Vibration analysis of a cylinder with slight diameter and thickness variations

2.1. Asymmetry examination of a typical cylinder

Fig. 1 shows a typical circular cylinder used for this study. The length, maximum inner and outer diameters, and wall thickness of the cylinder are, respectively, 1.2 m, 0.156 m, 0.170 m and 7 mm. The inner and outer diameters are not constant. The diameters were measured using a pair of Vernier scale calipers from 0° (defined at the welding line) of the cylinder coordinates to 170°, with a 10° angular interval. Their diameter variations around the average values are plotted in Fig. 2. These are within ±0.6 mm, which is about 0.4 % of the average diameter. The different variations of the inner and outer diameters at the same angular positions indicate a variation in the cylinder’s wall thickness. The measured wall thickness variations at the top and bottom ends are shown in Fig. 3. They are within ±0.22 mm, which is about 3 % of the average thickness.

Fig. 1The cylinder used for the FEM and the experimental study, and its coordinates

Another important observation from Fig. 2 is that the maximum diameter variation at the top end ($z=$0 m) occurs at a different angular location from that at the bottom end ($z=$1.2 m). At the bottom end, it leads by approximately 20°. Based on this observation, we postulate that the diameter variation pattern gradually shifts, along the negative $\theta$ direction, from the bottom end of the cylinder to the top end, and that the angle of shift per unit length along the axial direction is 20/1.2°/m. Although this assumption could be justified by a direct diameter measurement after cutting the cylinder at many cross sections along its length, as will be shown, the angular shift of measured axial nodal lines of some mode shapes supports this assumption.

Fig. 2Measured variation of diameters at a) the top end, z= 0 m, and b) the bottom end (z= 1.2 m) of the cylinder

Fig. 3Measured variations of the cylinder wall thickness

2.2. Modeling of cylinder vibration

Although the vibration of a cylinder with slight variations has no general analytical solution, it can be readily modeled and solved by numerical schemes such as the FEM. In the FEM, the cylinder is divided into a number of elements. Each element is connected to adjacent elements at its corners, known as nodal points. The displacement vector of the element is described by the nodal displacements and a set of coefficients using a shape function [18, 19]:

where, $u_i$, $v_i$ and $w_i$are nodal displacement of node $i$ in the Cartesian coordinate system, the relationship between stress and strain in the element can be written as [19]:

where, $\sigma$ is stress vector $\left\{{\sigma }_x,{\sigma }_{y,}{\sigma }_{z,}{\sigma }_{xy},{\sigma }_{xz},{\sigma }_{yz}\right\}$, $D$ is elasticity or elastic stiffness matrix, $\epsilon$ is total strain vector $\left\{{\epsilon }_x,{\epsilon }_{y,}{\epsilon }_{z,}{\gamma }_{xy},{\gamma }_{xz},{\gamma }_{yz}\right\}$, which is described as[19]:

The element stiffness matrix can be derived and written as [19]:

where [$J$] is Jacobian matrix, other definitions of matrix can be found in reference [19]. Accordingly, the mass matrix is calculated and shown as:

The element stiffness and mass matrices are assembled in to get global matrices. The equation of motion of the cylinder can be written as:

To calculate the vibration characteristics of the cylinder, 10 nodes tetrahedral element is selected for meshing the geometry. For exist of the middle nodes in every edge of element, internal shear stress can be calculated by Eq. (2) and Eq. (3). Internal moment is also achieved by the follow equation:

Fig. 4FEM model of the free cylinder

In this paper, the geometric model of a cylinder with slight thickness and diameter variation is established in SOLIDWORKS by data of Fig. 2 and 3 with consideration of assumptions in Section 2.1. Then, it is imported into FEM software ANSYS and meshed by 10 nodes tetrahedral element. The finite element model of the cylinder with free boundary condition is obtained and shown in Fig. 4. Through mode analysis module in the software, the natural frequencies and mode shapes of the free cylinder are calculated by solving eigenvalue problem of Eq. (6).

3.1. Characteristics of simplified models

To study the sensitivity of the vibration characteristics of the cylinder to each variation, the diameter and thickness variations are studied separately using two simplified models.

