Effects of Material Characteristics on Nonlinear Dynamics of Viscoelastic Axially Functionally Graded Material Pipe Conveying Pulsating Fluid
https://doi.org/10.1007/s11804023003288

Abstract
The nonlinear dynamic behaviors of viscoelastic axially functionally graded material (AFG) pipes conveying pulsating internal flow are very complex. And the dynamic behavior will induce the failure of the pipes, and research of vibration and stability of pipes becomes a major concern. Considering that the elastic modulus, density, and coefficient of viscoelastic damping of the pipe material vary along the axial direction, the transverse vibration equation of the viscoelastic AFG pipe conveying pulsating fluid is established based on the EulerBernoulli beam theory. The generalized integral transform technique (GITT) is used to transform the governing fourthorder partial differential equation into a nonlinear system of fourthorder ordinary differential equations in time. The time domain diagram, phase portraits, Poincaré map and power spectra diagram at different dimensionless pulsation frequencies, are discussed in detail, showing the characteristics of chaotic, periodic, and quasiperiodic motion. The results show that the distributions of the elastic modulus, density, and coefficient of viscoelastic damping have significant effects on the nonlinear dynamic behavior of the viscoelastic AFG pipes. With the increase of the material property coefficient k, the transition between chaotic, periodic, and quasiperiodic motion occurs, especially in the highfrequency region of the flow pulsation.Article Highlights• The transverse vibration equation is established for the viscoelastic axially functionally graded material (AFG) pipe conveying pulsating fluid.• The generalized integral transform technique is developed to analyze the nonlinear dynamics of AFG pipes.• The effects of the elastic modulus, density, and coefficient of viscoelastic damping on nonlinear dynamics of AFG pipes are discussed. 
1 Introduction
Pipe conveying fluid is an important structure in a number of engineering fields, and is widely used in heat exchangers, nuclear industry, petroleum transportation, aviation and other industries. Due to the excitation of the external and internal fluids, pipe structural properties, and other factors, pipes conveying fluid will have complex dynamic behaviors (Jin, 1997; Jin and Song, 2005; Wang and Chen, 2019). The vibration and stability of pipes becomes a major concern over the past decades. Paidoussis and collaborators (Paidoussis and Moon, 1988; Paidoussis et al., 1989; Paidoussis et al., 1992; Paidoussis and Semler, 1993) investigated the nonlinear dynamic behavior of cantilever pipes through experiments and numerical simulations, the chaotic motion behavior through phase portraits, bifurcation diagrams and Poincaré maps, and found that chaos occurred through perioddoubling bifurcation. Tang and Dowell (1988) studied the chaotic phenomenon of cantilevered pipes using experimental method and numerical technique, and the results showed that chaotic motion would occur when the flow rate exceeded a critical value. The nonlinear dynamic behavior of pipe conveying pulsating fluid with high average velocity and spring constraint were investigated by Wang and his collaborators (Wang, 2009; Wang et al., 2017), and they found that the average velocity and spring constraints have significant effects on the nonlinear dynamic response of the system. Ni et al. (2014) studied the nonlinear dynamic behavior of two coupled pipe system conveying pulsating fluids, and the results showed that the connection stiffness had a significant effect on the dynamics of the coupled pipe system. For some parameters, the motion types of the two pipes may be out of synchronization. Oyelade and Oyediran (2020) studied the nonlinear dynamics of pipes under thermal loads, and effects of the initial curvature, temperature, axial force, pressure, longitudinal and transverse strain were took into account. The nonlinear analysis results showed that the amplitude of pipe motion was significantly increased due to the influence of thermal load.
Functional gradient materials (FGM) are new heterogeneous composite materials made of two or more homogeneous materials, and it has been widely used in various fields due to their excellent properties. By varying the volume fraction of each homogeneous material, the properties of AFG pipes vary continuously along the pipe from one end to the other. Comparing with the traditional composite pipe (Nikbakht et al., 2019; Wang and Soares, 2021; Fu et al., 2022a; Fu et al., 2022b; Wang et al., 2022), the AFG pipe can eliminate the interface effects of traditional uniform composites. For instance, the AFG pipe can be used in optimized design of control lines in subsea engineering and aerospace engineering by varying the properties along the axial direction (Gupta and Talha, 2015). The dynamic behaviour of functionally gradient pipes become a major concern over the past decade. However, most of the literature mainly focuses on the linear dynamics behavior of the AFG pipes. An and Su (2017) studied the linear vibration characteristics of AFG pipes with simple support, and analyzed the influence of the material elastic modulus, density distribution and flow rate on the free vibration characteristics of AFG pipes. Zhou et al. (2018)discussed the linear dynamics of cantilever AFG pipes, and analyzed the influence of the elastic modulus and density distribution on the critical flow rate. Heshmati (2020) analyzed the influence of different eccentricity and power rate exponents on the critical flow rate of the radially functional