Nonlinear analysis of L-shaped pipe conveying fluid with the aid of absolute nodal coordinate formulation

By adopting the absolute nodal coordinate formulation, a novel and general nonlinear theoretical model, which can be applied to solve the dynamics of combined straight-curved fluid-conveying pipes with arbitrary initially configurations and any boundary conditions, is developed in the current study. Based on this established model, the nonlinear behaviors of a cantilevered L-shaped pipe conveying fluid with and without base excitations are systematically investigated. Before starting the research, the developed theoretical model is verified by performing three validation examples. Then, with the aid of this model, the static deformations, linear stability and nonlinear self-excited vibrations of the L-shaped pipe without the base excitation are determined. It is found that the cantilevered L-shaped pipe suffers from the static deformations when the flow velocity is subcritical, and will undergo the limit-cycle motions as the flow velocity exceeds the critical value. Subsequently, the nonlinear forced vibrations of the pipe with a base excitation are explored. It is indicated that period-n, quasi-periodic and chaotic responses can be detected for the L-shaped pipe, which has a strong relationship with the flow velocity, excitation amplitude and frequency.


Introduction
The system of pipes conveying fluid, as a one of the typical and simplest fluidstructure interaction system, always appears in various engineering fields, including the nuclear industry, marine oil extraction, aerospace engineering, and so on. However, due to the properties, working environment of the fluid-conveying pipe, it may suffer from the flow-induced vibrations or the forced vibrations under the action of the internal flow, external flow or external excitation. And these vibrations may cause catastrophic damage, so it is quite necessary to understand the mechanism of the pipes conveying fluid. In addition, the system of the fluid-conveying pipe can display rich dynamical behaviors and has become a new paradigm in the field of dynamics, which has been pointed out by Païdoussis [1]. Thus, due to these facts, the literature concerned with the dynamics of pipes conveying fluid has emerged in the past few decades [2][3][4][5][6][7][8][9].
The existing literature in this topic was mainly concerned with fluid-structure interaction of the straight or curved pipes conveying fluid. To predict the stability of the straight fluid-conveying pipes, the linear theoretical model was first developed. It was found that the cantilevered pipes would lose stability by flutter [10], while the buckling instability would occur in the system of the fluid-conveying pipes with both ends supported [11,12]. Because of this found, the researchers hope to further determine the dynamic responses of the pipe when the flow velocity become sufficiently high.
Inspired by this, the famous nonlinear theoretical model for the pipes conveying fluid, including the cantilevered pipes and pipes supported at both ends, was established in the study of Semler et al. [13], with the aid of the extended Hamilton's principle. For a long time, this theoretical model has been widely used by researchers to study the dynamics of the fluid-conveying pipe system. Unfortunately, this pipe model could only deal with the situation when the deformation of the pipe was considered to be small. To solve this problem, several large-deformation-based theoretical models were developed based on the absolute nodal coordinate formulation (ANCF) [14,15] or the geometrically accurate beam model [16,17]. As another kind of pipe commonly used in engineering, the curved fluid-conveying pipe, has also been received extensive attention from scholars [18][19][20][21][22][23][24][25][26][27]. In these studies, three different theories, including the conventional inextensible theory, the extensible theory and the modified inextensible theory, were mainly employed to develop the different theoretical models that could predict the stability and dynamics of a semi-circular pipe conveying fluid with both ends supported. However, it should be pointed out that these models can only simulate the pipe with an initially circular configuration, and are not suitable for the pipe with arbitrary initially configurations. This kind of pipe are also common in engineering. As a consequence, a few theoretical models, which could deal with the pipes with arbitrary shapes, were proposed [28][29][30][31][32][33]. For instance, based on the Hamilton's extended principle, a nonlinear theoretical model was proposed by Sinir [28] to investigate the nonlinear dynamics of a slightly curved fluid-conveying pipe with both ends supported.
It was found that the periodic and chaotic motions could be observed in this considered pipe system. To explore the three-dimensional dynamics of the curved fluid-conveying pipes, Łuczko and Czerwiński [29] developed a model, which is general and well applicable to analysis of a wide variety of systems. In addition, this proposed model could handle various boundary conditions easily. In quite recently, the statics and dynamics of the slightly curved cantilevered fluid-conveying pipes with four different initial shapes were first explored in the study of Zhou et al. [30] by employing the absolute nodal coordinate formulation. Some interesting and sometimes unexpected results were found based on their numerical calculations.
