ANALYSIS OF THE DEFORMATION STATE OF A TRANSVERSE ISOTROPIC CYLINDRICAL SHELL UNDER LONGITUDINAL-RADIAL VIBRATIONS

ABSTRACT

The article analyzes the deformation state of a transversely isotropic cylindrical shell under longitudinal-radial vibrations interaction with an internal viscous fluid. The problem is formulated on the basis of the theory of three-dimensional elasticity, and a numerical solution is obtained using the finite difference method. In the calculations, various torsional moments act on the free end of the aluminum cylindrical shell, and the other end is considered to be rigidly fixed. The results show that with an increase in the torsional moment, the longitudinal and radial displacements also increase, which is consistent with the nature of the physical process. The obtained solutions were compared with the results obtained in the Ansys program, and a high agreement between them was found. This confirms the accuracy and reliability of the proposed method.

АННОТАЦИЯ

В статье анализируется деформированное состояние трансверсально-изотропный цилиндрической оболочки при взаимодействии продольно-радиальных колебаний с внутренней вязкой жидкостью. Задача формулируется на основе теории трехмерной упругости, а численное решение получается с помощью метода конечных разностей. В расчетах на свободный конец алюминиевой цилиндрической оболочки действуют различные моменты кручения, а другой конец считается жестко закрепленным. Результаты показывают, что с увеличением момента кручения, увеличиваются также продольные и радиальные смещения, что соответствует природе физического процесса. Полученные решения были сопоставлены с результатами, полученными в программе Ansys, и было обнаружено высокое согласие между ними. Это подтверждает точность и надежность предложенного метода.

Keywords: finite difference method, longitudinal-radial vibration, stress-strain analysis, transverse-isotropic shell, viscous fluid.

Ключевые слова: метод конечных разностей, продольно-радиальная вибрация, анализ напряжений и деформаций, трансверсально-изотропная оболочка, вязкая жидкость.

Introduction

The widespread use of thin-walled cylindrical shells in construction and industry requires a thorough study of their dynamic behavior, in particular, their longitudinal-radial vibrations. In [1], a numerical analytical algorithm was developed to calculate the dynamics of infinitely long rotating cylindrical shells, and it was shown that it can be used to model the longitudinal-radial vibrations of transversely isotropic circular shells.

In [2-4], the three-dimensional vibration equations of transversely isotropic cylindrical shells were simplified using an analytical approach based on potential functions. This method allows the decomposition of differential equations into second-order equations and is used to determine the influence of anisotropy parameters on frequency and deformation models. Also, in [5], the problem of wave propagation in transversely isotropic shells was studied using the Fourier collocation method. This approach allows for a high degree of determination of the influence of boundary conditions and material anisotropy on the properties.

Longitudinal vibrations in a circular cylindrical shell filled with a viscous fluid [6] were caused by a longitudinal moment, and the lateral surface of the shell was assumed to be free from external loads.

Accurate three-dimensional mathematical models have been created that take into account the interaction between the fluid and the shell [7,8], and solutions have been found based on classical and improved approximate equations of vibration. In this work, the unsteady longitudinal-radial torsional vibrations of a transversely isotropic cylindrical shell interacting with an internal viscous fluid are expressed using Boltzmann-Volterra integral operators. The equations of motion of the layer and fluid, as well as the contact conditions, are presented in cylindrical coordinates [9,10].

Materials and methods

The area of the transversely isotropic cylindrical shell under study , outer radius , inner radius , let the thickness be . We consider the problem of the unsteady interaction of the cylindrical shell under consideration with an internal viscous fluid. For this, we use the system of basic equations given in [11] (1)

  (1)

Here  are differential operators, depends on the physical and geometric parameters of the cylindrical shell under consideration,  and  are the longitudinal and radial components of the displacement in the cross section of the cylindrical shell,  and  are the forces acting on the cylindrical shell.

