Study of vibration characteristics of the short thin cylindrical shells and its experiment

1. Introduction

Short thin cylindrical shell usually means that the axial half-wave number is 1, and the ratios of thickness and other minimum parameter (i.e. diameter and length) is between 1/80 and 1/5 [1]. The short thin cylindrical shell has grown in importance with the increasing use of shell structures for a wide variety of applications in aerospace, shipbuilding, chemical machinery and other branches of engineering. It is important component used in rotating machinery and its function is to connect shafts and transmitted torque. It is subjected to centrifugal force, aerodynamic force, and vibration alternating force and other load during its working. The kind of components is always destroyed due to vibrational state. So it is necessary to further research on the vibration characteristics.

Currently numerous studies have been made to understand the vibration characteristic of the short thin cylindrical shell. In order to calculate the inherent characteristic of cantilever cylindrical shell, the Rayleigh-Ritz method, the axis-symmetry finite element method and the exact analytic method are employed by C. B. Sharma and D. J. Johns [2-3]. By using of the Love’s shell equations, the influence of boundary conditions for a thin rotating stiffened cylindrical shell is researched by Lee [4]. Vibration characteristics of the laminated cylindrical shell are studied, then influence of boundary conditions and geometric parameters for system dynamics characteristics is analyzed by Zhang [5]. Studies for dynamic characteristic of boundary conditions for thin cylindrical shells have also been carried out by Lam and Loy [6]. Li Xuebin presents a new approach of separation of variables in circular cylindrical shell analysis for arbitrary boundary conditions [7]. The instant response of the laminated shell is subject to radial impacting loads and axial pressure load, and the boundary condition investigated in the study is clamped-simply. The analysis is carried out using Love-type shell theory and the first order shear deformation theory by Jafari and Hamid [8, 9]. However, many of these studies are restricted to theory analyses. Studies for the contrast test of different analyze methods are seldom attempted, especially the experimental tests.

In this paper, the vibration characteristics of short thin cylindrical shells are solved using three analytical methods which is the beam function method, the transfer matrix method and the finite element method respectively, and then using the results of experiment confirmed the correctness and validity of analysis methods.

2. Analysis based on the beam function method

Figure 1 shows the nomenclature of a short thin cylindrical shell. The reference surface of the cylinder is taken at the middle surface where an orthogonal coordinate system ($Ox\theta z$) is fixed. The coordinates system ($Ox\theta z$) is the coordinates system ($O\text{'}x\text{'}\theta \text{'}r$) translate $R$ from origin $O$ along $r$ direction to one point of the middle surface. $x$-axis and $x^{\text{'}}$-axis are parallel and in the same direction. $\theta$-axis and ${\theta }^{\text{'}}$-axis are overlapping and in the same direction. $z$-axis and $r$-axis are overlapping. In the Figure 1, $R$ is the radius, $L$ is the length, $h$ is the thickness, $\rho$ is the material density, $E$ is the Young's modulus and $\mu$ is the Poisson's ratio. The deformations of the cylindrical shell in the $x$, $\theta$ and $z$ directions are denoted by $u(x,\theta ,t)$, $v(x,\theta ,t)$ and $w(x,\theta ,t)$, respectively.

Fig. 1The structures of the thin-walled cylindrical shells

3. Analysis based on the transfer matrix method

3.1. Fundamental equations

Figure 2 shows the nomenclature of a short thin cylindrical shell. The reference surface of the cylinder is taken at the middle surface where an orthogonal coordinate system ($Ox\theta z$) is fixed. The coordinates system ($Ox\theta z$) is the coordinates system ($O\text{'}x\text{'}\theta \text{'}r$) translate $R$ from origin $O$ along $r$ direction to one point of the middle surface. $x$-axis and $x^{\text{'}}$-axis are parallel and in the same direction. $\theta$-axis and ${\theta }^{\text{'}}$-axis are overlapping and in the same direction. $z$-axis and $r$-axis are overlapping. In the figure, $R$ is the radius, $L$ is the length, $h$ is the thickness, $\rho$ is the material density, $E$ is the Young's modulus and $\mu$ is the Poisson's ratio. The deformations of the cylindrical shell in the $x$, $\theta$ and $z$ directions are denoted by $u(x,\theta ,t)$, $v(x,\theta ,t)$ and $w(x,\theta ,t)$, respectively. The short thin cylindrical shell is divided into $N$ sections along its length. The length of every section is $L_1,L_2,...,L_{k-1},L_k\left(k\le N\right)$, respectively.