Although the above-mentioned literature covers a wide range of pipe models, including the straight-shaped pipes, circular-shaped pipes and even the pipes with special or arbitrary configurations, they all appeared singly and the combination of them was not taken into account. In fact, due to some site restrictions or special requirements, the straight-curved combination pipes may often be used in engineering, such as L-, U-, Z-, J-shaped pipes, etc. Thus, it is worth to explore the dynamics of the straight-curved combination fluid-conveying pipe. Indeed, a few researchers have studied this kind of pipe [34][35][36][37][38][39][40][41][42][43]. In 1990, a linear analytical model that include the Poisson coupling was proposed by Lesmez et al. [34] to perform the modal analysis of vibrations in liquid-filled piping system. Two examples, including single pipe bend and piping system with U-bend, were given to verify this proposed model. The dynamic stiffness method of the wave approach was employed by Koo and Yoo [36] to determine the natural frequencies, frequency response functions, and the stability of the Korea Advanced Liquid Metal Reactor (KALIMER) IHTS hot leg piping system. A 3D straight-curved combination pipe conveying fluid was taken into account in the study of Dai et al. [38], who mainly explored the influence of the internal flow velocity on the natural frequencies of the considered pipe system with the aid of transfer matrix method. In addition, it was found that the steady combined force could have a great impact on the vibration characteristics of the curved-shaped pipe. As pointed out by Wen et al. [40], the nonlinear force caused by the deformation of the straight pipe segment, static deformation and geometrical non-linearity of the pipe could have a considerable influence on the dynamics of the straight-curved combination pipe. Unfortunately, this effect was not taken into account in the study of Koo and Yoo [36], and only the static axial force caused by static deformation of the pipe was contained in the Dai et al.'s model [38]. Thus, a modified four segmental kinetic theoretical model was established by Wen et al. [40] to explore the nonlinear static deformation and the linear stability of the straight-curved pipe conveying fluid. Their numerical results indicated that the effect of the static deformation of the pipe on the natural frequencies of the pinned-pinned pipe or the pinned-sliding bearing-pinned pipe was pronounced, while for pinned-pinned pipe, this effect could be ignored. Based on the active learning Kriging model, Zhao et al. [41] first investigated the resonance failure of the straight-curved combination pipe conveying fluid and a failure performance function was built. The finite volume method was applied by Guo et al. [42] to study fluid-induced vibrations of the Z-shaped pipe with different supports and the effects of the supports on the vibration amplitude of the pipeline. The corresponding results demonstrated that the elastic support could effectively reduce the vibration amplitude and was also relatively safe. More recently, the vibration and in-plane wave propagation analysis of a L-shaped fluid-conveying pipe with multiple supports was performed in the study of Wu et al. [43] with the help of impedance synthesis method. This method had been proved by the corresponding experiments. It was noted that the periodic supports could effectively suppress the vibration level of the considered pipe system for a given frequency window.
In this paper, a novel and general theoretical model is proposed to explore the static deformations, linear stability, nonlinear self-excited vibrations and nonlinear forced vibrations of the straight-curved combination fluid-conveying pipe with the aid of the absolute nodal coordinate formulation. It should be pointed out that the model developed in the present work has four advantages compared with those models mentioned in the Ref. [31][32][33][34][35][36][37][38][39][40]: (i) it can determine the extremely large-amplitude vibrations of the soft straight-curved combination pipes conveying fluid; (ii) it can be applied to the straight-curved combination pipes with arbitrary initially shapes, such as L-, Z-, U-, J-shaped pipes, etc.; (iii) it can be applied to any boundary conditions, such as pinned-pinned, clamped-free, pinned-pinned-free and so on; (iv) it can handle not only the self-excited vibrations but also the forced oscillations. However, the present developed model can only analyze the planar vibration of the pipe, which will be improved in the future. Considering the length of this paper, only the system of cantilevered L-shaped fluid-conveying pipe is considered in this present work since the L-shaped pipe is most especially applied in engineering. The length of the straight pipe segment near the clamped end is L1, and the length near the free end is L2. As for the curved pipe segment, it is a 1/4 arc of radius R. It should be mentioned that, in this paper, the three pipe segments are assumed to be equal in length, which means L1=L2=πR/2=L/3, where L is the length of the whole L-shaped pipe. Moreover, the mass per unit length of the L-shaped pipe is m and the flexural rigidity is EI. The fluid flowing in the pipe has mass per unit length M and mean velocity

