**1. Introduction**

[16] Andrieux, S., & Varé, C. (2002). A 3D cracked beam model with unilateral contact.

[17] Nandi, A., & Neogy, S. (2002). Modelling of a beam with a breathing edge crack and some observations for crack detection. *Journal of Vibration and Control*, 8, 673-693.

[18] Andreaus, U., Casini, P., & Vestroni, F. (2007). Nonlinear dynamics of a cracked can‐ tilever beam under harmonic excitation. *International Journal of Nonlinear Mechanics*,

[19] Bachschmid, N., Tanzi, E., & Audebert, S. (2008). The effect of helicoidal cracks on the behavior of rotating shafts. *Engineering Fracture Mechanics*, 75, 475-488.

[20] Bathe, K. J. (1996). *Finite Element Procedures*, Prentice-Hall, Upper Saddle River, NJ.

*plication*, Englewood Cliffs, NJ, Prentice Hall.

*applications*, Reading, MA, Addison Wesley Longman.

ments. *Engineering Fracture Mechanics*, 5, 193-205.

bending. *Journal of Sound and Vibration*, 91, 583-593.

[21] Brigham, E. O. (1973). *The Fast Fourier Transform: An Introduction to Its Theory and Ap‐*

[22] Rao, R. M., & Bopardikar, A. S. (1998). *Wavelet transforms- introduction to theory and*

[23] Nandwana, B. P., & Maiti, S. K. (1997). Modelling of vibration of beam in presence of inclined edge or internal crack for its possible detection based on frequency measure‐

[24] Kisa, M., & Brandon, J. (2000). The effects of closure of cracks on the dynamics of a

[25] Chasalevris, A. C., & Papadopoulos, C. A. (2006). Identification of multiple cracks in beams under bending. *Mechanical Systems and Signal Processing*, 20, 1631-1673.

[26] Bush, A. J. (1976). Experimentally determined stress-intensity factors for single-edge-

[27] Zou, J., Chen, J., & Pu, Y. P. (2004). Wavelet time-frequency analysis of torsional vi‐ brations in rotor system with a transverse crack. *Computers and Structures*, 82,

[28] Dimarogonas, A. D., & Papadopoulos, C. A. (1983). Vibration of cracked shaft in

[29] Grabowski, B. (1979). The vibrational behavior of a turbine rotor containing a trans‐ verse crack. *ASME Design Engineering Technology Conference* [79-DET], Paper.

[30] Bachschmid, N., & Tanzi, E. (2004). Deflections and strains in cracked shafts due to rotating loads: a numerical and experimental analysis. *International Journal of Rotating*

cracked cantilever beam. *Journal of Sound and Vibration*, 238, 1-18.

crack round bars loaded in bending. *Experimental Mechanics*, 16-249.

Application to rotors. *European Journal of Mechanics A/Solids*, 21, 793-810.

42, 566-575.

204 Advances in Vibration Engineering and Structural Dynamics

1181-1187.

*Machinery*, 10, 283-291.

Beams, plates and shells are the most commonly-used 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 in-plane deformations together as manifested in the fact that all three displacement components si‐ multaneously appear in each of the governing differential equations and boundary condi‐ tions. 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 stud‐ ied 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 [2-9]. Wave propagation approach was employed by several researchers [10-13] 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 Rayleigh-Ritz methods or equivalent energy methods have been widely used for solving shell problems [14-18]. In the Rayleigh-Ritz methods, the characteristic functions for a "similar" beam problem are typi‐ cally used to represent all three displacement components, leading to a characteristic equa‐ tion 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 natu‐ ral frequencies directly from the shell parameters and the frequency parameters for the anal‐ ogous 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 con‐

© 2012 Yang et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Yang et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ditions, it is mathematically difficult to access or ensure the accuracy and convergence of such a solution.

where *ρ* is the mass density of the shell material, and *N*1, *N*12, *N*, *Q*1 and *Q* denote the resul‐

5 *k*

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 207

8 *k*

6 *k*

(2)

7 *k*

*l* 

In terms of the shell displacements, the force and moment components can be expressed as

(1 ) (c) <sup>2</sup>

(a)

(b)

(d)

(e)

(g)

(h)

tant forces acting on the mid-surface.

q

*h*

*R* 

4 *k*

*x* 

**Figure 1.** A circular cylindrical shell elastically restrained along all edges.

1

2

12

12

2 2 1 2 22


æ ö ¶ ¶ =- + ç ÷ ¶ ¶ è ø æ ö ¶ ¶ =- + ç ÷ ¶ ¶ è ø

s

*R x*

q

q

s

 s q

*u vw N K xR R*

s

æ ö ¶ ¶ = ++ ç ÷ è ø ¶ ¶

æ ö ¶ ¶ =+ + ç ÷ è ø ¶ ¶

*vw u N K RR x u v N K*

q

s

*w w M D x R*

*w w M D R x*

2 2 2 22 2

2

q

¶ ¶ ¶ ¶ = + =- + ¶ ¶ ¶ 2 2

2 33 2

¶ ¶ æ ö ¶ ¶ = + =- + ç ÷ ¶ ¶ ¶ ¶¶ è ø

*M M w w Q D R x R Rx*

s q

*R x M M w w Q D x R <sup>x</sup>*

1 12 1 3

2 12

q s

¶ =- - ¶ ¶

q

*<sup>w</sup> M D*

3 3

æ ö ç ÷ ¶ ¶ è ø

3 3

*R x*

q

 q

(1 ) (f)

q

1 *k*

*k*<sup>2</sup> *r* 

3 *k*

where

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 flexi‐ bility 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 non-uniform boundary constraints were studied by Amabili and Gar‐ ziera [21] using the artificial spring method in which the modes for the corresponding lessrestrained problem were used to expand the displacement solutions. The non-uniform 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 re‐ view 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 pro‐ cedure 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 chap‐ ter will be devoted to developing a general analytical method for solving shell problems in‐ volving 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*, cir‐ cumferential *θ* and radial *r* directions, respectively. The equations of the motions for the shell can be written as

$$\begin{aligned} \frac{\partial \mathcal{N}\_1}{\partial \mathbf{x}} + \frac{\partial \mathcal{N}\_{12}}{R \partial \theta} &= \rho h \frac{\partial^2 u}{\partial t^2} \\ \frac{\partial \mathcal{N}\_{12}}{\partial \mathbf{x}} + \frac{\partial \mathcal{N}\_2}{R \partial \theta} &= \rho h \frac{\partial^2 v}{\partial t^2} \\ \frac{\partial \mathcal{Q}\_1}{\partial \mathbf{x}} + \frac{\partial \mathcal{Q}\_1}{R \partial \theta} - \frac{\mathcal{N}\_2}{R} &= \rho h \frac{\partial^2 w}{\partial t^2} \end{aligned} \tag{1}$$

(2)

where *ρ* is the mass density of the shell material, and *N*1, *N*12, *N*, *Q*1 and *Q* denote the resul‐ tant forces acting on the mid-surface.

**Figure 1.** A circular cylindrical shell elastically restrained along all edges.

In terms of the shell displacements, the force and moment components can be expressed as

$$\mathcal{N}\_1 = \mathcal{K}\left(\frac{\partial u}{\partial \alpha} + \sigma \frac{\partial v}{R \partial \theta} + \sigma \frac{w}{R}\right) \tag{a}$$

$$\mathcal{N}\_2 = \mathcal{K}\left(\frac{\partial \mathcal{v}}{R \partial \mathcal{\partial}} + \frac{\mathcal{w}}{R} + \sigma \frac{\partial \mathcal{u}}{\partial \mathcal{\chi}}\right)\_+ \tag{b}$$

$$N\_{12} = K \frac{(1 - \sigma)}{2} \left(\frac{\partial u}{R \partial \theta} + \frac{\partial v}{\partial x}\right) \tag{c}$$

$$\mathcal{M}\_1 = -D\left(\frac{\partial^2 w}{\partial x^2} + \sigma \frac{\partial^2 w}{\partial x^2 \partial \theta^2}\right) \tag{d}$$

$$\mathcal{M}\_2 = -D\left(\frac{\partial^2 w}{\partial^2 \partial \theta^2} + \sigma \frac{\partial^2 w}{\partial x^2}\right) \tag{e}$$

$$\mathcal{M}\_{12} = -D(1 - \sigma) \frac{\partial^2 w}{\partial \vec{\alpha} \vec{\alpha} \partial \theta} \tag{f}$$

$$Q\_1 = \frac{\partial \mathcal{M}\_1}{\partial \mathbf{x}} + \frac{\partial \mathcal{M}\_{12}}{R \partial \theta} = -D \left( \frac{\partial^3 w}{\partial \mathbf{x}^3} + \frac{\partial^3 w}{R^2 \partial \mathbf{x} \partial \theta^2} \right) \tag{\text{g}}$$

$$Q\_2 = \frac{\partial \mathcal{M}\_2}{R \partial \theta} + \frac{\partial \mathcal{M}\_{12}}{\partial \chi} = -D \left( \frac{\partial^3 w}{R^3 \partial \theta^3} + \frac{\partial^3 w}{R \partial \chi^2 \partial \theta} \right) \tag{h}$$

where

(1)

ditions, it is mathematically difficult to access or ensure the accuracy and convergence of

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 flexi‐ bility 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 non-uniform boundary constraints were studied by Amabili and Gar‐ ziera [21] using the artificial spring method in which the modes for the corresponding lessrestrained problem were used to expand the displacement solutions. The non-uniform 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 re‐ view 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 pro‐ cedure 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 chap‐ ter will be devoted to developing a general analytical method for solving shell problems in‐

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*, cir‐ cumferential *θ* and radial *r* directions, respectively. The equations of the motions for the

2

2 2

> 2 2

r

2

(a)

(b)

(c)

1 12

*N N <sup>u</sup> <sup>h</sup> x R t N N <sup>v</sup> <sup>h</sup> x R t Q QN <sup>w</sup> <sup>h</sup> xR R t*

¶ ¶ ¶ + = ¶ ¶ ¶

r q

> r q

12 2

1 12

¶ ¶ ¶ + = ¶ ¶ ¶

q

¶ ¶ ¶ + -= ¶ ¶ ¶

such a solution.

206 Advances in Vibration Engineering and Structural Dynamics

volving general elastically restrained ends.

shell can be written as

**2. Basic equations and solution procedures**

$$\begin{aligned} \text{K} &= \text{Eh} \,/\, (1 - \sigma^2) , & \text{(a)}\\ \text{D} &= \text{Eh}^3 \,/\, 12 (1 - \sigma^2) & \text{(b)} \\ \text{K} &= \text{D} \,/\, \text{K} = \text{h}^2 \,/\, 12 & \text{(c)} \end{aligned} \tag{3}$$

2

2

q

 s

*R xR R R x*

 s

 s

*R x R Rx R x*

æ ö - ¶¶ - ¶ ¶ æ ö = ++ - ç ÷ ç ÷ è ø ¶ ¶ ¶ ¶¶ è ø

<sup>æ</sup> - ¶¶ - ¶ ¶ æ ö = ++ ç ÷ <sup>ç</sup> <sup>+</sup> è ø ¶ ¶ <sup>ç</sup> ¶ ¶ ¶ <sup>è</sup>

2

 q

<sup>1</sup> (1 ) (b) 2 2

q

 q

(1 ) (f)

1 1 (1 ) (h) 2 2

2 2 32 2

*xR R x* s

q

A shell problem can be solved either exactly or approximately. An exact solution usually im‐ plies that both the governing equations and the boundary conditions are simultaneously sat‐ isfied exactly on a point-wise basis. Otherwise, a solution is considered approximate in which one or more of the governing equations and boundary conditions are enforced only

Approximate methods based on energy methods or the Rayleigh-Ritz procedures are widely used for the vibration analysis of shells with various boundary conditions and/or complicat‐

> q

 q

> q

dary conditions. Although characteristic functions generally exist in the forms of trigono‐ metric and hyperbolic functions, they also include some integration and frequency constants that have to be determined from boundary conditions. Consequently, each boundary condi‐

*<sup>m</sup>*(*x*), *α* =*u*, *v*, and *w*, are the characteristic functions for beams with *similar* boun‐

ing factors. In such an approach, the displacement functions are usually expressed as

( , ) ( ) cos (a)

*m m u m*

*m m v m*

*m m w m*

( , ) ( ) sin (b)

( , ) ( ) cos (c)

(a)

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816

(c)

(d)

(e)

(6)

209

(7)

(g)

2

ö ÷ ÷ ø

1 1 (i) 2 2

q

*u v*

 s

 q

1 2

æ ö ¶¶ ¶ = + +- ç ÷ è ø ¶ ¶ ¶

 s q

*u v wDw N K <sup>K</sup> x R RR x u vD v w N K*

q

s

s

q

s

2 22 2

æ ö æ ö ¶ ¶¶ = ++ + + ç ÷ ç ÷ è ø ¶ ¶ ¶ è ø

2 2

æ ö ¶ ¶¶ =- - + - ç ÷ ¶ ¶ ¶ ¶ è ø

s q

*R x R R x*

in an approximate sense. Both solution strategies will be used below.

**2.1. An approximate solution based on the Rayleigh-Ritz procedure**

0

qj

å

=

=

=

¥ = ¥ = ¥ =

qj

å

*ux a x n*

*vx b x n*

*wx c x n*

0

qj

å

0

q

æ ö -¶ + ¶ ç ÷ + - ¶ ¶ ¶ ¶ è ø

*R x Rx*

q

q

æ ö ¶ ¶¶ ¶ =- + - - ç ÷ ¶¶ ¶ ¶ è ø

2 21 2

3 32

q

q

1

*w wu Q D x R xR*

¶ ¶¶ =- + - ¶ ¶¶

21 2

*w wu v M D xR R R x*

q

2 2 1 2 22 2 2

s

æ ö ¶ ¶ =- - - ç ÷ ¶¶ ¶ è ø

æ ö ¶ ¶ =- + + ç ÷ ¶ ¶ è ø

2 2 22 2

*ww w M D RR x w uv M D*

s

*w v M D*

s

1 3 22

1 (1 ) 2 2

*vw uD w w N K R R xR R R u vD u w N K*

s

12

12

where *φα*

*E* and *σ* are respectively the Young's modulus and Poisson ratio of the material; *M*1, *M*<sup>2</sup> and *M*12 are the bending and twisting moments.

The boundary conditions for an elastically restrained shell can specified as:

at *x*=0,

$$\begin{cases} N\_x - k\_1 u = 0 & \text{(a)}\\ N\_{x\theta} - k\_2 v = 0 & \text{(b)}\\ Q\_x + \frac{\partial M\_{x\theta}}{R \partial \theta} - k\_3 w = 0 & \text{(c)}\\ M\_x + k\_4 \frac{\partial w}{\partial \alpha} = 0 & \text{(d)} \end{cases} \tag{4}$$

at *x*=*L*,

5 6 7 8 0 (a) 0 (b) 0 (c) 0 (d) *x x x x x N ku N kv <sup>M</sup> Q kw <sup>R</sup> <sup>w</sup> M k <sup>x</sup>* q q q + = + = ¶ + += ¶ ¶ - = ¶ (5)

where *k*1, *k*2, …, *k*8 are the stiffnesses for the restraining springs. The elastic supports repre‐ sent 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 Donnell-Mushtari 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)

$$\mathbf{N}\_1 = \mathbf{K} \left( \frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \sigma \frac{\partial \mathbf{v}}{R \partial \theta} \right) + \mathbf{K} \sigma \frac{\mathbf{w}}{R} - \frac{D}{R} \frac{\partial^2 \mathbf{w}}{\partial \mathbf{x}^2} \tag{a}$$

2 3 2 2

s

*K Eh D Eh*

k

*M*12 are the bending and twisting moments.

208 Advances in Vibration Engineering and Structural Dynamics

at *x*=0,

at *x*=*L*,

infinity.

