On mathematical modeling,analysis and vibration control of a submarine pipeline conveying fluid and subjected to a regular wave excitation

Abstract

We consider the problem of nonlinear vibration response analysis and control of a submarine pipeline model conveying fluid and subjected to regular wave excitation. Wave excitation is modeled as a hydrodynamic force consisting of inertial and Morison-type drag components. After a brief description of the physical model, the governing equations of the system, with and without control, are established using the generalized Hamiltonian variational principle. A nonlinear energy sink (NES)-type controller is employed to reduce the vibration amplitude of the pipeline model. An analytical technique based on the multiple time scale method is applied to the linearized system to illustrate how parameters such as the internal fluid flow velocity, controller parameters, and wave amplitude affect the system’s response. The results indicate that as the internal fluid flow velocity increases, the resonant amplitude of the pipeline vibration also increases. This means that a high value of this velocity can contribute to destabilizing the system. Similarly, increasing the wave excitation intensity leads to an increase in the resonance (peak amplitude) of the pipeline. This important finding demonstrates that the highest wave excitation intensity can cause large displacements and contribute to destabilizing the structure. To highlight what we consider an original contribution, the fluctuations generated by the approximation of the drag force after approximation show a notable increase in the peak resonance amplitude of the structure. This load parameter can cause large displacements of the pipeline, potentially leading to significant consequences for the structure. Overall, the study demonstrates that the NES-type controller provides a considerable vibration attenuation rate, which is beneficial for system safety. Numerical simulations applied to the reduced nonlinear governing equations of the system are carried out to confirm the analytical predictions.

Keywords

submarine pipeline wave resonance NES-type controller vibration fluid flow velocity fluctuations of the drag force

1. Introduction

The exploitation of marine energy resources and the development of offshore renewable energies are major strategic challenges worldwide today. In this context, submarine pipelines represent critical infrastructure, serving as vital arteries for the transport of hydrocarbons. In general, a pipeline is a complex technical structure designed primarily for the continuous transport of hydrocarbons or other substances in liquid and gaseous states over vast geographical areas^1,2. Indeed, this fluid transport system is implemented in various engineering fields, including aerospace, machinery, civil engineering, marine engineering, biology, nuclear energy, oil and gas, hydroelectricity, and large-scale hydraulic engineering projects³. Therefore, the operational reliability and structural integrity of pipelines are of paramount importance, as any failure can have dramatic economic, environmental, and human consequences. The fundamental complexity of analyzing these structures arises from the fact that they are subjected to a dual dynamic load whose interdependence cannot be neglected. On the one hand, the marine environment exerts considerable external stresses, with wave action generating complex hydrodynamic forces (drag, lift, and inertia) that vary cyclically and potentially nonlinearly. On the other hand, the flow of the internal fluid introduces dynamic Coriolis effects, centrifugal forces, and pressure variations that lead to different types of instability, such as static divergence (commonly called buckling), flutter instabilities, or self-excited oscillations.^4–6

To prevent instabilities, certain viscoelastic materials, such as neoprene, are generally placed in the contact zones between the pipeline and its supports. This provides some isolation from hydrodynamic forces and minimizes excessive pipeline displacement. Furthermore, other passive control techniques are also used, such as mass-spring systems, which offer a considerable reduction in vibration amplitudes. However, for submarine pipelines exhibiting nonlinear behavior, the use of simple mass-spring systems is not sufficiently effective in reducing vibrations induced by wave impact, likely due to nonlinear effects that produce excessive system displacements. As a result, the analysis and control of the dynamic behavior of fluid-conveying pipelines have attracted the attention of several researchers. For example, Reis et al.⁷ conducted a thorough analysis of the stability and dynamics of submarine pipelines subjected to irregular seabed and ocean current conditions. They showed that natural frequencies decrease with increasing pipeline length. In a similar vein, Ali Karrech et al.⁸ worked on the nonlinear vibrations of free-span submarine pipelines with multidimensional stretching of the midplane. They demonstrated that lateral stretching of the midplane reduces the overall deflection of the free span, while increases in damping and nonlinear stiffness lead to a reduction in deflection. Tang et al.⁹ worked on the dynamic stability of a viscoelastic pipeline conveying fluid with fractional-order properties and time-dependent flow velocity. In this work, the effects of the viscoelastic properties of the structure characterized by fractional-order derivatives on its dynamic stability are investigated. Various parametric resonances induced by these properties are demonstrated. Tang et al.¹⁰ worked on the post-buckling behavior and nonlinear free vibration analysis of a functionally graded material (FGM) pipeline conveying fluid. They showed that increasing the internal fluid flow velocity beyond the critical value creates a fork-like bifurcation of the pipeline. Xia Tan et al.¹¹ studied the primary and super-harmonic resonance responses of forced vibrations in Timoshenko pipes conveying high-velocity fluid. We draw the reader’s attention to the fact that some general and recent discussions on vibration analysis of nonlinear systems, including fractional or fractal oscillators and fractional MEMS systems, are useful for this work^12–15. In fact, these references help us to better understand the analysis of the reduced-order response of our system. In addition, Ze-Qi Lu et al.¹⁶ worked on the internal resonance and stress distribution of pipes conveying supercritical fluid around curved pipelines. They showed that resonance can worsen the tensile and bending stress distributions in the pipeline at supercritical speeds. Yu Dai et al.¹⁷ performed a dynamic analysis of a deepwater mining pipeline system, taking into account internal and external flows. They demonstrated that the dynamic performance of the pipeline system is greatly affected by the ocean current angle, sea depth, and buoyancy modulus distribution. Jijun Gu et al.¹⁸ conducted a dynamic analysis of a pipeline conveying fluid under axial tension and thermal loads, showing that increasing the fluid velocity leads to an increase in the maximum deflection. Javadi et al.¹⁹ studied the primary and secondary resonances of a viscoelastic fractional pipeline model subjected to dynamic excitation, and showed how the fractional order of the derivative and time delay have significant effects on resonance amplitudes for different velocity values. The vibration of mechanical structures conveying fluid can be controlled by suitable structural design or by using active/passive devices. The Nonlinear Energy Sink (NES) controller is an interesting choice when the behaviour of the structure is nonlinear. Rony Philip et al.²⁰ worked on the possibility of using an NES-type controller with nonlinear damping for vibration limitation in a fluid-conveying pipeline. They showed that the external excitation required to trigger a bifurcation is greatly reduced for an NES with nonlinear damping. Duan et al.²¹ performed a stability analysis based on Lyapunov theory for a pipeline conveying fluid equipped with an NES-type vibration controller. The results showed that the vibrational energy of the original system decays exponentially, leading to the conclusion that the pipeline is exponentially stable.

Despite these advances, a rigorous modeling and analysis of a submarine pipeline conveying fluid that takes into account a fluctuation parameter of the drag force arising from the Taylor expansion of the Morison-type drag force and which considerably affects the system response remains unexplored. Thus, the main motivation of this work is to provide solutions to problems regularly encountered in piping engineering by developing a mathematical model to analyze pipeline vibrations induced by fluid velocity, fluid pressure, and sea waves, and by designing or using a control system to mitigate these vibrations. We believe that progress has been made in three directions: (a) A rigorous mathematical model of wave motion in a submarine environment is proposed as an unexplored configuration that takes into account the fluctuating part of the drag force; (b) The influence of key parameters, namely, the internal fluid flow velocity (from subcritical to supercritical cases) and the fluctuation parameter of the drag force is analyzed using analytical and numerical approaches; (c) The effect of the NES controller on the system’s response is analyzed. The first and second results are, in our view, the most striking findings of this study.

The remainder of the paper is organized as follows. Section 2 presents the physical description of the model system. Section 3 explores the influence of the internal fluid flow velocity, structural damping, wave amplitude, and fluctuation parameter of the drag force on system stability. Section 4 proposes a nonlinear controller and analyzes its effects on the dynamic behavior of the pipeline model. The conclusion is given in Section 5.

2. Model description and equations of motion

Based on Olunloyo Vincent²², the simplified physical model of a submarine pipeline conveying fluid on rigid supports and subjected to regular waves and marine currents is shown in Figure 1(a). An equivalent simplified model of the submarine pipeline, considered as a uniform structure conveying fluid with both ends simply supported, is shown in Figure 1(b). The system consists of a pipeline of length $L$ , the neutral axis coordinate $x$ , time $t$ , fluid density $ρ_{f}$ , pipeline density $ρ_{p}$ , cross-sectional area of the flow $A_{f}$ , and cross-sectional area of the pipeline $A_{p}$ . In order to derive the equations of motion, the following assumptions are made: (i) the deformation of the pipeline conveying fluid is confined to a plane; (ii) the Euler–Bernoulli beam theory with geometric nonlinearities is adopted to account for large deformations, while the effects of rotary inertia and shearing deformation are neglected; (iii) the fluid in the pipeline is incompressible and satisfies a plug-flow condition; (iv) variations in cross-sectional dimensions are negligible; (v) the fluid inside the pipeline is in steady flow with constant speed; (vi) the thermal effects of the fluid and the velocity of the marine current are neglected. The variables $w (x, t)$ and $u (x, t)$ represent the transverse and longitudinal displacements, respectively, which characterize the coupled planar motion of the pipeline.

