Abstract
Comparison study on free vibration of circular cylindrical shells between thin and moderately thick shell theories when using the exact dynamic stiffness method (DSM) formulation is presented. Firstly, both the thin and moderately thick dynamic stiffness formulations are examined. Based on the strain and kinetic energy, the vibration governing equations are expressed in the Hamilton form for both thin and moderately thick circular cylindrical shells. The dynamic stiffness is assembled in a similar way as that in classic skeletal theory. With the employment of the Wittrick-Williams algorithm, natural frequencies of circular cylindrical shells can be obtained. A FORTRAN code is written and used to compute the modal characteristics. Numerical examples are presented, verifying the proposed computational framework. Since the DSM is an exact approach, the advantages of high accuracy, no-missing frequencies, and good adaptability to various geometries and boundary conditions are demonstrated. Comprehensive parametric studies on the thickness to radius ratio () and the length to radius ratio () are performed. Applicable ranges of are found for both thin and moderately thick DSM formulations, and influences of on frequencies are also investigated. The following conclusions are reached: frequencies of moderately thick shells can be considered as alternatives to those of thin shells with high accuracy where is small and is large, without any observation of shear locking.
1. Introduction
Free vibration analysis of shell structures is important for further dynamic analysis in engineering. Among all shell types, circular cylindrical shells are the most widely used in a good variety of applications, for example, tubes, containment vessels, and aircraft fuselages. Therefore, exact modal characteristic analysis of circular cylindrical shells is of considerable scientific and engineering significance.
It is well known that shell theories can be categorised into thin, moderately thick, and thick forms. Since the Kirchhoff-Love hypothesis for thin elastic shell theory was established [1], much effort was spent on its vibration development. Arnold and Warburton [2] analysed the flexural vibration of thin cylindrical shells with general boundary conditions using Rayleigh’s principle. Leissa [3] provided a systematic review and summary on the vibration of different shell theories and shell types. Bardell et al. [4] investigated the free vibration of isotropic open cylindrical shell panels by applying an version finite element method. Tan [5] introduced a substructuring method for predicting the natural frequencies of circular cylindrical shells with arbitrary boundary conditions.
On the other hand, moderately thick shell theory which derives from the Reissner-Mindlin hypothesis is less frequently used in analysing the free vibration behaviour of shells due to its complexity. Naghdi and Cooper [6] conducted vibration analysis on circular cylindrical shells with the consideration of shear deformation and rotary inertia by employing elastic wave method. Mirsky and Herrmann [7] studied the nonaxisymmetric vibration of moderately thick circular cylindrical shells. Tornabene and Viola [8] proposed an approach to analyse free vibration of parabolic shells using Generalised Differential Quadrature method (GDQ) based on the first-order shear deformation theory. Mantari et al. [9] applied a new accurate higher-order shear deformation theory to perform free vibration analysis of multilayered shells.
Although the free vibration of shells has been extensively examined, there are continuing research efforts made with new methods and shell types, for example, -version mixed finite element by Kim et al. [10]; GDQ method for doubly curved shells of revolution by Tornabene et al. [11, 12]; higher-order theory for thin-walled beams by Pagani et al. [13]; and three-dimensional analysis method by Ye et al. [14, 15].
The dynamic stiffness method (DSM) is an exact method since the dynamic stiffness is computed directly from the exact vibration governing equations. It is suitable for computational analysis of free vibration of continuous systems with infinite degrees of freedom. In conjunction with the Wittrick-Williams (W-W) algorithm [16, 17] and the recursive Newton’s method [18, 19], the DSM has been employed in free vibration of skeletal structures, for example, nonuniform Timoshenko beams [20], three-layered sandwich beams [21], and functionally graded beams [22]. Pagani et al. [23] employed the exact dynamic stiffness elements to investigate the free vibration of thin-walled structures. A recent paper [24] applied the DSM to the free vibration analysis of thin circular cylindrical shells. The present authors used to employ the DSM to investigate the free vibration of shells of revolution [25], and the complete theory was given in [26] based on the thin shell assumption. Further extension of the DSM to moderately thick circular cylindrical shells [27] was made with the consideration of transverse shear deformation and rotary inertia.
