Analysis of Vibration Characteristics of Viscoelastic Slurry Pipe Considering Fluid–Structure Interaction Effects

Abstract

To study the vibration characteristics of viscoelastic slurry pipe structures under fluid–structure interaction (FSI), we constructed a three-dimensional FSI pipe model based on the finite element method to systematically investigate the effects of fluid effects, pipe length, and wall thickness on the vibrational characteristics of viscoelastic slurry pipes. A modal analysis demonstrated that fluid effects not only significantly reduced the natural frequency of the pipe but also disrupted the symmetry of the vibration modes and eliminated the phenomenon of frequency degeneracy. The frequency reduction caused by FSI reached 54%, which was dominant compared with the water-attached effects, and its impact intensified with the increasing vibration order. The water-attached effect exhibited differences between odd and even orders, attributed to the influence of vibration modes on the distribution of fluid inertial forces, with a contribution of 45.07% to 55.24% in the odd orders and of only 37.69% to 38.93% in the even orders. When the FSI and water-attached effects acted together, the frequency reduction was further aggravated, but the reduction ratio did not follow a simple linear superposition. The parametric analysis of the pipe showed that when the pipe length increased from 1 m to 3 m, the growth rate of its natural frequency was only 26.52% that of the shorter pipe, indicating that the longer the pipes, the slower the growth rate of frequency. When the wall thickness increased from 5 mm to 11 mm, the growth rate of the first-order natural frequency decreased from 15.43% to 7.44%, suggesting that the frequency improvement effect caused by the stiffness augmentation diminished with the increase in wall thickness. The research results hold significant guiding significance for the structural design of slurry pipe systems in practical engineering and the safe operation of pipe systems.

1. Introduction

Hydraulic mechanical excavation is a construction method that utilizes the scouring force of water flow to liquefy soil into slurry, which is then transported over long distances by slurry pumps and slurry transport pipes. It is widely applied in ground excavation projects. During construction, to avoid affecting traffic and municipal facilities, the slurry pipes can be laid within watercourse. These pipes typically employ viscoelastic materials such as polyethylene (PE) pipes. During pumping operations, conditions like slurry pump startup and shutdowns or accidental power outages can generate water hammer waves, to varying degrees, in the pipe system. The resulting water hammer pressure fluctuations induce pipe vibration, which, in turn, alters the flow state of the slurry. There exists a complex coupling relationship between them, commonly referred to as the fluid–structure interaction (FSI) effect [1]. This unstable state leads to increased frictional head loss and local head losses of the pipe, elevates the lift of the slurry pump, and consequently increases pumping energy consumption and wear. Meanwhile, when the natural frequency of the pipe is close to the frequency of external excitation forces, resonance will occur, leading to a significant increase in amplitude. This causes energy to be concentrated and transmitted to the pipe and its connecting components such as flanges, ultimately resulting in structural damage or even safety accidents. Therefore, studying the vibration characteristics of viscoelastic slurry pipes under FSI, analyzing the mechanism of pipe structural vibration, and optimizing the pipe structural design hold important guiding significance for the safe operation of pipe systems.

The problem of FSI vibration in pipe systems exists across various engineering fields. Pipe vibration characteristics manifest not only in the action of fluid pressure on the pipe structure but also in the influence of pipe deformation on the fluid flow state. Research focusing on the mutual coupling between the two is namely pipe FSI [2]. Chen et al. [3] established a hydraulic transient model for gas–liquid–solid three-phase flow with FSI effects to address multiphase flow problems in engineering. Dai [4] investigated various factors affecting the natural frequency of pipe structures and showed that stable binding forces significantly impact the natural frequency of pipes. Zhou et al. [5] used a finite element FSI model to analyze the influence of flow conditions and geometric changes in vertical transport pipes on self-excited vibration frequencies induced by internal or external pipe fluids. Bao [6] simulated the pipe as a common bearing beam, considered the interaction between the internal and external pipe fluids and the elastic coefficient of the supporting soil, and studied the influence of factors such as the soil elastic coefficient on the natural frequency of free-span submarine pipes through the differential orthogonal method. Mikota et al. [7] studied FSI in bend pipelines using a modal method and conducted a two-degree-of-freedom study on resonance coupling using the modal coordinates of hydraulic and mechanical subsystems. Experimental modal analysis of the coupled hydraulic–mechanical system confirmed the predicted resonance splitting. Ferras [8] studied the FSI effect in the hydraulic transients of straight pipes under different anchoring conditions by constructing a four-equation model of the pipe, realizing three main interaction mechanisms: Poisson, friction, and binding coupling. Yang et al. [9] considered geometric and hydrodynamic nonlinear factors, established a three-dimensional nonlinear dynamic model, and described the behavior of flexible fluid transport pipes under eddy current-induced vibration through the extended Hamilton principle. Lannes et al. [10] found that the vibration acceleration of horizontal pipes exhibited a positively exponential relationship with the volume flow rate of gas–liquid two-phase flow and the gas content in the pipe. Zhao et al. [11] analyzed the natural frequency of T-shaped pipes under fluid action based on bidirectional FSI and studied the influence of fluid pressure, fluid velocity, and fluid density on the modal of T-shaped pipes. Zhu et al. [12] studied the vibration caused by the pressure fluctuations of gas–liquid two-phase flow within pipes and found that with the increase in the specific gravity of the gas–liquid two-phase flow, the vibration of the flexible risers became more intense. A.O. Mohmmed et al. [13] compared the influence of liquid flow velocity on the pipe stress between unidirectional FSI and the structural interaction of gas–liquid two-phase flow in horizontal pipes through experimental and numerical simulation methods, finding excellent agreement between the numerical simulation and the experimental measurement results of stress, with a maximum error of 10.2%.