Figure 1.

Sketch of (a) the submarine pipeline conveying fluid system, and (b) an equivalent simplified model under sea wave excitation, modeled as a Morison force (inertia force and drag force).

2.1. Mathematical modeling: System without control

The generalized Hamiltonian principle^23,24 can be applied to establish the mathematical model of the coupled planar motion of the submarine pipeline in the following form:

\int_{t_{1}}^{t_{2}} δ (T - U) d t + \int_{t_{1}}^{t_{2}} δ W_{n c} d t = 0

(1)

Where

T, U, and W_{n c}

represent the total kinetic energy, the total potential energy, and the work done by the internal non-conservative and external forces of the submarine pipeline system conveying fluid, respectively. The total kinetic energy is the sum of the kinetic energy of the fluid

T_{f}

and the kinetic energy of the pipeline

T_{p}

T = T_{p} + T_{f}

(2)

According to assumptions (i), (iii), and (v) listed above, the kinetic energies $T_{f}$ and $T_{p}$ can be expressed as follows:^23,24

{\begin{cases} T_{p} = \frac{1}{2} ρ_{p} A_{p} \int_{0}^{L} [{(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial w}{\partial t})}^{2}] d x \\ T_{f} = \frac{1}{2} ρ_{f} A_{f} \int_{0}^{L} [{(V + \frac{\partial u}{\partial t} + V \frac{\partial u}{\partial x})}^{2} + {(\frac{\partial w}{\partial t} + V \frac{\partial w}{\partial x})}^{2}] d x \end{cases}

(3)

The total potential energy is the sum of the strain energy caused by a variation of the stress with respect to the initial configuration of the pipeline, denoted $U_{p}$ , and the strain energy related to the subsequent stretching of the pipeline $U_{t}$ :

U = U_{p} + U_{t}

(4)

According to assumptions (i) and (iv) listed above, $U_{p}$ and $U_{t}$ can be expressed as follows:^23,24

{\begin{cases} U_{p} = \frac{1}{2} \int_{0}^{L} \int_{A_{p}} E {(- z \frac{\partial^{2} w}{\partial x^{2}} + \sqrt{{(1 + \frac{\partial u}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}} - 1)}^{2} d A d x \\ U_{t} = \int_{0}^{L} [P \sqrt{{(1 + \frac{\partial u}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}} - 1] d x \end{cases}

(5)

Where Z is the coordinate measured from the plane of the neutral fibers, E is the Young’s modulus of the pipeline, and P is the axial pressure caused by the flow of the fluid. The virtual work done by the internal non-conservative forces and external forces can be expressed as follows:

\int_{t_{1}}^{t_{2}} \int_{0}^{L} δ W_{n c} d x d t = \int_{t_{1}}^{t_{2}} \int_{0}^{L} (f_{M} (x, z, t) - c \frac{\partial w}{\partial t}) δ w d x d t

(6)

f_{M} (x, z, t)

represents the sea wave excitation, and

c

is the viscosity coefficient of the pipeline. Substituting equations (2), (3), (4), (5), and (6) into equation (1), and then applying integration by parts, we obtain the dynamic equations:

{\begin{cases} (ρ_{p} A_{p} + ρ_{f} A_{f}) \frac{\partial^{2} u}{\partial t^{2}} + 2 ρ_{f} A_{f} V \frac{\partial^{2} u}{\partial x \partial t} + ρ_{f} A_{f} V^{2} \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial}{\partial x} (\frac{(σ_{N} A_{p} + P) (1 + \frac{\partial u}{\partial x})}{\sqrt{{(1 + \frac{\partial u}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}}}) = 0 \\ (ρ_{p} A_{p} + ρ_{f} A_{f}) \frac{\partial^{2} w}{\partial t^{2}} + 2 ρ_{f} A_{f} V \frac{\partial^{2} w}{\partial x \partial t} + ρ_{f} A_{f} V^{2} \frac{\partial^{2} w}{\partial x^{2}} - (\frac{(σ_{N} A_{p} + P) (\frac{\partial^{2} w}{\partial x^{2}})}{\sqrt{{(1 + \frac{\partial u}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}}}) + c \frac{\partial w}{\partial t} + E I \frac{\partial^{4} w}{\partial x^{4}} = f_{M} (x, z, t) \end{cases}

(7)

where

σ_{N} = E (\sqrt{{(1 + \frac{\partial u}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2}} - 1)

(8)

Since the nonlinear vibration involves small but limited stretching, the coupling between longitudinal and transverse motion is neglected, and only the lowest-order nonlinear terms are retained. Moreover, the displacements both transverse and longitudinal are assumed to be much smaller than the pipe length. We assume that $w (x, t)$ and $u (x, t)$ , as well as their spatial derivatives, are small quantities. Considering that the nonlinear oscillation is a problem of low but limited stretching, we neglect the coupling between axial and transverse motion and retain only the lowest-order terms²⁵. Accordingly, the Taylor expansion is applied as follows:

{({(1 + \frac{\partial u}{\partial x})}^{2} + {(\frac{\partial w}{\partial x})}^{2})}^{- \frac{1}{2}} \approx 1 - \frac{\partial u}{\partial x} - \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}

(9)

and

σ_{N} = E (\frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2})

(10)

Considering that the initial axial tension $P$ is constant, substituting equations (9) and (10) into equation (7), and neglecting the influence of longitudinal inertia and other higher-order nonlinear terms, we obtain:

\frac{\partial}{\partial x} [\frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}] = 0

(11)

(ρ_{p} A_{p} + ρ_{f} A_{f}) \frac{\partial^{2} w}{\partial t^{2}} + c \frac{\partial w}{\partial t} + 2 ρ_{f} A_{f} V \frac{\partial^{2} w}{\partial x \partial t} + (ρ_{f} A_{f} V^{2} - P) \frac{\partial^{2} w}{\partial x^{2}} + E I \frac{\partial^{4} w}{\partial x^{4}} - E A_{p} \frac{\partial}{\partial x} [\frac{\partial w}{\partial x} (\frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2})] = f_{M} (x, z, t)

(12)

By integrating the equation (11) with respect to x twice, and considering the initial and boundary conditions $(u = 0 a t x = 0, x = L)$ , we obtain:

u (x) = \frac{x}{2 L} \int_{0}^{L} {(\frac{\partial w}{\partial x})}^{2} d x - \frac{1}{2} \int_{0}^{x} {(\frac{\partial w}{\partial x})}^{2} d x

(13)

Then, by substituting the equation (13) into the governing equation (12), we obtain:

(ρ_{p} A_{p} + ρ_{f} A_{f}) \frac{\partial^{2} w}{\partial t^{2}} + c \frac{\partial w}{\partial t} + 2 ρ_{f} A_{f} V \frac{\partial^{2} w}{\partial x \partial t} + (ρ_{f} A_{f} V^{2} - P) \frac{\partial^{2} w}{\partial x^{2}} + E I \frac{\partial^{4} w}{\partial x^{4}} - \frac{E A_{p}}{2 L} \frac{\partial^{2} w}{\partial x^{2}} \int_{0}^{L} {(\frac{\partial w}{\partial x})}^{2} d x = f_{M} (x, z, t)

(14)

The sea wave excitation is formulated using Morison’s equation and Airy theory^26,27 and is given by:

f_{M} (x, z, t) = ρ_{f_{w}} c_{M} A_{P} {\dot{u}}_{f} (t) - ρ c_{a} A \frac{\partial^{2} w}{\partial t^{2}} + ρ_{f_{w}} c_{D} \frac{D}{2} (u_{f} (t) - \frac{\partial w}{\partial t}) | (u_{f} (t) - \frac{\partial w}{\partial t}) |

(15)

where

c_{M}

c_{D}

and

c_{a}

are the coefficients of the drag force, the inertia force of the beam, and the added mass, respectively.

