vibration analysis of partially cracked thin cylindrical

Journal of Mechanical Engineering Research and Developments

ISSN: 1024-1752

CODEN: JERDFO

Vol. 44, No. 6, pp. 95-110

Published Year 2021

95

Vibration Analysis of Partially Cracked Thin Cylindrical Shell Immersed in

Fluid

Mustafa A. Mohammed†*, Wael Najm Abdullah‡, Marwah Ali Husain††

†Alsalam University College, Department of Medical Instrumentation Techniques Engineering, Iraq

‡Al-Mustansiriyah University, College of Engineering, Department of Mechanical Engineering, Baghdad, Iraq

††Alsalam University College, Department of Refrigeration and Air Conditioning Engineering Techniques, Iraq

*Corresponding Author Email: [email protected]

ABSTRACT: One of the complex fluid structure interaction dynamic problems is the vibration analysis related to thin

cracked cylindrical shells, that is of high importance to detect any shell damage and monitor the condition of the shell.

In the present study, governing equations with regard to transverse deflection regarding cracked shells immersed in

fluid were obtained with the use of the conventional shell theory. In addition, fluid forces that are related to the inertial

effects have been added in governing differential equation for incorporating fluid structure interaction effects, while

the flexural rigidity equation was utilized for formulating crack coefficients for the purpose of accommodating the

impact of a crack in governing equation, while the governing equation was Transformed into the simpler form with

the use the theory of Donell-Mushtari-Vlasov (DMV) for getting fundamental frequency. The impact of parameters

as crack position, crack depth, crack length, and shell geometry like thickness, radius, and the shell’s length was

examined on natural frequency, also the simply supported boundary conditions have been specified in this study.

Furthermore, this work is not just making significant contributions to the field of researches concentrating on vibration

analysis related to cylindrical shells immersed in fluid, yet also offering a significant crack damage detection approach

for the structures.

KEYWORDS: Crack, fundamental frequency, flexural rigidity, immersed in fluid, cylindrical shell.

INTRODUCTION

Recently, the thin cylindrical shells were designed as structural components of high importance in the applications of

marine engineering. Therefore, the knowledge related to the dynamic properties of the thin structures immersed in

fluid was significant for their designing purposes. The study is showing that the existence of cracks is affecting the

stiffness and thus change the vibration behavior regarding the cylindrical shells in a vacuum. Also, it has been

indicated that the vibrations regarding immersed intact shells were distinctive compared to those in a vacuum. Yet,

the cracks might be appearing in immersed structures because of the considerable fluid’s pressure fluctuations. The

impact of crack on the behavior related to the immersed cylindrical shells becomes considerable. Analytical study of

the dynamic features is conducted for avoiding upcoming risks prior to their practical use.

It is indicated in literature that the existence of cracks is affecting the shell’s vibration properties. Numerical

approaches are majorly used to examine the static solutions for cracked shells. Some of the published studies on

analytical modeling related to the cracked shells were provided and the majority of them take into account analysis in

the nonexistence of fluid medium. For the purpose of examining the vibration properties related to the cylindrical shell

with cracks, a study conducted by Yin and Lam [1] created an analytical approach to determine the natural frequencies

regarding finite length circular cylindrical shells which contain circumferential surface cracks. In addition, they

obtained the governing equations, integrate the line spring model with classical thin shell theory, and lastly utilized

effective numeric that is calculating the natural frequencies regarding the shell via the 1D process of optimization. A

study conducted by Moradi and Tavaf [2] utilized the line spring approach for finding the formulations regarding free

vibration of the shell that contains circumferential surface cracks and deriving the equations of motion, the researchers

indicated that the natural frequencies utilizing differential quadrature approach and suggested evolutionary

mailto:[email protected]

Vibration Analysis of Partially Cracked Thin Cylindrical Shell Immersed in Fluid

96

optimization algorithm with regard to the detection of cracks in cylindrical shell structures. A study conducted by

Tiwari and Joshi [3] examined the impact of cracks in the homogeneous isotropic partially longitudinal cracked

cylindrical shell on natural frequency. The association between bending stress and tensile stress at the crack location

and at the far side was utilized with the using line spring approach. A study conducted by (Moazzez, Saeidi Googarchin

& Sharifi [4] and Saeidi & Moazzez [5]) examined the natural frequencies related to orthotropic (Moazzez and Saeidi

2019) and isotropic (Moazzez et al. 2018) cylindrical shell involving the surface crack impact. The researchers

prepared the governing equations related to the cracked shells with the use of classical shell theory with regard to

simply supported (S-S) boundary conditions and clamped-clamped (C-C). A study conducted by Husain & Al-

shammari [6, 7] provided numerical and analytical study regarding the vibration analysis of cylindrical shells which

contains circumferential partially cracks is on the basis of flexural rigidity with many crack sizes and locations with

regarded to the simply-supported boundary conditions.

All researchers indicated till now are concentrating on the un-damaged shells. Also, the analysis related to the cracked

cylindrical shells was more complex, and just extremely little researches were conducted in such a field. Most of the

existing investigations have relied on the finite element method. For instance, a study conducted by Javidruzi et al.

[8] provided a model of the finite element for studying the dynamic stability behaviors regarding fixed supported

cracked cylindrical shell that is constrained to the periodic load of the edge. A study conducted by Xin et al. [9]

examined the vibration behaviors regarding cracked cylindrical shells with the time-varying rotating speed via finite

element method, while the impacts of orientation, crack length, constant speed of rotation, also the length diameter

ratio regarding the shell on free vibration behavior was examined.

It has been indicated that the existence of fluid medium reduces the cylindrical shell’s frequencies compared to those

estimated in a vacuum, such reduction in the natural frequency was because of the presence of fluid around the

cylindrical shell that result in increasing the kinetic energy related to the whole system with no equivalent increment

in the strain energy. Most works were conducted on free as well as forced vibration regarding the intact cylindrical

shells along with the fluid. A study conducted by Zhang [10] provided a wave propagation method for frequency

investigation related to the thin cylindrical shell sub-merged in the acoustic high-density medium. In addition, an

analytical study was done via Naeem et al. [11] for evaluating the vibration properties of immersed cylindrical shell

formed from the functionally-graded materials. The problem has been framed through combining the shell dynamical

equations on the basis of Love first order thin shell theory with acoustic wave equation.

The literature is showing some works considering the impact of fluid and crack on the vibration regarding cylindrical

shells. A study conducted by Soni et al. [12] examined the impact of the fluidic medium on the vibration properties of

cracked isotropic plates through the incorporation of the inertial impact of fluid forces on formerly created models. A

study conducted by Jain et al. [13] provided analytical solutions with regard to vibration analysis of orthotropic as

well as FGM immersed cylindrical shell which contain surface crack regarding variable angular orientation.