The above-described research achievements are primarily based on the assumption that the wall of the water transport pipe is elastic, without involving the influence of wall viscosity, and are mainly applicable to elastic pipes such as cast iron pipes, steel pipes, and alloy pipes. With the rapid development of polymer materials, plastic pipes such as PE, polyvinyl chloride (PVC), and polypropylene (PP) pipes are now being widely used due to their advantages of good durability, low cost, and convenient production and installation. These are categorized as viscoelastic pipes. Viscoelastic pipes refer to pipes whose wall materials not only exhibit the characteristic of instantaneous deformation like elastic solids but also demonstrate deformation behavior that changes over time. Similar to viscous fluids, they possess the dual mechanical properties of elastic recovery and viscous dissipation. For viscoelastic pipes, their water hammer vibration characteristics become considerably more complex, and the fundamental theories of elastic pipes are no longer applicable, requiring various improvements.

Soares et al. [14] measured the creep function of PVC pipes and confirmed that the viscoelastic effect of PVC pipes has a significant impact on hydraulic transients. Evangelista and Leopardi et al. [15] studied the vibration characteristics of high-density polyethylene (HDPE) pipes under water hammer action. Mousavifard [16] established a two-dimensional model of cavitation in the transient flow distribution of low-density polyethylene (LDPE) pressure pipes combined with the k-ω turbulence model. Wahba [17] simulated the viscoelastic effect of viscoelastic pipes by combining a two-dimensional transient flow model with Kelvin–Voigt, and their findings indicated that viscoelastic effects became more pronounced with the increase in pipe length, diameter, and wave speed. Zhu [18], through experimental data calibration and validation, showed that as the gas content in the liquid increased, the influence of the viscoelastic effect of the viscoelastic pipe wall on transient amplitude damping exhibited a decreasing trend. Urbanowicz and Firkowski [19] considered the delayed deformation of viscoelastic pipes and found that for pressure decay, the importance of frictional resistance was far less than the viscoelastic effect of the pipe wall. Andrade et al. [20] compared the changes in pipe energy and fluid energy between elastic and viscoelastic pipes under different conditions, concluding that pipe viscoelasticity not only delayed pressure oscillations but also made the pressure decay more rapid.

Although certain progress has been made in current FSI research on viscoelastic pipes, most of it focuses on gas–liquid two-phase flow, and there are few studies on solid–liquid two-phase flow, such as that of transported slurry. Therefore, this study constructs an FSI modal analysis model of PE slurry pipes and, based on this, analyzes the influence laws of pipe length, wall thickness parameters, and FSI effects on the vibration characteristics of the system.

2. Materials and Methods

2.1. Fluid–Structure Interaction Mathematical Model

Viscous incompressible fluids possess a constant density and have internal frictional properties. Utilizing the law of mass conservation and the law of momentum conservation, along with the constitutive relation of fluid viscosity, the inter-relations between physical quantities such as velocity, pressure, and density are formulated, namely, Navier–Stokes equations [21].

The mass control equation is as follows:

$${\displaystyle \frac{\partial \mathit{\rho }}{\partial \mathit{t}}} + \nabla ·\mathit{\rho }{\mathit{u}}_{\mathit{f}} = 0$$

(1)

where t is the time (s); ρ is the fluid density (kg/m³); and u_f denotes the fluid velocity vector.

For incompressible fluids, $\partial \mathit{\rho }/\partial \mathit{t}=0$, and this equation can be simplified to the following:

$$\nabla ·{\mathit{u}}_{\mathit{f}} = 0$$

(2)

The momentum control equation is as follows:

$${\displaystyle \frac{\partial {\mathit{u}}_{\mathit{f}}}{\partial \mathit{t}}}+\left({\mathit{u}}_{\mathit{f}}·\nabla \right){\mathit{u}}_{\mathit{f}}=-{\displaystyle \frac{1}{\mathit{\rho }}}\nabla \mathit{p}+\mathit{f}+\mathit{\nu }{\nabla }^2{\mathit{u}}_{\mathit{f}}$$

(3)

where p is the fluid pressure (Pa); f is the body force vector of the fluid medium; v is the kinematic viscosity (m²/s); and $\nabla$ is the Hamilton operator.

In the calculation of the vibration characteristics of slurry pipes, accounting for FSI, the realizable k-ε model [22] more realistically computes the amplitude and frequency distribution of turbulent fluctuation pressure by accurately resolving the turbulent fluctuation velocity field. Consequently, the realizable k-ε turbulence model is applied for the fluid flow equations, with the equations for the turbulent kinetic energy (k) and the turbulent energy dissipation rate (ε) being specified as follows:

k:

$${\displaystyle \frac{\partial \left(\mathit{\rho }\mathit{k}\right)}{\partial \mathit{t}}}+{\displaystyle \frac{\partial \left(\mathit{\rho }\mathit{k}{\mathit{u}}_{\mathit{i}}\right)}{\partial {\mathit{x}}_{\mathit{i}}}}={\displaystyle \frac{\partial }{\partial {\mathit{x}}_{\mathit{j}}}}\left[\left(\mathit{\mu }+{\displaystyle \frac{{\mathit{\mu }}_{\mathit{t}}}{{\mathit{\sigma }}_{\mathit{k}}}}\right){\displaystyle \frac{\partial \mathit{k}}{\partial {\mathit{x}}_{\mathit{j}}}}\right]+{\mathit{G}}_{\mathit{k}}+{\mathit{G}}_{\mathit{b}}-\mathit{\rho }\mathit{\epsilon }-{\mathit{Y}}_{\mathit{m}}+{\mathit{S}}_{\mathit{k}}$$