ρ_{f_{w}}

is the density of sea water, while

u_{f}

and

{\dot{u}}_{f}

are the velocity and acceleration of the waves, respectively, such that:

{\begin{cases} u_{f} = \frac{π H}{T} \frac{\cos h [k (z + d)]}{\sin h (k d)} \cos (k x - ω t) \\ {\dot{u}}_{f} = \frac{2 π^{2} H}{T^{2}} \frac{\cos h [k (z + d)]}{\sin h (k d)} \sin (k x - ω t) \end{cases}

(16)

At the bottom of the sea, $z = - d$ and for $x = 0$ , we have:

u_{f} = \frac{π H}{T s i n h (k d)} \cos (ω t) and {\dot{u}}_{f} = \frac{2 π^{2} H}{T^{2} \sin h (k d)} \sin (ω t)

(17)

Also, based on the work of Michael Isaacson²⁸, and other authors^29,30, the nonlinear expression of the wave drag force in Morison’s equation is obtained through its Taylor expansion as follows:

Let u_{f} (t) - \frac{\partial w}{\partial t} = u_{r} then, (u_{f} (t) - \frac{\partial w}{\partial t}) | (u_{f} (t) - \frac{\partial w}{\partial t}) | = u_{r} | u_{r} |

(18)

Considering the Taylor expansion for $u_{f} = u_{r}$ , we obtain:

u_{r} | u_{r} | = u_{f} | u_{f} | - 2 | u_{f} | \frac{\partial w}{\partial t} + O ({(\frac{\partial w}{\partial t})}^{2})

(19)

Hence, by substituting the equation (19) into the equation (15), we finally obtain the load expression below:

f_{M} (x, z, t) \approx ρ_{f_{w}} c_{M} A_{P} {\dot{u}}_{f} (t) - ρ c_{a} A \frac{\partial^{2} w}{\partial t^{2}} + ρ_{f_{w}} c_{D} \frac{D}{2} (u_{f} (t) | u_{f} (t) | - 2 | u_{f} (t) | \frac{\partial w}{\partial t})

(20)

hence,

\begin{array}{l} (ρ_{p} A_{p} + ρ_{f} A_{f} + ρ_{f_{w}} c_{a} A_{p}) \frac{\partial^{2} w}{\partial t^{2}} + (c + \frac{π H ρ_{f_{w}} c_{D} D ζ}{T \sin h (k d)} \cos (ω t)) \frac{\partial w}{\partial t} + 2 ρ_{f} A_{f} V \frac{\partial^{2} w}{\partial x \partial t} + \\ (ρ_{f} A_{f} V^{2} - P) \frac{\partial^{2} w}{\partial x^{2}} + E I \frac{\partial^{4} w}{\partial x^{4}} - \frac{E A_{p}}{2 L} \frac{\partial^{2} w}{\partial x^{2}} \int_{0}^{L} {(\frac{\partial w}{\partial x})}^{2} d x = \frac{π^{2} H^{2} ρ_{f_{w}} c_{D} D ζ}{4 {(T \sin (k d))}^{2}} (1 + \cos (2 ω t)) \\ + \frac{2 π^{2} H ρ_{f_{w}} c_{M} A_{P}}{T^{2} \sin (k d)} \sin (ω t) \end{array}

(21)

with

ζ = \pm 1

Equation (21) is the mathematical equation governing the transverse displacement response of the submarine pipeline conveying fluid under wave load excitation. Introducing the following dimensionless variables into equation (21):

\begin{array}{l} w = r η, x = L ξ, τ = ω_{0} t, Ω = \frac{ω}{ω_{0}}, ω_{0} = \frac{1}{L^{2}} \sqrt{\frac{E I}{M_{e}}}, r = \sqrt{\frac{I}{A_{p}}}, γ = \frac{P L^{2}}{E I} \\ M_{e} = ρ_{p} A_{p} + ρ_{f} A_{f} + ρ_{f} c_{a} A_{p}, β = \frac{ρ_{f} A_{f}}{M_{e}}, μ_{t} = \frac{L^{2} ρ_{f_{w}} c_{D} D π H}{T \sin (k d) \sqrt{E I M_{e}}}, χ = V L \sqrt{\frac{ρ_{f} A_{f}}{E I}} \\ \begin{array}{l} μ = \frac{L^{2} c}{\sqrt{E I M_{e}}}, f_{1} = \frac{2 π^{2} H ρ_{f_{w}} c_{M} L^{4} {(\sqrt{A_{p}})}^{3}}{T^{2} E \sin (k d) {(\sqrt{I})}^{3}}, f_{0} = \frac{{(π H)}^{2} ρ_{f_{w}} c_{D} D L^{4} \sqrt{A_{p}}}{4 E {(T \sin (k d))}^{2} {(\sqrt{I})}^{3}} . \end{array} \end{array}

(22)

The dimensionless equation is obtained as:

\frac{\partial^{2} η}{\partial τ^{2}} + \frac{\partial^{4} η}{\partial ξ^{4}} + (μ + μ_{t} \cos (Ω τ)) \frac{\partial η}{\partial τ} + 2 \sqrt{β} χ \frac{\partial^{2} η}{\partial τ \partial ξ} + (χ^{2} - γ) \frac{\partial^{2} η}{\partial ξ^{2}} - \frac{1}{2} \frac{\partial^{2} η}{\partial ξ^{2}} {\int_{0}^{1} (\frac{\partial η}{\partial ξ})}^{2} d ξ = f_{0} + f_{0} \cos (2 Ω τ) + f_{1} \sin (Ω τ)

(23)

2.2. Modal equation

To perform the analytical analysis, we resort to an assumed mode expansion. Specifically, it is assumed that $η$ can be expressed as the infinite series:

η (ξ, τ) = \sum_{n = 1}^{\infty} q_{n} (τ) \sin (n π ξ)

(24)

where

q_{n} (τ)

is the time response of the transverse displacement of the submarine pipeline conveying fluid, and

ϕ_{n} (ξ)

is the modal shape of the system, defined as follows:

ϕ_{n} (ξ) = \sin (n π ξ)

Figure 2 presents the amplitude of deformation of the first three mode shapes of the pipeline for a length $L = 6 m$ . We note that for the first three mode shapes, the amplitude of the dimensionless displacement of the pipeline is constant and equal to 1. The analysis is limited to the first mode, since the first mode of vibration is expected to carry most of the energy, and thus it should suffice to obtain a first estimate of the system behavior^31,32. After substituting equation (24) into equation (23), multiplying both sides of the resulting equation by the shape function, and integrating with respect to the beam axis x over the length L, the single one-dimensional modal equation is given as follows:

{\ddot{q}}_{1} (τ) + (μ + μ_{t} \cos (Ω τ)) {\dot{q}}_{1} (τ) + ϖ^{2} q_{1} + β_{1} q_{1}^{3} = P_{0} + P_{0} \cos (2 Ω τ) + P_{1} \sin (Ω τ)

(25)

with

ϖ^{2} = π^{4} - π^{2} (χ^{2} - γ)

and

β_{1} = \frac{π^{4}}{4}, P_{0} = \frac{4 f_{0}}{π}, P_{1} = \frac{4 f_{1}}{π}

Figure 2.

Mode shapes of the pipeline for $L = 6 m$ .

3. Natural frequency of the structure versus critical flow velocity of the internal fluid

In this section, we determine the critical flow velocity as function of the natural frequency. To do so, the dimensionless equation (23) is rewritten in absence of external loads and the nonlinear term (as shown in Refs. 10 and 15):

\frac{\partial^{2} η}{\partial τ^{2}} + \frac{\partial^{4} η}{\partial ξ^{4}} + μ \frac{\partial η}{\partial τ} + 2 χ \sqrt{β} \frac{\partial^{2} η}{\partial τ \partial ξ} + (χ^{2} - γ) \frac{\partial^{2} η}{\partial ξ^{2}} = 0

(26)

It is well known that the solution of equation (26) can be taken as $η (ξ, τ) = ϕ_{n} (ξ) e^{i ω τ}$ , where $ϕ_{n} (ξ)$ represents the eigenmodes of vibration and $ω$ represents the complex angular velocities. By multiplying by an orthogonal mode to $ϕ_{m} (ξ)$ and integrating, we can express the dimensionless frequency as a function of the dimensionless critical velocity as follows:

{\begin{cases} χ_{c r} = \sqrt{{(n π)}^{2} + γ} \\ f (χ_{c r}) = \frac{1}{2 π} \sqrt{{(n π)}^{4} - {(n π)}^{2} (χ_{c r}^{2} - γ)} \end{cases}

(27)

with X = V L \sqrt{\frac{ρ_{f} A_{f}}{E I}}

;

γ = \frac{P L^{2}}{E I}

μ = \frac{L^{2} c}{\sqrt{E I M_{e}}}

The pipeline properties, sea wave characteristics, and the characteristics of the internal fluid used for numerical analysis are given in Tables 1–3²²

Table 1.

