pipes with different boundary conditions.Equation (4) can be discretized using the Galerkin method to solve the related partial differential equation. The dimensionless lateral deflection is represented as follows:(5)ηξ,τ=∑i=1Nφiξqiτ+ε,where φiξ is the shape function that satisfied all considered boundary conditions for the pipe (see Table 1), and qiτ is the generalized coordinate of the discretized pipe structure, and ε is the order of the Galerkin truncation error (N is the number of mode shapes).Table 1
Mode shape functions for a Bernoulli beam in the different boundary conditions [45].
Boundary conditionsExpressionPinned-pinnedϕnξ=2sinλnξλn=nπClamped-pinnedϕnξ=e−λnξ−cosλnξ+1+e−λn/coshλnsinλnξ−e−λn/coshλnsinhλnξλ1=3.9266,λ2=7.0685,λ3=10.2102,λ4=13.3518Cantileveredϕnξ=e−λnξ−cosλnξ+1+e−λn−cosλn+sinλn/sinhλn−sinλnsinλnξ−−e−λn−cosλn−sinλn/coshλn+cosλnsinhλnξλ1=1.8751,λ2=4.6941,λ3=7.8547,λ4=10.9955Clamped-clampedϕnξ=e−λnξ−cosλnξ+1+e−λn−cosλn+sinλn/sinhλn−sinλnsinλnξ−e−λn−cosλn+sinλn/sinhλn−sinλnsinhλnξλ1=4.7300,λ2=7.8532,λ3=10.9995,λ4=14.1372The convergence can be obtained when the condition ofN ≥ 4 is satisfied [9]. In this study, it is considered N = 4. Therefore, we have(6)ηξ,τ≅∑i=14φiξqiτ,≅φ1ξq1τ+φ2ξq2τ+φ3ξq3τ+φ4ξq4τ.By considering theϕ=φ1φ2φ3φ4 and Q=q1q2q3q4 and introducing the parameters presented in (7), (8), and (5) is changed into the matrix form as shown in (9),(7)ηξ,τ=ϕTQ=QTϕ∂4η∂ξ4=ϕ4TQ∂2η∂ξ2=ϕ″TQ∂2η∂ξ∂τ=ϕ′TQ˙∂2η∂τ2=ϕTQ¨,(8)2βu=Λ,u2+P1−2μ=Π,(9)ϕTQ¨+Λ,ϕ′TQ˙+ϕ4T+Πϕ″TQ=0.By multiplyingϕ=φ1φ2φ3φ4 in both sides of (9) and integration respect to ξ on the interval [0, 1], and considering the orthogonality of the mode shape functions, the discrete form of dynamic equation is obtained as follows:(10)∫01ϕϕTdξQ¨+∫01Λ,ϕϕ′TdξQ˙+∫01ϕϕ4T+Πϕϕ″TdξQ=0.Equation (10) can be written as(11)MQ¨+CQ˙+KQ=0,where M, C, and K are the matrices of mass, damping, and stiffness, respectively. Q¨,Q˙, and Q are the generalized acceleration, velocity, and displacement. The elements of above matrices are presented in Appendix, equations (A1) to (A3). The dimensionless natural frequency of the pipe (ω) depends on Λ and Π parameters, as implied in equations (A2) and (A3).Then, (11) is converted to the problem of solving the eigenvalues of the matrix A, as follows:(12)A=04×4I4×4−M−1K−M−1C8×8=04×4I4×4−K−C8×8.The complex eigenvalues ofA can be calculated by using software, such as the MATLAB, where the coefficient of imaginary part represents the dimensionless natural frequency.


