1 Introduction

The forced vibration of pipe conveying fluid has attracted increasing attention in engineering practice such as natural gas transportation system, hot leg piping system for nuclear reactor, and heating system. As pointed out by Paidoussis and Li in Ref. [1], the dynamics of pipe conveying fluid has become a typical dynamical problem. The study on the dynamic behavior of pipe conveying fluid has been becoming an important issue not only from the view of improving the service life of piping system but also from that the experience gained in studying this problem can be radiated into other areas of applied mechanics, especially the fluid-structure interaction related slender system with flowing media. Paidoussis ^[2], Wang et al. ^[3], Ghavanloo et al. ^[4], Jung and Chung ^[5], Wang et al. ^[6], and Karami and Farid ^[7] give some available information for the vibration of pipe conveying fluid.

Since the fluid-structure interaction vibration plays a great role in the leakage of the inner medium in piping system, attaining more accurate vibration characteristics of piping system is indispensable to prevent the undesirable responses and to enhance the system safety. Curved pipe conveying fluid is one of the important components in piping system, and its vibration is more complex than straight pipe conveying fluid because it has four displacement variables. Generally speaking, there are three available theories on curved pipes conveying fluid, i.e., extensible, modified in-extensible, and in-extensible. Misra et al. ^[8] reported that the results of the former two theories were remarkably close to each other if both ends of the pipe were supported. Therefore, the extensible or modified in-extensible theory taking the steady combined force into account is more reliable.

Due to the vibration complexity, finding analytical methods for curved pipes may be unpractical or completely inaccessible. Therefore, several numerical methods have been sprung up with the development of computing technology in recent decades, e.g., finite element method (FEM), Galerkin method, transfer matrix method (TMM) ^[9-10], differential transformation method (DTM) ^[11], differential quadrature method (DQM) ^[12], and the generalized form of DQM (GDQM) ^[13].

In the Green function method (GFM), the superposition principle through integration is the main characteristic, which determines whether it can output an accurate solution corresponding to the given governing equation in closed form or not. Therefore, it belongs to an analytical method. The GFM was usually adopted to solve the problems of beams including Euler-Bernoulli and Timoshenko models. Foda and Abduljabbar ^[14] utilized the GFM to study the response of a simply supported Euler-Bernoulli beam of finite length subjected to a moving mass traversing its span. Abu-Hilal ^[15-16] solved the forced vibration related problem of Euler-Bernoulli beam by the GFM. Li et al. ^[17] studied the forced vibration of a Timoshenko beam with damping effect by use of the GFM.

To the best of the authors' knowledge, analytical studies on the forced vibration of curved pipe conveying fluid have seldom been reported, let alone the effects of damping, added mass, steady combined force, and the distribution of the excitation on the dynamic response. In fact, the above parameters are existent, and play important roles in engineering applications. There is no work published yet in the literature concerning the GFM of curved pipe conveying fluid with the consideration of the above parameters.

In this paper, the GFM is used to study the in-plane forced vibration of the curved pipe conveying fluid considering the modified in-extensible theory, which is governed by the 6th-order differential equation, and the excitation is introduced by its first derivate with respect to the angle coordinate. The method can also be radiated to out of plane or more complex problems. Three supporting types, i.e., clamped-clamped, clamped-pinned, and pinned-pinned, are considered. The effects of the aforementioned parameters on the displacement along both the tangential direction and the radial direction are studied in detail. Besides, the forced vibration problems of curved pipe with both ends supported under the motion constraints are briefly discussed at last with the consideration of the GFM.

2 Steady state response of a fluid conveying curved pipe vibrating in plane 2.1 Governing equation

Consider a fluid conveying curved pipe subjected to radial excitation when both ends of the pipe, which can be fixed or simply supported, are supported (see Fig. 1).

