Bifurcations and chaos of a two-degree-of-freedom dissipative gyroscope

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Bifurcations and chaos of a two-degree-of-freedom

dissipative gyroscope

Hsien-Keng Chen

a,*

, Zheng-Ming Ge

b

a

Department of Industrial Management, Hsiuping Institute of Technology, 11 Gungye Road, Dali City, Taichung, Taiwan, Republic of China

b_{Department of Mechanical Engineering, National Chiao Tung University Hsinchu, Taiwan, Republic of China}

Accepted 6 July 2004

Abstract

The dynamic behaviors of a dissipative gyroscope mounted on a vibrating base are investigated qualitatively and numerically. It is shown that the nonlinear system can exhibit regular and chaotic motions. The qualitative behaviors of the system are studied by the center manifold theorem and the normal form theorem. The co-dimension one bifur-cation analysis for the Hopf bifurbifur-cation is carried out. The pitchfork, Hopf, and saddle connection bifurbifur-cations for co-dimension two bifurcation are also found in this study. Regular and chaotic motions are shown to be possible in the parameter space. Numerical methods are used to obtain the time histories, the Poincare´ maps, the Liapunov exponents, and the Liapunov dimensions. The eﬀect of the spin speed of the gyroscope on its dynamic behavior is also studied by numerical simulation in conjunction with the Liapunov exponents, and it has been found that the higher spin speed of the gyroscope can quench the chaotic motion.

1. Introduction

The motion of the gyro is an interesting problem in classical mechanics. Research in the area of gyro dynamics dates back about 100 years, whereas the pioneering paper on the concept of chaotic motion in gyros was already presented in

1981 by Leipnik and Newton[1], showing the existence of two strange attractors. Recently much interest has been

fo-cussed on these types of problems. The chaotic motions of a rate gyro and a symmetric heavy gyroscope with harmonic

excitation have been found by Ge et al.[2–5]. In 2001, the motion of a symmetric gyro which is subjected to a harmonic

vertical base excitation has been studied by Tong et al.[6], with particular emphasis on its nonlinear dynamic behavior

without taking into account the damping eﬀect. Very recently, on studying anti-control of chaos in rigid body motion,

Chen and Lee[7]introduced a new chaotic attractor. Their studies have proved that chaotic motion can be found in

rigid gyro. This paper presents a variety of interesting dynamic phenomena of a dissipative gyroscope excited by a har-monic force with emphasis on its chaotic motion.

*

Corresponding author.

E-mail address:kanechen@giga.net.tw(H.-K. Chen).

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The concept of chaos has been ﬁrst introduced by Poincare´[8]to describe orbits in space mechanics. The chaotic

behaviors of ﬂuids and gases have been given by Lorenz[9]. It is well-known that the sensitivity of the system to initial

conditions is necessary for chaotic motion. In other words, it may happen that small diﬀerences in the initial conditions produce very great ones in ﬁnal phenomena. Therefore prediction becomes impossible. Recently, many excellent books

were given by Moon[10], Thompson and Stewart[11], Chen and Dong[12], and Kapitaniak[13]. Moreover, some

out-standing works were also presented by El Naschie[14]and Kapitaniak[15].

Various interesting dynamic behaviors of a symmetric gyroscope mounted on a vibrating base have been found by

Ge and Chen[4,5]. In the past study, the gyroscope was assumed to be rigid. A single equation of motion is used to

analyze the dynamic behavior of the system; the system is viewed as a single-degree-of-freedom system. The gyroscope may not always be assumed to be a rigid body. In this paper, a two-degree-of-freedom system of a dissipative gyroscope mounted on a vibrating base is considered. Qualitative behaviors of the system are studied by center manifold theorem

[16]and normal form theorem[17]. The co-dimension one and co-dimension two bifurcation analyses are presented.

Further, numerical techniques are used to analyze the dynamic of this typical gyro. The time evolutions of the nonlinear dynamical system responses are described in phase portraits via the Poincare´ maps. The occurrence and the nature of chaotic attractors are veriﬁed by evaluating the Liapunov exponents, and the Liapunov dimensions. Besides, the eﬀect of the spin speed of the gyroscope is studied by numerical simulation in conjunction with Liapunov exponents, and it is shown that a chaotic motion will become regular as the spin speed of the gyroscope increases.