(4)

ε:

$${\displaystyle \frac{\mathit{\text{∂}}\left(\mathit{\rho }\mathit{\epsilon }\right)}{\partial \mathit{t}}}+{\displaystyle \frac{\partial \left(\mathit{\rho }\mathit{\epsilon }{\mathit{u}}_{\mathit{i}}\right)}{\partial {\mathit{x}}_{\mathit{i}}}}={\displaystyle \frac{\partial }{\partial {\mathit{x}}_{\mathit{j}}}}\left[\left(\mathit{\mu }+{\displaystyle \frac{{\mathit{\mu }}_{\mathit{t}}}{{\mathit{\sigma }}_{\mathit{\epsilon }}}}\right){\displaystyle \frac{\partial \mathit{\epsilon }}{\partial {\mathit{x}}_{\mathit{j}}}}\right]+\mathit{\rho }{\mathit{C}}_1{\mathit{S}}_{\mathit{\epsilon }}-{\displaystyle \frac{\mathit{\rho }{\mathit{C}}_2{\mathit{\epsilon }}^2}{\mathit{k}+\sqrt{\mathit{\nu }\mathit{\epsilon }}}}+{\mathit{C}}_{1\mathit{\epsilon }}{\displaystyle \frac{\mathit{\epsilon }}{\mathit{k}}}{\mathit{C}}_{3\mathit{\epsilon }}{\mathit{G}}_{\mathit{b}}+{\mathit{S}}_{\mathit{\epsilon }}$$

(5)

where µ is the dynamic viscosity (N·s/m²); µ_t is the turbulent viscosity coefficient; u_i and u_j is the velocity component; x_i and x_j denote the coordinate component; σ_k and σ_ε denote the turbulent Prandtl number; G_k is the turbulent kinetic energy generation term caused by average velocity gradient; G_b is the turbulent kinetic energy term produced by buoyancy; Y_m is the influence of compressible turbulent fluctuation; ${\mathit{C}}_1=\max \left[0.43,\mathit{\eta }/\left(\mathit{\eta }+5\right)\right]$; $\mathit{\eta }={\mathit{S}}_{\mathit{k}}/\mathit{\epsilon }$; C₂, C_1ε and C_3ε are constant; and S_k and Sε denote the source item.

FSI needs to conform to the principle of conservation [23]. At the fluid–structure interface, the fluid and structure stresses (τ) and displacements (d) are conserved, meaning they adhere to the following conditions:

$${\mathit{\tau }}_{\mathit{f}}{\mathit{n}}_{\mathit{f}}={\mathit{\tau }}_{\mathit{s}}{\mathit{n}}_{\mathit{s}}$$

(6)

$${\mathit{d}}_{\mathit{f}}={\mathit{d}}_{\mathit{s}}$$

(7)

where τ_f is the shear vector of fluid; n_f is the number of fluid nodes; τ_s is the shear force vector of solid; n_s is the number of solid nodes; d_f is the fluid displacement (m); and d_s is the displacement of solid (m).

By establishing an FSI model, the pressure fluctuation information within the pipe is acquired and utilized as the boundary conditions for modal analysis to investigate the vibration characteristics of the pipe. The fundamental dynamic equations are as follows:

$$\left[\mathit{M}\right]\left\{\ddot{\mathit{x}}\right\}+\left[\mathit{C}\right]\left\{\dot{\mathit{x}}\right\}+\left[\mathit{K}\right]\left\{\mathit{x}\right\}=\left\{\mathit{F}\left(\mathit{t}\right)\right\}$$

(8)

where [M] is the mass matrix; [C] is the damping matrix; [K] is the stiffness matrix; {$\ddot{\mathit{x}}$} is the acceleration vector; {$\dot{\mathit{x}}$} is the velocity vector; and {x} is the displacement vector.

To study the most essential natural vibration characteristics of pipe structures, a free modal analysis is required. Therefore, neglecting the effects of damping and external forces, meaning both [C] and [F(t)] become zero. Equation (8) can be simplified to the following:

$$\left[\mathit{M}\right]\left\{\ddot{\mathit{x}}\right\}+\left[\mathit{K}\right]\left\{\mathit{x}\right\}=0$$

(9)

Assuming a solution in the form of simple harmonic vibration $\left\{\mathit{x}\right\}={\left\{\mathit{\phi }\right\}}_{\mathit{i}}\mathit{cos}{\mathit{\omega }}_{\mathit{i}}\mathit{t}$ and substituting it into Equation (9) yields the following:

$$\left(\left[\mathit{K}\right]-{\mathit{\omega }}_{\mathit{i}}^2\left[\mathit{M}\right]\right){\left\{\mathit{\phi }\right\}}_{\mathit{i}}=0$$

(10)

where {φ}_i is the Eigenvector corresponding to the mode shape of the i-th order mode, and ω_i is the natural frequency of the i-th order mode.

Since {φ}_i cannot be identically zero, the determinant of the coefficient matrix must be zero:

$$\mathit{det}\left|\left[\mathit{K}\right]-{\mathit{\omega }}_{\mathit{i}}^2\left[\mathit{M}\right]\right|=0$$

(11)

In Equation (11), the vibration frequency ω_i. of the structure can be obtained from the stiffness matrix and mass matrix of the structure, and the equation for calculating the natural frequency is as follows:

$${\mathit{f}}_{\mathit{i}}={\displaystyle \frac{{\mathit{\omega }}_{\mathit{i}}}{2\mathit{\pi }}}$$

(12)

This demonstrates that the natural frequency is primarily determined by the stiffness and mass of the structure.