As an exact method, the dynamic stiffness method has the following advantages:(1)The method is exact and theoretically solutions can be obtained with any desired accuracy.(2)The original eigenvalue problems of two-dimensional partial differential equations (PDEs) are degraded to the eigenvalue problems of a set of one-dimensional ordinary differential equations (ODEs) by introducing the circumferential vibration modes. Since the method is exact, frequencies and vibration modes of any order can be calculated with a small number of elements.(3)The Wittrick-William algorithm is mathematically strict and ensures that no eigenvalue will be omitted.(4)The DSM formulation is general and versatile, making it capable of solving the free vibration of circular cylindrical shells with a variety of geometries and arbitrary boundary conditions.Though the free vibration formulation using the DSM has been established for both thin and moderately thick circular cylindrical shells, there is no comprehensive comparison study available currently to address the applicability of DSM formulations between the two different shell theories. This paper attempts to show the compatibility on the free vibration of circular cylindrical shells between thin and moderately thick DSM formulations. The DSM formulations of both thin and moderately thick circular cylindrical shells are well compared in Sections 2 and 3. Afterwards, the Wittrick-Williams algorithm is presented, providing a solution to computing the number of clamped-end natural frequencies . Four numerical examples are given in Section 5, demonstrating the capability and reliability of the proposed method. Extensive parametric studies are performed in Section 6 on the ratios of thickness to radius () and length to radius () to find out the applicable ranges of each theory. Differences between frequencies from thin and moderately thick DSM formulations are analysed. The following conclusions are reached in the final: the compatibility of DSM between thin and moderately thick formulations is proven without shear locking.
2. Vibration Equations
This section theoretically presents a comparison study on DSM formulations for free vibration of both thin and moderately thick circular cylindrical shells. Figure 1 shows the coordinate system of a homogeneous, isotropic, and circumferentially closed circular cylindrical shell. The axial and circumferential directions of the shell are termed as and , respectively. The length of the shell is , radius is , and the thickness is . Young’s modulus is , Poisson’s ratio is , and the density is .
For thin shells, the vibration can be represented by the translational displacements of the middle surface along , , and directions, while, for moderately thick shells, the angular displacements of the middle surface along and directions are independent, which results in an expansion of the displacement vector to .
The strain-displacement relationship at the middle surface of a circular cylindrical shell can be written in the form ofwhere is the strain vector and is the displacement vector which can be or as mentioned above. The expressions of the differential matrix for both thin and moderately thick circular cylindrical shells can be found in [3].
Similarly, the internal force-strain relationship at the middle surface of a circular cylindrical shell can also be expressed aswhere is the internal force vector and is the stiffness matrix which can also be found in [3].
Denote as the area of an infinitesimal portion in the middle surface of a circular cylindrical shell (see Figure 2), and the area of this infinitesimal portion is . The total strain energy for free vibration of the circular cylindrical shell can be integrated along the whole middle surface asand the total free vibration kinetic energy is obtained asfor thin shell, orfor moderately thick circular cylindrical shells.
The dynamic displacement functions and are expressed differently as is the circumferential wave number; is a circular frequency; , , , , and are axial vibration functions.
The Hamilton principle suggestswhere is an arbitrary time interval during vibration. The equations of motion can be derived from (7). Substituting the dynamic displacement functions (6) into equations of motion, the free vibration governing differential equations can be obtained. By rewriting the governing differential equations into the Hamilton form, generalised displacements and forces are obtained, where and are given in the Appendix.
The vibration governing equations can be written in the Hamilton form ofwhere denotes the differentiation with respect to and . and I is an identity matrix with the order of four (or five) for thin (or moderately thick) shells. The coefficient matrix is a symmetric eighth- (or tenth-) order matrix for thin (or moderately thick) circular cylindrical shells. Details of can also be found in the Appendix.
3. Dynamic Stiffness Matrix
Should the circumferential wave number be a specific value, (8) is one-dimensional with respect to only. Therefore, a circular cylindrical shell can be analysed as a general one-dimensional skeleton-like structure with four (or five) degrees of freedom (or ) for thin (or moderately thick) shells. Accordingly, the internal force vectors for both thin and moderately thick shell theories are and , respectively. The dynamic stiffness of free vibration of a circular cylindrical shell can now be set up in a similar way as that in classic one-dimensional skeletal theory.