## 2.3. Simplified Method Based on Galerkin Discretization (SMGD)

In this method, the nth mode shape of the Galerkin discretization is used to determine the nth natural frequency of the fluid conveying pipe. The components of the diagonal matrices M, C, and K, which are calculated based on the Galerkin method, are used to calculate the natural frequencies, orders of the matrices of mass, damping, and stiffness being equal to 1. Hence, the matrix form is converted to an algebraic form. The characteristic equation for the nth dimensionless natural frequency can be expressed as follows:(13)Kn−Mnωn2+iCnωn=0,where Mn, Cn, and Kn are the nth components of the mass, damping, and stiffness matrices, respectively, as follows:(14)Mn=mcofn,(15)Cn=Λ,ccofn,(16)Kn=kcofn1+Πkcofn2,where mcofn, ccofn, kcofn1,kcofn2 are the components of the diagonal matrices of Mcof,Ccof, Kcof1 and Kcof2, respectively.By substituting (8) into equation (13) and using the components of the diagonal matrices of Mcof,Ccof, Kcof1, and Kcof2, the dimensionless form of nth natural frequency, as a function of u,P, and β, and the dimensionless critical velocities for the different types of the boundary conditions (pinned-pinned, clamped-pinned, cantilevered and clamped-clamped), can be obtained as shown in Table 2 and 3. We explicitly derive the dynamic matrices of the system, including the mass matrix, the damping matrix, and the stiffness matrix. Considering only one mode shape to determine the natural frequency is the main idea of the research. Next, by considering only one mode shape, the characteristic equation is obtained only to determine the natural frequency. The relevant relationships were listed in Table 2. According to the relationships of the mode shapes described in Table 1, the integral multiplication of the mode shape with its derivative, which is described in equation (12), becomes zero for pinned-pinned, clamped-pinned and clamped-clamped conditions. Therefore, the damping term does not appear in the dynamic equations. Furthermore, the relationships presented in Tables 2 and 3 show that increasing the mean internal pressure reduces the natural frequencies and critical flow velocities.Table 2
Formulas of SMGD for the first four dimensionless natural frequencies of the fluid conveying pipes with internal pressure for the different boundary conditions.
Boundary conditionsExpressionPinned-pinnedω1=97.4091−9.8696u2+Pω2=1558.5−39.4784u2+Pω3=7890.1−88.8264u2+Pω4=24937−157.9137u2+PClamped-pinnedω1=237.722−11.5125u2+Pω2=2496.5−42.8964u2+Pω3=10868−94.0377u2+Pω4=31780−164.9183u2+PCantileveredω1=12.3622+0.8583u2+P−4βuω2=485.5182−13.2943u2+P−4βuω3=3806.6−45.9043u2+P−4βuω4=14617−98.9169u2+P−4βuClamped-clampedω1=500.563−12.3026u2+P1−2μω2=3803.5−46.0501u2+P1−2μω3=14618−98.9047u2+P1−2μω4=39944−171.5861u2+P1−2μTable 3
Formulas of SMGD for the first four dimensionless critical velocities of the fluid conveying pipes with internal pressure for the different boundary conditions.
Boundary conditionsExpressionPinned-pinneduc1=9.8696−Puc2=39.4784−Puc3=88.8264−Puc4=157.9137−PClamped-pinneduc1=20.649−Puc2=58.1984−Puc3=115.571−Puc4=192.701−PCantilevereduc1=4β2−40.858312.3622+0.8583P/20.8583uc2=4β2−4−13.2943485.5182−13.2943P/2−13.2943uc3=4β2−4−45.90433806.6−45.9043P/2−45.9043uc4=4β2−4−98.916914617−98.9169P/2−98.9169Clamped-clampeduc1=20.649−P1−2μuc2=58.1984−P1−2μuc3=115.571−P1−2μuc4=192.701−P1−2μThe simplified method, as shown in Figure1, is fundamentally different from the Galerkin method. In the Galerkin method, the dynamic components of the system are matrices. But in the simplified method, the dynamic components of the system are algebraic.


