solution for the motion of a symmetric euler gyroscope for ... · solution for the motion of a...

Advances in Theoretical and Applied Mechanics, Vol. 7, 2014, no. 2, 91 - 111

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/atam.2014.4816

Solution for the Motion of a Symmetric Euler

Gyroscope for Arbitrary Initial Values of the Euler

Angles using Eру Kinematic

Differential Poisson Equations

P. K. Plotnikov

Yu. Gagarin State Technical University of Saratov.77 Politechnicheskaya

Street 410054 Saratov, Russia

Copyright © 2014 P. K. Plotnikov. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Abstraсt

It is proved that the regular precession in symmetric Euler gyroscope (SEG) is not

the unique type of motion, it corresponds only to well-known initial Euler angles

slaving between them. For any other initial angles arise movement different from

the regular precession. The paper solved analytically, as a result of integration of

the differential kinematic equations Poisson with periodic coefficients, the

problem of motion of the SEG with arbitrary initial Euler angles. Periodic

coefficients – are SEG angular velocity, found as a solution independent from

angles of its dynamical equations. Derived from the general formulas obtained for

private, agreed initial Euler angles formulas for regular precession, coinciding

with the well-known. For some changed in relation to them the initial angle are

received formulas for irregular precession. In addition to the solutions of the Euler

angles obtained the solutions in Euler - Krylov angles. The analytical results are

supported by the mathematical modeling. In particular, we find conditions

("whack") when the irregular SEG precession at the Euler-Krylov angles occurs

in the direction of the rotational impulse and the sign of the angular velocity of

proper rotation is reversed. In Euler angles of the motion of irregular precession

92 P. K. Plotnikov

at a «strong» and «weak» impact are qualitatively the same. In relation to the

case of regular precession the changes are significant: the angles of precession and

nutation are oscillatory, and the angular velocity and the angle of proper rotation

change their sign to the opposite.

Keyword: symmetric Euler gyroscope, precession, Euler angles, Poisson

equations, rotation, mathematical modeling, nutation, Euler-Krylov angles,

angular velocity, impact, drift

Statement of the problem. Solution of the problem of the inertial motion of a

symmetric Euler gyroscope (SEG) is well known and is described in many works,

in particular, in [1,2,5]. This movement is a regular precession, nutation

characterized by a constant angle between the axis of angular momentum,

combined with the axis inertial basis, and the axis of own rotation of the SEG. In

this case the angular velocity of precession and nutation are constant.

These properties have been used in [4] in the preparation of the experiment on

tests of general relativity using SEG and telescope on the satellite when deciding

on the choice of the relations between the principal moments of inertia, ensuring a

very small angular velocity of precession. In the experiment [7-8] angular velocity

drifts amounted magnitude less than 10-11 deg / h, which provided the Einstein

theory true with an error of less than 1%.

Note that the solution of the problem of regular precession became possible

under the following restrictions on the initial Euler angles [5, formulas (2.39),

(2.41)]: G

Crconstconst 0

000 cos;0;

where G - the angular momentum; r - the component of the angular velocity SEG

on proper rotation axis; C- SEG principal moment of inertia around the same axis.

In contrast to [1, 2, 5], the problem is solved in two stages. The first stage of the

problem is to determine the angular velocity of the gyroscope, what is achieved by

integrating of the dynamic equations of motion of the SEG, around the equatorial

axis which is acted rotary pulse (shock). Initial Euler angles - zero, because one of

the axes of the inertial system is directed along the vector of the angular

momentum of the SEG and the other axis at the initial time were combined with

the other two axes of the inertial coordinate system. The angular SEG momentum

its size and direktion are changed as the result of shock. At the same stage for zero

initial conditions are determined by the current Euler angles after integration of

kinematic Poisson equations whose coefficients are found by the angular velocity

of the SEG. Formulas for the Euler angles found in the first stage, show the

irregular precession. In the second phase are taken in consideration nonzero initial

Euler angles. By using the matrix method of addition of turns are taken the

formulas for the resulting Euler angles of rotation, as well as the Euler-Krylov. As

a special case, for known, mutually initial conditions (see. Previous paragraph) are

Solution for the motion of a symmetric Euler gyroscope 93

the expressions for the regular precession. Of these same formulas for irregular

precession angles for some other arbitrary initial angles are received too.

1. About the problem of influence of initial conditions of the

kinematic equations on the character of motions of a

symmetric Euler gyroscope

In this section the following task is set: to define exactly the range of values of

initial Euler angles for Euler kinematic equations symmetric gyroscope (SEG) at

which they become identities- after the substituting them their analytical solutions

given in [1], as well as the range of solutions of dynamic equations given in [5].