2. Formulation

The gyroscope contains a mechanical vibration absorber in the interior in the form of a spring-mass-dashpot. The absorber mass (m) is centered on the z axis and position parallel to z axis. The spring has constant k, and the dashpot

has damping constant C. The geometry of the problem under consideration is depicted inFig. 1. The motion of a

sym-metric gyroscope mounted on a vibrating base can be described by EulerÕs angles h, / and w. It is not diﬃcult to show that the Lagrangian can be expressed as

L¼1 2ðI1þ mz 2_{Þð _/}2_{þ _h}2_sin2_{hÞ þ}1 2I3ð _/ cos h þ _wÞ 2 þ1 2m_z 2_{Mgð‘ þ}

‘sin xtÞ cos h mgðz þ ‘sin xtÞ cos h k

2ðz ‘0Þ 2

; ð1Þ

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where I1, and I3are the polar and equatorial moments of inertia of the typical gyroscope, Mg is the gravity force, ‘is the amplitude of the external excitation disturbance, x the frequency of the external excitation disturbance, m is the mass of the damper, and k the spring constant. It is clear that coordinates / and w are cyclic, which provides us with two ﬁrst integrals of the motion expressing the conjugate momenta. The momentum integrals are

P/¼ oL o _/¼ ðI1þ mz 2_{Þ _/sin}2 hþ I3ð _/ cos h þ _/Þ cos h ¼ b/; ð2Þ Pw¼ oL o _w¼ I3ð _/ cos h þ _wÞ ¼ I3xz¼ bw; ð3Þ

where xzis the spin speed of the gyroscope.

Using the RouthÕs procedure via Eqs.(2) and (3), the Routhian of the system becomes

R¼ L b//_ bww_ ¼1 2ðI1þ mz 2_{Þ _h}2 ðb/ bwcos hÞ 2 2ðI1þ mz2Þsin2h þb 2 / 2I3 " # þ1 2m_z

2_{Mgð‘ þ}_‘_{sin xtÞ cos h mgðz þ}_‘_{sin xtÞ cos h}

1

2kðz ‘0Þ 2

: ð4Þ

For the trivial solution, from Eqs.(2) and (3), b/= bwis automatically satisﬁed and is assumed to hold afterwards[18].

The dissipation function is given by

F ¼1

2C _z 2

: ð5Þ

The equations of motion describing the system can be obtained from d dt oR o _qi oR o _qi þoF o _qi ¼ 0; fqijh; zg; ð6Þ

The system is viewed as a two-degree-of-freedom system. The equations governing the gyroscope are given by € hþ b 2 / ðI1þmz2Þ2 ð1cos hÞ2 sin3_h Mgð‘þ‘Þ sin xt ðI1þmz2Þ h i sin h¼ 0; €zþC m_zþ kðz ‘0Þ zð _h 2 Þ þ g cos h zb 2 / ðI1þmz2Þ2 ð1cos hÞ2 sin2_h ¼ 0: 8 > < > : ð7Þ

The equilibrium point is found to be

h¼ 0; _h ¼ 0; z ¼ ‘0

mg k

; _z¼ 0: ð8Þ

For convenient analysis, the ﬁxed point is shifted to the trivial one. The equation in ﬁrst order form can be rewritten as _x1¼ x2; _x2¼ b2 / ½I1þmðx3þp2Þ2 ð1cos x1Þ sin3_x 1 þ ½ðMg‘þmgpÞþmgx3þðMþmÞg‘ sin xt sin x1 ½I1þmðx3þpÞ2 ; _x3¼ x4; _x4¼ b2 / ½I1þmðx3þp2Þ2 ð1cos x1Þ2 sin2_x 1 ðx3þ pÞ þ gð1 cos x1Þ _mkx3þ ðx3þ pÞx22 2cx4; 8 > > > > > > > < > > > > > > > : ð9Þ where x1¼ h; x2¼ _h; x3¼ z p; x4¼ _z; p¼ ‘0 mg k ; 2c¼ C m: ð10Þ