gradient material pipes with eccentric defects. Deng et al. (2017) studied the dynamic characteristics of multispan viscoelastic FGM pipes by the dynamic stiffness method, and analyzed the influence of the volume fraction index, flow velocity, internal pressure and damping on pipe stability and frequency effects. Recently, there are few studies discussing the nonlinear dynamics of FGM pipes. Lu et al. (2020) studied the fatigue life of AFG pipes under the conditions of primary resonance and 1:3 internal resonance. Tang and Yang (2018) applied the homotopy analysis method to study the postbuckling behavior and nonlinear vibration characteristics of radially functional gradient pipes with steady internal flow. The results showed that the statics and dynamics of pipe can be significantly changed by the material properties. Khodabakhsh et al. (2020) used the Timoshenko beam model to study the postbuckling behavior and nonlinear vibration characteristics of radially FGM pipes considering shear deformation and moment of inertia, and discussed the effects of moment of inertia and shear deformation on buckling behavior, critical flow rate, nonlinear frequency and dimensionless amplitude. Shafiei and She (2018) predicted the vibration characteristics of a twodimensional FGM based on the higherorder theory. The generalized differential quadrature method (GDQM) was employed in his studies, and the influences of nonlocal parameters, strain gradient parameters, temperature and material variations on the vibration characteristics of FGM pipes under different boundary conditions are discussed. EbrahimiMamaghani et al. (2020) investigated the thermomechanical vibration of AFG pipe conveying fluid. Using Rayleigh beam theory and considering the linear and nonlinear stresstemperature relationship, the govern equation of the system was estabulished. The effects of the material gradient, power index, boundary conditions, moment inertia factor, temperature and boundary conditions on the dynamic structure of the system are discussed. Dai et al. (2019) investigated the thermoelastic vibration of AFG pipe conveying fluid considering temperature variation effects. The differential quadrature method was employed to solve the govern equations based on EulerBernoulli beam theory. The influence of the elasticity and thermoelasticity gradients on the material performance were discussed. Most recently, Tuo et al. (2022) investigated the linear stability of axially functionally graded pipe conveying fluid using the generalized integral transform technique (GITT) for four combinations of boundary conditions, and the results showed that the complex frequency, critical velocity of buckling instability and coupled flutter of the axially functionally graded pipe are significantly influenced by the elastic modulus gradient.
The present work aims to investigate the nonlinear dynamic behavior of AFG pipe conveying pulsating fluid. Based on the EulerBernoulli beam theory, the nonlinear vibration equation of an AFG pipe conveying pulsating flow with simply supported boundary condition was estabulished. Using the GITT technique (Cotta et al., 2019; Fu et al., 2019; Li et al., 2020; Tuo et al., 2022; Fu et al., 2022c), the governing nonlinear partial differential equations were transformed into a nonlinear system of coupled ordinary differential equations, which were solved numerically using the buildin function NDsolve of the software Mathematica. Furthermore, the GITT technique intrinsically keeps the property of the partial differential equations according to eigenvalues, which is different from the way adding auxiliary variable to construct the physicalpreserving scheme (Zhao et al., 2017; Shen et al., 2018; Jiang et al., 2022, 2023). With this model, The effects of elastic modulus, density and viscoelastic damping coefficient in the axial direction on the nonlinear dynamic behavior were studied. The bifurcation, phase portrait, and energy spectra diagrams were obtained to investigate the nonlinear dynamic behavior characteristics of the AFG pipe.
2 Theoretical model
Considering a viscoelastic AFG pipe conveying a pulsating fluid, as illustrated in Figure 1. The pipe has a length L, inner radius r_{o}, outer radius r_{1}, and crosssectional area A_{p}. The fluid has a mass per unit length m_{f} and an average velocity u (t). The elastic modulus E (x), coefficient of viscoelastic damping E^{∗} (x), and density of the AFG pipe are given by Shariati et al. (2020), Loghman et al. (2021), and Babilio (2014) :
$$ E(x)=E_0 f(x) $$ (1a) $$ E^*(x)=E_0^* f(x) $$ (1b) $$ \rho(x)=\rho_0 g(x) $$ (1c) $$ f(x)=1+\left(\alpha_{\mathrm{E}}1\right)\left(\frac{x}{L}\right)^k $$ (2a) $$ h(x)=1+\left(\alpha_{\mathrm{E}^*}1\right)\left(\frac{x}{L}\right)^k $$ (2b) $$ g(x)=1+\left(\alpha_\rho1\right)\left(\frac{x}{L}\right)^k $$ (2c) where α_{E} = E_{L}E_{O} is the elastic modulus ratio, α_{E}_{*} = E_{L}^{∗}E_{O}^{*} the coefficient of the viscoelastic damping ratio, and αρ = ρ_{L}ρ_{O} the density ratio.E_{1}, E_{1}^{∗}, ρ_{1} and E_{O}, E_{O}^{*} and ρ_{O} represent the elastic modulus, coefficient of viscoelastic damping and density at the left and right ends of the pipe, respectively. Figure 2 shows the variation of the material properties along the axial under different exponents k when α_{E}=2.
The stressstrain relationship of the KelvinVoigt type of viscoelastic material is given by:
$$ \sigma=\left(E(x)+E^*(x) \frac{\partial}{\partial t}\right) \varepsilon $$ (3) Bending moment is expressed as:
$$ M=\left(E(x)+E^*(x) \frac{\partial}{\partial t}\right) I \frac{\partial^2 w}{\partial x^2} $$ (4) where w (x, t) is the transversal deflection of the pipe.