As these solutions describe the regular precession, then the question is, at what

initial conditions, it is observed, and on what conditions it is not.

Dynamic equations for symmetric Euler gyroscope form [1, p. 126]

.;0;0

;0

;0

0 constrrdt

dr

dt

drC

rpCAdt

dqA

qrACdt

dpA

(А.1)

Kinematic Euler equations [1, p. 115]

.cos

;sincossin

;cossinsin

r

q

p

(А.2)

Solutions of these equations, obtained in [1, § 37]:

.cos;

;;cos

;;

00101

0

0

0

nrntn

constG

Cr

G

Cr

AGnnt

(А.3)

Here Gr ,0 ; 2

0

222222 rCqApAG - constants.

94 P. K. Plotnikov

Equations (A.2), solved with respect to [5, p. 46]:

.cossin

;sincos

;sin

cossin

qpctgrdt

d

qpdt

d

qp

dt

d

(А.4)

According to solutions of equations (A.1) and the notation of [5, p. 88], we have:

.tan;sin;

;;;

;;cos;sin

30

10

0

10

0

1

2

30

22

10

22

300102101

C

A

G

An

nHGCAG

rrtqtp

(А.5)

Substituting (A5) in (A4), and using (A3) for the third equation in (A.4):

;coscossin

;coscossinsin

cos

0100100

0

00

10000

ttA

G

tg

ttttr

A

Gr

.coscos 01010010

10

G

A

A

G (А.6)

Equality (A.6) is an identity in

00 ; ,...3,2,12 mm , (А.7)

I.e. the angle should be zero. For the equation (A.2) of (A.4) we have

;sincoscossin0 0010 tttt

.sinsin0 010010 tt (А.8)

Equality (A8) becomes an identity at angles 0 =0; ,...3,2,1 mm .

For the angle nt0 of the first equation in (A.4) we have

.

sincossin

AGqp

n

(А.9)


Considering (A3) and (A5), we obtain:

;coscossin 0100100 ttA

G

01010010

10 coscos

G

A

A

G,

I.e. we obtain again correlations (A.7). This means that equation (A9) becomes an

identity when 0 ; tn1 ; for any values 0 . From (A9), it also follows that

the variation of the angle 0 of the initial value 0 , i.e. 00 the

equality (A.9) will not turn into an identity.

From these calculations, we conclude that the regular precession of a gyroscope in

symmetric Euler gyroscope is only possible under the following initial values of

the angles:

.cos;0; 0000 G

Crconstconst (А.10)

In any other initial values of the Euler angles the equations (A.4) will be identical

with other solutions which do not coincide with the functions of (A3).

The actuality of the article is confirmed by publications [4,7,8].

2. Solution of Problem using the Poisson equation for arbitrary

initial Euler angles

Given in [9] the solutions of the Poisson equations for zero initial angles as

matriciant (7) [9] containing the periodic final relations, and the results of the

mathematic simulation listed in the annex to this article, also prove the result that

- in different from (A.10) of the initial values of the angle there is an irregular

precession. There are also analytical solutions for

,cos;0; 0000 G

Crconst

also for 000 ;0; ,

that in the first variant of the initial angles describe regular, and in the second

version irregular precession.

Above we show that the regular precession of SEG, which is characterized by

(A.5), occurs only when the initial values of the angles are described by (A.10).

With any other initial values of these angles occurs irregular precession. In

particular, for zero initial angles of the problem is solved in [9, 10] analytically:

matriciant is found for differential Poisson kinematic equations, i.e. the

transforming matrix with formulas for making the identity matrix of the direction

96 P. K. Plotnikov

cosines of the initial angles. In the same article are given formulas for determining

Euler-Krylov angles for the indicatid matriciant.

Below we give the analytical solution of the same problem for arbitrary initial

Euler angles. Following [5], we present the scheme of the Euler angles of rotation

with the image of freewheeling framework gimbal of FIG. 1. As in [9], with the

body gyroscope we associate the moving coordinate system Oxyz (corresponding

coordinate system O1'2'3 'in [5]), as well as the inertial coordinate systems: the

first Oξηζ, with which the system coordinate Oxyz at the initial time coincides,

and second Oξnηnζn in respect to which the coordinate system Oξηζ is developed

to initial angles Ψo, Θo, Φo.

FIG. 1

Scheme of turns of introduced coordinate systems 1 and represented by Ishlinskii

A.Y. [3], relations (1)

xyzАА

ННН

111

1

000

0

~ xyz

АННН

(1)

,0

1AAA


where Ψo, Θo, Φo, Ao - initial angles of rotation SEG and the corresponding SЕG

to them transforming matrix (direction cosine matrix); Ψ1, Θ1, Φ1, A1 - angles of

rotation corresponding matriciant A1 when Ao = E (E - the identity matrix); Ψ, Θ,

Φ, A - resulting rotation angles and the corresponding matrix transforms.