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3. Stability analysis

3.1. The case of a pair of pure imaginary eigenvalues

In this subsection, the qualitative behaviors of the dissipative gyroscope system in the case, where the disturbance is

absent, i.e., ‘¼ 0 will be investigated. The study is concentrated on a co-dimension one bifurcation problem of the

sys-tem. By TaylorÕs expansion, the equation of motion can be rewritten as _x1¼ x2; _x2¼ b2/ 4ðI1þmp2Þ2 Mg‘þmgp ðI1þmp2Þ x1þ a1x1x3þ a2x13þ a3x1x23þ Oðx4iÞ; _x3¼ x4; _x4¼mkx3 2cx4þ a4x21þ px22þ px3x22þ a5x21x3þ Oðx4iÞ; 8 > > > > > < > > > > > : ð11Þ where a1¼ mgp ðI1þ mp2Þ ðMg‘ þ mgpÞð2mpÞ ðI1þ mp2Þ 2 þ b2_/ ðI1þ mp2Þ 3; a2¼ ðMg‘ þ mgpÞ 6ðI1þ mp2Þ b 2 / 12ðI1þ mp2Þ3 ; a3¼ 2m2_gp ðI1þ mp2Þ2 þðMg‘ þ mgpÞð3m 2_p2_I 1mÞ ðI1þ mp2Þ3 ð5m 2_p2_2mI 1Þ 2ðI1þ mp2Þ4 ; a4¼ b2_/p 4ðI1þ mp2Þ2 g 2; a5¼ ðI1 3mp2Þ 4ðI1þ mp2Þ : ð12Þ

Eq.(11)is then rewritten in vector form as follows:

_ X ¼ AX þ F ðX Þ þ Oð4Þ; ð13Þ where X ¼ ½x1; x2; x3; x4T; F¼ ½0; F1;0; F2T; A¼ 0 1 0 0 x2 1 0 0 0 0 0 0 1 0 0 k m 2c 2 6 6 6 4 3 7 7 7 5; ð14Þ and x2 1¼ b2_/ 4ðI1þ mp2Þ2 Mg‘þ mgp 4ðI1þ mp2Þ2 ; F1¼ a1x1x3þ a2x31þ a3x1x23; F2¼ a4x21þ px 2 2þ px3x 2 2þ a5x21x3: ð15Þ

From conventional linear stability analysis, one knows that in a certain parametric range the linearized system can be stable or unstable. However, the linearized system can only provide qualitative information on this nonlinear system in some cases, namely, when the eigenvalues of matrix A has no zero or purely imaginary values. A pair of purely

imag-inary eigenvalues is the major focus in the following analysis. The matrix A has complex eigenvalues k1= x1i and

k2=c + x2i (as well as k1¼ x1i and k2¼ c x2i), where

c2k

m¼ x

2

2: ð16Þ

To transform Eq.(13)into the form in which the center manifold theorem can be applied, the following linear

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B¼ 0 1 0 0 x1 0 0 0 0 0 0 1 0 0 x2 c 2 6 6 6 4 3 7 7 7 5 ð17Þ

is used. The matrix B is built from the eigenvectors of the matrix A. Let the coordinate transformation X¼ Bq; where q¼ ½q1; q2; q3; q4

T

: ð18Þ

Then Eq.(13)is transformed as

_q¼ B1_ABq_{þ B}1_F_{ðqÞ þ Oð4Þ:} _ð19Þ

The detailed expression is _q1 _q₂ _q₃ _q4 2 6 6 6 4 3 7 7 7 5¼ 0 x1 0 0 x1 0 0 0 0 0 c x2 0 0 x2 c 2 6 6 6 4 3 7 7 7 5 q1 q₂ q3 q4 2 6 6 6 4 3 7 7 7 5þ F1ðqiÞ 0 F2ðqiÞ 0 2 6 6 6 4 3 7 7 7 5þ Oð4Þ; ð20Þ where F1¼ 1 x1 ða1q2q4þ a2q32þ a3q2q24Þ; F2¼ 1 x2 ða4q22þ px 2 1q 2 1þ px 2 1q 2 1q4þ a5q22q4Þ: ð21Þ