*DK h*

The boundary conditions for an elastically restrained shell can specified as:

1 2


*x x*

q

*N ku N kv*

*x*

*x*

4

*<sup>w</sup> M k <sup>x</sup>*

5 6

+ = + =

*x x*

q

*N ku N kv*

*x*

*x*

corresponding force and moment components are written as

8

*<sup>w</sup> M k <sup>x</sup>*

*x*

q

q

¶ + += ¶

*<sup>M</sup> Q kw <sup>R</sup>*

¶ - = ¶

7

0 (a) 0 (b)

0 (c)

0 (d)

where *k*1, *k*2, …, *k*8 are the stiffnesses for the restraining springs. The elastic supports repre‐ sent 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

The above equations are usually referred to as Donnell-Mushtari equations. Flügge's theory is also widely used to describe vibrations of shells. In terms of the shell displacements, the

*x*

q

q

¶ + -= ¶

*<sup>M</sup> Q kw <sup>R</sup>*

¶ + = ¶

3

= - = - = =

/ (1 ), (a) / 12(1 ) (b) / / 12 (c)

*E* and *σ* are respectively the Young's modulus and Poisson ratio of the material; *M*1, *M*<sup>2</sup> and

0 (a) 0 (b)

0 (c)

0 (d)

(3)

(4)

(5)

s

$$N\_{12} = \frac{1-\sigma}{2} K \left(\frac{\partial u}{R\partial\theta} + \frac{\partial v}{\partial x}\right) + \frac{D}{R} \frac{(1-\sigma)}{2} \left(\frac{\partial v}{R\partial x} - \frac{\partial^2 w}{R\partial\theta\partial x}\right) \tag{9}$$

$$\mathcal{N}N\_2 = K\left(\frac{\partial \upsilon}{R\partial \theta} + \frac{\upsilon \upsilon}{R} + \sigma \frac{\partial u}{\partial x}\right) + \frac{D}{R}\left(\frac{\partial^2 \upsilon \upsilon}{R^2 \partial \theta^2} + \frac{w}{R^2}\right) \tag{c}$$

$$N\_{21} = \frac{1-\sigma}{2} K \left( \frac{\partial u}{R \partial \theta} + \frac{\partial v}{\partial x} \right) + \frac{D}{R} \frac{(1-\sigma)}{2} \left( \frac{\partial u}{R^2 \partial \theta} + \frac{\partial^2 w}{R \partial \theta \partial x} \right) \tag{d}$$

$$\mathcal{M}\_1 = -D\left(\frac{\hat{\sigma}^2 w}{\hat{\alpha}\pi^2} + \sigma \frac{\hat{\sigma}^2 w}{R^2 \hat{\alpha}\theta^2} - \frac{\hat{\alpha}u}{R\hat{\alpha}\chi} - \sigma \frac{\hat{\sigma}v}{R^2 \hat{\alpha}\theta}\right) \tag{6}$$

$$M\_{12} = -D(1 - \sigma) \left( \frac{\partial^2 w}{R \partial \theta \partial \mathbf{x}} - \frac{\partial w}{R \partial \mathbf{x}} \right) \tag{6}$$

$$\mathcal{M}\_2 = -D\left(\frac{\varpi}{R^2} + \frac{\hat{\sigma}^2 w}{R^2 \hat{\sigma} \theta^2} + \sigma \frac{\hat{\sigma}^2 w}{\hat{\sigma} \alpha^2}\right) \tag{9}$$

$$\left(\begin{array}{cccc}\varphi\_{2\varpi} & \mathbf{1} & \varphi\_{2\mathbf{u}} & \mathbf{1} & \varphi\_{2\mathbf{u}}\end{array}\right)$$

$$M\_{21} = -D(1 - \sigma) \left( \frac{\partial^2 w}{\partial \mathcal{E} \partial \mathcal{E} \alpha} + \frac{1}{2} \frac{\partial u}{\partial \mathcal{E} \partial \theta} - \frac{1}{2} \frac{\partial v}{\partial \mathcal{E} \alpha} \right) \tag{4}$$

$$Q\_1 = -D\left(\frac{\partial^3 w}{\partial \mathbf{x}^3} + \frac{1}{R^2} \frac{\partial^3 w}{\partial \theta^2 \partial \mathbf{x}} - \frac{\partial^2 u}{R \partial \mathbf{x}^2} + \frac{1 - \sigma}{2R^3} \frac{\partial^2 u}{\partial \theta^2} - \frac{1 + \sigma}{2R^2} \frac{\partial v}{\partial \mathbf{x} \partial \theta}\right) \tag{i}$$

A shell problem can be solved either exactly or approximately. An exact solution usually im‐ plies that both the governing equations and the boundary conditions are simultaneously sat‐ isfied exactly on a point-wise 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 Rayleigh-Ritz procedure**

Approximate methods based on energy methods or the Rayleigh-Ritz procedures are widely used for the vibration analysis of shells with various boundary conditions and/or complicat‐ ing factors. In such an approach, the displacement functions are usually expressed as

$$u(\mathbf{x}, \theta) = \sum\_{m=0}^{\phi} a\_m \phi\_u^m(\mathbf{x}) \cos n\theta \tag{\text{a}}$$

$$v(\mathbf{x}, \theta) = \sum\_{m=0}^{\phi} b\_m \phi\_v^m(\mathbf{x}) \sin n\theta \tag{\text{b}} \tag{7}$$

$$w(\mathbf{x}, \theta) = \sum\_{m=0}^{\phi} c\_m \phi\_w^m(\mathbf{x}) \cos n\theta \tag{\text{c}}$$

where *φα <sup>m</sup>*(*x*), *α* =*u*, *v*, and *w*, are the characteristic functions for beams with *similar* boun‐ dary conditions. Although characteristic functions generally exist in the forms of trigono‐ metric and hyperbolic functions, they also include some integration and frequency constants that have to be determined from boundary conditions. Consequently, each boundary condi‐ tion basically calls for a special set of modal data. In the literature the modal parameters are well established for the simplest homogeneous boundary conditions. However, the determi‐ nation of modal properties for the more complicated elastic boundary supports can become, at least, a tedious task since they have to be re-calculated each time when any of the stiffness values is changed. It should also be noted that calculating the modal properties will typical‐ ly involve seeking the roots of a nonlinear transcendental equation, which mathematically requires an iterative root searching scheme and careful numerical implementations to en‐ sure no missing data. To overcome this problem, a unified representation of the shell solu‐ tions will be adopted here in which the displacements, regardless of boundary conditions, will be invariably expressed as

$$\begin{aligned} \mu(\mathbf{x},\ \Theta) &= \left(\sum\_{m=0}^{\infty} a\_m \cos \lambda\_m \mathbf{x} + p\_u(\mathbf{x})\right) \cos m\Theta, \ (\lambda\_m = \frac{m\pi}{L}) \text{(a)}\\ \nu(\mathbf{x},\ \Theta) &= \left(\sum\_{m=0}^{\infty} b\_m \cos \lambda\_m \mathbf{x} + p\_v(\mathbf{x})\right) \sin m\Theta \ \mathbf{\hat{z}} \ \mathbf{\hat{z}} \\ \nu(\mathbf{x},\ \Theta) &= \left(\sum\_{m=0}^{\infty} c\_m \cos \lambda\_m \mathbf{x} + p\_w(\mathbf{x})\right) \cos m\Theta \mathbf{\hat{z}} \end{aligned} $$

(8)

1 12 2 1 32 4

*px x x x x*

=+++

 b

 b

> bz

*px x x px x x*

= + = +

z bz

z bz

z bz

1

z

z

z

z

2

*T*

an explicitly defined function [26-28].

derived either exactly or approximately.

tions, that is,

where

*u v w*

where

1 52 63 74 8

4 3 22 4 <sup>3</sup> 4 22 4 <sup>4</sup>

*x x Lx L L*

This alternative form of Fourier series recognizes the fact that the conventional Fourier ser‐ ies 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 ser‐ ies 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

The coefficients *β<sup>i</sup>* 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 trigono‐ metric terms. The relationships between the constants and the expansion coefficients can be

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 pa‐ rameters *β<sup>i</sup>* in Eqs. (10) will be here determined from a simplified set of the boundary condi‐

1,5

*<sup>w</sup> k w <sup>x</sup>*

*w w <sup>k</sup> <sup>x</sup> <sup>x</sup>*

m

¶ ¶ <sup>=</sup> ¶ ¶

*<sup>u</sup> k u <sup>x</sup>*

m

¶ <sup>=</sup> ¶

m

¶

¶

k

k

By substituting Eqs. (8) and (10) into (12), one will obtain

2,6

*<sup>v</sup> k v <sup>x</sup>*


m

± =

(1 ) <sup>ˆ</sup> 0 (b) <sup>2</sup>

ˆ 0 (a)

ˆ 0 (c)

ˆ 0 (d)

<sup>ˆ</sup> / . *i i k kK* <sup>=</sup> (13)

ì ü ì ü - - ï ï ï ï ï ï - = = í ýí <sup>ý</sup> ï ïï-- + - <sup>ï</sup> ï ïï <sup>ï</sup> î þ - + î þ

( ) (6 2 3 ) / 6 ( ) (3 ) / 6 ( ) () (15 60 60 8 ) / 360 ( ) (15 30 7 ) / 360

*x Lx L x L <sup>x</sup> xL L <sup>x</sup> <sup>x</sup> x Lx L x L L*

 bz

2 2

2 2

 b

z (11)

(10)

211

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816

(12)

() () () (a) () () () (b) ( ) ( ) ( ) ( ) ( ) (c)

where *pα*(*x*), *α* =*u*, *v*, and *w*, denote three auxiliary polynomials which satisfy

$$\begin{aligned} &\left.\frac{\partial p(x,0)}{\partial x}\right|\_{x=0} = \left.\frac{\partial p(x,0)}{\partial x}\right|\_{x=0} = \beta\_1, \qquad \text{(a)}\\ &\left.\frac{\partial p(x,0)}{\partial x}\right|\_{x=L} = \left.\frac{\partial u(x,0)}{\partial x}\right|\_{x=L} = \beta\_2, \qquad \text{(b)}\\ &\left.\frac{\partial p(x,x)}{\partial x}\right|\_{x=0} = \left.\frac{\partial v(x,x,\tau/2)}{\partial x}\right|\_{x=0} = \beta\_3, \qquad \text{(c)}\\ &\left.\frac{\partial p(x,x)}{\partial x}\right|\_{x=L} = \left.\frac{\partial v(x,x,\tau/2)}{\partial x}\right|\_{x=L} = \beta\_4, \qquad \text{(d)}\\ &\left.\frac{\partial p(x,x)}{\partial x}\right|\_{x=0} = \left.\frac{\partial w(x,0)}{\partial x}\right|\_{x=0} = \beta\_5, \qquad \text{(e)}\\ &\left.\frac{\partial p(x,x)}{\partial x}\right|\_{x=L} = \left.\frac{\partial w(x,0)}{\partial x}\right|\_{x=L} = \beta\_6, \qquad \text{(f)}\\ &\left.\frac{\partial^2 w(x,0)}{\partial x^2}\right|\_{x=0} = \left.\frac{\partial^2 w(x,0)}{\partial x^3}\right|\_{x=0} = \beta\_7, \qquad \text{(g)}\\ &\left.\frac{\partial^3 w(x,0)}{\partial x^3}\right|\_{x=L} = \left.\frac{\partial^3 w(x,0)}{\partial x^3}\right|\_{x=L} = \beta\_8 \qquad \text{(h)}\end{aligned}$$

It is clear from Eqs. (9) that these auxiliary polynomials are only dependent on the first and third derivatives *β<sup>i</sup>* , (*i*=1,2,…,8) of the displacement solutions on the boundaries. In terms of boundary derivatives, the lowest-order polynomials can be explicitly expressed as [24, 25]

$$\begin{aligned} p\_u(\mathbf{x}) &= \mathcal{L}\_1(\mathbf{x})\beta\_1 + \mathcal{L}\_2(\mathbf{x})\beta\_2 & \quad \text{(a)}\\ p\_v(\mathbf{x}) &= \mathcal{L}\_1(\mathbf{x})\beta\_3 + \mathcal{L}\_2(\mathbf{x})\beta\_4 & \quad \text{(b)}\\ p\_w(\mathbf{x}) &= \mathcal{L}\_1(\mathbf{x})\beta\_5 + \mathcal{L}\_2(\mathbf{x})\beta\_6 + \mathcal{L}\_3(\mathbf{x})\beta\_7 + \mathcal{L}\_4(\mathbf{x})\beta\_8 & \text{(c)}\end{aligned} \tag{10}$$

(8)

(9)

tion basically calls for a special set of modal data. In the literature the modal parameters are well established for the simplest homogeneous boundary conditions. However, the determi‐ nation of modal properties for the more complicated elastic boundary supports can become, at least, a tedious task since they have to be re-calculated each time when any of the stiffness values is changed. It should also be noted that calculating the modal properties will typical‐ ly involve seeking the roots of a nonlinear transcendental equation, which mathematically requires an iterative root searching scheme and careful numerical implementations to en‐ sure no missing data. To overcome this problem, a unified representation of the shell solu‐ tions will be adopted here in which the displacements, regardless of boundary conditions,

*<sup>L</sup>* )(a)

where *pα*(*x*), *α* =*u*, *v*, and *w*, denote three auxiliary polynomials which satisfy

( ) ( ,0) , (a)

( ) ( ,0) (b)

( ) ( , / 2) , (c)

( ) ( , / 2) , (d)

( ) ( ,0) , (e)

( ) ( ,0) , (f)

0 0

= =

*x L x L*

= =

¶ ¶ = = ¶ ¶

¶ ¶ = = ¶ ¶

*x x*

¶ ¶ = = ¶ ¶

¶ ¶ = = ¶ ¶

*p x u x x x p x u x x x p x v x x x p x v x x x p x w x x x p x w x x x*

0 0

= =

p

p

*x x*

0 0

= =

*x L x L*

= =

3

0 0

= =

*w x x*

¶ = = ¶

3 3 8

*x L x L*

= =

*x x*

¶ ¶ = = ¶ ¶

¶ ¶ = = ¶ ¶

*x L x L*

= =

1

b

2

b

3

b

4

b

5

b

6

( ,0) , (g)

b

b

It is clear from Eqs. (9) that these auxiliary polynomials are only dependent on the first and

boundary derivatives, the lowest-order polynomials can be explicitly expressed as [24, 25]

, (*i*=1,2,…,8) of the displacement solutions on the boundaries. In terms of

b

3 7

( ) ( ,0) (h) *x*

will be invariably expressed as

210 Advances in Vibration Engineering and Structural Dynamics

*am*cos*λm<sup>x</sup>* <sup>+</sup> *pu*(*x*)) cos*nθ*, (*λ<sup>m</sup>* <sup>=</sup> *<sup>m</sup><sup>π</sup>*

*bm*cos*λmx* + *pv*(*x*)) sin*nθ* , (b)

*cm*cos*λmx* + *pw*(*x*)) cos*nθ*(c)

*u*

*u*

*v*

*v*

*w*

*w*

3 3

¶ ¶ ( )

*x*

3 3

*p x w x x x*

¶ ¶ = = ¶ ¶

*w*

*w*

*p x x*

*u*(*x*, *θ*)=(∑

*v*(*x*, *θ*)=(∑

*w*(*x*, *θ*)=(∑

*m*=0 *∞*

*m*=0 *∞*

> *m*=0 *∞*

third derivatives *β<sup>i</sup>*

$$\mathbf{Q}(\mathbf{x})^{\mathsf{T}} = \begin{pmatrix} \boldsymbol{\zeta}\_{1}(\mathbf{x}) \\ \boldsymbol{\zeta}\_{2}(\mathbf{x}) \\ \boldsymbol{\zeta}\_{3}(\mathbf{x}) \\ \boldsymbol{\zeta}\_{4}(\mathbf{x}) \end{pmatrix} = \begin{pmatrix} (6L\mathbf{x} - 2L^{2} - 3\mathbf{x}^{2})/6L \\ (3\mathbf{x}^{2} - L^{2})/6L \\ -\left(15\mathbf{x}^{4} - 60L\mathbf{x}^{3} + 60L^{2}\mathbf{x}^{2} - 8L^{4}\right)/360L \\ \left(15\mathbf{x}^{4} - 30L^{2}\mathbf{x}^{2} + 7L^{4}\right)/360L \end{pmatrix} \tag{11}$$

This alternative form of Fourier series recognizes the fact that the conventional Fourier ser‐ ies 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 ser‐ ies 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 [26-28].

The coefficients *β<sup>i</sup>* 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 trigono‐ metric 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 pa‐ rameters *β<sup>i</sup>* in Eqs. (10) will be here determined from a simplified set of the boundary condi‐ tions, that is,

$$\begin{aligned} \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\alpha}} \pm \hat{k}\_{1,3} &= 0 & \text{(a)}\\ \frac{(1-\mu)}{2} \frac{\partial \boldsymbol{v}}{\partial \boldsymbol{\alpha}} \mp \hat{k}\_{2,6} &= 0 & \text{(b)}\\ \kappa \frac{\partial^3 \boldsymbol{w}}{\partial \boldsymbol{\alpha}^3} \pm \hat{k}\_{3,7} \boldsymbol{w} &= 0 & \text{(c)}\\ \kappa \frac{\partial^2 \boldsymbol{w}}{\partial \boldsymbol{\alpha}^2} \mp \hat{k}\_{4,8} \frac{\partial \boldsymbol{w}}{\partial \boldsymbol{\alpha}} &= 0 & \text{(d)} \end{aligned} \tag{12}$$

where

$$
\hat{k}\_i = k\_i \nmid \mathbb{K}.\tag{13}
$$

By substituting Eqs. (8) and (10) into (12), one will obtain

$$\begin{Bmatrix} \beta\_{\mathbf{1}}\\ \beta\_{\mathbf{2}} \end{Bmatrix} = \sum\_{m=0}^{\circ} \mathbf{H}\_{u}^{-1} \mathbf{Q}\_{u}^{m} a\_{m} \tag{a}$$

$$\begin{Bmatrix} \beta\_{\mathbf{3}}\\ \beta\_{\mathbf{4}} \end{Bmatrix} = \sum\_{m=0}^{n} \mathbf{H}\_{v}^{-1} \mathbf{Q}\_{v}^{w} b\_{m} \tag{b} \tag{14}$$

$$\left(\begin{array}{cccc}\boldsymbol{\beta\_{\mathsf{B}}} & \boldsymbol{\beta\_{\mathsf{O}}} & \boldsymbol{\beta\_{\mathsf{T}}} & \boldsymbol{\beta\_{\mathsf{O}}} \end{array}\right)^{\mathsf{T}} = \sum\_{m=0}^{\mathsf{e}} \mathbf{H}\_{\mathsf{w}}^{-1} \mathbf{Q}\_{\mathsf{w}}^{\mathsf{m}} \boldsymbol{c\_{\mathsf{w}}} \quad \text{(c)}$$

$$\mathbf{H}\_{u} = \begin{bmatrix} \hat{k}\_{1}L & \hat{k}\_{1}L \\ \hline 3 & 6 \\ \hat{k}\_{5}L & \hat{k}\_{5}L + 1 \\ \hline 6 & 6 \end{bmatrix} \tag{a)$$
 
$$\mathbf{Q}\_{u}^{m} = \left\langle \hat{k}\_{1} \quad \left(-1\right)^{m+1} \hat{k}\_{5} \right\rangle^{\mathrm{T}} \tag{b)$$

$$\mathbf{Q}\_{\mu}^{m} = \begin{Bmatrix} k\_1 & (-1)^{m+1} k\_5 \end{Bmatrix} \tag{b}$$

$$\begin{bmatrix} \hat{k}\_2 L & 1 - \mu & \hat{k}\_2 L \\ \frac{\hat{k}\_2 L}{\sigma} + \frac{1 - \mu}{\sigma} & \frac{\hat{k}\_2 L}{\sigma} \end{bmatrix} \tag{b}$$

$$\mathbf{H}\_{\upsilon} = \begin{bmatrix} \overline{3} + \overline{2} & \overline{6} \\ \hat{k}\_{\text{6}} L & \hat{k}\_{\text{6}} L + \frac{1-\mu}{2} \\ \overline{6} & (-1)^{m+1} \hat{k}\_{\text{6}} \end{bmatrix} \tag{c}$$
 
$$\mathbf{O}^{m} = \begin{bmatrix} \hat{k} & (-1)^{m+1} \hat{k}\_{\text{6}} \end{bmatrix}^{\mathsf{T}} \tag{d}$$