For the first simplified model, the cross-section profile at the axial coordinate $z$ of the cylinder is described by an ellipse with a constant wall thickness. The inner $r_i\left(\theta ,z\right)$ and $r_o\left(\theta ,z\right)$ outer radii are expressed by:

where $2\mathrm{\Delta }r$ is the maximum diameter variation from the average diameter ($2\overline{r_{i,o}}$) of the cylinder and $\mathrm{\Phi }\left(z\right)$ is the angular shift of the cross-section profiles along the axial direction. $r_{i,o}\left(\theta ,z\right)$ is common function to express $r_i\left(\theta ,z\right)$ or $r_o\left(\theta ,z\right)$. If the variation profile at the top end shifts with respect to that at the bottom end by a twist angle $\mathrm{\Phi }\left(z\right)=20/180\pi$ (as determined experimentally) and the shift is linear along the $z$-direction, then $\mathrm{\Phi }\left(z\right)$ is expressed as:

where ${\mathrm{\Phi }}_o=\pi /3$ is an initial phase angle at the top end.

Based on the measured inner and outer diameters shown in Fig. 2, the major and minor axes of the outer ellipse are selected, respectively, as 0.085 m and 0.0845 m. Those of the inner ellipse are 0.078 m and 0.0775 m. For this case, the wall thickness is 7 mm. Fig. 5 shows the diameter variations at the bottom and top ends of the cylinder.

The second simplified model describes the wall thickness variation by a slight shift of the inner circular wall from the center. The thickness of the cylinder is then expressed by:

where $∆t$ is the maximum thickness variation, ${\overline{r}}_o$ and ${\overline{r}}_i$ are the outer and inner radii and $\delta \left(z\right)$ defines the shift angle of the thickness variation profile along the $z$-direction. If the maximum thickness variation at the top end leads that at the bottom by a twist angle ${\delta }_T$ and the twist of the thickness variation profile is linear along the $z$-direction, then:

where ${\delta }_0$ is an initial angle of the thickness variation at the top end.

Based on the measured wall thickness, the maximum thickness variation $∆t$ from the average value is 0.22 mm. For this model the inner and outer diameters are 0.156 m and 0.170 m respectively. Using the angular positions of the first peaks in the thickness variation curves in Fig. 3, we obtained ${\delta }_T=–80/180\pi$ and ${\delta }_0=-2/9\pi$. Fig. 6 shows the thickness variations used in the second simplified model.

Fig. 5Diameter variation used by the first simplified model

Fig. 6Thickness variation used by the second simplified model

For comparison purposes, the model characteristics of a circular free cylinder without diameter and thickness variation are also analyzed. The length, inner and outer diameters, and wall thickness of this cylinder are, respectively, 1.2 m, 0.156 m, 0.170 m and 7 mm.

The first ten natural frequencies of the free cylinder described by the two simplified models are obtained using FEM and listed in Table 1, along with those of the circular cylinder without any variation. The free-free boundary conditions are assumed for all of the models. The corresponding modal shapes are described by circumferential (m: number of node pairs) and axial (n: number of nodes) node numbers [20] in an ascending order of the natural frequencies. Also, the mode shape of the free cylinder is expended in planar style [20] to show their changes clearly.

It is well known that the modes of the cylinders exist in pairs, in which each pair has the same node numbers and two spatially orthogonal mode shapes. In the Table 1, the natural frequencies of the modal pairs of the cylinder with the specified thickness variation are identical except for a small difference existing in the (1, 2) mode pair.

On the other hand, the natural frequencies of the mode pairs of the free cylinder with diameter variation can be quite different from each other. In comparison with the natural frequency of the circular cylinder, one mode in the pair shifts to the lower frequency and the other shifts to the higher frequency. The shift of some natural frequencies of the mode pairs is so significant that the mode orders change. For example, the second natural frequency of the (2, 0) mode pair becomes higher than the first natural frequency of the (2, 1) mode pair. As a result, the mode order of the latter is ahead of that of the former.