To enhance the universality of the model and eliminate dependence on specific geometric dimensions, the following non-dimensional quantities are introduced:

$$\left\{{\mathit{x}}^{*}\right\}={\displaystyle \frac{\left\{\mathit{x}\right\}}{\mathit{L}}}$$

(13)

$$\left[{\mathit{K}}^{*}\right]={\displaystyle \frac{\left[\mathit{K}\right]{\mathit{L}}^2}{\mathit{EI}}}$$

(14)

$$\left[{\mathit{M}}^{*}\right]={\displaystyle \frac{\left[\mathit{M}\right]}{\mathit{\rho }\mathit{AL}}}$$

(15)

$${\mathit{\omega }}_{\mathit{i}}^{*}={\mathit{\omega }}_{\mathit{i}}\mathit{L}\sqrt{{\displaystyle \frac{\mathit{\rho }\mathit{A}}{\mathit{EI}}}}$$

(16)

where L is the pipe length (m); E is the elastic modulus (Pa); I is the moment of inertia of cross-section (m²); and A is the cross-sectional area (m²).

Equation (9) can then be expressed in the following non-dimensional form:

$$\left[{\mathit{M}}^{*}\right]\left\{{\ddot{\mathit{x}}}^{*}\right\}+\left[{\mathit{K}}^{*}\right]\left\{{\mathit{x}}^{*}\right\}=0$$

(17)

The conversion relationship between the non-dimensional natural frequency ω_i ^∗ and the dimensional frequency f_i is as follows:

$${\mathit{f}}_{\mathit{i}}={\displaystyle \frac{{\mathit{\omega }}_{\mathit{i}}^{*}}{2\mathit{\pi }\mathit{L}}}\sqrt{{\displaystyle \frac{\mathit{\rho }\mathit{A}}{\mathit{EI}}}}$$

(18)

In practical engineering, based on the natural frequencies obtained from the above free modal analysis, damping effects and external excitations are reintroduced for modal analysis. In the modal analysis method, the dry modal method separates the structural system from the fluid, disregarding the fluid’s coupling effects. Dry modal analysis can ascertain the natural frequencies, mode shapes, and associated amplitude distributions of fluid-conveying pipes in spite of fluid action, thereby reflecting the natural frequencies determined solely by the structural characteristics and being independent of external excitation conditions.

The wet modal method initially transforms thee coordinates via the natural modal matrix, converts the system into a single-degree-of-freedom system described by the principal coordinates of the solution, and subsequently applies the superposition principle to obtain the solution of the multiple-degree-of-freedom system. This method is appropriate for FSI systems but requires that we account for the influence of the surrounding fluid. Wet modes not only reflect the structural dynamic characteristics but also the frequency drift and damping attributes induced by the fluid medium.

2.2. Calculation of Immersion Depth

In actual construction, to avoid affecting traffic and municipal facilities, the slurry pipe is laid in a watercourse. Given that the slurry pipe density is 950 kg/m³, which is less than the density of water (998.2 kg/m³), the pipe will exhibit a floating state when empty, as illustrated in Figure 1. Based on the buoyancy principle, the immersion depth, h, of the pipe is determined by balancing the pipe’s self-weight with the volume of displaced water. The derivation process is as follows.

We calculate the pipe mass m as follows:

$${\mathit{A}}_{\mathit{p}}=\mathit{\pi }\left({\mathit{R}}^2-{\mathit{r}}^2\right)$$

(19)

$$\mathit{m}={\mathit{\rho }}_{\mathit{p}}{\mathit{A}}_{\mathit{p}}\mathit{L}$$

(20)

From the balance between gravity and buoyancy,

$${\mathit{\rho }}_{\mathit{w}}{\mathit{A}}_{\mathit{s}}\mathit{L}\mathit{g}=\mathit{mg}$$

(21)

$${\mathit{A}}_{\mathit{s}}={\mathit{\rho }}_{\mathit{p}}/{\mathit{\rho }}_{\mathit{w}}{\mathit{A}}_{\mathit{p}}$$

(22)

The immersion area A_s is given by the following:

$${\mathit{A}}_{\mathit{s}}={\mathit{R}}^2{\mathit{cos}}^{-1}\left({\displaystyle \frac{\mathit{R}-\mathit{h}}{\mathit{R}}}\right)-\left(\mathit{R}-\mathit{h}\right)\sqrt{2\mathit{Rh}-{\mathit{h}}^2}$$

(23)

Simultaneously solving the above equations yields the immersion depth h.

Here, A_p is the pipe cross-sectional area (m²); ρ_p is the pipe density (kg/m³); ρ_w is the water density (kg/m³); R is the pipe outside radius (m); r is the pipe inside radius (m); and g is the gravitational acceleration (m/s²).

During the slurry pumping process, the PE pipe will undergo bending deflection under the influence of the slurry’s gravity, causing its actual shape to deviate from the initial state. Thus, the immersion depth in the empty pipe state is no longer applicable. At this point, the pipe is regarded as a beam with both ends fixed, and the self-weight of the pipe and slurry is treated as a uniform load, Q, applied to the beam, as illustrated in Figure 2.

The deflection w is calculated based on the deflection curve equation concerning mechanics of materials, which allows us to determine the immersion depth of different cross-sections of the pipe filled with slurry.

$$\mathit{w}={\displaystyle \frac{1}{\mathit{EI}}}\left(-{\displaystyle \frac{\mathit{QL}}{24}}{{\mathit{l}}_{\mathit{x}}}^4+{\displaystyle \frac{\mathit{QL}}{12}}{{\mathit{l}}_{\mathit{x}}}^3-{\displaystyle \frac{\mathit{Q}{\mathit{L}}^2}{24}}{{\mathit{l}}_{\mathit{x}}}^2\right)$$

(24)

where D is the pipe diameter (m), and l_x is the distance from the pipe cross-section to the left end (m).