Furthermore, the governing equations were obtained utilizing classical shell theory.

After reviewing the literature on cylindrical shells’ vibration, it might be indicated that the analysis related to cracked

cylindrical shells was examined in a vacuum, while the intact shells in the fluid were examined. Therefore, it is useful

for extending the present research field regarding the vibration analysis of cracked cylindrical shells in the case when

the shell was immersed in a fluid medium. In addition, this work aims on performing analytical investigation related

to the vibration characteristics of the partially-cracked isotropic cylindrical shells in the existence of a fluid medium.

Also, in this work, the analytical model of Ref. [6] was modified rigorously via taking into account the impact of a

fluid medium, while the fluid forces via using Bernoulli equation and velocity potential function, which act on the

cylindrical shell were incorporated in governing equation regarding the cracked cylindrical shell that has been derived

analytically with the use of classical shell theory, also being simplified through Donell–Mushtari–Vlasov (DMV)

theory. The impacts of many parameters like crack depth, crack length, and it’s the position, shell length, shell radius,

and shell thickness were examined on fundamental natural frequency values, the simply supported (S-S) boundary

conditions were taken into account in this study. Lastly, the results acquired for fundamental natural frequencies were

provided along with validation from published literature.


97

MATHEMATICAL MODEL

The governing equation related to the vibration analysis of anisotropic cracked cylindrical shell immersed in a fluid

was obtained based upon the classical theory of the thin shells. Also, the shell has been indicated for being

homogenous, elastic and the shell material was isotropic. Also, the cylindrical shell includes a part through

circumferential surface crack with length (𝑎) and located at (𝑥𝑐) distance from shell’s edge end and was parallel to

longitudinal direction as can be seen in figure1.

Figure 1. Circumferential partially crack on an immersed cylindrical shell in the fluid.

The governing equations which are related to partially-cracked isotropic cylindrical shell was extensively examined

in Marwah and Mohsin [6]. In the presented section, novel governing equations for the partially-cracked shell under

fluidic mediums were obtained on a basis of the equilibrium concept regarding classical shell theory. Furthermore,

assumptions that have been specified in derivation were taken customarily based on classical shell theory for the thin

shells Rao [14].

Considering a small element of a cracked immersed cylindrical shell as can be seen in the two figures 2 and 3.

Figure 2. Analysis for resultants of forces and transverse shear in a small element of the shell.


98

Figure 3. Analysis for resultants of moment in small element of the shell.

The forces as well as bending moments act on shell’s mid-plane were specified as per classical shell theory. Next, are

the equations of the force equilibrium that might describe the classical shell theory in 𝑥, 𝜃, and 𝑧 direction;

𝜕𝑁𝑥𝑥

𝜕𝑥+

1

𝑅 𝜕𝑁𝜃𝑥

𝜕𝜃+ 𝑓𝑥 = 𝜌ℎ

𝜕2𝑢

𝜕𝑡2 (1)

𝜕𝑁𝑥𝜃

𝜕𝑥+

1

𝑅 𝜕𝑁𝜃𝜃

𝜕𝜃+

𝑄𝜃𝑧

𝑅+ 𝑓𝜃 = 𝜌ℎ

𝜕2𝑣

𝜕𝑡2 (2)

𝜕𝑄𝑥𝑧

𝜕𝑥 +

1

𝑅 𝜕𝑄𝜃𝑧

𝜕𝜃−

𝑁𝜃𝜃

𝑅+ 𝑓𝑧 = 𝜌ℎ

𝜕2𝑤

𝜕𝑡2 + ∆𝑃 (3)

Where 𝑢 , 𝑣 and 𝑤 are the displacements in the axial, circumferential and normal directions, respectively.

(𝑁𝑥𝑥 , 𝑁𝜃𝑥 , 𝑁𝑥𝜃 , 𝑁𝜃𝜃) and (𝑄𝜃𝑧 , 𝑄𝑧𝜃) represent the resultant forces and the transverse shear forces on mid-surface,

respectively. ρ represents mass density, h represents the thickness of shell and ( 𝑓𝑥 , 𝑓𝜃 , 𝑓𝑧 ) represent the external

forces along the 𝑥, θ, and 𝑧 directions, respectively.

∆P = 𝑃𝑖 − 𝑃𝑒 is the fluid dynamic pressure that acts on shell’s top and bottom surfaces.

The moment equilibrium equations can be written as;

𝜕𝑀𝑥𝑥

𝜕𝑥+

1

𝑅 𝜕𝑀𝜃𝑥

𝜕𝜃= 𝑄𝑥𝑧 (4)

𝜕𝑀𝑥𝜃

𝜕𝑥+

1

𝑅 𝜕𝑀𝜃𝜃

𝜕𝜃 = 𝑄𝜃𝑧 (5)

(𝑀𝑥𝑥 , 𝑀𝜃𝑥 , 𝑀𝑥𝜃 , 𝑀𝜃𝜃) represent the bending moment resultants.

In terms of shell displacements 𝑢 , 𝑣 and , the force and bending moment components may be represented as:

𝑁𝑥𝑥 = 𝐶 (𝜕𝑢

𝜕𝑥+

𝜈

𝑅 𝜕𝑣

𝜕𝜃+

𝜈

𝑅 𝑤) (6)


99

𝑁𝑥𝜃 = 𝑁𝑥𝜃 = 𝐶 (1 − 𝜈

2) (

𝜕𝑣

𝜕𝑥+

1

𝑅 𝜕𝑢

𝜕𝜃) (7)

𝑁𝜃𝜃 = 𝐶 ( 1

𝑅 𝜕𝑣

𝜕𝜃+

𝑤

𝑅+ 𝜈

𝜕𝑢

𝜕𝑥 ) (8)

𝑀𝑥𝑥 = 𝐷 (− 𝜕2𝑤

𝜕𝑥2+

𝜈

𝑅2 𝜕𝑣

𝜕𝜃−

𝜈

𝑅2 𝜕2𝑤

𝜕𝜃2) (9)

𝑀𝜃𝜃 = 𝐷 (1

𝑅2 𝜕𝑣

𝜕𝜃−

1

𝑅2 𝜕2𝑤

𝜕𝜃2− 𝜈

𝜕2𝑤

𝜕𝑥2 ) (10)

𝑀𝑥𝜃 = 𝑀𝑥𝜃 = 𝐷 (1 − 𝜈

2) (

1

𝑅 𝜕𝑣

𝜕𝑥−

2

𝑅

𝜕2𝑤

𝜕𝑥𝜕𝜃 ) (11)

Where 𝐶 , and 𝐷 are the torsional stiffness and flexural rigidity of the shell. These terms can be expressed as:

𝐶 = 𝐸ℎ

1−𝜈2 𝐷 =

𝐸ℎ3

12(1−𝜈2)