Shear force is given by:
$$ S=\frac{\partial M}{\partial x}=\frac{\partial}{\partial x}\left[\left(E(x)+E^*(x) \frac{\partial}{\partial t}\right) I \frac{\partial^2 w}{\partial x^2}\right] $$ (5) The lateral vibration equation of an AFG pipe is written as (Paidoussis and Issid, 1974; Paidoussis, 1987; Paidoussis and Li, 1993),
$$ \begin{array}{r} \frac{\partial^2}{\partial x^2}\left[\left(E(x)+E^*(x) \frac{\partial}{\partial t}\right) I \frac{\partial^2 w}{\partial x^2}\right]+\frac{\partial}{\partial x}\left[\left(TA_f p_1\right) \frac{\partial w}{\partial x}\right]= \\ m_f\left(\frac{\partial^2 w}{\partial t^2}+2 u \frac{\partial^2 w}{\partial x \partial t}+u^2 \frac{\partial^2 w}{\partial x^2}+\frac{\mathrm{d} u}{\mathrm{~d} t} \frac{\partial w}{\partial x}\right)+m_f(x) \frac{\partial^2 w}{\partial t^2} \end{array} $$ (6) where, T is the axial force, A_{f} is the crosssectional flow area and p_{1} is the fluid pressure.
Considering that the extension of the pipe due to bending deformation will produce additional axial force:
$$ \begin{aligned} TA_f p_1= & T_0A_f p_o(12 v \delta)+m_f \frac{\mathrm{d} u}{\mathrm{~d} t}(xL) \\ & +\left[E(x)+E^*(x) \frac{\partial}{\partial t}\right] \frac{A_p}{2 L} \int_0^L\left(\frac{\partial w}{\partial x}\right)^2 \mathrm{~d} x \end{aligned} $$ (7) where, T_{0} is static tension, and p_{0} is the pressure of internal flow at x = L.
Substituting Eq7 into the lateral vibration equation (Eq6) of the AFG pipe, we have:
$$ \begin{aligned} & \frac{\partial^2}{\partial x^2}\left[E(x) I \frac{\partial^2 w}{\partial x^2}+E^*(x) I \frac{\partial^3 w}{\partial x^2 \partial t}\right] \\ &+\left[m_{\mathrm{f}} u^2T_0+A_f p_0(12 v \delta)+m_{\mathrm{f}} \frac{\mathrm{d} u}{\mathrm{~d} t}(Lx)\right] \frac{\partial^2 w}{\partial x^2} \\ &\frac{\partial}{\partial x}\left[\left(E(x)+E^*(x) \frac{\partial}{\partial t}\right) \frac{A_p}{2 L} \int_0^L\left(\frac{\partial w}{\partial x}\right)^2 \mathrm{~d} x \frac{\partial w}{\partial x}\right] \\ &+2 m_{\mathrm{f}} u \frac{\partial^2 w}{\partial x \partial t}+\left[m_{\mathrm{f}}+\rho(x) A_{\mathrm{p}}\right] \frac{\partial^2 w}{\partial t^2}=0 \end{aligned} $$ (8) Neglecting the pressure p_{0} and static tension T_{0}, we have:
$$ \begin{aligned} & \frac{\partial^2}{\partial x^2}\left[E(x) I \frac{\partial^2 w}{\partial x^2}+E^*(x) I \frac{\partial^3 w}{\partial x^2 \partial t}\right] \\ &+\left[m_{\mathrm{f}} u^2+m_{\mathrm{f}} \frac{\mathrm{d} u}{\mathrm{~d} t}(Lx)\right] \frac{\partial^2 w}{\partial x^2} \\ &\frac{\partial}{\partial x}\left[\left(E(x)+E^*(x) \frac{\partial}{\partial t}\right) \frac{A_p}{2 L} \int_0^L\left(\frac{\partial w}{\partial x}\right)^2 \mathrm{~d} x \frac{\partial w}{\partial x}\right] \\ &+2 m_{\mathrm{f}} u \frac{\partial^2 w}{\partial x \partial t}+\left[m_{\mathrm{f}}+\rho(x) A_{\mathrm{p}}\right] \frac{\partial^2 w}{\partial t^2}=0 \end{aligned} $$ (9) The following dimensionless parameters are introduced:
$$ \begin{gathered} \xi=\frac{x}{L}, w^*=\frac{w}{L}, \tau=\left(\frac{E_o I}{m_f+\rho_o A_p}\right)^{1 / 2} \frac{t}{L^2}, \\ \alpha_0=\frac{E_0^*}{L^2} \sqrt{\frac{I}{E_0\left(m_f+\rho_0 A_p\right)}}, \\ Q=\frac{A_p L^2}{2 I}, \beta=\frac{m_{\mathrm{f}}}{m_{\mathrm{f}}+\rho_o A_p}, \\ \alpha(\xi)=\frac{\rho(x) A_p}{m_{\mathrm{f}}+\rho_o A_p}, U=u L\left(\frac{m_{\mathrm{f}}}{E_o I}\right)^{1 / 2} \end{gathered} $$ (10) The dimensionless governing equation is then obtained as (dropping the asterisks of w^{*} for simplicity):
$$ \begin{aligned} f(\xi) & w^{\prime \prime \prime \prime}+2 f^{\prime}(\xi) w^{\prime \prime \prime}+\left[f^{\prime \prime}(\xi)+U^2+\sqrt{\beta} \dot{U}(1\xi)\right] w^{\prime \prime} \\ & +\alpha_o h^{\prime \prime}(\xi) \dot{w}^{\prime \prime}+2 \alpha_o h^{\prime}(\xi) \dot{w}^{\prime \prime \prime}+\alpha_o h(\xi) \dot{w}^{\prime \prime \prime \prime} \\ & Q\left[f(\xi) w^{\prime \prime}+f^{\prime}(\xi) w^{\prime}\right] \int_0^1\left(w^{\prime}\right)^2 \mathrm{~d} \xi \\ & +2 \sqrt{\beta} U \dot{w}^{\prime}2 \alpha_o Q\left[h(\xi) w^{\prime \prime}+h^{\prime}(\xi) w^{\prime}\right] \int_0^1 w^{\prime} \dot{w}^{\prime} \mathrm{d} \xi \\ & +[\alpha(\xi)+\beta] \ddot{w}=0 \end{aligned} $$ (11) Here, the prime "'" denotes the spatial derivative in ξ and the dot "∙" denotes the time derivative inτ.