(15)

( ) cos ( ) , ( , , ) *<sup>m</sup> T m*


a

ì ü ï ï <sup>=</sup> í ý ï ï î þ

**Q**

*m m*

**Q HQ**

(a)

 0 (b) 0

( ) (c)

Since the boundary conditions are not exactly satisfied by the displacements such construct‐ ed, the Rayleigh-Ritz procedure will be employed to find a weak form of solution. The cur‐ rent solution is noticeably different from the conventional Rayleigh-Ritz 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 ex‐ pansion constitutes a complete set so that the convergence of the Rayleigh-Ritz solution is guaranteed mathematically, and c) it does not suffer from the well-known numerical insta‐ bility 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 con‐

The potential energy consistent with the Donnell-Mushtari theory can be expressed from

qqq

 q

from

( )

¶ ¶

4 0

( )

 q

By minimizing the Lagrangian *L=V-T* against all the unknown expansion coefficients, a final

n

q

q

q

]

*w x Rd*

*x*

=

*x L*

=

 q

(a)

(b)

2 2 2

ïæ ö æ öæ ö ¶ ¶ ¶¶ -¶ ¶ = + + -- + + + + íç ÷ ç ÷ç ÷ è ø è øè ø ¶ ¶ ¶ ¶ ¶ ¶ ïî

*x R R xR R x R*

*w w ww w Rdxd*

2 2 2 2 22 2

é ù æ ö <sup>ü</sup> æ ö ¶ ¶ ¶¶ ¶ æ ö ê ú ç ÷ <sup>ï</sup> ç ÷ + -- - ç ÷ <sup>ý</sup> ê ú ¶ ¶ ¶¶ ç ÷ ç ÷ ¶ ¶ è ø è ø <sup>ï</sup> ê ú ë û è ø <sup>þ</sup>

<sup>2</sup> <sup>2</sup>

<sup>2</sup> <sup>2</sup> 22 2 567 8

ated

*k u k v k w k w x Rd*

12 [ ]

*<sup>L</sup> K u vw uvw v u <sup>V</sup>*

n

2 22 222

+ ++ +

( ) ( ) ( )

<sup>2</sup> 22 2

*T h u t v t w t Rdxd*

é ù = ¶ ¶ +¶ ¶ +¶ ¶ ê ú ë û

system of linear algebraic equations can be derived

12 [

0

ò

p

p

0

ò

and the total kinetic energy is calcul

kn

q

2(1 )

22 2 123

*ku kv kw k*

+ + + + ¶¶

*x R x R R x*

(1 ) 2(1 ) 2 2

 a

*x x x uvw* =+ = z **Q** (17)

(18)

213

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816

(19)

 a

*m*

*m m w w m*

% %

=

**Q Q**

*m*

a

**Q**

%

a aa

=

jl

a

where

ditions.

0 0

0 0

ò ò

1

*L*

p

r

2

ò ò

p ì

$$\mathbf{Q}\_{\nu}^{m} = \begin{Bmatrix} \hat{k}\_{2} & (-1)^{m+1} \hat{k}\_{6} \end{Bmatrix}, \tag{d}$$

$$\mathbf{H}\_{\text{av}} = \begin{bmatrix} -\hat{k}\_{3}L & -\hat{k}\_{3}L & \hat{k}\_{3}L^{3} + \kappa & \frac{7\hat{k}\_{3}L^{3}}{360} \\ -\frac{\hat{k}\_{7}L}{3} & -\frac{\hat{k}\_{7}L}{3} & \frac{7\hat{k}\_{7}L^{3}}{360} & \frac{\hat{k}\_{7}L^{3}}{45} + \kappa \\ -\frac{\hat{k}\_{7}L}{6} & -\frac{\kappa}{L} & \frac{\kappa L}{3} & \frac{\kappa L}{6} \\ \hat{k}\_{4} + \frac{\kappa}{L} & -\frac{\kappa}{L} & \frac{\kappa L}{3} & \frac{\kappa L}{6} \\ -\frac{\kappa}{L} & \hat{k}\_{8} + \frac{\kappa}{L} & \frac{\kappa L}{6} & \frac{\kappa L}{3} \end{bmatrix} \tag{e}$$

*m m*

+

and

$$\mathbf{Q}\_{\mu}^{\prime\prime\prime} = \begin{pmatrix} -\hat{k}\_3 & (-1)^{\prime\prime}\hat{k}\_7 & -\kappa \lambda\_m^2 & (-1)^{\prime\prime}\kappa \lambda\_m^2 \end{pmatrix}^T \tag{16}$$

In light of Eqs. (13), Eqs (8) can be reduced to Eqs. (7) with the axial functions being defined as

$$\boldsymbol{\phi}\_{\boldsymbol{\alpha}}^{\boldsymbol{m}}(\mathbf{x}) = \cos \boldsymbol{\lambda}\_{\boldsymbol{m}} \mathbf{x} + \mathbf{f}(\mathbf{x})^{\top} \boldsymbol{\tilde{\mathbf{Q}}}\_{\boldsymbol{\alpha}}^{\boldsymbol{m}}, \left(\boldsymbol{\alpha} = \boldsymbol{\mu}, \boldsymbol{\upsilon}, \boldsymbol{w}\right) \tag{17}$$

{ }

5 6 7 8

bbbb

*m*

å

b

212 Advances in Vibration Engineering and Structural Dynamics

ì ü ï ï í ý <sup>=</sup> ï ï î þ ì ü ï ï í ý <sup>=</sup> ï ï î þ

b

b

b

{ }

*k k k L k L*

*m m*

= -

*m m*

= -

1 5 2 2

ˆ ˆ 1

m

é ù - ê ú <sup>+</sup>



8

 kk

ˆ

*L L*

*L L*

kk

+ -

+

<sup>T</sup> <sup>1</sup>

1

<sup>T</sup> <sup>1</sup>

6 32

3 33 3

*kL kL kL kL*

é ù ê ú

ˆ ˆˆ ˆ 7 3 6 45 360 ˆ ˆ ˆˆ 7 6 3 360 45

7 7 77

*kL kL kL kL*


*L L <sup>k</sup>*


3 7

 k

1 1

*kL kL*

é ù ê ú + = ê ú ê ú ê ú <sup>+</sup> ë û

ˆ ˆ 1

5 5

*kL kL*

*u*

**H**

*u*

**Q**

*v*

**H**

=

*v*

**Q**

*w*

**H**

=

6 3

{ }

*k k*

4

k

ˆ

*k*

2 6

6 6

*kL kL*

+

where

and

as

*m*

å

**H Q**

**H Q**

¥ - = ¥ - =

*m u um*

*a*

*m v vm*

*b*

(c)

=

1 0

**H Q**

*c*

*T m w wm*

¥ - =

*m*

3 6 (a) ˆ ˆ

ˆ ˆ ( 1) (b)

32 6 (c) ˆ ˆ <sup>1</sup>

m

3 3

ˆ ˆ ( 1) (d)

k

3 3

3 6

*L L*

 k

 k

6 3

{ } 2 2

In light of Eqs. (13), Eqs (8) can be reduced to Eqs. (7) with the axial functions being defined

ˆ ˆ ( 1) ( 1) *<sup>T</sup> mm m <sup>w</sup> m m* **Q** =- - - - *k k* kl

k

 kl (e)

(16)

å

(a)

(b)

(14)

(15)

$$\begin{aligned} \tilde{\mathbf{Q}}\_w^m &= \tilde{\mathbf{Q}}\_w^m \\ \tilde{\mathbf{Q}}\_\alpha^m &= \begin{bmatrix} \tilde{\mathbf{Q}}\_\alpha^m \\ 0 \\ 0 \end{bmatrix} & & \text{(b)} \\ \tilde{\mathbf{Q}}\_\alpha^m &= (\mathbf{H}\_\alpha)^{-1} \mathbf{Q}\_\alpha^m & \text{(c)} \end{aligned} \tag{18}$$

Since the boundary conditions are not exactly satisfied by the displacements such construct‐ ed, the Rayleigh-Ritz procedure will be employed to find a weak form of solution. The cur‐ rent solution is noticeably different from the conventional Rayleigh-Ritz 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 ex‐ pansion constitutes a complete set so that the convergence of the Rayleigh-Ritz solution is guaranteed mathematically, and c) it does not suffer from the well-known numerical insta‐ bility 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 con‐ ditions.

The potential energy consistent with the Donnell-Mushtari theory can be expressed from

$$V = \frac{K}{2} \frac{1}{l} \int\_{0}^{2\pi} \left[ \left( \frac{\partial u}{\partial x} + \frac{\partial v}{R \partial \theta} + \frac{w}{R} \right)^2 - 2(1 - \nu) \frac{\partial u}{\partial x} \left( \frac{\partial v}{R \partial \theta} + \frac{w}{R} \right) + \frac{(1 - \nu)}{2} \left( \frac{\partial v}{\partial x} + \frac{\partial u}{R \partial \theta} \right)^2 + \cdots \right] \tag{3}$$

$$\kappa \left[ \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{R^2 \partial \theta^2} \right)^2 - 2(1 - \nu) \left( \frac{\partial^2 w}{\partial x^2} \frac{\partial^2 w}{R^2 \partial \theta^2} - \left( \frac{\partial^2 w}{R \partial x \partial \theta} \right)^2 \right) \right] \text{Rd}xd\theta \tag{4}$$

$$\begin{split} + 4l \frac{2\pi}{\ell} \left[ k\_1 w^2 + k\_2 v^2 + k\_3 w^2 + k\_4 \left( \text{Cov/fcn} \right)^2 \right]\_{x=0} \text{Rd}d\theta\\ + 4l \frac{2\pi}{\ell} \left[ \left( k\_5 w^2 + k\_6 v^2 + k\_7 w^2 + k\_8 \left( \text{Cov/fcn} \right)^2 \right) \right]\_{x=L} \text{Rd}d\theta \end{split} \tag{19}$$

and the total kinetic energy is calcul ated from

$$T = \frac{1}{2} \int\_0^{1.2} \rho h \left[ \left( \left< \mathbf{\hat{u}} \mu \middle| \mathbf{\hat{\varepsilon}} t \right>^2 + \left( \mathbf{\hat{c}} \mathbf{w} \middle| \mathbf{\hat{\varepsilon}} t \right)^2 + \left( \mathbf{\hat{c}} \mathbf{w} \middle| \mathbf{\hat{\varepsilon}} t \right)^2 \right] R \mathbf{d} \mathbf{x} d\theta \tag{9}$$

By minimizing the Lagrangian *L=V-T* against all the unknown expansion coefficients, a final system of linear algebraic equations can be derived

$$
\begin{bmatrix}
\mathbf{A}^{\omega} & \mathbf{A}^{\omega \theta} & \mathbf{A}^{\omega} \\
\mathbf{A}^{\theta \top} & \mathbf{A}^{\theta \theta} & \mathbf{A}^{\theta \prime} \\
\mathbf{A}^{\omega \top} & \mathbf{A}^{\theta \top} & \mathbf{A}^{\prime \prime}
\end{bmatrix}
\begin{bmatrix}
\overline{\mathbf{a}} \\
\overline{\mathbf{b}} \\
\overline{\mathbf{c}}
\end{bmatrix} - \alpha^{2} \begin{bmatrix}
\mathbf{M}^{\omega} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{M}^{\theta \theta} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{M}^{\prime \prime}
\end{bmatrix} \begin{bmatrix}
\overline{\mathbf{a}} \\
\overline{\mathbf{b}} \\
\overline{\mathbf{c}}
\end{bmatrix} = \mathbf{0} \tag{20}
$$

'

 b

0 ' ' ' '

 a

<sup>2</sup> ( ) cos *L m T cw m x x dx*

l

*m*

4 66

4 127 2 0 945 302400 4725 *L LL*


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 condi‐ tions in a weak sense via the use of the Rayleigh-Ritz procedure. To overcome this problem,

(,) cos ( ) cos (a)

æ ö ç ÷ è ø æ ö ç ÷ è ø æ ö ç ÷ è ø

*u x A px n*

= +

= +

= +

*v x B px n*

*w x C px n*

*mn m n*

l

*x*

*x*

*x*

*u*

 q

 q

> q

*v*

*w*

(,) cos ( ) sin (b)

l

*mn m n*

(,) cos ( ) cos (c)

*mn m n*

l

(b) (c)

(1 ) (a)

 d = + = Z =

**P Q Q Q**X

*m m mm m m c mm mT m*

*S*a

e

ab a b

0

1

+ <sup>ì</sup> <sup>=</sup> <sup>ï</sup> <sup>ï</sup> <sup>=</sup> íì ü ï ï -- - ïí ý <sup>¹</sup> <sup>ï</sup>

2 2

7 2 .

*L L sym*

é ù ê ú

4 31 2 945 7560 4725

*LL L*

2

*L*

 l

2 45

*<sup>T</sup> m m*

*L m*

 0 0 0 0 , for 0 <sup>2</sup> (a) 1 ( 1) 1 ( 1) , for 0

0 44 6

31 756


4


*L*

*L* =

*x x xx x x x x dx*

 l

<sup>2</sup> (cos cos cos ( ) cos ( ) ( ) ( ) )

a ' '

 b

(26)

215

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816

(28)

(29)

<sup>z</sup> <sup>z</sup> <sup>z</sup> <sup>z</sup> (25)

ò **<sup>P</sup>** <sup>z</sup> (27)

(b)

*T m T m mT T m*

**Q QQ Q**

ba

'

*T m T m*

**Q Q**

*x x x x dx*

 l

' '

*mm m m*

= +++

 l

<sup>2</sup> (cos ( ) )(cos ( ) )

z z

a

' ''

*m m mm mm*

*SSZ*

{ }

*L T*

ò <sup>X</sup> <sup>z</sup> <sup>z</sup>

= =

*L x x dx*

lll

224 4

*mmm m*

îï ï î þ

*T*

180 45 2 () ()

**2.2. A strong form of solution based on Flügge's equations**

the displacement expressions, Eqs. (8), will now be generalized to

0 0

*n m*

¥ ¥ = = ¥ ¥ = = ¥ ¥ = =

å å

q

q

q

*n m*

å å

0 0

0 0

*n m*

å å

=+ +

*m m*

0

ò

*L*

*L*

*L*

0

ò

*L*

e d

where

and

*m mm*

'

a b ab

= +++

ll

l

where

$$\begin{aligned} \overline{\mathbf{a}} &= \left\{ \left. a\_{00}, a\_{01}, \dots, a\_{mn}, \dots \right\}^{\mathsf{T}} \right. \\ \overline{\mathbf{b}} &= \left\{ \left. b\_{01}, b\_{02}, \dots, b\_{mn}, \dots \right\}^{\mathsf{T}} \right. \\ \left. \overline{\mathbf{c}} = \left\{ \left. c\_{00}, c\_{01}, \dots, c\_{mn}, \dots \right\}^{\mathsf{T}} \right. \\ & \qquad \text{(c)} \end{aligned} \tag{21}$$

$$
\boldsymbol{\Lambda}\_{nm,w^{\*}n^{\*}}^{\;\;\ast} = \delta\_{nn^{\*}} \boldsymbol{I}\_{uu,11}^{\;\!\ast} + \frac{(1-\mu)n^{2}}{2\mathcal{R}^{2}} \boldsymbol{I}\_{uu,00}^{\;\ast} + \frac{2}{L} \hat{k}\_{1} \boldsymbol{\phi}\_{u}^{\;\ast}(0) \boldsymbol{\phi}\_{u}^{\;\ast}(0) + \frac{2}{L} \hat{k}\_{5} \boldsymbol{\phi}\_{u}^{\;\ast}(L) \boldsymbol{\phi}\_{u}^{\;\ast}(L) \tag{a}
$$

$$\Lambda\_{nm,m'n'}^{s\theta} = \mathcal{S}\_{nn'} \{ \frac{\mu n}{\mathcal{R}} I\_{uv,10}^{nm'} - \frac{(1-\mu)\mu}{\mathcal{Z}\mathcal{R}} I\_{uv,01}^{nm'} \} \tag{9}$$

$$\Lambda\_{mn,m'n'}^{sr} = \delta\_{mn'} \frac{\mu}{R} I\_{nm',10}^{mm'} \tag{c}$$

$$\Lambda\_{mn}^{n'} = \dots, \quad \Lambda\_{mn}^{n'} = \dots, \quad \mathcal{T} \sim \dots, \quad \mathcal{T} \sim \dots \dots, \quad \mathcal{T} \sim \dots, \dots, \dots, \dots, \dots, \tag{5}$$