Table 1 demonstrates that different variation affects the natural frequencies of the free cylinder differently. For the selected cylinder, the diameter variation seems to be responsible for the large split of the natural frequencies in the modal pairs. Accordingly, the mode shapes of the first ten modes of the free cylinders with diameter or thickness variations are plotted in Table 3, together with those of the cylinder without variation. Compared with the mode shapes of the free cylinder without variation, the changes of the mode shapes of the cylinders with variations can be summarized as follows.

Table 3The first10 mode shapes of the two simplified models compared with that with no variation.

Mode order (free-free)	($m$, $n$)	No Variation	Thickness Variation	Diameter Variation
1	(1,2)
2	(1,2)
3	(2,0)
4	(2,0)
5	(2,1)
6	(2,1)
7	(2,2)
8	(2,2)
9	(2,3)
10	(2,3)

The angular locations of the axial nodal lines shift from that of the free cylinder without variation; and the axial nodal lines of some modes are twisted. In which the angular shift of the axial nodal lines is measured by the difference between the angular position of the first nodal line of the mode of the free cylinder with variation and that of the cylinder without variation. The positions of the two ends of the first nodal lines are marked by red dots on model shapes in Table 3. The twist angle (${\psi }_T$) of each axial nodal line is measured by the difference of the angular position of the first axial nodal line at $Z=L$ (the ending angle ${\psi }_e$) and that at $Z=$0 (the starting angle ${\psi }_s$). For the three cylinder models, the starting and ending angles of the first axial node lines of the first ten modes are listed in Table 2. This illustrates that the mode shapes of the cylinders are sensitive to both thickness and diameter variations.

Clearly, the asymmetry created by the variations causes the shift and twist of the angular positions of the axial nodal lines from that of the cylinder without variations in Table 3. It is also revealed from Table 3 that angular shift or twist of the axial nodal lines appear more obviously in diameter variation model, especially at third ,4th, 5th mode shape. It is seem that the diameter variation described by the simplified model captures the key influencing elements that affect the modal characteristics of the cylinder. To validate above conclusion and reveal the relationship between initial angle Φ0 or the twist angle $\mathrm{\Phi }\mathrm{T}$ and thickness or diameter variations, the effects of a slight variation in its diameter and thickness on structural dynamic properties are studied as follows.

To demonstrate the dependence of the shift and twist of the axial nodal lines on ${\mathrm{\Phi }}_0$, ${\mathrm{\Phi }}_T$, ${\delta }_0$and ${\delta }_T$, four cases of parameter study are carried out using the two simplified variation models.

3.1.1. Case 1: 90° $\le {\mathit{\delta }}_0\le \mathbf{}$180° and ${\mathit{\delta }}_{\mathit{T}}=$ –80° with thickness variation

Fig. 7 shows the starting angle ${\psi }_S$ of the first ten modes as a function of the initial angle ${\delta }_0$ in the thickness variation. For this case, the twist angle of the thickness profile is kept as a constant. Results for the same circumferential and axial node numbers are grouped together for comparison. A simple linear relationship between ${\psi }_S$ and ${\delta }_0$ is found except for a 180° jump in ${\psi }_S$ for the first mode in Fig. 7(a). There are two axial nodal lines for each of the (1, 2) modes and they are 180° apart. An increase of ${\delta }_0$ increases both angular positions of the two axial nodal lines at $z=$0. When the angular position of the first axial nodal line exceeds 180^o, the starting angle of the second axial nodal line passes 360° and becomes ${\psi }_S$, which therefore results in the jump.

Table 2 shows that the difference between the first node lines of the modes in the modal pair is either 90° ($m=$1) or 45° ($m=$2). However, Fig. 7 shows that, due to thickness variation, the difference for the modes in the (2, 1) modal pair is 48° (see Fig. 7(c)), while the difference for the modes in (2, 0) pair is 36° (see Fig. 7(b)).