Search for a two-dimensional center manifold: q3¼ h1ðq1; q2Þ ¼ S1q12þ S2q1q2þ S3q22þ Oð3Þ; q4¼ h2ðq1; q2Þ ¼ T1q12þ T2q1q2þ T3q22þ Oð3Þ:

ð22Þ According to the center manifold theorem, the following:

oh1 oq1½x1q2þ F1ðq1; q2; h1; h2Þ þ oh1 oq2ðx1q1Þ ðch1 x2h2Þ F2ðq1; q2; h1; h2Þ ¼ 0; oh2 oq1ðx1q2Þ þ oh2 oq2ðx1q1Þ ðch2 x2h1Þ ¼ 0 8 < : ð23Þ

must be satisﬁed. Equating powers of q2

1, q1q2, q 2

2, one obtains that

c x1 0 x2 0 0 0 x1 c 0 0 x2 2x1 c 2x1 0 x2 0 x2 0 0 c x1 0 0 x2 0 2x1 c 2x1 0 0 x2 0 x1 c 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 S1 S2 S3 T1 T2 T3 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ¼ px2 1=x2 a4=x2 0 0 0 0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 : ð24Þ

By CramerÕs rule, the solution of the non-homogeneous linear set of equations can be obtained. We deﬁne

S1¼ S1; S2¼ S2; S3¼ S3; T1¼ T1; T2¼ T2; T3¼ T3; ð25Þ

thus one has

q3¼ h1ðq1; q2Þ ¼ S1q 2 1þ S 2q1q2þ S3q 2 2þ Oð3Þ; q4¼ h2ðq1; q2Þ ¼ T1q 2 1þ T 2q1q2þ T3q 2 2þ Oð3Þ: ð26Þ

Substituting Eq.(26)into(20), the reduced system, which determines stability, is therefore given by

_q1 _q2 ¼ 0 x1 x1 0 _q 1 q2 þ x 1 1 ½a1T1q 2 1q2þ a1T2q 2 2q1þ ða1T3þ a2Þq32 0 þ Oð5Þ: ð27Þ

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The center manifold theorem has been applied to this system. Up to now, the stability of the system is undetermined, thus the normal form theorem will be adopted to study this problem afterwards. The normal form theorem implies that

many nonlinear terms in Eq.(27)can be removed by a nonlinear transformation and this transformation does not aﬀect

the qualitative behavior of the system. The normal form theorem gives a coordinate transformation which transforms

the system(27)into the following system:

_y1 _y2 ¼ 0 x1 x1 0 _y 1 y2 þ ðK1y1þ K2y2Þðy 2 1þ y 2 2Þ ðK1y2 K2y1Þðy21þ y22Þ " # þ Oð5Þ; ð28Þ where K1¼ a1T2 8x1 ; K2¼ 1 8 a1T1 x1 þ3ða1T 3þ a2Þ x1 : ð29Þ

In the absence of the terms of order O(5), the local family(29)is more tractable in plane polar coordinates (r, H). It is

easily shown that

_r¼ rðl þ K1Þr2; H_ ¼ x1 K2r2: ð30Þ

Let us assume that K1< 0. The phase portrait of the system(30)for l < 0 consists of a hyperbolic, stable focus at the

origin. When l¼ 0, _r ¼ K1r3and the origin is still asymptotically stable, though it is no longer hyperbolic. For l > 0,

_r¼ 0 for r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffil=jK1j p

as well as for r = 0. It follows that for l > 0 there is a stable limit cycle, of radius proportional to ffiffiffi

l p

, surrounding a hyperbolic, unstable focus at the origin. This is called a supercritical Hopf bifurcation; the authors

refer the reader to Thompson and Stewart[11]. If K1> 0, then the limit cycle occurs for l < 0: it is unstable and

sur-rounds a stable ﬁxed point. As l increases, the radius of the limit cycle decreases to zero at l = 0, where the ﬁxed point

at the origin becomes a weakly unstable focus. For l > 0, (y1, y2)T= 0 is unstable and hyperbolic. This is known as a

subcritical Hopf bifurcation.

3.2. The case of a double zero eigenvalues

When x1= 0 the matrix A has a double zero eigenvalues, and a pair of complex eigenvalues with negative real parts.