Geometrics parameters of the pipeline.

Parameters	Symbols	Values
External diameter $(m)$	$D_{e x}$	$0.4064$
Internal diameter $(m)$	$D_{i n}$	$0.394$
Length of pipeline $(m)$	$L$	$6$
Young’s modulus $(N / m^{2})$	$E$	$200 \times 1 0^{9}$
Moment of inertia $(m^{4})$	$I$	$1.56 \times 1 0^{- 4}$
Structural damping coefficient $(N . s / m^{2})$	$c$	$1.5$
Density $(k g / m^{3})$	$ρ_{p}$	$7850$

Table 2.

Parameters of the internal fluid.

Parameters	Symbols	Values
Density $(k g / m^{3})$	$ρ_{f}$	$850$
Velocity $(m / s)$	$V$	$3$
Pressure $(N / m^{2})$	$P$	$5 \times 1 0^{5}$

Table 3.

Parameters of the external fluid.

Parameters	Symbols	Values
Drag coefficient	$C_{D}$	$0.8$
Coefficient of inertia	$C_{M}$	$1.7$
Added mass coefficient	$C_{a}$	$1$
Wave height $(m)$	$H$	$2$
Wave Period $(s)$	$T$	$25$
Wave number	$k$	$0.1$
Density $(k g / m^{3})$	$ρ_{f_{w}}$	$980$

As one can observed in Figure 3, by increasing the dimensionless flow velocity of the fluid, the natural frequency decreases. We obtain two values of the critical velocity: one for the first mode of vibration, denoted $χ_{c r 1} = 3.14$ , and one for the second mode, denoted $χ_{c r 2} = 6.28$ . Considering the first mode, we observe that internal flow velocities below the critical value the so-called subcritical velocities $(χ ≺ χ_{c r 1} or χ ≺ 3.14)$ correspond to stable system behavior, while supercritical velocities $(χ ≻ χ_{c r 1} or χ ≻ 3.14)$ correspond to unstable behavior. This analysis is clearly verified and confirmed by the plots in Figures 4 and 5.

Figure 3.

Natural frequency of the pipeline versus the flow velocity of the internal fluid for the first and second vibration modes.

Figure 4.

(a) Time series of the displacement of the pipeline and (b) the phase portrait in plane (q, y) for $Ω = 10.15$ and $χ = 0.0328$ . Stable behaviour of the pipeline.

Figure 5.

(a) Time series of the displacement of the pipeline and (b) the phase portrait in plane (q, y) for $Ω = 10.15$ and $χ = 3.14$ : Unstable behaviour of the pipeline.

4. Effect of the critical flow velocity of the fluid on the amplitude response of the pipeline

Figure 4 presents the time series of the displacement (Figure 4(a)) and the phase portrait in the plane (q, y) (Figure 4(b)) of the pipeline response for a dimensionless internal fluid flow velocity $χ = 0.0328$ . It is observed that when the dimensionless velocity of the internal fluid is less than the critical value, the pipeline vibrates around a stable equilibrium point (0,0), and the structure exhibits stable periodic dynamic behavior. This result is confirmed by the phase portrait, which shows a limit cycle. In contrast, when the flow velocity is greater than or equal to the critical value, the structure vibrates around an unstable equilibrium point, as shown in Figure 5(a) for $χ_{c r} = 3.14$ . This indicates that under such conditions, the structure displays unstable dynamic behavior, such as chaotic motion¹¹, which is further confirmed by the phase portrait in Figure 5(b). Generally, such unstable vibrations can lead to rapid fatigue of the structure.

5. Effects of the wave frequency and the internal fluid flow velocity on the vibration amplitude of the pipeline

In this section, a multiple time scale method¹¹ is employed to obtain an analytical approximate solution. Two-time scales are defined: one fast time and one slow time. The time differentiation is expanded as follows:

\frac{d}{d t} = D_{0} + ε D_{1} + . . . ., \frac{d^{2}}{d t^{2}} = D_{0}^{2} + 2 ε D_{0} D_{1} + \dots .,

(28)

and

{\ddot{q}}_{1} (τ) + ϖ^{2} q_{1} = ε [- (μ + μ_{t} \cos (Ω τ)) {\dot{q}}_{1} (τ) - β_{1} q_{1}^{3} + P_{0} \cos (2 Ω τ) + P_{1} \sin (Ω τ) + P_{0}]

(29)

where

ε

refers to a very small perturbation parameter,

D_{0}

and

D_{1}

are the operators defined as follows:

D_{0} = \frac{\partial}{\partial T_{0}}

and

D_{1} = \frac{\partial}{\partial T_{1}}

. The expansion of the solution

q_{1} (τ, ε)

is in the form:

q_{1} (τ, ε) = q_{00} (T_{0}, T_{1}) + ε q_{01} (T_{0}, T_{1}) + \dots \dots \dots

(30)

With T_{m} = ε^{m} τ, m = 0, 1, . . .

After this expansion, the different types of resonance of the system are analyzed.

5.1. Harmonic resonance of the sub-marine pipeline conveying fluid

When the wave frequency is equal to the natural frequency or equal to half of the natural frequency of the fluid-conveying pipeline, resonance phenomena generally occur, known as primary and secondary resonances, respectively. We will then examine each of these resonance types case by case. By substituting the equations (28) and (30) into (29) and equating the coefficients of the same powers of $ε$ , we obtain:

ε^{0} : D_{0}^{2} q_{00} + ϖ^{2} q_{00} = 0

(31)

ε^{1} : D_{0}^{2} q_{01} + ϖ^{2} q_{01} = - 2 D_{0} D_{1} q_{00} - (μ + μ_{t} \cos (Ω T_{0})) D_{0} q_{00} - β_{1} q_{00}^{3} + P_{0} \cos (2 Ω T_{0}) + P_{1} \sin (Ω T_{0}) + P_{0}

(32)

The solving equation (31) gives a solution in the form;

q_{00} (T_{0}, T_{1}) = A (T_{1}) e^{i ϖ T_{0}} + \bar{A} (T_{1}) e^{- i ϖ T_{0}}

(33)

By substituting equation (33) into equation (32), we obtain:

D_{0}^{2} q_{01} + ϖ^{2} q_{01} = [- 2 i ϖ A^{'} - i ϖ μ A - 3 β_{1} A^{2} \bar{A}] e^{i ϖ T_{0}} - \frac{i ϖ μ_{t} A}{2} (e^{i (ϖ + Ω) T_{0}} + e^{i (ϖ - Ω) T_{0}}) + \frac{i ϖ μ_{t} \bar{A}}{2} e^{- i (ϖ - Ω) T_{0}} - β_{1} A^{3} e^{3 i ϖ T_{0}} - \frac{i P_{1}}{2} e^{i Ω T_{0}} + \frac{P_{0}}{2} e^{2 i Ω T_{0}} + P_{0} + c c

(34)

Where

c c

denotes the complex conjugate terms. From the above equation, two types of resonance may occur during the vibration of the system.

➢ First case: $ϖ \approx Ω$ (primary or fundamental resonance).

The deviations of $Ω$ and $ϖ$ are represented by the following definition:

Ω = ϖ + ε σ; T_{0} Ω = ϖ T_{0} + σ T_{1}; e^{i (ϖ - Ω) T_{0}} = e^{- i σ T_{1}}; e^{i Ω T_{0}} = e^{i σ T_{1}} e^{i ϖ T_{0}}

(35)

Where

σ

is the detuning parameter. After considering the above definition and eliminating the secular terms from the equation, we obtain:

2 i ϖ A^{'} + i μ ϖ A + 3 β_{1} A^{2} \bar{A} + \frac{i P_{1}}{2} e^{i σ T_{1}} = 0

(36)

By using the polar notation $A (T_{1}) = \frac{1}{2} a (T_{1}) e^{i θ (T_{1})}$ in equation (36) and separating the real and imaginary parts, we obtain the following system of amplitude equations:

{\begin{cases} - a ϖ θ^{'} + \frac{3 β_{1}}{8} a^{3} = \frac{P_{1}}{2} \sin (ψ) \\ ϖ a^{'} + \frac{μ ϖ}{2} a = - \frac{P_{1}}{2} \cos (ψ) \end{cases}

(37)

where

ψ = σ T_{1} - θ

For a steady-state response, the solution is given $a^{'} = 0, ψ^{'} = 0$ . Using these conditions, and after some mathematical expansion and reduction, we finally obtain the amplitude equation for primary resonance:

\frac{9 β_{1}^{2}}{16} a^{6} - 3 β_{1} σ ϖ a^{4} + [(μ^{2} + 4 σ^{2}) ϖ^{2}] a^{2} = P_{1}^{2}

(38)

➢ Second case: $ϖ \approx 2 Ω$ (secondary resonance).