On substituting 𝑄𝜃𝑧 and 𝑄𝑥𝑧 from Eq4 and Eq5 into the force equilibrium equations and employing Eq6 to Eq11 in

the governing equation, may be represented in terms of displacement components 𝑢 , 𝑣 and 𝑤 as:

𝐶 𝜕2𝑢

𝜕𝑥2+

1 − 𝑣

2𝑅2 𝜕2𝑢

𝜕𝜃2 +

𝜈

𝑅

𝜕𝑤

𝜕𝑥+

1 + 𝑣

2𝑅

𝜕2𝑣

𝜕𝑥𝜕𝜃 + 𝑓𝑥 = 𝜌ℎ

𝜕2𝑢

𝜕𝑡2 (12)

𝐶 (1−𝑣

2

𝜕2𝑣

𝜕𝑥2 +1

𝑅2

𝜕2𝑣

𝜕𝜃2 +1

"𝑅2 𝜕𝑤

𝜕𝜃+

1+𝑣

2𝑅

𝜕2𝑢

𝜕𝑥𝜕𝜃) + 𝐷 (

1−𝑣

2𝑅2 𝜕2𝑣

𝜕𝑥2 +1

𝑅4 𝜕2𝑣

𝜕𝜃2 −1

𝑅4 𝜕3𝑤

𝜕𝜃3 −1

𝑅2 𝜕3𝑤

𝜕𝑥2𝜕𝜃) + 𝑓𝜃 =

𝜌ℎ 𝜕2𝑣

𝜕𝑡2 (13)

𝐷 (−𝜕4𝑤

𝜕𝑥4 +1

𝑅2 𝜕3𝑣

𝜕𝑥2𝜕𝜃

2

𝑅2 𝜕4𝑤

𝜕𝑥2𝜕𝜃2 −1

𝑅4 𝜕4𝑤

𝜕𝜃4 +1

𝑅4 𝜕3𝑣

𝜕𝜃3) − 𝐶 (1

𝑅2

𝜕𝑣

𝜕𝜃+

𝑤

𝑅+

𝜈

𝑅

𝜕𝑢

𝜕𝜃) + 𝑓𝜃 = 𝜌ℎ

𝜕2𝑤

𝜕𝑡2 +

∆𝑃 (14)

EQUATION OF CRACK

The effect of a crack on the cylindrical shell and calculating the flexural rigidity, 𝐷𝑐𝑠, for a cylindrical shell to involve

an exponential function given by:

𝐷𝑐𝑠 = 𝐷 𝑓(𝑥) = 𝐷

1 + 𝑆. 𝑒(−2𝛼|𝑥−𝑥𝑐|/ℎ ) (15)

Where 𝑓(𝑥) =1

1+𝑆.𝑒(−2𝛼|𝑥−𝑥𝑐|/ℎ )

Where 𝜶 : is a non-dimensional value that has been derived by Marwah [6], can be expressed as:

𝛼 = −0.02391 + 0.027616 𝑥𝑐

𝑙+ 0.002666

ℎ𝑐

ℎ+ 0.00415

𝑎

2𝜋𝑅 (16)

x is the location along the shell, and 𝑥𝑐 is the position of the crack, 𝑙 is the length of the shell, h represents the shell

thickness, ℎ𝑐 represents the depth of crack, 𝑎 is the crack length, as shown in figure 4.

𝑆 =𝐼𝑜−𝐼𝑐

𝐼𝑐 (17)

Where 𝐼𝑜 , 𝐼𝑐 : are the second moment of areas of the un-cracked and cracked cylindrical shell, respectively. These

terms can be expressed as:


100

𝐼𝑜 =𝜋

4(𝑅𝑜

4 − 𝑅𝑖4) , 𝐼𝑐 =

𝜋

4(𝑅𝑜

4 − 𝑅𝑖4) − [

𝜋

4(𝑅𝑜

4 − 𝑅𝑐4) ∗

𝑎

2𝜋 ]

𝑅𝑜, 𝑅𝑖 are the outer and inner shell radius, respectively. 𝑅𝑐 represents the radius of cracked shell.

Figure 4. The element of crack

The force, moment, and transverse shear force resultants of the cracked shell can be presented as:

𝑁𝑥𝑥 = 12𝐷

ℎ2 𝑓(𝑥) [

𝜕𝑢

𝜕𝑥 +

𝜈

𝑅 𝜕𝑣

𝜕𝜃 +

𝜈

𝑅𝑤] (18)

𝑁𝑥𝜃 = 𝑁𝜃𝑥 =12(1 − 𝜈)𝐷

2ℎ2𝑓(𝑥) [

𝜕𝑣

𝜕𝑥+

1

𝑅 𝜕𝑢

𝜕𝜃] (19)

𝑁𝜃𝜃 = 12𝐷

ℎ2𝑓(𝑥) [

1

𝑅 𝜕𝑣

𝜕𝜃+

𝑤

𝑅+ 𝜈

𝜕𝑢

𝜕𝑥] (20)

𝑀𝑥𝑥 = 𝐷. 𝑓(𝑥) [−𝜕2𝑤

𝜕𝑥2+

𝜈

𝑅 𝜕𝑣

𝜕𝜃−

𝜈

𝑅2 𝜕2𝑤

𝜕𝜃2] (21)

𝑀𝜃𝜃 = 𝐷. 𝑓(𝑥) [1

𝑅2 𝜕𝑣

𝜕𝜃−

1

𝑅2 𝜕2𝑤

𝜕𝜃2− 𝜈

𝜕2𝑤

𝜕𝑥2] (22)

𝑀𝑥𝜃 = 𝐷. 𝑓(𝑥) (1 − 𝜈

2) [

1

𝑅 𝜕𝑣

𝜕𝑥−

2

𝑅

𝜕2𝑤

𝜕𝑥𝜕𝜃] (23)

Employing Eq21 to Eq23 in the Eq4 and Eq5 can be expressed as:

𝑄𝑥𝑧 = D [

𝑓ˊ(𝑥) (−𝜕2𝑤

𝜕𝑥2 + 𝜈

𝑅

𝜕𝑉

𝜕𝜃−

𝜈

𝑅2 𝜕2𝑤

𝜕𝜃2)

+𝑓(𝑥) (− 𝜕3𝑤

𝜕𝑥3 + 1+𝜈

2𝑅2 𝜕2𝑣

𝜕𝑥𝜕𝜃−

1

𝑅2 𝜕3𝑤

𝜕𝑥𝜕𝜃2)

] (24)

𝑄𝑥𝑧 = D [

𝑓ˊ(𝑥) (1−𝜈

2𝑅 𝜕𝑣

𝜕𝑥−

1−𝜈

𝑅

𝜕2𝑤

𝜕𝑥𝜕𝜃)

