Transformed form of boundary condition for differential transform method.

## Abstract

Knowledge of natural frequency of pipeline conveying fluid has relevance to designer to avoid failure of pipeline due to resonance. The damping characteristics of pipe material can be increased by using smart materials like magnetostrictive namely, TERFENOL-D. The objective of the present chapter is to investigate vibration and instability characteristics of functionally graded Terfenol-D layered fluid conveying pipe utilizing Terfenol-D layer as an actuator. First, the divergence of fluid conveying pipe is investigated without feedback control gain and thermal loading. Subsequently, the eigenvalue diagrams are studied to examine methodically the vibrational characteristics and possible flutter and bifurcation instabilities eventuate in different vibrational modes. Actuation of Terfenol-D layer shows improved stability condition of fluid conveying pipe with variation in feedback control gain and thermal loading. Differential quadrature and differential transform procedures are used to solve equation of motion of the problem derived based on Euler-Bernoulli beam theory. Finally, the effects of important parameters including the feedback control gain, thermal loading, inner radius of pipe and density of fluid on vibration behavior of fluid conveying pipe, are explored and presented in numerical results.

### Keywords

- control gain
- isothermal load
- flutter
- bifurcation instability
- differential quadrature and differential transform method

## 1. Introduction

Composite fluid-conveying pipes have become a practicable substitute to metallic pipes in several engineering applications such as oil and gas transport lines, hydraulic and pneumatic systems, thermal power plants, heat transfer equipment, petroleum and chemical process industries, underground refueling pipelines in airports, hospitals, medical devices, municipal sewage and drainage, corporation water supply and many more. Divergence and flutter instabilities are illustrious in fluid-conveying pipe due to fluid–structure interaction. One type of instability encountered in cantilever fluid-conveying pipes is called bifurcation, when the imaginary portion of the complex frequency disappears and the real portion splits into two branches. Fundamental concepts and early development in fluid structure interaction of fluid conveying pipes have been complied and studied by [1] systematically. A few more specialized topics are briefly discussed and well documented in Ref. [2, 3, 4]. Remarkable contributions in the area of fluid-conveying pipe vibrations also include the works of Chen [5].

In the meantime, performing a review on literature, it can be seen that a few studies have been carried out in the several field of vibrations such as in-depth nonlinear dynamics [6, 7, 8, 9, 10], vibration control [11, 12, 13, 14, 15, 16, 17, 18], microtubes or nanotubes in microfluidic devices [19, 20, 21, 22], and pipes using functionally graded materials [23, 24, 25, 26].

The pseudo excitation method in conjunction with the complex mode superposition method was deduced to solve dynamic equation of Timoshenko pipeline conveying fluid [6]. The post-buckling and closed-form solutions to nonlinear frequency and response [8] of a FG fluid-conveying pipe have been investigated using analytical homotopy analysis method. Natural frequencies and critical flow velocities has been obtained for free vibration problem of pipes conveying fluid with several typical boundary conditions using DTM [11]. Dynamics and pull-in instability of pipes conveying fuid with nonlinear magnetic force have been investigated by [13], for clamped-clamped and clamped-free boundary conditions. The conclusion of investigation is that, location of magnets has a great impact on the static deflection and stability of the pipe. Wavelet based FEM has been used to examine the effect of internal surface damage [14] on free vibration behavior of fluid-conveying pipe. The natural frequencies of pipe conveying fluid has been determined by [15], using Muller’s bisection method.

Failure due to filament wound with consideration of production process inconsistencies have been assessed by Rafiee et al. [16]. Vibration and instability response of magnetostrictive sandwich cantilever fluid-conveying micro-pipes is investigated utilizing smart magnetostrictive layers as actuators by [18].