2.3. Pipe Geometry Model

The selection of model dimensions refers to the national standard of the People’s Republic of China “Polyethylene (PE) Pipe Systems for Water Supply-Part 2: Pipes” (GB/T 13663.2-2018), with comprehensive consideration of engineering practicability, adaptability to research objectives, and effectiveness of numerical calculations [24]: A nominal outer diameter of 125 mm meets the design requirements in the standard, ensuring that the research results have reference value for practical engineering. A wall thickness of 9 mm falls within the common range (5–14 mm) for PE slurry transportation pipes, which not only guarantees the basic structural stability of the pipe but also allows setting up a wall thickness gradient of 5 mm, 7 mm, and 11 mm centered around it, helping to reveal the attenuation law of wall thickness on frequency growth. A length of 3 m forms a gradient with pipe lengths of 1 m and 2 m, which can clearly reflect the influence of pipe length on the rate of frequency growth. Meanwhile, this length setting also balances calculation accuracy and efficiency.

Therefore, a polyethylene pipe with the above geometric parameters (nominal outer diameter of 125 mm, wall thickness of 9 mm, and length of 3 m) was selected for modal analysis, with both ends of the pipe fixed and constrained. Its geometric schematic diagram is shown in Figure 3.

ANSYS-Mesh (Version 2022 R1) was employed for mesh generation, and mesh independence was verified based on the maximum equivalent elastic strain and maximum equivalent stress. When the number of grids reaches 250,000, the data accuracy tends to be stable. The final mesh delineation results are shown in Figure 4 and the final number of grids is 296,666, the minimum value of grid orthogonal quality is 0.50. The verification results of grid independence are shown in Figure 5.

Other material parameters are listed in

Table 1. Physical parameters of materials.

	Parameter	Value
Slurry	Density (kg/m3)	1338.56
	Viscosity (kg/m·s)	1.5045 × 10−3
	Poisson Ratio	0.33
	Bulk Modulus (Pa)	2.61 × 109
	Sound Velocity (m/s)	1396
Polyethylene	Density (kg/m3)	950
	Poisson Ratio	0.42
	Shear Modulus (Pa)	3.8732 × 108
	Damping Factor	0.005

.

2.4. Model Validation

The Timoshenko beam model [25] is an advanced form of the Euler–Bernoulli beam model. It accounts for the effects of shear deformation and rotational inertia, making it more accurate for the analysis of stubby beams, thick-walled pipes, and high-order vibration modes. In this study, the Timoshenko beam theory was used to theoretically calculate the natural frequencies of the pipe in the empty state, and the results were compared with those from the finite element method to ensure the accuracy of the model.

The governing equations for its free vibration are as follows:

$$\mathit{\rho }\mathit{A}{\displaystyle \frac{{\partial }^2\mathit{y}}{{\partial \mathit{t}}^2}}-\mathit{\kappa }\mathit{AG}{\displaystyle \frac{\partial }{\partial \mathit{x}}}\left({\displaystyle \frac{\partial \mathit{y}}{\partial \mathit{x}}}-\mathit{\psi }\right)=0$$

(25)

$$\mathit{E}\mathit{I}{\displaystyle \frac{{\partial }^2\mathit{\psi }}{{\partial \mathit{x}}^2}}-\mathit{\kappa }\mathit{AG}\left({\displaystyle \frac{\partial \mathit{y}}{\partial \mathit{x}}}-\mathit{\psi }\right)-\mathit{\rho }\mathit{I}{\displaystyle \frac{{\partial }^2\mathit{\psi }}{{\partial \mathit{t}}^2}}=0$$

(26)

where y is the lateral displacement of the beam (m); ψ is the rotation angle of the beam cross-section due to bending (°); and I is the shear modulus (pa).

κ is the shear correction factor (Timoshenko shear factor), which is used to correct the non-uniformity of the shear stress distribution across the cross-section. Its calculation formula is as follows:

$$\mathit{\kappa }={\displaystyle \frac{6\left(1+\mathit{\nu }\right){\left(1+{\mathit{m}}^2\right)}^2}{\left(7+6\mathit{\nu }\right){\left(1+{\mathit{m}}^2\right)}^2+\left(20+12\mathit{\nu }\right){\mathit{m}}^2}}$$

(27)

where m = r/R (the ratio of inner diameter to outer diameter), and ν is Poisson’s ratio.

Using the method of separation of variables, substitute y = y(x)e^iωt and ψ = ψ(x)e^iωt into Equations (25) and (26). By eliminating e^iωt, the frequency equation (characteristic equation) of the Timoshenko beam can be derived. This equation determines the natural frequencies of the beam.

$$\mathit{\rho }\mathit{A}{\mathit{\omega }}^2\mathit{y}\left(\mathit{x}\right)+\mathit{\kappa }\mathit{AG}{\displaystyle \frac{\partial }{\partial \mathit{x}}}\left[{\displaystyle \frac{\partial \mathit{y}\left(\mathit{x}\right)}{\partial \mathit{x}}}-\mathit{\psi }\left(\mathit{x}\right)\right]=0$$