The internal fluid velocity of the AFG pipe is assumed to be sinusoidal and given in the following dimensionless form:
$$ U=U_0(1+\psi \sin (\omega \tau)) $$ (12) whereU_{0} is the average flow velocity, ψ is the pulsation amplitude, and ω is the dimensionless pulsation frequency.
The boundary conditions for simplysupported ends are:
$$ w(0, \tau)=w(1, \tau)=w^{\prime \prime}(0, \tau)=w^{\prime \prime}(1, \tau)=0 $$ (13) 3 Integral transform solution
Following the formalism of the generalized integral transform technique, the governing equation of the following auxiliary eigenvalue problem is employed (Cotta et al., 2019; Fu et al., 2019; Li et al., 2020) :
$$ \frac{\mathrm{d}^4 X_i(\xi)}{\mathrm{d} \xi^4}=\mu_i^4 X_i(\xi), 0<\xi<1 $$ (14) with simply supported (SS) boundary conditions:
$$ X_i(0)=X_i^{\prime \prime}(0)=X_i(1)=X_i^{\prime \prime}(1)=0 $$ (15) where the eigenfunctions X_{i}(ξ) and eigenvalues μ_{i} are given by:
$$ X_i(\xi)=\sqrt{2} \sin \left(\mu_i \xi\right), i=1, 2, 3 \cdots $$ (16) $$ \mu_i=i \pi, i=1, 2, 3 \cdots $$ (17) The eigenfunctions satisfy the following orthonormal property:
$$ \int_0^1 X_i(\xi) X_j(\xi) \mathrm{d} \xi=\delta_{i j}=\left\{\begin{array}{l} 1, i=j \\ 0, i \neq j \end{array}\right. $$ (18) The integral transform and inverse pairs are defined by:
$$ \bar{w}_i(\tau)=\int_0^1 w(\xi, \tau) X_i(\xi) \mathrm{d} \xi $$ (19a) $$ w_i(\xi, \tau)=\sum\limits_{i=1}^{\infty} X_i(\xi) \bar{w}_i(\tau) $$ (19b) Multiplying the governing equation (Eq11) by the eigenfunction X_{i}(ξ) and integrating in ξ ∈ [ 0, 1], the gonvern equation can be rewritten:
$$ \begin{aligned} & \sum\limits_{j=1}^{\infty}\left[L_{i j} \frac{\mathrm{d}^2 \bar{w}_j(\tau)}{\mathrm{d} t^2}\right]+\sum\limits_{j=1}^{\infty}\left[\alpha_o\left(A_{i j}^* \mu_j^4+C_{i j}\right)+2 U \sqrt{\beta} H_{i j}\right] \frac{\mathrm{d} \bar{w}_j(\tau)}{\mathrm{d} t} \\ & 2 \alpha_o Q \sum\limits_{j=1}^{\infty}\left[N_{i j} \bar{w}_j(\tau)\right] \sum\limits_{j=1}^{\infty} \sum\limits_{k=1}^{\infty}\left[G_{j k} \bar{w}_j(\tau) \frac{\mathrm{d} \bar{w}_k(\tau)}{\mathrm{d} t}\right] \\ & Q \sum\limits_{j=1}^{\infty}\left[F_{i j} \bar{w}_j(\tau)\right] \sum\limits_{j=1}^{\infty} \sum\limits_{k=1}^{\infty}\left[G_{j k} \bar{w}_j(\tau) \bar{w}_k(\tau)\right] \\ & +\sum\limits_{j=1}^{\infty}\left[\left(A_{i j} \mu_j^4+B_{i j}+U^2 D_{i j}+\sqrt{\beta} \dot{U} E_{i j}\right) \bar{w}_j(\tau)\right]=0 \end{aligned} $$ (20) where these coefficients defined by:
$$ A_{i j}=\int_0^1 f(\xi) X_i X_j \mathrm{~d} \xi $$ (21a) $$ A_{i j}^*=\int_0^1 h(\xi) X_i X_j \mathrm{~d} \xi $$ (21b) $$ B_{i j}=2 \int_0^1 f^{\prime}(\xi) X_i X_j^{\prime \prime \prime} \mathrm{d} \xi+\int_0^1 f^{\prime \prime}(\xi) X_i X_j^{\prime \prime} \mathrm{d} \xi $$ (21c) $$ C_{i j}=2 \int_0^1 h^{\prime}(\xi) X_i X_j^{\prime \prime \prime} \mathrm{d} \xi+\int_0^1 h^{\prime \prime}(\xi) X_i X_j^{\prime \prime} \mathrm{d} \xi $$ (21d) $$ D_{i j}=\int_0^1 X_i X_j^{\prime \prime} \mathrm{d} \xi $$ (21e) $$ E_{i j}=\int_0^1(1\xi) X_i X_j^{\prime \prime} \mathrm{d} \xi $$ (21f) $$ G_{i j}=\int_0^1 X_i^{\prime} X_j^{\prime} \mathrm{d} \xi $$ (21g) $$ F_{i j}=\int_0^1 f(\xi) X_i X_j^{\prime \prime} \mathrm{d} \xi+\int_0^1 f^{\prime}(\xi) X_i X_j^{\prime} \mathrm{d} \xi $$ (21h) $$ H_{i j}=\int_0^1 h^{\prime}(\xi) X_i X_j^{\prime} \mathrm{d} \xi $$ (21i) $$ N_{i j}=\int_0^1 h(\xi) X_i X_j^{\prime \prime} \mathrm{d} \xi+\int_0^1 h^{\prime}(\xi) X_i X_j^{\prime} \mathrm{d} \xi $$ (21j) $$ L_{i j}=\int_0^1[\beta+g(\xi)(1\beta)] X_i X_j \mathrm{~d} \xi $$ (21k) 4 Results and discussion
4.1 Convergence analysis and results verification
As no previous study on nonlinear vibration results of the AFG pipe is encountered in the literature, the hybrid analyticalnumerical solution is verified by analyzing the nonlinear vibration of the simply supported homogeneous pipe. Figure 3 shows the bifurcation diagram by the present study compared with that by Ni et al. (2014), for mass ratio β = 0.6, average flow velocityU_{0} = 4.5, viscoelastic damping coefficientα_{0} = 0.005, pulsation amplitude ψ = 0.4, and slenderness ratio parameter Q= 5 000. It can be seen that both the displacement amplitude and the bifurcation point are in good agreement with the results presented by Ni et al. (2014). It can be concluded that the proposed GITT solution for the nonlinear dynamics analysis of pipe was successfully verified.