Fig. 2The segmented mode of the thin-walled cylindrical shells

The bigger of $N$, the more accurate of the result is, but it spends more time on achieving the result. Thus, the $N$ can’t be too large or too small. As shown in Table 2, the natural frequencies are very close when $N$ is 3, 4 and 5. Considering the time and accuracy, the $N$ is 3 in this paper.

Table 2The natural frequencies with different N (m=1, n=6)

$N$	1	2	3	4	5
Natural frequencies	14045.8	1377.44	1369.03	1369.02	1369.02

Based on the Kirchhoff theory, the shear $V_x$ and transversal shear $S_x$ are the following [13]:

10

$V_x=Q_x+\frac{1}{R}\frac{\partial M_{x\theta }}{\partial \theta },$

11

$S_x=N_{x\theta }+\frac{1}{R}{M}_{x\theta }.$

Using the Love shell theory, the shell's equilibrium equations are expressed as vibration differential equation [14]:

12

$\left.\begin{array}{l}\frac{\partial N_x}{\partial s}+\frac{\partial N_{x\theta }}{R\partial \theta }-\rho H\frac{{\partial }^{2}u}{\partial t^2}=0,\\ \frac{\partial S_x}{\partial x}-\frac{\partial M_{x\theta }}{R\partial x}+\frac{\partial N_{\theta }}{R\partial \theta }+\frac{Q_{\theta }}{R}-\rho H\frac{{\partial }^{2}v}{\partial t^2}=0,\\ \frac{\partial V_x}{\partial x}-\frac{\partial M_{x\theta }}{R\partial x\partial \theta }+\frac{\partial Q_{\theta }}{R\partial \theta }-\frac{N_{\theta }}{R}-\rho H\frac{{\partial }^{2}w}{\partial t^2}=0,\end{array}\right\},$

The relationship of generalized force, corner and middle surface displacement are:

13

$N_x=K\left[\frac{\partial u}{\partial x}+\mu \left(\frac{1}{R}\frac{\partial v}{\partial \theta }+\frac{w}{R}\right)\right],$

14

$M_x=D\left(\frac{\partial {\phi }_x}{\partial x}-\frac{\mu }{R^2}\frac{{\partial }^{2}w}{\partial {\theta }^2}\right),$

15

$V_x=\frac{\partial M_x}{x}-\frac{2}{R^2}D\left(1-\mu \right)\frac{{\partial }^{3}w}{\partial x\partial {\theta }^2},$

16

$S_x=K\frac{1-\mu }{2}\left(\frac{1}{R}\frac{\partial u}{\partial \theta }+\frac{\partial v}{\partial x}\right)-\frac{1}{R^2}D(1-\mu )\frac{{\partial }^{2}w}{\partial x\partial \theta },$

17

${\phi }_x=-\frac{\partial w}{\partial x},$

where $K$, $D$ are the shell’s membrane stiffness, bending respectively.

