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 TITLE: The motion of a high-speed rotor under the influence of a moment perpendicular to the axes of precession and nutation
 AUTHOR: Ben-Amots, N.
 SOURCE: D.Sc. thesis, Technion, Haifa, Israel (1975).
 LANGUAGE: Hebrew.
 SUPERVISORS: I. Porat (1934-2012), D. Bousso (1933-1971).
 ADVISORS: M. Shmuely (1935-1980), B. Popper (1927-2012).
 ABSTRACT: The purpose of this research is to investigate the laws that determine the stability of a symmetrical rotor rotating about a fixed point, with the resultant of the external moments acting on the rotor having components both in the direction perpendicular to the axes of precession and rotation, and in the direction perpendicular to the axes of nutation and spin.

Such moments exist in cases where the driving moment and the load moment are not coaxial. Engineering examples are: drive by a flexible shaft; certain joints in special gyroscope linkages; belt suspension-drive, in which a rotor is suspended in and driven by a belt transmission only. A number of other examples are given in this report.

The experimental data are based on experiments with a belt suspension-drive, which were carried out partially during previous research and partially in the course of this research. Among other things, the experimental observations indicate non-linear influence of the nutation angle of rotor deviation on rotor motion in the case of belt suspension-drive.

Both approximate linear equations and the exact non-linear equations for a high-speed symmetrical rotor about a fixed point were developed for certain external moments. These moments included perpendicular moment, moment about the nutation axis, with both symmetrical and asymmetrical dependence of these moments on the precession angle; external and internal damping moments, and, in the exact non-linear equations - moment about the spin axis.

The linear equations are solved by previously known analytical methods. For the non-linear equations, new approximate methods were developed for obtaining analytical solutions and for numerical simulations.

In the analytical methods solutions and the corresponding stability criteria were obtained, for both slow and fast precession. These general solutions and stability criteria were specialized for certain engineering examples, including those mentioned above. For the simpler cases whose analytical solutions are known beforehand, the new approximate analytical solution yields results that are the same as the known solutions.

The solutions and the stability criteria made it possible to determine the influence of all three external moments acting on the rotor on the stability of slow and fast precession. In particular, the influence of the perpendicular moment has been determined, to which previous authors have not given sufficient consideration. The stability was also found for conditions of variable speed, and/or non-linear external moments, for which the known methods of solution are inappropriate. Because of the great number of cases that were analyzed under various conditions, no general rule could be formulated to simply describe the influence of the perpendicular moment and the other moments on rotor stability.

The analytical method uses a coordinate system and definition of angles that are very convenient for describing and following the rotor motion in space. The combination of the analytical method developed in this research, together with the appropriate coordinate system and definition of angles, facilitates the identification of the various expressions of the solution with the corresponding physical phenomena. Thus the influence of the various external moments on rotor behavior and stability, and the explanation of experimental phenomena are easily demonstrated.

In this manner, good control of rotor stability was achieved in the experiments with the belt suspension-drive. For example, the new analytical solution and stability criteria make it possible to achieve the fastest possible rotor braking without causing divergence of the slow precession.

The exact non-linear equations of motion, and the way of developing their analytical solution, promote also a good understanding of the physical phenomena represented by each of the different terms of the required moments.

In order to verify the approximate analytical solution of the exact non-linear equations, numerical simulation for these equations was run on a digital computer. A number of numerical integration methods were tested for exact non-linear equations of motion of a rotor about a fixed point, for cases whose analytical closed-form solutions were known beforehand. The numerical simulation showed that the standard computer programs for the integration methods are not accurate enough for the stability investigation of the above-mentioned equations. Double-precision computer programs were written and tested, until two integration methods were found sufficiently accurate for the simulation results to closely approximate the known analytical solution.

The accurate simulation methods were then applied to the equations of motion for the cases of this research, and were found to yield results that are essentially the same as the results of the analytical method, thus verifying the analytical methods.

 REVIEWERS: R. Cohen, I. Porat (1934-2012)
 REVIEW SOURCE: Influence of load torque on stability of rotor driven by flexible shaft,
Journal of Sound and Vibrations, v. 95, pp. 151-160 (1984). See pp. 151, 159. 
Ben-Amots, in his study of the influence of the transverse moments on a symmetric rotor rotating about a fixed point in space, used a model which includes a constant-velocity joint and found stability conditions for the two precessions.

 53 references:
  1. Arnold, R.N. (1908-1963), Maunder, L.M., "Gyrodynamics and its engineering applications," Academic Press, London (1961).
  2. Bousso, D. (1933-1971), "A stability criterion for rotating shafts," D.Sc. thesis, Technion, Haifa, Israel (1963) (In Hebrew) synopsis in English.
  3. Leimanis, E. (1905-1992), "The general problem of the motion of coupled rigid bodies about a fixed point," Springer, Berlin (1965).
  4. Porat, I. (1934-2012), "Optimal damping of flexibly-mounted rotor," Israel J. Technol, v. 7, pp. 235-245 (1969) Abstract in English.
  5. Ben-Amots, N., "The dynamical behavior of a rotor on a belt suspension drive," M.Sc. thesis, Technion, Haifa, Israel (1969) (In Hebrew) Abstract in English.
  6. Klajn, M. (1937-2008), "An approximate solution for the motion of a spinning projectile with a non linear Magnus moment," Israel J. Technol., v. 8, pp. 181-188 (1970) Abstract in English.
  7. Bousso, D. (1933-1971), "A stability criterion for rotating shafts," Israel J. Technol., v. 10, pp. 409-423 (1972) Abstract and reviews in English.
  8. Bousso, D.(1933-1971), Ben-Amots, N., "A simple means for attaining high centrifugal acceleration," J. Phys. E: Sci. Inst., v. 5, pp. 291-295 (1972) Abstract in English
  9. Ben-Amots, N., "Approximate analytical solution for high-speed spin-axisymmetric rotor, using coordinate system linked to precession and nutation," Acta Mechanica, v. 25, No. 1-2, pp.111-119 (March 1976)
    Abstract and reviews in English.
and more 44 references.

 First page

Cohen, R.,
"Effect of the coupling between flexural and torsional vibrations on rotors driven by a flexible shaft,"
DSc. thesis, Technion, Haifa, Israel (1982). In Hebrew. See p. 12, 203 Abstract

Cohen, R., Porat, I. (1934-2012)
"Influence of load torque on stability of rotor driven by flexible shaft,"
J. Sound Vib., v. 95, pp. 151-160 (1984). See p. 151, 159

● COMMENT: Dedicated to the memory of Professor Dino Bousso (1933-1971) who acted as supervisor in the first stage of the study.

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