(28)

$$\mathit{EI}{\displaystyle \frac{{\partial }^2\mathit{\psi }\left(\mathit{x}\right)}{{\partial \mathit{x}}^2}}+\mathit{\kappa }\mathit{AG}\left[{\displaystyle \frac{\partial \mathit{y}\left(\mathit{x}\right)}{\partial \mathit{x}}}-\mathit{\psi }\left(\mathit{x}\right)\right]+\mathit{\rho }\mathit{I}{\mathit{\omega }}^2\mathit{\psi }\left(\mathit{x}\right)=0$$

(29)

Without loss of generality, the following non-dimensional quantities are defined:

$$\mathit{X}={\displaystyle \frac{\mathit{x}}{\mathit{L}}}$$

(30)

$$\mathit{Y}\left(\mathit{X}\right)={\displaystyle \frac{\mathit{y}\left(\mathit{x}\right)}{\mathit{L}}}$$

(31)

$$\mathit{\psi }\left(\mathit{X}\right)={\displaystyle \frac{\mathit{\psi }\left(\mathit{x}\right)}{\mathit{L}}}$$

(32)

$${\mathit{\Omega }}^2={\displaystyle \frac{\mathit{\rho }\mathit{A}{\mathit{\omega }}^2{\mathit{L}}^4}{\mathit{EI}}}$$

(33)

$$\mathit{\eta }={\displaystyle \frac{\mathit{I}}{\mathit{A}{\mathit{L}}^2}}$$

(34)

$$\mathit{\xi }={\displaystyle \frac{\mathit{\kappa }\mathit{GA}{\mathit{L}}^2}{\mathit{EI}}}$$

(35)

Substituting the above non-dimensional quantities into Equations (28) and (29) allows for the numerical solution of the frequency equation for the beam with fixed ends, yielding the first six natural frequencies of the empty pipe as shown in

Table 2. Comparison between Timoshenko theory and simulation results.

Order	Timoshenko Solution/Hz	Finite Element Method/Hz	Relative Error/%
1	16.21	17.16	5.54
2	16.85	17.16	1.81
3	42.13	45.94	8.29
4	43.76	45.94	4.75
5	79.34	86.89	8.69
6	80.81	86.89	7.00

.

As can be seen from Table 2, the simulation results from ANSYS Workbench (Version 2022 R1) are in close agreement with the numerical calculation results using the Timoshenko method, with relatively small relative errors. Thus, it can be concluded that the finite element simulation method used in this study is reliable.

3. Results

3.1. Comparison of Modal Analysis in Different States

The natural frequencies were solved for four distinct configurations: the empty pipe, the pipe considering the FSI effect, the pipe considering the water-attached effect, and the pipe considering both the FSI and the water-attached effects. The results are presented in

Table 3. Comparison of natural frequencies (Hz).

Order	Empty Pipe	FSI Pipe	Water-Attached Pipe	All Considered
1	17.16	7.80	7.68	6.72
2	17.16	7.80	10.48	7.31
3	45.94	20.96	22.80	18.10
4	45.94	20.96	28.13	20.02
5	86.89	39.89	47.73	34.50
6	86.89	39.89	54.14	38.41

. The line chart for the natural frequency variation is shown in Figure 6.

Table 3 and Figure 6 illustrate that the natural frequencies of each of the two orders of the empty pipe are equal (i.e., the first and second orders, the third and fourth orders, and the fifth and sixth orders). Due to the symmetry of the pipe geometry and the constraint conditions, there is a frequency degeneracy phenomenon in the empty pipe. Different vibration modes “share” the same frequency during natural vibration, resulting in vibration modes in different directions having the same natural frequency; that is, the frequencies are equal, and the vibration modes are orthogonal. Figure 7 illustrates the first six natural vibration modes of the empty pipe. Nevertheless, the degeneracy is eliminated when the water-attached effect is taken into account. This suggests that the fluid alters the vibration form or mass distribution of the structure, thereby disrupting the structural symmetry.

The FSI dominates the natural frequency reduction, with the reduction ratio of FSI remaining stable at around 54%, which is significantly higher than that of the water-attached. For example, the sixth-order natural frequency diminishes by 54.09% under FSI, while it only reduces by 37.69% under the water-attached effect. The FSI enhances the effective mass of the pipe, with the impact becoming more pronounced as the order increases. For instance, the difference between the higher-order natural frequency under FSI and that of the empty pipe reaches 47.00 Hz, whereas the difference between the lower-order natural frequency and that of the empty pipe is only 9.36 Hz.

The reduction ratios for the water-attached pipe are higher (45.07~55.24%) for odd orders (1, 3, 5) and lower (37.69~38.93%) for even orders (2, 4, 6). The reduction ratio for the odd order is approximately 1.20 to 1.42 times that of the even order. For example, the first-order natural frequency of the water-attached pipe is 7.68 Hz, which differs from the empty pipe’s frequencies of 17.16 Hz by 9.48 Hz. The second-order natural frequency of the water-attached pipe is 10.48 Hz, with only a 6.68 Hz difference from that of the empty pipe. In even-order vibrations, the vibrational modes mostly exhibit symmetric bending, with the fluid motion directions on either side being antagonistic. The symmetric flow field causes the inertial forces to be partially offset, thus diminishing the equivalent mass impact.

When both effects are taken into account simultaneously, the natural frequency is diminished compared to the individual effects; however, the reduction ratio is not a simple sum of the two. Using the first-order natural frequency as a reference, FSI results in a 54.55% decrease, the water-attached effect leads to a 55.24% decrease, and the theoretical superposition reduction should amount to 79.66%; nevertheless, the actual reduction is merely 60.84%. This signifies that when the two effects operate concurrently, the coupling effect offsets a portion of the increment in equivalent mass.

