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Nonlinear forced vibration of rotating composite laminated cylindrical shells under hygrothermal environment

  • Xiao Li , Wentao Jiang , Xiaochao Chen and Zhihong Zhou EMAIL logo
Published/Copyright: June 30, 2021
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 76 Issue 9

Abstract

This work aims to study nonlinear vibration of rotating composite laminated cylindrical shells under hygrothermal environment and radial harmonic excitation. Based on Love’s nonlinear shell theory, and considering the effects of rotation-induced initial hoop tension, centrifugal and Coriolis forces, the nonlinear partial differential equations of the shells are derived by Hamilton’s principle, in which the constitutive relation and material properties of the shells are both hygrothermal-dependent. Then, the Galerkin approach is applied to discrete the nonlinear partial differential equations, and the multiple scales method is adopted to obtain an analytical solution on the dynamic response of the nonlinear shells under primary resonances of forward and backward traveling wave, respectively. The stability of the solution is determined by using the Routh–Hurwitz criterion. Some interesting results on amplitude–frequency relations and nonlinear dynamic responses of the shells are proposed. Special attention is given to the combined effects of temperature and moisture concentration on nonlinear resonance behavior of the shells.

Keywords: composite laminated cylindrical shell; hygrothermal environment; nonlinear forced vibration; radial harmonic excitation; rotating
Corresponding author: Zhihong Zhou, Department of Mechanics & Engineering, Sichuan University, Chengdu 610065, PR China, E-mail:

Funding source: Natural Science Foundation of Fujian Province

Award Identifier / Grant number: 2020J05103

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12002225

Award Identifier / Grant number: 12002088

Award Identifier / Grant number: 12072213

Acknowledgments

This research work was supported by the National Natural Science Foundation of China (Grant Nos. 12002225, 12072213, 12002088) and Natural Science Foundation of Fujian Province (Grant No. 2020J05103).

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

The coefficients L ij and A non in Eq. (12) are given as

(A.1) L 11 = A 11 ∂ 2 ∂ x 2 + A 66 R 2 ∂ 2 ∂ θ 2 + 2 A 16 R ∂ 2 ∂ x ∂ θ + ρ t Ω 2 ∂ 2 ∂ θ 2 − ρ t ∂ 2 ∂ t 2

(A.2) L 12 = A 16 + 2 B 16 R ∂ 2 ∂ x 2 + A 26 R 2 + B 26 R 3 ∂ 2 ∂ θ 2 + A 12 + A 66 R + B 12 + 2 B 66 R 2 ∂ 2 ∂ x ∂ θ

(A.3) L 13 = A 12 R ∂ ∂ x + A 26 R 2 ∂ ∂ θ − B 11 ∂ 3 ∂ x 3 − B 12 + 2 B 66 R 2 ∂ 3 ∂ x ∂ θ 2 − 3 B 16 R ∂ 3 ∂ x 2 ∂ θ − B 26 R 3 ∂ 3 ∂ θ 3

(A.4) L 14 = A 11 ∂ 2 ∂ x 2 + A 66 R 2 ∂ 2 ∂ θ 2 + 2 A 16 R ∂ 2 ∂ x ∂ θ

(A.5) L 15 = A 16 R ∂ 2 ∂ x 2 + A 26 R 3 ∂ 2 ∂ θ 2 + A 12 + A 66 R 2 ∂ 2 ∂ x ∂ θ

(A.6) L 21 = L 12

(A.7) L 22 = A 66 + 2 D 66 R 2 + 3 B 66 R ∂ 2 ∂ x 2 + A 22 R 2 + D 22 R 4 + 2 B 22 R 3 ∂ 2 ∂ θ 2 + 2 A 26 R + 6 B 26 R 2 + 4 D 26 R 3 ∂ 2 ∂ x ∂ θ + ρ t Ω 2 ∂ 2 ∂ θ 2 − ∂ 2 ∂ t 2

(A.8) L 23 = A 26 R + 2 B 26 R 2 ∂ ∂ x + A 22 R 2 + B 22 R 3 ∂ ∂ θ − B 16 + 2 D 16 R ∂ 3 ∂ x 3 − D 22 R 4 + B 22 R 3 ∂ 3 ∂ θ 3 − D 12 + 4 D 66 R 2 + B 12 + 2 B 66 R ∂ 3 ∂ x 2 ∂ θ − 3 B 26 R 2 + 4 D 26 R 3 ∂ 3 ∂ x ∂ θ 2 + 2 ρ t Ω 2 ∂ ∂ θ − Ω ∂ ∂ t

(A.9) L 24 = L 21

(A.10) L 25 = A 22 R 3 + B 22 R 4 ∂ 2 ∂ θ 2 + A 66 R + B 66 R 2 ∂ 2 ∂ x 2 + 2 A 26 R 2 + 2 B 26 R 3 ∂ 2 ∂ x ∂ θ