## 3. Validations

### 3.1. Numerical and Analytical Validation

In this section, the equations presented in Tables2 and 3 are validated using several numerical and analytical methods and experimental data. The structure of the pipe can be considered as a straight beam when the fluid velocity is equal to zero, and internal pressure does not exist. Hence, the natural frequencies can be obtained analytically. At the first step, the case of u=0andP=0 is considered to compare the results of the explicit formulas with those of the DTM, DQM, and the exact solutions. Data in Table 4 show that the results of the explicit formulas are in good agreement with the exact solutions for the different boundaries. More importantly, the results indicate the higher accuracy of explicit formulas in comparison with the DTM and DQM. To investigate the precision of the presented method, the natural frequencies of fluid conveying pipes with four considered boundary conditions are calculated using the equations defined in Table 2. The observed results imply that the parameter of the fluid velocity plays a vital role in the natural frequencies. The results of the explicit formulas are compared for the case of a cantilevered pipe, β=0.1 and the case of clamped-clamped, μ=0.3 with those predicted by DTM, DQM, and exact solutions. A good agreement in the results is observed, and the relative errors are less than 0.01 percent for all boundary conditions. Moreover, the explicit formulas would be beneficially easier and computational cost lower than the other methods.Table 4
The first four dimensionless natural frequencies of the fluid conveying pipes for the value ofu=0 and P=0 for the different boundary conditions.
Boundary conditionsResultω1ω2ω3ω4Pinned-pinnedSMGD9.870039.480088.8300157.9000DTM [26]9.866939.478488.8264157.9137Relative error (%)0.03140.00410.0041−0.0087DQM [26]9.871639.486388.8442157.9454Relative error (%)−0.0162−0.0160−0.0160−0.0287Exact [46]9.869639.478488.8264157.9137Relative error (%)0.00410.00410.0041−0.0087Clamped-pinnedSMGD15.420049.9600104.2000178.3000DTM [26]15.418249.9649104.2477178.2697Relative error (%)0.0117−0.0098−0.04580.0170DQM [26]15.422849.9799104.2790178.3234Relative error (%)−0.0182−0.0398−0.0758−0.0131Exact [46]15.418249.9649104.2477178.2697Relative error (%)0.0117−0.0098−0.04580.0170CantileveredSMGD3.516022.030061.7000120.9000DTM [26]3.516022.034561.6972120.9019Relative error (%)0.0000−0.02040.0045−0.0016DQM [26]3.516722.038961.7096120.9260Relative error (%)−0.0199−0.0404−0.0156−0.0215Exact [46]3.516022.034561.6972120.9019Relative error (%)0.0000−0.02040.0045−0.0016Clamped-clampedSMGD22.370061.6700120.9000199.9000DTM [26]22.373361.6728120.9034199.8594Relative error (%)−0.0147−0.0045−0.00280.0203DQM [26]22.377861.6852120.9276199.8995Relative error (%)−0.0349−0.0246−0.02280.0003Exact [46]22.373361.6728120.9034199.8594Relative error (%)−0.0147−0.0045−0.00280.0203Another important issue for the problems of the fluid conveying pipe is the determination of dimensionless critical velocities obtained by the equations presented in Table3. As reported by Paidoussis [2], the fluid conveying pipe system is stable for dimensionless velocities in the range less than dimensionless critical velocities and loses stability at dimensionless velocities equal to dimensionless critical velocities. For a pipe with positively supported ends, the system will lose stability at dimensionless critical velocities through divergence. However, the flutter is the expected form of instability for a cantilevered system because the cantilever system is nonconservative. In this study, the dimensionless critical velocities of fluid conveying pipe are obtained using the explicit formulas (SMGD) for different boundary conditions. They are listed in Table 5 to compare with the results of the DTM, DQM, and given by Paidoussis [2]. As can be seen, the relative errors are less than 2.08 percent for all boundary conditions. Also, Table 5 shows that the internal pressure leads to a reduction in the dimensionless critical velocities, except for the first mode of cantilevered cases.Table 5
The first four dimensionless critical velocities of the fluid conveying pipes for the different values of P in the different boundary conditions.