$$
\Lambda\_{mn,m'n'}^{\theta\theta} = \delta\_{nn'} \frac{n^2}{R^2} I\_{vv,00}^{mm'} + \frac{(1-\mu)}{2} I\_{vv,11}^{mm'} + \frac{2}{L} \hat{k}\_2 \rho\_v^m(0) \rho\_v^{m'}(0) + \frac{2}{L} \hat{k}\_6 \rho\_v^{m'}(L) \rho\_v^{m'}(L) \text{ (d)}\tag{22}
$$

$$
\Lambda\_{mn,m'n'}^{\theta\sigma} = \delta\_{nn'} \frac{n}{\mathbf{p}^2} I\_{vv,00}^{mm'} \tag{23}
$$

$$\begin{split} & \mathbb{R}^{2} \\ \Lambda^{0\theta}\_{uu,w^{i}u^{i}} &= \delta\_{ww^{i}} \{ \frac{1}{R^{2}} I^{uu^{i}}\_{uw,00} + \kappa [I^{uu^{i}}\_{uw,22} + \frac{n^{4}}{R^{4}} I^{uu^{i}}\_{uw,00} + 2(1-\mu) \frac{n^{2}}{R^{2}} I^{uu^{i}}\_{uw,11} \\ & \qquad \quad - \frac{\mu n^{2}}{R^{2}} (I^{uu^{i}}\_{uw,02} + I^{uu^{i}}\_{uw,20}) \} + \frac{2}{L} \hat{k}\_{3} \phi^{m}\_{w}(0) \phi^{m}\_{w}{}^{m}(0) + \frac{2}{L} \hat{k}\_{7} \phi^{m}\_{w}(L) \phi^{m}\_{w}(L) + \\ & \qquad \quad + \frac{2}{L} \hat{k}\_{4} \frac{\partial \phi^{m}\_{w}(0)}{\partial \mathbf{x}} \frac{\partial \phi^{m}\_{w}(0)}{\partial \mathbf{x}} + \frac{2}{L} \hat{k}\_{8} \frac{\partial \phi^{m}\_{w}(L)}{\partial \mathbf{x}} \frac{\partial \phi^{m}\_{w}(L)}{\partial \mathbf{x}} \end{split} \tag{f}$$

$$\begin{aligned} \mathbf{M}\_{mn,m'n'}^{ss} &= \boldsymbol{\mathcal{S}}\_{nn'} \rho h I\_{nn',00}^{mm'} & \quad \text{(a)}\\ \mathbf{M}\_{mn,m'n'}^{\theta\theta} &= \boldsymbol{\mathcal{S}}\_{nn'} \rho h I\_{\nu\phi,00}^{mm'} & \quad \text{(b)}\\ \mathbf{M}\_{mn,m'n'}^{rr} &= \boldsymbol{\mathcal{S}}\_{nn'} \rho h I\_{nn',00}^{mm'} & \quad \text{(c)} \end{aligned} \tag{23}$$

and

$$I\_{\alpha\beta\ldots\eta}^{mm'} = 2\oint \mathcal{L} \int\_0^\mathcal{L} \frac{\partial^p \phi\_\alpha^m}{\partial \mathbf{x}^p} \frac{\partial^q \phi\_\beta^{m'}}{\partial \mathbf{x}^q} d\mathbf{x} \quad (\alpha, \beta = u, v, w) \tag{24}$$

The integrals in Eq. (23) can be calculated analytically; for instance

$$I\_{\alpha\beta,00}^{\;\,\mu m^{\ast}} = \frac{\mathcal{D}}{L} \int\_{0}^{L} \rho\_{\alpha}^{\;\,\mu m} \rho\_{\beta}^{\;\,\,\mu m^{\ast}} \,d\alpha$$

#### Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 215

$$\begin{split} &= \frac{2}{L} \Big[ \hat{\mathbf{c}} (\cos \boldsymbol{\lambda}\_{m} \mathbf{x} + \mathbf{Q}(\mathbf{x})^{T} \tilde{\mathbf{Q}}\_{a}^{m}) (\cos \boldsymbol{\lambda}\_{m} \mathbf{x} + \mathbf{Q}(\mathbf{x})^{T} \tilde{\mathbf{Q}}\_{\beta}^{m}) \, \mathrm{d}x \\ &= \frac{2}{L} \Big[ (\cos \boldsymbol{\lambda}\_{m} \mathbf{x} \cos \boldsymbol{\lambda}\_{m} \mathbf{x} + \cos \boldsymbol{\lambda}\_{m} \mathbf{x} \mathbf{Q}(\mathbf{x})^{T} \tilde{\mathbf{Q}}\_{a}^{m} + \cos \boldsymbol{\lambda}\_{m} \mathbf{x} \mathbf{Q}(\mathbf{x})^{T} \tilde{\mathbf{Q}}\_{\beta}^{m'} + \tilde{\mathbf{Q}}\_{a}^{mT} \mathbf{Q}(\mathbf{x}) \mathbf{Q}(\mathbf{x})^{T} \tilde{\mathbf{Q}}\_{\beta}^{m'} \, \mathrm{d}x \\ &= \boldsymbol{\kappa}\_{m} \boldsymbol{\delta}\_{m m'} + \boldsymbol{S}\_{a}^{m'm} + \boldsymbol{S}\_{\beta}^{m m'} + \boldsymbol{Z}\_{a\beta}^{m m'} \end{split} \tag{25}$$

where

T 2

*ss s sr ss*

*sr r rr rr*

{ } { } { }

*aa a bb b cc c*

00 01

= = =

, ' ' ' ,11 2 ,00 1 5 ' '

 m

m

*ss mm mm m m m m mn m n nn uu uu u u u u*


, ' ' ' ,00 ,22 2 42 ,00 ,11

*mn m n nn ww ww ww ww*

4 8

**M M M**

2 2 (0) (0) ( ) ( ) ˆ ˆ }

<sup>1</sup> { [ 2(1 )

, ' ' ' ,10 ,01 '

*n n I I R R*

m


*s mm mm mn m n nn uv uv*

2 '

*I R n I R*

*n I R*

*R*

m

m


m d

(1 [

'

2 '

dk

*mm ww ww*

*<sup>n</sup> I I*

2 ,02 ,2

jj

'

ab

L = + + +-

*mm*

, ' ' ' ,10

*sr mm mn m n nn uw*

d

d

q

qq

q

qq

and

*<sup>I</sup>αβ*,00 *mm*' <sup>=</sup> <sup>2</sup>

*<sup>L</sup> ∫* 0

*L φα <sup>m</sup>φβ m*' *dx*

L =

L =

*mn m n nn vv*

*r mm mn m n nn vw*

d

(

d

, ' ' ' ,00 2

, ' ' ' ,00 2

**a b c**

01 02

00 01

w

é ùé ù ì ü ì ü ê úê ú ï ï ï ï í ý - = í ý ï ï ï ï î þ î þ ë ûë û

qq

**a a M 00 b 0M 0 b c c 0 0M**

T

 , ,..., ,.. , (a) , ,..., ,... , (b) , ,..., ,... , (c)

*mn mn mn*

<sup>2</sup> ' '' '

jj

*<sup>n</sup> I Ik kL L R L L*

4 2 '' ' '

*mm mm mm mm*


 jj

, ' ' ' ,00

*ss mm mn m n nn uu*

d r

= = =

*mn m n nn vv rr mm mn m n nn ww*

d r

d r

, ' ' ' ,00

, ' ' ' ,00

*n n II I I R RR*

> *mm m m ww w w*

*L L k k L x xL x x*

¶¶ ¶¶ + + ¶¶ ¶¶

qq

*<sup>p</sup> <sup>m</sup> <sup>q</sup> <sup>m</sup> <sup>L</sup> mm*

The integrals in Eq. (23) can be calculated analytically; for instance

f f T

T

(1 ) 2 <sup>2</sup> ˆ ˆ [ (0) (0) ( ) ( )] (a) <sup>2</sup>

 jj

> jj

2 2 ˆ ˆ )] (0) (0) ( ) ( ) (f)

 jj

(a) (b) (c)

(1 ) [ ] (b) <sup>2</sup>

,11 2 6

*mm m m m m*

'

'

*mm*

'

a b

¶ ¶ <sup>=</sup> <sup>=</sup> ¶ ¶ <sup>ò</sup> (24)

, <sup>0</sup> <sup>2</sup> ( , , , )

*pq p q I L dx u v w x x* a b

*hI hI hI*

'

*L L*

jj

jj

+ +

)2 2 ˆ ˆ (0) (0) ( ) ( )] (d) <sup>2</sup>

 m

*ww w w*

*k kL L*

++ +

*I k kL L L L*

*mm m m m m vv v v v v*

0

(20)

(21)

(22)

(23)

(c)

(e)

T T

q qq q

214 Advances in Vibration Engineering and Structural Dynamics

where

*s r*

L L L L L L L L L

q

q

$$\begin{aligned} \mathcal{a}\_{m} &= (1 + \delta\_{m0}) & \quad \text{(a)}\\ \mathcal{S}\_{a}^{m'm} &= \mathbf{P}\_{c}^{m'} \overline{\mathbf{Q}}\_{a}^{m} & \quad \text{(b)}\\ \mathbf{Z}\_{a\boldsymbol{\rho}}^{m\boldsymbol{m}'} &= \overline{\mathbf{Q}}\_{a}^{m'} \mathbf{B} \overline{\mathbf{Q}}\_{\boldsymbol{\rho}}^{\boldsymbol{m}'} & \text{(c)} \end{aligned} \tag{26}$$

and

$$\mathbf{P}\_c^m = \frac{2}{L} \int\_0^L \mathbf{f}\_m(x)^T \cos \lambda\_m x \, dx \tag{27}$$

$$\begin{aligned} &=\frac{2}{L}\left\{ \left< \frac{0}{\lambda\_m^4} \quad \frac{(-1)^m}{\lambda\_m^2} \quad \frac{1}{\lambda\_m} \quad \frac{(-1)^{m+1}}{\lambda\_m} \right>^T, \qquad \text{for } m \neq 0 \right. \\ &\left. \left( \frac{2L^2}{45} \right) \left< \mathbf{E}(\mathbf{x}) \right> \right| \quad \text{for } m \neq 0 \right. \\ &\left. \left( \mathbf{E} \right)^L \mathbf{E}(\mathbf{x})^\dagger \mathbf{Q}(\mathbf{x}) dx = \begin{bmatrix} 2L^2 \\ 45 \\ \frac{7L^2}{180} & \frac{2L^2}{45} \end{bmatrix} \begin{bmatrix} 28 \\ 56 \\ -42 \end{bmatrix} \tag{28} \\ &= \begin{array}{c} -4L^4 & -\frac{31L^4}{7560} & \frac{2L^6}{4725} \\ -\frac{41L^4}{945} & -\frac{127L^6}{7560} & \frac{2L^6}{4725} \\ -\frac{31L^4}{7560} & -\frac{41L^4}{945} & \frac{127L^6}{92400} & \frac{2L^6}{4725} \end{bmatrix} \end{aligned} \tag{39}$$

#### **2.2. A strong form of solution based on Flügge's equations**

0 0

= =

*n m*

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 condi‐ tions in a weak sense via the use of the Rayleigh-Ritz procedure. To overcome this problem, the displacement expressions, Eqs. (8), will now be generalized to

$$\begin{aligned} u(\mathbf{x},\theta) &= \sum\_{n=0}^{\alpha} \left( \sum\_{m=0}^{\alpha} A\_{nm} \cos \hat{\mathcal{A}}\_{n} \mathcal{X} + p\_{n}^{\mu}(\mathbf{x}) \right) \cos n\theta \quad \text{(a)}\\ v(\mathbf{x},\theta) &= \sum\_{n=0}^{\alpha} \left( \sum\_{m=0}^{\alpha} B\_{mn} \cos \hat{\mathcal{A}}\_{n} \mathcal{X} + p\_{n}^{\nu}(\mathbf{x}) \right) \sin n\theta \quad \text{(b)}\\ w(\mathbf{x},\theta) &= \sum\_{n=0}^{\alpha} \left( \sum\_{m=0}^{\alpha} C\_{mn} \cos \hat{\mathcal{A}}\_{n} \mathcal{X} + p\_{n}^{\nu}(\mathbf{x}) \right) \cos n\theta \quad \text{(c)}\end{aligned} \tag{29}$$

which represent a 2-D version of the improved Fourier series expansions, Eqs. (8).

To demonstrate the flexibility in choosing the auxiliary functions*pn <sup>u</sup>*(*x*), *pn v* (*x*)and*pn <sup>w</sup>*(*x*), an alternative set is used below:

$$\begin{cases} p\_n^{\mu}(\mathbf{x}) = \mathbf{A}\_n^{\mu}\mathbf{a}(\mathbf{x}) & \text{(a)}\\ p\_n^{\nu}(\mathbf{x}) = \mathbf{A}\_n^{\nu}\mathbf{a}(\mathbf{x}) & \text{(b)}\\ p\_n^{w}(\mathbf{x}) = \mathbf{A}\_n^{w}\mathbf{f}(\mathbf{x}) & \text{(c)}\end{cases} \tag{30}$$

( )

*m*

l g

s s pg

p

3 3

 s

p

*mn mn m mn*

2 2

*<sup>n</sup> <sup>k</sup> A C*

*n n lk a b c e fg lR lR R R <sup>D</sup>*


2 3

3

*mn mn*

n

 p p

2 3 2

2 3 2

*n n n n*

 g

> p

*m mm*

sg

s

*m*

p l

p s

s

as

2 2 0 0

pl

*R R*

æ ö =- - + ç ÷

*m m*

p

¥ ¥ = =

å å

1 3 7n 4 n 4 3 3

æ ö - <sup>=</sup> -- + ç ÷ è ø

s

¥ ¥¥ = ==

å åå

*KR R*

5 2 0 00


7 2 2

2 3 2 8 2 3 2 <sup>2</sup> <sup>2</sup>

*mn m mn*

 s

è ø

2 4 (3 ) 7 4 (2 ) 2 3

*n n lk l k ab d e f lR lR R R <sup>D</sup>*

2 2

¥ ¥ = =

å å

<sup>3</sup> 0 0

*R D R*

cos( ) cos( )

p

*nn n n n*

*m m*

s

*n n n n*

æ öæ ö + + - ++ ç ÷ç ÷ è øè ø

*n n m B m C*

*<sup>l</sup> <sup>k</sup> l l b ef g Rl D R R*


<sup>2</sup> <sup>2</sup>

 s l

è ø

*l l ln <sup>k</sup> <sup>a</sup> f he R l R R <sup>D</sup>*

 s

*nn n n nn*

4 2 (3 ) (2 ) 7

s

<sup>3</sup> 0 0

*R D R*

1 3 7n 4 4 3 3

*n n B C R R*

p

2

3 3

cos( ) cos( ) cos( )

*<sup>k</sup> <sup>n</sup> m A m B R mC*

p

> s

*<sup>m</sup> m n <sup>k</sup> <sup>A</sup> m C*

 p *n*

*g*

<sup>1</sup> (1 ) (1 ) <sup>1</sup> cos( ) cos( ) (f) 2 22

 s

<sup>3</sup> (g) cos( ) (1 )cos( ) cos( ) <sup>2</sup>

s

 p p

> p

Equations (31) represent a set of constraint conditions between the unknown (boundary) constants, *an*, *bn*,..., *gn* and *hn*, and the Fourier expansion coefficients *Amn*, *Bmn*, and *Cmn* ( *m*, *n* = 0, 1, 2,... ). The constraint equations (31a-h) can be rewritten more concisely, in matrix form,

*n n <sup>k</sup> d f m A m B R K*

*n n mn mn*

*mn mn m mn*

 s

 s

æ ö ç ÷ è ø æ ö <sup>=</sup> <sup>+</sup> + + ç ÷ è ø

p

*n n mn mn*

*n n <sup>k</sup> c e AB*

 s

 g

> p

 s l g

> 2 0 0

<sup>3</sup> (c) (1 )

3 3 3

p

*l k <sup>h</sup> D*

6

*n n*

+

*g h <sup>D</sup>*

p

 p

> p

**Ly Sx** = (34)

0 0

7 3

lg

*m m*

¥ ¥ = =

å å

*mn mn*

p

p

2 3

7

*n*

(d)

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 217

(e)

(33)

(h)

4

3

4

p

<sup>1</sup> (1 ) (1 ) <sup>1</sup> (b) 2 22

*R K*

*m m*

¥ ¥ = =

å å

3 4 <sup>3</sup> (a)

1 2 0 00

æ ö = - -+ ç ÷ è ø

sg


*<sup>k</sup> <sup>n</sup> A B R C*

*n nn*

æ ö æ ö -+ - + ç ÷ ç ÷ è ø è ø

*l R l lR a fh R l <sup>R</sup>*

73 4

*m mm*

s

2 2

æ ö - =- + ç ÷ è ø

2

*m m*

s

¥ ¥ = =

å å

2 2 0 0

æ ö = ++ ç ÷

*m m*

pg

( )

g

s

s

p n

¥ ¥ = =

å å

*n n*

37 4 4 3 3

*R l lR l b e l R R*

 s

æ ö -- + - + ç ÷ è ø

p p

*mn m mn*

æ öæ ö - + + ++ - ç ÷ç ÷ è øè ø

*KR R*

¥ ¥¥ = ==

å åå

here*Λ<sup>n</sup> <sup>u</sup>* <sup>=</sup> *an bn* , *Λ<sup>n</sup> <sup>v</sup>* <sup>=</sup> *cn dn* , *Λ<sup>n</sup> <sup>w</sup>* <sup>=</sup> *en <sup>f</sup> <sup>n</sup> gn <sup>h</sup> <sup>n</sup>* with *an*, *bn*,..., *gn* and *hn* being the unknown coef‐ ficients to be determined; *α*(*x*)={*α*1(*x*) *α*2(*x*)}Tand *β*(*x*)={*β*1(*x*) *β*2(*x*) *β*3(*x*) *β*4(*x*)}T and with their elements being defined as

$$\begin{aligned} \alpha\_1(\mathbf{x}) &= \mathbf{x}(\frac{\mathbf{x}}{l} - 1)^2 & \text{(a)}\\ \alpha\_2(\mathbf{x}) &= \frac{\mathbf{x}^2}{l}(\frac{\mathbf{x}}{l} - 1) & \text{(b)} \end{aligned} \tag{31}$$

and

$$\begin{aligned} \beta\_1(\mathbf{x}) &= \frac{9l}{4\pi} \sin(\frac{\pi x}{2l}) - \frac{l}{12\pi} \sin(\frac{3\pi x}{2l}) & \text{(a)}\\ \beta\_2(\mathbf{x}) &= -\frac{9l}{4\pi} \cos(\frac{\pi x}{2l}) - \frac{l}{12\pi} \cos(\frac{3\pi x}{2l}) & \text{(b)}\\ \beta\_3(\mathbf{x}) &= \frac{l^3}{\pi^3} \sin(\frac{\pi x}{2l}) - \frac{l^3}{3\pi^3} \sin(\frac{3\pi x}{2l}) & \text{(c)}\\ \beta\_4(\mathbf{x}) &= -\frac{l^3}{\pi^3} \cos(\frac{\pi x}{2l}) - \frac{l^3}{3\pi^3} \cos(\frac{3\pi x}{2l}) & \text{(d)} \end{aligned} \tag{32}$$

In Eqs. (27), the sums of x-related terms are here understood as the series expansions in x-direction, 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 point-wise basis; that is, the solution can be found in strong form, as described below.

 p

Substituting Eqs. (6) and (27) into (4) and (5) will lead to

4 3 3

p

(33)

$$\begin{aligned} &a\_n - \left(\frac{7\sigma l}{3\pi R} + \frac{3\pi R\gamma}{4l}\right) f\_n - \left(\frac{4\sigma l^3}{3\pi^3 R} + \frac{l\gamma R}{\pi}\right) h\_n\\ &= \frac{k\_1}{K} \sum\_{m=0}^{\psi} A\_{mn} - \frac{\sigma n}{R} \sum\_{m=0}^{\psi} B\_{mn} - \sum\_{m=0}^{\psi} \left(\frac{\sigma}{R} + \lambda\_m^{-2}\gamma R\right) \mathbb{C}\_{mn} \end{aligned} \tag{a}$$

which represent a 2-D version of the improved Fourier series expansions, Eqs. (8).