Convergence analysis has been carried out to analyze the dynamic behavior of the simply supported AFG pipe under pulsating flow. Bifurcation diagrams of the transient displacement at the center of the pipe versus the dimensionless pulsation frequency are shown in Figure 4, with α _{E} = α_{E}∗ = α_{ρ} = 2, β= 0.6, U_{0} = 4.5, α_{0} = 0.005, ψ = 0.4, and Q = 5 000, calculated with NW= 2, 4, 6, and 8. As can be seen, when the dimensionless pulsation frequency of internal fluid is less than 40, there is no difference between the results with truncation orders NW=2, 4, 6 and 8. However, when the dimensionless pulsation frequency is greater than 40, there is the obvious difference between the bifurcation diagram with NW=2 and higher truncation orders (NW=4, 6 and 8). For truncation orders NW=4, 6, and 8, the variation of the bifurcation diagrams can be negleced. It can be concluded that the present hybrid numericalanalytical model (GITT mdoel) with NW=8 can predict the dynamic behavior of AFG pipes within the selected dimensionless frequency rangeω∈ [ 0, 90 ]. It can be seen from Figure 4 (d) that an AFG pipe (k = 1, α_{E} = α_{E}∗ = α_{ρ} = 2) has a complex dynamic behavior in the range of pulsation frequency from 0 to 90, presenting periodic, quasiperiodic, and even chaotic motions.
Figure 5 shows the phase portraits, Poincaré map, power spectra diagram, and time domain diagram for different dimensionless pulsation frequencies shown in Figure 4(d), which can reflect the characteristics of chaotic, quasiperiodic, and periodic motion. Within frequency range ω ∈ [ 0, 10 ], the pipe shows aperiodic (chaotic) motion in a large frequency range, and the power spectra of chaotic motion show a broadband phenomenon without obvious main peaks, as shown by Figures 5 (a) and(b). There is also a small region of periodic motion, in the range of ω ∈ [ 5, 6 ]. In the range of ω ∈ [10, 15 ], the chaotic motion becomes period1 motion through the perioddoubling bifurcation. From ω = 15 to 36.7, the pipe vibration is mainly manifested as period1 motion. The power spectrum of period1 motion is a smooth curve with obvious main peaks, shown by Figure 5 (c), but there is a perioddoubling motion near ω= 16. For ω ∈ [ 36.7, 52.6 ], the periodic motion is observed. For ω ∈ [ 52.6, 54.7 ] and[ 60.8, 63.6 ], the AFG pipe is mainly manifested as chaotic motion, but between the two ranges, periodic motion is observed. In the range of ω ∈ [ 63.6, 90 ], except for period3 motion in ω ∈ [ 70.2, 80.7 ], all the other frequencies induce quasicperiodic motion.The power spectrogram of quasicperiodic motion has a wide frequency phenomenon at the same time and an obvious main peak is shown in Figure 5e.
