Abstract
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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
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. [7] 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. [13] investigated the vibration of simply supported cylindrical shell containing an incompressible fluid using Donnell shallow shell theory. Fluidfilled cylindrical shell subjected to lateral harmonic loading is studied by Del Prado et al. [14]. In addition, Zhang et al. [15] considered vibration of composite cylindrical shells subjected to axial pressure and radial load in both side of the cylinder. Bolotin [16] focused on the natural frequency distribution and the intervals in cylindrical shells and Fung et al. [17] examined the effects of shell thickness and internal pressure on the vibration of cylinders. Analytical and experimental study is done by Stillman [18] to describe the vibration behavior of liquidfilled pressure vessels. Selmane and Lakis [19] 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 [20] to evaluate fluid pressure effect on vibratory response of cylindrical shells with infinite rigid extensions. Chiba and Abe [21] analyzed hydroelastic vibration of cylindrical vessel containing liquid and Gupta and Hutchinson [22] studied the free vibration of liquid storage tanks. Krishna and Ganesan [23] introduced an approach based on polynomial terms for calculating the added mass for fluidfilled cylindrical shells. The vibration of partially filled laminated composite cylindrical shells is studied in ref [24]. Isvandzibaei et al. [25] studied the effects of internal pressure in cylindrical shells made of functionally graded materials (FGMs) using first order shear deformation theory. Shakouri et al. [26] 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
Figure 1 illustrates the cylindrical pressure vessel considered in this paper. The (\(r,\theta ,z\)) represent the cylinder in radial, circumferential and longitudinal directions, respectively. It is assumed that the cylinder remains circular in all internal pressures. Thus, the displacements of the shell can be shown as [27]:
where \(\delta r,\delta \theta ,\delta z\) denotes radial, circumferential and axial displacement, L represents length, r is the radius, t is the thickness and \(A\) is the amplitude of radial displacement which is periodic over time.
In addition, \(\alpha ,\beta\) are axial half wave and circumferential wave, respectively.
2.2 Governing equations
To extract the governing equation, bending and stretching energies are written for pressurized cylindrical shell. Fluid expansion energy is represented for indicating fluid effect. Tensile tension in cylindrical shell due to internal gas pressure is \(\frac{pr}{2}\) and \(pr\) in the axial and circumferential directions, respectively. For deformation of the middle surface in \(\theta\) direction we have [27]
where Δθ denotes the circumferential expansion of the cylinder. Considering small displacements (i.e. δr/r ≪ 1,\(\partial \delta r/r\partial \theta \ll 1\)), the Eq. (3) can be written as
where Δz is the axial expansion of the cylindrical shell. The stretching energy (V_{S}) of the cylinder can be written as [27]
where p is internal pressure of the cylinder. In addition, the gas expansion energy (V_{E}) is
Combining Eqs. (6) and (7) results in
The bending energy (V_{B}) of cylinder can be defined as
where E is young modulus and ν is the Poisson’s ratio. In addition, ɛ_{r}, ɛ_{θ}, ɛ_{z} represent the strains in radial, circumferential and axial directions, respectively and τ_{θz} is the shear strain. According to [28], by thin shell assumption for the cylinder, the values of normal stress in radial direction and the shear strain can be ignored. Therefore, we have
Substituting Eqs. (12) and (13) into (11), the potential bending energy of cylinder can be expressed as
The kinetic energy of the cylinder is considered as
where \(T_{C}\) and ρ_{C} respectively denote the kinetic energy and density of the cylinder. The gas in the cylinder is assumed incompressible and nonrotating, only with small pressure variations. According to these assumptions, for kinetic energy of the pressurized gas, we have
where T_{G} represents the kinetic energy of gas, ρ_{G} represents the density of gas, V_{G} is the velocity potential and H is the unit vector, normal to the cylinder surface. For small displacements we have
To satisfy the Hamilton’s principle, the velocity potential can be assumed as
where I_{n} is the modified Bessel function. By substituting Eq. (20) into (19), the kinetic energy of the gas can be expressed as
where the expressions of Modified Bessel function can be explained as [29]
where n and y are general parameters and Γ denote the gamma function. According to Eqs. (23)–(24) and (22), the potential energy of gas can be obtained as
Therefore, the total kinetic energy of system is obtained as
Where T_{T} denotes total kinetic energy, ρ_{E} is an equivalent density as follows
Using Eq. (14) and (9) the total potential energy of the system can be written as
which V_{T} represents the total potential energy. Considering the Lagrange equation for investigating natural frequencies [30]
The natural frequency of cylinder can be expressed as
where ω represents natural frequency of pressurized cylindrical shell and ω_{0} is the natural frequency of the cylinder, without considering the gas effects.
