Optimizing Fundamental Frequencies in Axially Compressed Rotating Laminated Cylindrical Shells

Abstract

Optimizing composite materials, particularly in rotating structures, offers several practical benefits in the mechanical engineering and aerospace engineering industries. Improved material configurations, such as optimal fiber orientations, enhance the structural performance by maximizing stiffness-to-weight ratios and reducing vibrations. This study optimized the fundamental frequencies of rotating laminated cylindrical shells using the golden section method with respect to fiber orientations. The investigation explored the impact of various factors such as end conditions, shell length, axial compressive force, rotating speed, and the size of the cutout on the maximum fundamental frequencies. Additionally, the associated vibration modes and optimal fiber orientations were demonstrated in relation to these influencing parameters. Generally, it was observed that the optimal frequency decreased with increasing length-to-radius ratio and compressive force.

1. Introduction

Fiber-reinforced composite laminated materials (Figure 1) offer significant benefits to mechanical engineering and aerospace engineering due to their unique properties, such as their exceptional strength-to-weight ratio, providing a high level of structural integrity while increasing payload capacity. Moreover, these materials exhibit excellent resistance to corrosion, fatigue, and impact, resulting in improved durability and longevity of aerospace components. The tailored design flexibility of composite laminates allows for the optimization of structural performance, enabling engineers to meet specific requirements and address complex geometries more effectively. As a result, fiber-reinforced composite laminated materials play a crucial role in various applications in mechanical and aerospace structures, with one notable instance being their use in the construction of rotating cylindrical shells, such as aircraft jet engines and motors. These structures experience dynamic loads during operation. Therefore, understanding the dynamic characteristics of rotating cylindrical shells, including their fundamental natural frequencies, is crucial [1,2,3,4].

The fundamental natural frequency of rotating laminated cylindrical shells is significantly influenced by boundary conditions [2,5,6,7], axial force [4,8], rotation speed [8,9,10,11,12], ply orientation [9,10,13,14], and geometric factors such as thickness [5,10,11], length-to-radius ratio [12,13,15,16,17], and cutouts [18,19]. Thus, selecting the right lamination to optimize the fundamental frequency of rotating laminated cylindrical shells is a critical issue [20,21,22,23,24,25].

Structural optimization involves mechanics, mathematics, computer science, and other engineering fields. It has become a key advancement in modern designs [26,27]. Of the various optimization approaches, the golden section method is a simple technique that can be easily programmed for computer-based solutions [26,28]. This investigation sought to optimize the fundamental natural frequency of rotating, axially compressed laminated cylindrical shells by changing fiber angles through the golden section method. The fundamental frequencies of these shells were obtained using the Abaqus finite element program [29]. The material constitutive equations for fiber-reinforced lamina [22,30,31,32] and the principles of the golden section method [5,22,26,28] are well-established in textbooks and research articles; therefore, they are not included in this paper. Instead, the focus is on a brief review of the vibration analysis. The paper then discusses the effects of end conditions, rotational speed, axial compressive force, the length-to-radius ratio, and cutouts on the maximum fundamental natural frequency, optimal fiber orientations, and vibration modes of the laminated cylindrical shells, concluding with significant findings.

2. Vibration Analysis

For free vibration analysis, the motion of an undamped structure can be written in the following equation [33]:

$$[\mathrm{M}]\{\ddot{\mathrm{D}}\}+[\mathrm{K}\left]\right\{\mathrm{D}\}=\{0\}$$

(1)

The {0} is a zero vector. The displacement vector {D} can be expressed as a harmonic motion as

$$\left\{\mathrm{D}\right\}=\{\overline{\mathrm{D}}\}\sin \omega \mathrm{t};. \left\{\ddot{\mathrm{D}}\right\}=-{\omega }^2\left\{\overline{\mathrm{D}}\right\}\sin \omega \mathrm{t}$$

(2)

where $\omega$ is the nature frequency and $\left\{\overline{\mathrm{D}}\right\}$ is the amplitude of {D}. Then, Equation (1) becomes

$$([\mathrm{K}]-{\omega }^2[\mathrm{M}])\{\overline{\mathrm{D}}\}=\{0\}$$

(3)