Boundary conditionsPressureResultucr1ucr2ucr3ucr4Pinned-pinned0SMGD3.14166.28329.424812.5664DTM [26]3.14166.2832——Relative error (%)−0.0002−0.0002——DQM [26]3.14206.2838——Relative error (%)−0.0129−0.0098——Paidoussis [2]3.14006.2800——Relative error (%)0.05070.0507——1.57SMGD3.04006.23309.391412.5414Clamped-pinned0SMGD4.54417.628810.750213.8818DTM [26]4.4934———Relative error (%)1.1288———DQM [26]4.4937———Relative error (%)1.1220———Paidoussis [2]4.4900———Relative error (%)1.2054———1.57SMGD4.47457.587510.721013.8591Cantilevered0SMGD3.79516.04329.106212.1561DTM [26]——9.3224—Relative error (%)——−2.3187—DQM [26]——9.3233—Relative error (%)——−2.3281—Paidoussis [2]——9.3000—Relative error (%)——−2.0834—1.57SMGD3.99665.91199.019612.0913Clamped-clamped0SMGD6.37879.088212.157115.2576DTM [26]6.2832———Relative error (%)1.5196———DQM [26]6.2838———Relative error (%)1.5099———Paidoussis [2]6.2800———Relative error (%)1.5714———1.57SMGD6.32939.053612.131215.2370In Table6, the specifications of a pinned-pinned fluid conveying pipe are given to compare the results further. To verify the accuracy of the results, they are compared with those given by Xu et al. [21] and Housner [17].Table 6
Specifications considered for a pinned-pinned fluid conveying pipe [47].
Elasticity modulusOuter diameterWall thicknessPipe lengthDensity of the pipeDensity of the internal flow210Gpa324mm16mm32m8200kg.m−3908.2kg.m−3A homotopy perturbation method (HPM) was used to determine the dimensionless natural frequency of a pinned-pinned fluid conveying pipe [21]. Dimensionless natural frequencies of a simply supported pipe are obtained as follows:(17)ωnωN2=n4−n2VVc2+16n6βVVc2∑k≠nk2−1n+k−12n2+k2n2−k23,n=1,2,..,where ωN is the fundamental natural frequency of the pipe in the absence of the flow and can be expressed as follows:(18)ωN=π2L2EIMp,and Vc is the critical velocity of flow for static buckling of the pipe:(19)Vc=πLEIMf.Housner [17] suggested that the first two dimensionless natural frequencies of a pinned-pinned pipe conveying fluid can be computed by(20)ωnωN2=α±α2−41−VVc24−VVc2,n=1,2,where(21)α=8.5−VVc22.5+128β9π2.The SMGD shows that the dimensionless natural frequency of a simply supported pipe for the case ofP=0 is as follows:(22)ωn=nπ4−nπ2u2,n=1,2….Equation (22) is simpler than (17) and (20). The first four dimensionless natural frequencies of a pinned-pinned fluid conveying pipe are calculated by equations (18), (14), and (17) and listed in Table 7. The results of the explicit formulas (SMGD) are in good agreement with the results of the FEM, HPM, and Housner’s, and the relative errors are less than 1.83 percent for all cases.Table 7
The first four dimensionless natural frequencies (rad/s) of a pinned-pinned fluid conveying pipe for the different values ofu and without internal pressure.
Flow velocity (m/s)01525354555Dimensionless velocity00.60151.00251.40351.80452.20551st modeSMGD4.37334.29254.14493.91263.57993.1144FEM [21]4.37324.29214.14413.91163.35813.1115Relative error (%)0.00170.00940.01860.02546.60440.0934HPM [21]4.37324.28704.12933.88003.52223.0145Relative error (%)0.00170.12840.37710.84001.63773.3142Housner [17]4.37324.29714.15763.93723.61833.1660Relative error (%)0.0017−0.1070−0.3062−0.6250−1.0618−1.62962nd modeSMGD17.493117.412917.269217.051116.756116.3800FEM [21]17.492817.412317.268217.049916.754416.3775Relative error (%)0.00170.00340.00590.00700.01040.0153HPM [21]17.492817.417117.268217.076516.799116.4457Relative error (%)0.0017−0.02420.0062−0.1488−0.2557−0.3995Housner [17]17.492817.392217.212216.933016.568616.0855Relative error (%)0.00170.11890.33130.69741.13201.83083rd modeSMGD39.359539.279439.136438.920638.631338.2666FEM [21]39.358739.278339.135038.919038.629238.2638Relative error (%)0.00190.00270.00360.00410.00540.0073HPM [21]39.358739.285839.155938.960238.697838.3672Relative error (%)0.0019−0.0164−0.0498−0.1016−0.1718−0.26224th modeSMGD69.972469.892369.749669.534669.247268.8863To compare the results with available experimental data, a pinned-pinned fluid conveying pipe is considered based on the specifications in Table8 [48]. The changes of natural frequencies with increasing dimensionless flow velocity are shown in Figure 3. The results of the present study are compared to those of the experimental data [48], FEM [21], HPM (17), and Housner (20). Figure 3 confirms that the results of the presented study are significantly consistent with the experimental data and the results of the finite element method.Table 8
Specifications considered for a pinned-pinned fluid conveying pipe [48].
Elasticity modulusOuter diameterWall thicknessPipe lengthMass-per-unit-length of fluidMass-per-unit-length of fluid and pipe68.948Gpa2.54cm0.165cm3.2m0.383kg.m−10.7134kg.m−1Figure 3
The first third natural frequencies of a pinned-pinned fluid conveying pipe for the different values ofV/Vc and without internal pressure based on experimental data; (a) the first frequency, (b) the second frequency, and (c) the third frequency.
(a)(b)(c)