Figure 3 shows that a circular cylinder is divided into several shell segments along its axial direction. Taking element (), for example, the coordinates at its two ends are and , respectively. The segment-end displacement vector is defined asand the force vector isMathematically apply the boundary conditions in (11) or (12) to (8) in turn,where is a unit vector with the th element equal to one, and the corresponding element dynamic stiffness matrix is formulated byDue to the complexity, (8) is solved computationally using adaptive ordinary differential equations (ODEs) solver COLSYS [33, 34]. The global dynamic stiffness matrix K of a circular cylindrical shell is assembled in a regular FEM way.
4. Wittrick-Williams Algorithm
The dynamic stiffness method is used in conjunction with the Wittrick-Williams (W-W) algorithm. The W-W algorithm merely gives the number of frequencies below with the expression ofwhere is the total number of frequencies exceeded by the trial frequency ; is the sign count of the global dynamic stiffness matrix and equal to the number of negative elements on the diagonal of an upper triangular matrix obtained from by applying standard Gaussian elimination without row exchange; is the number of clamped-end frequencies exceeded by and can be accumulated from of each shell segment element over the whole meridian as is expressed inTo solve , a substructuring method is employed by taking advantage of the self-adaptability of COLSYS. When computing the dynamic stiffness matrix of a shell segment element, for example, element (), a submesh will be generated by COLSYS adaptively as is shown in Figure 4.
Applying the Wittrick-Williams algorithm again on the submesh, we havewhere notation “” stands for submesh. Since COLSYS is capable of controlling the error tolerance adaptively, the number of clamped-end natural frequencies exceeded by on the submesh has to be zero, suggesting . Otherwise, the clamped-end modes can be enlarged infinitely, resulting in the unsatisfaction of the given error tolerance.
The following will introduce the formulation of in (16). Taking a subelement with the coordinates () on element (), for instance, its corresponding dynamic stiffness matrix can be obtained by linearly combining the eight (or ten) solutions of (8) and (11) or (12). Note that the end displacement and force vectors for this subelement are and , respectively, and the subelement stiffness matrix satisfiesBy applying the boundary conditions in (11) or (12) in turn again for thin or moderately thick circular cylindrical shells, (18) is obtained.wherefor thin shells, orfor moderately thick shells.
Since , is nonsingular (otherwise is one of clamped-end frequencies, which is conflicted with ), and (21) is obtained.Thus can be assembled regularly from . Based on the above theories, a FORTRAN code was written and used to solve the free vibration of both thin and moderately thick circular cylindrical shells in this paper.
5. Validation
In this section, four examples on free vibration of circular cylindrical shells are presented to show the reliability and accuracy of the proposed method in the above sections. Examples in Sections 5.1 and 5.2 are calculated with thin shell theory; in Section 5.3, both thin and moderately thick shell theories are employed to study the free vibration of the same shell, and results are given opposite for comparison; examples in Section 5.4 are performed with the moderately thick shell theory.
5.1. Free Vibration of a Thin Clamped-Clamped Shell
Zhang et al. [28] and Xuebin [29] analysed the natural frequencies of a clamped-clamped (C-C) circular cylindrical shell based on the Flügge thin shell theory by employing the FEM code MSC/NASTRAN and wave propagation approach, respectively. The clamped boundary condition is considered asThe length of the shell is , radius , thickness . Density , Young’s modulus , Poisson’s ratio . In this example, thin shell theory is used in formulating the dynamic stiffness, and the first nine frequencies are tabulated in Table 1, where denotes the number of half-waves along the axial direction.
It can be observed from Table 1 that the frequencies obtained using the present method agree very well with that from the FEM (MSC/NASTRAN) and wave propagation approach. Further investigation into Table 1 shows that the lowest natural frequency of a circular cylindrical usually comes with a relatively larger ; that is, in this case.
5.2. Free Vibration of a Thin Clamped-Free Shell
In Sharma [30], the free vibration of a clamped-free (C-F) circular cylindrical shell is investigated based on the Rayleigh-Ritz method and Flügge shell theory. The geometry parameters of the shell are length m, radius m, and thickness m; and the material properties are Young’s modulus GPa, Poisson’s ratio , and density . A thin shell formulation is used in the present dynamic stiffness approach. Table 2 presents the lowest four frequencies of the shell with a relatively larger circumferential wave number and 6.
