**3. Results and discussion**

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

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

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‐ rameters, *Ω* =*ωR ρ*(1−*σ* <sup>2</sup> ) / *E*, calculated using different numbers of terms in the series ex‐ 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 mathematically from the fact that the simple sine function (in the axial direction) in the exact solution is actually expanded as a cosine series expansion in the current solution.

gle beam characteristic function in contrast to the three complete sets (of basis functions) in

2035.05 971.531 990.339 1600.90 2486.49

**Table 3.** The natural frequencies in Hz for a clamped-clamped shell; *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, *E*=2.1E+11,

**Mode** *n=2 n=3 n=4 n=5 n=6 n=7*

0.01792 0.01680 0.01679 0.02068 0.01734 0.01731 0.08343 0.07938

Another classical example involves a completely free shell. Vibration of a free-free shell is of particular interest as manifested in the debate between two legendary figures, Rayleigh and Love, about the validity of the inextensional theory of shells. The lower-order modes are typically related to the rigid-body motions in the axial direction. Theoretically, the H*<sup>w</sup>* ma‐ trix given in Eqs. (14) will become non-invertible for a completely free shell. However, this numerical irregularity can be easily avoided by letting one of the bending-related springs

parameters calculated using different techniques. While the results obtained from the cur‐ rent technique agree reasonably well with the other two reference sets, perhaps within the variance of different shell theories, the frequency parameters for the two lower order modes with rigid-body rotation (*n*=2 and 3) are clearly inaccurate which probably indicates that the inability of exactly satisfying the shell boundary conditions by the "beam functions" tends

^ <sup>4</sup> =10−<sup>6</sup> Current

3854.75 2039.66 1454.80 1769.54 2572.31

0.02830 0.02717 0.02714 0.02995 0.02774 0.02769 0.06292 0.05893

) / *E*, for a free-free shell; *R*=0.5 m, *L*=4*R*, *h*=0.002*R*, and μ=0.28.

. Table 4 shows a comparison of the frequency

*m=*<sup>2</sup> **Eq. (42)**

0.04099 0.03986 0.03980 0.04220 0.04045 0.04037 0.05906 0.05555 4302.05 2189.59 1500.07 1782.28 2578.07

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 225

0.05599 0.05487 0.05475 0.05713 0.05546 0.05533 0.06606 0.06332

the current method.

**Mode**

μ=0.28, ρ=7800 kg/m3.

Translation: current (2.130) [22] FEA Rotation: Current (2.132) [22] FEA *m*=1: Current FEA

Current

1886.74 934.220 982.265 1598.55 2484.78

0.00413 0.00310 0.00310 0.01907 0.00343 0.00343 0.24075 0.23810

**Table 4.** Frequency parameters, Ω =ω*R* ρ(1−σ <sup>2</sup>

have a very small stiffness, such as,*k*

*m=*<sup>1</sup> **Eq. (42)**

0.00986 0.00876 0.00876 0.01676 0.00924 0.00923 0.13190 0.12836


**Table 1.** Frequency parameters, Ω =ω*R* ρ(1−σ <sup>2</sup> ) / *E*, obtained using different numbers of terms in the displacement expansions.


**Table 2.** Frequency parameters, Ω =ω*R* ρ(1−σ <sup>2</sup> ) / *E*, for a simply-supported shell; *L*=4*R*, *h*/*R*=0.05 and μ=0.3.

Next, consider a cylindrical shell clamped at each end, that is,*u* =*v* =*w* =∂*w* / ∂ *x* =0. The clamped-clamped boundary condition is a case when the stiffnesses of the restraining springs all become infinitely large. The related shell and material parameters are as follows: *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, *E*=2.1×1011, *μ*=0.28, and *ρ*=7800. Listed in Table 3 are some of the lowest natural frequencies for this clamped-clamped shell. The reference results given there are calculated from

$$
\Omega \,\, \Omega^6 - A\_2 \Omega^4 + A\_1 \Omega^2 - A\_0 = 0 \tag{44}
$$

where*Ω* =*ωR ρ*(1−*σ* <sup>2</sup> ) / *E*, and the coefficients *A*0, *A*1 and *A*2 are the functions of the modal indices, shell parameters, and the boundary conditions [27]. Equation (42) can be derived from the Rayleigh-Ritz procedure by adopting the beam characteristic functions as the axial functions for all three displacement components. A noticeable difference between these two sets of results may be attributed to the fact that: a) Eq. (42) given in ref. [29] is based on the Flügge shell theory, rather than the Donnell-Mushtari theory, and b) Eq. (42) uses only a sin‐


gle beam characteristic function in contrast to the three complete sets (of basis functions) in the current method.