## 3.1. Numerical and Analytical Validation

In this section, the equations presented in Tables2 and 3 are validated using several numerical and analytical methods and experimental data. The structure of the pipe can be considered as a straight beam when the fluid velocity is equal to zero, and internal pressure does not exist. Hence, the natural frequencies can be obtained analytically. At the first step, the case of u=0andP=0 is considered to compare the results of the explicit formulas with those of the DTM, DQM, and the exact solutions. Data in Table 4 show that the results of the explicit formulas are in good agreement with the exact solutions for the different boundaries. More importantly, the results indicate the higher accuracy of explicit formulas in comparison with the DTM and DQM. To investigate the precision of the presented method, the natural frequencies of fluid conveying pipes with four considered boundary conditions are calculated using the equations defined in Table 2. The observed results imply that the parameter of the fluid velocity plays a vital role in the natural frequencies. The results of the explicit formulas are compared for the case of a cantilevered pipe, β=0.1 and the case of clamped-clamped, μ=0.3 with those predicted by DTM, DQM, and exact solutions. A good agreement in the results is observed, and the relative errors are less than 0.01 percent for all boundary conditions. Moreover, the explicit formulas would be beneficially easier and computational cost lower than the other methods.Table 4
The first four dimensionless natural frequencies of the fluid conveying pipes for the value ofu=0 and P=0 for the different boundary conditions.
Boundary conditionsResultω1ω2ω3ω4Pinned-pinnedSMGD9.870039.480088.8300157.9000DTM [26]9.866939.478488.8264157.9137Relative error (%)0.03140.00410.0041−0.0087DQM [26]9.871639.486388.8442157.9454Relative error (%)−0.0162−0.0160−0.0160−0.0287Exact [46]9.869639.478488.8264157.9137Relative error (%)0.00410.00410.0041−0.0087Clamped-pinnedSMGD15.420049.9600104.2000178.3000DTM [26]15.418249.9649104.2477178.2697Relative error (%)0.0117−0.0098−0.04580.0170DQM [26]15.422849.9799104.2790178.3234Relative error (%)−0.0182−0.0398−0.0758−0.0131Exact [46]15.418249.9649104.2477178.2697Relative error (%)0.0117−0.0098−0.04580.0170CantileveredSMGD3.516022.030061.7000120.9000DTM [26]3.516022.034561.6972120.9019Relative error (%)0.0000−0.02040.0045−0.0016DQM [26]3.516722.038961.7096120.9260Relative error (%)−0.0199−0.0404−0.0156−0.0215Exact [46]3.516022.034561.6972120.9019Relative error (%)0.0000−0.02040.0045−0.0016Clamped-clampedSMGD22.370061.6700120.9000199.9000DTM [26]22.373361.6728120.9034199.8594Relative error (%)−0.0147−0.0045−0.00280.0203DQM [26]22.377861.6852120.9276199.8995Relative error (%)−0.0349−0.0246−0.02280.0003Exact [46]22.373361.6728120.9034199.8594Relative error (%)−0.0147−0.0045−0.00280.0203Another important issue for the problems of the fluid conveying pipe is the determination of dimensionless critical velocities obtained by the equations presented in Table3. As reported by Paidoussis [2], the fluid conveying pipe system is stable for dimensionless velocities in the range less than dimensionless critical velocities and loses stability at dimensionless velocities equal to dimensionless critical velocities. For a pipe with positively supported ends, the system will lose stability at dimensionless critical velocities through divergence. However, the flutter is the expected form of instability for a cantilevered system because the cantilever system is nonconservative. In this study, the dimensionless critical velocities of fluid conveying pipe are obtained using the explicit formulas (SMGD) for different boundary conditions. They are listed in Table 5 to compare with the results of the DTM, DQM, and given by Paidoussis [2]. As can be seen, the relative errors are less than 2.08 percent for all boundary conditions. Also, Table 5 shows that the internal pressure leads to a reduction in the dimensionless critical velocities, except for the first mode of cantilevered cases.Table 5
The first four dimensionless critical velocities of the fluid conveying pipes for the different values of P in the different boundary conditions.
Boundary conditionsPressureResultucr1ucr2ucr3ucr4Pinned-pinned0SMGD3.14166.28329.424812.5664DTM [26]3.14166.2832——Relative error (%)−0.0002−0.0002——DQM [26]3.14206.2838——Relative error (%)−0.0129−0.0098——Paidoussis [2]3.14006.2800——Relative error (%)0.05070.0507——1.57SMGD3.04006.23309.391412.5414Clamped-pinned0SMGD4.54417.628810.750213.8818DTM [26]4.4934———Relative error (%)1.1288———DQM [26]4.4937———Relative error (%)1.1220———Paidoussis [2]4.4900———Relative error (%)1.2054———1.57SMGD4.47457.587510.721013.8591Cantilevered0SMGD3.79516.04329.106212.1561DTM [26]——9.3224—Relative error (%)——−2.3187—DQM [26]——9.3233—Relative error (%)——−2.3281—Paidoussis [2]——9.3000—Relative error (%)——−2.0834—1.57SMGD3.99665.91199.019612.0913Clamped-clamped0SMGD6.37879.088212.157115.2576DTM [26]6.2832———Relative error (%)1.5196———DQM [26]6.2838———Relative error (%)1.5099———Paidoussis [2]6.2800———Relative error (%)1.5714———1.57SMGD6.32939.053612.131215.2370In Table6, the specifications of a pinned-pinned fluid conveying pipe are given to compare the results further. To verify the accuracy of the results, they are compared with those given by Xu et al. [21] and Housner [17].Table 6
Specifications considered for a pinned-pinned fluid conveying pipe [47].
Elasticity modulusOuter diameterWall thicknessPipe lengthDensity of the pipeDensity of the internal flow210Gpa324mm16mm32m8200kg.m−3908.2kg.m−3A homotopy perturbation method (HPM) was used to determine the dimensionless natural frequency of a pinned-pinned fluid conveying pipe [21]. Dimensionless natural frequencies of a simply supported pipe are obtained as follows:(17)ωnωN2=n4−n2VVc2+16n6βVVc2∑k≠nk2−1n+k−12n2+k2n2−k23,n=1,2,..,where ωN is the fundamental natural frequency of the pipe in the absence of the flow and can be expressed as follows:(18)ωN=π2L2EIMp,and Vc is the critical velocity of flow for static buckling of the pipe:(19)Vc=πLEIMf.Housner [17] suggested that the first two dimensionless natural frequencies of a pinned-pinned pipe conveying fluid can be computed by(20)ωnωN2=α±α2−41−VVc24−VVc2,n=1,2,where(21)α=8.5−VVc22.5+128β9π2.The SMGD shows that the dimensionless natural frequency of a simply supported pipe for the case ofP=0 is as follows:(22)ωn=nπ4−nπ2u2,n=1,2….Equation (22) is simpler than (17) and (20). The first four dimensionless natural frequencies of a pinned-pinned fluid conveying pipe are calculated by equations (18), (14), and (17) and listed in Table 7. The results of the explicit formulas (SMGD) are in good agreement with the results of the FEM, HPM, and Housner’s, and the relative errors are less than 1.83 percent for all cases.Table 7
The first four dimensionless natural frequencies (rad/s) of a pinned-pinned fluid conveying pipe for the different values ofu and without internal pressure.
Flow velocity (m/s)01525354555Dimensionless velocity00.60151.00251.40351.80452.20551st modeSMGD4.37334.29254.14493.91263.57993.1144FEM [21]4.37324.29214.14413.91163.35813.1115Relative error (%)0.00170.00940.01860.02546.60440.0934HPM [21]4.37324.28704.12933.88003.52223.0145Relative error (%)0.00170.12840.37710.84001.63773.3142Housner [17]4.37324.29714.15763.93723.61833.1660Relative error (%)0.0017−0.1070−0.3062−0.6250−1.0618−1.62962nd modeSMGD17.493117.412917.269217.051116.756116.3800FEM [21]17.492817.412317.268217.049916.754416.3775Relative error (%)0.00170.00340.00590.00700.01040.0153HPM [21]17.492817.417117.268217.076516.799116.4457Relative error (%)0.0017−0.02420.0062−0.1488−0.2557−0.3995Housner [17]17.492817.392217.212216.933016.568616.0855Relative error (%)0.00170.11890.33130.69741.13201.83083rd modeSMGD39.359539.279439.136438.920638.631338.2666FEM [21]39.358739.278339.135038.919038.629238.2638Relative error (%)0.00190.00270.00360.00410.00540.0073HPM [21]39.358739.285839.155938.960238.697838.3672Relative error (%)0.0019−0.0164−0.0498−0.1016−0.1718−0.26224th modeSMGD69.972469.892369.749669.534669.247268.8863To compare the results with available experimental data, a pinned-pinned fluid conveying pipe is considered based on the specifications in Table8 [48]. The changes of natural frequencies with increasing dimensionless flow velocity are shown in Figure 3. The results of the present study are compared to those of the experimental data [48], FEM [21], HPM (17), and Housner (20). Figure 3 confirms that the results of the presented study are significantly consistent with the experimental data and the results of the finite element method.Table 8
Specifications considered for a pinned-pinned fluid conveying pipe [48].
Elasticity modulusOuter diameterWall thicknessPipe lengthMass-per-unit-length of fluidMass-per-unit-length of fluid and pipe68.948Gpa2.54cm0.165cm3.2m0.383kg.m−10.7134kg.m−1Figure 3
The first third natural frequencies of a pinned-pinned fluid conveying pipe for the different values ofV/Vc and without internal pressure based on experimental data; (a) the first frequency, (b) the second frequency, and (c) the third frequency.
(a)(b)(c)


