Explanation/supplement concerning the original paper by the author

The original paper : https://www.jstage.jst.go.jp/article/jsdd/4/1/4 1 142/ article/

Because I(=the author) am not an English native speaker, please permit that some expressions not

good might be included in the following English sentences.

Minimum basic knowledge necessary for understanding of the original paper.

1.Only the abc of the dynamics of the solid body in Newtonian mechanics.

Especially, it is enough only by the knowledge at the principal axis of inertia or the principal

moment of inertia for moment of inertia.

2.The knowledge of mathematics is enough by rudimentary knowledge of the ordinary differential

equation with complex variables.

(The knowledge of Laplace transform/inversion for solution of the differential equation is

preferable.)

3.Euler's formula : e iW t(∫ Exp[iW t]) = cos[W t]+ i sin[W t] AEq.(1)

( i is unit of imaginary number, t is time and W is a constant.velocity )

　A circular motion of the angular velocityW of the unit radius that centers on the origin of the

complex coordinates. This is the most important knowledge, and used many many times.

Sin and cos need not be used in the following explanations thanks to this expression any longer.

4.The knowledge of the analytical mechanics is quite unnecessary!.

In the following explanations, a lot of explanatory Figures and Equations not shown in the original

paper will be included. These new Figures and Equations are expressed by AFig. or AEq.

Figures and Equations without "A" are directly quoted from the original paper.

Refs. of This Explanation:

[1] Thomson, W.T. Introduction to Space Dynamics. (Dover: 1986 reprinting of Wiley 1963).

[2] F.Klein, A.Sommerfeld: Ueber die Theorie des Kreisels, Teubner 1910, Johnson Reprint 1965.

Chap.1 The most important coning motion : nutation.

Perhaps, in your textbook of the dynamics of the solid body in Newtonian mechanics, figures of

body cone, space cone will be included. In this explanation/supplement, only another cone is used

exclusively for explanation of nutation, free spinning motion of a symmetric body about an angular

momentum vector: H. The word: nutaton is used in space dynamics as free coning motion drawn

by spin-axis S about centering axis H, of an axisymmetric spinning satellite. Another word:

precession is defined as coning motion of S around the zenith axis Z due to the gravitational torque.

The former: nutation is free motion, on the contrary, the latter precession is forced motion. These

two definitions are different from them of the old text book or [Ref.2].

In this explanation, derivation of linearized equation of spinning tops is based on circular motion.

This method is by far simpler and intuitive, because no Euler's angle is used.

Expression of constant speed circular motion on the complex coordinate plane using Euler's

formula AEq.(1).

The left-hand figure shown below expresses nutation three-dimensionally, and the right-hand figure is

upper part (Z=1) of the left. Both Im- and Re- axes are parallel to the inertial Y- and X- axes, respectively. The underlined characters are the complex numbers, and subscript "0" means initial(t=0) value. R of the figures means a real number, magnitude of the rotating radius or half-cone angle(n ). From the figures:

�

R(t) = R0 Exp[iW t ] = R Exp[ij 0 ] ◊Exp[iW t ] = R Exp[i(j 0 + W t)] (

�

j is rotation angle)

�

S(t) ∫ S = C + R(t) = C + R0 Exp[iW t ] = C + R Exp[i(j 0 + W t)] AEq(2)

�

S0 ∫ S(t = 0) = C + R0

You will understand that Euler's formula is extremely convenient from these expressions where S draws

a circle around Hor C. If we use H and n , then:

�

S = H + n = H + n 0Exp[i Wt] = H + nExp[i(j0 + Wt)] (free circular motion) AEq.(3)

I want you completely to understand the meaning of this expression with the AFig.1 below.

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Wt

�

j 0

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ν

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AFig.1 Explanation of Euler's formula AEq.(1)

In this explanation, time derivative is expressed by dot or sometimes prime notation.

�����θ'∫ ˙ q , θ"∫ ˙ ̇ q , S'∫ - i˙ q , S" ∫ - i̇ ̇ q

Nearly horizontal components of angular velocity ω :w x and w y are quite equal to ˙ q x and ˙ q y ,respectively.

w ∫ w x + iw y = ˙ q x + i ˙ q y ∫˙ q The z-component of ω is wz ∫ n ª w and is named spin velocity.