mathematically from the fact that the simple sine function (in the axial direction) in the exact

0.257389 0.257386 0.257385 0.257385

0.257385 0.257385 0.574179 0.574176 0.764375 0.764355

Next, consider a cylindrical shell clamped at each end, that is,*u* =*v* =*w* =∂*w* / ∂ *x* =0. The clamped-clamped boundary condition is a case when the stiffnesses of the restraining springs all become infinitely large. The related shell and material parameters are as follows: *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, *E*=2.1×1011, *μ*=0.28, and *ρ*=7800. Listed in Table 3 are some of the lowest natural frequencies for this clamped-clamped shell. The reference results

indices, shell parameters, and the boundary conditions [27]. Equation (42) can be derived from the Rayleigh-Ritz procedure by adopting the beam characteristic functions as the axial functions for all three displacement components. A noticeable difference between these two sets of results may be attributed to the fact that: a) Eq. (42) given in ref. [29] is based on the Flügge shell theory, rather than the Donnell-Mushtari theory, and b) Eq. (42) uses only a sin‐

642

**Ω =ω***R* **ρ(1−σ <sup>2</sup>**

**Ω =ω***R* **ρ(1−σ <sup>2</sup>**

*n=0 n=1 n=2 n=3 n=4*

0.127128 0.127128 0.337652 0.337649 0.532951 0.532923

) / *E*, for a simply-supported shell; *L*=4*R*, *h*/*R*=0.05 and μ=0.3.

<sup>210</sup> W- W+ W- = *AAA* 0 (44)

) / *E*, and the coefficients *A*0, *A*1 and *A*2 are the functions of the modal

*n=0 n=1 n=2 n=3 n=4*

0.127132 0.127129 0.127128 0.127128 **) /** *E*

) / *E*, obtained using different numbers of terms in the displacement

**) /** *E*

0.143329 0.143327 0.143327 0.143327

0.143327 0.143327 0.248813 0.248810 0.399893 0.399865 0.234823 0.234822 0.234822 0.234822

0.234822 0.234822 0.285620 0.285619 0.383688 0.383667

solution is actually expanded as a cosine series expansion in the current solution.

0.464652 0.464649 0.464648 0.464648

0.464648 0.464648 0.928907 0.928907 0.948172 0.948172

**Number of terms used in the series**

**Mode**

**Table 1.** Frequency parameters, Ω =ω*R* ρ(1−σ <sup>2</sup>

224 Advances in Vibration Engineering and Structural Dynamics

**Table 2.** Frequency parameters, Ω =ω*R* ρ(1−σ <sup>2</sup>

given there are calculated from

where*Ω* =*ωR ρ*(1−*σ* <sup>2</sup>

*M=5 M=7 M=9 M=10*

expansions.

*m*=1, Current Exact m=2, Current Exact m=3, Current Exact

**Table 3.** The natural frequencies in Hz for a clamped-clamped shell; *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, *E*=2.1E+11, μ=0.28, ρ=7800 kg/m3.


**Table 4.** Frequency parameters, Ω =ω*R* ρ(1−σ <sup>2</sup> ) / *E*, for a free-free shell; *R*=0.5 m, *L*=4*R*, *h*=0.002*R*, and μ=0.28.

Another classical example involves a completely free shell. Vibration of a free-free shell is of particular interest as manifested in the debate between two legendary figures, Rayleigh and Love, about the validity of the inextensional theory of shells. The lower-order modes are typically related to the rigid-body motions in the axial direction. Theoretically, the H*<sup>w</sup>* ma‐ trix given in Eqs. (14) will become non-invertible for a completely free shell. However, this numerical irregularity can be easily avoided by letting one of the bending-related springs have a very small stiffness, such as,*k* ^ <sup>4</sup> =10−<sup>6</sup> . Table 4 shows a comparison of the frequency parameters calculated using different techniques. While the results obtained from the cur‐ rent technique agree reasonably well with the other two reference sets, perhaps within the variance of different shell theories, the frequency parameters for the two lower order modes with rigid-body rotation (*n*=2 and 3) are clearly inaccurate which probably indicates that the inability of exactly satisfying the shell boundary conditions by the "beam functions" tends to have more serious consequence in such a case. Amazingly enough, the inextensional theo‐ ry works very well in predicting the frequency parameters for the "rigid-body" modes (those with rigid-body motions in the axial direction). It is also seen that the frequency pa‐ rameters of the rigid-body modes increases monotonically with the circumferential modal index *n*.

