# Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulations

#### 2.1. Description of the Model

_{1}and L

_{2}denote the lengths of the two meridians. The thickness of the combined shell is uniformly set to h. φ represent the semi-vertex angle of conical shell. The symbols R

_{1}and R

_{2}represent the radii at both ends of the conical shell, respectively. The cylindrical shell is connected at the big end of the conical shell, so the radius of the cylindrical shell is also R

_{2}. The orthogonal curvilinear coordinate system (x, θ, z) is introduced into the middle surface of each substructure. The orthogonal coordinate system o-xθz is established on the middle surface of the substructure, then the radius R of the random position on the conical shell is as follows:

#### 2.2. Governing Equations and Boundary Conditions

_{x}, ψ

_{θ}) of the mid-plane.

**A**represents the tensile stiffness matrix,

**B**is the bending stiffness matrix, and

**D**is the coupled tensile bending stiffness matrix.

**A**denotes the shear stiffness matrix. Their specific form is:

_{c}_{c}= 5/6 is the shear correction coefficient and the symbol ${\overline{Q}}_{ij}^{k}$ denotes the elastic stiffness coefficient [38].

_{k}and α

_{ij}denote the fiber angle of the kth layer and linear thermal expansion coefficients along the principal axes of a layer, respectively.

^{3}.

**C**and

**M**are expressed as:

#### 2.3. Meshfree TRPIM Shape Function

_{i}(

**x**) is the radial basis function (RBFS), and n

_{r}is the number of nodes of the point x in the support domain. p

_{j}(

**x**) is the polynomial in the space coordinate

**x**

^{T}= (x, y), and n

_{p}represents the number of polynomials. If n

_{p}= 0, it is a single radial basis function (RBFS), otherwise it is an RBF with n

_{p}polynomial basis functions added. Generally, for a one-dimensional problem, the basis function of the polynomial is p

_{j}(x) = [1,x,...,x

^{np}]

^{T}, and in a two-dimensional problem, the polynomial basis is p

_{j}(x) = [1,x,y,...,x

^{np},xy

^{np}

^{−1},...,yx

^{np}

^{−1},y

^{np}]

^{T}. However, using a power function polynomial basis is often inaccurate in solving differential equations. Chebyshev polynomials have important applications in approximation theory. Corresponding interpolation polynomials minimize the Longo phenomenon and provide the best consistent approximation of polynomials in continuous functions. Therefore, this study uses Chebyshev polynomials as interpolation basis functions.

_{i}denotes the distance between the supported point x

_{J}(J = 1,2,n

_{r}) in the supported domain and calculated node x

_{I}. For the one-dimensional problem in this paper r

_{i}= |x

_{J}-x

_{I}|, d

_{c}is a characteristic length related to the node spacing in the support domain of the compute node. When the nodes are evenly distributed, d

_{c}is the distance between adjacent nodes. Otherwise, dc is the average node spacing within the node distribution domain.

_{s}represents the scale factor of the support domain.

**a**and

**b**in Equation (30), a support field for calculated node x

_{I}needs to be formed, which includes n

_{r}field nodes. Let Equation (30) satisfy the calculation of n node values around point x

_{I}, which yields n

_{r}linear equations. The matrix of these equations can be expressed as the following form.

**R**

_{0}represents the RBFs matrix and

**T**

_{nt}is the Chebyshev polynomial matrix [30]. The coefficient vector a of RBFs is expressed as follow.

**b**of the Chebyshev polynomial basis function is written as follow:

_{r}+ n

_{t}unknowns in Equation (35), a unique solution cannot be obtained, so it is necessary to add n

_{r}equations through the following constraints to make the coefficient matrix of the equation system full rank.

#### 2.4. Discretization of Governing Equations and Boundary Conditions

_{I}is represented by a Chebyshev-RPIM shape function.

_{s}represents the number of nodes covered by the support domain,

**I**

_{5}represents a 5 × 5 identity matrix.

**K**

_{xI},

**C**

_{xI}and

**m**

_{I}are as follows.

#### 2.5. Continuous Condition

_{b}denotes the connection stiffness between substructures, and symbols co and cy denote conical and cylindrical shells, respectively. The matrix form of the continuous condition can be written as follows.

**U**

_{sco}and

**U**

_{scy}are the displacement vectors of the nodes of the cylindrical shell and the conical shell on the coupling interface, respectively. The coupled stiffness matrices

**K**

_{12}and

**K**

_{21}are as follows.