Nonlinear vibration of a carbon nanotube conveying fluid with piezoelectric layer lying on Winkler-Pasternak foundation under the influence of thermal effect [21] and magnetic field [22] have been investigated using Galerkin and multiple scale method. The in-plane free vibration frequency of a zirconia-aluminum functionally graded curved pipe conveying fluid have been explored by the complex mode method [23]. The effect of axial variations of elastic modulus and density on dynamical behavior of an axially functionally graded cantilevered pipe conveying fluid has been analyzed by [24]. Dai et al. [25] studied the thermo-elastic vibration of axially functionally graded pipe conveying fluid considering temperature changes. Heshmati [26] studied the stability and vibration behaviors of functionally graded pipes conveying fluid considering the the effect of eccentricity imperfection induced by improper manufacturing processes. Xu Liang et al. [27] have used differential quadrature method (DQM) and the Laplace transform and its inverse, to analyze the dynamic behavior of a fluid-conveying pipe with different pipe boundary conditions. Huang Yi-min et al. [28] used the separation of variables method and the derived method from Ferrari’s method to decouple the the natural frequency and the critical flow velocity equations of fluid-conveying pipe with both ends supported. Planar and spatial curved fluid-conveying pipe [29] have been investigated for their free vibration behavior with Timoshenko beam model and B-spline function used as the shape function in Galerkin method.

There are few investigations in the literature on fluid-conveying pipes containing Terfenol-D layers. Certainly, a study on the mechanical behavior of functionally graded Terfenol-D layered fluid conveying pipe will contribute to the understanding for future design engineers, hence an attempt on the vibration and stability of functionally graded Terfenol-D layered fluid conveying pipe. Inherent features of the Terfenol-D layer to regulate the vibration instabilities and critical flow velocity of a FGMT pipe are attempted numerically. Terfenol-D is a popular magnetostrictive material exhibiting force output for a corresponding magnetic field input and produces magnetic field for mechanical force as an input. Every term in Terfenol-D has a meaning (see Figure 1), for example, Ter means Terbium, Fe signifies chemical symbol for iron, Nol stands for Naval Ordnance Laboratory, and D stands for Dysprosium [30]. Terfenol-D has numerous distinguish characteristics, including a high electromechanical coupling coefficient (0.73), a high magnetostrictive strain (800–1600 ppm), a fast response, a high energy density, and a large output force. The total stiffness of the pipe is affected by actuation of the Terfenol-D layer due to the creation of tensile forces with a change in feedback control gain and temperature change in the fluid-conveying pipe. The governing equation of motion for FGMT fluid-conveying pipe is derived based on Euler-Bernoulli’s theory. Differential quadrature and differential transform approaches are used to obtain the frequency of boundary value problem. Critical velocities of the FGMT pipe are also determined for various boundary conditions, feedback control gain, and thermal loading. Validation of frequencies and critical velocities is accomplished using accessible analytical relations.

## 2. Functionally graded fluid conveying pipe

Powder metallurgy is considered as manufacturing process for present functionally graded Terfenol-D layered fluid-conveying pipe. The functioanlly graded pipe is assumed to compose of aluminum (as metallic) and aluminum oxide (as ceramic). In between the graded composition of aluminum and aluminum oxide Terfenol-D layer is included. The material properties, volume fraction and expression for calculation of properties is given in [31]. Figure 2 shows the layout of FGMT fluid-conveying pipe.

### 2.1 Derivation of governing equation

Considering the FGMT fluid-conveying pipe as an Euler-Bernoulli beam, the equation for the motion of the pipe can be derived using Hamilton’s principle. The kinetic energy of the internal fluid is appended to the kinetic energy of the pipe to obtain total kinetic energy of FGMT pipe, and is described by the equation

Where,

Where,

Where

where

Where,

Also, the axial moment produced by Terfenol-D layer is,

Applying the Hamilton’s principle, one can write the functional of FGMT pipe as,

Where,

The governing equation for FGMT fluid-conveying pipe with thermal loading making use of Ref. [33] can be obtained as:

Where,

Where,

## 3. Transformation of PDE into a sets of ODEs

Authors used the differential quadrature method to solve the free vibration equation of FGMT fluid-conveying pipe as given in Eq. (9). Here, the Eq. (9) is transformed into sets of ordinary differential equations. The standard eigenvalue form [34, 35] of the Eq. (9) can be obtained by assuming:

Now, substitute the analog form of differential quadrature for respective derivative (first, second, third and fourth) such as:

Now, Eq.15 becomes,

Now separate the terms associated with

Where,

Where,

Where

## 4. Application of differential transform method to FGMT fluid-conveying pipe

Differential transform technique (DTM) may be used to solve integral equations, ordinary partial differential equations, and differential equation systems. Using this approach, a polynomial solution to differential equations may be derived analytically. For large orders, the Taylor series approach is computationally time-consuming. This method is appropriate for linear and nonlinear ODEs since it does not need linearization, discretization, or perturbation. It is also possible to significantly reduce the amount of computing labour required while still precisely delivering the series solution and rapidly converging. The DTM has several disadvantages, though. Using the DTM, a truncated series solution may be obtained. This truncated solution does not display the actual behavior of the problem, but in the vast majority of situations it offers a good approximation of the actual solution in a relatively limited area. Solutions are expressed as convergent series with components that may be readily computed using the differential transform technique. The linear equation of motion for free vibration of FGMT fluid-conveying pipe is given by,

The differential transformation form of Eq. (22) can be written as

Rearranging Eq. (23), one will get a simple recurrence relation as:

Similarly, analogous form of original boundary conditions for the differential transformation can be done using Table 1, where

Original Form | DTM Form | Original Form | DTM Form |
---|---|---|---|

Where

Therefore, the eigenvalues

## 5. Results and discussion

In the following section, the numerical results are proposed to investigate the free vibration behavior of FGMT fluid-conveying pipe subjected to control gain and thermal loading. Since there is no published research on the subject of free vibration of FGMT fluid-conveying pipes in the open literature, a differential quadrature and differential transform approach is used to conduct a condensed analysis of the current study. The imaginary component (

### 5.1 Validation of present study

The current MATLAB code for the differential quadrature and transform technique is validated using Ref. [28], as shown in Table 2. The validation for FGMT fluid conveying pipe is also given by the author in Ref. [36]. Furthermore, the solution obtained using the differential quadrature approach corresponds well with the solution acquired using the differential transform method.

It has been identified that, the differential transform method requires the 58 number of terms to get the converged solution whereas 19 grid points used to obtain the convergence solutions shown. The natural frequencies of pipes conveying fluid depend on the fluid velocity

Boundary | Nodes | Mode | |||
---|---|---|---|---|---|

Im( | Im( | Im( | Im( | ||

7 | 18.4437 | 58.7726 | 111.9680 | — | |

11 | 18.4530 | 76.9717 | 166.6046 | 255.9565 | |

S-S | 15 | 18.4529 | 77.4675 | 175.6156 | 331.6998 |

17 | 18.4529 | 77.4688 | 175.7221 | 313.2417 | |

19 | 18.4529 | 77.4688 | 175.7221 | 313.2417 | |

DTM | 18.4765 | 77.4445 | 175.6849 | 313.2167 | |

7 | 44.6652 | 94.0809 | 149.5829 | — | |

11 | 43.8719 | 120.7255 | 224.0424 | 324.0114 | |

C-C | 15 | 43.8700 | 121.9308 | 239.5009 | 435.8037 |

17 | 43.8700 | 121.9365 | 239.7773 | 396.9056 | |

19 | 43.8700 | 121.9365 | 239.7780 | 396.9056 | |

DTM | 43.9087 | 122.1424 | 240.0518 | 397.2302 | |

7 | 29.5136 | 75.3884 | 133.9371 | — | |

11 | 29.8002 | 97.5593 | 196.8662 | 272.6251 | |

S-C | 15 | 29.8007 | 98.5000 | 206.4495 | 391.9600 |

17 | 29.8006 | 98.5037 | 206.5287 | 353.8381 | |

19 | 29.8006 | 98.5037 | 206.5287 | 353.8381 | |

DTM | 29.8321 | 98.6121 | 206.6739 | 354.0290 | |

7 | 5.1817 | 41.1890 | 112.4169 | — | |

11 | 7.1259 | 43.1741 | 128.5438 | 207.9994 | |

C-F | 15 | 7.1260 | 43.1894 | 122.0607 | 241.8281 |

17 | 7.1260 | 43.1894 | 121.9887 | 239.7690 | |

19 | 7.1260 | 43.1894 | 121.9887 | 239.7690 | |

DTM | 7.1487 | 43.5897 | 122.3256 | 240.9832 |

One of the key concerns for fluid conveyance pipes to be of significant importance is stability. The natural frequencies decrease with higher flow rates for pipelines with supported ends. The system destabilizes by diverging (buckling) when the natural frequencies fall to zero, and the resulting flow velocity is known as the critical flow velocity. In the case of

