1. Introduction
Beams, plates and shells are the most commonlyused structural components in industrial applications. In comparison with beams and plates, shells usually exhibit more complicated dynamic behaviours because the curvature will effectively couple the flexural and inplane deformations together as manifested in the fact that all three displacement components simultaneously appear in each of the governing differential equations and boundary conditions. Thus it is understandable that the axial constraints can have direct effects on a predominantly radial mode. For instance, it has been shown that the natural frequencies for the circumferential modes of a simply supported shell can be noticeably modified by the constraints applied in the axial direction [1]. Vibrations of shells have been extensively studied for several decades, resulting in numerous shell theories or formulations to account for the various effects associated with deformations or stress components.
Expressions for the natural frequencies and modes shapes can be derived for the classical homogeneous boundary conditions [29]. Wave propagation approach was employed by several researchers [1013] to predict the natural frequencies for finite circular cylindrical shells with different boundary conditions. Because of the complexity and tediousness of the (exact) solution procedures, approximate procedures such as the RayleighRitz methods or equivalent energy methods have been widely used for solving shell problems [1418]. In the RayleighRitz methods, the characteristic functions for a “similar” beam problem are typically used to represent all three displacement components, leading to a characteristic equation in the form of cubic polynomials. Assuming that the circumferential wave length is smaller than the axial wave length, Yu [6] derived a simple formula for calculating the natural frequencies directly from the shell parameters and the frequency parameters for the analogous beam case. Soedel [19] improved and generalized Yu’s result by eliminating the short circumferential wave length restriction. However, since the wavenumbers for axial modal function are obtained from beam functions which do not exactly satisfy shell boundary conditions, it is mathematically difficult to access or ensure the accuracy and convergence of such a solution.
The free vibration of shells with elastic supports was studied by Loveday and Rogers [20] using a general analysis procedure originally presented by Warburton [3]. The effect of flexibility in boundary conditions on the natural frequencies of two (lower order) circumferential modes was investigated for a range of restraining stiffness values. The vibrations of circular cylindrical shells with nonuniform boundary constraints were studied by Amabili and Garziera [21] using the artificial spring method in which the modes for the corresponding lessrestrained problem were used to expand the displacement solutions. The nonuniform spring stiffness distributions were systematically represented by cosine series and their presence was accounted for in terms of maximum potential energies stored in the springs.
A large number of studies are available in the literature for the vibrations of shells under different boundary conditions or with various complicating features. A comprehensive review of early investigations can be found in Leissa’s book [22]. Some recent progresses have been reviewed by Qatu [23]. Regardless of whether an approximate or an exact solution procedure is employed, the corresponding formulations and implementations usually have to be modified or customized for different boundary conditions. This shall not be considered a trivial task in view that there exist 136 different combinations even considering the simplest (homogeneous) boundary conditions. Thus, it is useful to develop a solution method that can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures. Mathematically, elastic supports represent a general form of boundary conditions from which all the classical boundary conditions can be readily derived by simply setting each of the spring stiffnesses to either zero or infinity. This chapter will be devoted to developing a general analytical method for solving shell problems involving general elastically restrained ends.
2. Basic equations and solution procedures
Figure 1 shows an elastically restrained circular cylindrical shell of radius R, thickness h and length L. Each of the eight sets of elastic restraints shall be understood as a distributed spring along the circumference. Let u, v, and w denote the displacements in the axial x, circumferential θ and radial r directions, respectively. The equations of the motions for the shell can be written as
where ρ is the mass density of the shell material, and N_{1}, N_{12}, N_{}, Q_{1} and Q_{} denote the resultant forces acting on the midsurface.
In terms of the shell displacements, the force and moment components can be expressed as
(2) 
where
E and σ are respectively the Young’s modulus and Poisson ratio of the material; M_{1}, M_{2} and M_{12} are the bending and twisting moments.
