^{1}

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In the present study, a hybrid ?nite element method is applied to investigate the dynamic behavior of a spherical shell partially filled with fluid and subjected to external supersonic airflow. The structural formulation is a combination of linear spherical shell theory and the classic finite element method. In this hybrid method, the nodal displacements are derived from exact solution of spherical shell theory rather than approximated by polynomial functions. Therefore, the number of elements is a function of the complexity of the structure and it is not necessary to take a large number of elements to get rapid convergence. Linearized first-order potential (piston) theory with the curvature correction term is coupled with the structural model to account for aerodynamic loading. It is assumed that the fluid is incompressible and has no free surface effect. Fluid is considered as a velocity potential at each node of the shell element where its motion is expressed in terms of nodal elastic displacements at the ?uid-structure interface. Numerical simulation is done and vibration frequencies are obtained. The results are validated using numerical and theoretical data available in literature. The investigation is carried out for spherical shells with different boundary conditions, geometries, filling ratios, flow parameters, and radius to thickness ratios. Results show that the spherical shell loses its stability through coupled-mode flutter. This proposed hybrid finite element method can be used efficiently for analyzing the flutter of spherical shells employed in aerospace structures at less computational cost than other commercial FEM software.

Shells of revolution, particularly spherical shells are one of the primary structural elements in high speed aircraft. Their applications include the propellant tank or gas-deployed skirt of spacecraft. Due to the aerodynamic shape combined with thin wall thicknesses, spherical shells are more disposed to dynamic instability or flutter induced by high Mach number gas flow. It is therefore important to understand the effect of different flow parameters and loadings on their aeroelastic response.

Aeroelastic analysis of shells and plates has been studied by numerous researchers experimentally and analytically [

There are also some researchers who focused their efforts on the numerical study of this problem. The equations of virtual displacements were solved using the finite elements method. Aeroelastic governing equations were formulated by applying classical shell theory coupled with the piston theory for evaluation of aerodynamic forces. For example, Bismarck-Nasr [

Aeroelasticity of conical shells has also been investigated by few researchers. The leading work in this field was conducted by Shulman [

An analytical approach to the supersonic flutter of spherical shell becomes very complicated if one wishes to include different parameters. Therefore, the efficiency of numerical methods such as the finite element method (FEM) is an advantage for cases involving changes to all factors affecting flutter boundaries. The aim of the present study is to develop a hybrid finite element method in order to predict the aeroelastic behavior of isotropic spherical shells with different parameters as boundary conditions, geometries, flow parameters, filling ratios and radius to thickness ratios. The finite element is a spherical frustum instead of the usual rectangular shell element. Linear thin shell theory is coupled with linear piston theory. In the case of a fluid filled shell the effect of dynamic pressure acting on the wall is modeled based on a velocity potential formulation and Bernoulli’s equation. It is assumed that the fluid is incompressible and has no free surface effect. The linear mass, damping and stiffness matrices are obtained. The aeroelastic equation of motion is reduced to a standard eigenvalue problem. The flutter boundary is found by analyzing the real and imaginary parts of the eigenvalues as freestream pressure is varied.

In this study the structure is modeled using hybrid finite element method which is a combination of spherical shell theory and classical finite element method. In this hybrid finite element method, the displacement functions are found from exact solution of spherical shell theory rather than approximated by polynomial functions as done in classical finite element method. In the spherical coordinate system (R, θ, ϕ) shown in

where_{ϕ}, M_{θ}, M_{ϕθ} the bending stress resultants and Q_{ϕ}, Q_{θ} the shear forces (

Strains and displacements in axial,

Geometry of the spherical shell

Stress resultants and stress couple

Displacements

The stress vector

where

Spheical frustum element

Upon substitution of Equations (2), (4) and (5) into Equation (1), a system of equilibrium equations can beobtained as a function of displacements:

These three linear partial differential operators

where

The element is a circumferential spherical frustum shown in

For motions associated with the circumferential wave number n, we may write:

The transversal displacement

where

and where

The expression of the axial displacement u_{ϕν}(ϕ) is:

where the coefficient E_{i} is given by:

The auxiliary function ψ is given by the expression:

Finally the circumferential displacement u_{ϕν}(ϕ) can be expressed as:

The degree

where

and where

with

The above equation has three roots with one real root and two others complex conjugates.

