Precessional vibrations and resonances of compound shells during complex rotation

Gulyaev, V. L.; Solov’ev, I. L.

doi:10.1007/bf02682184

Cited by 7 publications

(7 citation statements)

References 2 publications

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“…In the case of bifurcational instability, the same algorithm is used with the only difference that the terms with c and c 2 are omitted in (5) and combinations of T and w that cause the matrix D to degenerate. The instability mode can be found as described above for natural modes [4]. Since the ratio of the length to the diameter of the shell is relatively large, we can use the theories of shells and beams.…”

Section: Problem Formulation Vibration Equations For Rotating Thin Smentioning

confidence: 99%

“…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.…”

Section: Problem Formulation Vibration Equations For Rotating Thin Smentioning

confidence: 99%

“…We choose the following right-handed coordinate systems: (i) an inertial coordinate system OXYZ with the origin at the center of the support section of the shell and the OZ-axis aligned with the Oz-axis and (ii) in the mid-surface of the shell, an orthogonal curvilinear coordinate system Ox 1 x 2 x 3 with the x 1 -axis directed along the generatrix, the x 2 -axis in the circumferential direction, and the x 3 -axis along the inward normal. The dynamic equilibrium of an element of the shell is described by the following equations in the curvilinear orthogonal coordinate system Ox 1 x 2 x 3 with basis vectors r e a on the surface [4,5]: …”

Section: Problem Formulation Vibration Equations For Rotating Thin Smentioning

confidence: 99%

“…Numerical Technique. The numerical technique is based on the method of initial parameters and Godunov's orthogonalization [3,4]. The system of differential equations of the eighth order (3) is reduced to a system of eight equations of the first order: …”

Section: Problem Formulation Vibration Equations For Rotating Thin Smentioning

confidence: 99%

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

2010

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Section: Problem Formulation Vibration Equations For Rotating Thin Smentioning

confidence: 99%

“…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.…”

Section: Problem Formulation Vibration Equations For Rotating Thin Smentioning

confidence: 99%

Section: Problem Formulation Vibration Equations For Rotating Thin Smentioning

confidence: 99%

Section: Problem Formulation Vibration Equations For Rotating Thin Smentioning

confidence: 99%

See 2 more Smart Citations

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

2010

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“…Let us examine the possibility of instability of dynamic equilibrium or forced precessions in the form of a harmonic wave precessing with a frequency r ω relative to the initial stress state. Let us use the geometrically nonlinear equations of dynamic equilibrium of a shell element in general form [3,12]:…”

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confidence: 99%

Bifurcations of a Single-Support Elastic Thin-Walled Rotor

2005

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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.

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Vibrations of Shells of Revolution with Branched Meridian

Bespalova

Urusova

2016

Int Appl Mech

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Precessional vibrations and resonances of compound shells during complex rotation

Cited by 7 publications

References 2 publications

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

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

Bifurcations of a Single-Support Elastic Thin-Walled Rotor

Vibrations of Shells of Revolution with Branched Meridian

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