( ) ( ) (a) ( ) ( ) (b) ( ) ( ) (c)

ficients to be determined; *α*(*x*)={*α*1(*x*) *α*2(*x*)}Tand *β*(*x*)={*β*1(*x*) *β*2(*x*) *β*3(*x*) *β*4(*x*)}T and with

2

2

= -

= -

*x x <sup>x</sup> l l*

*<sup>x</sup> x x <sup>l</sup>*

( ) ( 1) (a)

( ) ( 1) (b)

9 3 ( ) sin( ) sin( ) (a) 4 2 12 2

 p

 p  p

 p

In Eqs. (27), the sums of *x*-related terms are here understood as the series expansions in *x*direction, 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 point-wise basis; that is, the solution can be

 p

 p

9 3 ( ) cos( ) cos( ) (b) 4 2 12 2

 p

<sup>3</sup> ( ) sin( ) sin( ) (c) 2 2 <sup>3</sup>

<sup>3</sup> ( ) cos( ) cos( ) (d) 2 2 <sup>3</sup>

 p

L a L a L b

*u u n n v v n n w w n n*

1

a

2

3 3 3 3 3 3 3 4 3 3

p

*l xl x <sup>x</sup> l l*

p

*l xl x <sup>x</sup> l l*

*l xl x <sup>x</sup> l l*

p

p

= -

=- -

= -

=- -

p

p

p

p

*l xl x <sup>x</sup> l l*

a

*px x px x px x*

= = = *<sup>u</sup>*(*x*), *pn v*

*<sup>w</sup>* <sup>=</sup> *en <sup>f</sup> <sup>n</sup> gn <sup>h</sup> <sup>n</sup>* with *an*, *bn*,..., *gn* and *hn* being the unknown coef‐

(*x*)and*pn*

*<sup>w</sup>*(*x*), an

(30)

(31)

(32)

To demonstrate the flexibility in choosing the auxiliary functions*pn*

alternative set is used below:

216 Advances in Vibration Engineering and Structural Dynamics

*<sup>u</sup>* <sup>=</sup> *an bn* , *Λ<sup>n</sup>*

their elements being defined as

*<sup>v</sup>* <sup>=</sup> *cn dn* , *Λ<sup>n</sup>*

1

b

2

b

b

b

found in strong form, as described below.

Substituting Eqs. (6) and (27) into (4) and (5) will lead to

here*Λ<sup>n</sup>*

and

$$\frac{1-\sigma}{2}(1+\gamma)c\_n + \frac{(1-\sigma)\gamma n}{2}e\_n = \frac{(1-\sigma)n}{2R}\sum\_{m=0}^{\sigma} A\_{nm} + \frac{k\_2}{K}\sum\_{m=0}^{\sigma} B\_{nm} \tag{b}$$

$$\begin{aligned} & -\frac{4}{lR}a\_u - \frac{2}{lR}b\_u + \frac{(3-\sigma)\mu}{2R^2}c\_u + \frac{(2-\sigma)\mu^2}{R^2}c\_u + \frac{7lk\_3}{3\pi D}f\_u - g\_u + \frac{4l^3k\_3}{3\pi^3 D}h\_n\\ &= \sum\_{m=0}^{\phi} \left(\frac{\lambda\_m}{R} - \frac{(1-\sigma)\mu^2}{2R^3}\right)A\_{mu} + \frac{k\_3}{D}\sum\_{m=0}^{\phi}C\_{mu} \end{aligned} \tag{c}$$

$$\begin{aligned} & -\frac{1}{R}a\_{nl} + \left(\frac{3\pi}{4l} + \frac{7\nu l\ln^2}{3\pi R^2}\right)f\_n + \left(\frac{l}{\pi} + \frac{4\nu l^3 n^2}{3\pi^3 R^2}\right)h\_n - \frac{k\_4}{D}c\_n\\ & = \frac{\sigma n}{R^2} \sum\_{m=0}^{\psi} B\_{mn} + \sum\_{m=0}^{\psi} \left(\lambda\_m^{-2} + \frac{\sigma n^2}{R^2}\right) \mathbb{C}\_{mn} \end{aligned} \tag{d}$$

$$\begin{aligned} &-b\_{\text{n}} - \left(\frac{3\pi\gamma\text{R}}{4l} + \frac{7\sigma l}{3\pi\text{R}}\right)e\_{\text{n}} - \left(\frac{l\gamma\text{R}}{\pi} + \frac{4\sigma l^3}{3\pi^3\text{R}}\right)g\_{\text{n}}\\ &= \frac{k\_5}{K}\sum\_{m=0}^{\infty}\cos(m\pi)A\_{mn} + \frac{m\sigma}{R}\sum\_{m=0}^{\infty}\cos(m\pi)B\_{mn} + \sum\_{m=0}^{\infty}\left(\frac{\sigma}{R} + \lambda\_m^{-2}\gamma R\right)\cos(m\pi)\mathcal{C}\_{mn} \end{aligned} \tag{e}$$

$$-\frac{1-\sigma}{2}(1+\gamma)d\_u - \frac{(1-\sigma)n\gamma}{2}f\_u = -\frac{(1-\sigma)n}{2R}\sum\_{m=0}^{\sigma}\cos(m\pi)A\_{mn} + \frac{k\_{\tilde{\mathbf{g}}}}{K}\sum\_{m=0}^{\sigma}\cos(m\pi)B\_{mn} \tag{f}$$

$$\begin{split} & -\frac{2}{lR}a\_{n} - \frac{4}{lR}b\_{n} - \frac{(3-\sigma)n}{2R^{2}}d\_{n} - \frac{7lk\_{7}}{3\pi D}c\_{n} - \frac{(2-\sigma)n^{2}}{R^{2}}f\_{n} - \frac{4l^{3}k\_{7}}{3\pi^{3}D}g\_{n} + h\_{n} \\ & = -\sum\_{m=0}^{\omega} \left( \frac{\cos(m\pi)\lambda\_{m}^{2}}{R} - \frac{(1-\sigma)\cos(m\pi)n^{2}}{2R^{3}} \right) \mathbf{A}\_{nm} + \frac{k\_{7}}{D} \sum\_{m=0}^{\omega} \cos(m\pi)\mathbf{C}\_{nm} \\ & \frac{1}{R}b\_{n} + \left(\frac{3\pi}{4l} + \frac{7\sigma l\ln^{2}}{3\pi R^{2}}\right) \mathbf{c}\_{n} - \frac{k\_{8}}{D}f\_{n} + \left(\frac{l}{\pi} + \frac{4\sigma l^{3}n^{2}}{3\pi^{3}R^{2}}\right) \mathbf{g}\_{n} \\ \end{split} \tag{9}$$

$$\mathbf{E} = -\frac{\sigma n}{R^2} \sum\_{m=0}^{\infty} \cos(m\pi) \mathbf{B}\_{mn} - \sum\_{m=0}^{\infty} \left( \lambda\_m \mathbf{A}\_m^2 + \frac{\sigma n^2}{R^2} \right) \cos(m\pi) \mathbf{C}\_{mn} \tag{\text{h}}$$

Equations (31) represent a set of constraint conditions between the unknown (boundary) constants, *an*, *bn*,..., *gn* and *hn*, and the Fourier expansion coefficients *Amn*, *Bmn*, and *Cmn* ( *m*, *n* = 0, 1, 2,... ). The constraint equations (31a-h) can be rewritten more concisely, in matrix form, as

$$\mathbf{L}\mathbf{y} = \mathbf{S}\mathbf{x} \tag{34}$$

The elements of the coefficient matrices can be readily derived from Eqs. (31); for example, Eq. (31a) implies

$$\left\{\mathbf{L}\_{31}\right\}\_{m,n'} = \mathcal{S}\_{mn'} \tag{\text{a}}$$

$$
\left< \mathbf{L}\_{36} \right>\_{n,n'} = -\left( \frac{7\sigma l}{3\pi R} + \frac{3\pi\eta' R}{4l} \right) \delta\_{nn'} \tag{9}
$$

$$
\left< \mathbf{L}\_{38} \right>\_{u,n'} = -\left( \frac{4\sigma l^3}{3\pi^3 R} + \frac{l\chi R}{\pi} \right) \delta\_{uu'} \tag{3}
$$

$$\begin{aligned} \left\{ \mathbf{L}\_{32} \right\}\_{n,n'} &= \left\{ \mathbf{L}\_{33} \right\}\_{n,n'} = \left\{ \mathbf{L}\_{34} \right\}\_{n,n'} = \left\{ \mathbf{L}\_{35} \right\}\_{n,n'} = \left\{ \mathbf{L}\_{37} \right\}\_{n,n'} = 0 \end{aligned} \tag{d}$$
  $\left\{ \mathbf{S}\_{11} \right\}\_{1,\dots,1} = k\_1 \delta\_{nn'} / K$ 

(35)

2 2 <sup>2</sup>

æ öæ ö - + - + ç ÷ç ÷ - - + - ¢¢ è øè ø

(1 )(1 ) (1 )(1 ) cos ( ) 2 2

*R R*

*x*

(1 )(1 3 ) ( ) ( ) <sup>2</sup> (b)

*v*

2

*u*

L a

*x*

L a

*u*

*n n A x*

L a

s g

(a)

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 219

(37)

(c)

2 2 0 0 <sup>0</sup>

 l *x*

*m m mn n*

*x*

s g

*m m mn n*

*m m m m mn*

(1 ) () () ( ) <sup>2</sup>

æ ö - ç ÷ - - ¢¢¢ ¢ è ø

*<sup>n</sup> x Rx <sup>x</sup> <sup>R</sup>*

cos () 0

*n A nx*

*m m mn*

 l

*v*

L a

*m m*

*m m mn*

 l *x*

g

*<sup>n</sup> R C*

*w*

L b

 g in

l

*m mn*

*x*

*A*

*m m m mn*

 g

 ll

*n x*

*m mn n*

*m m mn n*

l

 g

*x*

(1 ) sin 2

 ll

s g

*u*

L a

*m m mn*

 l

*<sup>n</sup> R C*

b b

*n B nx*

0 0 0 <sup>2</sup> <sup>3</sup>

(1 ) sin ( ) <sup>2</sup>

å å å

<sup>+</sup> æ ö + -ç ÷ <sup>+</sup> ¢ è ø

 l

*m n n*

0 0 0

*<sup>h</sup> A x*

*x*

*m n n*

l

¥ ¥ ¥ = = = ¥ ¥ ¥ = = =

åå å

æ ö <sup>+</sup> ç ÷ + = è ø <sup>+</sup> æ ö + - ç ÷ ¢ è ø

g

*m n n*

s

s g

0 0 0

*n*

*<sup>h</sup> B x*

*x*

*n n x x <sup>R</sup>*

*m n n*

l

¥ ¥ ¥ = = =

åå å

æ ö <sup>+</sup> ç ÷ + = è ø

> sg

*R R*

æ ö - + - ¢¢ ç ÷ è ø

L a a

s g

æ ö - -- + ç ÷ è ø

æ ö - - - ¢¢ ç ÷ è ø

L b b

*R R*

æ ö - + + ç ÷ è ø

l

æ ö - + - + ç ÷ è ø

0 0 0 <sup>2</sup> <sup>2</sup>

(1 )(1 3 ) cos <sup>2</sup>

*<sup>n</sup> <sup>B</sup>*

2

*<sup>n</sup> <sup>x</sup>*

(3 ) cos <sup>2</sup>

l

*x x <sup>R</sup> <sup>n</sup> <sup>C</sup>*

s

æ ö - + ç ÷ - ¢¢ è ø

a a

(3 ) ( ) ( ) <sup>2</sup>

s g

<sup>2</sup> <sup>2</sup>

*R*

2

*n*

(1 ) ( ) () () <sup>2</sup>

*x x Rx*

2 2 2 4 22

2 2 2 2

g

*<sup>n</sup> x xR x*

*n*

1 ( 1) () 2 () ()

cos () 0

*m mn n*

l

1 ( 1) 2 cos

g

æ ö + + <sup>+</sup> ç ÷ - + ¢¢ ¢¢¢¢ è ø

L b b b

l l

*m mn n*

(1 ) <sup>s</sup> <sup>2</sup>

g

2

(3 ) cos <sup>2</sup>

 sg

L a a a

æ ö - +- - ç ÷ ¢ ¢ ¢¢¢ è ø

*n n <sup>B</sup>*

l

(3 ) ( ) ( ) <sup>2</sup>

s g

*n n x x <sup>R</sup>*

æ ö + + <sup>+</sup> ç ÷ + + è ø

0 0 0

*<sup>h</sup> C x*

*x*

*m n n*

l

¥ ¥ ¥ = = =

åå å

æ ö - ç ÷ + = è ø

cos () 0

 g

(1 ) sin ( ) <sup>2</sup>

 l

å å å

*m n n*

¥ ¥ ¥ = = = ¥ ¥ ¥ = = =

å å å

s g

l

*R R*

æ ö - - +- ç ÷ è ø

g

ll

0 0

¥ =

å

0 2

*K*

*n*

w r

*R*

¥ ¥ = = T

å å

L

s

<sup>2</sup> 0 0

<sup>2</sup> 0 0

*R*

s

s

<sup>2</sup> <sup>0</sup> 2

*R n*

2 <sup>2</sup> <sup>0</sup>

æ ö - - + ç ÷ è ø

*m n v n*

¥ = ¥ ¥ = = ¥ <sup>T</sup> =

å


*n*

*m n w n n*

å å

*K*

¥ ¥ = = ¥ <sup>T</sup> = ¥ ¥ = = ¥ <sup>T</sup> = ¥ = =

å å

0 0

<sup>2</sup> 0 0

*R*

<sup>2</sup> <sup>0</sup>

å

*w n n*

¥ <sup>T</sup> =

<sup>2</sup> <sup>0</sup>

<sup>2</sup> <sup>0</sup>

g

*R*

g

*R*

0

å å

å

*m n u n n*

å

¥

å

*m n*

2

w r

å

*K*

0

*m n u n n*

w r

å

*R*

s

l

*w n*

+ ¢

T

L b

s

*R*

s

*m n*

¥ ¥ = =

å å

$$\begin{aligned} \left< \mathbf{S}\_{11} \right>\_{mm,n'} &= k\_1 \delta\_{nn'} / \,\mathrm{K} \\ \left< \mathbf{S}\_{12} \right>\_{mm,n'} &= -\sigma n \delta\_{nn'} / \,\mathrm{R} \\ &\qquad \ldots \end{aligned} \tag{f}$$

$$\left< \left< \mathbf{S}\_{13} \right>\_{nm,u'} = -\delta\_{nm'} \left( \frac{\sigma}{R} + \frac{m^2 \pi^2 \gamma R}{l^2} \right) \tag{9}$$