stiffness *k*

support *k* ^

mode for *k*

^

**Mode** *k*

*E*=2.1E+11, μ=0.28, ρ=7800 kg/m3;*k*

*^*

404.108 487.598 865.603 1003.38

7=0.1 m-1 and the second mode for *k*

*<sup>7</sup> = 0 k*

^ <sup>5</sup> =*k* ^ <sup>6</sup> =*k* ^ <sup>8</sup> =0.

**Figure 2.** First four modes for the clamped-elastically supported shell; *k*

*^*

451.242 513.222 886.656 1023.61

^

7 is increased from 0.01 to 0.1 m-1, the third natural frequency goes from 886.66 to

7 (from 0.01 to 0.1 m-1) has significantly raised the natural frequencies for the first

*^*

627.345 679.082 926.173 1200.88

^

<sup>7</sup> =0.01 <sup>m</sup>-1and*<sup>k</sup>*

^ <sup>5</sup> =*k* ^ <sup>6</sup> =*k* ^ <sup>8</sup> =0.

<sup>7</sup>=1 m-1, as shown in Figs. 3 and 4.

**^**

729.593 935.745 1084.91, 1319.99, ,

**<sup>7</sup> =1** *k*

**^ <sup>7</sup> =10<sup>10</sup>**

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 227

742.920 936.719 1269.58 1333.37

*<sup>7</sup> = 0.1 k*

926.17 Hz. However, this frequency drift may not necessarily reflect the direct effect of the stiffness change on the (original) third mode. It is evident from Figs. 2 and 3 that the third and fourth modes are actually switched in these two cases: the original third mode now be‐ comes the fourth at 1200.88 Hz. It is also interesting to note that while stiffening the elastic

two modes, the fourth mode is adversely affected: its frequency has actually dropped from 1023.61 to 926.17 Hz (see Figs. 2 and 3). A similar trend is also observed between the fourth

^

*<sup>7</sup> = 0.01 k*

**Table 6.** Natural frequencies in Hz for a clamped-elastically supported shell; *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m,

After it has been adequately illustrated how the classical boundary conditions can be easily and universally dealt with by simply changing the stiffness values of the restraining springs, we will direct our attention to shells with elastic end restraints. For the purpose of compari‐ son, the problems previously studied in ref. [20] will be considered here. It was observed in that study that the tangential stiffness had the greatest effect on the natural frequency of the cylinder supported at both ends while the axial boundary stiffness had the greatest influ‐ ence on the natural frequency of the cylinder supported at one end. It was also determined that natural frequencies varied rapidly with the boundary flexibility when the non-dimen‐ sionalized stiffness is between 10-2 and 102 .

The frequency parameters for the "clamped"-free shell are shown in Table 5 for the reduced axial stiffness *k* ^ <sup>1</sup>*<sup>L</sup>* (1−*<sup>μ</sup>* <sup>2</sup> )=1 (corresponding to *ku* \* =1 in ref. [20]). It is seen that the current results are slightly larger than those taken from ref. [20]. The possible reasons include: 1) the difference in shell theories (the Flügge theory, rather than the Donnell-Mushtari, was used there), and 2) different Poisson ratios may have been used in the calculations.


Note: the numbers in parentheses are taken from ref. [20]

**Table 5.** Frequency parameters, Ω =ω*R* ρ(1−σ <sup>2</sup> ) / *E*, for a "clamped"-free shell; *R*=0.00625 m, *L*=*R*, *h*=0.1*R*, μ=0.28, and*k* ^ <sup>1</sup>*<sup>L</sup>* (1−<sup>μ</sup> <sup>2</sup> )=1.

Although all eight sets of springs can be independently specified here, for simplicity we will only consider a simple configuration: a cantilevered shell with an elastic support being attach‐ ed to its free (right) end in the radial direction. Listed in Table 6 are the four lowest natural fre‐ quencies for several different stiffness values. Obviously, the cases for *k* ^ <sup>7</sup> =0 and *∞* represent the clamped-free and clamped-simply supported boundary conditions, respectively.