Similarly, the center manifold theorem is used to reduce the original-dimensional equation to a two-dimensional one to simplify the analysis. The center manifolds are found as follows:

x3¼ h1ðx1; x2Þ ¼ a4 c2_{þ x}2 2 x2 1 4a4c ðc2_{þ x}2 2Þ 2x1x2þ 1 c2_{þ x}2 2 pþ 8a4c 2 ðc2_{þ x}2 2Þ 2 2a4 c2_{þ x}2 2 ! x2 2; x4¼ h2ðx1; x2Þ ¼ 2a4 c2_{þ x}2 2 x1x2 4a4c ðc2_{þ x}2 2Þ 2x 2 2: ð31Þ

Then, the reduced system is obtained as _x1¼ x2; _x2¼ a1a4 c2_{þ x}2 2 x3₁ 4a1a4c ðc2_{þ x}2 2Þ 2x 2 1x2þ a1 c2_{þ x}2 2 pþ 8a4c 2 ðc2_{þ x}2 2Þ 2 2a4 c2_{þ x}2 2 ! x1x22; ð32Þ

Next, the above reduced system will be transformed to the simplest form, known as the normal form. This is given by _y1¼ y2; _y₂¼ a1y31þ a2y21y2; ð33Þ where a1¼ a1a4 c2_{þ x}2 2 ; a2¼ 4a1a4c ðc2_{þ x}2 2Þ 2: ð34Þ

The unfolding of the normal form is given as _y₁¼ y₂;

_y2¼ l1y1þ l2y2þ a1y31þ a2y21y2:

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It is easy to find that Eq.(35)has fixed points at A :ðy1; y2Þ ¼ ð0; 0Þ; B :ðy1; y2Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1 a1 ;0 r : ð36Þ

It is not diﬃcult to determine the stabilities of these ﬁxed points. For point A, when l2> 0 and l22þ 4l1>0, the

equi-librium is a saddle. If l2> 0 and l22þ 4l1<0, then the point A is a source. If u2< 0 and l22þ 4l1>0, it is known that

this point is a saddle point. If l2< 0 and l22þ 4l1<0, then this point A is a sink. Further, the supercritical pitchfork

bifurcations occur on the line l1= 0, the Hopf bifurcations occur when l1< 0 and l2= 0. For point B, when

l2 l1> 0 and l1> 0 the equilibrium is a source. When l2 l1> 0 and l1< 0 the equilibrium is a saddle point. If

l2 l1< 0 and l1> 0, then the point B is a sink. If l2 l1< 0 and l1< 0, then the point B is a saddle point. Besides,

the supercritical pitchfork bifurcation occurs on the line l1= 0. Again it is found that a secondary bifurcation cannot

occur.

Further, the so-called saddle connection bifurcation must be examined by the rescaling and the Melnikov method. The rescaling transformation is introduced as follows:

y1¼ el; y2¼ e 2 m; l1¼ e 2 m1; l2¼ e 2 m2: ð37Þ

This brings the system to the form of _

l¼ m;

_

m¼ m1Mþ a1M3þ eðm2mþ a2l2mÞ:

ð38Þ For e = 0, the Hamiltonian function is

Hðl; mÞ ¼m 2 2 m1m2 2 a1l4 4 : ð39Þ

Simply, let m1= a2, the system(39)has a pair of saddle points (l, m) = (± 1,0), and the heteroclinic orbits are

l₀¼ pffiffiffi2sech ffiffiffiffiffiffia1t p ; m0¼ ffiffiffiffiffiffiffi 2a1 p ðsech ffiffiffiffiffiffia1t p Þ tanh ffiffiffiffiffiffia1t p : ð40Þ

From the Melnikov function, when the saddle connection is preserved, the following condition:

Mðm2Þ ¼

Z 1

1

m0ðm2m0þ a2l20m0Þ dt ¼ 0: ð41Þ

must be satisﬁed. From the above condition, one obtains m2¼

a2 5 ffiffiffiffiffia1

p : ð42Þ

Returning the unsealed parameters, the final criterion for the saddle connection is l1 l2 ¼5 ffiffiffiffiffi a2 1 3 p a2 : ð43Þ

This completes the local analysis of the system in the neighborhood of the bifurcation.