When a laminated cylindrical shell rotates about its axial direction, centrifugal forces develop in the radial direction. Furthermore, axial compression induces axial compressive forces. The centrifugal and axial compressive forces act as initial loads, producing initial stresses in the shell. As a result, the stiffness matrix [K] in Equation (3) can be decomposed into two distinct matrices as follows:

$$\left[\mathrm{K}\right] ={ [\mathrm{K}}_{\mathrm{L}}] +{ [\mathrm{K}}_{\sigma }]$$

(4)

where $[{\mathrm{K}}_{\mathrm{L}}]$ is the linear stiffness matrix and ${ [\mathrm{K}}_{\sigma }]$ is the geometric stress stiffness matrix generated by the initial stresses. Now, Equation (3) becomes

$$([{\mathrm{K}}_{\mathrm{L}}]+[{\mathrm{K}}_{\sigma }]-{\omega }^2[\mathrm{M}])\left\{\overline{\mathrm{D}}\right\}=\left\{0\right\}$$

(5)

In the above equation, if $\left\{\overline{\mathrm{D}}\right\}$ is not a zero vector, we must set

$$\left|[{\mathrm{K}}_{\mathrm{L}}]+[{\mathrm{K}}_{\sigma }]-{\omega }^2[\mathrm{M}]\right|=0$$

(6)

In this study, the Abaqus program with the subspace iteration procedure [29] was employed to determine $\omega$ and the corresponding vibration modes. The resulting fundamental frequency was then used as the objective function for maximization.

3. Numerical Analysis

In the numerical analysis, laminated cylindrical shells were modeled with eight-node S8R shell elements. Each node of the element had three degrees of freedom for displacements and three for rotations. The element stiffness matrix was computed using a reduced integration rule, with hourglass stiffness control applied for stability [29].

The accuracy of the shell element in the Abaqus program for frequency analysis has been validated by the authors [5,21,34], showing a high level of agreement between the numerical results and both analytical solutions and experimental data. Thus, the shell element in Abaqus was deemed sufficiently precise for the vibration analysis of laminated cylindrical shells.

3.1. Laminated Cylindrical Shells with Various End Conditions, Axial Compressive Forces, and Rotating Speeds

This section analyzes composite laminated cylindrical shells with two types of end conditions (shown in Figure 2): fixed at both ends (denoted FF) and simply supported at both ends (denoted SS). These boundary conditions were defined in the cylindrical coordinate system, where x denotes the longitudinal direction, y the circumferential direction, and z the radial direction (Figure 1).

The radius R of the shell was 10 cm, while the length L was either 10 cm or 40 cm. The laminate layup of the shells was ${[\pm \theta /90/0]}_{2\mathrm{s}}$, with each ply having a thickness of 0.125 mm. The axial compressive force ranged from 0 to $0.8 {\mathrm{N}}_{\mathrm{c}\mathrm{r}}$, where ${\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ is the critical buckling load of the laminated cylindrical shell. The rotational speed $\Omega$ was either 0 or 20,000 RPM.

The lamina in analysis was composed of graphite/epoxy. Its material properties were taken from Crawley [35], namely, ${\mathrm{E}}_{11}=128\hspace{0.33em}\mathrm{GPa},$ ${\mathrm{E}}_{22}=11\hspace{0.33em}\mathrm{GPa},$ ${\mathrm{\nu }}_{12}=0.25,$ ${\mathrm{G}}_{12}={\mathrm{G}}_{13}=4.48\hspace{0.33em}\hspace{0.33em}\mathrm{GPa},$ ${\mathrm{G}}_{23}=1.53\hspace{0.33em}\hspace{0.33em}\mathrm{GPa},$ and $\rho =1500\hspace{0.33em}\mathrm{K}\mathrm{g}/{\mathrm{M}}^3$. In this investigation, no symmetry simplifications were applied to the shells. Based on prior experience and current results of convergence studies [3,5,36], 160$(32\times 5)$, 320$(32\times 10)$, 480$(32\times 15)$, and 640$(32\times 20)$ shell elements were chosen to model the laminated cylindrical shells with L/R ratios equal to 1, 2, 3, and 4, respectively, in the subsequent analyses.