According to Ref. [8], if the modified in-extensible theory is used, the dimensionless in-plane motion of the pipe can be generally expressed in a real domain as follows:

(1)

The dimensionless parameters can be defined as follows:

(2)

where EI denotes the flexural rigidity, w is the tangential displacement, R is the uniform radius of the pipe, m_f and m_p are the mass per unit length of fluid and pipe, respectively, U is the constant flowing velocity of the plug flow model, Θ and t denote the angle and the time coordinate, respectively, L is the arc length of the pipe, θ_op is the opening angle of the pipe, l is the curvilinear coordinate along the centerline, m_t and c_t are the added mass per unit length and the coefficient of viscous damping due to the surrounding fluid associated with the transverse motion, m₁ and c₁ are the added mass per unit length and the coefficient of viscous damping due to the surrounding fluid associated with the longitudinal motion, Π₀ is the steady combined force related to the fluid flow, and F denotes the external excitation.

2.2 Derivation of the solution

Under most circumstances, the excitation can be treated to be periodic. Therefore, we consider a time-varying periodic function. According to the Fourier series, the function can be expanded as follows:

(3)

where h is a half of the cycle, and

(4)

Then, the excitation can be written in the complex form as follows:

(5)

If the former N terms are utilized to approximate the periodic function and introduced into Eq. (1), the steady state dynamic response will be denoted by the sum of N terms where the nth term corresponds to the nth frequency according to the superposition principle. From this perspective, the investigation of time harmonic excitation with single frequency is the basis of the complex situation. Therefore, when a harmonic excitation along the radial direction is considered, since the pipe is damped, things will be convenient if the excitation is written in the complex form as follows:

(6)

where is a force that can be concentrated or distributed but conclusive.

Then, the complex form of Eq. (1) is

(7)

where

(8)

Then, the solution of Eq. (7) can be assumed to be

(9)

With the combination of Eqs. (8) and (9), the dimensionless tangential displacement in the real domain can be expressed as follows:

(10)

Substituting Eq. (9) into Eq. (7) yields

(11)

According to the superposition principle, the solution of Eq. (11) can be written as follows:

(12)

where G(θ, θ₀) is the Green function of the curved pipe. According to its definition, G(θ, θ₀) is physically the response caused by a unit concentrated force acting at an arbitrary position θ₀, i.e., the solution of the following equation:

(13)

There are many methods for solving Eq. (13), e.g., the method of undetermined coefficients and the Galerkin method. A comparatively efficient way for a linear problem is the Laplace transform, with which the accurate solution can be generated in closed form. By virtue of the Laplace transform and after the elementary transformation, Eq. (13) becomes

(14)

where s is generally a complex variable in the transformed domain corresponding to θ, and the value in the bracket of the superscript denotes the order of the derivative hereafter in the paper. Meanwhile, some parameters in Eq. (14) are defined as follows:

With the denominator factorization in Eq. (14), we have

(15)

With the introduction of Eq. (15), Eq. (14) can be rewritten as follows:

(16)

Some parameters can be defined by

(17)

where λ_{q-1, j} denotes the result of substituting s with s_j into λ_q-1.

By means of the inverse Laplace transform, the solution of Eq. (13) will be obtained, i.e.,

(18)

where H(*) represents the Heaviside function.

With the combination of Eqs. (12) and (18), the solution of Eq. (11) is

(19)

Substituting Eq. (19) into Eq. (10) yields the steady state dynamic response.

In Eq. (14), y^(q-1)(0) (q=1, 2, 3, 4, 5, 6) are variables that can be determined by various boundary conditions. To obtain them, the former five order derivatives of y(θ, θ₀) when θ >θ₀ should be calculated, i.e.,

(20)

Substituting θ =1 into Eqs. (18) and (20), after further manipulation, we have

(21)

where φ_j-1^(i-1)(1) (i, j=1, 2, 3, 4, 5, 6) are the elements of Φ(1), and

For convenience, Eq. (21) can be defined as the conjunct equation in this paper, since it connects both ends.