+ 𝑓(𝑥) (1−𝜈

2𝑅

𝜕2𝑣

𝜕𝑥2 −1

𝑅

𝜕3𝑤

𝜕𝑥2𝜕𝜃+

1

𝑅3 𝜕2𝑣

𝜕𝜃2 −1

𝑅3 𝜕3𝑤

𝜕𝜃3)

] (25)


101

Finally, using Eq18 to Eq20 and Eq24 & Eq25, the governing equations of motion for a cracked cylindrical shell

immersed in fluid is given by:

12𝐷

ℎ2 [𝑓ˊ(𝑥) (𝜕𝑢

𝜕𝑥+

𝜈

𝑅

𝜕𝑣

𝜕𝜃+

𝜈

𝑅𝑤) + 𝑓(𝑥)(

𝜕2𝑢

𝜕𝑥2 +𝜈

𝑅

𝜕2𝑣

𝜕𝑥𝜕𝜃+

𝜈

𝑅 𝜕𝑤

𝜕𝑥+

1−𝜈

2𝑅2 𝜕2𝑢

𝜕𝜃2)] + 𝑓𝑥 = 𝜌ℎ𝜕2𝑢

𝑑𝑡2 (26)

12𝐷

ℎ2 [𝑓ˊ(𝑥) (1−𝜈

2 𝜕𝑣

𝜕𝑥+

1−𝜈

2𝑅 𝜕𝑢

𝜕𝜃) + 𝑓(𝑥)(

1−𝜈

2

𝜕2𝑣

𝜕𝑥2 +1+𝜈

2𝑅

𝜕2𝑢

𝜕𝑥𝜕𝜃+

1

𝑅2 𝜕2𝑣

𝜕𝜃2 +1

𝑅2 𝜕𝑤

𝜕𝜃)] + 𝐷 [𝑓ˊ(𝑥) (

1−𝜈

2𝑅2 𝜕𝑣

𝜕𝑥−

1−𝜈

𝑅2 𝜕2𝑤

𝜕𝑥𝜕𝜃) +

𝑓(𝑥)(1−𝜈

2𝑅2 𝜕2𝑣

𝜕𝑥2 −1

𝑅2 𝜕3𝑤

𝜕𝑥2𝜕𝜃+

1

𝑅4 𝜕2𝑣

𝜕𝜃2 −1

𝑅4 𝜕4𝑤

𝜕𝜃4)] + 𝑓𝜃 = 𝜌ℎ𝜕2𝑣

𝜕𝑡2 (27)

𝐷 [𝑓˝(𝑥) (−𝜕2𝑤

𝜕𝑥2 +𝜈

𝑅2

𝜕𝑣

𝜕𝜃−

𝜈

𝑅2 𝜕2𝑤

𝜕𝜃2 ) + 𝑓ˊ(𝑥) ( −2𝜕3𝑤

𝜕𝑥3 +1+𝜈

𝑅2

𝜕2𝑣

𝜕𝑥𝜕𝜃−

2

𝑅2

𝜕3𝑤

𝜕𝑥𝜕𝜃2 )] + [𝑓(𝑥) (−𝜕4𝑤

𝜕𝑥4 + 1

𝑅2

𝜕3𝑣

𝜕𝑥2𝜕𝜃−

2

𝑅2

𝜕4𝑤

𝜕𝑥2𝜕𝜃2 +1

𝑅4

𝜕3𝑣

𝜕𝜃3 −1

𝑅4

𝜕4𝑤

𝜕𝜃4 )] −12𝐷

ℎ2 [ 𝑓(𝑥) (1

𝑅2

𝜕𝑣

𝜕𝜃+

𝑤

𝑅2 +𝜈

𝑅

𝜕𝑢

𝜕𝑥)] + 𝑓𝑧 = 𝜌ℎ

𝜕2𝑤

𝜕𝑡2 +

∆𝑃 (28)

DONNELL–MUSHTARI–VLASOV (DMV) THEORY

The governing equations for the cylindrical shell, may be represented simpler with the use of the DMV theory.

Assumptions that have been included in the theory of the DMV in a context of the cylindrical shells vibrations are:

1. The in-plane displacement contributions u & v to bending moment that has resulted from the Eq9 to Eq11) is

negligible.

2. The impact of shear term (1/R) 𝑄𝜃𝑧 in motion equation that corresponds to v has been negligible and it indicates the

terms which include 𝐷 in Eq13.

The motion equations for a immersed cylindrical shell Eq12 to Eq14 which correspond to DMV theory may be

represented as:

𝐶 𝜕2𝑢

𝜕𝑥2+

1 − 𝑣

2𝑅2 𝜕2𝑢

𝜕𝜃2 +

𝜈

𝑅

𝜕𝑤

𝜕𝑥+

1 + 𝑣

2𝑅

𝜕2𝑣

𝜕𝑥𝜕𝜃 + 𝑓𝑥 = 𝜌ℎ

𝜕2𝑢

𝜕𝑡2 (29)

𝐶 (1−𝑣

2

𝜕2𝑣

𝜕𝑥2 +1

𝑅2

𝜕2𝑣

𝜕𝜃2 +1

"𝑅2 𝜕𝑤

𝜕𝜃+

1+𝑣

2𝑅

𝜕2𝑢

𝜕𝑥𝜕𝜃) + 𝑓𝜃 = 𝜌ℎ

𝜕2𝑣

𝜕𝑡2 (30)

𝐷 (−𝜕4𝑤

𝜕𝑥4 +2

𝑅2 𝜕4𝑤

𝜕𝑥2𝜕𝜃2 −1

𝑅4 𝜕4𝑤

𝜕𝜃4) − 𝐶 (1

𝑅2

𝜕𝑣

𝜕𝜃+

𝑤

𝑅+

𝜈

𝑅

𝜕𝑢

𝜕𝜃) + 𝑓𝜃 = 𝜌ℎ

𝜕2𝑤

𝜕𝑡2 + ∆𝑃 (31)

The governing equations for a immersed cylindrical shell and which contain a partially crack Eq26 to Eq28 which

correspond to DMV theory may be represented as:

𝑓ˊ(𝑥) (𝜕𝑢

𝜕𝑥+

𝜈

𝑅

𝜕𝑣

𝜕𝜃+

𝜈

𝑅 𝑤) + 𝑓(𝑥) (

𝜕2𝑢

𝜕𝑥2 +1+𝜈

2𝑅

𝜕2𝑣

𝜕𝑥𝜕𝜃+

𝜈

𝑅

𝜕𝑤

𝜕𝑥+

1−𝜈

2𝑅2 𝜕2𝑢

𝜕𝜃2) =(1−𝜈2)𝜌

𝐸

𝜕2𝑢

𝜕𝑡2 (32)