Fig. 1
Schematic of cantilevered L-shaped fluid-conveying pipe subjected to a base excitation.
The absolute nodal coordinate formulation (ANCF) is employed here to establish a nonlinear theoretical model for the considered L-shaped fluid-conveying pipe. The Lshaped pipe in this paper is considered to be slender and can only vibrate in the X-Y plane, hence, the 2-node planar curved ANCF elements [44] are chosen to discretize this pipe system. Since we have chosen the ANCF to deal with this problem, correspondingly, the extended Lagrange equation introduced by Irschik and Holl [45] is required to derive the nonlinear governing equations of this L-shaped pipe system.
And this equation can be written as follows [45]: where T denotes the total kinetic energy of the system, q and q & are respectively the generalized coordinate vector and velocity vector, and Q represents the vector of After determining all the terms shown in Eq. (1) and make a series of operations, we can obtain the nonlinear dynamic equation of the pipe element subjected to a base excitation, which can be found in Eq. (2). For the sake of brevity, we put the detailed process of formula derivation in Appendix A, and the interested readers can refer to it.
In addition, in order to obtain the above expressions, we have introduced the following quantities: According to the concept of the traditional finite element method, these matrices and vector for those pipe elements can be assembled into the corresponding global matrices and vector, and then we can have the nonlinear governing equation of the whole L-shaped pipe system Then, based on this equation and with the aid of fourth-order Runge-Kutta integration algorithm, the nonlinear dynamic behaviors of the L-shaped pipe subjected to base excitation can be easily determined.
In addition to the nonlinear dynamic responses, the static deformations of the Lshaped pipe under the action of the internal flow also need to be determined since this considered pipe contains the curved pipe segments. According to the suggestion in Ref. [30], to determine the static deformation, we can divide the generalized coordinate vector e into two parts: a static part and a perturbation about the static part, leading to the following expression where es is the generalized coordinate vector of the static equilibrium configuration, and Δe denotes the perturbation about the static equilibrium position. By substituting Eqs. (6) into Eqs. (5), and deleting all the time-dependent terms, the static equilibrium equation of the L-shaped pipe system can be obtained Based on the equilibrium equation shown in Eq. (7), we can determine the static deformations of the L-shaped pipe with the help of Newton-Raphson method.

Results of the L-shaped pipe without the base excitation
In this section, the cantilevered L-shaped pipe without the base excitation will be investigated first, since the results of such a pipe system are rarely reported in the existing literature. In addition, the nonlinear mechanism of forced vibrations of the Lshaped pipe with a base excitation can be better understood based on the results shown in this section. Unless otherwise stated, several key system parameters are chosen to be: taken into account in this section, the amplitude of the excitation is set to be zero, i.e.
d0=0. Moreover, it should be pointed out that, 12 finite ANCF pipe elements will be employed in the current work to discrete the considered L-shaped fluid-conveying pipe.
The Appendix B will demonstrate that 12 elements are sufficient to predict the nonlinear responses of the pipe system under consideration.