From the system of equations (1) given above, we can find the following functions  and . These functions are the main kinematic quantities characterizing the dynamic interaction between the fluid and the cylindrical shell, which are of great importance in the analysis of the vibration processes of the cylindrical shell. Using these found functions, it is possible to find the displacements and stresses that arise at the points of their cross sections during unsteady vibrations of circular cylindrical layers and shells interacting with an internal viscous fluid. Using the found functions, the unsteady vibration states of circular cylindrical layers and shells interacting with an internal viscous fluid are analyzed in depth. For this, we first introduce the following dimensionless quantities into the system of equations (1)

                   (2)

As a result, we can write (1) in the following form:

 (3)

Here  are the differential operators transformed to the dimensionless state, depending on the physical and geometric parameters of the cylindrical shell under consideration. (3) we use the finite difference method to solve the system of equations. We solve the problem using the finite difference method. In this case, each equation of the original system of differential equations is expressed in terms of finite differences, resulting in a new system of four algebraic equations with four unknowns. Using the definitions introduced in this way, the system is simplified structurally and then brought into a convenient form for the calculation and analysis of the results. As a result of using this approach, the efficiency of the calculation processes increases, and the reliability and accuracy of the solutions are ensured to a sufficient extent..

Results and discussion

The geometric dimensions of the circular cylindrical shell for solving the system of equations using the finite difference method using the Maple program are as follows:   We apply torques to the free end of a circular cylindrical shell at . The end at  is rigidly fixed. Let's assume the circular cylindrical shell material is aluminium. For aluminium material        . The cylindrical shell thickness is . As a result of the calculations, Figures 2.3 are obtained..

From the graph in Fig. 1, it can be seen that when a torque  is applied to the  end of a circular cylindrical shell interacting with an internal viscous fluid, the maximum value of the displacement is equal to . When a torque of  is applied to the end of a cylindrical shell , the maximum value of the displacement  is  and when a torque of  is applied, the maximum value of the displacement  is . As the value of the torque increases, the value of the displacement  also increases. This is consistent with the physical nature of the issue.

The graph in Fig. 2 shows that when a torque of  is applied to the end of the cylindrical shell , the maximum value of the displacement  is equal to  at the section  and it starts to decay from this section. When a torque of  is applied to the end of the cylindrical shell , the maximum value of the displacement  is equal to  at the section  and it also starts to decay from this section. When the value of the torque is , the displacement reaches its maximum value at this section  and it also starts to decay from this section. Here we can also see that as the value of the torque increases, the value of the displacement  also increases. This corresponds to the physical essence of the problem.

We compare the main results obtained in the longitudinal-radial vibrations of a transversely-isotropic circular cylindrical shell interacting with an internal viscous fluid with the results obtained using the Ansys program. For comparison, we enter the same geometric parameters as above in this program. We apply a torque  to the free end of the circular cylindrical shell  and obtain the results using the program.

We first assume that the material of the cylindrical shell interacting with the internal viscous fluid is aluminium and we apply longitudinal and radial forces to the side surface of the circular cylindrical shell and torques to the ends.

Fig. 3 shows a three-dimensional graph of the shell point displacement  for a circular cylindrical shell interacting with an internal viscous fluid, the material of which is aluminium. This graph shows that the longitudinal displacement of the shell cross-section points  is significant in some places and zero in others.

It can be seen that this situation repeats itself along the entire length of the shell. This shows that the longitudinal displacement  obtained using the Ansys programme coincides with the longitudinal displacement  obtained using the finite difference method, shown in Fig. 1.

Fig. 4 shows the displacement  of the cylindrical shell points when a torque  is applied to the free end  of a circular cylindrical shell interacting with an internal viscous fluid. It can be seen from these figures that the displacements of the shell points reach maximum values up to the section  and then decrease. This corresponds to the shape of the displacement graph  obtained in Fig. 2.

Conclusion

In this study, the deformation state of transversely-isotropic cylindrical shell acting with an internal viscous fluid under longitudinal-radial vibrations was studied on the basis of numerical analysis. In the studies, a system of equations based on the theory of three-dimensional elasticity was solved using the finite difference method and numerical calculations were performed in the Maple program. The results show that with an increase in the value of the turning moment, the longitudinal-radial displacements in the cylindrical shell also increase, which fully corresponds to the physical nature of the process. When comparing the obtained numerical solutions with the results obtained in the Ansys program, a high degree of agreement is observed. This confirms the reliability, accuracy and practical applicability of the proposed mathematical model and numerical approach. Also, the results of the study create an important theoretical basis for improving the dynamic analysis of transversely-isotropic shells filled with internal fluids and optimizing vibration processes in the fields of mechanical engineering, aerospace and hydraulic engineering.

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