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The fundamental principles of the gyroscope theory contain the system of the inertial torques generated by the rotating mass of the spinning disc that interrelated by the ratio of its angular velocities rotation around axes. The action of the centrifugal, common inertial Coriolis forces and the change in the angular momentum generate the system of inertial torques. These four dynamical components make up the system of the eight torques acting simultaneously on the spinning disc. They manifest their action in gyroscopic effects. The ratio of the precessed motions of the gyroscope presents one of the gyroscopic effects around axes of rotation. The known mathematical model for this ratio contains an error that was corrected in this work.

Gyroscopic effects attract many physicists and mathematicians for over two centuries, as far as researchers cannot find true mathematical models and cannot explain their physics. The term “gyroscopic effects” was derived because of unexplainable motions in space and the action of the inertial forces on the spinning objects [

It is necessary to note that the famous physicists and mathematicians of past times could not develop the gyroscope theory in principle. The chronicle dates of the physical concepts and the properties of the matter are evidence that scientists and researchers of the 17 - 19 centuries did not have in full scale the mathematical tools to solve problems of gyroscopic effects [

The physics of gyroscopic effects are far more complicated than presented in the simplified theories [

The system of the interrelated inertial torques produced by the external torque T acting on the spinning disc is represented in

Inertial torques acting around axes generated by | Equation |
---|---|

Centrifugal forces | T c t . i = T i n . i = 4 π 2 J ω ω i |

Inertial forces | |

Coriolis forces | T c r . i = ( 8 / 9 ) J ω ω i |

Change in angular momentum | T a m . i = J ω ω i |

Where ω_{i} is the angular velocity of the spinning disc around axis i; ω is the angular velocity of the spinning disc around axis oz; J is the mass moment of inertia of the spinning disc; other components are as specified above.

Practical tests of the action of the external torque on the gyroscope show the different angular velocities of the gimbals around two axes. The minor turn of the external gimbal leads to an intensive turn of the inner gimbal. The gyroscopic effects are a manifestation of the principle of the conservation of mechanical energy for the spinning disc described in publications [

The kinetic energy of the spinning disc rotation around axes remains constant according to the principle of conservation of mechanical energy. This statement enables the combination of the mechanical energies of the rotating disc around two axes to be described. The inertial torques acting on the spinning disc expresses the kinetic energy of a rotation around axes.

The spinning disc with the inclined axis on the angle γ and the action of the external torque (

− T c t . x − T c r . x − T i n . y − T a m . y = T i n . x cos γ + T a m . x cos γ − T c t . y cos γ − T c r . y cos γ (1)

where the angle γ is the inclination of the disc’s axel, other components are as specified in

Substituting expressions of the torques (

− 4 π 2 9 J ω ω x − 8 9 J ω ω x − 4 π 2 9 J ω ω y − J ω ω γ = 4 π 2 9 J ω ω x cos γ + J ω ω x cos γ − 4 π 2 9 J ω ω y cos γ − 8 9 J ω ω y cos γ (2)

Simplification of Equation (2) yields the following:

ω y = − [ 4 π 2 + 8 + ( 4 π 2 + 9 ) cos γ 4 π 2 + 9 − ( 4 π 2 + 8 ) cos γ ] ω x (3)

where the sign (−) means the direction of the action of the inertial torque that can be omitted from the following analytical considerations.

Equation (3) presents the variable ratio of the angular velocities for the precessions of the spinning disc around two axes depending on the angle γ. The ratio ω_{y}/ω_{x} of the angular velocities of the gyroscope around axis oyand ox is changed with the change in the angle γ (

The diagram of _{y}/ω_{x} is changed from maximal value for the disc horizontal to zero, with the change in its angular disposition.