## 4. Results and Discussions

For the sake of the argument, the material properties of the studied cases are chosen based on the literature as follows:E (the elastic modulus of the pipes), μ (the Poisson ratio), ρp (the density of the pipe), D (the outer diameter of the pipes), d (the internal diameter of the pipes), L (length of the pipes), ρf (the density of the fluid), and p¯ (the mean internal pressure) are considered as 68 GPa, 0.3, 2700 kg.m−3, 46 mm, 40 mm, 2 m, 870 kg.m−3, and 2 MPa, respectively. At the first step, three cases including u = 0, 1, and 2, and P = 1.57 are considered to compare the results with the Galerkin method (with N = 4, for all cases β = 0.5). Table 9 shows that the results of the explicit formulas (SMGD) are in good agreement with the Galerkin method for the different boundary conditions. Furthermore, the acceptable accuracy of the results is obtained by the SMGD. For all cases, relative errors are less than 2.60 percent (except for the first frequency of the cantilever boundary condition, with u = 2). At high velocities, the system practically fails, and there is no pipe. Since the behavior of the system tends to be nonlinear at higher critical velocities and there are complex issues related to instability, the accuracy of relationships in these areas is reduced.Table 9
The first four dimensionless natural frequencies of the pipes conveying fluid for the different values ofu and with internal pressure p=1.57 in the different boundary conditions.
Boundary conditionsDimensionless velocityResultω1ω2ω3ω4Pinned-pinned0SMGD9.050038.690088.0400157.1000N = 49.050038.680088.0400157.1300Relative error (%)0.00000.02590.0000−0.01911SMGD8.480038.170087.5300156.6000N = 48.440038.220087.6100156.7100Relative error (%)0.4739−0.1308−0.0913−0.07022SMGD6.510036.590086.0000155.1000N = 46.370036.810086.3100155.4600Relative error (%)2.1978−0.5977−0.3592−0.2316Clamped-pinned0SMGD14.820049.2900103.5000177.5000N = 414.820049.2900103.5400177.5500Relative error (%)0.00000.0000-0.0386-0.02821SMGD14.430048.8500103.1000177.1000N = 414.360048.8900103.1500177.1600Relative error (%)0.4875−0.0818−0.0485−0.03392SMGD13.180047.5000101.7000175.7000N = 412.940047.6900101.9800176.0000Relative error (%)1.8547−0.3984−0.2746−0.1705Cantilevered0SMGD3.70321.5661.11120.3N = 43.71621.552761.1105120.2569Relative error (%)−0.34980.0339-0.00080.03581SMGD3.4221.1860.71119.8N = 43.395721.300760.786120.3519Relative error (%)0.7156−0.5666−0.1250−0.45862SMGD2.412059.49118.6N = 42.2820.535259.8251120.6216Relative error (%)5.7018−2.6063−0.5601−1.6760Clamped-clamped (μ = 0.3)0SMGD22.200061.4400120.6000199.6000N = 422.230061.4700120.6900199.6300Relative error (%)−0.1350−0.0488−0.0746−0.01501SMGD21.920061.0600120.2000199.2000N = 421.870061.1300120.3300199.2700Relative error (%)0.2286−0.1145−0.1080−0.03512SMGD21.060059.9200119.0000197.9000N = 420.790060.1000119.2700198.2000Relative error (%)1.2987−0.2995−0.2264−0.1514