Figure 8 illustrates the initial six natural vibration modes of the pipe, accounting for both FSI and water-attached effects.

It can be seen from Figure 8 that the amplitudes of the lower-order natural frequencies are larger. For example, the maximum deformations of the first and second orders are 16.38 mm and 15.35 mm, which are significantly higher than those of the higher order (15.55–19.24 mm). This indicates that the fluid is more likely to induce large-amplitude vibrations at the lower-order natural frequencies. By comparing these results with the natural vibration modes of the empty pipe, it can be found that the maximum deformations of each order in the empty pipe are quite close (15.53–16.37 mm), suggesting that the energy distribution is more balanced when there is no fluid.

3.2. Effect of Pipe Length on FSI Vibration Characteristics

Pipe lengths of 1 m, 2 m, and 3 m are taken to carry out a comparative analysis of the pipe vibration conditions. Other structural parameters and fluid parameters remain unchanged.

Table 4. Comparison of natural frequencies for different pipe lengths (Hz).

Order	1 m	2 m	3 m
1	51.89	14.74	6.72
2	95.43	21.06	7.31
3	120.15	38.65	18.10
4	123.88	50.21	20.02
5	160.64	60.69	34.50
6	171.40	71.28	38.41

displays the first six natural frequency values for various pipe lengths. The line chart of natural frequency variation with different pipe lengths is shown in Figure 9.

The data in Table 4 indicates that the discrepancy between the first-order and sixth-order natural frequencies of the 3 m pipe is 31.69 Hz, but for the 2 m pipe, it is 56.54 Hz, which is 1.78 times that of the 3 m pipe. The discrepancy between the first-order and sixth-order natural frequencies of the 1 m pipe is 119.51 Hz, which is 3.77 times that of the 3 m pipe. This suggests that the shorter pipe correlates with a more drastic growth rate. The extension of the pipe length results in a reduction in the overall structural stiffness while concurrently increasing the mass. The combined effect of these two factors leads to a decrease in the stiffness–mass ratio, thereby causing a reduction in the natural frequency. It can also be seen in Figure 9 that the frequency increment slows down with the increase in the order; for example, the natural frequency of the 1 m pipe increases by 43.54 Hz from the first order to the second order, whereas it only increases by 10.76 Hz from the fifth order to the sixth order.

As the length increases from 1 m to 3 m, the frequency is expected to diminish to 11.11% of its initial value, representing an 88.89% reduction. The actual reduction ratios in the lower-order natural frequencies range from 87.05% to 92.34%, which are close to the theoretical value, exhibiting a deviation of −1.84% to 3.45%. The divergence of the higher-order frequencies is quite substantial. The actual reduction of the sixth order is 77.59%, with a deviation of −11.31%. This is 6.14 times the deviation value of the first-order natural frequency. The lower-order modes correspond to overall bending vibrations, whose vibration energy is mainly concentrated in the overall deformation of the structure. The frequency is directly determined by the bending stiffness, thus being most sensitive to changes in pipe length. The increase in pipe length results in a sharp reduction in bending stiffness, significantly influencing the lower-order frequencies. The higher-order modes not only depend on the overall stiffness but also correlate with local stiffness and mass distribution. Consequently, the influence of the additional mass and boundary conditions on the higher-order modes diminishes the effect of length, leading to a reduction in the actual stiffness that is lower than the theoretical value.

Figure 10 illustrates the first six natural vibration modes of the 1 m pipe.

It can be seen from Figure 10 that the amplitude of the 1 m pipe increases with the frequency, which is the opposite to that of the long pipe (the long pipe has a larger amplitude at lower-order natural frequencies). For example, the amplitude of the sixth order is 42.00 mm, which is 48% higher than that of the first order (28.44 mm). Meanwhile, the deformation is concentrated at the fifth- and sixth-order natural frequencies, reflecting that the fluid additional mass is sensitive to the boundary effect.

3.3. Effect of Pipe Wall Thickness on FSI Vibration Characteristics

The vibration conditions of a pipe with wall thicknesses of 5 mm, 7 mm, 9 mm, and 11 mm are studied. Other structural parameters, fluid parameters, and constraint conditions remain unchanged.

Table 5. Comparison of natural frequencies for different wall thickness (Hz).

Order	5 mm	7 mm	9 mm	11 mm
1	5.25	6.06	6.72	7.22
2	5.71	6.59	7.31	7.88
3	14.11	16.29	18.10	19.48
4	15.49	17.99	20.02	21.69
5	26.70	30.98	34.50	37.19
6	29.32	34.33	38.41	41.80

presents the first six natural frequency values for various pipe wall thicknesses. The line chart for the natural frequency variation with different wall thicknesses is shown in Figure 11.

It can be seen from the data in Table 5 that the disparity between the first-order and sixth-order natural frequencies of the 6 mm pipe is 24.07 Hz, whereas for the 11 mm pipe, it is 34.58 Hz, representing 143.66% of the 5 mm pipe. This phenomenon demonstrates that an increase in wall thickness amplifies the differences between natural frequencies of various orders. This is because the increase in wall thickness primarily boosts the bending stiffness of the pipe cross-section, thereby strengthening the structural stiffness, and consequently dominating the frequency variation.

The growth rate between adjacent wall thicknesses exhibits a trend of gradual decline. Using the first order as a reference, when the wall thickness increases from 5 mm to 7 mm, the growth rate of natural frequency is 15.43%; when the wall thickness increases from 7 mm to 9 mm, this yields a 10.89% increase; and when it increases from 9 mm to 11 mm, the growth rate is 7.44%. This suggests that the impact of increasing wall thickness on frequency enhancement gradually weakens, and the effect of additional mass strengthens with the increase in wall thickness.