For the angle γ = 0 the ratio ω_{y}/ω_{x} is maximal as is follows:

ω y ω x = [ 4 π 2 + 8 + ( 4 π 2 + 9 ) cos 0 4 π 2 + 9 − ( 4 π 2 + 8 ) cos 0 ] = 8 π 2 + 17 = 95 .956835 (4)

For the turn of the spinning disc axle on the angle, φ max = 168 .342302 ∘ the ratio ω_{y}/ω_{x} is null, as is follows:

ω y ω x = [ 4 π 2 + 8 + ( 4 π 2 + 9 ) cos 168 .342302 ∘ 4 π 2 + 9 − ( 4 π 2 + 8 ) cos 168 .342302 ∘ ] = 0 (5)

The turn of the spinning disc around axis ox is active by the ratio ω_{y}/ω_{x} until the turn of the spinning disc around axis oy on the angle that is defined by the following expression:

γ = 168 .342302 ∘ 8 π 2 + 17 = 1 .754354 ∘ (6)

The angles γ = 1 .754354 ∘ and φ max = 168 .342302 ∘ are maximal for the spinning disc turn around two axes simultaneously that maintain their ratio. Practically, the outer gimbal of the gyroscope turns on the angle γ max = 1 .754354 ∘ around the vertical axis yields the turn of its internal gimbal with the spinning disc turns on the angle φ max = 168 .342302 ∘ . The following turn of the outer gimbal does not lead to the turn of the inner gimbal.

Practically, the angular motions of the gyroscopic gimbals are visible on its

stand. This gyroscope property is validated by the test of the gyroscope with the horizontal location of the spinning disc axis and the turn of the outer and inner gimbals around axes oy and ox. The defined angles of the gyroscope turn around axes are demonstrated on the Super Precision Gyroscope “Brightfusion LTD” (

The interrelated motions of the spinning disc around axes ox and oy result from the action of its inertial torques generated by one rotating mass. The spinning disc with the angular velocity ω, located symmetrically regarding its supports of the gimbal 1, which rotates on supports of the gimbal 2, and later one rotates on pivots of the platform 4 (

The practical test of the gimbal motions was conducted for the horizontal location of the spinning disc and its following turn until vertical. At the starting

condition, the disposition of the spinning disc and the outer gimbal is showed in

The Mitutoyo Universal Bevel Protractor was used to conduct the measurement of the angles of the turn for the gimbals. The angle of the turn of the inner gimbal on ω_{y}t = 90˚ from horizontal to vertical location implements by the minor turn of the outer gimbal on the angle ω_{x}t = γ that defined by Equation (3).

90 ∘ = [ 4 π 2 + 8 + ( 4 π 2 + 9 ) cos 0 ∘ 4 π 2 + 9 − ( 4 π 2 + 8 ) cos 0 ∘ ] γ and γ = 0.937921 ∘ = 0 ∘ 5 6 ′ 1 6 ″ ^{ }

The measurements of the angle γ = 0.937921 ∘ were problematic due to the high sensitivity on the manual turn of the outer gimbal that yields the turn of the inner gimbal on 90˚. The obtained result of practical motions of the gimbals is the validation of the correctness of Equations (4)-(6) that matched theoretical results. The turn of the inner gimbal around axis ox on φ max = 168 .342302 ∘ was confirmed approximately because the measurement by the tool was problematic due to the constructional properties of the gyroscope cage.

The mathematical model for the ratio of the angular velocities of gyroscope motions describes their physics. These gyroscope properties can be useful for modeling the behavior for the gyroscopic devices that is routing work for practitioners.

The planned fundamental principles of the theory of gyroscopic effects contain minor inevitable errors of the first edition. The author should correct the gyroscope theory to avoid criticism of users. One of the mathematical errors of the theory of gyroscopic effects is the ratio of the precessed motions of the gyroscope around axes of rotation. This error results from the incomplete presentation of the angular motions of the spinning disc around axes of rotation. The corrected ratio of the angular velocities of the gyroscope around axes of rotation is observed and confirmed practically.

The theory of gyroscopic effects for spinning objects opens a new direction in classical mechanics for computing inertial forces and motions of the objects in space. Science and engineering receive a new powerful analytical tool and methods that enable solving gyroscopic effects in engineering. The first publication of the theory of gyroscopic effects contains some errors in the analytical formulation of the acting inertias tortures that must be corrected for sake of wide use in practice. One error in the mathematical model for the ratio of the precessed motions of the gyroscope around axes of rotation is corrected and can be used for many mechanical gyroscopic devices in engineering.

The author declares no conflicts of interest regarding the publication of this paper.

Usubamatov, R. (2021) Interrelation of Precessed Motions of the Gyroscope. World Journal of Mechanics, 11, 177-185. https://doi.org/10.4236/wjm.2021.119013