Though they are complex numbers, they have a dimension: T-1. They are non-dimensionalized and

normalized by divided by nutation velocity:W . Then:

n = -w / W = -˙

q / W = -Hr / Hp (distance or small angle from H to S) Eq(3.7)

Subscript p means polar or z-direction and r means radial or orthogonal to polar direction.

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AFigure 2 shown below is main attitude vectors of free two axi-symmetric rotors having different

moment inertia ratio: s ∫ C / A >1 ands <1. Because of axi-symmetricity, the angular velocity�w is

included in a plane constructed by S and H vectors. Note: small angle:n of AFig.2 is rather exaggerated. Free

means ˙ H = 0 or no external torque. In this case, two angular velocity w s are decomposed to W and ns along with each different parallelogram, or w =.W�+ns� ��Note their magnitudes and directions. The ns is

called nutation velocity viewed by observer on spinner to S-H plane.

Let's differentiate AEq(3) of previous page:

˙ S = ˙ n = n 0

d

dtExp[iW t] = iWn 0 Exp[iWt] = iWn Æ ˙ n - iWn = 0 Dn = i W D t ◊n 　　　　　　Eq(5,2)

or from Eq(3.7) : ˙ H r = -Hp˙ n = -Hp i Wn = i W Hr Æ D Hr = iWDt Hr

AFig.2 Decomposition of angular velocity ω to Ω and ns and explanation of n

What is free coning motion or nutation of axi-symmetric rotor?

Free means� ˙ H = 0 ����Let's differentiate AEq(3) of previous page:

������� ˙ S = ˙ n = n 0ddt Exp[iWt] = iWn 0 Exp[iWt] = iWn Æ ˙ n - iWn = 0 ��� ˙ S = ˙ H + ˙ n � ˙ H r = -Hp ˙ n

or from Eq(3.7) : ˙ H r = -Hp˙ n = -Hp i Wn = i W Hr �Æ ��

�

DH r = iWDtH r �

　 　 AFig.3 Internal torque (left two figs, H unchanged) and external torque (right fig. H changed).

Because of free motion, position of H should be unchanged. Therefore, left figure of AFig.3 is not

correct, but center figure is correct. On the other hand, according to Newton's law, free motion is:

Free motion : A ˙ ̇ q = externaltorque= 0 or ˙ n = 0

However, results gained at former page:

Free motion : ˙ n = iWn or ˙ ̇ q = iW ˙ q

The right-hand sides of these formulas are not zero. Of course, they are not external torque. I think they should be called internal torque. We can see this term in Eq.(4.4) or Eq.(5.3) of the original paper. They comes from x- and y- components of Eq.(4.2) and called as inter-axis-crosscoupling term. This cross coupling term is not external torque but internal one, i.e. this term changes only direction of radial component of H∫ Hr . As having mentioned above, head of the Hr -arrow is not moved

but tail is moved to the opposite direction. This action makes circular motion of S ! i.e. nutation.

The right figure of AFig.3 is the case of the external torque. In this case, the position of H0 takes a

sudden change to H�at the moment when an external angular impulse torque was given,and S moves to

a new circle around H (new nutation)�from the old nutation circle. If no torque is applied, S continues

drawing a circle around H0.(old nutation). Note that S does change not�position but direction of

motion at that moment. If, you open the applet below on the top page:

Applet for Training of Control for Spin Axis/

and you move the H-position, perhaps not only you understand above-mentioned phenomenon, but

also you understand the method of attitude control of spinning bodies.

�*#2���4#+0+0)�1(��10641.�(14��2+0��:+5�75+0)��22.'6�51(69#4'� Using the above-mentioned applet-software, many examples will be shown here.

(a)cycloid (Locus of S of a horizontally suspended spinning wheel: Starting point of H accords with S )

AFig.4 Cycloid of Horizontally suspended spinning wheel

(b)trochoid (Starting point of H is right under S0)

AFig.5 Trochoid of Horizontally suspended spinning wheel

(c)parallel lines (Starting point of H is right above two scales )

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In this figure, locus of S becomes wave, In this figure, position of H0 is shifted 1/2-scale right,

this comes from applet software, i.e. locus of S becomes nearly a straight line.

every click makes 30-deg. nutation.