3 Green functions of the curved pipe with different boundary conditions

Once the Green function of the pipe is obtained, the steady state dynamic response will be easily solved by integration. According to the modified in-extensible theory proposed by Misra et al. ^[8], if both ends of the pipe are supported and the gravity effect and orientation of the pipe are neglected, Π₀ =-u². The radial displacement r (see Fig. 1) equals the 1st-order derivative of w with respect to Θ. If the dimensionless form of the radial displacement is denoted by η with the definition η =rθ_op/R, the differential relation will be naturally determined.

Since there are totally twelve unknown variables and six equations in Eq. (21), six boundary conditions are necessary for a statically determinate solution. For the three typical kinds of curved pipes, i.e., clamped-clamped, clamped-pinned, and pinned-pinned, the boundary conditions are as follows:

(ⅰ) Clamped-clamped pipe

(22)

(ⅱ) Clamped-pinned pipe

(23)

(ⅲ) Pinned-pinned pipe

(24)

For a conveying fluid curved pipe with both ends fixed, Eq. (22) means that

Substituting the above equation into Eq. (21), we can rewrite the conjunct equation as follows:

(25)

which can be simplified as

(26)

When Eq. (18) demonstrates, y⁽³⁾(0), y⁽⁴⁾ (0), and y⁽⁵⁾ (0) are needed for obtaining the Green function, and they are just parts of the solution of Eq. (25). By virtue of the elementary linear algebra, we have

(27)

(28)

(29)

where A_{i, j} is the element of A in the ith row and the jth column. Substituting y⁽³⁾ (0), y⁽⁴⁾ (0), and y⁽⁵⁾ (0) into Eq. (18), we obtain the Green function of the clamped-clamped curved pipe as follows:

(30)

Consider a concentrated time harmonic force shown in Fig. 1, if the angle between the horizontally rightward axis and the force is A₀, the excitation can be written in the complex form as follows:

(31)

where F₀ is the excitation amplitude, and Ω is the exciting frequency.

Then, the corresponding dimensionless form is

(32)

where

Substituting Eq. (32) into Eq. (12) and after the elementary transformation, we have

(33)

Substituting Eq. (33) into Eq. (10), we can obtain the displacement along the tangential direction in the real domain.

For clamped-pinned and pinned-pinned curved pipes, the same procedures can be applied. For a cantilevered curved pipe, the effect of the "steady combined force" is less pronounced, and it can be neglected (see Ref. [12]), which is not included in this paper.

4 Numerical results 4.1 Validity of the current method

The concentrated force and a clamped-clamped pipe are utilized for the validity in this subsection. As usual, the most important characteristics concerning the forced vibration are the natural frequency and the displacement response. Thus, the validity can be divided into these two aspects, where the DTM and Galerkin method are introduced for comparison.

4.1.1 Natural frequency

The DTM has been verified to be a highly accurate method in calculating free vibration problems ^[18-20].

The general expression of the DTM used in the present problem is

(34)

where N is the differential order, and Y(n)=y⁽ⁿ⁾(0)/n!.

The aim of the DTM is to construct a characteristic equation written as follows:

(35)

where B_ij is the element of B in the ith row and the jth column, and Y=(Y(3), Y(4), Y(5))^T.

To obtain a non-zero Y, the determinant of the coefficient matrix in Eq. (35) should equal zero. When (B)=0 is solved, each order of the natural frequency will be obtained. Obviously, the accuracy and efficiency depend on N.

If the exciting frequency equals the natural frequency, the pipe will resonate, and the vibration will surely be standout. If the parameters are selected as follows:

the results of the GFM at θ =0.8 are shown in Fig. 2, where ξ_max and η_max are separately used to denote the amplitudes or the maximum tangential and radial displacements hereafter.

As anticipated, there appear peaks for both ξ_max and η_max at the same exciting frequency (see Fig. 2). According to the resonance principle, such peaks correspond to the natural frequency. The detailed values of both the GFM and DTM are listed in Table 1.