Figure 3 shows the fiber angle $\theta$ and the associated fundamental frequency $\omega$ for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with L/R = 1, $\Omega =0 \mathrm{RPM}$, different boundary conditions, and various axial compressive forces. From the figure, we can observe that the fundamental frequencies of the laminated cylindrical shells with FF ends were usually higher than those of the shells with SS ends. The fundamental frequencies of the laminated cylindrical shells generally decreased as the axial compressive forces increased. Additionally, the maximum fundamental frequencies of the ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells appeared to occur when the value of $\theta$ fell within the range of $40°$ to $60°$.

Figure 4 shows the fiber angle $\theta$ and the associated fundamental frequency $\omega$ for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with L/R = 1, $\Omega =20,000\hspace{0.33em}\mathrm{r}P\mathrm{M}$, different boundary conditions, and various axial compressive forces. From the figure, we can observe similar trends as those in Figure 3. Comparing Figure 4 with Figure 3, we can see that the fundamental frequencies of the laminated cylindrical shells increased as the rotating speeds $\Omega$ increased.

Figure 5 shows the fiber angle $\theta$ and the associated fundamental frequency $\omega$ for ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with L/R = 4, $\Omega =0 \mathrm{RPM}$, different boundary conditions, and various axial compressive forces. From the figure, we can observe that the fundamental frequencies of the laminated cylindrical shells with FF ends were again higher than those of the shells with SS ends. However, when the L/R ratio increased to 4, the differences in the fundamental frequencies between the shells with FF and SS ends were significantly reduced. Again, the fundamental frequencies of the laminated cylindrical shells generally decreased as the axial compressive forces increased. Furthermore, the maximum fundamental frequencies of the ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells seemed to occur in the region $40°\le \theta \le 60°$ when the axial compressive force was lower than $0.4 {\mathrm{N}}_{\mathrm{c}\mathrm{r}}$. The maximum fundamental frequencies tended to occur in the region $70°\le \theta \le 90°$ when the axial compressive force was larger than $0.6 {\mathrm{N}}_{\mathrm{c}\mathrm{r}}$.

Figure 6 shows the fiber angle $\theta$ and the associated fundamental frequency $\omega$ for ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with L/R = 4, $\Omega =20,000\hspace{0.33em}\mathrm{r}P\mathrm{M}$, different boundary conditions, and various axial compressive forces. Again, the fundamental frequencies of the laminated cylindrical shells decreased as axial compressive forces increased. The differences in the fundamental frequencies between the shells with FF and SS ends were significantly reduced. The maximum fundamental frequencies of these shells seemed to occur either in the region $20°\le \theta \le 40°$ or at $\theta$ of around $70°$.

As the influence of the boundary conditions between FF ends and SS ends diminished with a larger L/R ratio, subsequent studies focused exclusively on maximizing the fundamental frequency of composite shells with fixed ends.

3.2. Maximization of the Fundamental Frequency of ${\mathit{[}\mathit{\pm }\mathit{\theta }\mathit{/}\mathit{90}\mathit{/}\mathit{0}\mathit{]}}_{\mathit{2}s}$ Laminated Cylindrical Shells with Various Lengths, Various Axial Compressive Forces, and $\mathit{\Omega } \mathit{=} \mathit{0} RPM$

In this section, the optimization of ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends, various lengths (R = 10 cm, $1\le \mathrm{L}/\mathrm{r}\le 4),$ different axial compressive forces $(0\le \mathrm{N}\le 0.8 {\mathrm{N}}_{\mathrm{c}\mathrm{r}})$, and $\Omega =0 \mathrm{RPM}$ is described. To determine the maximum fundamental frequency $\omega$ and the corresponding optimal fiber angle $\theta$, the optimization problem can be formulated as follows:

$$\mathrm{Maximize}: \omega (\theta )$$

(7)

$$\mathrm{Subjected} \mathrm{to}: 0°\le \theta \le 90°$$

(8)

Before applying the golden section method [5,22,28,36], the fundamental frequency $\omega$ of the laminated cylindrical shell was computed by the Abaqus program for each $10°$ increment in the $\theta$ angle, allowing for an approximate identification of the maximum point, as illustrated in Figure 3, Figure 4, Figure 5 and Figure 6. Next, appropriate upper and lower bounds were chosen, and the golden section method was applied. The optimization process ended when an absolute tolerance (the difference between the two intermediate points) of $\Delta \theta \le 0.5°$ was achieved.