## 3. Numerical Results and Discussions

_{1}= 175 GPa, E

_{2}= 32 GPa, μ = 0.25, G

_{12}= G

_{13}= 12 GPa, G

_{23}= 5.7 GPa, ρ = 1760 kg/m

^{3}, α

_{11}= 1.2 × 10

^{−6}, α

_{22}= 2.3 × 10

^{−}

^{6}and α

_{12}= 0. The symbols C, F, and S are used to represent the tightened boundary conditions, free boundary conditions and simply supported boundary conditions, respectively. The corresponding boundaries are described as follows: C: k = k

_{v}= k

_{w}= k

_{x}= k

_{θ}= 10

^{14}. S: k

_{u}= k

_{v}= k

_{w}= k

_{θ}= 10

^{14}. k

_{x}= 0, F: k

_{u}= k

_{v}= k

_{w}= k

_{x}= k

_{θ}= 0. Then define boundary rules. For example, CF represents that the boundary of the conical shell segment is a fixed boundary, and the boundary of the cylindrical shell segment is a free boundary.

#### 3.1. Verification and Convergence Study

_{1}= 0.5 m, L

_{1}= 1 m, L

_{2}= 2 m, h = 0.05 m, φ = 30°, △T = 0 K; the lamination scheme is δ

_{k}= [0°/90°]. The research results show that, no matter what kind of boundary, when N ≥ 9 (N is node number), the numerical results are stable and the convergence speed is faster.

_{1}= 0.4226 m, φ = π/6, L

_{2}= R

_{2}= 1 m and h = 0.01 m. The material properties are: E = 211 GPa, ρ = 7800 kg/m

^{3}, μ = 0.3. The dimensionless frequency of a non-rotating combined conical-cylindrical shell structure is defined as: ${\omega}^{*}=\omega {R}_{2}\sqrt{\rho \left(1-{\mu}^{2}\right)/E}$. The results obtained by the meshless method are compared with the published literature [10] and [15], and the difference between the results obtained by the meshless method and the literature is very small. Table 3 compares the frequency results obtained by different numerical methods for rotating isotropic combined conical-cylindrical shells. The boundary conditions, geometry and Poisson’s ratio of the combined structure are the same as those in Table 2, and the rotational speeds considered are 0.01 rad/s, 100 rad/s, and 500 rad/s, respectively. The comparison results show that the method in this paper is in good agreement with the results in the literature. Finally, it is verified that the model established in this paper can be applied to the structural vibration solution in a thermal environment. In Table 4, the vibration frequency of the non-rotating laminated combined conical-cylindrical shell structure in a thermal environment is analyzed using the finite element software ABAQUS and the method in this paper, respectively. The considered structural geometry is: R

_{1}= 0.5 m, R

_{2}= 1.5 m, L

_{2}= 2 m, h = 0.1 m, N

_{k}= [0°/90°/0°]. The temperature change is 50K. The frequencies obtained by these two methods are in good agreement. Figure 2 and Figure 3 represent the mode shapes of laminated combined conical-cylindrical shell, corresponding to the natural frequencies from Table 4. Meanwhile, it is necessary to point out the fact that the following numerical discussion illustrates that this method can be used to analyze structural vibration behavior in thermal environments. All in all, after sufficient comparison, it is proved that the method established in this paper can be applied to the vibration analysis of the rotating composite conical shell and cylindrical shell in a thermal environment.

#### 3.2. Numerical Examples

_{1}= 0.5 m, L

_{1}= 2 m, L

_{2}= 1 m, h = 0.05 m, δ

_{k}= [0°/90°/0°/90°]. It can be seen from Figure 4a,c,d that as the half-apex angle of the conical shell increases, the frequency of the combined structure gradually increases slightly first and then decreases significantly under the CC, SS and CS boundary conditions, respectively. At the same time, the difference between the forward wave frequency and the backward wave frequency of the conical-cylindrical composite shell with different rotational speeds is getting smaller and smaller, and the influence of rotational speed is also weakened. As on can see from Figure 4b, under the CF boundary condition, the backward wave frequency of the composite structure corresponding to Ω = 150 rad/s and Ω = 200 rad/s will decrease first and then increase with the increase of the half apex angle, and the rest of the natural frequency change curves all decrease. Likewise, with the same rotational speed, the gap between the forward traveling wave and the backward traveling wave of the structure also decreases.