4.2 Influence of elastic modulus distribution on system dynamic behavior
Figures 6‒8 show the parametric analysis of the elastic modulus distribution on the dynamic response of the AFG pipe, with the homogeneous density and coefficient of viscoelastic damping (α_{E}∗ = α_{ρ} = 1). The elastic modulus ratio α _{E} is taked as 2, and k is takend as 0.1, 0.5, 1, 2, 5 and 10. Figure 6 (a) shows the bifurcation diagram of AFG pipe withk=0.1. In the range of ω ∈ [ 0, 11.7 ], the pipe is mainly manifested as chaotic motion, and there are also areas of partial periodic motion, for instance ω ∈ [ 7.5, 8 ]. In the range of ω ∈[11.7, 15 ], chaotic motion is changed into period1 motion. In the range of ω ∈[15, 28 ], periodic motion is mainly observed. From ω = 28 to 41.5, it is manifested as period1 motion. In the range of ω ∈ [ 42.6, 70 ], it is mainly manifested as multiperiod motion. From the range of ω= 70 to 90, the static equilibrium state is transformed into periodic motion through Hopf bifurcation. Figure 6(b) shows the bifurcation diagram for k= 0.5. For dimensionless frequencyω=0 to 11.7, it is mainly manifested as chaotic motion, and there is also periodic motion, such as in the range of ω ∈ [ 5, 5.5 ]. Within the range of ω ∈ [11.7, 63.5 ], it is mainly manifested as periodic ω = 10, 50, motion. For dimensionless pulsation frequency ω =63.6 to 72, chaotic motion is transformed into periodic motion. For dimensionless frequency ω=72 to 76.8, it is mainly manifested as quasiperiodic motion. In the frequency range ofω∈ [ 76.85, 90 ], chaotic motion is transformed into periodic motion. For k= 1, as shown in Figure 6 (c), in the range of ω ∈ [ 45.5, 82 ], there exists a large range of chaotic motion, as well as a small range of periodic motion (ω∈ [ 75, 76.3]) and quasiperiodic motion (ω∈ [ 68.8, 69.8 ]). Within the range of ω ∈ [ 82, 90 ], the motion is quasiperiodic. As seen in Figure 6 (c)‒(f), when k ≥ 1the motion state in the lowfrequency region (ω ∈ [ 0, 45.5 ]) is stable and does not change with changing k, while there is a large difference in the highfrequency region. With increasing is squeezed, while the periodic and quasiperiodic regions will be extended.
4.3 Influence of distribution coefficients of viscoelastic damping and density on the system dynamic behavior
It is assumed that the density and elastic modulus of the functionally gradient material pipe are uniform in all directions (α_{E} = α_{ρ} = 1), the coefficient of viscoelastic damping ratio α_{E}∗ = 2, and k is taken as 0.1, 0.5, 1, 2, 5, and 10 to analyze the influence of the coefficient of viscoelastic distribution on the dynamic response of AFG pipes.
As shown in Figures 9 (a) and (b), when k < 1, the bifurcation diagram of dimensionless pulsation frequency changes little in the range ofω∈ [ 0, 90 ], and there are great differences mainly in the range of frequency ω ∈ [ 67, 76 ]. When k = 0.1, it shows period1 motion, and there is also a small range of period3 motion, as shown in Figure 9 (a). When k = 0.5, period3 motion is transformed into period1 motion, as shown in Figure 9 (b). When k= 1, the motion is transformed from multiperiodic motion to period1 motion, as shown in Figure 9 (c). When k > 1, the region with great variation in the bifurcation diagram is within the range of frequency ω ∈ [ 50, 60 ] andω∈ [ 70, 90 ]. Withinω∈ [ 50, 60 ], when k = 2, it mainly manifests as multiperiodic motion. When k= 5 and k = 10, chaotic motion is mainly manifested in a larger range. When the dimensionless frequency increases from ω=70 to 90, the range of quasiperiodic motion will increase gradually with increasing k. It can be seen that the k has a great influence on the motion state at high frequency, and its influence on the motion state at low frequency can be neglected.
It is assumed that the elastic modulus and coefficient of viscoelastic damping of AFG pipe are nonuniform (α_{E}= α_{E}∗ = 1), the density ratio α_{ρ} = 2, and k is taken as 0.1, 0.5, 1, 2, 5, and 10 to analyze the influence of density distribution on the dynamic response of the pipe. Figure 10 (a) shows the bifurcation diagram for AFG pipe with k= 0.1. In the range ofω∈ [ 0, 10 ], the pipe is in chaotic motion, but there are also a few pulsation frequencies such as ω ∈ [ 4.5, 6 ] that are in periodic motion. Within frequency range ω ∈ [ 9.6, 11.6 ], it appears as period3 motion. When the frequencyωincreases from 11.6 to 34.4, the period1 motion appears, but a sudden change occurs at the dimensionless pulsation frequency of 12.6, changing from period1 motion to period2 motion and lasting to the frequency of 13.6. In the range of ω ∈ [ 34.4, 61.5 ], it is mainly manifested as chaotic motion, but periodic motion and quasiperiodic motion can also be observed. Within frequency range ω ∈ [ 61.5, 90 ], it is mainly manifested as quasiperiodic motion, and there is also a small range of periodic motion.
It can be seen from Figure 10 that when the dimensionless pulsation frequencyω is less than 34.4, the effects of the material variation index k on the dynamic response of AFG pipes can be ignored. However, when ω > 34.4, with increasingk, the frequency range of period1 motion will gradually extended, and the chaotic and quasiperiodic motion frequency regions will not be changed, but the frequency of chaotic and quasiperiodic motion will gradually increase. Therefore, it can be concluded that the k has almost no effect on the motion state under low frequency range (ω ∈ [ 0, 34.4 ]). However, in the high frequency region, with increasing k, the motion state will not change, but the starting frequency at which each motion state occurs is increased.
5 Conclusion
The nonlinear dynamics of AFG pipe conveying pulsating flow is studied by using the generalized integral transform technique. The effects of the elastic modulus, density and coefficient of viscoelastic damping distribution on the dynamic behavior of the pipe are investigated, with the following main conclusions.
The distribution of the elastic modulus has an important influence on the motion state of the system. When the index of material variation k < 1, it has a greater influence on the motion in both lowfrequency and highfrequency regions. However, when k > 1, it has a greater effect on the motion only at high frequencies, with negligible effects in the low frequency region.