Fig. 7Starting angles of the first ten modes as functions of initial angle in the thickness variation model: a) (1, 2) modal pair; b) (2, 0) modal pair; c) (2, 1) modal pair; d) (2, 2) and (2, 3) modal pairs

3.1.2. Case 2: ${\mathit{\delta }}_0=\mathbf{}$140° and –90°$\mathbf{}\le {\mathit{\delta }}_{\mathit{T}}\le \mathbf{}$0° with thickness variation

Fig. 8 shows the starting angle ${\psi }_s$of the modes as a function of ${\delta }_T$, while ${\delta }_0$ is kept as a constant. For the (1, 2), (2, 2) and (2, 3) modes, ${\psi }_s$ and ${\delta }_T$ are linearly related. For the other modes, ${\psi }_s$ fluctuates around a straight line of best fit. The maximum fluctuation angle is 6°. As ${\delta }_T$, varies from 0° to –90°, the difference in the starting angles between the two modes in the (2, 0) mode pair decreases from 45° to 36°, and that of the (2, 1) mode pair increase from 45° to 48°.

Fig. 8Starting angles of the first ten modes as functions of twist angle in the thickness variation model: a) (1, 2) modal pair; b) (2, 0) modal pair; c) (2, 1) modal pair; d) (2, 2) and (2, 3) modal pairs

For thickness variation, the axial nodal lines of the 3rd, 4th, 5th and 6th modes in the (2, 0) and (2, 1) modal pairs are twisted slightly. Depending on ${\delta }_T$, the magnitudes of the twist angles $\left|{\psi }_T\right|$ are less than or equal to 6° as shown in Fig. 9.

Fig. 9Twist angles of the 3rd, 4th, 5th and 6th modes as a function of twist angle in the thickness variation model

3.1.3. Case 3: 90°$\mathbf{}\le {\mathbf{\Phi }}_0\le \mathbf{}$180° and ${\mathbf{\Phi }}_{\mathit{T}}=$ 20° with diameter variation

Fig. 10 illustrates ${\psi }_s$ as a function of the initial angular position ${\mathrm{\Phi }}_0$ of the diameter variation and when ${\mathrm{\Phi }}_T=$ 20°. A linear relationship exists between ${\psi }_s$ and ${\mathrm{\Phi }}_0$. The differences between the starting angles of the modes in the (2, 0) and (2, 1) modal pairs become 24° and 42°, respectively.

Fig. 10Starting angles of the first ten modes as functions of initial angle in the diameter variation model: (a) (1, 2) modal pair; (b) (2, 0) modal pair; (c) (2, 1) modal pair; (d) (2, 2) and (2, 3) modal pairs

3.1.4. Case 4: ${\mathbf{\Phi }}_0=\mathbf{}$60° and 0°$\mathbf{}\le {\mathbf{\Phi }}_{\mathit{T}}\le \mathbf{}$90° with diameter variation

Fig. 11 shows that ${\psi }_s$ is linearly related to ${\mathrm{\Phi }}_T$ of the modes in the (1, 2), (2, 2) and (2, 3) modal pairs. However, no exact linear relation between ${\psi }_s$ and ${\mathrm{\Phi }}_T$ can be found for the (2, 0) and (2, 1) modal pairs.

For diameter variation, the twist of the axial nodal lines of the 3rd, 4th, 5th and 6th modes in the (2, 0) and (2, 1) modal pairs are significantly affected by the twisted angle ${\mathrm{\Phi }}_T$ in the diameter variation model. The ${\psi }_T$ of the 3rd and 4th modes increases linearly while ${\mathrm{\Phi }}_T$ increases from 0° to 90° and are summarized by the following expressions:

For the 3rd mode:

12

${\psi }_T=\frac{{\varphi }_T}{2}.$

For the 4th mode:

13

${\psi }_T=\frac{{\varphi }_T}{2}-\frac{3\pi }{4}.$

However, ${\psi }_T$ and ${\mathrm{\Phi }}_T$ for the 5th and 6th modes in Fig. 12 are not linearly related. Comparing the twist angles in Fig. 8 with those in Fig. 12, a value of ${\psi }_T$ caused by ${\mathrm{\Phi }}_T$ of diameter variation can be much larger than that caused by ${\delta }_T$ of thickness variation. Thus, when both thickness and diameter variations are present, the large twist of the mode shape is caused by the twist in diameter variation.