The first four natural frequencies of the simply supported fluid-conveying FGMT pipe with

Figures 7 and 8 shows the first four natural frequencies of the C-C fluid-conveying pipe with

Figures 9 and 10 presents the first four natural frequency of the S-C fluid-conveying FGMT pipe with

BC | Mode | Velocity ( | Instability Form |
---|---|---|---|

S-S | 1st Mode | 15 | Divergence |

Navier Solution [37] | 15 | — | |

2nd Mode | 30 | Divergence | |

1st & 2nd Combined | 31 | Paidoussis coupled mode flutter | |

C-C | 1st Mode | 30 | Divergence |

Navier Solution [37] | 30 | — | |

1st & 2nd Combined | 43 | Coupled mode flutter | |

C-F | 3rd Mode | 42 | Flutter |

S-C | 1st Mode | 22 | Divergence |

1st & 2nd Combined | 37 | Coupled mode flutter |

The relationships between the imaginary component of frequency of the FGMT pipe and the fluid density for different boundary conditions are plotted in Figure 11. Because the inertial and Coriolis forces were stronger with increasing fluid density, it was more simpler for the pipe to lose its stability. This led to a lower natural frequency. The changes of imaginary component of frequency with inner radius of the FGMT pipe for different boundary conditions are shown in Figure 12. For very small values of the inner radius, an increase in the inner radius has a considerable impact on frequency; nevertheless, when the inner radius value is near to the outer radius, the frequency increases. In the boundary conditions clamped-clamped, simply supported-simply supported, and simply supported-clamped, the imaginary component of frequency drops as the feedback control gain rises. Imaginary component of the eigenvalue for a clamped-free frequency becomes zero for 3000 feedback control gain,

It is worth pointing out that the important aspect of present research work is maneuvering the use of Terfenol-D layers attached on the top FGMT fluid-conveying pipe to control the critical flow velocity and also improve the stability region. When Terfenol-D layer actuates tensile forces are generated in FGMT fluid-conveying pipe which affects the stiffness of fluid-conveying pipe. In order to evaluate this objective, Figure 14 shows the real part (

Figure 16 depicts the coupled effect of feedback control gain along with thermal loading. It is inferred that, there is decreasing effect of critical flow velocity as thermal loading increases. The reduction in overall stiffness of pipe is the reason for instability of FGMT pipe at lower flow velocity with thermal loading. Therefore, critical flow velocity condition under thermal loading can be amplified through imposing higher feedback control gain. The control gain varies between 0 and 2000 as the temperature of the fluid conveying pipe changes. It is inferred that, with a zero control gain and

## 6. Concluding remarks

In this chapter, the differential quadrature and differential transform method is applied to analyze the free vibration of FGMT pipes conveying fluid with different boundary conditions. Boundary value problem of FGMT fluid-conveying pipe is solved straightforwardly using DQM and DTM. Close agreement is established for critical velocity and frequencies results generated by DQM, DTM with those of Navier and Galerkin solution. Eigenvalue diagrams are detailed enough to shows the illustration about the effects of feedback control gain, density of fluid, inner radius of pipe and thermal loading on the vibrational and instability characteristics. To attenuate the amplitude of vibration or displacement, inherent damping property of the material cannot be sufficient. To dampen out large amplitude vibration during resonance, special techniques have been explored, like using sandwich pipes namely, viscoelastic layer placed between two layers of the parent pipe material. This approach is called passive damping. Viscoelastic materials like, natural rubber, and synthetic rubber like nitrile butadine rubber and styrene butadine rubber, silicone rubber can be proposed. Sophisticated technique is the active vibration. This method involves use of materials like, piezoelectric, magnetostrictive, magnetorehology, electrostricitve and shape memory alloys. Magnetostrictive material presented in this chapter works on the ability of the material to respond mechanically to the presence of magnetic field. The magnetic field is produced using a coil with passage of time dependent current. A magnetostrictive material responds with a force, hence magnetostrictive actuator. The force produced should be used to counteract the forces due to vibration. Thus, damping is introduced. The idea of incorporting Terfenol-D layer facilitates the best control of the fluid conveying FGMT pipe to avoid the bifurcation and flutter instabilities and achieve more adaptive and efficient system. Additionally increasing or decreasing effect of feedback control gain and thermal loading on critical flow velocity and instabilities have been addressed.