(d)epi-cycloid, epi-trochoid (In this case G is equivalent to the zenith,loci of H are circles around G..)

Three examples below are very near to the ordinary spinning top. An assumption here is that

magnitude of the gravity torque is constant regardless of distance from G to S.

　

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(e) two-impulse method (A method to move S at any position, and to make agree with a position

of H, or no-nutation condition.) This method is used in spade engineering attitude control of spinning satellites with two impulsive gas jets.

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(f) trial of example shown in Ref.[1] Thomson, p.161�(Horizontally suspended top beaten by vertical

downwards angular impulse at top end of the spin axis or horizontal angular impulse )

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The beginning potion is similar to (c)parallel lines:

in this case, however, this part is small. Height of the starting blue line is only 1/4 scale above x-axis.This is 1/8 to (c) case. Therefore, number of click in one scale is 8 times. Unfortunately, the locus ofS did not become the straight line because eight positions to click were incorrect. For the same reason, as for the right part of the Y-axis, the loci of the red circles by the click became incorrect, too.

(f) trial of automatic control: (automatic control means two actions i.e. by moving H, to decay two

modes :N-mode or fast mode and P-mode or slow one.) Specifically, to decay N-mode is damping, or to bring H towards S, and to decay P-mode is to bring H towards the origin or vertical attitude. These actions will be explained in the next chapter. As shown in below left figure, after 9th control, S accords with H or N-mode disappears. In the right figure, N-mode damping action is added, and we can see the effect of this action. These actions are done by KN or kN and KP or kP terms of Eqs.(4.4)~(4.6) in the original paper.

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��"�����'��&�! �!��&�$�����%�%����������������� �������You can test effects of the Gravitational torque on behavior of the spinning top using the next html:

https://main-equationspin-top.ssl-lolipop.jp/Kg-change.html

In this html, at first shown page, you must click your mouse at any position of this page, if not, you cannot go to next page (this additional action is caused by my lack of knowledge for Java programming language. Next required action is to click up-ward arrow key of your real keyboard. The next page willbe open. This up-key action is reset action for computation of Fig.4 model of the original paper. I re-commend you to click or push your right-arrow key in order to compute the model. Continuous push ofthis right-key simulate the attitude motion of the spinning top. Loci of S and H will be shown nearly real time if your machine is fast enough. You can change the gravitational coefficient Kg, +, - or zero by selecting the arrow-keys. Selection of Kg , reset and computation are done by clicking your real keyboard: up-key : reset action. Initial conditions are set to an ordinary spinning top shown in Fig.4 of the original paper. down-arrow : hanging down top (statically stable) or reverse mark of above case (any time).

left-arrow : Kg = KN = KP=0 or spinning satellite�(any time).

Computation is done by clicking right-arrow key.

AFig.3.1 Two examples of computation results by above applet software.

���Perhapps, you will recognize large difference between the above examples by sign of Kg.. The right

case shows slow decay of N-mode, on the contrally fast decay of P-mode. The main reason is whether H is inside or outside to S. Please read the Section 6 of the original paper. In short, directions of both terms of KN and KP are opposite in left. On the contrally, in right, direction of KN term is to S,

and this time, at the same time, this direction is to the origin, i.e.the both terms do collaboration in P-mode decay.

��� ��"�����'��&�! �!����� �� ��� �&�����! ��&�! %�&!�&���%�����!����In the previous chapter, the intial conditions i.e. H0 and S0 are fixed. I tried to make a new applet

program to the same dynamical model:

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Methode of the changing the initial condition is shown in the left down part or the third quadrant of

opened applet complex coordinate system as shown in the next page, though the method.is'nt shown.

Unfortunately, changing is done to either H0 or S0. Simultaneous change is impossible due to my poor ability

of Java programming

AFig.4.1 Two examples : the case only S0 is changed(left) and the case only H0 is changed(right).