As in [9], we find an analytical solution for the matrix A1; [1] we use the notation

U = A1. In this article the finding of angular velocities p, q, r is treated as a

solution of the dynamical equations SEG, who had the initial angular velocity r(o)

= R, where the impact into the axis of the gyroscope body in the form of rotational

momentum Mo about Ox (hereinafter denoted Mo = Hx - the angular momentum

of the impact) was applied.

Dynamic Euler equations for gyroscopes with dynamic symmetry axis, including

gyro ball (C = A) are as follows:

,;0

;0

;0

A

AOr

dt

dr

pdt

dq

tIdt

d

A

Mq

dt

dp

(2)

p, q, r - the components of the angular velocity vector of the gyroscope

rotation in the related with it axes; I (t) - the identity function; Mo - angular

momentum, given to the gyroscope as the result of the impact.

For the initial conditions

t = 0; p (0) = 0; q (0) = 0; r (0) = R (3)

solving of system of differential equations (2) with initial conditions (3)

are as follows respectively [5,9]:

.;;sin;cos 0

A

MaRrtaqtap (4)

In particular, for ball gyro

.;0; Rrqap (5)

Converting of coordinate system Oxyz from the initial position Oξηζ

characterized by the formulas:

,; 11 AAAAAxyzT

(6)

where AAA ,, - coordinate transformation matrix of simple turns. On

the other hand, this matrix may be determined by integrating of the kinematic

equation matrix Poisson

98 P. K. Plotnikov

;; 111

EtAAtPdt

dA (7)

.

0

0

0

;1

33

1

32

1

31

1

23

1

22

1

21

1

13

1

12

1

11

1

pq

pr

qr

tP

aaa

aaa

aaa

A (8)

Matrix of the direction of cosines of the Euler angles to FIG. 1 with a combination

of coordinate systems and:

.

cossinsinsincos

sinsincoscossincossincossinsincoscos

cossinsincoscoscossinsinsincoscoscos

11111

111111111111

11111111111

1

A (9)

Matrix of the direction of cosines of the Euler - Krylov angles (2) analogous to

the matrix [3], has the form:

.

coscoscossinsin

cossinsinsincoscoscossinsinsincossin

sinsincossincossincoscossinsincoscos

kA (10)

FIG. 2


Tensor of angular velocities for gyroscopes with dynamic symmetry axis

has the form:

0cossin

cos0

sin0

tata

taR

taR

tP (11)

i.e. it satisfies to the condition:

2

;tPtP .

Because of this condition, the system (8) is driven by Lyapunov [6]. Indeed, the

substitution

1)( АtФZ (12)

system (8), (11) is reduced to the matrix equivalence of differential equations with

constant coefficients

BZdt

dZ , (13)

3;2;1,;;;

00

0

00

;

100

0cossin

0sincos

)( 11

1

jiRRZZ

a

aR

R

Btt

tt

tФ ij

. (14)

Equivalence of equations (8) and (13) is confirmed by the implementation of

identity ВtФtPФtФ )()()( 11 .

Solution of a linear homogeneous differential equation (13) is a formula Cauchy

001 ZNtNtZ , (15)

where N (t) - the fundamental matrix of solutions; Z (0) - matrix of the initial

values of the direction cosines, and on the condition Z (0) = E. After finding the

fundamental matrix and a number of transformation the solution (15) takes the

form:

100 P. K. Plotnikov

.;

;

cossincos1

sincossin

cos1sincos

1

2

1

22

2

2

1

2

2

2

1

1

2

11

2

2

2

2

1

A

CRRRan

n

Rnt

n

ant

n

ant

n

aR

ntn

antnt

n

R

ntn

aRnt

n

R

n

ant

n

R

Z

(16)

From (12) it follows that, so that ZtФА )(11 a solution of equation (8) for a

gyroscope with dynamic symmetry axis is the matrix (matriciant)

2

2

1

2

2

2

1

2

1

2

11

1

2

2

2

2

1

2

11

1

2

2

2

2

1

1

coscos1cos1

cossin

sincos1

coscos

sinsin

cossin

cossin

sinsin

coscos1

cossin

sincos

sinsin

coscos

n

Rnt

n

ant

n

aRnt

n

aR

tntn

a

tntn

aR

ntt

nttn

R

tntn

R

n

ant

n

Rt

tntn

a

tntn

aR

ntt

nttn

R

tntn

R

n

ant

n

Rt

A

. (17)