Figure 7 shows the optimal fiber angle ${\theta }_{\mathrm{opt}}$ versus the L/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends. From Figure 7a–c, we can see that when $\mathrm{L}/\mathrm{R}\le 3$, the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the cylindrical shell seemed to be less influenced by the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio and oscillated between $45°$ and $60°.$ When L/R = 4, the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the cylindrical shell seemed to be influenced by $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio significantly and scattered between $35°$ and $80°$.

Figure 8 shows the maximum fundamental frequency $\omega$ versus the L/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends. From Figure 8a, we can observe that the optimal frequency $\omega$ decreased with increasing of L/R ratios. From Figure 8b, we can observe that the optimal frequency $\omega$ also decreased with increasing of $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios. From Figure 8c, we can see that the laminated cylindrical shells with larger L/R ratios and larger $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios had the lowest optimal frequency $\omega$. In addition, the influence of the L/R ratio on the optimal frequency $\omega$ was more significant than that of the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio.

Figure 9 displays the fundamental vibration modes of thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with fixed–fixed ends under optimal fiber orientation with L/R ratios of 1 and 4. We can see that as the L/R ratio increased, the fundamental vibration modes of these cylindrical shells exhibited fewer waves in the circumferential direction. Additionally, when the L/R ratio was low (e.g., L/R = 1), the fundamental vibration modes of shells under higher axial compressive forces showed fewer waves in the circumferential direction than those under lower axial compressive forces.

3.3. Maximization of the Fundamental Frequency of ${\mathit{[}\mathit{\pm }\mathit{\theta }\mathit{/}\mathit{90}\mathit{/}\mathit{0}\mathit{]}}_{\mathit{2}s}$ Laminated Cylindrical Shells with Various Lengths, Various Axial Compressive Forces, and $\mathit{\Omega }\mathit{=}\mathit{10}\mathit{,}\mathit{000} RPM$

In this session, optimization of ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends, various lengths (R = 10 cm, $1\le \mathrm{L}/\mathrm{r}\le 4),$ various axial compressive forces $(0\le \mathrm{N}\le 0.8 {\mathrm{N}}_{\mathrm{c}\mathrm{r}}),$ and $\Omega =10,000$ RPM is described.

Figure 10 shows the optimal fiber angle ${\theta }_{\mathrm{opt}}$ versus the L/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends. From Figure 10a–c, we can see that when $\mathrm{L}/\mathrm{R}\le 3$, the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the cylindrical shell seemed to be less influenced by the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio and oscillated between $40°$ and $60°.$ When L/R = 4, the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the cylindrical shell seemed to be influenced by the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio significantly for high $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios.

Figure 11 shows the maximum fundamental frequency $\omega$ versus the L/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends. From Figure 11a, we can observe that the optimal frequency $\omega$ decreased with increasing L/R ratios. From Figure 11b, we can observe that the optimal frequency $\omega$ also decreased with increasing $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios. From Figure 11c, we can see that the laminated cylindrical shells with larger L/R ratios and larger $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios had the lowest optimal frequency $\omega$. In addition, the influence of the L/R ratio on the optimal frequency $\omega$ was more significant than that of the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio.

Figure 12 presents the fundamental vibration modes of thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with fixed–fixed ends under the optimal fiber orientation with L/R ratios of 1 and 4. We can find that as the L/R ratio increased, the fundamental vibration modes of these cylindrical shells exhibited fewer waves in the circumferential direction. Furthermore, when the L/R ratio was small (e.g., L/R = 1), the fundamental vibration modes of shells under higher axial compressive forces exhibited fewer waves in the circumferential direction than shells under lower axial compressive forces.