## 4. Conclusions

- (1)
- The meshless Chebyshev-PRIM technique is effective and has relatively high accuracy in the vibration solution of rotating structures. This method has the advantage of fast convergence, and relatively accurate results can be obtained with a smaller number of nodes.
- (2)
- The increase of the half-apex angle of the conical shell reduces the structural rigidity, so the structural frequency decreases. For the combined structure under the CC boundary, after the cone angle increases to a certain extent, the effect of the rotational speed will decrease, and the frequencies corresponding to different rotational speeds will gradually approach.
- (3)
- If the temperature is too high, thermal stress is accumulated inside the structure, the stiffness of the structure is reduced, and the frequency of the combined structure will also decrease. For the boundary conditions with weakened constraints, such as the CF boundary, thermal buckling also occurs with the increase of the temperature difference.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Mode shapes of laminated combined conical-cylindrical shell with CC boundary condition (m = 1, φ = π/6) (

**a**) ABAQUS (

**b**) Present.

**Figure 3.**Mode shapes of laminated combined conical-cylindrical shell with FC boundary condition (m = 1, φ = π/4) (

**a**) ABAQUS (

**b**) Present.

**Figure 4.**Variation of dimensionless frequencies ${\omega}^{*}$ of rotating laminated combined conical-cylindrical shell with different semi-vertex angle (m = 1).

**Figure 5.**Variation of dimensionless frequencies ${\omega}^{*}$ of non-rotating laminated combined conical-cylindrical shell subjected to thermal effect (m = 1).

**Figure 6.**Variation of dimensionless frequencies ${\omega}^{*}$ of rotating laminated combined conical-cylindrical shell in the thermal environment.

N | CC | SS | FC | CF | ||||
---|---|---|---|---|---|---|---|---|

n = 1 | n = 2 | n = 1 | n = 2 | n = 1 | n = 2 | n = 1 | n = 2 | |

5 | 0.2328 | 0.1812 | 0.2232 | 0.1754 | 0.1171 | 0.0803 | 0.0494 | 0.0339 |

6 | 0.2284 | 0.1790 | 0.2262 | 0.1761 | 0.1127 | 0.0825 | 0.0464 | 0.0317 |

7 | 0.2306 | 0.1805 | 0.2255 | 0.1754 | 0.1142 | 0.0833 | 0.0457 | 0.0317 |

8 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0464 | 0.0324 |

9 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

10 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

11 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

12 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

13 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

14 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

15 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

16 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

17 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

18 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

19 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

20 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

21 | 0.2299 | 0.1798 | 0.2262 | 0.1754 | 0.1135 | 0.0840 | 0.0457 | 0.0317 |

**Table 2.**Comparison of dimensionless frequencies for non-rotating isotropic combined conical-cylindrical shell with F-C boundary condition (μ = 0.3).