The mode shapes for the three intermediate stiffness values are plotted in Figs. 2-4. It is seen that the modal parameters can be significantly modified by the stiffness of the restraining springs. The four modes in Fig. 2 for *k* ^ <sup>7</sup> =0.01 <sup>m</sup>-1 closely resemble their counterparts in the clamped-free case, even though the natural frequencies have been modified noticeably. While all the first four natural frequencies happen to increase, more or less, with the spring stiffness, the modal sequences are not necessarily the same. For example, when the spring stiffness *k* ^ 7 is increased from 0.01 to 0.1 m-1, the third natural frequency goes from 886.66 to 926.17 Hz. However, this frequency drift may not necessarily reflect the direct effect of the stiffness change on the (original) third mode. It is evident from Figs. 2 and 3 that the third and fourth modes are actually switched in these two cases: the original third mode now be‐ comes the fourth at 1200.88 Hz. It is also interesting to note that while stiffening the elastic support *k* ^ 7 (from 0.01 to 0.1 m-1) has significantly raised the natural frequencies for the first two modes, the fourth mode is adversely affected: its frequency has actually dropped from 1023.61 to 926.17 Hz (see Figs. 2 and 3). A similar trend is also observed between the fourth mode for *k* ^ 7=0.1 m-1 and the second mode for *k* ^ <sup>7</sup>=1 m-1, as shown in Figs. 3 and 4.

to have more serious consequence in such a case. Amazingly enough, the inextensional theo‐ ry works very well in predicting the frequency parameters for the "rigid-body" modes (those with rigid-body motions in the axial direction). It is also seen that the frequency pa‐ rameters of the rigid-body modes increases monotonically with the circumferential modal

After it has been adequately illustrated how the classical boundary conditions can be easily and universally dealt with by simply changing the stiffness values of the restraining springs, we will direct our attention to shells with elastic end restraints. For the purpose of compari‐ son, the problems previously studied in ref. [20] will be considered here. It was observed in that study that the tangential stiffness had the greatest effect on the natural frequency of the cylinder supported at both ends while the axial boundary stiffness had the greatest influ‐ ence on the natural frequency of the cylinder supported at one end. It was also determined that natural frequencies varied rapidly with the boundary flexibility when the non-dimen‐

.

)=1 (corresponding to *ku*

there), and 2) different Poisson ratios may have been used in the calculations.

0.514686 1.12788

quencies for several different stiffness values. Obviously, the cases for *k*

the clamped-free and clamped-simply supported boundary conditions, respectively.

^

The frequency parameters for the "clamped"-free shell are shown in Table 5 for the reduced

results are slightly larger than those taken from ref. [20]. The possible reasons include: 1) the difference in shell theories (the Flügge theory, rather than the Donnell-Mushtari, was used

**Mode** *n=0 n=1 n=2 n=3 n=4 n=5*

0.32866 (0.315\*) 1.08573

Although all eight sets of springs can be independently specified here, for simplicity we will only consider a simple configuration: a cantilevered shell with an elastic support being attach‐ ed to its free (right) end in the radial direction. Listed in Table 6 are the four lowest natural fre‐

The mode shapes for the three intermediate stiffness values are plotted in Figs. 2-4. It is seen that the modal parameters can be significantly modified by the stiffness of the restraining

clamped-free case, even though the natural frequencies have been modified noticeably. While all the first four natural frequencies happen to increase, more or less, with the spring stiffness, the modal sequences are not necessarily the same. For example, when the spring

0.361036 1.10467

) / *E*, for a "clamped"-free shell; *R*=0.00625 m, *L*=*R*, *h*=0.1*R*, μ=0.28,

\*

=1 in ref. [20]). It is seen that the current

0.532604 (0.498) 1.16021

^

<sup>7</sup> =0.01 <sup>m</sup>-1 closely resemble their counterparts in the

0.782661 1.432

<sup>7</sup> =0 and *∞* represent

index *n*.

axial stiffness *k*

*m*=1 m=2

and*k* ^ <sup>1</sup>*<sup>L</sup>* (1−<sup>μ</sup> <sup>2</sup>

sionalized stiffness is between 10-2 and 102

226 Advances in Vibration Engineering and Structural Dynamics

<sup>1</sup>*<sup>L</sup>* (1−*<sup>μ</sup>* <sup>2</sup>

0.9752 1.22044

**Table 5.** Frequency parameters, Ω =ω*R* ρ(1−σ <sup>2</sup>

springs. The four modes in Fig. 2 for *k*

)=1.