3.4. Maximization of the Fundamental Frequency of ${\mathit{[}\mathit{\pm }\mathit{\theta }\mathit{/}\mathit{90}\mathit{/}\mathit{0}\mathit{]}}_{\mathit{2}s}$ Laminated Cylindrical Shells with Various Lengths, Various Axial Compressive Forces, and $\mathit{\Omega }\mathit{=}\mathit{20}\mathit{,}\mathit{000} RPM$

In this session, optimization of ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends, various lengths (R = 10 cm, $1\le \mathrm{L}/\mathrm{R}\le 4),$ various axial compressive forces $(0\le \mathrm{N}\le 0.8 {\mathrm{N}}_{\mathrm{c}\mathrm{r}}),$ and $\Omega =20,000$ RPM is described.

Figure 13 shows the optimal fiber angle ${\theta }_{\mathrm{opt}}$ versus the L/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends. From Figure 13a–c, we can see that when $\mathrm{L}/\mathrm{R}\le 3$, the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the cylindrical shell seemed to be less influenced by the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio and oscillated between $30°$ and $50°.$ When L/R = 4, the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the cylindrical shell seemed to be influenced by the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio significantly for high $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios.

Figure 14 shows the maximum fundamental frequency $\omega$ versus the L/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends. From Figure 14a, we can observe that the optimal frequency $\omega$ decreased with increasing L/R ratios. From Figure 14b, we can observe that the optimal frequency $\omega$ also decreased with increasing $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios. From Figure 14c, we can see that the laminated cylindrical shells with larger L/R ratios and larger $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios had the lowest optimal frequency $\omega$. In addition, the influence of the L/R ratio on the optimal frequency $\omega$ was more significant than that of the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio.

Figure 15 illustrates the fundamental vibration modes of thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with fixed–fixed ends under the optimal fiber orientation with L/R ratios of 1 and 4. Once again, we can observe that as the L/R ratio increased, the fundamental vibration modes of these cylindrical shells exhibited fewer waves in the circumferential direction.

3.5. Effect of the Rotating Speed on the Optimal Fiber Angle and the Maximum Fundamental Frequency of ${\mathit{[}\mathit{\pm }\mathit{\theta }\mathit{/}\mathit{90}\mathit{/}\mathit{0}\mathit{]}}_{\mathit{2}s}$ Laminated Cylindrical Shells

In this section, the data obtained from Section 3.2, Section 3.3 and Section 3.4 are replotted to study the effect of the rotating speed. From Figure 16, we can see that the cylinder shells with higher rotating speeds had smaller optimal fiber angles than those with lower rotating speeds. In addition, the rotating speed had more effect on the maximum fundamental frequency of the laminated cylindrical shells when $\mathrm{L}/\mathrm{R}>2$. From Figure 17, it can be seen that the maximum fundamental frequency of the cylinder shells increased with increasing rotation speed. In addition, the maximum fundamental frequency of the cylinder shells decreased with increasing L/R ratios. From Figure 18, we can see that the cylinder shells with higher rotating speeds had smaller optimal fiber angles than those with lower rotating speeds. From Figure 19, it can be seen that the maximum fundamental frequency of the cylinder shells increased with increasing rotation speed. In addition, the maximum fundamental frequency of the cylinder shells decreased with increasing $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios. Finally, comparing Figure 15 with Figure 9, we can see that when rotating speed was increased, the fundamental vibration modes of the shells with high rotating speed had fewer waves in the circumferential direction.

3.6. Maximization of the Fundamental Frequency of ${\mathit{[}\mathit{\pm }\mathit{\theta }\mathit{/}\mathit{90}\mathit{/}\mathit{0}\mathit{]}}_{\mathit{2}s}$ Laminated Cylindrical Shells with Various Lengths, Various Cutouts, Various Axial Compressive Forces, and $\mathit{\Omega }\mathit{=}\mathit{20}\mathit{,}\mathit{000} RPM$

In this section, optimization of ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends, various lengths (R = 10 cm, $\mathrm{L}/\mathrm{r}=2,\hspace{0.33em}4),$ various axial compressive forces $(0\le \mathrm{N}\le 0.8 {\mathrm{N}}_{\mathrm{c}\mathrm{r}}),$ and $\Omega =20,000$ RPM is described. In addition, the cylinder shells contained circular cutouts at the center (Figure 2), and the radius d of the cutout was either 0 cm, 2 cm, or 4 cm, i.e., d/R = 0, 0.2, 0.4.