As shown in Table 1, the first five orders of the natural frequency obtained by the GFM and the DTM are more and more close to each other when N increases. This not only means that the GFM can be regarded as an efficient tool in obtaining natural frequency, avoiding resonance, and improving the service life of pipes, but also reveals the disadvantage of the DTM that larger N is needed in calculating high order natural frequency, which is just the shortcoming of almost all numerical methods.

4.1.2 Displacement response

The Galerkin method is introduced to check the validity of the GFM in solving the steady state dynamic response, of which the details can be referred to Ref. [21]. The general expression of the Galerkin method used in the present problem is

(36)

where φ_n (θ) is the nth shape function, q_n (τ) is the corresponding time-dependent term, and N is the number of the shape function.

The aim of the Galerkin method is to obtain an equation written as follows:

(37)

where M is the mass matrix, G is the gyroscopic matrix, K is the structural stiffness matrix, F is the external force vector, and q is the time-dependent vector.

For a concentrated time harmonic excitation as shown in Eq. (31), if the parameters are selected as follows:

the maximum tangential displacement at θ =0.8 of the GFM and Galerkin method can be calculated. The results are listed in Table 2.

If N is large enough, the results of the Galerkin method are closer to those of the GFM as anticipated. According to Table 2, the results of the Galerkin method are nearly equal to those of the GFM when N=12, which confirms the validity of the present method in calculating the steady state dynamic response. To obtain an intuitive understanding, Fig. 3 shows the maximum displacements of the whole curved pipe obtained by the GFM and Galerkin method, where u=1, N=12, and the other parameters are the same as before.

From Fig. 3, we can see that, since the force is implemented in the middle of the pipe, both ξ_max and η_max of the curved pipe with both ends fixed are symmetric with respect to the exciting position (θ =0.5). The dynamic responses of the GFM and Galerkin method are quite close to each other, which further confirms the validity of the present method.

4.2 Effects of some key parameters on the steady state dynamic response

The steady state dynamic response of a semi-circular clamped-clamped curved pipe conveying fluid is studied in this subsection with external excitation implemented in its middle.

The effects of damping, added mass, and distribution of excitation on the steady state response are studied in detail. The parameters listed in Table 3 are fixed hereafter for further investigations unless there are particular illustrations.

4.2.1 Damping effects

The displacement response under the concentrated force as a function of θ is solved by the GFM in this subsection (see Fig. 4), where α ranges from 0.0 to 0.6 and β_l=β_t=0.

Both ξ_max and η_max decrease with the increase in α, and the results agree well with engineering practice due to that the damping absorbs energy and thus restrains the vibration. Figure 5 shows the comprehensive effects of α_t and α_l on ξ_max and η_max, respectively.

As shown in Fig. 5, the gradient weights of ξ_max and η_max tend obviously to α_l, which means that the longitudinal motion related damping α_l has a more pronounced effect than that pertinent to the transverse displacement characterized by α_t. Moreover, since the results without the consideration of damping are larger than those with the consideration of damping, the reliability design based on the neglect of damping will be partial to safety.

4.2.2 Added mass effects

By the same means aforementioned, Fig. 6 shows the displacement response as a function of θ under the concentrated force, where β₀ ranges from 0.0 to 0.6 and α_l=α_t=0.

As Fig. 6 shows, both ξ_max and η_max decrease with the increase in β₀. Furthermore, the decreasing degree seems larger when β₀ is comparatively small. Figure 7 shows the comprehensive effects of β_t and β_l on ξ_max and η_max, respectively.

As shown in Fig. 7, the gradient weights of ξ_max and η_max tend obviously to β_l, which means that the longitudinal motion related mass ratio β_l has a more pronounced effect than that pertinent to the transverse displacement characterized by β_t. Besides, whether β_l or β_t helps to weaken the vibration, the effect seems more obvious when they are comparatively little, e.g., β_l or β_t is less than 0.1. The similar conclusion as mentioned in the above subsection can be drawn, i.e., the reliability design based on the neglect of added mass will be partial to safety.

