Gas pressure and density effects on vibration of cylindrical pressure vessels: analytical, numerical and experimental analysis
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The present research deals with the natural frequencies and mode shapes of cylindrical pressure vessels filled with different gasses in various internal pressures. The Hamilton’s principle is employed to derive the governing equations, which are solved using analytical and finite element analyses, and a closed form relation as the equivalent density is obtained to include the gas properties. The experimental modal analysis is performed and the obtained results are compared and validated with analytical and finite element results. The results show that the type of the gas, as well as the gas pressure have significant effect on vibrational behavior of the structure and should be accounted in the design and analysis of vessels.
KeywordsFree vibration Modal testing Pressure vessel
Pressure vessels have many applications in different areas of engineering like mechanical, aerospace, marine, etc. Since the pressure vessels should work with various gas types and pressures during the working life, the effects of gas properties on the operation of the vessel should be considered in the design process.
Many researchers previously studied the vibration characteristics of cylindrical pressure vessels with internal and external pressure [1, 2, 3, 4, 5, 6]. Li et al.  analyzed the effects of hydrostatic pressure on the vibration of piezoelectric laminated cylindrical shell. Senjanović et al. [8, 9, 10] studied the effects of internal pressure on vibrational behavior of rotating cylindrical shell. Arnold and Warburton [11, 12] employed Hamilton’s principle to derive equations of motion of the cylindrical shell. Amabili et al.  investigated the vibration of simply supported cylindrical shell containing an incompressible fluid using Donnell shallow shell theory. Fluid-filled cylindrical shell subjected to lateral harmonic loading is studied by Del Prado et al. . In addition, Zhang et al.  considered vibration of composite cylindrical shells subjected to axial pressure and radial load in both side of the cylinder. Bolotin  focused on the natural frequency distribution and the intervals in cylindrical shells and Fung et al.  examined the effects of shell thickness and internal pressure on the vibration of cylinders. Analytical and experimental study is done by Stillman  to describe the vibration behavior of liquid-filled pressure vessels. Selmane and Lakis  presented the vibration of anisotropic cylindrical shell subjected to internal and external flow using Sanders shell theory. The generalized Fourier series are used by Stepanishen  to evaluate fluid pressure effect on vibratory response of cylindrical shells with infinite rigid extensions. Chiba and Abe  analyzed hydroelastic vibration of cylindrical vessel containing liquid and Gupta and Hutchinson  studied the free vibration of liquid storage tanks. Krishna and Ganesan  introduced an approach based on polynomial terms for calculating the added mass for fluid-filled cylindrical shells. The vibration of partially filled laminated composite cylindrical shells is studied in ref . Isvandzibaei et al.  studied the effects of internal pressure in cylindrical shells made of functionally graded materials (FGMs) using first order shear deformation theory. Shakouri et al.  studied the Effects of imperfection shapes on buckling of conical shells under compression.
According to the above papers, many of previous investigations of cylindrical shell vibrations have been limited to unpressurized vessels. In addition, in the case of pressurized vessels, the pressure is applied as the internal loading on the shell and the effects of the gas properties on the vibration of the pressure vessel are not considered.
In this paper, the natural frequencies and mode shapes of pressurized cylinders containing various types of the gasses (air and oxygen) with internal pressures ranges from zero to 100 bar is investigated. The total kinematic energy of the cylinder and internal gasses has been written and the governing equation is obtained using Hamilton’s equation and the relation for natural frequencies of the pressure vessel including gas effects is obtained. The results are compared and validated with finite element analysis and modal testing. Finally, the effects of mass and pressure of the gas on the frequency response function of the pressurized cylinder is studied.
2 Governing equations for pressure vessel containing gas
2.1 Displacements and strains
In addition, \(\alpha ,\beta\) are axial half wave and circumferential wave, respectively.
2.2 Governing equations
3 Experimental study
Measured geometry and material properties of pressure vessel
Mass of cylinder
Molar mass of oxygen and air
The cylinder is free in both sides and the first six frequencies belongs to the rigid body motions (i.e. the frequency is zero). To obtain the free boundary condition, the pressure vessel is hanged with a long rope, so that the frequencies implemented from the boundary conditions are far from the vessel frequencies .
3.2 Excitation and accelerometers
The 4-input channel analyzer is used and the motion is detected and measured by using piezoelectric accelerometer. Number of points which be created in cylinder for modal testing is 49 points and the roving hammer method is used to excite the cylinder. Each frequency response function (FRF) is evaluated after averaging over several measurements (three times) in order to reduce noise.
4 Finite element analysis
For finite element (FE) analysis, the traditional ANSYS software is used and the 4-node SHELL181 element is employed to model the vessel. The Lanczos method is employed to obtain the natural frequencies and mode shapes. The pressure is applied as internal distributed load on the shell and the effects of the gas mass is modeled by using the equivalent density as presented in Eq. (27). After the mesh study and convergence check, the total number of mesh elements in cylindrical shell is 55376, which is the optimum conditions between solution time and the accuracy of the results.
5 Results and discussion
Comparison of natural frequencies of cylinder which filled by air in various pressures
Cylinder filled with air
FE with mass effect (Hz)
FE without mass effect (Hz)
Equation (30) (Hz)
Comparison of natural frequencies of cylinder which filled by oxygen in various pressures
Cylinder filled with oxygen (Hz)
FE with mass effect (Hz)
FE without mass effect (Hz)
Equation (30) (Hz)
It can be seen that each gas causes different slope in frequency-pressure relation.
The gas density has significant effect on the natural frequency of the system. The natural frequency of cylindrical shell increases with increase in the pressure.
For the gas with higher density, the frequencies gradient decreases with respect to pressure. It is mainly because of the fact that the mass of the system has reverse effect on the natural frequency.
Considering both pressure and gas effect on the frequency, the effect of pressure is more than the mass, so the frequencies of system increases by increasing pressure and mass.
The mode shapes of the pressure vessel do not change with the pressure and density of the gas.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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