The deviations of $Ω$ and $ϖ$ are represented by the following definition:

Ω = \frac{1}{2} ϖ + ε σ; T_{0} Ω = \frac{1}{2} ϖ T_{0} + σ T_{1}; e^{i 2 Ω T_{0}} = e^{2 i σ T_{1}} e^{i ϖ T_{0}}

(39)

As before, by considering the same hypotheses and the same steady-state response conditions, and after mathematical expansion, we obtain the following amplitude equation for secondary resonance:

\frac{9 β_{1}^{2}}{16} a^{6} - 6 β_{1} σ ϖ a^{4} + [(μ^{2} + 16 σ^{2}) ϖ^{2}] a^{2} = P_{0}^{2}

(40)

Equations (38) and (40) represent the nonlinear amplitude responses for the primary and secondary resonances of the system. The oscillation amplitudes are evaluated using the dichotomy method¹¹, and compared with numerical simulations obtained via the fourth-order Runge–Kutta method^11,32, as shown in Figures 6 and 7. For the investigated parameters, the numerical results are found to be in good agreement with the analytical predictions. Both figures show that a resonance peak exists for a specific value of the detuning parameter $(σ = 0)$ , at which the vibration amplitude reaches a maximum. It is also observed that the peak amplitude of the primary resonance is larger than that of the secondary resonance. Such resonance phenomena can induce large vibrations and thereby contribute to structural failure and damage.

Figure 6.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for primary resonance $(Ω = ϖ + σ)$ . The dotted line represents the numerical results obtained by simulating the full Eq. (29) while the curves with circles are the analytical results from Eq. (38). The simulation parameters are given in Tables 1–3.²²

Figure 7.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for secondary resonance $(Ω = \frac{ϖ}{2} + σ)$ The dotted line represents the numerical results obtained by simulating the full Eq. (29) while the curves with circles are the analytical results from Eq. (40). The simulation parameters are given in Tables 1–3.²²

Figures 8 and 9 show the effect of the internal fluid flow velocity (subcritical, critical, and supercritical) on the steady-state amplitude a and the detuning parameter $σ$ (related to the wave frequency) for the primary and secondary resonances of the system. The data reported in Figures 8 and 9 demonstrate that the resonance peak appears as the flow velocity increases. In short, the resonance of the pipeline system strongly depends on the internal fluid flow velocity. The peak amplitude is maximal for supercritical flow velocity in both primary and secondary resonance cases, indicating that supercritical velocity can reduce the life expectancy of the pipeline. Such effects of flow velocity in fluid–structure interaction have been extensively discussed in¹¹. Figure 10 reports the effect of wave amplitude on the steady-state vibration amplitude a of the pipeline versus the detuning parameter $σ$ for primary resonance, considering a subcritical internal fluid velocity. It is observed that the resonance becomes more pronounced as the wave amplitude increases, meaning that a high wave excitation amplitude appears to be dangerous for system safety. This interesting result confirms the observations made by Xia Tan¹¹.

Figure 8.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for primary resonance $(Ω = ϖ + σ)$ . The effect of the flow velocity of the internal fluid (in terms of sub-critical, critical and super critical velocity). The simulation parameters are given in Tables 1–3.²²

Figure 9.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for secondary resonance $(Ω = \frac{ϖ}{2} + σ)$ . The effect of the flow velocity of the internal fluid (in terms of sub-critical, critical and super critical velocity). The simulation parameters are given in Tables 1–3.²²

Figure 10.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for primary resonance $(Ω = ϖ + σ)$ , showing the effect of wave amplitude $P_{1}$ . The simulation parameters are given in Tables 1–3.²²

5.2. Superharmonic, subharmonic, and combined harmonic resonance of the submarine pipeline

When the wave amplitudes become very large, the behavior of the system shifts to other types of vibrational resonance, namely superharmonic, subharmonic, and combined harmonic resonance. By applying the same method and development as previously described, we obtain the following amplitude equations:

ε^{0} : D_{0}^{2} q_{00} + ϖ^{2} q_{00} = P_{0} \cos (2 Ω T_{0}) + P_{1} \sin (Ω T_{0}) + P_{0}

(41)

ε^{1} : D_{0}^{2} q_{01} + ϖ^{2} q_{01} = - 2 D_{0} D_{1} q_{00} - (μ + μ_{t} \cos (Ω T_{0})) D_{0} q_{00} - β_{1} q_{00}^{3}

(42)

Hence, a solution to (41) is in the form:

q_{00} (T_{0}, T_{1}) = A (T_{1}) e^{i ϖ T_{0}} + α_{0} + α_{1} e^{2 i Ω T_{0}} + α_{2} e^{i Ω T_{0}} + c c

(43)

Avec α_{0} = \frac{P_{0}}{ϖ^{2}}; α_{1} = \frac{P_{0}}{2 (ϖ^{2} - 4 Ω^{2})}; α_{2} = - \frac{i P_{1}}{2 (ϖ^{2} - Ω^{2})}; i^{2} = - 1

Substituting equation (43) into equation (42), and after mathematical expansion and simplification, the following expression is obtained:

\begin{array}{l} (- 2 i ϖ A^{'} - i ϖ μ A - 3 β_{1} A^{2} \bar{A} - 3 β_{1} α_{0}^{2} A) e^{i ϖ T_{0}} - β_{1} A^{3} e^{3 i ϖ T_{0}} - 3 β_{1} α_{0} A^{2} e^{2 i ϖ T_{0}} - \frac{i Ω μ_{t} α_{2}}{2} \\ - β_{1} α_{0}^{3} - 6 β_{1} α_{0} A \bar{A} + (- i μ Ω α_{2} - i μ_{t} Ω α_{1} - 6 β_{1} α_{2} A \bar{A} - 3 β_{1} α_{0}^{2} α_{2}) e^{i Ω T_{0}} \\ \begin{array}{l} + (- 2 i μ Ω α_{1} - \frac{i Ω μ_{t} α_{2}}{2} - 6 β_{1} α_{1} A \bar{A} - 3 β_{1} (α_{0} α_{2}^{2} + α_{0}^{2} α_{1})) e^{i 2 Ω T_{0}} \\ + (- i μ_{t} Ω α_{1} - β_{1} (6 α_{0} α_{1} α_{2} + α_{2}^{3})) e^{i 3 Ω T_{0}} - 3 β_{1} (α_{0} α_{1}^{2} + α_{1} α_{2}^{2}) e^{i 4 Ω T_{0}} - 3 β_{1} α_{1}^{2} α_{2} e^{i 5 Ω T_{0}} \\ \begin{array}{l} - β_{1} α_{1}^{3} e^{i 6 Ω T_{0}} + (- \frac{i ϖ μ_{t} A}{2} - 6 β_{1} A α_{0} α_{2}) e^{i (ϖ + Ω) T_{0}} - \frac{i ϖ μ_{t} A}{2} e^{i (ϖ - Ω) T_{0}} - (- \frac{i ϖ μ_{t} \bar{A}}{2} + 6 β_{1} \bar{A} α_{0} α_{2}) e^{- i (ϖ - Ω) T_{0}} \\ - 3 β_{1} α_{1} A^{2} e^{i 2 (ϖ + Ω) T_{0}} - 3 β_{1} α_{1} {\bar{A}}^{2} e^{- i 2 (ϖ - Ω) T_{0}} - 3 β_{1} α_{2} A^{2} e^{i (2 ϖ + Ω) T_{0}} - 3 β_{1} α_{2} {\bar{A}}^{2} e^{- i (2 ϖ - Ω) T_{0}} \\ \begin{array}{l} - 3 β_{1} α_{1}^{2} A e^{i (ϖ + 4 Ω) T_{0}} - 3 β_{1} A (α_{2}^{2} + 2 α_{0} α_{1}) e^{i (ϖ + 2 Ω) T_{0}} - 6 β_{1} A α_{1} α_{2} e^{i (ϖ + 3 Ω) T_{0}} - 3 β_{1} α_{1}^{2} \bar{A} e^{- i (ϖ - 4 Ω) T_{0}} \\ - 3 β_{1} \bar{A} (α_{2}^{2} + 2 α_{0} α_{1}) e^{- i (ϖ - 2 Ω) T_{0}} - 6 β_{1} \bar{A} α_{1} α_{2} e^{- i (ϖ - 3 Ω) T_{0}} + c c \end{array} \end{array} \end{array} \end{array}

(44)

where

c c

represents the complex conjugates of the preceding terms. Note that the system can exhibit two superharmonic resonant states (for

6 Ω = ϖ + ε σ; 3 Ω = ϖ + ε σ

), two subharmonic resonant states (for

Ω = 3 ϖ + ε σ; Ω = \frac{3}{2} ϖ + ε σ

), and two combined harmonic resonances.

a. Investigation on super-harmonic resonance of the system response.