4.2.3 Excitation distribution effects

In general, the pipe excitation is not concentrated, but is distributed under most circumstances such as forces originating from the intermediate supports or shock due to the contact with other parts. Therefore, the steady state dynamic response with the consideration of distribution definitely needs to be valued. Hence, a local uniformly distributed force is introduced in this paper to capture the main character.

Under this circumstance, the excitation can be written in the complex form as follows:

(38)

Then, the corresponding dimensionless form is

(39)

where c and d are dimensionless forms of the angle boundary. The angle between the midpoint of the acting force and the horizontally rightward axis is also denoted by a, i.e., a=(c+d)/2. It will be convenient if γ is used to denote the span angle of the interval, i.e., γ =d-c. Obviously, if the excitation is uniformly implemented on the whole pipe, c and d will equal 0 and 1, respectively, and the force is implemented intensively on both ends from the dimensionless form view. Therefore, the response will equal zero.

According to Eq. (12) and after elementary transformation, y(θ) becomes

(40)

where

(41)

Particularly, if γ tends to zero infinitely, Eq. (39) will become

(42)

which is identical to Eq. (32). It reveals that the concentrated force is just the exception of the distributed one when γ tends to zero. The conclusion is not only suitable for uniformly distributed force, but also suitable for arbitrarily but conclusively distributed force according to the differential principle.

An approximate model named the local uniformly distributed force is considered in this subsection. For comparison, the equivalent force is set to be the same value f₀ and the concentrated force is included in the investigation. To obtain an intuitive understanding, Fig. 8 shows the maximum displacement response of the whole pipe as a function of θ, where

As anticipated, both ξ_max and η_max of the clamped-clamped curved pipe are symmetric with respect to the exciting position (see Fig. 8), which has nothing to do with the opening angle. In addition, there appears a peak for η_max at the exciting position, while things are different in the tangential direction since there appears sinking near the exciting position. It will be of significance to calculate the maximum displacement difference under the concentrated force and the local uniformly distributed force to study the effect of the force distribution on the curved pipe. Due to the symmetry, it is convenient to use the former half of the pipe, i.e., θ ∈ [0, 0.5], to continue the investigation. The results in the other half interval will be obtained immediately by the symmetry with respect to θ =0.5.

From Fig. 9, we can see that there are no values below zero, which means that ξ_max and η_max under uniformly distributed force are always less than those under concentrated one. The maximum displacement under local uniformly distributed force decreases with the increase in γ, which means that the more dispersive the excitation is, the more placid the vibration will become. Particularly, if γ reaches one, there will be no vibration, just as Eq. (39) shows. If γ is less than a threshold, the amplitude along both the tangential direction and the radial direction will be nearly equal to that under the concentrated force, and the distributed force can be treated to be concentrated, where the threshold depends on the design requirement. Such treatment during the designing process is partial to the safety from engineering perspective of view. Hence, to improve the service life of the pipe, the external excitation should be dispersed as far as possible, but too excessive dispersion may aggravate the complexity of the piping system. This belongs to an optimization problem, which leaves a subject worthy of studying in the future.

4.2.4 Flowing velocity of the inner fluid

As mentioned in almost all literatures, the flowing velocity plays a great role in the dynamics of curved pipe conveying fluid. Hence, a remarkable thing concerning it when an external excitation is implemented on the curved pipe needs to be emphasized. If other parameters are the same as those listed in Table 3, and the curved pipe is subjected to a concentrated force, Fig. 10 demonstrates the maximum displacements of the pipe as a function of θ under different flowing velocities, where β_l, β_t, α_l, and α_t are neglected.

As Fig. 10 shows, whether the flowing velocity equals zero or not, there are three and two peaks along the pipe in the radial and tangential directions, respectively. For η_max, one of three peaks appears at the exciting position, while things are different in the tangential direction, if the flowing velocity is zero, the pipe can be treated as a curved slender beam, and ξ_max equals zero at the exciting position, which absolutely has the same effect when the mass ratio is 0 according to Eq. (11). However, if the flowing velocity is nonzero, there is a slight sinking near the exciting position.