Matrix 1A in totality with Ao=E is the solution of the first step of the problem [9]

- irregular precession. For initial Euler matrix 0A is so:

.

cossinsinsincos

sinsincoscossincossincossinsincoscos

cossinsincoscoscossinsinsincoscoscos

00000

000000000000

000000000000

0

A

(18)

Formulas to determine the Euler angles (Solution of the 2 step of problem) are so:

3

1

0

1

1

1

3

1

0

3

1

2

11

233

1

0

3

1

3333

1

0

1

1

3

3

1

0

2

1

3

31

32 ;cos;

k

kk

k

kk

k

kk

k

kk

k

kk

aa

aa

a

atgaaa

aa

aa

a

atg . (19)


Poisson equations correspond to the following kinematic Euler equations:

.;;;

;sin

;cos

;cossin

;cossin

;sin

cossin

0000000

ttttt

Rr

taq

tap

ctgpqr

qp

pq

; (20)

3. Application of the obtained formulas to the case of regular

precession

Let us use initial values appropriate to this type of precession

CR

Aatg 000 ;0 the matrix Ao:

.

cos0sin

010

sin0cos

00

00

0

A (21)

With this in mind, we obtain:

;

cossinsincos

cossinsincos

cossinsincos

cos0sin

010

sin0cos

0

1

330

1

31

1

320

1

330

1

31

0

1

230

1

21

1

220

1

230

1

21

0

1

130

1

11

1

120

1

130

1

11

00

00

1

33

1

32

1

31

1

23

1

22

1

21

1

13

1

12

1

11

aaaaa

aaaaa

aaaaa

aaa

aaa

aaa

A

(22)

0

1

330

1

31

1

32

31

32

sincos

aa

a

a

atg . (23)

After transformations we receive

;

cossincos1cos

sin

2

2

1

2

2

02

1

0

n

Rnt

n

ant

n

aR

ntn

a

tg (24)

;sin;cos 00H

Aa

H

CR (25)

102 P. K. Plotnikov

nA

Ht

A

Hnttgnttg ;; (26)

- decision coincided with a classic.

Determine the value of the nutation angle Θ:

.cossincos 0

1

330

1

3133 aaa

After calculations

.coscos 0H

RC (27)

Decision on Θ by (27) also coincided with the classical solution for regular

precession. Consider the solution to the angle of proper rotation Φ.

0

1

130

1

11

0

1

230

1

21

11

23

cossin

cossin

aa

aa

a

atg . (28)

.*,*

*

t

ttgtg (29)

The formulas coincide with the formulas of the classical solution, but with zero

initial precession angles and proper rotation.

4. Lets examine now an option for solving the problem of

irregular precession

For initial angles CR

Aatg 000 ;0 differing from the angles (25),

generating regular precession, by sign nutation angle only. After the

transformation formulas for determining the Euler angles for JEG have the form:

tntt

tntttg

nttg

nt

nttg

coscos2sin2cos2cossin

cossin2sinsin2cossin

coscossin22coscos

cos

cos2coscos2

sin

000

000*

00

2

0

2

00*

00

2

*

(30)


Expression (30) suggests that only by changing the sign of the initial nutation

angle - at constant initial two other corners in the Euler gyroscope having

irregular motion of precession.

5. Mathematical modeling

FIG. 3 - 8 shows the results of mathematical modeling of the kinematic Euler

equations, which confirm the analytical obtained results FIG. 3 and 4 are graphs

modeling of processes of change of the Euler angles and Euler-Krylov,

respectively, for the initial angles

,0;)0(;)0( 0000000 (М.1)

ie corresponding to the conditions (20) regular precession in Euler angles. The

relationship of the Euler angles and Euler-Krylov set by the equality of the

corresponding elements of the matrices (9) and (10). Options SEG

radRC

aAsradR

AC

sradRsradassmsCssmsA

,308.0arctan;/,1057.11

;/,1570;/,10;,2.0;,1.0

0

3

322

(М.2)

Graphics fig.3 depict the change of the Euler angles of the regular

precession, which is not possible for the charts fig.4 for Euler – Krylov angles .

On it in the angles and observed harmonic oscillations with a frequency of

several more 500Hz, and in the angle - it rise- FIG. 4.