Model validation
Before embarking some results of the cantilevered L-shaped pipe conveying fluid, two validation examples will be given first in this subsection to demonstrate the reliability of the proposed theoretical pipe model in simulating the dynamics of the Lshaped pipe without the base excitation.
The first validation example is to reproduce the natural frequencies of an empty cantilevered L-shaped pipe, which have been reported in researches of Jong [46] and Wu et al. [43]. For the convenience of comparison, the system parameters utilized in this example are selected to be the same as those applied in Refs. [46] and [43]: the length of the straight pipe segments is L1=L2=0.9m, the radius of curvature of the curved pipe segment is R=0.127m, Young's modulus is E=210Gpa, mass density of the pipe is ρp=7800kg/m 3 , outer diameter of the pipe cross section is ro=0.1m, inner diameter of the pipe cross section is ri=0.09m. Based on the present ANCF model and with these parameters, the first four natural frequencies of the empty L-shaped pipe are obtained, which are summarized in given. By comparing these results, it is found that the results of the present model are larger than those of other models, but the error is acceptable, within 10%. This can be understood since the Poisson effect is taken into account in these existing models but not in the present ANCF model. Besides, the first four vibration modes of the considered L-shaped pipe are plotted in Fig. 2 to further verify the present ANCF model.
It is obvious that the mode shapes shown in Fig. 2 are consistent with those reported in Ref. [43].  Before leaving this subsection, it should be pointed out that the present ANCF model is superior to the simulation model in two aspects. The first is the calculation time. To determine the static deformation of pipe at one flow velocity, the calculation time required by the present ANCF model in this paper is 0.035835 seconds, while that required by the simulation model is about 8 hours. In addition, the simulation model takes up ten cores of CPU when calculating, while the ANCF model only takes up one.
The second advantage of the present ANCF model is that it can predict the extremely large static deformations and the nonlinear dynamic responses of the considered Lshaped fluid-conveying pipe, but the simulation model is hard to realize. This is due to that when flutter instability occurs in the pipe system, the deformation of the pipe is relatively large, and at this time, the grid distortion problem is prone to appear in the simulation model, which leads to the divergence of the simulation results. To solve the grid distortion problem, we can only increase the number of grids, however, this will greatly increase the simulation time. Thus, based on the above considerations, we believe that the present ANCF model has more advantages in dealing with the problem of the pipes conveying fluid.

Linear stability analysis around the static equilibrium configuration
As suggested by the previous study [30], the linear stability analysis of the fluidconveying pipe should be performed around the corresponding static equilibrium configurations in the case of the considered pipe initially curved. Thus, the linear stability analysis in this subsection will be performed around the static equilibrium configuration since the L-shaped pipe contains the curved pipe segment. To achieve this goal, the static deformations of the L-shaped pipe must be determined first.
where KT represents the tangential stiffness matrix at the static equilibrium configuration and can be given by [30]  Then, based on Eq. (8), the eigenvalues of the considered L-shaped pipe without base excitation can be easily obtained. Thus, Fig. 5(b) is illustrated, where the dimensionless complex frequencies of the lowest four modes of the pipe as a function of the dimensionless flow velocity are given. In addition, it should be pointed out that the xcoordinate in Fig. 5(b) is the dimensionless frequency of the pipe, while the ycoordinate denotes the dimensionless damping. It is well known that when the damping of the fluid-conveying pipe is negative, the flutter instability occurs, and when the damping is zero, the corresponding flow velocity is the critical flow velocity. Bering this in mind and by inspecting Fig. 5(b), it is easy to find that the flutter instability occurs in the second mode of the considered L-shaped pipe and the corresponding critical flow velocity is ucr=7.28.

Nonlinear dynamics
In this subsection, the self-excited vibrations of the L-shaped pipe without base excitation will be examined. To this end, the initial tip-end displacement of the pipe in the Y direction is assumed to be 0.001 and the dimensionless flow velocity gradually increases from 7 to 13, in the progress of numerical calculations. Then, based on Eq.
(5) and applying the fourth-order Runge-Kutta integration algorithm, the bifurcation diagram of the dimensionless tip-end displacements in X-direction of the cantilevered L-shaped pipe versus internal flow velocity is obtained, which can be found in Fig. 6.
From this figure, it is immediately seen that the flutter instability occurs in the system of cantilevered L-shaped pipe, and the critical flow velocity is ucr=7.28, which is identical to the results predicted by the linear stability analysis. When the flow velocity is below the critical value, the considered pipe will only suffer from the static deformation, which corresponds the single points in the bifurcation diagram. Once the flow velocity is beyond the critical value, the limit cycle oscillations take place, which leads to two points corresponding to one flow velocity in the bifurcation diagram. In order to further understand the nonlinear dynamic responses of the L-shaped pipe, the oscillating shapes of the pipe for two typical flow velocities, including u=7.1 and 12, are added in Fig. 6, where the initially curved shapes of the pipe are highlighted in red, the static equilibrium configurations are marked in black and the blue lines denote the oscillating shapes of the pipe. It is found that in the case of u=7.1, the pipe is stabilized at a static equilibrium position. However, when the flow velocity is beyond the critical flow velocity, such as u=12, the pipe will vibrate around the corresponding static equilibrium configuration rather than the initially curved shape.   Fig. 7. When the flow velocity is below the critical flow velocity (ucr=5.6), the results obtained by the two different models are almost the same, except for the difference in vibration amplitude when the excitation frequency is high, which can be found in Fig. 7(a). When the flow velocity is beyond the critical value, it can be clearly seen from Fig. 7(b) that the bifurcation trends obtained by the two different models are basically the same, but the vibration amplitudes are slightly different in the considered rang of excitation frequency. Based on the discussions in Ref. [30], this difference is easy to be understood. Thus, it can be concluded that the present ANCF model can also deal with problems of the fluidconveying pipe under base excitations.