Other sub-matrices 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)(3*N*+2)+8*N*+6 unknown expansion coeffi‐ cients in the displacement functions. Since Eq. (33) represents a set of 8*N*+6 equations, addi‐ tional (*M*+1)(3*N*+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

$$\frac{\partial \mathcal{N}\_1}{\partial \chi} + \frac{\partial \mathcal{N}\_{21}}{\mathcal{R} \partial \theta} = \rho h \frac{\hat{\sigma}^2 u}{\hat{\sigma} \mathbf{f}^2} \tag{a}$$

$$\frac{\partial \mathcal{N}\_{12}}{\partial \mathbf{x}} + \frac{\partial \mathcal{N}\_2}{R \partial \theta} + \left(\frac{\partial \mathcal{M}\_2}{R^2 \partial \theta} + \frac{\partial \mathcal{M}\_{12}}{R \partial \mathbf{x}}\right) = \rho h \frac{\partial^2 \upsilon}{\partial t^2} \tag{36}$$

$$\frac{\partial^2 M\_1}{\partial \mathbf{x}^2} + \frac{\partial^2 M\_{12}}{R \partial \mathbf{x} \partial \boldsymbol{\theta}} + \frac{\partial^2 M\_{21}}{R \partial \mathbf{x} \partial \boldsymbol{\theta}} + \frac{\partial^2 M\_2}{R^2 \partial \boldsymbol{\theta}^2} - \frac{N\_2}{R} = \rho h \frac{\partial^2 w}{\partial t^2} \tag{c}$$

#### Substituting Eqs. (6) and (37) into Eqs. (34) results in

2 2 <sup>2</sup> 2 2 0 0 <sup>0</sup> 0 0 0 <sup>2</sup> <sup>3</sup> 0 0 (1 )(1 ) (1 )(1 ) cos ( ) 2 2 (1 ) sin ( ) <sup>2</sup> (1 ) sin 2 *u m m mn n m n n v m m mn n m n n m m m m mn m n w n n n A x R R n B nx R <sup>n</sup> R C R R R x x x* s g s g s s s g g s l l l l ll ll ¥ ¥ ¥ = = = ¥ ¥ ¥ = = = ¥ ¥ = = T æ öæ ö - + - + ç ÷ç ÷ - - + - ¢¢ è øè ø <sup>+</sup> æ ö + -ç ÷ <sup>+</sup> ¢ è ø æ ö - - +- ç ÷ è ø + ¢ å å å å å å å å L a L a L b 2 0 2 0 0 0 0 0 0 <sup>2</sup> <sup>2</sup> <sup>2</sup> 0 0 (a) (1 ) () () ( ) <sup>2</sup> cos () 0 (1 ) sin ( ) <sup>2</sup> (1 )(1 3 ) cos <sup>2</sup> *n u m mn n m n n u m m mn n m n n m m mn m n v n <sup>n</sup> x Rx <sup>x</sup> <sup>R</sup> <sup>h</sup> A x K n A nx R <sup>n</sup> <sup>B</sup> R n x x x* s g g w r s s g l l l l l ¥ = ¥ ¥ ¥ = = = ¥ ¥ ¥ = = = ¥ ¥ = = T æ ö - ç ÷ - - ¢¢¢ ¢ è ø æ ö <sup>+</sup> ç ÷ + = è ø <sup>+</sup> æ ö + - ç ÷ ¢ è ø æ ö - + - + ç ÷ è ø å åå å å å å å å b b L a L a L 2 <sup>2</sup> <sup>0</sup> 2 <sup>2</sup> 0 0 <sup>2</sup> <sup>0</sup> 2 0 0 0 <sup>2</sup> <sup>2</sup> (1 )(1 3 ) ( ) ( ) <sup>2</sup> (b) (3 ) cos <sup>2</sup> (3 ) ( ) ( ) <sup>2</sup> cos () 0 (1 ) <sup>s</sup> <sup>2</sup> *n m m mn m n w n n v m mn n m n n m m x x <sup>R</sup> <sup>n</sup> <sup>C</sup> R n n x x <sup>R</sup> <sup>h</sup> B x K R R R <sup>n</sup> <sup>x</sup> x n* s g s g s g w r s sg g l l l l l ¥ = ¥ ¥ = = ¥ <sup>T</sup> = ¥ ¥ ¥ = = = æ ö - + ç ÷ - ¢¢ è ø æ ö - - + ç ÷ è ø æ ö - - - ¢¢ ç ÷ è ø æ ö <sup>+</sup> ç ÷ + = è ø æ ö - -- + ç ÷ è ø å å å å åå å a a L b b L a 0 0 2 0 2 <sup>2</sup> 0 0 <sup>2</sup> <sup>0</sup> 2 2 2 4 22 <sup>2</sup> <sup>0</sup> in (1 ) ( ) () () <sup>2</sup> (3 ) cos <sup>2</sup> (3 ) ( ) ( ) <sup>2</sup> 1 ( 1) 2 cos *m mn m n u n n m m mn m n u n n m m m mn m n A x x Rx R R n n <sup>B</sup> R n n x x <sup>R</sup> <sup>n</sup> R C R x n x n x* s sg g s g s g g g g l l l l ll ¥ ¥ = = ¥ <sup>T</sup> = ¥ ¥ = = ¥ <sup>T</sup> = ¥ = = æ ö - +- - ç ÷ ¢ ¢ ¢¢¢ è ø æ ö - + + ç ÷ è ø æ ö - + - ¢¢ ç ÷ è ø æ ö + + <sup>+</sup> ç ÷ + + è ø å å å å å å å L a a a L a a 0 2 2 2 2 <sup>2</sup> <sup>0</sup> 2 0 0 0 (c) 1 ( 1) () 2 () () cos () 0 *w n n w m mn n m n n <sup>n</sup> x xR x R <sup>h</sup> C x K n x* g g g w r l ¥ ¥ <sup>T</sup> = ¥ ¥ ¥ = = = æ ö + + <sup>+</sup> ç ÷ - + ¢¢ ¢¢¢¢ è ø æ ö - ç ÷ + = è ø å å åå å L b b b L b

The elements of the coefficient matrices can be readily derived from Eqs. (31); for example,

7 3 (b) 3 4

<sup>4</sup> (c) <sup>3</sup>

/ (e) / (f)

(a)

(35)

(36)

0 (d)

(a)

(b)

(c)

Eq. (31a) implies

{ } { }

**L L**

31 36

, ,

218 Advances in Vibration Engineering and Structural Dynamics

*n n nn*

=

d

¢ ¢

{ }

**L**

38

{ } { }

**S S**

11 12

{ }

13

{ } { } { } { } { }

3

*n n nn*

¢ ¢

*k K n R*

d

¢ ¢ ¢ ¢

s d

d

¢ ¢

*n n nn*

æ ö =- + ç ÷ è ø æ ö =- + ç ÷ è ø

¢ ¢

s pg d

p

s

p

*l R R l l lR R*

 g d

p

**LLLLL**

, 2

<sup>æ</sup> =- + <sup>ç</sup> è

s

, 3

1 , ,

= = -

*mn n nn mn n nn*

*mn n nn*

32 33 34 35 37

2 2

 pg

<sup>÷</sup> ç ÷ø

Accordingly, we will turn to the governing differential equations.

2

12 2 2 12

q

r q

2

*N N MM <sup>v</sup> <sup>h</sup> x R R t R x*

¶¶ ¶¶ æ ö ¶ ++ + = ç ÷ ¶¶ ¶ è ø ¶ ¶

q

2 2 22 2 2 2 1 12 21 2 2 2 2 2 2

*M M M MN <sup>w</sup> <sup>h</sup> x Rt Rx Rx R*

 q

¶¶ ¶ ¶ ¶ + + + -= ¶ ¶¶ ¶¶ ¶ ¶

In Flügge's theory, the equations of motion are given as

1 21

Substituting Eqs. (6) and (37) into Eqs. (34) results in

*N N <sup>u</sup> <sup>h</sup> x R t*

¶ ¶ ¶ + = ¶ ¶ ¶

q

*m R R l*

,,,,,

¢¢¢¢¢

=====

**S** (g) ö

Other sub-matrices 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)(3*N*+2)+8*N*+6 unknown expansion coeffi‐ cients in the displacement functions. Since Eq. (33) represents a set of 8*N*+6 equations, addi‐ tional (*M*+1)(3*N*+2) equations are needed to be able to solve for all the unknown coefficients.

2

r

r

q

*nn nn nn nn nn*

(37)

By expanding all non-cosine terms into Fourier cosine series and comparing the like terms, the following matrix equation can be obtained

$$\mathbf{E}\mathbf{x} + \mathbf{F}\mathbf{y} - \frac{\rho h o \rho^2}{K} (\mathbf{P}\mathbf{x} + \mathbf{Q}\mathbf{y}) = 0. \tag{38}$$

where E, F, P and Q are coefficient matrices whose elements are given as:

$$
\left\langle \left( \mathbf{E}\_{11} \right)\_{mm,nn'n'} = -\left( \mathcal{A}\_{in'} + \frac{(1-\sigma)(1+\gamma)n'^2}{\mathfrak{D}\mathcal{R}^2} \right) \delta\_{mm'} \delta\_{nn'} \tag{a}
$$

$$
\left\langle \mathbf{E}\_{12} \right\rangle\_{mn, m'n'} = -\frac{(1+\sigma)m'n\pi}{2lR} \mathcal{X}\_{m}^{m'} \delta\_{mm'} \delta\_{nn'} \tag{9}
$$

$$
\left< \mathbf{E}\_{13} \right>\_{mu, m'n'} = -\left( \frac{\sigma m' \pi}{lR} + \chi \left( R \lambda\_{m'} ^{\;3} - \frac{(1 - \sigma) m' n^2 \pi}{2lR} \right) \right) \mathcal{X}\_{m}^{m'} \delta\_{mm'} \delta\_{nn'} \tag{c}
$$

$$
\begin{pmatrix} \mathbf{E}\_{21} \end{pmatrix}\_{nm, m'n'} = \frac{(1 + \sigma)m'n\pi}{2IR} \mathcal{Z}\_{m}^{m'} \delta\_{mm'} \delta\_{nn'} \tag{d}
$$

$$
\begin{pmatrix} \mathbf{v}^2 & (1 - \mathbf{v})(1 + \Im \mathbf{v})\mathcal{Z} \ \mathbf{J} \end{pmatrix}
$$

$$
\left< \mathbf{E}\_{22} \right>\_{mu, m'n'} = -\left( \frac{n^2}{R^2} + \frac{(1 - \nu)(1 + \Im \gamma)\lambda\_m}{2} \right) \delta\_{mm'} \delta\_{nn'} \tag{e}
$$

$$
\left< \left( \mathbf{E}\_{23} \right)\_{nm, m'n'} = -\left( \frac{n}{R^2} + \frac{(3-\nu)\chi n \lambda\_{nn}^{\prime}}{2} \right) \delta\_{mm'} \delta\_{nn'} \tag{f}
$$

$$
\left< \mathbf{E}\_{31} \right>\_{mm, m'n'} = \left( -\frac{\nu \lambda\_{m'}}{R} + \gamma \left( \frac{(1-\sigma)\lambda\_{m'}m^2}{2R} - R\lambda\_{m'} \right) \right) \mathcal{Z}\_{m}^{m'} \delta\_{mm'} \delta\_{nn'} \tag{\text{g}}
$$

$$
\left\langle \left\{ \mathbf{E}\_{32} \right\} \right\rangle\_{nm, m'n'} = \left( \frac{n}{R^2} + \frac{(3-\sigma)\lambda\_m^2 m\gamma}{2} \right) \delta\_{nm'} \delta\_{nn'} \tag{\text{h}}
$$

$$\left\{ \left( \mathbf{E}\_{33} \right)\_{mm, m'n'} = \left( \frac{1}{R^2} + \mathcal{V} \left| \left( R\mathcal{A}\_m + \frac{n^2 - 1}{R} \right)^2 + \mathcal{Z}\mathcal{A}\_m \right) \right| \delta\_{mm'} \delta\_{nn'} \tag{i}$$

(39)

{ }

2

f d

f d

*n R n R*

(1 )(1 ) (a)

(1 )(1 ) (b)

2 2

(1 ) 9 (1 ) (g) 8 8 42 4

 p

 s

 p

(1 )(1 3 ) (k) <sup>2</sup>

(1 )(1 3 ) (l) <sup>2</sup>

k d

9 9 (3 ) 3 (3 ) (n)

3

k d

(3 ) 3(3 ) (p) 8 8 <sup>3</sup>

<sup>6</sup> (r)

(3 ) (s) <sup>2</sup> (3 ) (t) <sup>2</sup>

g

1 1 1 99 (w) 4 23 4 2

g

g

2 2 22 2 22 2

9 1 <sup>1</sup> 1 9 19 (v) <sup>4</sup> 4 2 48 8 3 2

 p

p

<sup>æ</sup> <sup>æ</sup> ö æ æ öö <sup>ö</sup> æ ö <sup>ç</sup> <sup>ç</sup> ÷ ç ç ÷÷ <sup>÷</sup> ç ÷ <sup>ç</sup> <sup>ç</sup> ÷ ç ç ÷÷ <sup>÷</sup> ç ÷ <sup>ç</sup> <sup>ç</sup> ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> è ø <sup>è</sup> <sup>è</sup> ø è è øø <sup>ø</sup>

p


*m nn*

¢

(1 ) 6 (o) <sup>2</sup>

) 3 (3 ) (m) <sup>32</sup> <sup>12</sup> <sup>32</sup>

¢

 s

9 (1 ) (e) <sup>8</sup> <sup>32</sup> <sup>16</sup>

¢

¢

¢

(h)

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 221

(40)

k d

k d

> k d

 k

*m nn*

(o) <sup>8</sup>

<sup>2</sup> 22 2 3 2 2


 p

> p

 ¢ -

 p k d

> p

8 8 2

æ ö æ ö <sup>ö</sup> æ ö ç ÷ ç ÷ <sup>÷</sup> ç ÷ <sup>÷</sup> è ø è ø è ø <sup>ø</sup>

*R n l l R*

p

 p

2 4 2 2 <sup>1</sup> 9 9 (x) 34 2 *<sup>m</sup> m nn*

*n l ll R R l R* p


p

1 9 (u)

k d

k d

> k d

*m nn*

¢

d

(1 ) (c) <sup>2</sup> (1 ) (d) <sup>2</sup>

2 4 2

 ps


 ps

*m m nn*

*m m nn*

*R n R R l*

9 9 9(1 ) 9 (1 ) (f) <sup>8</sup> <sup>32</sup> <sup>16</sup> <sup>8</sup> <sup>32</sup> <sup>16</sup>

(1 ) (i) <sup>2</sup> (1 ) (j) <sup>2</sup>

2 2 2 2

 s

s

 g

> g

2 2 2 2 2

 g

*R n Rn R RR R l l*

2 4 22 2 2 2 2 2 2 2 2 , 1 3 22 2 2

æ ö <sup>æ</sup> æö æ ö ö æ <sup>ö</sup> ç ÷ <sup>ç</sup> ç÷ ç ÷ ÷ ç <sup>÷</sup> ç ÷ <sup>ç</sup> ÷ ç <sup>÷</sup> è ø <sup>è</sup> èø è ø ø è <sup>ø</sup>

æ ö ç ÷ è ø

*R ln l R ln RR R R l R ln l R ln RR R R*

(1 ) 9 (1 )

kg

 p

1 3 2

p sg

> s

p

s g

 p *l*

g k d

> k d

p sg

*n nl n l l R nl n nl n R R l l nl nl nl n R R*

p

 p


æ ö <sup>æ</sup> ö æ <sup>ö</sup> ç ÷ <sup>ç</sup> ÷ ç <sup>÷</sup> è ø <sup>è</sup> ø è <sup>ø</sup>

æ ö <sup>æ</sup> ö æ <sup>ö</sup> ç ÷ <sup>ç</sup> ÷ ç <sup>÷</sup> è ø <sup>è</sup> ø è <sup>ø</sup>

æ ö æ ö ç ÷ ç ÷ è ø è ø æ ö æ öæ <sup>ö</sup> ç ÷ ç ÷ç <sup>÷</sup> <sup>÷</sup> è ø è øè <sup>ø</sup>

 p

 p

g

2 0 2


*R l*

g

<sup>2</sup> 22 2 , 1 22 2 2


*l Rn l Rl l R R*

9 1 1 1 9 4 3 424

{ } ( ) ( )

1 1 4 2

**<sup>F</sup>** ( ) <sup>2</sup> <sup>2</sup> <sup>2</sup>

 p

*l Rn l Rn*


*n l n l ll R ll R R R l R l R*

pp

k

= ++ - - + + -

 p ¢

kg

 p

2 4 , 22 2 2 2 <sup>2</sup>

k

*<sup>f</sup> <sup>m</sup> m nn mn n*

2 2 2 2 2 2 1 3 , 2 2 2 2

*<sup>g</sup> <sup>m</sup> m nn mn n*


k


*Rl l R l l R R*

<sup>æ</sup> <sup>æ</sup> æ öæ ö ö æ <sup>ö</sup> <sup>ö</sup> æ ö æ ö <sup>ç</sup> <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> ç ÷ ç ÷ <sup>ç</sup> <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> ç ÷ ç ÷ <sup>ç</sup> <sup>ç</sup> ÷ ç <sup>÷</sup> <sup>÷</sup> è ø è ø <sup>è</sup> <sup>è</sup> è øè ø ø è <sup>ø</sup> <sup>ø</sup>

 p

<sup>æ</sup> <sup>æ</sup> æ öæ ö ö æ <sup>ö</sup> <sup>ö</sup> æ ö æ ö <sup>ç</sup> <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> ç ÷ ç ÷ <sup>ç</sup> <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> è ø è ø <sup>è</sup> <sup>è</sup> è øè ø ø è <sup>ø</sup> <sup>ø</sup>

 dd

*m m nn*

 dd

*m m nn*

æ ö ö æ æ öö ç ÷ <sup>ç</sup> ÷ ç ç ÷÷ ç ÷ <sup>ç</sup> ÷ ç <sup>÷</sup> è ø <sup>è</sup> ø è è øø