𝑓ˊ(𝑥) (1−𝜈

𝑅 𝜕𝑣

𝜕𝑥+

1−𝜈

2𝑅

𝜕𝑢

𝜕𝜃) + 𝑓(𝑥) (

1−𝜈

2

𝜕2𝑣

𝜕𝑥2 +1+𝜈

2𝑅

𝜕2𝑢

𝜕𝑥𝜕𝜃+

1

𝑅2 𝜕2𝑣

𝜕𝜃2 +1

𝑅2 𝜕𝑤

𝜕𝜃) =

(1−𝜈2)𝜌

𝐸 𝜕2𝑢

𝜕𝑡2 (33)

− [ 𝑓(𝑥) (1

𝑅2 𝜕𝑣

𝜕𝜃+

𝑤

𝑅2 + 𝜈

𝑅 𝜕𝑢

𝜕𝑥)] −

ℎ2

12 [𝑓˝(𝑥) (

𝜕2𝑤

𝜕𝑥2 +𝜈

𝑅2 𝜕2𝑤

𝜕𝜃2 ) + 𝑓ˊ(𝑥) (2 𝜕3𝑤

𝜕𝑥3 +2

𝑅

𝜕3𝑤

𝜕𝑥𝜕𝜃2) + 𝑓(𝑥) ( 𝜕4𝑤

𝜕𝑥4 +

2

𝑅2 𝜕4𝑤

𝜕𝑥2𝜕𝜃2 +1

𝑅4

𝜕4𝑤

𝜕𝜃4 )] =(1−𝜈2)𝜌

𝐸 𝜕2𝑤

𝜕𝑡2 + ∆𝑃 (34)

FLUID DYNAMIC PRESSURE MODELING

For the purpose of formulating governing equation of the cylindrical shell with the interaction of fluid structure, the

pressure of the fluid that acts at the cylindrical shell surface has been represented as the acceleration of the function.

Which is why, there have been some assumptions made for expressing the fluid’s dynamic behaviors.


102

Those assumptions can be summarized as: (a) Linear Vibration (which means small deformations), (b) The fluid is

not compressible, and (c) Interactions between the fluid and the crack if any is disregarded. Taking under consideration

the assumptions that have been stated above, the Laplace equation may be represented in the cylindrical coordinates,

satisfied by the potential function of the velocity:-

∇2𝜙 =1

𝑟

𝜕

𝜕𝑟(𝑟

𝜕𝜙

𝜕𝑟) +

1

𝑟2 𝜕2𝜙

𝜕𝜃2+

𝜕2𝜙

𝜕𝑥2 (35)

Utilizing the equation of Bernoulli, the fluid pressure in the shell at all points can be represented as:

𝑃 = 𝜌1 𝜕𝜙

𝜕𝑡 (36)

Which is why, for external and internal pressure of the fluid may be represented as:

𝑃𝑖 = 𝜌𝑙𝑖 (

𝜕𝜙𝑖

𝜕𝑡)𝑅𝑖

𝑅𝑖 = 𝑅 −ℎ

2 (For the internal pressure) (37)

𝑃𝑒 = 𝜌𝑙𝑒 (

𝜕𝜙𝑒

𝜕𝑡)𝑅𝑒

𝑅𝑒 = 𝑅 +ℎ

2 (For the external pressures) (38)

here 𝜌𝑙 represents the density of the fluid for each unit of volume.

Now the variable separation approach will be applied for separating the function of the essential velocity potential,

satisfying Laplace equation as;

𝜙(𝑥, 𝜃, 𝑟, 𝑡) = ∑ 𝑆(𝑟)𝑄(𝑥. 𝜃. 𝑡)

8

𝑗=1

(39)

where 𝑆(𝑟) and Q(𝑥, 𝜃, 𝑡) represent the two distinctive functions that need being obtained from the boundary

conditions of the fluid. For a long-term contact between the peripheral fluid layer and the surface of the shell, the

condition of the impermeability will be expressed as:

( 𝜕𝜙

𝜕𝑟)

𝑟=𝑅𝑖

= 𝜕𝑊

𝜕𝑡 (40)

( 𝜕𝜙

𝜕𝑟)

𝑟=𝑅𝑒

= 𝜕𝑊

𝜕𝑡 (41)

On the introduction of Eq39 to Eq40 and Eq41, the result will bet:

𝑄(𝑥, 𝜃, 𝑡) = ∑1

(𝜕𝑆(𝑟)

𝜕𝑟)

𝑟=𝑅𝑖

8

𝑗=1

𝜕𝑊𝑛

𝜕𝑡 (42)

𝑄(𝑥, 𝜃, 𝑡) = ∑1

(𝜕𝑆(𝑟)

𝜕𝑟)

𝑟=𝑅𝑒

8

𝑗=1

𝜕𝑊𝑛

𝜕𝑡 (43)

Through the substitution of Eq42 & Eq43 into Eq39 the potential of the velocity on the interfaces of the fluid–shell

(in other words, lower and upper shell surfaces) may be represented as


103

∅(𝑥, 𝜃, 𝑟, 𝑡) = ∑𝑆(𝑟)

(𝜕𝑆(𝑟)

𝜕𝑟)

𝑟=𝑅𝑖

8

𝑗=1

𝜕𝑊𝑛

𝜕𝑡 (44)

∅(𝑥, 𝜃, 𝑟, 𝑡) = ∑𝑆(𝑟)

(𝜕𝑆(𝑟)

𝜕𝑟)

𝑟=𝑅𝑒

8

𝑗=1

𝜕𝑊𝑛

𝜕𝑡 (45)

From the shell theory, the transverse defection may be represented as (Montre [15]);

𝑊(𝑥, 𝜃, 𝑡) = ∑ 𝐶𝑗𝑒𝜂𝑗𝜃 sin (𝑚𝜋𝑥

𝑙) 𝑒𝑖𝜔𝑡

8

𝑗=1

(46)

By the substitution of the Eq44 or Eq45 in Eq35 and the expression of 𝑊 from Eq46 the result will be the Laplace 2nd

order equation in the following form:

𝑟2𝜕2 𝑆𝑛 (𝑟)

𝜕𝑟2+ 𝑟

𝜕 𝑆𝑛 (𝑟)

𝜕𝑟+ 𝑆𝑛(𝑟) [(

𝑖𝑚𝜋

𝑙)

2

𝑟2 − (𝑖𝜂𝑛)2] = 0 (47)

where 𝑖 represents the complex number, 𝑖2 = 1 , and 𝜂𝑛 represents complex solution for Characteristic equation. The

general equation solution may be represented as:

𝑆𝑛(𝑟) = 𝐴𝐽𝑖𝜂𝑛 (𝑖𝑚𝜋

𝑙𝑟) + 𝐵𝑌𝑖𝜂𝑛 (

𝑖𝑚𝜋

𝑙𝑟) (48)

Here, 𝐽𝑖𝜂𝑛 & 𝑌𝑖𝜂𝑛 represent the Bessel function of 1st & 2nd kind, respectively.