The boundary conditions for an elastically restrained shell can specified as:
at x=0,
at x=L,
where k_{1}, k_{2}, …, k_{8} are the stiffnesses for the restraining springs. The elastic supports represent a set of general boundary conditions, and all the classical boundary conditions can be considered as the special cases when the stiffness for each spring is equal to either zero or infinity.
The above equations are usually referred to as DonnellMushtari equations. Flügge’s theory is also widely used to describe vibrations of shells. In terms of the shell displacements, the corresponding force and moment components are written as
(6) 
A shell problem can be solved either exactly or approximately. An exact solution usually implies that both the governing equations and the boundary conditions are simultaneously satisfied exactly on a pointwise basis. Otherwise, a solution is considered approximate in which one or more of the governing equations and boundary conditions are enforced only in an approximate sense. Both solution strategies will be used below.
2.1. An approximate solution based on the RayleighRitz procedure
Approximate methods based on energy methods or the RayleighRitz procedures are widely used for the vibration analysis of shells with various boundary conditions and/or complicating factors. In such an approach, the displacement functions are usually expressed as
where
where
(9) 
It is clear from Eqs. (9) that these auxiliary polynomials are only dependent on the first and third derivatives β_{i}, (i=1,2,…,8) of the displacement solutions on the boundaries. In terms of boundary derivatives, the lowestorder polynomials can be explicitly expressed as [24, 25]
where
(11) 
This alternative form of Fourier series recognizes the fact that the conventional Fourier series for a sufficiently smooth function f(x) defined on a compact interval [0, L] generally fails to converge at the end points. Introducing the auxiliary functions will ensure the cosine series in Eqs. (8) to converge uniformly and polynomially over the interval, including the end points. As a matter of fact, the polynomial subtraction techniques have been employed by mathematicians as a means to accelerate the convergence of the Fourier series expansion for an explicitly defined function [2628].
The coefficients β_{i} represent the values of the first and third derivatives of the displacements at the boundaries, and are hence related to the unknown Fourier coefficients for the trigonometric terms. The relationships between the constants and the expansion coefficients can be derived either exactly or approximately.
In seeking an approximate solution based on an energy method, the solution is not required to explicitly satisfy the force or natural boundary conditions. Accordingly, the derivative parameters β_{i} in Eqs. (10) will be here determined from a simplified set of the boundary conditions, that is,
where
By substituting Eqs. (8) and (10) into (12), one will obtain
where
(15) 
and
In light of Eqs. (13), Eqs (8) can be reduced to Eqs. (7) with the axial functions being defined as
where
Since the boundary conditions are not exactly satisfied by the displacements such constructed, the RayleighRitz procedure will be employed to find a weak form of solution. The current solution is noticeably different from the conventional RayleighRitz solutions in that: a) the shell displacements are expressed in terms of three independent sets of axial functions, rather than a single (set of) beam function(s), b) the basis functions in each displacement expansion constitutes a complete set so that the convergence of the RayleighRitz solution is guaranteed mathematically, and c) it does not suffer from the wellknown numerical instability problem related to the higher order beam functions or polynomials. More importantly, the current method is that it provides a unified solution to a wide variety of boundary conditions.
The potential energy consistent with the DonnellMushtari theory can be expressed from
(19) 
By minimizing the Lagrangian L=VT against all the unknown expansion coefficients, a final system of linear algebraic equations can be derived
where
(22) 
and
The integrals in Eq. (23) can be calculated analytically; for instance
(25) 
where
and
2.2. A strong form of solution based on Flügge’s equations
As aforementioned, the displacement expressions in terms of beam functions cannot exactly satisfy the shell boundary conditions; instead they are made to satisfy the boundary conditions in a weak sense via the use of the RayleighRitz procedure. To overcome this problem, the displacement expressions, Eqs. (8), will now be generalized to
(29) 
which represent a 2D version of the improved Fourier series expansions, Eqs. (8).