æ ö <sup>æ</sup> æ öæ ö ö æ <sup>ö</sup> ç ÷ <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> ç ÷ <sup>ç</sup> ÷ ç <sup>÷</sup> è ø <sup>è</sup> è øè ø ø è <sup>ø</sup>

1 3 , 2 2

ss

pp

ss

k

24 2 4 8 8


y d

y d

k

2 4 , 2 2 3 3 <sup>1</sup> , 3 2 3 2

*<sup>f</sup> <sup>m</sup> m nn mn n*


4 32 12 32 (3 ) 3(3 ) 8 3

k

*h mn n m m nn*


*nl nl nl nl R R n R R R l n*


k


k k


k

*<sup>f</sup> <sup>m</sup> m nn mn n*


<sup>æ</sup> æö æ ö ö æ <sup>ö</sup> <sup>ç</sup> ç÷ ç ÷ ÷ ç <sup>÷</sup> <sup>ç</sup> ÷ ç <sup>÷</sup> <sup>è</sup> èø è ø ø è <sup>ø</sup>


2

2 2

 s

> s

*R n R R l*

, 1 1 2



*a mn n m m nn*


s g

s g

2

æ ö ç ÷ è ø æ ö ç ÷ è ø

2

9 9 9(1 ) 8 32 16

> p

*h m mn n*

, 1 1 2



*c mn n m m nn*


s g

æ ö ç ÷ è ø æ ö ç ÷ è ø

s g

s

p

2 , 1 0 2 2

æ ö æ ö ç ÷ ç ÷ è ø è ø

æ ö <sup>ö</sup> ç ÷ ç ÷ è ø è ø æ ö ç ÷ è ø æ ö ç ÷ è ø

*e m mn n*

p

<sup>æ</sup> æ ö æ ö æ ö <sup>ç</sup> ç ÷ ç ÷ ç ÷ <sup>ç</sup> è ø <sup>è</sup> è ø è ø

p

p

<sup>2</sup> <sup>2</sup> <sup>2</sup>

p g

 p


æ ö æ ö ç ÷ ç ÷ ç ÷ çè ø è ø

*n l ll R R l R*

=- + + -

s

p

*a mn n m m nn*

s g

> s g


3 3 , 2 4 3 2 3 2


j

j

 yd

> yd

s g

, 2 2 2

*d mn n m m nn*

p sg

æ ö ç ÷ è ø


p

9 9 (3 4

pp

g

g

*n R n R n R n R nl R*

s j d

s j d

f

f

p

*g m mn n*

(1 ) 2


*R R*

, 1 1 2 , 21 2 2

*c mn n m m nn d mn n m m nn*


*n n*

g

g

g

*n n*

=- +

p

p

p

s sg

s sg

æ

*R*

*R*

p

p

p

{ } ( )

p

, 2 2

p

p


f


f

<sup>æ</sup> æ ö ç ÷ è ø

, 2 1 2

*n R n R*

s j d

s j d

*b mn n m m nn*


, 1 , 2


y

y

<sup>+</sup> <sup>=</sup> <sup>+</sup> <sup>=</sup>

*c mn n m nn d mn n m nn*

, 2

s

2

*l*

s

p

s

, 1 , 2 2


<sup>+</sup> = - <sup>+</sup> = -

*a mn n m nn b mn n m nn*

2

, 2

2

sp


g

g

11

**F**

{ }

11

**F**

{ } { } { }

{ }

13

**F**

{ }

{ <sup>13</sup> }

{ } { } { }

21 21 22

**F F F**

{ }

22

**F**

{ }

{ }

{ } 23

{ }

23

**F**

{ }

31

**F**

{ }

{ } { }

32 32

**F F**

{ }

{ <sup>33</sup> }

33

**F**

33

*h mn n*


**F**


33

**F**

,

*b mn n*


31

**F**

23

**F**

*e mn n*



23

**F**

**F**

**F**

13

**F**

*e mn n*

,

=

*<sup>g</sup> mn n*




12 12 13

**F F F**

(h)

$$
\left\langle \left\{ \mathbf{F}\_{1\rightarrow s} \right\} \right\rangle\_{\text{inv}, \mathbf{x}'} - \left( \left\| \mathbf{v}\_{1\text{in}} - \frac{(1-\sigma)(1+\gamma)\mathbf{u}^2}{2R^2} \phi\_{1\text{in}} \right\| \right) \delta\_{\mathbf{x}\mathbf{u}'} \tag{4}
$$

$$
\delta\_{\mathbf{x}\mathbf{u}'} = \int\_{\mathbf{u}\mathbf{u}'} \dots \quad (1-\sigma)(1+\gamma)\mathbf{u}^2 \Big|\_{\mathbf{x}} \tag{5}
$$

By expanding all non-cosine terms into Fourier cosine series and comparing the like terms,

+ + = 0. 2 ( ) *<sup>h</sup>*

**Ex Fy Px Qy** - (38)

*K* r w

where E, F, P and Q are coefficient matrices whose elements are given as:

<sup>2</sup> <sup>2</sup>

¢ ¢ ¢ ¢ ¢

s g

*m*

*R*

*m*

2 2

¢ ¢ ¢ ¢

*mn m n mm nn*

ng l d d

¢ ¢ ¢ ¢

n

æ ö - + =- + ç ÷ è ø æ ö - =- + ç ÷ è ø

2

*mn m n m m mm nn <sup>n</sup> <sup>R</sup> R R*

¢ ¢ ¢ ¢ æ ö æ ö ç ÷ æ ö - ç ÷ =+ + + ç ÷ è ø è ø è ø

 s l

æ ö æ ö - =- + - ç ÷ ç ÷ è ø è ø

*m*

(3 ) 2

æ ö - = + ç ÷ è ø

¢ ¢ ¢

g

*R R n n*

sl g d

*mn m n mm*

g l

*mn m n mm nn*

*n n*

¢

*mn m n m m mm nn*

2

 gl d d

*m*

2

*<sup>n</sup> <sup>R</sup>*

*m m m mn m n m m mm nn*

<sup>2</sup> <sup>2</sup>

¢ ¢ ¢ ¢ ¢¢ ¢ ¢

*m*

¢ ¢ ¢ ¢ ¢

*<sup>m</sup> m n <sup>R</sup> lR lR*

æ ö ¢ ¢ æ ö - =- + - ç ÷ ç ÷ è ø è ø

*n*

*mn m n m mm nn*

æ ö - + =- + ç ÷ è ø

<sup>2</sup> <sup>3</sup>

s

d d

(1 )(1 ) (a) <sup>2</sup>

(1 ) (c) <sup>2</sup>

*m*

¢

cd d

(1 ) (b) <sup>2</sup>

(1 ) (d) <sup>2</sup>

(1 )(1 3 ) (e) <sup>2</sup>

(3 ) (f) <sup>2</sup>

(1 ) (g) <sup>2</sup>

(h)

(39)

3

1 1 <sup>2</sup> (i)

l dd

l cd d

*nn*

¢

d

 p

, 2

s p cd d

l

s p

<sup>+</sup> ¢ = -

*mn m n m mm nn*

*m n lR*

¢ ¢ ¢ ¢

g l

*mn m n m mm nn*

s p cd d

<sup>+</sup> ¢ <sup>=</sup>

*n R*

*R*

nl

, 2

, 2

, 2

, 2

*R*

*m n lR*

¢ ¢ ¢ ¢

the following matrix equation can be obtained

220 Advances in Vibration Engineering and Structural Dynamics

{ }

**E**

11

{ }

**E**

12

,

,

,

{ }

**E**

13

{ }

**E**

21

{ }

{ }

**E**

23

{ }

**E**

31

,

{ }

{ <sup>33</sup>}

**E**

**E**

32

**E**

22

$$
\left\{ \left\{ \mathbf{F}\_{11-\alpha} \right\}\_{nm,n'} - \left( \boldsymbol{\nu}\_{2m} - \frac{(1-\sigma)(1+\gamma)u^2}{2R^2} \boldsymbol{\phi}\_{1m} \right) \boldsymbol{\delta}\_{nn'} \right. \tag{6}
$$

$$
\left\{ \left\{ \mathbf{F}\_{12-\alpha} \right\}\_{nm,n'} - \frac{(1+\sigma)u}{2R} \boldsymbol{\phi}\_{1m} \boldsymbol{\delta}\_{nn'} \right. \tag{7}
$$

$$
\left\{ \left\{ \mathbf{F}\_{12\cdots l} \right\}\_{nm,n'} - \frac{(1+\sigma)n}{2R} \boldsymbol{\rho}\_{1m} \boldsymbol{\mathscr{S}}\_{nn'} \right. \tag{6}
$$

$$
\left\{ \left\{ \mathbf{F}\_{12\cdots l} \right\}\_{nm,n'} - \frac{(1+\sigma)n}{2R} \boldsymbol{\rho}\_{2m} \boldsymbol{\mathscr{S}}\_{nn'} \right. \tag{7}
$$

{ } 13 2 2 , 2 9 9 9(1 ) 8 32 16 *e mn n R R n R R l* sp s g - ¢ - =+ <sup>æ</sup> æ ö ç ÷ è ø **F** 2 2 2 4 2 9 (1 ) (e) <sup>8</sup> <sup>32</sup> <sup>16</sup> *m m nn R n R R l* s ps k g k d ¢ - -+ æ ö ö æ æ öö ç ÷ <sup>ç</sup> ÷ ç ç ÷÷ ç ÷ <sup>ç</sup> ÷ ç <sup>÷</sup> è ø <sup>è</sup> ø è è øø

$$
\begin{split}
\left<\mathbf{F}\_{1i-f}\right>\_{\mathrm{inv},n'}-\left<\left(\frac{9\sigma}{8R}+\gamma\left(\frac{9R\pi^{2}}{32I^{2}}-\frac{9(1-\sigma)n^{2}}{16R}\right)\right)\kappa\_{1n}+\left<\frac{\sigma}{8R}+\gamma\left(\frac{9R\pi^{2}}{32I^{2}}-\frac{(1-\sigma)n^{2}}{16R}\right)\right>\kappa\_{3n}\right>\delta\_{nn'} \\
\left<\mathbf{F}\_{1i-f}\right>\_{\mathrm{inv},n'}-\left(\left(\frac{\sigma\mathrm{I}^{2}}{\gamma}+\gamma\left(\frac{R}{\gamma}-\frac{(1-\sigma)\mathrm{I}^{2}n^{2}}{\gamma}\right)\right)\kappa\_{2n}-\left(\frac{\sigma\mathrm{I}^{2}}{\gamma}+\gamma\left(\frac{9R}{\gamma}-\frac{(1-\sigma)\mathrm{I}^{2}n^{2}}{\gamma}\right)\right)\kappa\_{4n}\right>\delta\_{nn'}
\end{split}
\tag{9}
$$

$$
\begin{split} \left\{ \left\{ \mathbf{F}\_{1-\frac{\sigma}{\sigma}} \right\}\_{\mathrm{nm},\mathrm{s}'} - \left\{ \left( \frac{\sigma\boldsymbol{\sigma}^{2}}{2\pi^{2}\mathcal{R}} + \boldsymbol{\gamma} \left( \frac{\boldsymbol{R}}{8} - \frac{(1-\sigma)\boldsymbol{\sigma}^{2}\boldsymbol{n}^{2}}{4\pi^{2}\mathcal{R}} \right) \right) \kappa\_{\mathrm{nm}} - \left( \frac{\sigma\boldsymbol{\sigma}^{2}}{2\pi^{2}\mathcal{R}} + \boldsymbol{\gamma} \left( \frac{9\boldsymbol{R}}{8} - \frac{(1-\sigma)\boldsymbol{\sigma}^{2}\boldsymbol{n}^{2}}{4\pi^{2}\mathcal{R}} \right) \right) \kappa\_{\mathrm{nm}'} \right\} \delta\_{\mathrm{nm}'} \\ \left\{ \left\{ \mathbf{F}\_{1\sim\mathbf{h}} \right\}\_{\mathrm{nm},\mathrm{s}'} - \left( \left( \frac{\sigma\boldsymbol{\sigma}^{2}}{2\pi^{2}\mathcal{R}} + \boldsymbol{\gamma} \left( \frac{\boldsymbol{R}}{8} - \frac{(1-\sigma)\boldsymbol{\sigma}^{2}\boldsymbol{n}^{2}}{4\pi^{2}\mathcal{R}} \right) \right) \kappa\_{\mathrm{nm}} + \left( \frac{\sigma\boldsymbol{\sigma}^{2}}{2\pi^{2}\mathcal{R}} + \boldsymbol{\gamma} \left( \frac{9\boldsymbol{R}}{8} - \frac{(1-\sigma)\boldsymbol{\sigma}^{2}\boldsymbol{n}^{2}}{4\pi^{2}\mathcal{R}} \right) \right) \kappa\_{\mathrm{nm}} \right) \delta\_{\mathrm{nm}'} \end{split} \tag{9}
$$

$$\left\{ \left\{ \mathbf{F}\_{21-a} \right\}\_{mn,a'} - \frac{(1+\sigma)\mu}{2R} \boldsymbol{\varrho}\_{1m} \boldsymbol{\delta}\_{m'} \right. \tag{6}$$
 
$$\left\{ \left\{ \mathbf{F}\_{21-b} \right\}\_{mn,a'} - \frac{(1+\sigma)\mu}{2R} \boldsymbol{\varrho}\_{2m} \boldsymbol{\delta}\_{m'} \right. \tag{7}$$

$$\begin{aligned} \left(\mathbf{F}\_{21\to\ell}\right)\_{nn,n'} &= 2R \left. \begin{array}{c} \mathbf{r}\_{21\to\ell} \\ \left(\mathbf{F}\_{22\to\ell}\right)\_{nn,n'} - \left(\frac{n^2}{R^2}\phi\_{1m} - \frac{(1-\sigma)(1+3\gamma)}{2}\psi\_{1m}\right)\delta\_{nn'} \\ \mathbf{r}\_{21\to\ell} & \end{array} \right) \delta\_{nn'} \tag{k} \end{aligned} \tag{k}$$

$$\begin{aligned} \left(\mathbf{F}\_{22\cdots d}\right)\_{mn,n'} &= -\left(\frac{n^2}{R^2}\phi\_{2m} - \frac{(1-\sigma)(1+3\gamma)}{2}\nu\_{2m}\right)\delta\_{nn'}\\ \left(\begin{array}{c} \text{ $\mu\_{ll'}$ } \end{array}\right)\_{0\pi/2} & \left(\begin{array}{c} \text{ $\mu\_{ll'}$ } \end{array}\right)\_{0\pi/2} \left(\begin{array}{c} \text{ $\mu\_{ll'}$ } \end{array}\right)\_{m'} \end{aligned} \tag{1}$$

$$
\begin{split} \left\{ \left\{ \mathbf{F}\_{21-\delta} \right\}\_{\mathbf{m},\mathbf{x}'} - \left( \left\{ \frac{9nl}{4\pi R^{2}} + \frac{9\pi (3-\sigma)n\eta}{32l} \right\} \kappa\_{1m} - \left( \frac{nl}{12\pi R^{2}} + \frac{3\pi (3-\sigma)n\eta}{32l} \right) \kappa\_{3m} \right) \delta\_{\mathbf{m}'} \\ \left\{ \mathbf{F}\_{21-\delta} \right\}\_{\mathbf{m},\mathbf{x}'} - \left( \left( \frac{9nl}{4\pi R^{2}} + \frac{9\pi (3-\sigma)n\eta}{32l} \right) \kappa\_{2m} + \left( \frac{nl}{12\pi R^{2}} + \frac{3\pi (3-\sigma)n\eta}{32l} \right) \kappa\_{4m} \right) \delta\_{\mathbf{m}'} \end{split} \tag{\text{m}}
$$

$$
\begin{pmatrix} 2\pi \left< -f \right>\_{mm'} \right> \\ \left( \pi \mathbb{1}\_{2^{1-\mathfrak{T}}} \right)\_{mm'} & \left< \mathbb{1} \left< \pi \mathbb{1}^2 \right> \\ \left( \mathbb{1}\_{2^{1-\mathfrak{T}}g} \right)\_{mm'} & -\left( \left( \frac{nl^3}{\sigma^3 \mathbb{R}^2} + \frac{(3-\sigma)nl}{8\pi} \right) \kappa\_{1m'} - \left( \frac{nl^3}{3\sigma^3 \mathbb{R}^2} + \frac{2(3-\sigma)nl\eta}{8\pi} \right) \kappa\_{3m'} \right) \delta\_{m'} \\ \left( \mathbb{1}\_{2^{1-\mathfrak{T}}g} \right)\_{mm'} & -\left( \left( \frac{nl^3}{3\sigma^3} + \frac{(3-\sigma)nl}{8} \right) \kappa\_{2m'} + \left( \frac{nl^3}{3\sigma^3 \mathbb{R}^2} + \frac{2(3-\sigma)nl\eta}{8\pi} \right) \kappa\_{3m'} \right) \delta\_{m'} \end{pmatrix} \tag{9}
$$

$$\left\{ \left( \mathbf{F}\_{2i-h} \right)\_{nm,i'} - \left| \left( \frac{n\mathbf{r}^\*}{\pi^3 R^2} + \frac{(3-\sigma)n\mathbf{l}}{8\pi} \right) \kappa\_{2m} + \left( \frac{n\mathbf{r}^\*}{3\pi^3 R^2} + \frac{3(3-\sigma)n\mathbf{l}}{8\pi} \right) \kappa\_{4m} \right| \delta\_{nn'} \tag{9}$$
 
$$\left\{ \mathbf{F}\_{1i-s} \right\}\_{nm,i'} - \left( \left( \frac{\sigma}{R} - \frac{(1-\sigma)\eta n^2}{2R} \right) \phi\_{1m} - \frac{6R\chi}{l^2} \delta\_{m0} \right) \delta\_{nn'} \tag{10}$$

$$\begin{aligned} \left\{ \begin{array}{cccc} \dots & \dots & \dots & \dots & \dots\\ \left(\right)^{K} & & & \end{array} \right\} & \left( \begin{array}{cccc} K & & & \vert & & & \vert\\ \left(\right)^{K} & & & & \vert & & & \vert\\ \left(\right)^{\sigma} & & & & \vert \left(\frac{\sigma}{R} - \frac{(1-\sigma)\chi n^{2}}{2R}\right) \vert \sigma\_{2m} - \frac{6R\chi}{l^{2}}\mathcal{S}\_{m0} \right) \delta\_{m'} & &\\ \left(\dots & & \cap & \neg \text{when} & \right) & & \end{array} \right\} \end{aligned} \tag{1}$$

$$
\begin{split} \left\{ \left\{ \mathbf{F}\_{12\sim d} \right\}\_{nn,u'} - \left( \frac{n}{R^2} \phi\_{1m} - \frac{(3-\sigma)n\eta}{2} \boldsymbol{\nu}\_{1m} \right) \delta\_{nn'} \\ \left\{ \mathbf{F}\_{12\sim d} \right\}\_{nn,u'} - \left( \frac{n}{R^2} \phi\_{21m} - \frac{(3-\sigma)n\eta}{2} \boldsymbol{\nu}\_{2m} \right) \delta\_{nn'} \end{split} \tag{4}
$$