By the substitution of Eq45 & Eq46 in Eq37 & Eq38, the pressure of the fluid in the terms of 𝑆(𝑟) and 𝑊 may be

represented as:

𝑃𝑖 = 𝜌𝑙 ∑𝑆(𝑟)

(𝜕𝑆(𝑟)

𝜕𝑟)

𝑟=𝑅𝑖

8

𝑗=1

𝜕2𝑊𝑛

𝜕𝑡2 (49)

𝑃𝑒 = 𝜌𝑙 ∑𝑆(𝑟)

(𝜕𝑆(𝑟)

𝜕𝑟)

𝑟=𝑅𝑒

8

𝑗=1

𝜕2𝑊𝑛

𝜕𝑡2 (50)

On the expression of 𝑆(𝑟) from Eq48 in Eq49 & Eq50, the dynamic pressure of the fluid on the surface of the

cylindrical shell may be represented as:

𝑃 = −𝜌𝑙 ∑ 𝑈𝑛

8

𝑛=1

(𝑖𝑚𝜋𝑅𝑖/𝑒

𝑙) (

𝜕2𝑊𝑛

𝜕𝑡2 ) (51)

𝑈𝑛 (𝑖𝑚𝜋𝑅𝑖

𝑙) =

𝑅𝑖

𝑖𝜂𝑛 − 𝑖𝑚𝜋𝑅𝑖

𝑙 𝐽𝑖𝜂𝑛+1(𝑖𝑚𝜋𝑅𝑖/𝑙)

𝐽𝑖𝜂𝑛(𝑖𝑚𝜋𝑅𝑖/𝑙)

(For the internal pressure) (52)

𝑈𝑛 (𝑖𝑚𝜋𝑅𝑒

𝑙) =

𝑅𝑒

𝑖𝜂𝑛 − 𝑖𝑚𝜋𝑅𝑖

𝑙 𝐽𝑖𝜂𝑛+1(𝑖𝑚𝜋𝑅𝑒/𝑙)

𝐽𝑖𝜂𝑛(𝑖𝑚𝜋𝑅𝑒/𝑙)

(For the external pressures) (53)


104

Where 𝑖𝜂 represents the characteristic equation roots of empty shell, 𝐽𝑖𝜂𝑛 & 𝑌𝑖𝜂𝑛 represent 1st and 2nd kind Bessel

function, respectively, 𝑚 represents the axial mode number 𝑅 represents the average shell radius and 𝑙 represents the

shell length.

The net dynamic pressure that acts on the cylindrical shell surface may be specified with the use of the 2 expressions

above, which is why the obtained pressure will be:-

∆𝑃 = 𝑃𝑒 − 𝑃𝑖 = −𝜌𝑙 [∑ 𝑈𝑛

8

𝑛=1

(𝑖𝑚𝜋𝑅𝑒

𝑙) − ∑ 𝑈𝑛

8

𝑛=1

(𝑖𝑚𝜋𝑅𝑖

𝑙)] (

𝜕2𝑊𝑛

𝜕𝑡2) (54)

Or

∆𝑃 = 𝑚𝑎𝑑𝑑 (𝜕2𝑊𝑛

𝜕𝑡2)

Where

𝑚𝑎𝑑𝑑 = −𝜌𝑙 [∑ 𝑈𝑛

8

𝑛=1

(𝑖𝑚𝜋𝑅𝑒

𝑙) − ∑ 𝑈𝑛

8

𝑛=1

(𝑖𝑚𝜋𝑅𝑖

𝑙)]

is the virtual added shell mass due to the surrounding fluids.

GENERAL SOLUTION OF GOVERNING EQUATION

The governing equation solutions of the motion for the simply supported boundary conditions that correspond to the

DMV theory may be represented as (Marwah [6]):

𝑢(𝑥, 𝜃) = ∑ ∑ 𝐴𝑚𝑛 cos𝑚𝜋𝑥

𝑙 cos 𝑛𝜃 cos 𝜔𝑡

𝑛 m

(55)

𝑣(𝑥, 𝜃) = ∑ ∑ 𝐵𝑚𝑛 sin𝑚𝜋𝑥

𝑙 sin 𝑛𝜃 cos 𝜔𝑡

𝑛 m

(56)

𝑤(𝑥, 𝜃) = ∑ ∑ 𝐶𝑚𝑛 sin𝑚𝜋𝑥

𝑙 cos 𝑛𝜃 cos 𝜔𝑡

𝑛 m

(57)

Here, 𝑚 represents the axial mode in the shell length direction, 𝑛 represents circumferential mode in the shell’s radial

direction, and 𝐴𝑚𝑛 , 𝐵𝑚𝑛 & 𝐶𝑚𝑛 stand for the constants.

Putting 𝑢 , 𝑣 , & 𝑤 values in governing equations for the intact immersed cylindrical shells in the fluid (Eq29 to Eq31),

the equations below may be obtained:

(− 𝜆2 − 𝑎1𝑛2 + 𝛺)𝐴𝑚𝑛 + (𝑎2𝜆𝑛) 𝐵𝑚𝑛 + (𝜈𝜆) 𝐶𝑚𝑛 = 0 (58)

(𝑎2𝜆𝑛) 𝐴𝑚𝑛 + (−𝑎1𝜆2 − 𝑛2 + 𝛺) 𝐵𝑚𝑛 + (−𝑛) 𝐶𝑚𝑛 = 0 (59)

(𝜈𝜆) 𝐴𝑚𝑛 + (−𝑛)𝐵𝑚𝑛 + (−1 − 𝜆4𝜇 − 2𝜆2𝑛2𝜇 − 𝑛4𝜇 + 𝛺 + 𝑚𝑎𝑑𝑑 𝜔2) 𝐶𝑚𝑛 = 0 (60)

here the value of constant are clarified as below:

𝜆 = 𝑚𝜋𝑅

𝑙 , 𝜇 =

ℎ2

12𝑅2 , 𝛺 = (1−𝜈2)𝑅2𝜌

𝐸𝜔2 , 𝑎1 =

1−𝜈

2 , 𝑎2 =

1+𝜈

2


105

For non-trivial 𝐴𝑚𝑛 , 𝐵𝑚𝑛 , and 𝐶𝑚𝑛 solutions, their coefficient determinants of a matrix will be equal to 0. Which

results in the equations of the frequency in the cubic forms that are solved for finding the basic natural frequency’s

numerical value of the intact cylindrical shell that has been submerged in the fluid.