### 4.1. The Effects of Mean Internal Pressure on Pinned-Pinned Fluid Conveying Pipe

Figure4 depicts how the first four dimensionless natural frequencies vary with dimensionless internal pressure for the three dimensionless flow velocity values. The results are obtained using the equations listed in Table 2. It can be seen that the mean internal pressure has more effect on the first frequency for the pinned-pinned boundary condition. Significantly, the value of the critical mean internal pressure (the case of uc=0) is equal to the dimensionless natural frequency of the pipe with u = 0 and P = 0.Figure 4
The first four dimensionless natural frequencies of a pinned-pinned pipe conveying fluid for the different values of internal pressure; (a) the first frequency, (b) the second frequency, (c) the third frequency, and (d) the fourth frequency.
(a)(b)(c)(d)It reveals that the fluid velocity affects the internal critical pressure besides the natural frequencies. The fluid velocity has more effect on the mean internal pressure for the first and second frequencies. Furthermore, the values of the mean internal pressure (which leads touc=0) for the first four mode shapes of the pinned-pinned fluid conveying pipe are equal to 9.8696, 39.4784, 88.8264, and 157.9137, respectively. The higher the average internal pressure of the pipe, the lower the natural frequency of the system. At high velocities, the safety expected for the pipe decreases, and the critical velocity, which is the limit of instability, decreases.Figures5–7 (and equations in Table 3) illustrate that natural frequencies and critical flow velocity gradually decrease as the mean internal pressure is increased (the inverse relationship, except at the first mode of the cantilever boundary condition).Figure 5
The first four dimensionless natural frequencies of a clamped-pinned pipe conveying fluid for the different values ofu.Figure 6
The first four dimensionless natural frequencies of a cantilevered pipe conveying fluid for the different values ofu and β=0.1.Figure 7
The first four dimensionless natural frequencies of a clamped-clamped pipe conveying fluid for the different values ofu and μ=0.3.


### 4.2. The Effects of the Flow Velocity on the Boundary Conditions

According to equations of clamped-pinned condition, presented in Table3, it shows that the values of the critical mean internal pressure (which leads to uc=0) for the first four mode shapes are equal to 15.42, 49.96, 104.2, and 178.3, respectively. The related values of the critical mean internal pressure are equal to dimensionless natural frequencies of the clamped-pinned pipe with u = 0 and P = 0. It can be comprehended that the mean internal pressure has more effect on the first mode natural frequency due to higher values of the critical mean internal pressures for the second, third, and fourth modes.Figure5 shows the first four dimensionless natural frequencies of a clamped-pinned pipe conveying fluid for the different values of u. The velocity values according to the Im (ω) = 0 are equal to critical velocity values. Also, the critical velocity decreases as the mean internal pressure increases.The values of the critical mean internal pressure for the first four mode shapes of the cantilevered fluid conveying pipe (β = 0.1) are equal to 3.51, 22.03, 61.7, and 120.9, respectively, which are equal to dimensionless natural frequencies for the condition of u = 0 and P = 0. In Figure 6, it can be observed that the mean internal pressure has less effect on the natural frequencies for the third and fourth frequencies and leads to a slight increase in the first frequency. Also, the values of the mean internal pressure for the first four mode shapes of the clamped-clamped fluid conveying pipe (μ=0.3) are equal to 22.37, 61.67, 120.9, and 199.9, respectively. They are equal to dimensionless natural frequencies for the condition of u = 0 and P = 0. Like the cantilevered condition, the mean internal pressure has less effect on the natural frequencies for the third and fourth frequencies.The values of the critical mean internal pressure (which leads touc=0) for the first four mode shapes of the pinned-pinned fluid conveying pipe are equal to 9.8696, 39.4784, 88.8284, and 157.9137, respectively. It can be concluded that mean internal pressure has more effect on the first mode natural frequency due to higher values of the critical mean internal pressures for the second, third, and fourth modes. The velocity values based on the Im (ω) = 0 are equal to critical velocity values, as shown in Figure 8.Figure 8
The first four dimensionless natural frequencies of a pinned-pinned pipe conveying fluid for the different values ofu.


## 4.1. The Effects of Mean Internal Pressure on Pinned-Pinned Fluid Conveying Pipe

Figure4 depicts how the first four dimensionless natural frequencies vary with dimensionless internal pressure for the three dimensionless flow velocity values. The results are obtained using the equations listed in Table 2. It can be seen that the mean internal pressure has more effect on the first frequency for the pinned-pinned boundary condition. Significantly, the value of the critical mean internal pressure (the case of uc=0) is equal to the dimensionless natural frequency of the pipe with u = 0 and P = 0.Figure 4
The first four dimensionless natural frequencies of a pinned-pinned pipe conveying fluid for the different values of internal pressure; (a) the first frequency, (b) the second frequency, (c) the third frequency, and (d) the fourth frequency.
(a)(b)(c)(d)It reveals that the fluid velocity affects the internal critical pressure besides the natural frequencies. The fluid velocity has more effect on the mean internal pressure for the first and second frequencies. Furthermore, the values of the mean internal pressure (which leads touc=0) for the first four mode shapes of the pinned-pinned fluid conveying pipe are equal to 9.8696, 39.4784, 88.8264, and 157.9137, respectively. The higher the average internal pressure of the pipe, the lower the natural frequency of the system. At high velocities, the safety expected for the pipe decreases, and the critical velocity, which is the limit of instability, decreases.Figures5–7 (and equations in Table 3) illustrate that natural frequencies and critical flow velocity gradually decrease as the mean internal pressure is increased (the inverse relationship, except at the first mode of the cantilever boundary condition).Figure 5
The first four dimensionless natural frequencies of a clamped-pinned pipe conveying fluid for the different values ofu.Figure 6
The first four dimensionless natural frequencies of a cantilevered pipe conveying fluid for the different values ofu and β=0.1.Figure 7
The first four dimensionless natural frequencies of a clamped-clamped pipe conveying fluid for the different values ofu and μ=0.3.