To demonstrate the flexibility in choosing the auxiliary functions
here
and
(32) 
In Eqs. (27), the sums of xrelated terms are here understood as the series expansions in xdirection, rather than characteristic functions for a beam with “similar” boundary condition. This distinction is important in that the boundary conditions and governing differential equations can now be exactly satisfied on a pointwise basis; that is, the solution can be found in strong form, as described below.
Substituting Eqs. (6) and (27) into (4) and (5) will lead to
(33) 
Equations (31) represent a set of constraint conditions between the unknown (boundary) constants, a_{n}, b_{n},..., g_{n} and h_{n}, and the Fourier expansion coefficients A_{mn}, B_{mn}, and C_{mn} ( m, n = 0, 1, 2,... ). The constraint equations (31ah) can be rewritten more concisely, in matrix form, as
The elements of the coefficient matrices can be readily derived from Eqs. (31); for example, Eq. (31a) implies
(35) 
Other submatrices can be similarly obtained from the remaining equations in Eqs. (31).
In actual numerical calculations, all the series expansions will have to be truncated to m=M and n=N. Thus there is a total number of (M+1)(3N+2)+8N+6 unknown expansion coefficients in the displacement functions. Since Eq. (33) represents a set of 8N+6 equations, additional (M+1)(3N+2) equations are needed to be able to solve for all the unknown coefficients. Accordingly, we will turn to the governing differential equations.
In Flügge’s theory, the equations of motion are given as
(36) 
Substituting Eqs. (6) and (37) into Eqs. (34) results in
(37) 
By expanding all noncosine terms into Fourier cosine series and comparing the like terms, the following matrix equation can be obtained
where E, F, P and Q are coefficient matrices whose elements are given as:
(39) 
(40) 
(41) 
The symbols
(42) 
All the unmentioned elements in matrices P and Q are identically equal to zero.
Equations (32) and (36) can be combined into
where
The final system of equations, Eq. (19) or (41), represents a standard characteristic equation for a matrix eigenproblem from which all the eigenvalues and eigenvectors can be readily calculated. It should be mentioned that the elements in each eigenvector are actually the expansion coefficients for the corresponding mode; its “physical” mode shape can be directly obtained from Eqs. (7) or (27).
In the above discussions, the stiffness distribution for each restraining spring is assumed to be axisymmetric or uniform along the circumference. However, this restriction is not necessary. For nonuniform elastic boundary restraints, the displacement expansions, Eq. (27), shall be used, and any and all of stiffness constants can be simply understood as varying with spatial angle θ. For simplicity, we can universally expand these functions into standard cosine series and modify Eq. (31) accordingly to reflect this complicating factor.
3. Results and discussion
Several numerical examples will be given below to verify the two solution strategies described earlier.
3.1. Results about the approximate RayleighRitz solution
We first consider a familiar simplysupported cylindrical shell. The simply supported boundary condition,
Next, consider a cylindrical shell clamped at each end, that is,
where
Mode  Current m=1  Eq. (42)  Current m=2  Eq. (42) 
n=1 2 3 4 5  1886.74 934.220 982.265 1598.55 2484.78  2035.05 971.531 990.339 1600.90 2486.49  3854.75 2039.66 1454.80 1769.54 2572.31  4302.05 2189.59 1500.07 1782.28 2578.07 
Another classical example involves a completely free shell. Vibration of a freefree shell is of particular interest as manifested in the debate between two legendary figures, Rayleigh and Love, about the validity of the inextensional theory of shells. The lowerorder modes are typically related to the rigidbody motions in the axial direction. Theoretically, the H_{w} matrix given in Eqs. (14) will become noninvertible for a completely free shell. However, this numerical irregularity can be easily avoided by letting one of the bendingrelated springs have a very small stiffness, such as,
After it has been adequately illustrated how the classical boundary conditions can be easily and universally dealt with by simply changing the stiffness values of the restraining springs, we will direct our attention to shells with elastic end restraints. For the purpose of comparison, the problems previously studied in ref. [20] will be considered here. It was observed in that study that the tangential stiffness had the greatest effect on the natural frequency of the cylinder supported at both ends while the axial boundary stiffness had the greatest influence on the natural frequency of the cylinder supported at one end. It was also determined that natural frequencies varied rapidly with the boundary flexibility when the nondimensionalized stiffness is between 10^{2} and 10^{2}.