3 Experimental study
To validate the accuracy of Eq. (30), the pressurized cylinder made of steel is tested in modal laboratory. Material and geometrical properties of the pressure vessel are presented in Table 1. Figure 2 shows the test equipment and pressure vessel in modal lab.
Oxygen and air gasses are filled separately into the cylinder. These gasses have different molar mass, which cause their densities to be different. Table 2 shows molar mass of oxygen and air.
3.1 Supports
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 [31].
3.2 Excitation and accelerometers
The 4input 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 4node 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
In this section, numerical, analytical and experimental results are presented and compared for the vibration of pressurized vessel. The pressure changes from zero to 100 bars and two types of gasses including oxygen and air are used. Effect of gas pressure on cylinder shell in the simulation is assumed steady. Tables 3 and 4 represent the first frequencies (excluding rigid body frequencies) of the system for air and oxygen, respectively. These frequencies are related to pressure and gas, which are calculated, by experiment, FE and analytical methods. In addition, the frequencies of pressurized vessels without considering the gas density effects is investigated. It can be seen that the results of Eq. (30) are in good accordance with finite element and experiment. In addition, the gas density has significant effect on the natural frequency, where the obtained results without gas density are far from the experiment results.
As the Tables 3 and 4 show, the density of the gas can change the value of frequencies, so in order to obtain exact frequency values of the pressurized vessel, the type of the gas must be denoted. Figure 3 shows the frequency response function as well as the phase angle of the pressurized cylinder with different pressures for both air and oxygen. As can be seen, except the third natural frequency, the natural frequencies of system increase with increase in the pressure. The remarkable phenomena, which is observed in both air and oxygen gasses, is that the third natural frequency decreases with increase in the internal pressure.
Figure 4 shows the frequency response function and corresponding phase angles of the pressurized cylinder filled with oxygen and air in the same pressure. This figure apparently describes the effects of gas density on the natural frequencies of the pressure vessel. As can be seen, since the stiffness of the vessel is the same for both gasses, it is expected that the air with less mass density has the higher natural frequency.
Results of Fig. 4 explain that density of gas can change natural frequencies and if type of gas is ignored in FE simulation, the results are different from the experiment. Figure 5 represents the variation of the frequency with respect to pressure, obtained from the experiment and corresponding linear fit (regression) between frequency and pressure. The linear function for both gasses are
It can be seen that each gas causes different slope in frequencypressure relation.
The first three mode shapes extracted from FE analysis and modal test are compared in Fig. 6. As can be seen the obtained mode shapes are in good accordance with the numerical results.
To ensure from the independency of the mode shapes from each other, the autoMAC study of the mode shapes are plotted Fig. 7. As seen, the mode shapes are independent from each other and none of them has significant effect on the others.
6 Conclusions
In this paper, the natural frequencies and mode shapes of pressurized cylinder considering the effects of pressure and the kind of the gas are studied. The modal analysis is done for two gasses (air and oxygen) in order to measure the effects of gas density on dynamic parameters of pressure vessel. The governing equations are obtained by evaluating kinetic and potential energies of system and substituting them into Lagrange equation. The results are compared to modal results and finally the effect of pressure and density of gas in are investigated. The major findings are:

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.