Figure 20 shows the optimal fiber angle ${\theta }_{\mathrm{opt}}$ versus the d/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends, a center cutout, L/R = 2, and $\Omega =20,000$ RPM. It can be seen that the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the shell fell between $40°$ and $50°.$ The optimal fiber angle ${\theta }_{\mathrm{opt}}$ slightly decreased with increasing d/R ratios and slightly increased with increasing $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios.

Figure 21 shows the maximum fundamental frequency versus the d/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends, a center cutout, L/R = 2, and $\Omega =20,000$ RPM. It can be seen that the maximum fundamental frequency of the shell increased with increasing d/R ratios and decreased with increasing $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios.

Figure 22 shows the optimal fiber angle ${\theta }_{\mathrm{opt}}$ versus the d/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends, a center cutout, L/R = 4, and $\Omega =20,000$ RPM. It can be seen that the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the shell usually fell between $30°$ and $40°.$ The exceptions were the shells with d/R = 0 and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}\ge 0.4$. Specially, when d/R = 0 and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}=0.8$, the optimal fiber angle ${\theta }_{\mathrm{opt}}$ of the shell was $70.7°$.

Figure 23 shows the maximum fundamental frequency versus the d/R ratio and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio for thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF ends, a center cutout, L/R = 4, and $\Omega =20,000$ RPM. It can be seen again that the maximum fundamental frequency of the shell increased with increasing d/R ratios and decreased with increasing $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios. Comparing Figure 23 with Figure 21, we can see that the increase in the maximum fundamental frequency of the short shell (L/R = 2) due to the cutout was more significant than that of the long shell (L/R = 4), especially when the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio was large.

Figure 24 and Figure 25 present the fundamental vibration modes of thin ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells with FF end conditions, a central cutout, and $\Omega =20,000$ RPM for L/R = 2 and 4, respectively. It can be seen that as the L/R ratio increased, the fundamental vibration modes exhibited fewer waves in the shell’s circumferential direction. Additionally, as the cutout size grew, the fundamental vibration modes of these cylindrical shells showed increased local disturbances near the cutout area. Finally, these local disturbances near the cutout area became more pronounced for the fundamental vibration modes when the L/R ratio was small and the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio was large.

4. Conclusions

Based on the numerical results of this investigation, the following conclusions may be drawn:

• The influence of boundary conditions between FF ends and SS ends on the fundamental frequency of ${[\pm \theta /90/0]}_{2\mathrm{s}}$ laminated cylindrical shells gradually decreased when the L/R ratio was large.
• The optimal frequency $\omega$ decreased with increasing L/R ratios and $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratios. In addition, the influence of the L/R ratio on the optimal frequency $\omega$ was more significant than that of the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio.
• With an increase in the L/R ratio, the fundamental vibration modes of the cylindrical shells displayed fewer waves in the circumferential direction. Additionally, when the L/R ratio was low, shells under higher axial compressive force exhibited fundamental vibration modes with fewer waves in the circumferential direction than those under lower axial compressive force.
• The maximum fundamental frequency of the cylinder shells increased with increasing rotation speed. When rotating speed was increased, the fundamental vibration modes of the shells with high rotating speed had fewer waves in the circumferential direction.
• The maximum fundamental frequency of the shell increased with increasing d/R ratios. In addition, the fundamental vibration modes of the cylindrical shells tended to have more local modes around the cutout when the L/R ratio was small, the d/R ratio was large, and the $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio was large.
• By understanding how the fiber orientation, L/R ratio, $\mathrm{N}/{\mathrm{N}}_{\mathrm{c}\mathrm{r}}$ ratio, and rotation speed affect optimal fundamental frequency and vibration mode, engineers can tailor laminated cylindrical shells to achieve desired vibrational characteristics. This can minimize resonance issues and enhance stability in applications in aerospace engineering.