m | n = 0 | n = 1 | n = 2 | |||
---|---|---|---|---|---|---|

FEM | Present | FEM | Present | FEM | Present | |

1 | 0.50375 | 0.50305 | 0.29287 | 0.29279 | 0.10203 | 0.09996 |

2 | 0.60986 | 0.60985 | 0.63581 | 0.63506 | 0.50290 | 0.50217 |

3 | 0.93092 | 0.93082 | 0.81123 | 0.81141 | 0.69148 | 0.69116 |

4 | 0.95632 | 0.95612 | 0.93088 | 0.93137 | 0.85890 | 0.85888 |

5 | 0.97160 | 0.97134 | 0.94850 | 0.95183 | 0.91607 | 0.91544 |

6 | 1.01188 | 1.01142 | 0.99145 | 0.99156 | 0.96048 | 0.96007 |

n = 3 | n = 4 | n = 5 | ||||

FEM | Present | FEM | Present | FEM | Present | |

1 | 0.09377 | 0.08750 | 0.14460 | 0.14441 | 0.20390 | 0.19930 |

2 | 0.39220 | 0.39115 | 0.33034 | 0.32996 | 0.29633 | 0.29579 |

3 | 0.51518 | 0.51434 | 0.39562 | 0.39537 | 0.37623 | 0.37013 |

4 | 0.75359 | 0.75289 | 0.64458 | 0.64594 | 0.58167 | 0.57874 |

5 | 0.79698 | 0.79629 | 0.69114 | 0.69248 | 0.61422 | 0.61285 |

6 | 0.91939 | 0.91893 | 0.87194 | 0.87098 | 0.81980 | 0.81642 |

**Table 3.**Comparison of dimensionless frequencies for a rotating isotropic combined conical-cylindrical shell.

Ω* | n | FEM | Present | ||
---|---|---|---|---|---|

w*_{b} | w*_{f} | w*_{b} | w*_{f} | ||

0.01 rad/s | 1 | 0.5264 | 0.5264 | 0.5267 | 0.5267 |

2 | 0.3769 | 0.3769 | 0.3774 | 0.3774 | |

3 | 0.2873 | 0.2873 | 0.2869 | 0.2869 | |

4 | 0.236 | 0.236 | 0.2363 | 0.2363 | |

5 | 0.2231 | 0.2231 | 0.2246 | 0.2246 | |

6 | 0.2474 | 0.2474 | 0.2469 | 0.2469 | |

100 rad/s | 1 | 0.5430 | 0.5097 | 0.5432 | 0.5103 |

2 | 0.3906 | 0.3648 | 0.3904 | 0.3645 | |

3 | 0.3005 | 0.2816 | 0.3010 | 0.2822 | |

4 | 0.2527 | 0.2383 | 0.2528 | 0.2387 | |

5 | 0.2455 | 0.234 | 0.2469 | 0.2352 | |

6 | 0.2747 | 0.2647 | 0.2740 | 0.2645 | |

500 rad/s | 1 | 0.6085 | 0.4422 | 0.6090 | 0.4421 |

2 | 0.4605 | 0.3308 | 0.4609 | 0.3304 | |

3 | 0.4174 | 0.322 | 0.4174 | 0.3222 | |

4 | 0.4484 | 0.3756 | 0.4480 | 0.3762 | |

5 | 0.5212 | 0.4629 | 0.5220 | 0.4633 | |

6 | 0.6157 | 0.5612 | 0.6161 | 0.5608 |

**Table 4.**Comparison of natural frequencies for non-rotating isotropic combined conical-cylindrical shell in thermal environment (ΔT = 50 K).

φ | Mode | CC | CS | FC | ||||||
---|---|---|---|---|---|---|---|---|---|---|

FEM | Present | Diff,% | FEM | Present | Diff,% | FEM | Present | Diff,% | ||

π/6 | 1 | 234.96 | 237.01 | 0.872 | 227.4 | 228.08 | 0.299 | 120.2 | 119.54 | −0.549 |

2 | 250.73 | 251.66 | 0.371 | 227.84 | 228.85 | 0.443 | 133.66 | 133.66 | 0 | |

3 | 252.19 | 254.21 | 0.801 | 244.34 | 243.77 | −0.233 | 234.47 | 234.16 | −0.132 | |

4 | 265.13 | 264.69 | −0.166 | 247.19 | 247.93 | 0.299 | 240.92 | 241.13 | 0.087 | |

5 | 272.86 | 274.39 | 0.561 | 265.13 | 264.69 | −0.166 | 270.66 | 270.72 | 0.022 | |

6 | 285.89 | 286.71 | 0.287 | 281.3 | 282.09 | 0.281 | 272.89 | 274.28 | 0.509 | |

π/4 | 1 | 270.23 | 272.39 | 0.799 | 239.64 | 239.73 | 0.038 | 145.93 | 145.81 | −0.082 |

2 | 281.53 | 283.4 | 0.664 | 250.99 | 250.54 | −0.179 | 152.71 | 152.7 | −0.007 | |

3 | 293.78 | 293.32 | −0.157 | 271.81 | 272.52 | 0.261 | 256.96 | 256.67 | −0.113 | |

4 | 294.13 | 296.48 | 0.799 | 293.78 | 293.32 | −0.157 | 268.27 | 270.41 | 0.798 | |

5 | 319.31 | 321.55 | 0.702 | 302.2 | 300.76 | −0.477 | 281.31 | 283.08 | 0.629 | |

6 | 328.18 | 329.1 | 0.28 | 312.49 | 314.21 | 0.55 | 305.66 | 306.68 | 0.334 |

**Table 5.**Dimensionless frequencies of rotating laminated combined conical-cylindrical shell with various length ratio in thermal environment. (L

_{1}= 1, R

_{1}= 0.5 m, h = 0.05 m, m = 1, φ = 30˚, ΔT = 50 K).