In left figure of above AFig.4.1, S0 was moved from 1st quadrant of the original program to the 2nd one (see

the former page). . In the right figure, H0 was moved from 1st quadrant of the original program to the 4th

one. In the left figure, we can see the first red curve loop includes the origin only once, which causes only

one small blue loop. The right figure shows the case where H-locus makes two loops due to S-locus's twice

inclusion of the origin. These three loops of H-loci are caused by N-mode which decay quickly and the

second loop of the blue curve disappeared. From this figure, we can imagine that the two eigen-values are

both stable, and N-mode has larger effect to S-curve, and small effect to H-curve. P-mode has rather small

decay effect due to relatively small Kp of this model.

Chap.5 Effects of Kg>0 or Kg<0 on the attitude motion (no friction).

AFig.4.2 Cross section of various spinning toys

Left figure shows cross section and notations

of typical spinning toys

h : height of G

P : contact point to the table surface

subscript : f: flat spheroid

e : boiled egg

m : mashroom-shaped top

t : ordinary spinning top

Note: Kg > 0 means ordinary spinning top, or

height of G = h > r (effective radius of each

contact point) means statically unstable.

The author intends to discuss these toys and

stone of Celt ?, mainly intuitive explanation

and partially with linear theory of linearized

equation the author derived.

At first, loci of S and H after N-mode decayed out in the case of Kg>0

is already shown many times. In the former page, you can see left

figure of AFig.4.1, two loci approaching to the origin. They are almost

parallel, and they are also shown in Chap.2. However, the case of

Kg<0, direction of the gravitational torque changed to the opposite

sign. So, they should be at opposite positions to the origin, each other.

The left figure is the case of the tippe-top, where H-locus is very small,

because the contact point:Pm is very near to the H-vector and slipping

velocity is small.. Perhaps, in the cases of boiled egg, or flat spheroid,

H-circles will be by far larger.

However, nearly vertical tippe-top produces only nearly horizontal

Im

Re

Z

H

S

AFig.4.3 H-S locus of tippe-top

due to only gravitational torque

torque. As deceleration of the spinning velocity is small, though not negligible, the influence on attitude

motion will be small. Then, we will apply the linearized equation to this tippe-top in this chapter. The boiled

egg will be discussed in the other chapter.

STABILITY from eigenvalues. (Tippe-top)

It is easy to show convergence of P-mode, if we use the characteristic root:lp:(Eq.(6.3) of the original paper)

lP ª -kP + i (kg /W) (Eq.(6.3)) , kP ∫ KP / A = mhr /(As ) = mhr /C (Eq.(3.16))As kP > 0, real part of lP is negative, then P-mode is convergent. On the other hand, from Eq.(6.2) real part of lN = -kN + kP = mhr /C - mh

2/ A = mh(r /C - h / A) > 0 Æ N-mode is unstable

:In case of the tippe-top, C ª A, and r > h , therefore above real part > 0, i.e. N-mode is unstable or divergent.

NOTE! Correction: in the original paper, Fig.3, (5) : kN > kP > 0 ( false) Æ kP > kN > 0 (correct)

Im Im

Re

Re

Enlarge

S Enlarged locus of H

H H0

S0

P-mode

Locus of

AFig.4.4 An example of

attitude motion of a tippe-top

(beginning part of spinning ,

initial attitude is near to the

AFig.4.2)

SIMULATION beginning part of spinning (Tippe-top)

W =50, q'' = (I*W - 1)*q' + (70*I - 100)*q, q[0] = 0.0 + 0.2*I, q [0] = -10.3 + 0*I}, {q, q' }, {t,0,2}

Pink curve written by hand is a locus of center of H-circle, showing convergence of P-mode. On the contrary,

locus of S is circular motion around the origin and increasing, showing divergence of not P-mode but N-mode

For your information, friction-free case is shown on the next page.

{t, 0., 2.8}

Enlarged locus of H

Im

ReH0

P-mode

to be continued, 2013 May/6 revised, author

SIMULATION of Friction-free Tippe-top case

{t, 0., 3.}

Im

Re

Enlarge

H

Locus of

q'' = (I*W)*q' + (- 100)*q, q[0] = 0.0 + 0.2*I, q [0] = -10.3 + 0*I} ,{q, q' }, W =50,{t,0,3)

AFig.4.4 An example of

attitude motion of a friction

-free tippe-top

At first, you should be aware that both H-circle and S-circle are not perfect circle because of unequal Kg-torque.