## Abbreviations

FGMT | Functionally graded material integrated with Terfenol-D |

BC | boundary conditions |

DQM | differential quadrature method |

DTM | differential transform method |

ODE | ordinary differential equation |

## References

- 1.
Tijsseling AS. Fluid-structure interaction in liquid-filled pipe systems: A review. Journal of Fluids and Structures. 1996; 10 (2):109-146 - 2.
Ibrahim RA. Overview of mechanics of pipes conveying fluids—Part I: Fundamental studies. Journal of Pressure Vessel Technology. 2010; 132 (3):034001-1-32 - 3.
Ibrahim RA. Mechanics of pipes conveying fluids—part ii: Applications and fluidelastic problems. Journal of Pressure Vessel Technology. 2011; 133 (2):024001-1-30 - 4.
Li S, Karney BW, Liu G. Fsi research in pipeline systems–a review of the literature. Journal of Fluids and Structures. 2015; 57 :277-297 - 5.
Chen S-S. Free vibration of a coupled fluid/structural system. Journal of Sound and Vibration. 1972; 21 (4):387-398 - 6.
Zhai H-b, Zi-yan W, Liu Y-s, Yue Z-f. Dynamic response of pipeline conveying fluid to random excitation. Nuclear Engineering and Design. 2011; 241 (8):2744-2749 - 7.
Cigeroglu E, Samandari H. Nonlinear free vibrations of curved double walled carbon nanotubes using differential quadrature method. Physica E: Low-dimensional Systems and Nanostructures. 2014; 64 :95-105 - 8.
Tang Y, Yang T. Post-buckling behavior and nonlinear vibration analysis of a fluid-conveying pipe composed of functionally graded material. Composite Structures. 2018; 185 :393-400 - 9.
Li B, Wang Z, Jing L. Dynamic response of pipe conveying fluid with lateral moving supports. Shock and Vibration. 2018; 2018 :1-17 - 10.
Wang Y, Zhang Q, Wang W, Yang T. In-plane dynamics of a fluid-conveying corrugated pipe supported at both ends. Applied Mathematics and Mechanics. 2019; 40 (8):1119-1134 - 11.
Ni Q, Zhang ZL, Wang L. Application of the differential transformation method to vibration analysis of pipes conveying fluid. Applied Mathematics and Computation. 2011; 217 (16):7028-7038 - 12.
Li Y-d, Yang Y-r. Forced vibration of pipe conveying fluid by the green function method. Archive of Applied Mechanics. 2014; 84 (12):1811-1823 - 13.
Dai HL, Wang L. Dynamics and stability of magnetically actuated pipes conveying fluid. International Journal of Structural Stability and Dynamics. 2016; 16 (06):1550026 - 14.
Oke WA, Khulief YA. Effect of internal surface damage on vibration behavior of a composite pipe conveying fluid. Composite Structures. 2018; 194 :104-118 - 15.
Sutar S, Madabhushi R, Chellapilla KR, Poosa RB. Determination of natural frequencies of fluid conveying pipes using muller’s method. Journal of The Institution of Engineers (India): Series C. 2019; 100 (3):449-454 - 16.
Rafiee R, Sharifi P. Stochastic failure analysis of composite pipes subjected to random excitation. Construction and Building Materials. 2019; 224 :950-961 - 17.
Rajidi SR, Gupta A, Panda S. Vibration characteristics of viscoelastic sandwich tube conveying fluid. Materials Today: Proceedings. 2020; 28 :2440-2446 - 18.
Amiri A, Masoumi A, Talebitooti R. Flutter and bifurcation instability analysis of fluid-conveying micro-pipes sandwiched by magnetostrictive smart layers under thermal and magnetic field. International Journal of Mechanics and Materials in Design. 2020; 16 (3):569-588 - 19.
Soltani P, Bahar P, Farshidianfar A. An efficient gdq model for vibration analysis of a multiwall carbon nanotube on pasternak foundation with general boundary conditions. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2011; 225 (7):1730-1741 - 20.