It can be concluded that the results agree very well with [30] and the errors are no larger than 0.4%. By comparing the results from the present method and the Rayleigh-Ritz method used in Sharma [30], the reliability of the present method is verified.
5.3. Free Vibration of a Shell with Different Boundary Conditions
Pagani et al. [23] applied the one-dimensional higher-order beam theory and exact dynamic stiffness elements to study the free vibration of solid and thin-walled structures, in which a thin-walled cylinder with a circular cross-section (Figure 5) is analysed. In a recent paper [13], they employed both the radial basis functions (RBFs) method and Carrera Unified Formulation (CUF) to examine the free vibration behaviour of both open and closed thin-walled structures.
The length of the examined cylinder is m, outer diameter is m, and thickness is m. The material properties are Young’s modulus GPa, Poisson’s ratio , and density kg/m3. Table 3 shows the natural frequencies of this thin-walled cylindrical beam with different boundaries, namely, both ends simply supported (S-S), both ends clamped (C-C), clamped-free (C-F), and free-free (F-F). The first and second flexural (bending), shell-like, and torsional natural frequencies are presented. Solutions from the present thin and moderately thick DSM, 2D MSC/NASTRAN, and [13, 23] are given for comparison.
It can be observed that the natural frequencies computed from the present dynamic stiffness method agree very well with that from 2D MSC/NASTRAN code for both thin and moderately thick shell theories. Comparing with the data from the higher-order beam theories [13, 23], the present method provides an excellent agreement for flexural and torsional modes. The differences of frequencies for the first and second shell-like modes are larger, and this can be largely attributed that the formulations established in [13, 23] are based on higher-order beam theories. Obviously, better agreement can be achieved if exact shell theories are employed by [13, 23]. Further investigation by the present DSM approach shows that corresponds to flexural (bending) mode, while and 2 correspond to torsional and shell-like vibration modes, respectively, which demonstrate that the current DSM approach is capable of showing the correct vibration modes as presented in existing literature.
5.4. Free Vibration of a Moderately Thick Shell
Bhimaraddi [32] analysed the free vibration of a simply supported (S-S) moderately thick circular cylindrical shell by using the higher-order displacement model with the consideration of in-plane inertia, rotary inertia, and shear effect. In [31], natural frequencies of the circular cylindrical shell subjects to different and are computed and compared with the solutions from 3D elasticity theory. Results from the moderately thick DSM approach are also presented and listed in Table 4. In this example, Poisson’s ratio is 0.3 and , where is the shear modulus.
According to Table 4, results from the present method agree very well with those from [31, 32], demonstrating the high accuracy and adaptability of the present method for circular cylindrical shells with different and .
6. Parametric Study
6.1. Ratio of Thickness to Radius ()
An important parameter to determine whether a shell is thin or moderately thick is the thickness to radius ratio (). According to the classic shell theory, a critical value for between thin and moderately thick shell theories is 0.05. In this section, influences of on natural frequencies are investigated extensively from 0.005 to 0.1 for both thin and moderately thick DSM formulations.
6.1.1. Low Order Frequencies
Consider a circular cylindrical shell with and Poisson’s ratio . To investigate the differences of natural frequencies between thin and moderately thick DSM formulations, a notation η in (23) is introduced aswhere and are the nondimensional frequency parameters defined asConsequently, log10(η) is the index of difference.
Figure 6 shows the for the first frequencies () under , 1, 2, and 3 between thin and moderately thick DSM formulations with a variety of boundary conditions, for example, clamped-clamped (C-C), clamped-free (C-F), clamped-simply supported (C-S), and both ends simply supported (S-S). In these cases, is investigated from 0.025 to 0.1.
(a) C-C
(b) C-F
(c) C-S
(d) S-S
If (which suggests the difference is no larger than 1%), it can be considered that results from the thin and moderately thick dynamic stiffness formulations are in good agreement with each other. Furthermore, frequencies computed from the moderately thick shell theory can be used as good alternatives to those computed from the thin shell theory. Referring to Figure 6, the following conclusions can be reached:(1)For low order frequencies, good agreement is reached between thin and moderately thick dynamic stiffness formulations with varying from 0.025 to 0.1. The smaller the , the smaller the difference.(2)The difference between frequencies from thin and moderately thick DSM formulations increases with the increase of , as always has a smaller magnitude under than those under , 2, and 3.(3)It can be argued that the difference is insensitive to boundary conditions. The differences are roughly within 10−5~10−2 in this investigation and the trends of curves are similar for different boundary conditions.
