Interconnection of critical states of parabolic shells in simple and compound rotations with values of their natural precession vibration frequencies

Quasistatic and dynamic instability of one-support cylindrical shells under gyroscopic and nonconservative forces

“…Therefore, we will approximate the unknown variables by basis functions sin(ct + x 2 ) and cos(ct + x 2 ), where c is the natural frequency [4,5,12,14]. Then Eqs.…”

Rotation of rigid and elastic cylindrical shells elastically coupled with a platform

“…A technique for elastic strain analysis of thin rotating shells without possible rotations through angles α and β at the elastic supports was discussed in [1,2,[6][7][8][9][10][11][12]. It is based on the general equations of motion of shells in a curvilinear orthogonal coordinate frame ox 1 x 2 x 3 with the basis vectors r e α on its mid-surface:…”

“…and sum them with the corresponding contravariant components of the accelerations of the elastic shell undergoing compound rotation [1,2,[6][7][8][9][10][11][12].…”

“…(14) about the state of simple rotation with angular velocity ω, assuming that the shell is prestressed by membrane forces T 11 and T 22 . The radius r, the angle ϕ, and the parameters of the second quadratic form b ii increase by ∆r, ∆ ∆ϑ ϕ = 1 , and ∆b ii , respectively, and the strains ε ij are defined by nonlinear relations [1,2,[6][7][8][9][10][11][12].…”

“…If ñ 1 > 0 and ñ 2 > 0 in (9), then, according to (7), at both frequencies, the gyroscope axis describes a cone in the rotating coordinate frame by moving in the opposite direction to self-rotation (regular retrograde precession). When one of the frequencies is negative, the corresponding motion (7) will be regular progressive precession.…”