The Legendre functions

(i = 2, 3) are complex functions. So we can put:

Setting

Substituting Equations (18), (19) and (20) in Equations (9), (11) and (14) we have:

In deriving the above relation we used the recursive relations:

Using matrix formulation, the displacement functions can be expressed as follows:

The vector

The elements of matrix

In the finite element method, the vector C is eliminated in favor of displacements at elements nodes. At each finite element node, the three displacements (axial, transversal and circumferential) and the rotation are applied. The displacement of node i are defined by the vector:

The finite element shown in

with

The terms of matrix, obtained from the values of matrix _{i} as a function of the degree of freedom:

Finally, one substitutes the vector

The strain vector

where matrix

This relation can be used to find the stress vector, Equation (4), in terms of the nodal degrees of freedom vector:

Based on the finite element formulation, the local stiffness and mass matrices are:

where

The surface element of the shell wall is

In the global system, the element stiffness and mass matrices are

where

From these equations, one can assemble the mass and stiffness matrices for each element to obtain the mass and stiffness matrices for the whole shell:

Piston theory, introduced by Ashley and Zartarian [

where

where

Finally, the aerodynamic pressure in terms of radial displacement is written:

and the pressure loading in terms of nodal degrees of freedom is written as:

where

The matrix

The matrix

where matrix

The general force vector due to a pressure field is written as:

The local damping matrix is given by:

Finally the local stiffness matrix is given by:

In the global system, the element damping and stiffness matrices are:

From these equations, one can assemble the damping and stiffness matrices for each element to obtain the damping and stiffness matrices for the whole shell:

The Laplace equation satisfied by velocity potential for inviscid, incompressible and irrotational fluid in the spherical system is written as:

where the velocity components are:

Using the Bernouilli equation, hydrodynamic pressure in terms of velocity potential

The impermeability condition, which ensures contact between the shell surface and the peripheral fluid, is written as:

with

Method of separation of variables for the velocity potential solution can be done as follows:

Placing this relation into the impermeability condition (50), we can find the function

Hence the equation becomes

With substitution of the above equation into Laplace Equation (47), the following second order equation in terms of

Solution of the above differential equation yields the following:

For internal flow

Finally, the hydrodynamic pressure in terms of radial displacement is written:

We put:

And the pressure loading in terms of nodal degrees of freedom is written as:

where matrix

where

The general force vector due the fluid pressure loading is given by:

After substituting for pressure field vector and matrix

In the global system the element stiffness and mass matrices are

From these equations, one can assemble the mass for each element to obtain the mass matrix for the whole shell:

The governing equation which accounts for fluid-shell interaction in the presence external supersonic airflow is derived as:

where subscripts s and f refer to shells in vacuo and fluid respectively.

The global fluid matrices mentioned in Equation (65) may be obtained, respectively, by superimposing the mass, damping and stiffness matrices for each individual fluid finite element. After applying the boundary conditions the global matrices are reduced to square matrices of order 4*(N + 1) − J, where N is the number of finite elements in the shell and J is the number of constraints applied. Finally, the eigenvalue problem is solved by means of the equation reduction technique. Equation (65) may be rewritten as follows:

where

where

An in house computer code based on the finite element method was developed as part of this work to establish the structural and fluid matrices of each element based on equations developed using the theoretical approach. The calculations for each finite element are performed in two stages: the first dealing with solid shell and the second with the effect of the flowing fluid. Aeroelastic stability will be examined by studying the eigenvalues in the complex plane. When the imaginary part of ω becomes negative the amplitude of the shell motion grows exponentially with time, thus indicating dynamic instability. The flutter boundary is obtained numerically by tracing the eigenvalues to see when the sign of imaginary part just changes from positive to negative. For the fixed value of circumferential wave number n, the onset of instability is determined by varying the value of freestream static pressure. This procedure is repeated for different values of n until the minimum critical pressure is obtained.

In this section numerical results are presented and compared with existing experimental, analytical and numerical data.

For the cases investigated in the present paper, the predicted dimensionless frequencies are expressed by the following relation:

where:

Results for different boundary conditions, geometries, flow parameters and radius to thickness ratios compared to experimental, theoretical and numerical analyses are presented (see

Case 1: clamped spherical shell with

Narassihan and Alwar [

Case 2: clamped spherical shell with

This case was investigated analytically by Kalnins [

Case 3: spherical shell with

Definition of angle

. Normalized natural frequencies for 10˚ clamped spherical shell with R/h = 200

Mode | Present theory | Sai Ram and Sreedhar babu [23] | Narassihan and Alwar [24] | |
---|---|---|---|---|