Note: the numbers in parentheses are taken from ref. [20]

^


**Table 6.** Natural frequencies in Hz for a clamped-elastically supported shell; *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, *E*=2.1E+11, μ=0.28, ρ=7800 kg/m3;*k* ^ <sup>5</sup> =*k* ^ <sup>6</sup> =*k* ^ <sup>8</sup> =0.

**Figure 2.** First four modes for the clamped-elastically supported shell; *k* ^ <sup>7</sup> =0.01 <sup>m</sup>-1and*<sup>k</sup>* ^ <sup>5</sup> =*k* ^ <sup>6</sup> =*k* ^ <sup>8</sup> =0.

**3.2. An exact solution based on the Flügge's equations**

To validate the exact solution method, the simply supported shell is considered again. Given

results agree well with the exact solutions based on Flügge's theory [30], solutions based on

**) /** *E*

**Ref. [30]a Ref. [31] Ref. [30]b Present**

 0.0929586 0.0929682 0.0929296 0.0929590 0.0161065 0.0161029 0.0161063 0.0161064 0.0393038 0.0392710 0.0392332 0.0393035 0.1098527 0.1098113 0.1094770 0.1098468 0.2103446 0.2102770 0.2090080 0.2103419

 0.0929296 0.0929298 0.0929296 0.0929299 0.0161011 0.0161011 0.0161011 0.0161023 0.0054532 0.0054530 0.0054524 0.0054547 0.0050419 0.0050415 0.0050372 0.0050427 0.0085341 0.0085338 0.0085341 0.0085344

*m = 1 m = 2* **FEM present difference (%) FEM present difference (%)**

 3229.8 3230.3 0.015% 5131.4 5131.1 0.006% 2478.6 2479.3 0.028% 4830.4 4830.6 0.004% 269.20 269.30 0.037% 276.62 278.58 0.704% 761.25 761.01 0.032% 770.99 771.62 0.082% 1459.2 1458.6 0.041% 1469.6 1469.3 0.020% 2359.4 2358.6 0.034% 2369.9 2369.0 0.038%

**Table 8.** Comparison of values of the natural frequency for a circular cylindrical shell with free-free boundary

conditions, *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, σ =0.28, *E*=2.1E+11 N/m3, ρ=7800 kg/m3.

) / *E*. The current

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 229

) / *E* for a circular cylindrical shell

in Table 7 are the calculated natural frequency parameters*Ω* =*ωR ρ*(1−*σ* <sup>2</sup>

beam functions [31] and three-dimensional linear elasticity solutions [30].

**h/R <sup>n</sup> <sup>Ω</sup> <sup>=</sup>ω***<sup>R</sup>* **<sup>ρ</sup>(1−<sup>σ</sup> <sup>2</sup>**

**Table 7.** Comparison of values of the natural frequency parameter Ω =ω*R* ρ(1−σ <sup>2</sup>

with simply supported boundary conditions, *m* = 1, *R/l* = 0.05, σ = 0.3.

0.05

0.002

a

*n*

 Exact solutions based on Flügge's theory. b Three-dimensional linear elasticity solutions.

**Figure 3.** First four modes for the clamped-elastically supported shell; *k* ^ <sup>7</sup> =0.1 <sup>m</sup>-1and*<sup>k</sup>* ^ <sup>5</sup> =*k* ^ <sup>6</sup> =*k* ^ <sup>8</sup> =0.

**Figure 4.** First four modes for the clamped-elastically supported shell; *k* ^ <sup>7</sup> =1 <sup>m</sup>-1and*<sup>k</sup>* ^ <sup>5</sup> =*k* ^ <sup>6</sup> =*k* ^ <sup>8</sup> =0.

## **3.2. An exact solution based on the Flügge's equations**

To validate the exact solution method, the simply supported shell is considered again. Given in Table 7 are the calculated natural frequency parameters*Ω* =*ωR ρ*(1−*σ* <sup>2</sup> ) / *E*. The current results agree well with the exact solutions based on Flügge's theory [30], solutions based on beam functions [31] and three-dimensional linear elasticity solutions [30].


a Exact solutions based on Flügge's theory.

**Figure 3.** First four modes for the clamped-elastically supported shell; *k*

228 Advances in Vibration Engineering and Structural Dynamics

**Figure 4.** First four modes for the clamped-elastically supported shell; *k*

^

^

<sup>7</sup> =1 <sup>m</sup>-1and*<sup>k</sup>*

^ <sup>5</sup> =*k* ^ <sup>6</sup> =*k* ^ <sup>8</sup> =0.

