Reprinted fromTHEORETICALAND
APPLIEDMECHANICS
Volume 48
Proceedings of the 48th japan National Congress on Theoretical and Applied Mechanics,1999

Edited by Yutaka MIYAKEfor Japan National Committee for Theoretical and Applied Mechanics, Science Council of Japan

A New Motional Equation of a Gyro Led by Centrifugal ForceToyoji BABA
Oita Menki, Oita

The author aims to analyze the movement direction of the material point of a gyroin precession and calculate the angular momentum by means of centrifugal force
caused by the crossing of the direction of centrifugal force and that of a.gyro.
１．I NTRODUCT ION
Until now the precession of a gyro has been explained by IωΩ＝ＷＬ ( I ，ω，Ω，Ｗ；I stands for inertia momentum, ωfor the speed of the spinning angle．Ωfor the speed ofprecession angle，W for the weight of the gyro．L for the length of the rotation axis) ，
but it does not make sense in that the angular momentum of I ωΩ wi11 increaseindefintely, if W and L are made to increase or Ω is made to increase by external force
under the condition of constant I ω．
This paper attempts to analyze the movement direction of the material point of a gyroin precession and derive the following equation by means of the couple caused by theinclination of centrifugal force and the momentum caused by the couple.

α=inclination angle of a rotation axis to perpendicularity
２．DISCUSSION
2.1 THE MOVEMENT DIRECTION OF TＨE MATERIAL POINT OF A GYRO IN PRECESSION

When a gyro is in precession, each material point works in the merging direction of thetangent of the spinning direction of the gyro and that of the movement direction of thegyro in precession.．Its tangent is at a right angle to the momentary spinning axis．
From the following conditions in Fig. 1：E：momentary spinning center． G：center of gravity of the gyro． A：fulcrum ofspinning axis, line AG：spinning axis, dotted lineAE:momentary spinning axis, angle
BDC=right,angle A' AG=right, AB=Gd, AG=Bd, lineA' B' parallel to E' D' ． line E'D' :central line which goes through the center of gravity of the gyro and is a right angle
to the spinning axis, V ：precession speed,
ω：angle speed of a gyro．Ｖa：spinning speed of the material point of radius rθ,θ：deviation angle (tentative name--the crossing angle EAG of the spinning axis andthe momentary central axis) , r：distance from E (momentary spinning point) to G
(central spinning point)

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T.BABA

regarding material point C, the following equation is derived：Vθsin b ：spinning speed parallel to E' D' in side view
V sec d ：precession speed in ground plan
V sec d sin d = V tan d：recession speed parallel to spinning axis
The front view is a composite picture of Vθsin b and Vtan d．
From the condition of X=angle GCF．The following equation is derived：

Therefore tan X=tanθ(Refer to Fig．1)． The crossing angle of the tangent of thematerial point and momentary spinning axis(AE) is right．

A New Motional Equation of a Gyro Led by Centrifugal Force

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2.2 DEFINITION OF INERTIA RADIUS AND
INERTIA QUANTITY
When a homogeneous disc of radius R is spininng on a fixed axis, the following
definitions are established：

From definitions (1) and (2), the sum total of centrifugal force=Mrω2 （ω＝angle speed）
2.3 GENERATION OF DEVIATION MOMENTUM( tentative name )Centrifugal force moves in the direction of the moving material point from the spinning
center． From the following condition：
θ=deviation angle (crossing angle) of momentary spinning central axis in precession
and spinning axis
couple (Mrω2 ・sinθ) is caused by centrifugal force (Mrω2 ),since each material point
moves at an angle of θto the spinning direction of the gyro.
Deviation momentum is Mrω2 sinθr = Mr2 ω2 sinθ= Ｉω2 sinθ（Ｉ＝Mr2 ）．
and its direction is opposite to the fulcrum of gravity (minus direction) ． since themomentary spinning central axis occurs on the spinning axis.
2.4 COMPARISON TO THE TRADITOONAL EQUATION
From the following condition：rθ= Iength from the spinning central point of the gyro to the momentary spinningcentral point(ω、Ｌ、α、Ω、θ、Ｉ) the same as above mentioned．The following equations are derived：

When a gyro of gravity W makes a precession at an angle of α to perpendicularity on
an axis of length L , gravity causes momentum of W L sinα to move in the direction ofgravity (plus direction) on the fulcrum. Supposing this momentum and deviationmomentum are equal, IωΩcosθ＝ＷＬ（from dividing both by sinα）
When a gyro is making a precession, each material point is making a circular
movement at a right angle to the momentary spinning central axis. Therefore, it isassumed that the angle momentum of Iω is that of the momentary spinning central
axis. Supposing the deviation angle is θ, the angle momentum of the gyro on thespinning axis is Iωcos θ． Supposing this is right, the equation derived from thetraditional angle momentum is the same as that in this paper, andＩω2 sinθ= W L sinα is defined as the momentum equation of the gyro.

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T.BABA

2. 5 LIMIT OF DEVIATION MOMENTUM

2. 6 PRECESSION ANGLE SPEEDΩ AND CONDITIONS OF PRECESSION

3. CONCLUSION
The precession movement equation of a gyro is Iω 2 sinθ=WLsinα．When thedirection of a spinning gyro is changed, the momentary spinning central axis crosses thespinning axis of the gyro. This crossing angle (deviation angle) produces the deviationmomentum.
Since centrifugal force works at a right angle to the momentary spinning central axis,
the deviation momentum occurs in a phase 90 degrees from the point where the force isapplied.
The reason a spinning gyro keeps precession on its fulcrum is supposed to be as
follows:
static momentum produced by the gravity of a gyro changes into precession direction
momentum due to the phase,deviation momentum begins to work as a result of precession,equilibrium is achieved between deviation momentum in a negative direction andmomentum in a positive direction which work on the fulcrum owing to gravity.
Since deviation momentum works to lessen the deviation angle, a gyro rigid .
condition is caused by deviation momentum. The characteristic movement of a gyro is
supposed to be caused by deviation momentum.
ACKNOWLEDGMENTS
The author is grateful to Professor Koichi Yoneda of General Education English．Oita
Medical University．for reviewing and improving the manuscript．
REFERENCES
Toyoji Baba ： About a new motional equation of a gyro led by centrifugal force,
Proceedings of The 29th Japan Society for Aeronautical and Space Sciences Annual
Meeting, (1998) , PP.．182-183．