The distribution of density has little effect on the motion state of the system, with k having a great influence on the motion state at high frequency, but almost no influence on the motion state at low frequency.
The distribution of the viscoelastic damping coefficient has an important effect on the motion state of the system. In the low frequency region, the k has almost no effect on the motion state. With the increase in k, the motion state at high frequency region will not change, but the starting frequency at which each motion state occurs is increased.
The present method can be extended to the analysis of on nonlinear dynamics of viscoelastic AFG pipe conveying pulsating fluid with other classical boundary conditions, such as clamped and free ends, or more general elastic boundary conditions such as rotationally restrained ends. It can also be extended to the analysis of linear and nonlinear stability of mooring cables or catenary risers using largedeflection EulerBernoulli beam models.
Competing interest The authors have no competing interests todeclare that are relevant to the content of this article. 

An C, Su J (2017) Dynamic behavior of axially functionally gradedpipes conveying fluid. Mathematical Problems in Engineering 2017: 6789634. https://doi.org/10.1155/2017/6789634 Babilio E (2014) Dynamics of functionally graded beams on viscoelastic foundation, International Journal of Structural Stability and Dynamics 14 (08): 1440014. https://doi.org/10.1142/S0219455414400148 Cotta RM, Lisboa KM, Curi MF, Balabani S, Quaresma JN, Perez Guerrero JS, Mac^edo EN, Amorim NS (2019) A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier – Stokes equations formulation. Numerical Heat Transfer, Part B: Fundamentals 76 (2): 60–87. https://doi.org/10.1080/10407790.2019.1642715 Dai J, Liu Y, Liu H, Miao C, Tong G (2019) A parametric study on thermomechanical vibration of axially functionally graded material pipe conveying fluid. International Journal of Mechanics and Materials in Design 15 (4): 715–726. https://doi.org/10.1007/s10999018094395 Deng J, Liu Y, Zhang Z, Liu W (2017) Dynamic behaviors of multispan viscoelastic functionally graded material pipe conveying fluid. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231 (17): 31813192. https://doi.org/10.1177/0954406216642483 EbrahimiMamaghani A, SotudehGharebagh R, Zarghami R, Mostoufi N (2022) Thermomechanical stability of axially graded rayleigh pipes. Mechanics Based Design of Structures and Machines 50(2): 412441. https://doi.org/10.1080/15397734.2020.1717967 Fu G, Li M, Yang J, Li S, Sun B, Estefen S F (2022a) A simplified equation for the collapse pressure of sandwich pipes with different core materials. Ocean Engineering 254: 111292. https://doi.org/10.1016/j.oceaneng.2022.111292 Fu G, Li M, Yang J, Sun B, Shi C, Estefen S F (2022b) The effect of eccentricity on the collapse behaviour of sandwich pipes. Applied Ocean Research 124(4): 103190. https://doi.org/10.1016/j.apor.2022.103190 Fu G, Peng Y, Sun B, An C, Su J (2019) An exact GITT solution for static bending of clamped parallelogram plate resting on an elastic foundation. Engineering Computations 36 (6): 2034–2047. https://doi.org/10.1108/EC1220180582 Fu G, Tuo Y, Sun B, Shi C, Su J (2022c) Bending of variable thickness rectangular thin plates resting on a doubleparameter foundation: integral transform solution. Engineering Computations 39 (7): 2689–2704. https://doi.org/10.1108/EC1120210692 Gupta A, Talha M (2015) Recent development in modeling and analysis of functionally graded materials and structures. Progress in Aerospace Sciences 79:1–14. https://doi.org/10.1016/j.paerosci.2015.07.001i Heshmati M (2020) Influence of an eccentricity imperfection on the stability and vibration behavior of fluidconveying functionally graded pipes. Ocean Engineering 203: 107192. https://doi.org/10.1016/j.oceaneng.2020.107192 Jiang M, Zhao J, Wang Q (2022) Linear energy stable numerical schemes for a general chemorepulsive model. Journal of Computational and Applied Mathematics, 114436. https://doi.org/10.1016/j.cam.2022.114436 Jiang M, Zhao J (2023) Linear relaxation schemes for the AllenCahntype and CahnHilliardtype phase field models. Applied Mathematics Letter 137: 108477. https://doi.org/10.1016/j.aml.2022.108477 Jin JD (1997) Stability and chaotic motions of a restrained pipe conveying fluid. Journal of Sound and Vibration 208 (3): 427 –439. https://doi.org/10.1006/jsvi.1997.1195 Jin J, Song Z (2005) Parametric resonances of supported pipesconveying pulsating fluid. Journal of Fluids and Structures 20 (6): 763–783. https://doi.org/10.1016/j.jfluidstructs.2005.04.007 Khodabakhsh R, Saidi A R, Bahaadini R (2020) An analytical solution for nonlinear vibration and postbuckling of functionally graded pipes conveying fluid considering the rotary inertia and shear deformation effects. Applied Ocean Research 101: 102277. https://doi.org/10.1016/j.apor.2020.102277 Li F, An C, Duan M, Su J (2020) Combined damping model for dynamics and stability of a pipe conveying twophase flow. Ocean Engineering 195: 