1 | 1.4861 | 1.4577 | 1.4588 | |

2 | 2.2498 | 2.2931 | 2.2999 | |

3 | 4.4779 | 4.5773 | 4.5461 | |

. Normalized natural frequencies for 30˚ clamped spherical shell with R/h = 20

Mode | Presenttheory | Kalnins [25] |
---|---|---|

1 | 1.169 | 1.168 |

2 | 2.224 | 2.589 |

3 | 3.303 | 3.230 |

4 | 4.200 | 4.288 |

5 | 4.923 | 4.683 |

Free axisymmetric vibration of the spherical shell in this case was studied by Kalnins [

Case 4: spherical shell with

Kraus [

The problem treated for validation is the flutter boundary of a simply–supported spherical shell subjected to external supersonic airflow. As there is no information available for flutter of spherical shells, this case has been compared with simply-supported cone studied by various authors. The conical shell has the following data: Young’s Modulus, E = 6.5 106 lb-in-2, Poisson’s ratio, ν = 0.29, material mass density, ρ = 8.33 10-4 lb-s2-in-4, shell thickness, h = 0.051 in, cone semi-vertex angle α = 5˚. The supersonic airflow has freestream Mach number, M∞ = 3, stagnation temperature, T∞ = 288.15 K. The results are shown in

where

When results are summarized and compared with other finite element and analytical solutions, this method shows good convergence using only 15 elements with small disagreements. It should be noted that the previous analytical methods [

Flutter which is observed in all the papers using piston theory is a coupled-mode flutter. Indeed, let us consider motion of the shell eigenvalues in the complex ω plane. If the freestream pressure is not very high, and the shell is stable, all complex frequencies are located in the top ω half-plane. Let us now increase freestream pressure. The first and the second complex frequencies move toward each other, almost merge, and then go away from

. Normalized natural frequencies for 60˚ clamped spherical shell with R/h = 20

Mode | Kalnins [26] | Navaratna [28] | Webster [29] | Tessler and Spiridigliozzi [31] | Gautham and Ganesan [32] | Buchanan and Rich [33] | Present theory |
---|---|---|---|---|---|---|---|

1 | 1.006 | 1.008 | 1.007 | 1.000 | 1.001 | 1.001 | 1.031 |

2 | 1.391 | 1.395 | 1.391 | 1.368 | 1.373 | 1.370 | 1.496 |

3 | - | 1.702 | 1.700 | 1.673 | 1.678 | 1.675 | 1.760 |

4 | - | 2.126 | 2.095 | - | - | 2.094 | 2.089 |

5 | 2.375 | 2.387 | 2.386 | 2.260 | - | 2.256 | 2.276 |

6 | 3.486 | 3.506 | 3.851 | 3.213 | - | 3.209 | 3.311 |

7 | 3.991 | 3.996 | 4.062 | 3.965 | - | 3.964 | 3.775 |

8 | - | 4.159 | 4.151 | - | - | 4.060 | 4.073 |

9 | 4.947 | 5.001 | 5.962 | 4.442 | - | 4.427 | 4.826 |

10 | - | 6.037 | 6.208 | 5.773 | - | 5.740 | 5.777 |

. Normalized natural frequencies for 60˚ simply supported spherical shell with R/h = 20

Mode | Kalnins [26] | Navaratna [28] | Greene et al. [30] | Cohen [27] | Gautham and Ganesan [32] | Buchanan and Rich [33] | Present theory |
---|---|---|---|---|---|---|---|

1 | 0.962 | 0.963 | 0.974 | 0.959 | - | 0.956 | 0.981 |

2 | 1.334 | 1.338 | 1.338 | 1.325 | 1.315 | 1.308 | 1.412 |

3 | - | 1.653 | 1.652 | 1.646 | 1.639 | 1.612 | 1.646 |

4 | 2.128 | 2.131 | 2.162 | - | - | 2.044 | 2.038 |

5 | - | 2.141 | - | - | - | 2.059 | 2.115 |

6 | 3.176 | 3.185 | - | - | - | 2.965 | 2.934 |

7 | 3.988 | 3.933 | - | - | - | 3.837 | 3.871 |

8 | - | 4.159 | - | - | - | 4.000 | 4.017 |

9 | 4.575 | 4.601 | - | - | - | 4.148 | 4.138 |

10 | - | 6.031 | - | - | - | 5.608 | 5.773 |

. Normalized natural frequencies for 90˚ clamped spherical shell with R/h = 10

Mode | Tessler and Spiridigliozzi [31] | Gautham and Ganesan [34] | Present theory |
---|---|---|---|