By considering $3 Ω = ϖ + ε σ$ and substituting to Eq. (44), then setting the secular terms to zero, we obtain:

- 2 i ϖ A^{'} - i ϖ μ A - 3 β_{1} A^{2} \bar{A} - 3 β_{1} α_{0}^{2} A + (- i μ_{t} Ω α_{1} - β_{1} (6 α_{0} α_{1} α_{2} + 6 \bar{A} α_{1} α_{2} + α_{2}^{3})) e^{i σ T_{1}} = 0

(45)

Using the polar notation $A (T_{1}) = \frac{1}{2} a (T_{1}) e^{i θ (T_{1})}$ in the equation (45), and separating the real and imaginary parts, we obtain the following system:

{\begin{cases} - a ϖ θ^{'} + \frac{3 β_{1}}{8} a^{3} + \frac{3 β_{1} α_{0}^{2}}{2} a + a Γ \sin (ψ) = (μ_{t} Ω α_{1} + ϒ) \sin (Φ) \\ ϖ a^{'} + \frac{ϖ μ}{2} a ‐ a Γ \cos (ψ) = ‐ (μ_{t} Ω α_{1} + ϒ) \cos (Φ) \end{cases}

(46)

with

ψ = σ T_{1} - 2 θ; Φ = σ T_{1} - θ; 3 β_{1} α_{1} α_{2} = - \underset{Γ}{\underset{⏟}{\frac{3 β_{1} P_{0} P_{1}}{4 (ϖ^{2} - 4 Ω^{2}) (ϖ^{2} - Ω^{2})}}} i = - i Γ β_{1} (6 α_{0} α_{1} α_{2} + α_{2}^{3}) = \underset{ϒ}{\underset{⏟}{[\frac{β_{1} P_{1}^{3}}{8 {(ϖ^{2} - Ω^{2})}^{3}} - \frac{3 β_{1} P_{0}^{2} P_{1}}{2 ϖ^{2} (ϖ^{2} - Ω^{2}) (ϖ^{2} - 4 Ω^{2})}]}} i = i ϒ

As before, considering the same steady-state response conditions, and after mathematical expansion, we obtain the following amplitude equation:

{(- \frac{2}{3} σ a ϖ + \frac{3 β_{1}}{8} a^{3} + \frac{3 β_{1} α_{0}^{2}}{2} a + a Γ \sin (σ T_{0} - 2 θ_{0}))}^{2} + {(\frac{ϖ μ}{2} a - a Γ \cos (σ T_{0} - 2 θ_{0}))}^{2} = {(μ_{t} Ω α_{1} + ϒ)}^{2}

(47)

where

T_{0} and θ_{0}

are fixed parameters in steady state.

Now, by considering $6 Ω = ϖ + ε σ$ and by substituting this condition into equation (44) and setting the secular terms to 0, we obtain:

- 2 i ϖ A^{'} - i ϖ μ A - 3 β_{1} A^{2} \bar{A} - 3 β_{1} α_{0}^{2} A - β_{1} α_{1}^{3} e^{i σ T_{1}} = 0

(48)

As before, by applying the polar form

A (T_{1}) = \frac{1}{2} a (T_{1}) e^{i θ (T_{1})}

and separating the real and imaginary parts, we obtain the following system:

{\begin{cases} - a ϖ θ^{'} + \frac{3 β_{1} α_{0}^{2}}{2} a + \frac{3 β_{1}}{8} a^{3} + β_{1} α_{1}^{3} \cos (ψ) = 0 \\ ϖ a^{'} + \frac{ϖ μ}{2} a + β_{1} α_{1}^{3} \sin (ψ) = 0 \end{cases}

(49)

with $ψ = σ T_{1} - θ$

By considering the same conditions for steady state response, and after mathematical expansion, we obtain the following amplitude equation:

{(\frac{ϖ μ}{2} a)}^{2} + {[(\frac{3 β_{1} P_{0}^{2}}{2 ϖ^{4}} - ϖ σ) a + \frac{3 β_{1}}{8} a^{3}]}^{2} = \frac{β_{1}^{2} P_{0}^{6}}{64 {(ϖ^{2} - 4 Ω^{2})}^{6}}

(50)

The effect of the internal fluid flow velocity is observed by representing the steady-state amplitude of the pipeline as a function of the detuning parameter in Figure 11 for (a) $3 Ω = ϖ + σ$ and for (b) $6 Ω = ϖ + ε σ$ . It is observed that for subcritical, critical, and supercritical flow velocities, the vibration amplitudes of the pipeline increase considerably as the frequency increases. Moreover, for the two resonance cases, the peak amplitudes are obtained at different values of the detuning parameter. This is due to the fact that the natural frequency of the pipeline is a function of the flow velocity; thus, modifying the flow velocity directly affects the resonance frequency of the waves. This occurs because the flow pressure of the internal fluid significantly contributes to the natural frequency of the pipeline.

Figure 11.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for superharmonic resonance: (a) $3 Ω = ϖ + σ$ ; (b) $6 Ω = ϖ + ε σ$ . The effect of the internal fluid flow velocity (subcritical, critical, and supercritical) is shown. The simulation parameters are given in Tables 1–3.²²

To highlight what we consider an original contribution, the fluctuations generated by the approximation of the drag force which, after approximation, presented a notable increase in the peak resonance amplitude of the structure are investigated. The data reported in Figure 12 show how an increase in this load parameter $μ_{τ}$ significantly increases the peak amplitude of the structure. This indicates that a high value of this parameter can cause large displacements and thereby lead to fatigue phenomena in the structure.

Figure 12.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for super harmonic resonance $3 Ω = ϖ + σ$ . The effect of the fluctuation parameter of the drag force $μ_{τ}$ . The simulation parameters are given in Tables 1–3.²²

b. Investigation on Subharmonic resonance of the system response.

In this part, two cases are treated: $Ω = 3 ϖ + ε σ; Ω = \frac{3}{2} ϖ + ε σ$ .

First case ( $Ω = 3 ϖ + ε σ$ ), the secular terms are eliminated when:

- 2 i ϖ A^{'} - i ϖ μ A - 3 β_{1} A^{2} \bar{A} - 3 β_{1} α_{0}^{2} A - 3 β_{1} α_{2} {\bar{A}}^{2} e^{i σ T_{1}} = 0

(51)

By inserting the polar form

A (T_{1}) = \frac{1}{2} a (T_{1}) e^{i θ (T_{1})}

and separating the real part form imaginary part, we obtain the following system:

{\begin{cases} - a ϖ θ^{'} + \frac{3 β_{1} α_{0}^{2}}{2} a + \frac{3 β_{1}}{8} a^{3} = - \frac{3 P_{1} β_{1}}{8 (ϖ^{2} - Ω^{2})} a^{2} \sin (ψ) \\ ϖ a^{'} + \frac{ϖ μ}{2} a = \frac{3 P_{1} β_{1}}{8 (ϖ^{2} - Ω^{2})} a^{2} \cos (ψ) \end{cases}

(52)

with $ψ = σ T_{1} - 3 θ$

Given that the steady-state response conditions are satisfied $ψ^{'}$ and $a^{'}$ , we obtain the following amplitude equation:

{((\frac{3 β_{1} P_{0}^{2}}{2 ϖ^{4}} - \frac{σ ϖ}{3}) + \frac{3 β_{1}}{8} a^{2})}^{2} + {(\frac{ϖ μ}{2})}^{2} = {(\frac{3 P_{1} β_{1}}{8 (ϖ^{2} - Ω^{2})} a)}^{2}

(53)

In the second case, we consider

Ω = \frac{3}{2} ϖ + ε σ

. Following the same development, the following amplitude equation is obtained:

{((\frac{3 β_{1} P_{0}^{2}}{2 ϖ^{4}} - \frac{2 σ ϖ}{3}) + \frac{3 β_{1}}{8} a^{2})}^{2} + {(\frac{ϖ μ}{2})}^{2} = \frac{9 {(β_{1} P_{0})}^{2}}{64 {(ϖ^{2} - 4 Ω^{2})}^{2}}

(54)

The data reported in Figure 13 show the effect of the internal fluid flow velocity on the steady-state vibration amplitude of the pipeline versus the detuning parameter $σ$ for the two cases of subharmonic resonance: Figure 13(a) $Ω = 3 ϖ + ε σ$ and Figure 13(b) $Ω = \frac{3}{2} ϖ + ε σ$ . The peak amplitude of the system increases as the flow velocity increases. It is also observed that for critical and supercritical velocities, this peak is more pronounced. However, for subcritical velocities, the influence is almost negligible.