106683. https://doi.org/10.1016/j.oceaneng.2019.106683 Loghman E, Kamali A, BakhtiariNejad F, Abbaszadeh M (2021) Nonlinear free and forced vibrations of fractional modeled viscoelastic FGM microbeam, Applied Mathematical Modelling 92: 297–314. https://doi.org/10.1016/j.apm.2020.11.011 Lu Z Q, Zhang K K, Ding H, Chen L Q (2020) Nonlinear vibration effects on the fatigue life of fluidconveying pipes composed of axially functionally graded materials. Nonlinear Dynamics 100 (2): 1091–1104. https://doi.org/10.1007/s11071020055778 Ni Q, Zhang Z, Wang L, Qian Q, Tang M (2014) Nonlinear dynamics and synchronization of two coupled pipes conveying pulsating fluid. Acta Mechanica Solida Sinic 27 (2): 162 – 171. https://doi.org/10.1016/S08949166(14)600264 Nikbakht S, Kamarian S, Shakeri M (2019) A review on optimization of composite structures part ii: Functionally graded materials. Composite Structures 214: 83102. https://doi.org/10.1016/j.compstruct.2019.01.105 Oyelade AO, Oyediran A A (2020) The effect of various boundary conditions on the nonlinear dynamics of slightly curved pipes under thermal loading. Applied Mathematical Modelling 87: 332–350. https://doi.org/10.1016/j.apm.2020.06.019 Paidoussis M (1987) Flowinduced instabilities of cylindrical structures 40(2): 163175.https://doi.org/10.1115/1.3149530 Paidoussis M, Cusumano J, Copeland G (1992) Lowdimensional chaos in a flexible tube conveying fluid. Journal of Applied Mechanics 59(1): 196–205. https://doi.org/10.1115/1.2899428 Paidoussis M, Issid N (1974) Dynamic stability of pipes conveying fluid. Journal of Sound and Vibration 33 (3): 267–294. https://doi.org/10.1016/S0022460X(74)800027 Paidoussis M, Li G, Moon F (1989) Chaotic oscillations of the autonomous system of a constrained pipe conveying fluid. Journal of Sound and Vibration 135(1): 119. https://doi.org/10.1016/0022460X(89)907505 Paıdoussis M, Li G (1993) Pipes conveying fluid: a model dynamical problem, Journal of Fluids and Structures 7(2): 137204. https://doi.org/10.1006/jfls.1993.1011 Paidoussis M, Moon F (1988) Nonlinear and chaotic fluid elastic vibrations of a flexible pipe conveying fluid, Journal of Fluids and Structures 2(6): 567591. https://doi.org/10.1016/S08899746(88)800239 Paidoussis M, Semler C (1993) Nonlinear and chaotic oscillations of a constrained cantilevered pipe conveying fluid: a full nonlinear analysis. Nonlinear Dynamics 4 (6): 655670. https://doi.org/10.1007/BF00162236 Shafiei N, She G L (2018) On vibration of functionally graded nanotubes in the thermal environment. International Journal of Engineering Science 133: 8498. https://doi.org/10.1016/j.ijengsci.2018.08.004 Shariati A, MohammadSedighi H, Zur K K, Habibi M, Safa M (2020) On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams. Materials 13 (7): 1707. https://doi.org/10.3390/ma13071707 Shen J, Xu J, Yang J (2018) The scalar auxiliary variable (SAV)approach for gradient flows. Journal of Computational Physics 353: 407416. https://doi.org/10.1016/j.jcp.2017.10.021 Tang D, Dowell E (1988) Chaotic oscillations of a cantilevered pipe conveying fluid. Journal of Fluids and Structures 2 (3): 263283. https://doi.org/10.1016/S08899746(88)800112 Tang Y, Yang T (2018) Postbuckling behavior and nonlinear vibration analysis of a fluid conveying pipe composed of functionally graded material. Composite Structures 185: 393400. https://doi.org/10.1016/j.compstruct.2017.11.032 Tuo Y, Fu G, Sun B, Lou M, Su J (2022) Stability of axially functionally graded pipe conveying fluid: Generalized integral transform solution. Applied Ocean Research 125: 103218. https://doi.org/10.1016/j.apor.2022.103218 Wang L (2009) A further study on the nonlinear dynamics of simply supported pipes conveying pulsating fluid. International Journal of NonLinear Mechanics 44 (1): 115121. https://doi.org/10.1016/j.ijnonlinmec.2008.08.010 Wang L, Liu Z, Abdelkefi A, Wang Y, Dai H (2017) Nonlinear dynamics of cantilevered pipes conveying fluid: towards a further understanding of the effect of loose constraints, International Journal of NonLinear Mecha 95: 1929. https://doi.org/10.1016/j.ijnonlinmec.2017.05.012 Wang Y, Chen Y (2019) Dynamic analysis of the viscoelastic pipeline conveying fluid with an improved variable fractional order model based on shifted Legendre polynomials. Fractal and Fractional 3 (4): 52. https://doi.org/10.3390/fractalfract3040052 Wang Z, Soares C G (2021) Upheaval thermal buckling of functionally graded subsea pipelines. Applied Ocean Research 116: 102881. https://doi.org/10.1016/j.apor.2021.102881 Wang Z, Chen B, Soares CG (2022) Analytical study on the upheaval thermal buckling of sandwich pipes. Marine Structures 85: 103245. https://doi.org/10.1016/j.marstruc.2022.103245 Zhao J, Wang Q, Yang X (2017) Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach. International Journal for Numerical Methods in Engineering 110: 279300. https://doi.org/10.1002/nme.5372 Zhou X W, Dai H L, Wang L (2018) Dynamics of axially functionally graded cantilevered pipes conveying fluid. Composite Structures 190: 112118. https://doi.org/10.1016/j.compstruct.2018.01.097