L_{2}/L_{1} | Ω, rad/s | n | Forward | Backward | ||||||
---|---|---|---|---|---|---|---|---|---|---|

CC | SS | CF | FC | CC | SS | CF | FC | |||

0.5 | 50 | 1 | 0.4723 | 0.4561 | 0.1127 | 0.2549 | 0.4826 | 0.4664 | 0.1230 | 0.2667 |

2 | 0.3809 | 0.3478 | 0.0847 | 0.1157 | 0.3883 | 0.3559 | 0.0928 | 0.1245 | ||

3 | 0.3544 | 0.3153 | 0.1341 | 0.1709 | 0.3595 | 0.3212 | 0.1400 | 0.1776 | ||

100 | 1 | 0.4671 | 0.4509 | 0.1083 | 0.2490 | 0.4877 | 0.4715 | 0.1282 | 0.2726 | |

2 | 0.3772 | 0.3448 | 0.0818 | 0.1120 | 0.3927 | 0.3595 | 0.0980 | 0.1297 | ||

3 | 0.3522 | 0.3139 | 0.1326 | 0.1702 | 0.3640 | 0.3249 | 0.1459 | 0.1820 | ||

1 | 50 | 1 | 0.3735 | 0.3618 | 0.0759 | 0.1835 | 0.3846 | 0.3728 | 0.0862 | 0.1945 |

2 | 0.3448 | 0.3161 | 0.0361 | 0.0921 | 0.3536 | 0.3249 | 0.0449 | 0.1009 | ||

3 | 0.3404 | 0.3028 | 0.0663 | 0.1687 | 0.3470 | 0.3087 | 0.0729 | 0.1754 | ||

100 | 1 | 0.3676 | 0.3559 | 0.0707 | 0.1776 | 0.3905 | 0.3780 | 0.0914 | 0.2004 | |

2 | 0.3411 | 0.3124 | 0.0332 | 0.0884 | 0.3581 | 0.3293 | 0.0508 | 0.1061 | ||

3 | 0.3382 | 0.3006 | 0.0670 | 0.1680 | 0.3507 | 0.3131 | 0.0803 | 0.1805 | ||

1.5 | 50 | 1 | 0.2896 | 0.2829 | 0.0553 | 0.1392 | 0.3006 | 0.2947 | 0.0663 | 0.1510 |

2 | 0.2505 | 0.2358 | 0.0258 | 0.0781 | 0.2593 | 0.2446 | 0.0346 | 0.0869 | ||

3 | 0.2218 | 0.2019 | 0.0597 | 0.1665 | 0.2291 | 0.2092 | 0.0670 | 0.1724 | ||

100 | 1 | 0.2837 | 0.2778 | 0.0501 | 0.1334 | 0.3065 | 0.2999 | 0.0715 | 0.1562 | |

2 | 0.2461 | 0.2313 | 0.0236 | 0.0744 | 0.2645 | 0.2498 | 0.0420 | 0.0928 | ||

3 | 0.2203 | 0.2004 | 0.0612 | 0.1650 | 0.2336 | 0.2137 | 0.0752 | 0.1776 | ||

2 | 50 | 1 | 0.2328 | 0.2284 | 0.0420 | 0.1105 | 0.2439 | 0.2402 | 0.0530 | 0.1223 |

2 | 0.1864 | 0.1783 | 0.0214 | 0.0700 | 0.1952 | 0.1871 | 0.0302 | 0.0788 | ||

3 | 0.1606 | 0.1496 | 0.0575 | 0.1540 | 0.1672 | 0.1569 | 0.0641 | 0.1606 | ||

100 | 1 | 0.2269 | 0.2225 | 0.0368 | 0.1046 | 0.2498 | 0.2453 | 0.0589 | 0.1282 | |

2 | 0.1820 | 0.1739 | 0.0192 | 0.0670 | 0.2004 | 0.1923 | 0.0376 | 0.0847 | ||

3 | 0.1584 | 0.1481 | 0.0589 | 0.1525 | 0.1724 | 0.1621 | 0.0729 | 0.1658 |

**Table 6.**Dimensionless frequencies of rotating laminated combined conical-cylindrical shell with various semi-vertex angle in thermal environment. (L

_{1}= 0.5 m, L

_{2}= 2 m, R

_{1}= 0.5 m, h = 0.1 m, m = 1, ΔT = 50 K).