6.1.2. High Order Frequencies
Investigation on the differences between high order frequencies computed from the thin and moderately thick DSM formulations is examined in this subsection. Since it is known that the DSM formulation is not sensitive to boundary conditions [26], a C-S circular cylindrical shell is studied. The geometry and material properties are the same to the example in Section 6.1.1 if not specifically stated.
For high order frequencies, is investigated from very small value 0.005 to a relatively large value 0.1, and the first ten frequencies under ranging from 0 to 10 from both thin and moderately thick DSM formulations are computed and compared. The differences are shown in Figure 7 and concluded as follows:(1)When , is smaller than 10−2 most of the time, which suggests that frequencies computed from moderately thick shell theory are well compatible with those from thin shell theory. No shear locking is observed.(2)When , only frequencies with a small (e.g., ) produce an acceptable difference (~10−2). With the increase of n, the difference becomes quite noticeable. The higher the , the more obvious the difference.(3)The order under a specific circumferential wave number does not have a significant impact on the differences, as most curves in Figure 7 are largely flat and stable with the increase of .
(a)
(b)
(c)
(d)
(e)
(f)
6.2. Ratio of Length to Radius ()
Parameter controls the relative length of a circular cylindrical shell. A larger indicates a long shell, and vice versa. In this subsection, the frequency trends and the compatibility between thin and moderately thick DSM formulations with respect to are studied. In this subsection, a C-C circular cylindrical shell with is examined. Figure 8 plots the trends of the first frequency parameter under each with from 0.2 to 10.
(a) , moderately thick shell
(b) , thin shell
(c) , moderately thick shell
(d) , thin shell
According to Figure 8, for a typical long circular cylindrical shell (e.g., ), the lowest natural frequency is obtained with ; with the decrease of , the circumferential wave number corresponding to the lowest natural frequency slightly increases to or 4; when is very small (e.g., and the shell is actually degraded to a ring), the first natural frequency under each increases monotonically and the smallest is reached at . The trends shown by Figure 8 are also in good agreement with other researchers’ work, for example, Markuš [35]. These obtained trends are applicable to both thin and moderately thick DSM formulations.
Figure 9 shows the differences of the first frequencies under each computed from the thin and moderately thick DSM formulations of the circular cylindrical shell with and 0.05. It can be understood from Figure 9 that the smaller the , the smaller the difference; the larger the , the smaller the difference; conditions of and need to be satisfied and a minor difference () is required for a normal length shell; that is, .
(a)
(b)
7. Conclusions
This paper conducts comparison study on the frequencies from thin and moderately thick circular cylindrical shells computed using the DSM formulations, and comprehensive parametric studies on and are performed. Based on these investigations, conclusions are outlined as follows:(1)Natural frequencies computed from the moderately thick DSM formulations can be considered as satisfactory alternatives to those from thin DSM formulations when and (differences are around or smaller than 10−2). Since the DSM is exact, no shear locking is observed.(2)Differences between frequencies from thin and moderately thick DSM formulations change with a variety of factors. The smaller , the smaller the difference; the smaller , the larger the difference; the larger the , the larger the difference.It is proved that moderately thick DSM formulations can be used to analyse the free vibration of normal length thin circular cylindrical shells without shear locking, demonstrating that the applicable range of moderately thick DSM formulations is expanded. These merits should be attributed to that the dynamic stiffness method is an exact method.
Though this comparison study is performed on circular cylindrical shells, similar conclusions can be extended to the free vibration of other shaped shells of revolution.
Appendix
where nonzero elements in the upper triangular domain arewhere nonzero elements in the upper triangular domain are
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors gratefully appreciate the financial support from National Natural Science Foundation of China (51078198), Tsinghua University (2011THZ03), Jiangsu University of Science and Technology (1732921402), and Shuang-Chuang Program of Jiangsu Province for this research.