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

Free nonlinear oscillations of cantilever circular cylindrical shells

“…Thus, the difference between the data of the harmonic balance method and the direct numerical integrations is not decreased. The same difference between the data of the harmonic balance method and the direct numerical integrations is reported in paper [35].…”

“…Thus, the eigenfrequencies are obtained with acceptable errors. Now the orthogonal polynomials are used as the trial functions ( ) , ( ) , ( ) for the expansion (21). The Donnell's shell theory is applied to describe the shell vibrations.…”

“…The outline of cantilever circular cylindrical shell carbon nanotube reinforced composite thin-walled structure in the supersonic flow are considered in the paper [21]. The properties of free linear vibrations and the self-sustained vibrations are discussed in detail.…”

Buckling of functionally graded carbon nanotubes reinforced composite joined spherical-cylindrical-spherical thin-walled structure

“…The projections of the shell displacements on the axes x,φ,z are u1(2),u2(2),u3(2). Using the higher- order shear deformation theory, 50 the cylindrical shell displacements take the following form:where R is radius of the cylindrical shell; ϕ1(2),normal ϕ2(2) are rotations angles of the normal to the middle surface; u(2), v(2),normal w(2) are projections of the middle surface displacements on the axes x,φ,z, respectively.…”

“…The general relations between the strains and the displacements projections, which are published in paper, 50 are used to calculate the strains of the cylindrical shell. The components of the strain tensor of the cylindrical shell take the following form:…”