Figure 13.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for sub harmonic resonance, (a) $Ω = 3 ϖ + ε σ$ ; (b) $Ω = \frac{3}{2} ϖ + ε σ$ . The effect of the internal fluid flow velocity (subcritical, critical, and supercritical) is shown. The simulation parameters are given in Tables 1–3.²²

c. Investigation on combined harmonic resonance of the system response.

In this part, the following situation is considered: $4 Ω = ϖ + ε σ; 5 Ω = ϖ + ε σ$

First case $4 Ω = ϖ + ε σ$ , the corresponding solvability condition is:

- 2 i ϖ A^{'} - i ϖ μ A - 3 β_{1} A^{2} \bar{A} - 3 β_{1} α_{0}^{2} A - 3 β_{1} (α_{0} α_{1}^{2} + α_{1} α_{2}^{2}) e^{i σ T_{1}} = 0

(55)

By inserting the polar form $A (T_{1}) = \frac{1}{2} a (T_{1}) e^{i θ (T_{1})}$ and proceeding as in the previous cases, we arrive at the following system of amplitude equations:

{\begin{cases} - a ϖ θ^{'} + \frac{3 β_{1} α_{0}^{2}}{2} a + \frac{3 β_{1}}{8} a^{3} = - 3 β_{1} (α_{0} α_{1}^{2} + α_{1} α_{2}^{2}) \cos (ψ) \\ ϖ a^{'} + \frac{ϖ μ}{2} a = - 3 β_{1} (α_{0} α_{1}^{2} + α_{1} α_{2}^{2}) \sin (ψ) \end{cases}

(56)

with $ψ = σ T_{1} - θ$

Given that the steady-state response conditions are satisfied $ψ^{'}$ and $a^{'}$ , we obtain the following amplitude equation:

{((\frac{3 β_{1} P_{0}^{2}}{2 ϖ^{4}} - σ ϖ) a + \frac{3 β_{1}}{8} a^{3})}^{2} + {(\frac{ϖ μ}{2} a)}^{2} = 9 β_{1}^{2} {(α_{0} α_{1}^{2} + α_{1} α_{2}^{2})}^{2}

(57)

In the second case, we consider

5 Ω = ϖ + ε σ

. Following the same development, the following amplitude equation is obtained:

{((\frac{3 β_{1} P_{0}^{2}}{2 ϖ^{4}} - σ ϖ) a + \frac{3 β_{1}}{8} a^{3})}^{2} + {(\frac{ϖ μ}{2} a)}^{2} = {(\frac{3 β_{1} P_{0}^{2} P_{1}}{8 {(ϖ^{2} - 4 Ω^{2})}^{2} (ϖ^{2} - Ω^{2})})}^{2}

(58)

For the combined harmonic resonance of the system illustrated in Figure 14, a notable and progressive increase in the peak resonance amplitudes of the structure is observed with respect to the different flow velocities of the internal fluid considered. Furthermore, for the same flow velocity conditions, the peak resonance amplitudes in the case of combined resonances are larger than those in the case of subharmonic resonances. This indicates that the flow velocity has a more considerable influence when the wave frequency is close to the combined resonance frequency than to subharmonic resonance. A control technique will be used to reduce this vibration amplitude.

Figure 14.

Steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for the combined harmonic resonance, (a) $4 Ω = ϖ + ε σ$ ; (b) $5 Ω = ϖ + ε σ$ . The effect of the internal fluid flow velocity (subcritical, critical, and supercritical) is shown. The simulation parameters are given in Tables 1–3.²²

6. Effect of the NES controller on the vibration amplitude of the submarine pipeline conveying fluid

6.1. Description of the control model

Since the vibration of the pipeline has been fully characterized as shown in Part 2 of this work, the next step is to minimize the dynamic deflection of the structure. To this end, an essentially nonlinear damped attachment is expected to irreversibly absorb the vibrational energy of the structure. A suitable way to reduce vibration in mechanical structures conveying fluid and subjected to regular wave excitation is to use a Nonlinear Energy Sink (NES) controller. The NES functions as an effective vibration absorber for nonlinear systems and has recently been reported to engage in resonance over a very broad frequency range³³. Figures 15–17 clearly illustrate the pipeline with the controller on the sea floor, followed by a mechanical model of the controller and finally a simplified model of the system.

Figure 15.

Schematic diagram of a sub-marine pipeline on the sea floor with integrated vibration control mechanism.

Figure 16.

Simplified model of the NES controller.

Figure 17.

Simplified model of the submarine pipeline with NES controller.

By taking into account the same considerations and hypotheses as before, the mathematical model describing the behavior of the submarine pipeline conveying fluid with an integrated vibration controller can be expressed as follows:

(ρ_{p} A_{p} + ρ_{f} A_{f}) \frac{\partial^{2} w}{\partial t^{2}} + c \frac{\partial w}{\partial t} + 2 ρ_{f} A_{f} V \frac{\partial^{2} w}{\partial x \partial t} + (ρ_{f} A_{f} V^{2} - P) \frac{\partial^{2} w}{\partial x^{2}} + E I \frac{\partial^{4} w}{\partial x^{4}} - \frac{E A_{p}}{2 L} \frac{\partial^{2} w}{\partial x^{2}} \int_{0}^{L} {(\frac{\partial w}{\partial x})}^{2} + f_{N E S} (x_{0}, t) δ (x - x_{0}) = f_{M} (x, z, t)

(59)

where

f_{N E S} (x_{0}, t)

represent the force generated by the controller defined as:³⁴

f_{N E S} (x_{0}, t) = k_{N E S} {(y - w)}^{3} + c_{N E S} (\frac{d y}{d t} - \frac{d w}{d t})

(60)

By substituting Eqs.(20) and (60) into (59), we obtain the following equation:

(ρ_{p} A_{p} + ρ_{f} A_{f} + ρ_{f_{w}} c_{a} A_{p}) \frac{\partial^{2} w}{\partial t^{2}} + (c + \frac{π H ρ_{f_{w}} c_{D} D}{T \sin h (k d)} \cos (ω t)) \frac{\partial w}{\partial t} + 2 ρ_{f} A_{f} V \frac{\partial^{2} w}{\partial x \partial t} +

(ρ_{f} A_{f} V^{2} - P) \frac{\partial^{2} w}{\partial x^{2}} - \frac{E A_{p}}{2 L} \frac{\partial^{2} w}{\partial x^{2}} \int_{0}^{L} {(\frac{\partial w}{\partial x})}^{2} d x + (k_{N E S} {(y - w)}^{3} + c_{N E S} (\frac{d y}{d t} - \frac{d w}{d t})) δ (x - x_{0}) + E I \frac{\partial^{4} w}{\partial x^{4}} = \frac{π^{2} H^{2} ρ_{f_{w}} c_{D} D}{4 {(T \sin (k d))}^{2}} \cos (2 ω t) + \frac{2 π^{2} H ρ_{f_{w}} c_{M} A_{P}}{T^{2} \sin (k d)} \sin (ω t) + \frac{π^{2} H^{2} ρ_{f_{w}} c_{D} D}{4 {(T \sin (k d))}^{2}}

(61)

By applying the Lagrangian method³⁴ on the discrete system of the controller (Figure 16), the equation of motion of the controller is expressed as:

m_{N E S} \frac{d^{2} y}{d t^{2}} + k_{N E S} {(y - w)}^{3} + c_{N E S} (\frac{d y}{d t} - \frac{d w}{d t}) = 0

(62)

Hence the governing equation of the pipeline conveying fluid under the action of wave loads with integrated vibration controller is given by:

{\begin{cases} \begin{array}{l} (ρ_{p} A_{p} + ρ_{f} A_{f} + ρ_{f_{w}} c_{a} A_{p}) \frac{\partial^{2} w}{\partial t^{2}} + (c + \frac{π H ρ_{f_{w}} c_{D} D}{T \sin h (k d)} \cos (ω t)) \frac{\partial w}{\partial t} + 2 ρ_{f} A_{f} V \frac{\partial^{2} w}{\partial x \partial t} \end{array} \\ \begin{array}{l} + (ρ_{f} A_{f} V^{2} - P) \frac{\partial^{2} w}{\partial x^{2}} - \frac{E A_{p}}{2 L} \frac{\partial^{2} w}{\partial x^{2}} \int_{0}^{L} {(\frac{\partial w}{\partial x})}^{2} d x + (k_{N E S} {(y - w)}^{3} + c_{N E S} (\frac{d y}{d t} - \frac{d w}{d t})) δ (x - x_{0}) \end{array} \\ \begin{array}{l} + E I \frac{\partial^{4} w}{\partial x^{4}} = \frac{π^{2} H^{2} ρ_{f_{w}} c_{D} D}{4 {(T \sin (k d))}^{2}} \cos (2 ω t) + \frac{2 π^{2} H ρ_{f_{w}} c_{M} A_{P}}{T^{2} \sin (k d)} \sin (ω t) + \frac{π^{2} H^{2} ρ_{f_{w}} c_{D} D}{4 {(T \sin (k d))}^{2}} \end{array} \\ \begin{array}{l} m_{N E S} \frac{d^{2} y}{d t^{2}} + k_{N E S} {(y - w)}^{3} + c_{N E S} (\frac{d y}{d t} - \frac{d w}{d t}) = 0 \end{array} \end{cases}