The absolute increase in higher-order frequencies is significant. As the wall thickness increases from 5 mm to 11 mm, the first-order natural frequency rises from 5.25 Hz to 7.22 Hz, resulting in an increment of 1.97 Hz. The sixth-order increment is more substantial, rising from 29.32 Hz to 41.80 Hz, resulting in an increment of 12.48 Hz. Nonetheless, its relative increment aligns with that of the lower-order natural frequencies. At a wall thickness of 11 mm, the sixth-order frequency of 41.80 Hz is 5.79 times that of the first-order frequency (7.22 Hz). In contrast, at a wall thickness of 5 mm, the frequency is 5.58 times larger. This demonstrates that an increase in wall thickness elevates the overall natural frequency, while the growth trend among the various frequency orders remains stable; that is, the wall thickness mainly affects the frequency magnitude rather than the growth pattern.

Figure 12 and Figure 13 illustrate the first six natural vibration modes of the pipe with wall thicknesses of 5 mm and 11 mm.

By observing the natural frequencies and vibration modes of each order in Figure 11 and Figure 12, it can be seen that as the pipe wall thickness increases, the amplitudes of all the orders decreases, and this decrease is more significant for higher-order modes. For example, the amplitude of the first order for a 5 mm pipe is 21.61 mm, while that for an 11 mm pipe is 14.95 mm, which is a decrease of 6.66 mm. For the sixth order, the decrease is 7.48 mm. This indicates that the increase in stiffness suppresses the vibration amplitude, and under the same excitation, thick-walled pipes are less prone to deformation.

4. Conclusions

This research methodically examined the vibrational characteristics of viscoelastic slurry pipes under FSI using a three-dimensional FSI model. The findings derived from the integration of modal analysis and parametric investigations are as follows:

(1)

The FSI effect resulted in a substantial reduction in the pipe’s natural frequency, decreasing by 54% for each order, which was more prominent compared with the water-attached effect. Specifically, the fifth- and sixth-order frequencies diminished from 86.89 Hz in the empty pipe to 39.89 Hz under FSI, reflecting a reduction of 54.09%, and verifying the increasing trend of the influence of FSI in higher-order modes. The phenomenon of frequency degeneracy caused by geometric symmetry in the empty pipe state, exemplified by the equal frequencies of the first and second orders, third and fourth orders, and fifth and sixth orders, was eliminated by fluid action, demonstrating that the fluid disrupted the structural symmetry by modifying the mass distribution and vibration mode.

(2)

The water-attached effect exhibited disparities between odd-order and even-order modes, owing to the limitations imposed by vibration mode geometries, accounting for 45.07% to 55.24% of the frequency reduction in odd-order modes and only 37.69% to 38.93% in even-order modes. This was because the symmetric bending of the pipe in even-order vibrations led to the mutual offsets of fluid inertia forces on both sides. For example, the symmetric fluid flow in second-order vibrations reduced the equivalent mass increase, whereas the uneven fluid inertial distribution from asymmetric vibrations in odd-order modes enhanced the frequency reduction. When FSI and water-attached effects operated concurrently, the frequency reduction did not show linear superposition. For instance, the actual reduction in the first-order frequency was 60.84%, which was less than the theoretical superposition value of 79.66%, indicating the nonlinear feature whereby the two effects partially offset the equivalent mass increment through coupling.

(3)

As the pipe length increased from 1 m to 3 m, the growth rate of natural frequency significantly diminished with each order. Specifically, the frequency difference between the first and sixth orders of the 1 m pipe was 119.51 Hz, while that of the 3 m pipe was only 31.69 Hz, resulting in a growth rate that diminished to 26.52% of that of the shorter pipe. This arose from the nonlinear alterations in structural bending stiffness and mass induced by the increase in pipe length, leading to a more significant decline in the stiffness–mass ratio of the longer pipe. Particularly in lower-order modes, the frequency predominantly influenced by overall bending vibrations showed heightened sensitivity to length alterations, resulting in an actual frequency reduction of 87.05% to 92.34%. In higher-order modes, the influence of length was weakened by the additional mass effect due to the involvement of local stiffness distribution, such as in the actual reduction in the sixth order being 77.59%.

(4)

As the pipe wall thickness increases, the natural frequencies of all orders gradually increased, though the rate of increase exhibited a declining trend. When the wall thickness increased from 5 mm to 11 mm, the growth rate of the first-order natural frequency diminished from 15.43% to 7.44%, signifying that the stiffness improvement effect brought about by the increase in wall thickness weakened as the thickness increased. In terms of absolute increment, the increment of the sixth-order frequency reached 12.48 Hz at 11 mm wall thickness compared to 5 mm, which was markedly greater than the 1.97 Hz increment of the first-order frequencies. However, its relative growth rate aligned with that of the lower-order frequencies, suggesting that wall thickness primarily influenced the absolute frequency value rather than changing the relative growth pattern of each frequency order.

In summary, by revealing the FSI vibration characteristics and influencing factors of viscoelastic slurry pipes, this study provides multi-dimensional theoretical support for the design, vibration control, and risk prevention of engineering slurry pipes. In practical engineering, if the frequency of external excitation can be adjusted, there is no need to modify the pipe itself; priority can be given to avoiding resonance by changing the excitation frequency. If the excitation frequency cannot be adjusted, such as in key equipment like fixed-speed pumps, optimization of the pipe structural design can be employed to specifically adjust stiffness or mass, thereby altering its natural frequency. The research results laying a foundation for the subsequent development of efficient and stable mud transportation technologies.