Even though the curve-fitted relationships, shown in Figs. 7 to 12, between the changes in mode shapes (in terms of starting and twist angles of the nodal lines) diameter and thickness variations (in terms of initial angles ${\delta }_0$, ${\mathrm{\Phi }}_0$ and twist angles ${\delta }_T$, ${\mathrm{\Phi }}_T$ of the variation) are obtained from the two simplified models, a simple and linear dependence of the changes in mode shapes on the characteristic angles of the variation is observed for most of the modes in the free cylinder. They are useful for an estimate of the effect of variations on the magnitude and direction of the changes in free mode shapes.

Fig. 11Starting angles of the first ten modes as functions of twist angle in the diameter variation model: a) (1, 2) modal pair; b) (2, 0) modal pair; c) (2, 1) modal pair; d) (2, 2) and (2, 3) modal pairs

Fig. 12Twist angles of the 3rd, 4th, 5th and 6th modes as a function of twist angle in the diameter variation model: a) (2, 0) modal pair; b) (2, 1) modal pair

3.2. Characteristics of a realistic model

In this section, a realistic model of the variation profiles of the cylinder is used in FEM simulations so that the predicted free modal characteristics of the cylinder shown in Fig. 1 can be compared with measured results. Fig. 2 shows two positive peaks in the measured diameter variation. In subsection above, only the first peak was described by Eq. (8) for simplification purposes. To include both peaks in the diameter variation profile, the inner $r_i\left(\theta ,z\right)$ and outer $r_o\left(\theta ,z\right)$ radii are expressed as follows:

For $0\le \frac{{\phi }_1\left(z\right)+{\phi }_2\left(z\right)}{2}\le \frac{\pi }{2}$:

For $\frac{\pi }{2}<\frac{{\phi }_1\left(z\right)+{\phi }_2\left(z\right)}{2}\le \pi$:

where $2∆r$ is the maximum variation in the diameters from that of the circular cylinder $2r_{i.o}$ and ${\phi }_1\left(z\right)$ and ${\phi }_2\left(z\right)$ are the relative angular shifts of the elliptical profiles along the axial direction. If the diameter variation profile at the top end shifts with respect to that at the bottom end by a twist angle ${\phi }_T$ and the shift is linear along the $z$-direction, then ${\phi }_1\left(z\right)$ and ${\phi }_2\left(z\right)$ are expressed as:

where ${\phi }_a$ and ${\phi }_b$ are the initial phase angles of the diameter profiles. According to the experimental data on the external profiles of the cross section, ${\phi }_T=20/180\pi$, ${\phi }_a=60/180\pi$ and ${\phi }_a=130/180\pi$. The major and minor axes of the external ellipses are, respectively, $r_0=$0.085 m and $r_0-∆r=$0.0845 m. Fig. 13 shows the diameter variation at the bottom and top ends of the external profiles of the cylinder.

For the internal profile of the cross section, ${\phi }_T=20/180\pi$, ${\phi }_a=70/180\pi$ and ${\phi }_a=140/180\pi$. The major and minor axes of the external ellipses are $r_0=$0.078 m and $r_0-∆r=$0.0775 m respectively. The diameter variation of the internal profile is plotted in Fig. 14. The 10° shift of the initial phase angles between the internal and external diameter variations describes the variation in wall thickness of this model.

Fig. 13Diameter variation of the external profile of a realistic cylinder model

Fig. 14Diameter variation of the internal profile of a realistic cylinder model

The natural frequencies, starting and ending angles of the first axial nodal line of each mode are listed in Table 4, together with the measured results. The corresponding FEA and measured mode shapes are plotted in Table 5. Although the realistic model includes more complex diameter variation and thickness variation, the general trends in the shift of natural frequencies and changes in starting and ending angles of the first nodal lines of the free modes are similar to those due to diameter variation in the simplified model. In summary:

Diameter variation plays a major role in generating large frequency shifts and large changes in the starting and ending angles. It is also responsible for the large twist angles of the first nodal lines of some modes. So, the diameter variation described by the simplified model captures the key influencing elements that affect the modal characteristics of the cylinder.