φ | Ω, rad/s | n | Forward | Backward | ||||||
---|---|---|---|---|---|---|---|---|---|---|

CC | SS | CF | FC | CC | SS | CF | FC | |||

π/6 | 50 | 1 | 0.2763 | 0.2645 | 0.0729 | 0.1260 | 0.2881 | 0.2763 | 0.0840 | 0.1378 |

2 | 0.2269 | 0.2085 | 0.0766 | 0.1444 | 0.2365 | 0.2173 | 0.0855 | 0.1532 | ||

3 | 0.2807 | 0.2652 | 0.2033 | 0.2763 | 0.2881 | 0.2726 | 0.2107 | 0.2829 | ||

100 | 1 | 0.2704 | 0.2593 | 0.0670 | 0.1201 | 0.2940 | 0.2822 | 0.0899 | 0.1437 | |

2 | 0.2225 | 0.2041 | 0.0729 | 0.1400 | 0.2417 | 0.2225 | 0.0914 | 0.1584 | ||

3 | 0.2785 | 0.2630 | 0.2019 | 0.2741 | 0.2925 | 0.2770 | 0.2159 | 0.2873 | ||

π/4 | 50 | 1 | 0.2859 | 0.2756 | 0.0589 | 0.1297 | 0.2969 | 0.2866 | 0.0700 | 0.1415 |

2 | 0.2306 | 0.2115 | 0.0597 | 0.1606 | 0.2402 | 0.2210 | 0.0685 | 0.1695 | ||

3 | 0.2549 | 0.2365 | 0.1599 | 0.2520 | 0.2616 | 0.2431 | 0.1665 | 0.2586 | ||

100 | 1 | 0.2800 | 0.2697 | 0.0538 | 0.1238 | 0.3028 | 0.2925 | 0.0752 | 0.1474 | |

2 | 0.2269 | 0.2078 | 0.0560 | 0.1562 | 0.2453 | 0.2255 | 0.0744 | 0.1746 | ||

3 | 0.2527 | 0.2343 | 0.1577 | 0.2498 | 0.2667 | 0.2476 | 0.1717 | 0.2638 | ||

π/3 | 50 | 1 | 0.2918 | 0.2822 | 0.0494 | 0.1319 | 0.3028 | 0.2940 | 0.0597 | 0.1437 |

2 | 0.2343 | 0.2144 | 0.0501 | 0.1857 | 0.2431 | 0.2232 | 0.0597 | 0.1945 | ||

3 | 0.2424 | 0.2218 | 0.1356 | 0.2394 | 0.2498 | 0.2291 | 0.1422 | 0.2468 | ||

100 | 1 | 0.2859 | 0.2770 | 0.0442 | 0.1260 | 0.3087 | 0.2999 | 0.0648 | 0.1496 | |

2 | 0.2299 | 0.2100 | 0.0472 | 0.1812 | 0.2483 | 0.2284 | 0.0656 | 0.1989 | ||

3 | 0.2402 | 0.2196 | 0.1341 | 0.2372 | 0.2542 | 0.2336 | 0.1481 | 0.2512 | ||

π/2 | 50 | 1 | 0.2962 | 0.2873 | 0.0427 | 0.1334 | 0.3080 | 0.2991 | 0.0516 | 0.1451 |

2 | 0.2380 | 0.2181 | 0.0457 | 0.2092 | 0.2476 | 0.2269 | 0.0545 | 0.2181 | ||

3 | 0.2372 | 0.2144 | 0.1194 | 0.2321 | 0.2439 | 0.2218 | 0.1267 | 0.2387 | ||

100 | 1 | 0.2903 | 0.2814 | 0.0383 | 0.1275 | 0.3139 | 0.3043 | 0.0560 | 0.1510 | |

2 | 0.2343 | 0.2137 | 0.0427 | 0.2048 | 0.2520 | 0.2321 | 0.0604 | 0.2232 | ||

3 | 0.2350 | 0.2129 | 0.1186 | 0.2299 | 0.2483 | 0.2262 | 0.1326 | 0.2439 |

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## Share and Cite

**MDPI and ACS Style**

Li, Z.; Hu, S.; Zhong, R.; Qin, B.; Zhao, X.
Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment. *Materials* **2022**, *15*, 6177.
https://doi.org/10.3390/ma15176177

**AMA Style**

Li Z, Hu S, Zhong R, Qin B, Zhao X.
Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment. *Materials*. 2022; 15(17):6177.
https://doi.org/10.3390/ma15176177

**Chicago/Turabian Style**

Li, Zhen, Shuangwei Hu, Rui Zhong, Bin Qin, and Xing Zhao.
2022. "Meshless Chebyshev RPIM Solution for Free Vibration of Rotating Cross-Ply Laminated Combined Cylindrical-Conical Shells in Thermal Environment" *Materials* 15, no. 17: 6177.
https://doi.org/10.3390/ma15176177