The differential equations consist of Eq. (10)-(12), which contain 8 equations and 8 unknown variables. The unknown variables include 3 elastic displacement components, 1 rotation angle and 4 generalized force components. The unknown variables are expressed state vector which are defined as [15]:

18

$\mathbf{Z}\left(x\right)={\left[\begin{array}{cccccccc}u& v& w& {\phi }_x& M_x& V_x& S_x& N_x\end{array}\right]}^T.$

Simplified Eq. (12) – Eq. (17) aiming at keeping only the state vector elements yields the following matrix equation:

19

$\frac{\text{d}\mathbf{Z}\left(x\right)}{dx}=\mathbf{U}\mathbf{Z}\left(x\right).$

The coefficient matrices $\mathbf{U}$ of 8×8 order can be expressed in a general form as:

20

$\mathbf{U}=\left[\begin{array}{cccccccc}0& -\frac{\mu }{R}\frac{\partial }{\partial \theta }& -\frac{\mu }{R}& 0& 0& 0& 0& \frac{1-{\mu }^2}{Eh}\\ \frac{1}{R}\frac{\partial }{\partial \theta }& 0& 0& -\frac{h^2}{6R^2}\frac{\partial }{\partial \theta }& 0& 0& \frac{2\left(1+\mu \right)}{Eh}& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& -\frac{\mu }{R^2}\frac{{\partial }^2}{\partial {\theta }^2}& 0& \frac{12\left(1-{\mu }^2\right)}{E{h}^3}& 0& 0& 0\\ 0& 0& 0& -\frac{E{h}^3}{6\left(1+\mu \right)R^2}\frac{{\partial }^2}{\partial {\theta }^2}& 0& -1& 0& 0\\ 0& \frac{Eh}{R^2}\frac{\partial }{\partial \theta }& \frac{E{h}^3}{12R^4}\frac{{\partial }^4}{\partial {\theta }^4}+\frac{Eh}{R^2}+\rho h\frac{{\partial }^2}{\partial t^2}& 0& \frac{\mu }{R^2}\frac{{\partial }^2}{\partial {\theta }^2}& 0& 0& \frac{\mu }{R}\\ 0& -\frac{Eh}{R^2}\frac{{\partial }^2}{\partial {\theta }^2}+\rho h\frac{{\partial }^2}{\partial t^2}& \frac{E{h}^3}{12R^4}\frac{{\partial }^3}{\partial {\theta }^3}-\frac{Eh}{R^2}\frac{\partial }{\partial \theta }& 0& \frac{\mu }{R^2}\frac{\partial }{\partial \theta }& 0& 0& \frac{\mu }{R}\frac{\partial }{\partial \theta }\\ \rho h\frac{{\partial }^2}{\partial t^2}& 0& 0& -\frac{E{h}^3}{12\left(1+\mu \right)R^3}\frac{{\partial }^2}{\partial {\theta }^2}& 0& 0& -\frac{1}{R}\frac{\partial }{\partial \theta }& 0\end{array}\right].$

According to the geometric characteristics of thin-walled cylindrical shell, the middle surface generalized displacement and generalized force internal force of the short thin cylindrical shell along the $\theta$ direction are written as:

21

$\mathbf{Z}\left(x,\theta ,t\right)=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\left[\begin{array}{c}\tilde{u}\left(x\right)\cos \left(n\theta \right)\\ \tilde{v}\left(x\right)\sin \left(n\theta \right)\\ \tilde{w}\left(x\right)\cos \left(n\theta \right)\\ {\tilde{\phi }}_x\left(x\right)\cos \left(n\theta \right)\\ {\tilde{M}}_x\left(x\right)\cos \left(n\theta \right)\\ {\tilde{V}}_x\left(x\right)\cos \left(n\theta \right)\\ {\tilde{S}}_x\left(x\right)\sin \left(n\theta \right)\\ {\tilde{N}}_x\left(x\right)\cos \left(n\theta \right)\end{array}\right]e^{j\omega t}.$

Substituting Eq. (21) into Eq. (19), the differential equation of modal function can be written as [16]:

22

$\frac{d\tilde{\mathbf{Z}}\left(x\right)}{dx}=\tilde{\mathbf{U}}\tilde{\mathbf{Z}}\left(x\right),$

where $\tilde{\mathbf{Z}}\left(x\right)={\left[\begin{array}{cccccccc}\tilde{u}& \tilde{v}& \tilde{w}& {\tilde{\phi }}_x& {\tilde{M}}_x& {\tilde{V}}_x& {\tilde{S}}_x& {\tilde{N}}_x\end{array}\right]}^T$.