(63)

Consider the following dimensionless variables:

w = r η; x = L ξ; τ = ω_{0} t; Ω = \frac{ω}{ω_{0}}; ϑ = \frac{y}{r}; ξ_{0} = \frac{x_{0}}{L}

ω_{0} = \frac{1}{L^{2}} \sqrt{\frac{E I}{M_{e}}}; r = \sqrt{\frac{I}{A_{p}}}; M_{e} = ρ_{p} A_{p} + ρ_{f} A_{f} + ρ_{f} c_{a} A_{p}

μ = \frac{L^{2} c}{\sqrt{E I M_{e}}}; μ_{t} = \frac{L^{2} ρ_{f_{w}} c_{D} D π H}{T \sin (k d) \sqrt{E I M_{e}}}; β = \frac{ρ_{f} A_{f}}{M_{e}}; γ = \frac{P L^{2}}{E I}; χ = V L \sqrt{\frac{ρ_{f} A_{f}}{E I}} C_{N E S} = \frac{2 c_{N E S} L^{4}}{E I}

f_{0} = \frac{{(π H)}^{2} ρ_{f_{w}} c_{D} D L^{4} \sqrt{A_{p}}}{4 E {(T \sin (k d))}^{2} {(\sqrt{I})}^{3}}; f_{1} = \frac{2 π^{2} H ρ_{f_{w}} c_{M} L^{4} {(\sqrt{A_{p}})}^{3}}{T^{2} E \sin (k d) {(\sqrt{I})}^{3}}; K_{N E S} = \frac{2 k_{N E S} L^{5}}{E A_{P}} λ_{1} = \frac{m_{N E S} E I}{L^{4} M_{e}}; λ_{2} = \frac{k_{N E S} I}{A_{P}}; λ_{3} = \frac{c_{N E S} \sqrt{E I}}{L^{2} \sqrt{M_{e}}}

(64)

By substituting (64) in (63) and after some mathematical expansion and reduction, we obtain the following dimensionless equations:

\frac{\partial^{2} η}{\partial τ^{2}} + \frac{\partial^{4} η}{\partial ξ^{4}} + (μ + μ_{t} \cos (Ω τ)) \frac{\partial η}{\partial τ} + 2 \sqrt{β} χ \frac{\partial^{2} η}{\partial τ \partial ξ} + (χ^{2} - γ) \frac{\partial^{2} η}{\partial ξ^{2}} - \frac{1}{2} \frac{\partial^{2} η}{\partial ξ^{2}} {\int_{0}^{1} (\frac{\partial η}{\partial ξ})}^{2} d ξ + (K_{N E S} {(ϑ - η)}^{3} + C_{N E S} (\frac{d ϑ}{d τ} - \frac{d η}{d τ})) δ (ξ - ξ_{0}) = f_{0} \cos (2 Ω τ) + f_{0} + f_{1} \sin (Ω τ)

(65a)

λ_{1} \frac{d^{2} ϑ}{d τ^{2}} + λ_{2} {(ϑ - η)}^{3} + λ_{3} (\frac{d ϑ}{d τ} - \frac{d η}{d τ}) = 0

(65b)

6 2. Modal equation and effectiveness of the controller

To obtain the modal equation, we apply Galerkin’s method as done in the first part of this work. Thus, using equation (24) and after some manipulations, the modal equation of motion for the fundamental vibration mode (n=1) is given by:

{\ddot{q}}_{1} (τ) + (μ + μ_{t} \cos (Ω τ)) {\dot{q}}_{1} (τ) + ϖ^{2} q_{1} + β_{1} q_{1}^{3} + K_{N E S} {(ϑ - q_{1})}^{3} + C_{N E S} (\dot{ϑ} - {\dot{q}}_{1}) = P_{0} \cos (2 Ω τ) + P_{1} \sin (Ω τ) + P_{0}

(66a)

λ_{1} \ddot{ϑ} + λ_{2} {(ϑ - q_{1})}^{3} + λ_{3} (\dot{ϑ} - {\dot{q}}_{1}) = 0

(66b)

The effectiveness of the NES coupled to the transverse vibration of the fluid-conveying pipeline is demonstrated. The numerical method based on the fourth-order Runge–Kutta algorithm is used to solve the coupled equations 66a and 66b in order to obtain the displacement amplitude of the pipeline with and without the NES controller.

Figure 18 illustrates the time displacement responses of the structure with and without the NES controller. It is observed that the viscoelastic effects and the mass of the controller are more pronounced, and the vibration of the system with the NES controller is drastically reduced compared to the system without the controller. This indicates that the NES controller is effective and ensures the safety of the structure. A reduction percentage of R=58% is observed for values of $M_{n e s} = 6$ , $K_{n e s} = 2 \times 1 0^{14}$ , and $C_{n e s} = 5$ , which is already very significant for the effectiveness of the controller. Figure 19 presents the influence of the controller on the primary resonance amplitude of the pipeline. A reduction of 6.47% in the peak resonance vibration amplitude of the pipeline is observed. Furthermore, the controlled amplitude response reaches its resonance peak at a slightly higher wave frequency than that of the uncontrolled response. This indicates that the controller parameters influence not only the natural frequency of the pipeline but also the wave frequency.

Figure 18.

Comparison of the pipeline response with and without NES (gray: pipeline coupled with NES; black: pipeline without NES) for $χ = 0.0328$ , $Ω = 10.15$ and $ξ_{0} = 0.5$ .

Figure 19.

Effectiveness of the controller on the steady-state vibration amplitude of the pipeline as a function of the detuning parameter $σ$ for primary harmonic resonance, showing the effect of the NES controller with $χ = 0.0328$ .

7. Conclusion

Summing up, we have investigated the dynamic behavior of a submarine pipeline conveying fluid, with and without a vibration controller, under wave excitation. The Euler–Bernoulli beam theory was used to model the dynamic behavior of the pipeline. Wave excitation was modeled as a hydrodynamic force consisting of inertial and Morison-type drag components. After a brief description of the physical model, the governing equations of the system, with and without control, were established using the generalized Hamiltonian variational principle. A nonlinear energy sink (NES)-type controller was employed to reduce the vibrations of the pipeline model. An analytical technique based on the multiple time scale method was applied to the linearized system to illustrate how the internal fluid flow velocity, controller parameters, wave amplitude, and fluctuation parameter of the drag force affect the system’s response. The results were compared with numerical solutions of the modal equation, showing good agreement. The effect of the internal fluid flow velocity on primary, secondary, superharmonic, subharmonic, and combined resonance states has been investigated. The analysis leads us to the conclusion that increasing the internal fluid flow velocity increases the peak amplitude response of the pipeline. This means that a high value of this velocity can contribute to destabilizing the system. Another interesting finding is that increasing the wave excitation intensity also increases the resonance (peak amplitude) of the pipeline. This important result demonstrates that the highest wave excitation intensity can cause large displacements and contribute to destabilizing the structure. Furthermore, the resonance phenomenon appears as the wave frequency increases, for all types of resonant states. To highlight what we consider as new, the fluctuations generated by the approximation of the drag force which, after approximation, presented a notable increase in the peak resonance amplitude have been investigated. This load parameter can cause large displacements of the pipeline and thus lead to significant consequences for the structure. Finally, a NES controller has been used to mitigate the vibrations of the pipeline. The results show an effective vibration reduction of 58%.

This exploratory analysis has, of course, its limitations. Most importantly, for more realistic simulations, one might consider different values for the pipeline parameters, chosen with greater realism rather than, as in the present work, to accelerate simulations and highlight qualitative results. For example, only the first mode was used in Galerkin’s method, and higher coupled modes were not included. It would be interesting to extend this work to incorporate these two main ideas. Nevertheless, we hope that the qualitative conclusions obtained in this study can provide reference values for such developments.

Footnotes

ORCID iDs

Barthelemy Roger Lekoa

Lionel Merveil Anague Tabejieu

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

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