References
Ross C, Haynes P, Richards WD (1996) Vibration of ringstiffened circular cylinders under external water pressure. Comput Struct 60(6):1013–1019
Ross CTF, Haynes P, Richards WD (1996) Vibration of ringstiffened circular cylinders under external water pressure. Comput Struct 60:1013–1019
Païdoussis MP, Suss S, Pustejovsky M (1977) Free vibration of clusters of cylinders in liquidfilled channels. J Sound Vib 55:443–459
Fuller CR, Fahy FJ (1982) Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid. J Sound Vib 81:501–518
Kabir MZ (2000) Finite element analysis of composite pressure vessels with a load sharing metallic liner. Compos Struct 49:247–255
Pavlou DG (2016) Dynamic response of a multilayered FRP cylindrical shell under unsteady loading conditions. Eng Struct 112:256–264
Li H, Lin Q, Liu Z, Wang C (2001) Free vibration of piezoelastic laminated cylindrical shells under hydrostatic pressure. Int J Solids Struct 38:7571–7585
Senjanović I, Alujević N, Ćatipović I, Čakmak D, Vladimir N (2018) Vibration analysis of rotating toroidal shell by the Rayleigh–Ritz method and Fourier series. Eng Struct 173:870–891
Senjanović I, Ćatipović I, Alujević N, Vladimir N, Čakmak D (2018) A finite strip for the vibration analysis of rotating cylindrical shells. ThinWalled Struct 122:158–172
Senjanović I, Áatipović I, Alujević N, Čakmak D, Vladimir N (2019) A finite strip for the vibration analysis of rotating toroidal shell under internal pressure. J Vib Acoust 141:021013
Arnold RN, Warburton GB (1949) Flexural vibrations of the walls of thin cylindrical shells having freely supported ends. Proc R Soc Lond Ser A Math Phys Sci 197:238–256
Arnold RN, Warburton GB (1953) The flexural vibrations of thin cylinders. Proc Inst Mech Eng 167:62–80
Amabili M, Pellicano F, PaÏDoussis MP (1998) Nonlinear vibrations of simply supported, circular cylindrical shells, coupled to quiescent fluid. J Fluids Struct 12:883–918
del Prado ZJ, Amabili M, Gonçalves PB (2017) Non linear vibrations of imperfect fluidfilled viscoelastic cylindrical shells. Procedia Eng 199:570–576
Zhang W, Yang S, Mao J (2018) Nonlinear radial breathing vibrations of CFRP laminated cylindrical shell with nonnormal boundary conditions subjected to axial pressure and radial line load at two ends. Compos Struct 190:52–78
Bolotin VV (1963) On the density of the distribution of natural frequencies of thin elastic shells. J Appl Math Mech 27:538–543
Fung YC, Sechler EE, Kaplan A (1957) On the vibration of thin cylindrical shells under internal pressure. J Aeronaut Sci 24:650–660
Stillman WE (1973) Free vibration of cylinders containing liquid. J Sound Vib 30:509–524
Selmane A, Lakis AA (1997) Vibration analysis of anisotropic open cylindrical shells subjected to a flowing fluid. J Fluids Struct 11:111–134
Stepanishen PR (1982) Modal coupling in the vibration of fluidloaded cylindrical shells. J Acoust Soc Am 71:813–823
Chiba M, Abe K (1999) Nonlinear hydroelastic vibration of a cylindrical tank with an elastic bottom containing liquidanalysis using harmonic balance method. ThinWalled Struct 34:233–260
Gupta RK, Hutchinson GL (1988) Free vibration analysis of liquid storage tanks. J Sound Vib 122:491–506
Krishna BV, Ganesan N (2006) Polynomial approach for calculating added mass for fluidfilled cylindrical shells. J Sound Vib 291:1221–1228
Saviz MR (2016) Coupled vibration of partially fluidfilled laminated composite cylindrical shells. J Solid Mech 8:823–839
Isvandzibaei MR, Jamaluddin H, Raja Hamzah RI (2016) Vibration analysis of supported thickwalled cylindrical shell made of functionally graded material under pressure loading. J Vib Control 22:1023–1036
Shakouri M, Spagnoli A, Kouchakzadeh M (2016) Effects of imperfection shapes on buckling of conical shells under compression. Struct Eng Mech 60:365–386
Warren CH, Barbin A (1988) Vibration of pressurized cylinders. In: Army Missile Command Redstone Arsenal Al Guidance And Control Directorate
Sadd MH (2009) Elasticity: theory, applications, and numerics. Academic Press, Cambridge
Baricz Á, Maširević DJ, Pogány TK (2017) Series of Bessel and Kummertype functions. Springer, Berlin
Craig RR, Kurdila AJ (2006) Fundamentals of structural dynamics. Wiley, New York
Ewins DJ (1984) Modal testing: theory and practice. Research Studies Press, Letchworth
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mir, O., Shakouri, M. & Ashory, M.R. Gas pressure and density effects on vibration of cylindrical pressure vessels: analytical, numerical and experimental analysis. SN Appl. Sci. 2, 134 (2020). https://doi.org/10.1007/s424520191916z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s424520191916z