Bahaadini R, Hosseini M. Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid. Computational Materials Science. 2016; 114 :151-159 - 21.
Askari H, Esmailzadeh E. Forced vibration of fluid conveying carbon nanotubes considering thermal effect and nonlinear foundations. Composites Part B: Engineering. 2017; 113 :31-43 - 22.
Kamali M, Mohamadhashemi V, Jalali A. Parametric excitation analysis of a piezoelectric-nanotube conveying fluid under multi-physics field. Microsystem Technologies. 2018; 24 (7):2871-2885 - 23.
Liang F, Yang XD, Bao RD, Zhang W. Frequency analysis of functionally graded curved pipes conveying fluid. Advances in Materials Science and Engineering. 2016; 2016 :1-9 - 24.
Zhou X-w, Dai H-L, Wang L. Dynamics of axially functionally graded cantilevered pipes conveying fluid. Composite Structures. 2018; 190 :112-118 - 25.
Dai J, Liu Y, Liu H, Miao C, Tong G. A parametric study on thermo-mechanical vibration of axially functionally graded material pipe conveying fluid. International Journal of Mechanics and Materials in Design. 2019; 15 (4):715-726 - 26.
Heshmati M. Influence of an eccentricity imperfection on the stability and vibration behavior of fluid-conveying functionally graded pipes. Ocean Engineering. 2020; 203 :107192 - 27.
Liang X, Zha X, Jiang X, Wang L, Leng J, Cao Z. Semi-analytical solution for dynamic behavior of a fluid-conveying pipe with different boundary conditions. Ocean Engineering. 2018; 163 :183-190 - 28.
Yi-min H, Ge Seng W, Wei, and He Jie. A direct method of natural frequency analysis on pipeline conveying fluid with both ends supported. Nuclear Engineering and Design. 2012; 253 :12-22 - 29.
Łuczko J, Czerwiński A. Three-dimensional dynamics of curved pipes conveying fluid. Journal of Fluids and Structures. 2019; 91 :102704 - 30.
Engdahl G, Mayergoyz ID. Handbook of Giant Magnetostrictive Materials. Vol. 386. San Diego: Academic Press; 2000 - 31.
Patil MA, Kadoli R. Differential quadrature solution for vibration control of functionally graded beams with terfenol-d layer. Applied Mathematical Modelling. 2020; 84 :137-157 - 32.
Reddy JN, Barbosa JI. On vibration suppression of magnetostrictive beams. Smart Materials and Structures. 2000; 9 (1):49-58 - 33.
Qian Q, Wang L, Ni Q. Instability of simply supported pipes conveying fluid under thermal loads. Mechanics Research Communications. Apr 2009; 36 (3):413-417 - 34.
Lin W, Qiao N. In-plane vibration analyses of curved pipes conveying fluid using the generalized differential quadrature rule. Computers & Structures. 2008; 86 (1–2):133-139 - 35.
Talebitooti M, Fadaee M. A magnetostrictive active vibration control approach for rotating functionally graded carbon nanotube-reinforced sandwich composite beam. Smart Materials and Structures. 2019; 28 (7):075007 - 36.
Patil MA, Kadoli R. Effect of two-parameter partial foundation and viscoelastic supports on free vibration of Terfenol-D layered functionally graded fluid conveying pipe using domain decomposition technique. Mechanics of Advanced Materials and Structures. 2022:1-13. DOI: 10.1080/15376494.2022.2103604 - 37.
Li Li, Yujin Hu. Critical flow velocity of fluid-conveying magneto-electro-elastic pipe resting on an elastic foundation. International Journal of Mechanical Sciences. Dec 2016; 119 :273-282 - 38.
Shamim M, Mehran S, Davood Y, Ebrahim E. Nonlinear vibration analysis of fluid-conveying microtubes. Nonlinear Dynamic. Mar 2016; 85 (2):1007-1021