Though locus of S shown above seems very thick, there is no enlargement, but each imperfect S-circle have each

phase angle, therefore, roughly speaking, each center of each imperfect S-circle moves on the pink P-mode

circle, and thickness of the red circle will be equal to the diameter of the P-mode circle.

You will recognize, the locus of S is very near to the initial circle of AFig.4.4. Of course, both N- and P-

mode exist due to initial conditions, but they donot diverge or converge.

Last Stage of the tippe-top: friction effects by sudden change of the contact point: from Pm to Pz.

Pz Pm

G

Z

Re

hm

tippe-top　

重心　

H

center of sphere　mass center　

ω

-S, -z

S, z

S

Im

ReZ

H

S

H Pz( )

Pm( )

Pzfriction torque at

AFig.4.5 Last stage of the tippe-top

ν

Above figure shows attitude of the tippe-top at the moment the knob touches to the table. The right picture

shows starting point of H by sudden change of friction torque, H-circle and shortened n. By this shortened n,

S stands up on the knob, or the tipp-top becomes ordinary top. However, this ordinary top has very small "r",

then N-mode becomes more stable, however, stability of P-mode becomes very weak. Of course, W becomes by

far smaller due to larger friction torque, then, duration of standing will become very short.

Chap.6 Attitude Motion of Spinning Boiled Egg.

★ Differences of the dynamic characteristic between the ordinary top and boiled egg.

ordinary top boiled egg ( ) means an example

Inertia Ratio : s = C/A (1.4) (1/1.4 ª 0.7) W is roughly a half

h/r (height of G / radius of P) (5) (1.6) makes kP larger, kg smaller

kN / kP = sh / r (7) (1.1) required value for N-mode stability: > 1

From above table, W of the boiled egg just after erected is more smaller, because of kinetic energy dissipation

due to friction and increasing of potential energy due to erection. In those cases, root-locus method shown in

Fig.3 -(3) of the original paper will be very helpful. Even in high speed region of N-mode, stability margin is

extremely small compared to the ordinary top shown in (2) of Fig.3. This shows a cause of the shortness of the

standing time.

★ Important relation between flat-spin of spinning satellite and rising of the spinning boiled egg.

The first satellite of both U.S.A and Soviet Union were spinning satellites who's inertia ratios:s = C/A were

smaller than 1. The results were "flat-spin".or spinning about unwanted maximum momentum of inertia axis.

If slender body axis is z-axis which is nominal spinning one, radial axis becomes a new spinning one. As H is

unchanged, coning angle n grows, and finally the cone becomes a flat circle drawn by z-axis. This will be the

origin of the name. At that time, it was said that it was caused by internal energy dissipation. Main internal

dissipation are: non-elastic distortion such as whip antenna and fluid sloshing.

In case of the boiled egg, attitude change is inverse, i.e. from flat-spin to thin-spin(?). However, the boiled

egg case is by far difficult, as not only T(kinetic energy or Poinsot)-ellipsoid but also H-sphereshrinks.

x- or y-direction

H

Nearly flat-spin of boiled egg.

±z-direction

H-sphere

H ±z-,S-

Nearly rised boiled egg.Beginning of rising boiled egg.

x- or y-direction

Ｔ-ellipsoid

HH

x- or y-directionz-direction

H

thin spin flat-spin

zH-sphere

Ｔ-ellipsoid

ellipsoid shrinks

Ｔ-ellipsoid

±z-direction

coning motion

ellipsoid shrinks

Both H-sphere and H-vector are unchangeable for inertial space, only T-ellipsoid shrinks.

H-sphere

H-sphere shrinks

H-sphere H

Ｔ-ellipsoid

shrinks

shrinks±z-directionn:decreases

direction

AFig.6.1 Thin spin to flat spin (satellite) and flat spin to thin spin of rised spinning boiled egg.