The case of the L-shaped pipe conveying subcritical fluid
First, we will pay our attention to the nonlinear forced vibrations of the L-shaped pipe conveying subcritical fluid under the base excitation. In the case of u=5, the bifurcation diagrams of the dimensionless tip-end displacements in X-direction of the considered pipe for three different values of excitation amplitude, including d0=0.001, 0.01 and 0.05, are given in Fig. 8. When the excitation amplitude is equal to 0.001, three peaks can be observed in the bifurcation diagram shown in Fig. 8(a), although the vibration amplitude is quite small. Considering that the L-shaped pipe considered in this section is subjected to a base excitation, it is easy to think that these three peaks are caused by the resonance of the pipe. Indeed, the dimensionless excitation frequencies corresponding to these three peaks are 17.5, 30 and 78.5, which are consistent with the second-, third-, and fourth-order natural frequencies of the pipe in the case of u=5. This indicates that the second-, third-, and fourth-mode resonance take place in the considered pipe system under the action of the base excitation. The oscillating shapes of the pipe for these three resonance frequencies are further illustrated in Fig. 9, where the static equilibrium configuration is highlighted in red and the blue lines denote the oscillating shapes. It should be mentioned that the deformations of the pipe shown in In addition, if we observe carefully, it can be found that each excitation frequency considered in Fig. 9 corresponds to different oscillating shapes of the pipe, which indicates that the main modes participating in the vibration of the pipe are different for these three excitation frequencies.
The bifurcation diagram shown in Fig. 8 (b) is the result for d0=0.01, which is very similar to that for d0=0.001. Two similarities can be summarized as: (i) the pipe always undergoes the period-1 motions within the excitation frequency under consideration; (ii) there are also three peaks in the bifurcation diagram, and the corresponding frequencies are also equal to the second-, third-, and fourth-order natural frequencies of the pipe. Moreover, compared with Fig. 8(a), it is also noted that the amplitudes of the forced vibrations of the considered L-shaped pipe are increased with the increase of excitation amplitude.
When the excitation amplitude is increased to 0.05, the situation is different, which can be immediately seen in Fig. 8(c). From this bifurcation diagram, it seems that with the increase of the excitation frequency, the pipe undergoes the period-1 (0<ω<27.5), period-n (28<ω<64), chaotic (64.5<ω<74.5), period-n (75<ω<77) and period-1 (77.5<ω<100) motions in sequence. This is not entirely true, however. To further analyze these dynamic behaviors, we plotted Fig. 10, in which the time histories, phase portraits and power spectra diagrams of the pipe for several typical values of excitation frequency, including ω=17.5, 35, 70.5 and 76.5, are given. It is noted that the pipe undergoes the period-1 motions when the excitation frequency equals to 17.5 or 35, chaotic motion occurs in the case of ω=70.5, and period-2 motion can be detected in the considered pipe system for ω=76.5. In the following, based on extensive calculations, it is found that the pipe always undergoes the period-1 motions in the range of 28<ω<64 instead of the period-n motions as originally thought. This means the pipe actually undergoes the period-1 (0<ω<64), chaotic (64.5<ω<74.5), period-n (75<ω<77) and period-1 (77.5<ω<100) motions, successively. In addition, it is indicated that when the L-shaped pipe is subjected to a base excitation with large excitation amplitude, it may display rich dynamic responses. Then, another subcritical flow velocity u=7 will be taken into account and the corresponding bifurcation diagrams are displayed in Fig. 11. When the excitation amplitude is relatively small, such as d0=0.001 and 0.01, the pipe always experiences the limit-cycle motions in the range of 0<ω<100, which is consistent with the results shown in Figs By further comparing Fig. 11 and Fig. 8, it can be concluded that when the flow velocity is below the critical value, increasing the flow velocity can make the dynamic behaviors of the considered L-shaped pipe under the same base excitation more simple.  Fig. 13 is given, where the time history, phase portrait, power spectra diagram and Poincare map of the oscillation for ω=41.5 are illustrated. When the excitation amplitude is increased to 0.05, the main dynamic behaviors of the considered pipe have changed from quasiperiodic motions to periodic, quasi-periodic and chaotic motions, which can be observed in Fig. 12(c). In other words, the strong base excitation can stimulate complex dynamic responses of the L-shaped pipe conveying the supercritical fluid.

