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Nonlinear Dynamics

, Volume 98, Issue 3, pp 1853–1876 | Cite as

Self-sustained vibrations of functionally graded carbon nanotubes-reinforced composite cylindrical shells in supersonic flow

  • K. V. AvramovEmail author
  • M. Chernobryvko
  • B. Uspensky
  • K. K. Seitkazenova
  • D. Myrzaliyev
Original paper
  • 61 Downloads

Abstract

Dynamic model of geometrical nonlinear deformations of functionally graded carbon nanotubes-reinforced composite cylindrical shell is obtained. Reddy higher-order shear deformation theory is used to derive this model. The finite-degree-of-freedom nonlinear system, which describes the structure nonlinear self-sustained vibrations, is obtained using the assumed-mode method. The linear piston theory is used to describe the supersonic flow. The loss of the cylindrical shell dynamic stability owing to the Hopf bifurcations is analyzed. The self-sustained vibrations, which describe the circumferential traveling waves flutter, occur due to this bifurcation. The harmonic balance method is applied to analyze these self-sustained vibrations. The properties of the circumferential traveling waves are analyzed.

Keywords

Functionally graded carbon nanotubes-reinforced material Cylindrical shell in supersonic flow Higher-order shear deformation theory Dynamic instability Self-sustained vibration Circumferential traveling waves flutter 

List of symbols

\(E_{11}^{\mathrm{CNT}},E_{22}^{\mathrm{CNT}}\)

Young’s modulus of CNTs

\(E^{m}\)

Young’s module of matrix

\(G_{12}^{\mathrm{CNT}} \)

Shear module of CNTs

\(G^{m}\)

Shear module of matrix

\(\mathbf{I}\)

Identity matrix

\(\mathbf{K}\)

Stiffness matrix

L

Length of cylindrical shell

\(\mathbf{M}\)

Mass matrix

\(M_X \)

Axial stress moment resultant per unit length

\(M_*\)

Mach number

\(N_X \)

Axial stress resultant per unit length

\(\mathbf{Q}\)

Vector of generalized forces

R

Radius of cylindrical shell

T

Kinetic energy of cylindrical shell

\(V_{\mathrm{CNT}} (z)\)

Part of volume, which is occupied by CNTs

\(V_{\mathrm{CNT}}^*\)

Part of volume, which is occupied by uniform distribution of CNTs

\(a_\infty \)

Free stream speed of sound

h

Constant thickness of cylindrical shell

n

Number of circumference waves

m

Number of longitudinal half-waves

p

Radial aerodynamic pressure

\(p_\infty \)

Free stream static pressure

\(\mathbf{q}\)

Vector of generalized coordinates

uvw

Three projections of the middle surface displacements

\(u_x \left( {x,\theta ,t,z} \right) ,u_\theta \left( {x,\theta ,t,z} \right) , u_z \left( {x,\theta ,t,z} \right) \)

Projections of shell points displacements, which are placed on the z distance from middle surface

\(x,\theta ,z\)

Curvilinear coordinate system

\(\mathbf{y}\)

Vector of phase coordinates

\(\prod \)

Potential energy of shell

\(\Omega \)

Frequency of self-sustained vibration

\(\alpha _{\nu j}^{\left( i \right) },\beta _{\nu jj_1 }^{\left( i \right) } \)

Numerical values, described structure geometrically nonlinear deformation

\(\gamma \)

Adiabatic exponent

\(\delta A\)

Virtual work of pressure

\(\delta w\)

Variation of radial displacements

\(\varepsilon _{XX},\varepsilon _{\theta \theta },\gamma _{\theta Z},\gamma _{XZ},\gamma _{X\theta } \)

Elements of the Green’s strain tensor

\(\eta _1,\eta _2,\eta _3\)

CNT/matrix efficiency parameters

\(\mu _{12}^{\mathrm{CNT}} \)

Poisson’s ratio of CNTs

\(\xi \)

Characteristic exponent

\(\rho ^{\mathrm{CNT}}\)

Density of CNTs

\(\rho ^{m}\)

Density of composite matrix

\(\sigma _{XZ},\sigma _{\theta Z} \)

Shear stresses

\(\sigma _{XX},\sigma _{\theta \theta },\sigma _{X\theta } \)

Elements of stress tensor

\(\phi _1,\,\phi _1 \)

Rotations of middle surface normal about \(\theta \) and x axes

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Vibrations, Podgorny Institute for Mechanical EngineeringNational Academy of Science of UkraineKharkivUkraine
  2. 2.Department of Mechanics and EngineeringM. Auezov South Kazakhstan State UniversityShymkentRepublic of Kazakhstan

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