1 | 0.8481 | 0.8439 | 0.8327 |

2 | 1.2328 | 1.2317 | 1.1919 |

3 | 1.5902 | 1.5808 | 1.5041 |

4 | 1.9435 | 1.9267 | 1.9161 |

each other in vertical directions (

A simply supported spherical shell with

. Normalized natural frequencies for 90˚ simply supported spherical shell

Mode | Kraus [22] R/h = 10 | Kraus [22] R/h = 50 | Ventsel et al. [35] R/h = 200 | Present theory R/h = 50 |
---|---|---|---|---|

1 | 0.8060 | 0.7548 | 0.7441 | 0.7579 |

2 | 1.2054 | 0.9432 | 0.9281 | 0.9034 |

3 | 1.6179 | 1.0152 | 0.9693 | 0.9499 |

4 | 1.9051 | 1.1082 | - | 1.1089 |

5 | 2.7205 | 1.2523 | - | 1.2759 |

6 | 2.9301 | 1.4576 | - | 1.4723 |

7 | 4.0274 | 1.6558 | - | 1.6237 |

8 | 5.5142 | 1.7636 | - | 1.7634 |

. Comparison of critical dynamical pressure parameter (simply supported case)

Present | Dixon and Hudson [15] | Udea et al. [14] | Pidaparti and Yang Henri [21] | Shulman [13] | Bismark-Nasr [17] |
---|---|---|---|---|---|

520(5)^{a} | 590(5) | 609(5) | 576(5) | 669(6) | 702(6) |

Trajectories of the complex frequencies loci in the complex ω plane during the changing of the dynamic pressure

values of dynamic pressure these real parts, representing the oscillation frequency, eventually coalesce into a single mode. Further increasing the dynamic pressure of the flow causes the shell to lose its stability at _{cr} = 410. This instability is due to coupled-mode flutter where the imaginary part of complex frequency (representing the damping term of the aeroelastic system) becomes zero for certain critical pressure (

The same behaviour is observed by real and imaginary parts of complex frequencies as the static pressure increases (_{cr} = 410 if the freestream static pressure is evaluated using Equation (37). Prediction of the critical freestream static pressure using Equation (36) provides approximately the same results when evaluating the pressure field using Equation (37). As expected, using the piston theory with the correction term to account for shell curvature produces a better approximation for the pressure loading acting on a curved shell exposed to supersonic flow.

In _{cr} with

The effect of radius to thickness ratio R/h is presented in _{cr} with an increase of radius to thickness ratio. This increase in _{cr} is attributed to the fact that the mass of shell is greater when the shell is thick, and the effect of pressure is less important for a thick shell than for a thin shell. On the other hand, when the shell is thin it becomes unstable at higher dynamical pressure levels due to an increase in stiffness because of a decrease in thickness. The same conclusion is reported in [

(a) Real part and (b) imaginary part of the complex frequencies versus the freestream static pressure parameter; static pressure evaluated by Equation (36)

cylindrical shells.

In order to study the effect of filling ratio,

The effect of boundary conditions on the flutter onset is presented in

An efficient hybrid finite element method is presented to investigate the aeroelastic stability of an empty or partially liquid filled spherical shell subjected to external supersonic flow. Linear shell theory is coupled with first order piston theory to account for aerodynamic pressure. The effect of curvature correction in piston theory was

(a) Real part and (b) imaginary part of the complex frequencies versus the freestream static pressure parameter; static pressure evaluated by Equation (37)

Variation of the critical freestream static pressure parameter with angle for simply supported shell

. Critical freestream pressure parameter for different boundary conditions

Boundary conditions | | Mode no. |
---|---|---|

Freely simply supported (v = w = 0) | 510.5 | Coupled 1^{st} and 2^{nd} |

Simply supported (u = v = w = 0) | 410 | Coupled 1^{st} and 2^{nd} |

Clamped | 410 | Coupled 1^{st} and 2^{nd} |

Variation of the critical freestream static pressure parameter with R/h for simply supported shell

Variation of the critical freestream static pressure parameter with R/H for simply supported shell

analyzed. Fluid structure interaction due to hydrodynamic pressure of internal fluid is also taken into account. The study has been done for shells with various geometries, radius to thickness ratios, filling ratios and boundary conditions. In all study cases one type of instability is found; coupled-mode flutter in the first and second mode. Increasing the radius to thickness ratio leads the onset of flutter to occur at higher dynamic pressure. Decreasing the angle