Spinning satellite

Spinning boiled egg

H-sphere Ｔ-ellipsoid

See documents [1], p-121 of the first page of this explanation about T-ellipsoid and H-sphere

Here, spinning boiled egg of AFig.6.1 will be explained. The down-left figure is the beginning attitude of a

boiled egg. Virtical H is selected, then horizontal z-axis cannnot be selected, as no external torque will be

generated. Therefore, the length of slightly declined T-ellipsoid becomes a little larger than the diameter of H-

sphere. The egg's Moment of Inertia Ratio (MOIR):s(∫ C / A) is:s ª 0.7as shown in previous page, then, w-

vector is between H and S(z-axis), or.a little right toH (see next AFig.6.2 ). The contact point P is outside to

w-vector from zenith- or Z-axis as shown in AFig.6.2. in which inclination angle q ª 75deg. Even if, q is nearly

90deg., P will be kept outside to w-vector due to more larger effective radius, r.

★Explanation to shrinking of H-sphere and T-ellipsoid.

The shrink of T-ellipsoid is easily predictable because of friction by P. However, the shrink of the H-sphere.

should be explained, using AFig.6.2. The P-point moves in the front direction of the space, and generates small

friction torque whose direction is perpendicular to G-P line as shown by a green arrow. The angle between H-

vector and the direction of the small friction torque is acute angle. The results are shrink of |H| and increase of

spinning velocity:n. At the beginning phase, the P-point or XP is very near to Z-axis, therefore decreasing |H|

is very small, on the contrally, as h is by far larger than XP, then the horizontal component of the friction

torque inclines H to S, or makes n smaller, at the same time, n larger. Initial inclination is very impotant for

increasing XP, or inclination of direction of friction torque. This means initial small inclination introduces

shrinking of H-sphere. The shrinkage of H-sphere introdeces decrease of n, or approach to S(z-axis) of H,

which is essential for nutation damping, or, in this case,standing up of the boiled egg.

If size of H is not enough, both H and w incline to S direction or decreases n easily on the way, and the

position of XP and Xw reverses, then direction of friction torque reverses also, which means increase of n, and

decrease of n, therefore the egg does not stand up.

To get large H-sphere is essential for beginning part of standing up of the boiled eggs.

AFig.6.2 Initial attitude relation between various vectors of boiled egg..

b ª Hz / Hy = sn /wy

wz /wy = n /wy = b /s > b

According to the table of former page

wz /wy ª 1.4b

Even if, H is vertical, w is declined to

S-direction: by initial spin velocity.

Direction of the friction torque is very

important. In this case, H decreases.

At the same time, n increases, and H

declines or decreases n, S stands up.

To be continued 2013 Aug.2, Author

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n

Revised 2013 Aug.25, Author

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0

SH

enlarged

H

S

P

G [rad]

[rad]

Middle stage ( q ª 50deg.) of the boiled egg, where Kg is nearly maximum, and Xp is outside to

Xw, then direction of friction torque has acute angle to

H-vector which means decrease of H-vector. A part of

friction torque increases spin-velocity = n. Anyway, H-vector approaches to S, meaning rising up of S.

S,z

HZ

x, y

-P

P

X

Xp

w

X

Direction of Torque

G

Torque increases nn

Direction of Torque

acute angle

w

�

h

★ Middle stage of the standing boiled egg.

AFig.6.3 An example of computer simulation results

of initial stage.

★ An example of computer simulation results of initial stage of boiled egg

The left figure is a case of the previous condition ;

initial S has small b, and H is vertical, or starting

point is the origin.

Note ! Locus of H is enlarged about 10 times.

We can see very slow standing up of S, and after this,

the rate of standing velocity increases due to P's move

towards outside ; increase of friction torque. From the

enlarged H, nearly the same move can be seen.

See the density variation of loci of H and S .

★ Last stage of the standing boiled egg.

AFig.6.4 Attitude relation between various

vectors of boiled egg at middle stage..

AFig.6.5 Attitude relation

between various vectors

of boiled egg at last stage.

(P is inner-most position)

on the slippy glass plate

H

S

P

G [rad]

[rad]

on the wooden plate

Friction torque coefficient ofwood is 5-times larger than theleft glass plate (for trial). P's real loci drawn on the plates including gravity center transition which are similar to S having opposite phase angle become by far larger due to smaller kg and

opposite direction of ns compared

to the ordinary tops.

2013 Sep. / 8 the Author(End of boiled egg, next item will be intuitive explanation of rattle back or stone of Celt )