On substituting 𝑢 , 𝑣 , and 𝑤 in the governing equations for the cracked immersed cylindrical shell in fluid Eq32 to

Eq34, yields the following equations:

[− 𝑓ˊ(𝑥)𝑅 𝜆 − 𝑓(𝑥) (𝜆2 + 𝑎1𝑛2) + 𝛺]𝐴𝑚𝑛 + [𝑓ˊ(𝑥)𝑅 𝜈 𝑛 + 𝑓(𝑥)(𝑎2𝜆𝑛)]𝐵𝑚𝑛 + [𝑓ˊ(𝑥)𝑅𝜈 + 𝑓(𝑥)(𝜈𝜆)]𝐶𝑚𝑛 =

0 (61)

[− 𝑓ˊ(𝑥)𝑅 (𝑎1𝑛) − 𝑓(𝑥) (𝑎2𝜆𝑛)]𝐴𝑚𝑛 + [𝑓ˊ(𝑥)𝑅 𝑎1𝜆 + 𝑓(𝑥)(−𝑎1𝜆2 − 𝑛2) + 𝛺]𝐵𝑚𝑛 − [𝑓(𝑥)𝑅 (𝑛)]𝐶𝑚𝑛 =

0 (62)

[𝑓(𝑥)(𝜈𝜆) ]𝐴𝑚𝑛 + [−𝑓(𝑥) (𝑛)] 𝐵𝑚𝑛 + {[𝑓(𝑥)(−1 − 𝜆4𝜇 − 2𝜆2𝑛2𝜇 − 𝑛4𝜇)] + [𝑓ˊ(𝑥)𝑅 (−2𝜆3𝜇 − 2𝑅𝜆𝑛2𝜇)] +

[𝑓˝(𝑥)𝑅2(−𝜆2𝜇 + 𝜈𝑛2𝜇)] + 𝛺 + 𝑚𝑎𝑑𝑑𝑅2𝜔2]} 𝐶𝑚𝑛 = 0 (63)

Where the constant values are represented below:

𝑓ˊ(𝑥) = 2𝑆𝛼/ℎ 𝑒(−2𝛼|𝑥−𝑥𝑐|/ℎ)

[1 + 𝑆. 𝑒(−2𝛼|𝑥−𝑥𝑐|/ℎ)]2

𝑓˝(𝑥) = −[1+𝑆.𝑒

(−2𝛼|𝑥−𝑥𝑐|

ℎ )]

2

.(2𝑠𝛼

ℎ)

2 𝑒


ℎ )+2(

2𝑠𝛼

ℎ)

2 𝑒


ℎ )2

[1+𝑆.𝑒(−

2𝛼|𝑥−𝑥𝑐|ℎ )

]

[1+𝑆.𝑒(−

2𝛼|𝑥−𝑥𝑐|ℎ )

]

4

Once more, for non-trivial 𝐴𝑚𝑛 , 𝐵𝑚𝑛 , & 𝐶𝑚𝑛 solutions of the Eq61 to Eq63, their coefficient determinants in a matrix

will be 0. Which will result in equations of frequency in the cubic forms that have been solved for the purpose of

finding the frequency parameter values, which are additionally substituted into the needed equation for the purpose of

finding the fundamental frequency numerical value of the cracked cylindrical immersed shell.

RESULTS AND DISCUSSIONS

This study suggests analytical framework for the cracked cylindrical shell vibrations, which have been submerged in

the fluid for the simply supported boundary conditions. The literature is lacking in the isotropic submerged cracked

cylindrical shell vibration results, therefore, the new results for the fundamental frequency as affected by the length

of the crack, the position of the crack, the depth of the crack, and the radius of the shell’s, thickness, and length have

been given in the present research. In (zhang [10]) the intact cylindrical shell results without influences of any crack

have been shown. As a result, for validating this framework, the submerged intact shell results which have been

produced from this model have been compared to results that have been stated in zhang [10] for the simply supported

boundary conditions only that have been illustrated in figure 5.


106

Figure 5. Comparison of frequency 𝛺 = (1 − 𝜈2)𝑅2𝜌𝜔2/𝐸 for a simply supported steel cylindrical shell in fluid

and vacuum medium

The shell material characteristics for the validation figure has been obtained from zhang [10]. It has been noticed from

figure5 that the results that have been obtained are a little bit higher compared to the existing (zhang [10]) results in

the vacuum as well as the fluid medium. Which results from the fact that the results have been obtained with the use

of the conventional shell theory and it includes no transverse shear deformation impact. None-the-less, the difference

in results by zhang [10] and this study is rather small and in 2%. Comparisons have shown that the approach which

has been proposed in this study is accurate, approximate, however, with sufficient level of the accuracy.

New fundamental frequency results of the cracked cylindrical shells have been computed for a variety of the crack

locations, crack depths, and crack length values under the submerged condition as well as the vacuum condition. For

the crack position variation (𝑥𝑐/𝑙) a nondimensional parameter has been taken and that may be characterized as a

crack position ratio along axial shell direction to the entire shell length. In a similar manner, the variation of the length

of the crack has been stated as (𝑎/2𝜋𝑅) a non-dimensional parameter which may be characterized as the partial crack

length ratio to the shell circumference, and crack depth variation (ℎ𝑐/ℎ) a nondimensional parameter has been obtained

and that may be characterized as the crack depth ratio on the external shell surface to the entire shell depth. The

material that has been taken under consideration for isotropic shell is a steel that has the constants of the material as;

elasticity Modulus 𝐸 =210GPa, steel density ρ = 7,850𝑘𝑔. 𝑚−3, the ratio of the Poisson 𝜐 = 0.30. The density of the

fluid 𝜌𝑙 is 1,000𝑘𝑔. 𝑚−3.

The fluid reservoir measurements has been assumed as (5x5x5) m. The mathematical formula has been performed for

the simply supported boundary conditions. The significant results have been stated below.

Figure6 illustrates the fundamental frequency variation for the cracked cylindrical shell as has been influenced by the

length of the crack (𝑎/2 𝜋𝑅), 𝑅/ℎ and 𝑙/𝑅 ratio. Figure6 a, c illustrate the fundamental frequency variations with

𝑅/ℎ and 𝑙/𝑅 constant value in the case of the fluid and vacuum mediums respectively. It was observed as in (Marwah

[6]) on the analyses of cracked cylindrical shells in the vacuum presence of the crack reduces natural frequency. It has

been observed from Fig6. A,c, for a certain crack length value, the natural frequency is dependent upon the 𝑙/𝑅 factor,

in a way that in the case of the increase in 𝑙/𝑅 value, then that results in the decrease in the fundamental frequency

value. Which holds as well for submerged shells. One more significant observation is that for the lower ratio of 𝑙/𝑅

the decrements of natural frequencies are more in the case where they are at a higher 𝑙/𝑅 value, such decrement is

lower. Which means that the length of the crack is noticeable for shorter cylinder shells in comparison with longer

shell, validating the physical understandings. It has been observed as well, that for a fixed 𝑙/𝑅 value, the increase in

the length of the crack the fundamental frequency is decreased and such decrement in fundamental frequency is of a

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7 8 9 10

No

n-d

ime

nsi

on

al f

req

ue

ncy

Circumferential mode n

Zhang [10] in vacuum

present work in vacuum

Zhang [10] in fluid

present work in fluid


107

higher significance in a case of the submerged shell. Such reduction in frequency results from a decreased shell

stiffness because of the cracks, and the virtual added mass (𝑚add) as a result of the surrounding fluid, increasing the

overall coupled system mass.