{ } 33 <sup>2</sup> 22 2 , 1 22 2 2 9 1 1 1 9 4 3 424 *e m mn n R l Rn l Rl l R R* p p g kg p p - ¢ - = + ++ - + <sup>æ</sup> æ ö æ ö æ ö <sup>ç</sup> ç ÷ ç ÷ ç ÷ <sup>ç</sup> è ø <sup>è</sup> è ø è ø **F** <sup>2</sup> 22 2 3 2 2 1 9 (u) 8 8 2 *m nn R n l l R* p p k d ¢ - + + æ ö æ ö <sup>ö</sup> æ ö ç ÷ ç ÷ <sup>÷</sup> ç ÷ <sup>÷</sup> è ø è ø è ø <sup>ø</sup>

{ <sup>33</sup> } 2 2 22 2 22 2 2 4 , 22 2 2 2 <sup>2</sup> 9 1 <sup>1</sup> 1 9 19 (v) <sup>4</sup> 4 2 48 8 3 2 *<sup>f</sup> <sup>m</sup> m nn mn n l Rn l Rn Rl l R l l R R* p p p p g k g k d p p - ¢ ¢ - - =- + + + - + + + <sup>æ</sup> <sup>æ</sup> æ öæ ö ö æ <sup>ö</sup> <sup>ö</sup> æ ö æ ö <sup>ç</sup> <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> ç ÷ ç ÷ <sup>ç</sup> <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> è ø è ø <sup>è</sup> <sup>è</sup> è øè ø ø è <sup>ø</sup> <sup>ø</sup> **F** { } ( ) ( ) { } ( ) 33 33 2 2 2 2 2 2 1 3 , 2 2 2 2 <sup>2</sup> <sup>2</sup> <sup>2</sup> , 2 2 1 1 1 99 (w) 4 23 4 2 1 1 4 2 *<sup>g</sup> <sup>m</sup> m nn mn n h mn n n l n l ll R ll R R R l R l R n l ll R R l R* p p g k g k d p pp p p p p g p p p - ¢ ¢ - ¢ - - = ++ - - + + - - =- + + <sup>æ</sup> <sup>æ</sup> æ öæ ö ö æ <sup>ö</sup> <sup>ö</sup> æ ö æ ö <sup>ç</sup> <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> ç ÷ ç ÷ <sup>ç</sup> <sup>ç</sup> ç ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> ç ÷ ç ÷ <sup>ç</sup> <sup>ç</sup> ÷ ç <sup>÷</sup> <sup>÷</sup> è ø è ø <sup>è</sup> <sup>è</sup> è øè ø ø è <sup>ø</sup> <sup>ø</sup> æ ö æ ö ç ÷ ç ÷ ç ÷ çè ø è ø **F <sup>F</sup>** ( ) <sup>2</sup> <sup>2</sup> <sup>2</sup> 2 4 2 2 <sup>1</sup> 9 9 (x) 34 2 *<sup>m</sup> m nn n l ll R R l R* p k g k d p p p ¢ - - ++ <sup>æ</sup> <sup>æ</sup> ö æ æ öö <sup>ö</sup> æ ö <sup>ç</sup> <sup>ç</sup> ÷ ç ç ÷÷ <sup>÷</sup> ç ÷ <sup>ç</sup> <sup>ç</sup> ÷ ç ç ÷÷ <sup>÷</sup> ç ÷ <sup>ç</sup> <sup>ç</sup> ÷ç ÷ ÷ ç <sup>÷</sup> <sup>÷</sup> è ø <sup>è</sup> <sup>è</sup> ø è è øø <sup>ø</sup>

(40)

$$\begin{cases} \left< \mathbf{P}\_{1} \right>\_{m,m',n'} - \left< \mathbf{P}\_{2} \right>\_{m,m,n'} - \left< \mathbf{P}\_{3} \right>\_{m,m,n'} - \left< \mathbf{P}\_{4} \right>\_{m,m,n'} \delta\_{m'} \end{cases} \tag{4}$$

$$\begin{cases} \left< \mathbf{Q}\_{11-x} \right>\_{m,m',n'} - \left< \mathbf{Q}\_{21-x} \right>\_{m,m'} - \delta\_{m} \delta\_{m'} \right> \\ \left< \mathbf{Q}\_{11-x} \right>\_{m,m',n'} - \left< \mathbf{Q}\_{22-x} \right>\_{m,m'} - \delta\_{m} \delta\_{m'} \delta\_{m'} \end{cases} \tag{5}$$

$$\left< \mathbf{Q}\_{11-x} \right>\_{m,m'} - \frac{l}{4\pi} \left| \mathbf{q} \kappa\_{1m} - \frac{\kappa\_{1m}}{3} \right| \delta\_{m'} \tag{4}$$

$$\left< \mathbf{Q}\_{11-x} \right>\_{m,m'} - \frac{l}{4\pi} \left( \delta \kappa\_{2m} + \frac{\kappa\_{1m}}{3} \right) \delta\_{m'} \tag{5}$$

$$\left< \mathbf{Q}\_{12-x} \right>\_{m,m'} - \frac{l^2}{4\pi} \left( \kappa\_{\ln} - \frac{\kappa\_{\ln}}{3} \right) \delta\_{m'} \tag{6}$$

$$\left< \mathbf{Q}\_{13-x} \right>\_{m,m'} - \frac{l^3}{x^3} \left( \kappa\_{\ln} - \frac{\kappa\_{\ln}}{3} \right) \delta\_{m'} \right>\_{m'} \tag{7}$$

All the unmentioned elements in matrices P and Q are identically equal to zero.

<sup>2</sup> *h K* æ ö r w ç ÷ è ø

*S*.

cosine series and modify Eq. (31) accordingly to reflect this complicating factor.

= 0,

The final system of equations, Eq. (19) or (41), represents a standard characteristic equation for a matrix eigen-problem from which all the eigenvalues and eigenvectors can be readily calculated. It should be mentioned that the elements in each eigenvector are actually the ex‐ pansion coefficients for the corresponding mode; its "physical" mode shape can be directly

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 neces‐ sary. For non-uniform 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

Several numerical examples will be given below to verify the two solution strategies descri‐

We first consider a familiar simply-supported cylindrical shell. The simply supported boun‐ dary condition, *Nx* =*Mx* =*v* =*w* =0at each end, can be considered as a special case when *k*2,6 =*k*3,7 =*∞* and *k*1,5 =*k*4,8 =0 (in actual calculations, infinity is represented by a sufficiently large number). To examine the convergence of the solution, Table 1 shows the frequency pa‐

pansions. It is seen that the solution converges nicely with only a small number of terms. In the following calculations, the expansions in axial direction will be simply truncated to *M*=15. Given in Table 2 are the frequencies parameters for some lower-order modes. Exact solution is available for the simply supported case and the results are also shown there for comparison. An excellent agreement is observed between these two sets of results. Although the simply supported boundary condition represents the simplest case in shell analysis, this problem is not trivial in testing the reliability and sophistication of the current solution method. From numerical analysis standpoint, it may actually represent a quite challenging case because of the extreme stiffness values involved. The non-trivialness can also been seen

) / *E*, calculated using different numbers of terms in the series ex‐

**K Mx** (43)

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 223

Equations (32) and (36) can be combined into

*S* and*M* =*P* + *QL* -1

**3.1. Results about the approximate Rayleigh-Ritz solution**

where *K* =*E* + *F L* -1

obtained from Eqs. (7) or (27).

**3. Results and discussion**

rameters, *Ω* =*ωR ρ*(1−*σ* <sup>2</sup>

bed earlier.

The symbols*κ*1*m*, *κ*2*m*, *κ*3*m*, *κ*4*m*, *ϕ*1*m*, *ϕ*2*m*, *φ*1*m*, *φ*2*m*, *ψ*1*m*, *ψ*2*m*, and *χ<sup>m</sup> i* in the above equations are defined as

p

$$\begin{aligned} \text{Solve}\left(\frac{2}{\pi}\right)&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}\left(\frac{2}{\pi}\right)&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}\left(\frac{2}{\pi}\right)&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}\left(\frac{2}{\pi}\right)&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}\left(\frac{2}{\pi}\right)&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}\left(\frac{2}{\pi}\right)&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}\left(\frac{2}{\pi}\right)&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Solve}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Explanation}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Examation}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Examation}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Examation}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Examation}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Examation}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \text{Examation}&=\sum\_{k=0}^{\infty}k\_{k0}\cos k\_{k0};\\ \$$

All the unmentioned elements in matrices P and Q are identically equal to zero.

Equations (32) and (36) can be combined into

{ } { } { } { } { } { } { }

**PP P Q Q Q Q**

1 ,

p

*l l l*

*e mn n m nn*

*<sup>f</sup> <sup>m</sup> nn mn n*

*h mn n m nn l*


<sup>2</sup> , 3 3 ,

p

3 2 ,

p

=

The symbols*κ*1*m*, *κ*2*m*, *κ*3*m*, *κ*4*m*, *ϕ*1*m*, *ϕ*2*m*, *φ*1*m*, *φ*2*m*, *ψ*1*m*, *ψ*2*m*, and *χ<sup>m</sup>*

p

11 22 33 11 22 11 22

,, , 1 , , 2 , , 3


*mn m n mn m n mn m n mm nn a mn n c mn n m nn b mn n d mn n m nn*

= =- =- = = - = = æ ö = - ç ÷ è ø æ ö =- + ç ÷ è ø

f d

¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢

f d

4

 d

> d

> > 2

2

<sup>ï</sup> <sup>¹</sup> ïî <sup>ì</sup> - = <sup>ï</sup>

*m*

p

*m*

p p

2 0, <sup>3</sup> s ; (e,f) 12 0, (9 4 )

<sup>ì</sup> <sup>=</sup> <sup>ï</sup>

p

1

*m+*

p

( )

<sup>ì</sup> <sup>=</sup> <sup>ï</sup>

p

 0, <sup>12</sup> (i,j) 2 6 6( 1) 0,

*<sup>l</sup> <sup>m</sup>*

*<sup>m</sup> <sup>m</sup> <sup>l</sup> <sup>m</sup>*

*m*

*<sup>m</sup> <sup>m</sup>*

0,

0,

*m*

0,

1 ( 1) 0,

<sup>ì</sup> <sup>=</sup> <sup>ï</sup> <sup>ï</sup><sup>ì</sup> - - <sup>ï</sup> <sup>=</sup> <sup>ï</sup>

p

*<sup>m</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>m</sup> m i* p

<sup>í</sup> <sup>¹</sup> <sup>ï</sup> é ù - - ïï ë û <sup>¹</sup> ïï

*i m i*

2 2

+

( )

î - î

(q,r) 12 1 ( 1) 0,

0 0,

(u,v) 0. 2 ( 1) 1 0,

*i*

0,

( )

<sup>ì</sup> - = <sup>ï</sup>

4 4

p

*m*

 0 0, ; 4 2 ( 1) (m,n) 0, 0 0,

*<sup>m</sup> <sup>m</sup> m*

*<sup>m</sup> <sup>m</sup>*

<sup>ï</sup> <sup>¹</sup> <sup>î</sup> -

<sup>ì</sup> <sup>=</sup> <sup>ï</sup>

<sup>í</sup> -+- <sup>ï</sup> <sup>¹</sup> <sup>ï</sup>

<sup>ì</sup> <sup>=</sup> <sup>ï</sup>

( )

2 2

p

íï

2 2

p

1 0,

*<sup>m</sup> lm*

*<sup>m</sup> lm*

*m*

2

2 2 4 4

p

<sup>í</sup> -+ - <sup>ï</sup> - ¹ ï

p

( )

<sup>ì</sup> <sup>=</sup> <sup>ï</sup> <sup>=</sup> <sup>í</sup> + <sup>ï</sup> <sup>¹</sup> <sup>î</sup>

p

*<sup>m</sup> <sup>m</sup>*

( )

p

1

*l m*

*m*

p

2 0,

<sup>ì</sup> <sup>=</sup> <sup>ï</sup>

p

<sup>ï</sup> <sup>¹</sup> ïî -

p

*m*

*m*

*m*

*m*

*m m*

2 0,

<sup>ì</sup> <sup>=</sup> <sup>ï</sup>

*m*

k k

*m*

 d

¢

 d

*m*

k k

3 1 3

k k

sin( ) cos ; (a,b) <sup>2</sup> 4 0, (1 4 )

k

k

3

 k

k

f

*m m m m m*

<sup>ï</sup> = = <sup>í</sup> - +- - <sup>ï</sup> <sup>¹</sup> <sup>ï</sup> î

f

1

j

j

y

y

cos( ) cos ; (c,d) <sup>2</sup> 4( 1) 0, (1 4 )

ï = í

2 0, <sup>3</sup> <sup>3</sup> cos( ) cos ; (g, h) <sup>2</sup> 12( 1) 0, (9 4 ) 

ï

î

î -

0, <sup>12</sup> ( ) cos ; (k, l) 2 ( 1) 6 6( 1)

( ) cos ; 4 1 2( 1) (o,p)

ï

î

( ) cos ; (s, t) 12 1 ( 1)

î

 c

î -

æ ö ç ÷ è ø æ ö =- + ç ÷ è ø

*m m nn m*

4

k k

<sup>9</sup> (d) 4 3 <sup>9</sup> (e) 4 3

(a)

*i*

in the above equations are

(41)

(42)

(b) (c)

d d

(f) <sup>3</sup> (g) <sup>3</sup>

{ } { } { }

**Q Q Q** { <sup>33</sup> }

222 Advances in Vibration Engineering and Structural Dynamics

**Q**

1 1 <sup>0</sup> <sup>2</sup>

<sup>ï</sup> = = <sup>í</sup>

*m m m*

*<sup>l</sup> <sup>m</sup>*

*<sup>l</sup> <sup>m</sup>*

*m m*

*<sup>l</sup> <sup>m</sup>*

<sup>ï</sup> = = <sup>í</sup> - <sup>ï</sup> <sup>¹</sup> <sup>ï</sup>

<sup>ï</sup> = = <sup>í</sup> - <sup>ï</sup> <sup>¹</sup> <sup>ï</sup>

2 2

4 4

*m m m*

2 2 2 2 2

<sup>0</sup> 2 2

<sup>0</sup> 2 2

å ( )

*<sup>m</sup> <sup>l</sup> x x*

<sup>ï</sup> ¢¢ = = <sup>í</sup> -+- <sup>ï</sup> <sup>¹</sup> <sup>ï</sup>

*m m m m*

*m ii m i i i mm m*

= == =

<sup>ì</sup> <sup>=</sup> <sup>ï</sup> ¢ = = <sup>í</sup> + -

*<sup>m</sup> m m <sup>m</sup>*

*x x l m*

= =

*m m m*

1 1 1

2 2 2

1 1 1

2 2 2

sin cos sin cos ;

*x xx x*

 l cl

*i m*

¥ ¥

å å

¢¢ = =

*m m m*

*<sup>l</sup> x x*

*x*

l

*m m m*

0

¥ =

å

*x x*

k l

*m*

¥ =

å

*m*

¥ =

å

p

> p

*x x*

k l

<sup>3</sup> sin( ) co <sup>2</sup>

=

0

0

 fl

*m*

¥ =

å

( ) cos

0

0

 yl

*m*

¥ =

å

 yl

*m*

( ) cos ;

 jl

*x x*

*x x*

 jl

*m m*

1 1

¢ =

*m*

*m*

=

¥ = ¥ =

å

¥

å

 fl

*m*

¥ =

å

a

a

a

a

a

a

l cl

( ) cos ;

*x x*

*x l*

p

> p

3 0

*x x*

k l

k

0

*m*

¥ =

å

*m m*

defined as

33 33 33

*<sup>g</sup> mn n*


$$\left(\mathbf{K} - \frac{\rho h o \boldsymbol{\sigma}^2}{\boldsymbol{\kappa}} \mathbf{M}\right) \mathbf{x} = \mathbf{0},\tag{43}$$

where *K* =*E* + *F L* -1 *S* and*M* =*P* + *QL* -1 *S*.

The final system of equations, Eq. (19) or (41), represents a standard characteristic equation for a matrix eigen-problem from which all the eigenvalues and eigenvectors can be readily calculated. It should be mentioned that the elements in each eigenvector are actually the ex‐ pansion 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 neces‐ sary. For non-uniform 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.