104 P. K. Plotnikov

FIG.3


FIG. 4

With applying of more strong rotational impulse around the axis Ox, when

Rs

a 1,4000 and other conditions for FIG. 3 and 4 the character constant of

movement is not changed (so the graphics are not shown), but we have the Euler

angles: .7.15)01.0(,905.0,50)01.0( 0 radconstradrad махмах

For Euler- Krylov angles FIG.6 and the amplitudes of fluctuations on , are

equal 0.5 rad, frequency about 530 Hz. Angle - increasing frequency oscillations

superimposed 530Hz. Also at constant simulation parameters SEG movements on

(M.1), (M.2) (Figures 3 and 4), but by changing of signs of initial nutation angles

on the back, ,308.000 rad we got pictures of movements 5 and 6. In Euler

angles FIG. 5 the movement became irregular nature of precession, namely by

and the oscillation frequencies above 500Hz bit of different amplitude with

offset approximately 0.3 rad centers hesitation. In angle the sign of velosity in

relation FIG.3 is reversed, and the angle became progressive. According to the

Euler-Krylov angles qualitatively analogical character.

106 P. K. Plotnikov

FIG.5


Фиг. 6

108 P. K. Plotnikov

FIG.7


FIG.8

Figures 7 and 8 shows the results of modeling SEG parameters by motion

corresponding to 5 and 6, with the only difference the angular velocity is provided

by equal Rs

a 1,4000 . As a result, the character of movements by the Euler

110 P. K. Plotnikov

angles (Fig. 7) has not changed qualitatively, quantitatively the centers of

fluctuations parted on the angles and to 0.45rad, the oscillations frequency

increased to 820 Hz. The angle is still increasing, with the superimposed

oscillations. At the same time, the movement in the Euler - Krylov angles changed

gardinally (FIG. 8). The angle became monotonically in crease in the direction

of the rotational impulse that is new. The angle still is oscillatory with a

frequency around 820 Hz displaced center of oscillation and angle the changed

the sign on the inverse of FIG.6.

Observation on the results of mathematical modeling: the movements

corresponding to the regular precession as in Euler angles so in the Euler-Krylov

angles qualitatively from the size of the angular velocity caused by the influence

of the rotational impulse is independent. Irregular movement of precession

depend cardinally from when the angle becomes monotonically increasing in the

direction of the momentum of impulse action and angle of proper rotation

changes the sign of its monotonous rotation.

Conclusion

1. It is Proved that the regular precession in the SEG is possible only if the

initial Euler angles are defined by the well-known formulas:

GCr

constconst 0000 cos;0; .

For any other initial angles the regular precession is impossible.

2. An analytical solution for the motion of the SEG by integrating of the

differential kinematic Poisson equations is found. The formulas for determining of

Euler angles and Euler – Krylov angles are done.

3. The derived formulas and the mathematical modeling confirmed that at other

initial angles different from angles indieated in the output the precession will be

different from the regular.

4. It is shown On the basis of analytical solutions and the results of mathematical

modeling, that movements corresponding to the regular precession as in Euler

angles – so it is shown, in the Euler-Krylov angles qualitatively from the angular

velocity caused by the influence of the rotational momentum are independent. At

the same time, the motion of irregular precession crucially depend from a: when

one angle velocity is beg (a>R) – the angle becomes monotonically increasing

in the direction of the momentum (with superimposed vibrational), and the angle

of of proper rotation changes the sign of its monotonic rotation (also with a

superimposed vibrational) on the reverse.


References

[1] Н.Н. Бухгольц, Основной курс теоретической механики. Часть 2, Москва,

1937.

[2] Р. Граммель, Гироскоп. Его теория и применения, Том 1, Издательство

иностранной литературы, Москва, 1952.

[3] А.Ю. Ишлинский, Механика гироскопических систем, Издательство АН

СССР, Москва, 1963.

[4] Р. Кеннон, Специальный гироскоп для измерения эффектов общей теории

относительности на борту астрономического спутника. Требования к

конструкции, в сборнике «Проблемы гироскопии», редактор Г. Циглер, Мир,

Москва, 1967.

[5] К. Магнус, Гироскоп. Теория и применение, Мир, Москва, 1974.

[6] И.Г. Малкин, Теория устойчивости движения, Физматгиз, Москва,1966.

[7] В.Г. Пешехонов, Уникальный гироскоп обеспечил проверку общей теории

относительности, Гироскопия и навигация, 4(2007), 111 – 114.

[8] В.Г. Пешехонов, Современное состояние и перспективы развития

гироскопических систем, Гироскопия и навигация, 1(2011), 3 – 16.

[9] П.К. Плотников, К вопросу о влиянии удара на движение гироскопа,

Саратовский политехнический институт, вып.55, г. Саратов, 1972, с. 53 – 61.

[10] П.К. Плотников, Измерительные гироскопические системы,

Издательство Саратовского университета, г. Саратов, 1976.

Received: August 24, 2014; Published: October 22, 2014

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