Conclusions
In

Acknowledgement
The financial support of the National Natural Science Foundation of China (Nos. 11972167, 11672115 and 11622216) and Alexander von Humboldt Foundation to this work is gratefully acknowledged.

Declaration of interest statement
The authors declare that they have no conflict of interest. All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. This article does not contain any studies with animals performed by any of the authors. Informed consent was obtained from all individual participants included in the study.

Appendix A
In this Appendix, the detailed derivations of Eqs. (2) and (3)  where S and q, respectively, denote the shape function and the nodal coordinate vector of the ANCF pipe element, which have been defined in Ref. [30]. Then, according to Eq. (A.1), the absolute velocity vector of the pipe element can be easily defined: According to the discussions in Ref. [30], the absolute velocity vector of the fluid element can be expressed as follows: Where f is the longitudinal deformation gradient and τ denotes the tangential unit vector along the deformed pipe axis. Their expression can be given as follows [30]:   T T  T  T  TT  F  F  F  TT  TT   T  T where a is a scalar and can be defined as a(x=0)=-1 and a(x=l)=1.
Finally, the vector of generalized forces of the system, Q, needs to be determined.
Since the influence of gravity and damping are not taken into account in this article, we only need to determine the generalized elastic forces vector of the system. Recalling that the L-shaped pipe in this paper is considered to be slender, and the Euler-Bernoulli beam theory is adopted. Due to this fact, the potential energy of the pipe element can be written as where κ0 denotes the initial curvature of the pipe element. For the straight pipe segments κ0=0, and for the curved pipe segment κ0≠0. Moreover, ε and κ are, respectively, the longitudinal strain and geometrical curvature of the deformed pipe element, which can be defined as follows [30]: According to Eq. (A.8), the vector of the generalized elastic forces is defined by

Appendix B
It is well known that if the elements used to discretize the pipe is not enough, the numerical results will not converge, and if the number of elements is too large, the calculation cost will increase. Due to this fact, it is necessary to determine a suitable number of pipe elements. To this end, the static equilibrium configurations of the Lshaped pipe for u=8 with four different numbers of pipe elements, including 6,9,12 and 15, are displayed in Fig. B1(a). From this figure, it is found that the results for 12 and 15 pipe elements are almost the same, indicating that 12 pipe elements are sufficient for predicting the static equilibrium configurations of the L-shaped pipe. Furthermore, Fig. B1(b) shows the bifurcation diagrams of dimensionless tip-end displacements in X-direction of the pipe without the base excitation for different numbers of the pipe elements. Again, it is easy to find that the results of 12 pipe elements are almost consistent with those of 15 pipe elements. According to these two figures, therefore, it is believed that 12 pipe elements are sufficient to predict the nonlinear statics and dynamics of the considered L-shaped pipe conveying fluid.
(a) (b)  Schematic of cantilevered L-shaped uid-conveying pipe subjected to a base excitation.   The static deformations of the cantilevered L-shaped uid-conveying pipe for two different dimensionless ow velocities: (a) u=1 and (b) u=3, obtained by the present ANCF model and the simulation model.  Bifurcation diagram of the dimensionless tip-end displacements in X-direction of the cantilevered Lshaped pipe versus internal ow velocity.

Figure 13
Dynamic responses of the cantilevered L-shaped pipe for ω=41.5 with d0=0.01 and u=8; (a) the time history, (b) phase portrait, (c) power spectra diagram, and (d) Poincare map.