The coefficient matrices $\tilde{\mathbf{U}}$ of 8×8 order can be expressed in a general form as:

23

$\tilde{\mathbf{U}}=\left[\begin{array}{cccccccc}0& U_{12}& U_{13}& 0& 0& 0& 0& U_{18}\\ U_{21}& 0& 0& U_{24}& 0& 0& U_{27}& 0\\ 0& 0& 0& U_{34}& 0& 0& 0& 0\\ 0& U_{42}& U_{43}& 0& U_{45}& 0& 0& 0\\ U_{51}& 0& 0& U_{54}& 0& U_{56}& U_{57}& 0\\ 0& U_{62}& U_{63}& 0& U_{65}& 0& 0& U_{68}\\ 0& U_{72}& U_{73}& 0& U_{75}& 0& 0& U_{78}\\ U_{81}& 0& 0& U_{84}& 0& 0& U_{87}& 0\end{array}\right].$

Suppose the cylinder is divided into $N$ sections, stress state of any point on the arbitrary cross-section of shell as shown in Figure 3.

Fig. 3Stress state of any point on the arbitrary cross-section

According to theory of solving differential equation, the general solution to state equation is the following:

24

$\tilde{\mathbf{Z}}\left(L_k\right)=\tilde{\mathbf{G}}\left(L_k\right)\tilde{\mathbf{Z}}\left(L_{k-1}\right),$

where $L_k$ is the length of $k$ th shell segment, $\mathbf{G}$ is transfer matrix and can be written as:

25

$\tilde{\mathbf{G}}\left(L_k\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(\tilde{\mathbf{U}}{L}_k\right).$

The modal function can be obtained by reusing transfer matrix and yields the following matrix equation:

26

$\tilde{\mathbf{Z}}\left(H\right)=\tilde{\mathbf{T}}\left(\omega \right)\tilde{\mathbf{Z}}\left(0\right),$

where:

27

$\tilde{\mathbf{T}}\left(\omega \right)={\tilde{\mathbf{G}}}_N\left(L_N\right){\tilde{\mathbf{G}}}_{N-1}\left(L_{N-1}\right)\cdots L{\tilde{\mathbf{G}}}_2\left(L_2\right){\tilde{\mathbf{G}}}_1\left(L_1\right).$

The coefficient matrices $\mathbf{U}$ are obtained by Eq. (22). Using the Eq. (24) and Eq. (26), the transitive relation of the state vectors can be obtained. The transfer matrix $\mathbf{T}$ is 8×8 matrix and can be written by calculating Eq. (27):

28

$\left\{\begin{array}{c}u\\ v\\ w\\ {\phi }_s\\ M_s\\ V_s\\ S_s\\ N_s\end{array}\right\}=⌈\begin{array}{cccccccc}{T}_{11}& T_{12}& T_{13}& T_{14}& T_{15}& T_{16}& T_{17}& T_{18}\\ T_{21}& T_{22}& T_{23}& T_{24}& T_{25}& T_{26}& T_{27}& T_{28}\\ T_{31}& T_{32}& T_{33}& T_{34}& T_{35}& T_{36}& T_{37}& T_{38}\\ T_{41}& T_{42}& T_{43}& T_{44}& T_{45}& T_{46}& T_{47}& T_{48}\\ T_{51}& T_{52}& T_{53}& T_{54}& T_{55}& T_{56}& T_{57}& T_{58}\\ T_{61}& T_{62}& T_{63}& T_{64}& T_{65}& T_{66}& T_{67}& T_{68}\\ T_{71}& T_{72}& T_{73}& T_{74}& T_{75}& T_{76}& T_{77}& T_{78}\\ T_{81}& T_{82}& T_{83}& T_{84}& T_{85}& T_{86}& T_{87}& T_{88}\end{array}⌉\left\{\begin{array}{c}u\\ v\\ w\\ {\phi }_s\\ M_s\\ V_s\\ S_s\\ N_s\end{array}\right\},$

where $T_{ij}(i=\mathrm{1,2},\cdots ,8;j=\mathrm{1,2},\cdots ,8)$ are the transfer matrix coefficients.