(A.11) L 31 = − L 13

(A.12) L 32 = − L 23

(A.13) L 33 = − A 22 R 2 − D 11 ∂ 4 ∂ x 4 − D 22 R 4 ∂ 4 ∂ θ 4 − 2 D 12 + 2 D 66 R 2 × ∂ 4 ∂ x 2 ∂ θ 2 − 4 D 16 R ∂ 4 ∂ x 3 ∂ θ − 4 D 26 R 3 ∂ 4 ∂ x ∂ θ 3 + 2 B 12 R − A x T H ∂ 2 ∂ x 2 + 2 B 22 R 3 − A θ T H R 2 ∂ 2 ∂ θ 2 + 4 B 26 R 2 − 2 A x θ T H R ∂ 2 ∂ x ∂ θ + ρ t Ω 2 ∂ 2 ∂ θ 2 − ∂ 2 ∂ t 2

(A.14) A non = A 11 ∂ u ∂ x ∂ 2 w ∂ x 2 + ∂ 2 u ∂ x 2 ∂ w ∂ x + 3 2 ∂ w ∂ x 2 ∂ 2 w ∂ x 2 + 2 A 12 + 2 A 66 R 2 ∂ w ∂ θ ∂ w ∂ x ∂ 2 w ∂ x ∂ θ + A 22 R 3 × 1 2 ∂ w ∂ θ 2 + ∂ 2 v ∂ θ 2 ∂ w ∂ θ + ∂ v ∂ θ ∂ 2 w ∂ θ 2 + w ∂ 2 w ∂ θ 2 + 3 2 R × ∂ w ∂ θ 2 ∂ 2 w ∂ θ 2 + A 12 R 2 ∂ 2 w ∂ θ 2 ∂ u ∂ x + A 12 + A 66 R 2 × ∂ w ∂ θ ∂ 2 u ∂ x ∂ θ + A 66 R 2 ∂ 2 u ∂ θ 2 ∂ w ∂ x + 2 ∂ u ∂ θ ∂ 2 w ∂ x ∂ θ + A 12 + A 66 R ∂ 2 v ∂ x ∂ θ ∂ w ∂ x + A 66 R ∂ 2 v ∂ x 2 ∂ w ∂ θ + 2 ∂ 2 w ∂ x ∂ θ ∂ v ∂ x + A 12 R ∂ v ∂ θ ∂ 2 w ∂ x 2 + 1 2 ∂ w ∂ x 2 + w ∂ 2 w ∂ x 2 + A 12 + 2 A 66 2 R 2 ∂ w ∂ x 2 ∂ 2 w ∂ θ 2 + ∂ w ∂ θ 2 ∂ 2 w ∂ x 2 + B 22 R 4 ∂ 2 v ∂ θ 2 ∂ w ∂ θ + ∂ v ∂ θ ∂ 2 w ∂ θ 2 + B 12 + 2 B 66 R 2 ∂ 2 v ∂ x ∂ θ ∂ w ∂ x + 2 B 12 + B 66 R 2 × ∂ 2 w ∂ θ 2 ∂ 2 w ∂ x 2 + 2 B 12 + B 66 R 2 ∂ 2 w ∂ x ∂ θ 2 + B 12 R 2 ∂ v ∂ θ ∂ 2 w ∂ x 2 + 2 B 66 R 2 ∂ 2 v ∂ x 2 ∂ w ∂ θ + 2 ∂ v ∂ x ∂ 2 w ∂ x ∂ θ + A 16 + 2 B 16 R ∂ 2 v ∂ x 2 ∂ w ∂ x + ∂ 2 w ∂ x 2 ∂ v ∂ x + A 16 R 2 ∂ 2 u ∂ x ∂ θ ∂ w ∂ x + 2 ∂ 2 w ∂ x ∂ θ ∂ u ∂ x + ∂ 2 u ∂ x 2 ∂ w ∂ θ + ∂ u ∂ θ ∂ 2 w ∂ x 2 + 3 ∂ w ∂ x 2 ∂ 2 w ∂ x ∂ θ + 3 ∂ w ∂ x ∂ w ∂ θ ∂ 2 w ∂ x 2 + A 26 R 2 + B 26 R 3 2 ∂ 2 v ∂ x ∂ θ ∂ w ∂ θ + 2 ∂ v ∂ θ ∂ 2 w ∂ x ∂ θ + ∂ 2 v ∂ θ 2 ∂ w ∂ x + ∂ v ∂ x ∂ 2 w ∂ θ 2 + A 26 R 3 ∂ 2 u ∂ θ 2 ∂ w ∂ θ + ∂ u ∂ θ ∂ 2 w ∂ θ 2 + 3 ∂ w ∂ θ 2 ∂ 2 w ∂ x ∂ θ + 3 ∂ w ∂ x ∂ w ∂ θ ∂ 2 w ∂ θ 2 + A 26 R 2 ∂ w ∂ x ∂ w ∂ θ + 2 w ∂ 2 w ∂ x ∂ θ

in which the stiffness components A ij , B ij and D ij in Eqs. (A.1)–(A.14) are defined as ( A i j , B i j , D i j ) = ∫ − h / 2 h / 2 Q ̄ i j 1 , z , z 2 d z i , j = 1,2,6 .

