Bifurcations of a Single-Support Elastic Thin-Walled Rotor

Abstract

The bifurcations of a thin-walled shell rotor during simple and complex rotation are analyzed. The similarity and difference of the problem formulations and solution techniques are pointed out. In both cases, the buckling mode is described by the first circumferential harmonic. The dependence of rotor bifurcations on natural frequencies is studied Keywords: shell rotor, bifurcation, instability, simple and complex rotations Introduction. It was established in [2-4, 10, 11, 15] that bifurcations (in the sense defined in [1, 6, 7]) of rotating conical, spherical, paraboloidal, and compound shells occur under centrifugal forces as static buckling if the rotation is simple and as precession resonance if the rotation is complex. It was shown that the critical states of simply rotating shells are due to instability of dynamic equilibrium, resulting in buckling caused by inertial forces, which depend on the position of the elastic element relative to the axis of rotation.The second type of bifurcations of a rotating shell mounted on a carrier body may occur when the body changes its attitude, forcing the rotation axis to move. In this case, the shell experiences precessions that may become resonant (bifurcation) at certain natural frequencies and angular velocity.We will demonstrate below that the natural frequencies and critical angular velocities of a thin-walled elastic rotor are in certain relationships during simple and complex rotations. The rotation of the rotor changes substantially the natural frequencies and modes, since multiple frequencies split and vibration modes transform into waves traveling (precessing) in the circumferential direction. In this case, one of the split frequencies corresponds to a wave running in the direction of rotation (direct regular precession) and the other to a wave running in the opposite direction (retrograde regular precession).The angular velocities at which the split frequencies combine and take zero values are the velocities of static (in the rotating coordinate system) buckling. During complex rotation, the rotor reaches critical states when the frequency of retrograde precession becomes equal to the velocity of rotation.It is of interest to find out which of the bifurcations of a compound shell occurs earlier and how they are related to the natural frequencies. The features of bifurcations of elastic shells during simple and complex rotation can be used to analyze the dynamic behavior of thin-walled rotors in the engine of a maneuvering aircraft.

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].…”

Stability of the equilibrium of rotating drillstrings

“…With large intervals of integration, the system of equations (12) becomes numerically stiff, adversely affecting the convergence of numerical integration methods. Here we will use the method of initial parameters in combination with the Runge-Kutta method and Godunov's orthogonalization [11][12][13][14]. Now, we will consider an algorithm to solve the system of equations (12).…”

“…It should be noted, however, that these tests are weak because Eqs. (12) are uncoupled, which means that they are much easier to integrate. A more rigorous test is to check the stability of a beam subjected at the ends to a constant torque M z and a constant longitudinal force T at ω = 0since Eqs.…”