The author would like to thank the reviewers that supported the original paper.1 They accepted the author’s previous publications with the correct mathematical models of gyroscope motions for its only horizontal location.2,3 All mathematical models for gyroscope motions for horizontal location were validated by practical tests.3

The mathematical models for the inclined gyroscope are inaccurate, but they are not fundamental.1

This erratum presents a logical error of the process of the analysis for the inertial torques and kinematic dependency of the gyroscope motions around the axes of rotation. The error is presented by the hiding processing of the inertial torques acting around axes that gives the distort result. The detailed explanation of this error is described as follows.

The action of inertial torques around axes and the spinning disk motions is derived on the principle of the kinetic energy equality of the gyroscope motions around axes (Fig. 1).2,3

FIG. 1.

External and inertial torques acting around axes on the spinning disk and its motions.

FIG. 1.

External and inertial torques acting around axes on the spinning disk and its motions.

Close modal

The dynamic parameters of the spinning disk are considered for the common case when its axle inclination is on the angle γ. The resultant torque acting around axis ox is presented by the following equation:

$Tt⋅x=Tct⋅x+Tcr⋅x+Tin⋅y+Tam⋅y.$
(1)

The resulting torque acting around axis oy is presented by the following equation:

$Tt,y=(Tin⋅x+Tam⋅x−Tct⋅y−Tcr⋅y)cos⁡γ.$
(2)

Equality of inertial torques indirectly expresses the kinetic energy equality of the gyroscope motions around the axes and is presented by the following equation:

$−2π29Jωωx−89Jωωx−2π29Jωωy−Jωωy=2π29Jωωx⁡cos⁡γ+Jωωx⁡cos⁡γ−2π29Jωωy⁡cos⁡γ−89Jωωy⁡cos⁡γ,$
(3)

where all components are presented in Fig. 1 and in Refs. 2 and 3.

Simplification of Eq. (3) yields the following result:

$ωy=−2π2+8+(2π2+9)cos⁡γ2π2+9−(2π2+8)cos⁡γωx.$
(4)

The inertial torques $−2π29Jωωy$ for the left- and right-hand sides of Eq. (3) for the horizontal location of the spinning disk (cos 0° = 1) are self-compensated by the rules of algebra because they have the minus sign (−). These torques are also compensated at the expressions for the inclined spinning disk on the angle γ. The right-hand side of Eq. (3) contains the expressions with the sign of cos γ of inertial torques acting around the axis oy. The self-compensation of torques has a force despite differences in expressions and rules of algebra. The kinetic energy of gyroscope motions is equal at the axes ox and oy inclined on the angle cos γ. The torque $−2π29Jωωy⁡cos⁡γ$ acts around axis oy and presents the part of the full value that acts around the inclined axis oy. This is the reason that two torques of one expression are compensated. This hidden self-compensation was not taken into account, which gives the subsequent errors in all equations.

1. In this connection, the precession torques $Tin⋅y=2π29Jωωy$ and Tamy = Jωωy generated at axis oy are acting around axis ox in full value. These accurate expressions should be used for all equations of gyroscope motions, kinematic dependencies around axes ox and oy, and kinetic energies around axis ox and corrected according to Eqs. (3) and (4) in Ref. 1. The corrected equations of gyroscope nutation will have different expressions.

2. Computing the gyroscope nutation by equations with precession torques $Tin⋅y=2π29Jωωy⁡cos⁡γ$, Tamy = Jωωy cos γ, and the equations of kinematic dependencies around axes ox and oy ωy = f(ωx)3 gives distorted results that are not acceptable for the given technical parameters of Ref. 1.

3. The method for the solution of gyroscope nutation is confirmed by the corrected math model for nutation in Ref. 4.

The corrected mathematical models for gyroscope motions and kinetic energies of its nutation do not change the conclusion of the paper.

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