Figure6 b,d show the impact of the length of the crack in the natural frequencies with 𝑅/ℎ at a constant 𝑙/ℎ value in

the fluid as well as the vacuum mediums respectively. From Figure6 b,d it can be observed that for a certain crack

length value, there is a natural decrease in the frequency with the increase in 𝑅/ℎ. It has been observed as well, that

for a certain value of 𝑅/ℎ, the increase in the length of the crack, results in the decrease of the fundamental frequency,

which results from stiffness decrease of submerged shells. Such decrease in frequency appears less at lower value of

𝑅/ℎ ratio whereas it is considerably higher at an increased 𝑅/ℎ ratio. Which indicates the fact that crack length impact

is greater for a very thin shell in comparison with the thin shells.

(a) In vacuum 𝑅/ℎ =30 , ℎ =1mm (b) In vacuum 𝑙/ℎ =1,500 , ℎ =1mm

0

30

60

90

120

150

180

210

30 35 40 45 50 55 60

Nat

ura

l fre

qu

en

cy (

rad

/s)

l/R

a/2πR=0

a/2πR=0.1

a/2πR=0.2

a/2πR=0.3

a/2πR=0.4

a/2πR=0.5

0

30

60

90

120

150

180

20 25 30 35 40 45 50

Nat

ura

l fre

qu

en

cy (

rad

/s)

R/h

a/2πR=0

a/2πR=0.1

a/2πR=0.2

a/2πR=0.3

a/2πR=0.4

a/2πR=0.5


108

(a) In fluid 𝑅/ℎ =30 , ℎ =1mm (b) In fluid 𝑙/ℎ =1,500 , ℎ =1mm

Figure 6. Fundamental frequency as a function of crack length (𝑎/2 𝜋𝑅), 𝑙/𝑅 and 𝑅/ℎ , (ℎ𝑐/ℎ=0.8, 𝑥𝑐/𝑙 = 0.5)

Figure7 illustrates the fundamental frequency variations for cracked cylindrical shells as influenced by the position of

the crack (𝑥𝑐/𝑙) and depth of the crack (ℎ𝑐/ℎ). It can be evident from figure7 a&b, at a constant crack position value

respectively in a case of the vacuum and fluid mediums. As projected, the fundamental frequency reductions are

increased with increasing the depth of the crack. The decreased value of shell stiffness as a result of the crack is the

cause of this frequency reduction. It has been seen from figure7 a&b, for constant crack depth value in cases of the

vacuum and fluid mediums respectively. The maximal natural frequency reduction happens in the case where the

crack is in middle along the shell’s longitudinal orientation. The reason behind those result is the fact that the

fundamental frequency mode shape is symmetrical about z plane at (𝑥/𝑙 = 0.50) under the simply supported

conditions of the boundary, in addition to that, due to the fact that the maximal deflection resides in the middle of

longitudinal cylindrical shell direction.

(a) In vacuum

0

15

30

45

60

75

90

30 35 40 45 50 55 60

Nat

ura

l fre

qu

en

cy (

rad

/s)

l/R

a/2πR=0

a/2πR=0.1

a/2πR=0.2

a/2πR=0.3

a/2πR=0.4

a/2πR=0.5

0

15

30

45

60

75

90

20 25 30 35 40 45 50N

atu

ral f

req

ue

ncy

(ra

d/s

)

R/h

a/2πR=0

a/2πR=0.1

a/2πR=0.2

a/2πR=0.3

a/2πR=0.4

a/2πR=0.5

0.8

0.84

0.88

0.92

0.96

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

fre

qu

en

cy r

atio

(ω

/ωn

o c

rack

)

Xc/l

hc/h=0.2

hc/h=0.4

hc/h=0.6

hc/h=0.8


109

(b) In fluid

Figure 7. Percentage decrease in the fundamental frequency as a crack depth function (ℎ𝑐/ℎ), and crack position

(𝑥𝑐/𝑙 ) , ( 𝑎/2𝜋𝑅=0.2, 𝑙/𝑅 = 1500 , 𝑅/ℎ =30)

Through the comparison between the frequency variation in the vacuum shell and in the fluid mediums, it has been

observed that the frequency reduction has higher significance in the fluid in comparison with the vacuum. Which

results from resistance which is offered by fluid mediums in a dynamic pressure form to the shell vibratory motions.

While immersed shell goes deep in fluid the fluid layer mass that vibrates with shell is increased, which results in the

increase of the virtual or the total masses of coupled systems. This variation phenomenon in the frequency has been

found as well in the literature (Soni etal. [12]) for the cracked and intact isotropic plates and (Jain et al. [13]) for the

FGM submerged and cracked orthotropic cylindrical shell types.

CONCLUSIONS

In this study, efforts were made for obtaining the basic isotropic cracked immersed cylindrical shell frequency. The

motion governing equation was obtained with the use of classical theory of the shell in the coupling with a flow theory

potential. The hydro-dynamic pressure that acts on the element of the shell as a result of the fluidic medium has been

utilized with a help from Bernoulli’s equation and the velocity potential. Results have been given as a fundamental

frequency with varying the crack depth, length, location, and shell size (𝑅/ℎ & 𝑙/𝑅). It has been stated that the shell

fundamental frequency is decreased with the existence of the crack and such reduction in the frequency is reduced

additionally with the existence of the fluid medium in this work. It has been seen that the cracked cylindrical shell

frequency is decreased in the case of the submerging into the fluid because of the dynamic pressure of the fluid. It has

been concluded as well that the existence of a crack results in decreasing shell stability as a result of stiffness decrease.

In this work, there is a comparison of the natural frequency between cracked shell in the fluid medium and in the

vacuum. Which has been the first attempt for the modeling of the cracked isotropic shell vibrations with the

consideration of the fluid-structure interaction effects and as a result, it would be instructive formulating analytical

model with the use of a theory of a higher order. In addition to that, the initial imperfection types, which are related

in the structures of the shell can be taken under consideration for more sufficient model analysis.

REFERENCES

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0.68

0.72

0.76

0.8

0.84

0.88

0.92

0.96

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

fre

qu

en

cy r

atio

(ω

/ωn

o c

rack

)

Xc/l

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hc/h=0.4

hc/h=0.6

hc/h=0.8


110

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vibration analysis of partially cracked thin cylindrical

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