4.2.5 Steady combined force

Obviously, previous investigations are all based on the existence of steady combined force. As a matter of fact, according to the modified inextensible theory, steady combined force should be involved if both ends of the curved pipe are supported, and the results will be more reliable. Figure 11 demonstrates the displacement response of the curved pipe with and without Π₀, where a concentrated time harmonic force is considered, and β_l, β_t, α_l, and α_t are absent or neglected.

As Fig. 11 shows, whether Π₀ is considered or not leads to different steady state dynamic responses of the curved pipe, and the difference is obvious under the given parameters listed in Table 3. Besides, the first five orders natural frequency without Π₀ are 4.07, 9.33, 17.38, 27.50, and 38.90 by the GFM, respectively, which are all slightly less than those listed in Table 1, while agree well with those in Ref. [9]. This further confirms the validity of the present method. Of course, all results are not reliable according to Ref. [8].

5 Discussion 5.1 Forced vibration of a curved pipe with both ends supported under linear motion constraints

Motion constraints often exist in reality, especially for a pipe buried underground. Consider the forced vibration problem of a curved pipe conveying fluid with both ends supported under linear motion constraints, if the constraints can be characterized by linear spring force, with the introduction of steady combined force, the present problem can be written as follows:

(43)

where k=KR⁴/(EIθ_op²) is the dimensionless anti-shifting stiffness of the linear spring, and other parameters have the same meanings with Eq. (2).

For a time harmonic excitation, the time-dependent term can be separated out. From Eqs. (13) and (18), with the Laplace and inverse Laplace transform method, we can obtain the explicit form of G(θ, θ₀). The only difference between G(θ, θ₀) and Eq. (18) is the change of λ₀ and λ₁, i.e.,

(44)

(45)

With the same manner mentioned in Section 3, the steady state dynamic response can be obtained subsequently.

5.2 Forced vibration of a curved pipe with both ends supported under nonlinear motion constraints

For the forced vibration problem of a curved pipe conveying fluid with both ends supported under nonlinear motion constraints, if the constraints can be characterized by cubic spring force ^[12], with the introduction of steady combined force, the present problem can be expressed as follows:

(46)

where k=KR⁶/(EIθ_op⁴) is the dimensionless anti-shifting stiffness of the cubic spring, and other parameters have the same meanings with Eq. (2).

The variable separation technology is hard to be performed since the time parameter is not able to be easily separated, and even though it is separated. The Laplace transform is unable to be performed, let alone obtain the analytical Green function.

To the best of the authors' knowledge, the GFM has not been used to solve nonlinear problems. Numerical methods such as the DQM ^[12] and the combination of the Galerkin method and the generalized-α method are often used to solve such problems ^[22]. However, the results of the DQM are highly restricted by the distance between the adjacent points at the two boundary ends, division of nodes, and weighting coefficients, and there often exist difficulties for the Galerkin method in defining the shape functions when they are adopted to solve other kinds of problems.

6 Conclusions

The forced vibration of curved pipe conveying fluid is investigated by the GFM. Three supporting types, i.e., clamped-clamped, clamped-pinned, and pinned-pinned, are considered, and the corresponding steady state responses are given in closed form. With given parameters, the effects of some other key parameters in the governing equation on the dynamic response are studied in detail, and some notable conclusions need to be emphasized.

Compared with other numerical methods, the GFM has unique advantages not only in calculating natural frequency but also in obtaining steady state response. It is an efficient and accurate method for solving problems with external disturbance. From this perspective, the hypothesis that the GFM can be utilized to solve nonlinear problems is theoretically tenable, but the Laplace transform is no longer suitable to obtain the Green function, which leaves an interesting topic in the future studies.

Acknowledgements The authors want to thank the anonymous referees for their valuable suggestions and comments.