Set H = (H x H θ H xθ ) t = ε T + ε H , then the stiffness terms A x T H , A θ T H and A x θ T H in Eq. (A.13) generated by hygrothermal expansion deformation are expressed as

A x T H = ∫ − h / 2 h / 2 Q ̄ 11 H x + Q ̄ 12 H θ + Q ̄ 16 H x θ d z A θ T H = ∫ − h / 2 h / 2 Q ̄ 12 H x + Q ̄ 22 H θ + Q ̄ 26 H x θ d z A x θ T H = ∫ − h / 2 h / 2 Q ̄ 16 H x + Q ̄ 26 H θ + Q ̄ 66 H x θ d z

Appendix B

The coefficients in Eq. (16) are expressed as

(B.1) M = ρ t π L 1 + α 11 2 + β 11 2 2 , C = − ρ t π L β 11 + β 11 2 , K = π L K t 2   K r = ρ t π L K i 2 , f n o n = A 11 λ 1 4 9 π L 64 + A 22 3 π L 64 R 4 + λ 1 2 π L A 12 + 2 A 66 32 R 2 F 1 = F 0 sin λ 1 x 0 cos θ 0 , F 2 = − F 0 sin λ 1 x 0 sin θ 0

where

(B.2) K t = α 11 2 A 11 λ 1 2 + A 66 R 2 − 2 λ 1 α 11 β 11 A 12 + A 66 R + B 12 + 2 B 66 R 2 − 2 α 11 A 12 R λ 1 + B 11 λ 1 3 + B 12 + 2 B 66 R 2 λ 1 + β 11 2 A 22 R 2 + D 22 R 4 + 2 B 22 R 3 + β 11 2 λ 1 2 A 66 + 2 D 66 R 2 + 4 B 66 R + 2 β 11 A 22 R 2 + B 22 R 3 + D 22 R 4 + B 22 R 3 + D 12 + 4 D 66 R 2 + B 12 + 2 B 66 R λ 1 2 + A 22 R 2 + D 11 λ 1 4 + D 22 R 4 + 2 D 12 + 2 D 66 R 2 λ 1 2 + 2 B 12 R − A x T H λ 1 2 + 2 B 22 R 3 − A θ T H R 2

(B.3) K i = α 11 2 + β 11 2 + 4 β 11 + 1

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Received: 2021-02-06
Accepted: 2021-06-07
Published Online: 2021-06-30
Published in Print: 2021-09-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Atomic, Molecular & Chemical Physics
  3. Chirp waveform control to produce broad harmonic plateau and single attosecond pulse
  4. Dynamical Systems & Nonlinear Phenomena
  5. Ion-acoustic stable oscillations, solitary, periodic and shock waves in a quantum magnetized electron–positron–ion plasma
  6. Nonlinear forced vibration of rotating composite laminated cylindrical shells under hygrothermal environment
  7. Vibration characteristics and stable region of a parabolic FGM thin-walled beam with axial and spinning motion
  8. Hydrodynamics
  9. Curvilinear flow of micropolar fluid with Cattaneo–Christov heat flux model due to oscillation of curved stretchable sheet
  10. Quantum Theory
  11. Accuracy of the typicality approach using Chebyshev polynomials
  12. Solid State Physics & Materials Science
  13. Impact of Sn ions on structural and electrical description of TiO2 nanoparticles
  14. The effect of non-bridging oxygen on the electrical transport of some lead borate glasses containing cobalt
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https://doi.org/10.1515/zna-2021-0029
Keywords for this article
composite laminated cylindrical shell; hygrothermal environment; nonlinear forced vibration; radial harmonic excitation; rotating

Articles in the same Issue

  1. Frontmatter
  2. Atomic, Molecular & Chemical Physics
  3. Chirp waveform control to produce broad harmonic plateau and single attosecond pulse
  4. Dynamical Systems & Nonlinear Phenomena
  5. Ion-acoustic stable oscillations, solitary, periodic and shock waves in a quantum magnetized electron–positron–ion plasma
  6. Nonlinear forced vibration of rotating composite laminated cylindrical shells under hygrothermal environment
  7. Vibration characteristics and stable region of a parabolic FGM thin-walled beam with axial and spinning motion
  8. Hydrodynamics
  9. Curvilinear flow of micropolar fluid with Cattaneo–Christov heat flux model due to oscillation of curved stretchable sheet
  10. Quantum Theory
  11. Accuracy of the typicality approach using Chebyshev polynomials
  12. Solid State Physics & Materials Science
  13. Impact of Sn ions on structural and electrical description of TiO2 nanoparticles
  14. The effect of non-bridging oxygen on the electrical transport of some lead borate glasses containing cobalt
  15. Thermodynamics & Statistical Physics
  16. Analytical solution for unsteady adiabatic and isothermal flows behind the shock wave in a rotational axisymmetric mixture of perfect gas and small solid particles
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