<sup>7</sup> =0.1 <sup>m</sup>-1and*<sup>k</sup>*

^ <sup>5</sup> =*k* ^ <sup>6</sup> =*k* ^ <sup>8</sup> =0.

b Three-dimensional linear elasticity solutions.

**Table 7.** Comparison of values of the natural frequency parameter Ω =ω*R* ρ(1−σ <sup>2</sup> ) / *E* for a circular cylindrical shell with simply supported boundary conditions, *m* = 1, *R/l* = 0.05, σ = 0.3.


**Table 8.** Comparison of values of the natural frequency for a circular cylindrical shell with free-free boundary conditions, *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, σ =0.28, *E*=2.1E+11 N/m3, ρ=7800 kg/m3.

The current solution method is also compared with the finite element model (ANSYS) for shells under free-free boundary condition. In the FEM model, the shell surface is divided in‐ to 8000 elements with 8080 nodes. The calculated natural frequencies are compared in Ta‐ bles 8. An excellent agreement is observed between these two solution methods.

the accuracy and convergence of the present solution. From practical point of view, the change of boundary conditions here is as simple as varying a typical shell or material pa‐ rameter (e.g., thickness or mass density), and does not involve any solution algorithm and procedure modifications to adapt to different boundary conditions. In addition, the pro‐ posed method does not require pre-determining any secondary data such as modal parame‐ ters for an "analogous" beam, or modifying the implementation algorithms to avoid the numerical instabilities resulting from computer round-off errors. It should be mentioned that the current method can be readily extended to shells with arbitrary non-uniform elastic restraints. The accuracy and reliability of the current solutions have been demonstrated

Vibrations of Cylindrical Shells http://dx.doi.org/10.5772/51816 231

The authors gratefully acknowledge the financial support from the National Natural Science

1 College of Power and Energy Engineering, Harbin Engineering University, Harbin, PR

[1] Influence of boundary conditions on the modal characteristics of thin cylindrical

[2] Forsberg, K. Axisymmetric and beam-type vibrations of thin cylindrical shells,.

[3] Warburton, G. B. Vibrations of thin circular cylindrical shell, J. Mech. Eng. Sci.

[4] Warburton, G. B., & Higgs, J. Natural frequencies of thin cantilever cylindrical shells.

[5] Goldman, R. L. (1974). Mode shapes and frequencies of clamped-clamped cylindrical

2 Department of Mechanical Engineering, Wayne State University, Detroit, USA

through numerical examples involving various boundary conditions.

and Lu Dai1

**Acknowledgments**

**Author details**

Tiejun Yang1

**References**

China

Foundation of China (No. 50979018).

, Wen L. Li2

shells,. AIAA Journal ; , 2-2150.

AIAA Journal (1969). , 7-221.

J. Sound Vib. (1970). , 11-335.

shells. *AIAA Journal*, 12-1755.

(1965). , 7-399.

In most techniques, such as the wave approach, the beam functions for the analogous boun‐ dary conditions are often used to determine the axial modal wavenumbers. While such an approach is exact for a simply supported shell, and perhaps acceptable for slender thin shells, it may become problematic for shorter shells due to the increased coupling of the ra‐ dial and two in-plane displacements. To illustrate this point, we consider relatively shorter and thicker shell (*l*=8*R* and *R* =39*h*). The calculated natural frequencies are compared in Ta‐ ble 9 for a clamped-clamped shell. It is seen that while the current and FEM results are in good agreement, the frequencies obtained from the wave approach (based on the use of beam functions) are significantly higher, especially for the lower order modes.


**Table 9.** Comparison of the natural frequencies for a circular cylindrical shell with clamped-clamped boundary conditions, *L*=0.502 m, *R*=0.0635 m, *h*=0.00163 m, σ=0.28, E=2.1E+11 N/m3, ρ=7800 kg/m3.

The exact solution method can be readily applied to shells with elastic boundary supports. Since the above examples are considered adequate in illustrating the reliability and accuracy of the current method, we will not elaborate further by presenting the results for elastically restrained shells. Instead, we will simply point out that the solution method based on Eqs. (27) is also valid for non-uniform or varying boundary restraint along the circumferential di‐ rection, which represents a significant advancement over many existing techniques.