The frequency parameters for the “clamped”free shell are shown in Table 5 for the reduced axial stiffness
Mode  n=0  n=1  n=2  n=3  n=4  n=5 
m=1 m=2  0.9752 1.22044  0.514686 1.12788  0.32866 (0.315*) 1.08573  0.361036 1.10467  0.532604 (0.498) 1.16021  0.782661 1.432 
[i]  Note: the numbers in parentheses are taken from ref. [20]
Although all eight sets of springs can be independently specified here, for simplicity we will only consider a simple configuration: a cantilevered shell with an elastic support being attached to its free (right) end in the radial direction. Listed in Table 6 are the four lowest natural frequencies for several different stiffness values. Obviously, the cases for
Mode 





1 2 3 4  404.108 487.598 865.603 1003.38  451.242 513.222 886.656 1023.61  627.345 679.082 926.173 1200.88  729.593 935.745 1084.91, 1319.99, ,  742.920 936.719 1269.58 1333.37 
The mode shapes for the three intermediate stiffness values are plotted in Figs. 24. It is seen that the modal parameters can be significantly modified by the stiffness of the restraining springs. The four modes in Fig. 2 for
3.2. An exact solution based on the Flügge’s equations
To validate the exact solution method, the simply supported shell is considered again. Given in Table 7 are the calculated natural frequency parameters
The current solution method is also compared with the finite element model (ANSYS) for shells under freefree boundary condition. In the FEM model, the shell surface is divided into 8000 elements with 8080 nodes. The calculated natural frequencies are compared in Tables 8. An excellent agreement is observed between these two solution methods.
In most techniques, such as the wave approach, the beam functions for the analogous boundary conditions are often used to determine the axial modal wavenumbers. While such an approach is exact for a simply supported shell, and perhaps acceptable for slender thin shells, it may become problematic for shorter shells due to the increased coupling of the radial and two inplane displacements. To illustrate this point, we consider relatively shorter and thicker shell (l=8R and R =39h). The calculated natural frequencies are compared in Table 9 for a clampedclamped shell. It is seen that while the current and FEM results are in good agreement, the frequencies obtained from the wave approach (based on the use of beam functions) are significantly higher, especially for the lower order modes.
The exact solution method can be readily applied to shells with elastic boundary supports. Since the above examples are considered adequate in illustrating the reliability and accuracy of the current method, we will not elaborate further by presenting the results for elastically restrained shells. Instead, we will simply point out that the solution method based on Eqs. (27) is also valid for nonuniform or varying boundary restraint along the circumferential direction, which represents a significant advancement over many existing techniques.
4. Conclusion
An improved Fourier series solution method is described for vibration analysis of cylindrical shells with general elastic supports. This method can be easily and universally applied to a wide variety of boundary conditions including all the 136 classical homogeneous boundary conditions. The displacement functions are invariantly expressed as series expansions in terms of the complete set of trigonometric functions, which can mathematically ensure the accuracy and convergence of the present solution. From practical point of view, the change of boundary conditions here is as simple as varying a typical shell or material parameter (e.g., thickness or mass density), and does not involve any solution algorithm and procedure modifications to adapt to different boundary conditions. In addition, the proposed method does not require predetermining any secondary data such as modal parameters for an “analogous” beam, or modifying the implementation algorithms to avoid the numerical instabilities resulting from computer roundoff errors. It should be mentioned that the current method can be readily extended to shells with arbitrary nonuniform elastic restraints. The accuracy and reliability of the current solutions have been demonstrated through numerical examples involving various boundary conditions.