## 4.2. The Effects of the Flow Velocity on the Boundary Conditions

According to equations of clamped-pinned condition, presented in Table3, it shows that the values of the critical mean internal pressure (which leads to uc=0) for the first four mode shapes are equal to 15.42, 49.96, 104.2, and 178.3, respectively. The related values of the critical mean internal pressure are equal to dimensionless natural frequencies of the clamped-pinned pipe with u = 0 and P = 0. It can be comprehended that the mean internal pressure has more effect on the first mode natural frequency due to higher values of the critical mean internal pressures for the second, third, and fourth modes.Figure5 shows the first four dimensionless natural frequencies of a clamped-pinned pipe conveying fluid for the different values of u. The velocity values according to the Im (ω) = 0 are equal to critical velocity values. Also, the critical velocity decreases as the mean internal pressure increases.The values of the critical mean internal pressure for the first four mode shapes of the cantilevered fluid conveying pipe (β = 0.1) are equal to 3.51, 22.03, 61.7, and 120.9, respectively, which are equal to dimensionless natural frequencies for the condition of u = 0 and P = 0. In Figure 6, it can be observed that the mean internal pressure has less effect on the natural frequencies for the third and fourth frequencies and leads to a slight increase in the first frequency. Also, the values of the mean internal pressure for the first four mode shapes of the clamped-clamped fluid conveying pipe (μ=0.3) are equal to 22.37, 61.67, 120.9, and 199.9, respectively. They are equal to dimensionless natural frequencies for the condition of u = 0 and P = 0. Like the cantilevered condition, the mean internal pressure has less effect on the natural frequencies for the third and fourth frequencies.The values of the critical mean internal pressure (which leads touc=0) for the first four mode shapes of the pinned-pinned fluid conveying pipe are equal to 9.8696, 39.4784, 88.8284, and 157.9137, respectively. It can be concluded that mean internal pressure has more effect on the first mode natural frequency due to higher values of the critical mean internal pressures for the second, third, and fourth modes. The velocity values based on the Im (ω) = 0 are equal to critical velocity values, as shown in Figure 8.Figure 8
The first four dimensionless natural frequencies of a pinned-pinned pipe conveying fluid for the different values ofu.


## 5. Conclusion

This paper presents a simplified approach to determine the dynamic behavior of fluid conveying pipes with different typical boundary conditions under the mean internal pressure. For this purpose, the method is suggested by reducing the order of problem, and the nth mode shape of the Galerkin discretization is used to determine the nth dimensionless natural frequency of the considered pipes. The presented method is validated by experimental data, FEM, HPM, DTM, DQM, and the exact solution methods. The following conclusions can be drawn:The results showed that the explicit formulas (SMGD) for the different boundary conditions are in good agreement with the exact solutions and have higher accuracy in comparison with the DTM and DQM. More importantly, the SMGD is more efficient and easier than the other methods. The results implied that the parameter of the fluid velocity plays a vital role in natural frequencies.The results are compared for the case of a cantilevered pipe,β = 0.1 and the case of clamped-clamped, μ = 0.3 with those predicted by DTM, DQM, and exact solutions. A good agreement is observed. The relative errors are less than 0.01 percent for all boundary conditions.The changes of natural frequencies with increasing dimensionless flow velocity are compared for the case of a pinned-pinned pipe. It was implied that the results of the presented method are significantly consistent with the experimental data and the results of the finite element method. For the cases of the pinned-pinned pipe and the clamped-pinned pipe, the mean internal pressure has more effect on the first frequency.Values of the critical mean internal pressure for the first four mode shapes of the cantilevered fluid conveying pipe are equal to dimensionless natural frequencies for the condition ofu = 0 and P = 0. Values of the mean internal pressure for the first four mode shapes of the clamped-clamped fluid conveying pipe are equal to 22.3700, 61.67, 120.9, and 199.9, respectively. Furthermore, it was observed that the critical mean internal pressure values of pipes are equal to dimensionless natural frequencies for the condition of u = 0 and P = 0 for all considered cases.Summarily, the simplified method analytically predicts the natural frequencies and critical velocities of the fluid-conveying pipes in different boundary conditions. It is less complex with a lower computational cost and higher accuracy than other methods.



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*Source: 5320019-2022-04-18.xml*