4. Numerical computation based on finite element method

In the classical vibration theory, the basic equation of dynamic problems based on finite element methods is presented as follows [17]:

where $\mathbf{M}$ is the total mass matrix, $\mathbf{C}$ is the total damping matrix, $\mathbf{K}$ is the total stiffness matrix, $\mathbf{F}$ is the total additional exciting force matrix, $\mathbf{\delta }$, $\dot{\mathbf{\delta }}$ and $\ddot{\mathbf{\delta }}$ are the joint displacement, velocity and acceleration column matrixes respectively.

As the natural vibration frequency of structural system is calculated with the consideration that the free vibration is conducted under undamped conditions. The equation of the natural vibration frequency is expressed as:

The solutions can be presumed as the following types:

where $\phi$ is the $n$ order vector quantity, $\omega$ is the natural frequency.

The Eq. (33) is substituted into Eq. (34), and then generalized eigenvalue problems are obtained as follows:

According to the linear algebra theory, the necessary and sufficient conditions which allow Eq. (35) with untrivial solution are as follows:

where $n$ characteristic solutions can be obtained by solving these equations, such as$\left({\omega }_1^2,{\phi }_1\right),\left({\omega }_2^2,{\phi }_2\right),\dots ,({\omega }_n^2,{\phi }_n),$ among which ${\omega }_1,{\omega }_2,\dots ,\mathrm{}{\omega }_n$ stand for $n$ natural frequencies of system, and correspond to $0\le {\omega }_1<{\omega }_2<\cdot \cdot \cdot <{\omega }_n$.

For every natural frequencies of structure, relative amplitudes of each node from one group can be concluded based on Eq. (35). Eigenvectors such as ${\mathbf{\phi }}_1,{\mathbf{\phi }}_2,\dots ,{\mathbf{\phi }}_n$ represent $n$ natural modes of vibration of structure. Their amplitudes can be set as follows:

Natural mode of vibration is orthogonal compared to matrix $\mathbf{M}$. Natural mode of vibration is defined as:

Then it can be obtained as follows:

The characters of characteristic solution can be also expressed as:

where $\mathbf{\Phi }$ and ${\mathbf{\Omega }}^2$ are the natural mode of vibration matrix and natural frequency matrix respectively.

5. The results of the three analytical methods

The natural frequencies of the static state of the cylindrical shell under clamped – free and clamped boundary conditions are calculated by the transfer matrix method, the beam function method and the finite element method, respectively, and make further verification of the natural frequencies. Table 3 shows the basic parameters of the thin-walled cylindrical shell.

The natural frequencies of the cylindrical shell under clamped – free are calculated by the transfer matrix method, the beam function method and the finite element method respectively. Table 4 and Fig. 4 shows that the results of the transfer matrix method and the finite element method calculation are very close, the result of beam function method calculation is close to those of transfer matrix method and the finite element method calculation only at the place of 6th order, the 8th order, the 9th order and the 11th order, the frequencies of other orders are quite different.

The natural frequencies of the cylindrical shell under both ends clamped boundary conditions are calculated by the transfer matrix method, the beam function method and the finite element method, respectively. Table 5 and Fig. 5 shows that the results of transfer matrix method and the finite element method calculation are very close, the result of beam function method calculation is close to those of transfer matrix method and the finite element method calculation only at the place of 8th order, the 10th order and the 12th order, the frequencies of other orders are quite different.

Fig. 4The frequencies comparison of the three methods in the clamped – free boundary condition