4. Numerical simulations and discussion

In order to simplify the analysis, most of the parameters are kept constant. The parameter values are: I1= 1.0,

k = 100, ‘ = 0.1, M = 0.5, m = 0.1, p = 0.1, b2_/¼ 100, x = 2.0, 2c = 0.5. The only varied parameter is ‘. Solutions of

Eq.(9)are obtained using a Runge–Kutta integration algorithm, with the time step size of 0.01, and diﬀerent initial

conditions. From the numerical integration of Eq.(9), phase plane plots and corresponding Poincare´ maps, can easily

be constructed and Liapunov exponents calculated. For some diﬀerent parameter values, the system under

considera-tion can be driven into chaos. In this study, the powerful Liapunov exponent tests are shown inFig. 2to conﬁrm the

chaotic motion. As ‘ <4:68, the system exhibits regular motion; when ‘ >4:68, it routes to chaotic motion.

The Poincare´ map inFig. 3is a closed curve, which indicates that the motion is quasi-periodic.Fig. 4shows the

Poincare´ map, which corresponds to a typical chaotic attractor. The Poincare´ maps provide an interesting strange attractor on which the motion is possibly unpredictable. It is found that a quasi-periodic behavior can be changed into

a chaotic motion as the forcing amplitude increases.Fig. 5also shows the time history of a quasi-periodic oscillation for

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The spectral analysis of the Liapunov exponents has proven to be the most useful dynamical diagnostic for chaotic system. The spectrum of the Liapunov exponents enables one to classify the system attractor, its dimension and the sensitivity of the system to initial conditions. The numerical calculations have been undertaken by using the method

described in Wolf et al.[19]. Time plays the role of one of the dimensions and in this direction the exponent is always

zero. The Liapunov exponents spectrum for any chaotic oscillation in four-dimensional phase space can be (+, 0,, ),

(+, +, 0,) or (+, 0, 0, ). The occurrence of two or more positive Liapunov exponents is called hyperchaos. In this

study, the Liapunov exponents spectrum for chaotic oscillation is the type (+, 0,, ). Hyperchaos is not found in this

research. The largest Liapunov exponents obtained for the system as a function of ‘are plotted inFig. 2. The sum of the

Fig. 2. The largest Liapunov exponent as a function of ‘for a dissipative gyro.

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Fig. 4. (a), (b) The phase trajectory for ‘¼ 5:0; (c), (d) The Poincare´ maps for ‘¼ 5:0.

Fig. 5. The time history for ‘¼ 2:0.

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four Liapunov exponents is negative and it approaches to0.5. For example, at the following parameters: I1= 1.0,

k = 100, ‘ = 0.1, ‘¼ 5:0, M = 0.5, m = 0.1, p = 0.1, b2

/¼ 100, x = 2.0, 2c = 0.5, the calculated Liapunov exponents

are k1= 0.072, k2= 0, k3= 0.277, k4= 0.295. The sum of the exponents is

P

k¼ 0:5. The dimension of an

attrac-tor reﬂects one of the essential aspects of dissipative dynamics; that is, the contraction of the phase volume. A chaotic attractor is federated by contraction accompanied by stretching and folding of the state trajectories, it has a non-integer

Fig. 7. The largest Liapunov exponent as a function of b2/ðb/¼ I3xz=I1) for ‘¼ 5:0.

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dimension. The Liapunov dimension dL can be obtained, according to the deﬁnition given by Kaplan and Yorke[20]. The calculated fractal dimension of the above chaotic attractor is 3.126.

The governing equations contain various different parameters. Obviously, it is not feasible to study the effects of the all these parameters. A few interesting cases are presented below. The associated parameter values are given in the figure captions. According to past experience in studying the gyroscope, the effect of the spin speed of the gyroscope on the system behavior plays an important role. Here the effect of the spin speed of the gyroscope on the dynamic behaviors of

the system is studied by numerical simulation in conjunction with the Liapunov spectrum analysis.Fig. 7shows that

when the spin speed of the gyroscope increased, the chaotic motion disappears, and regularity returns. Further a

para-meter diagram for x and ‘showing the regular and chaotic motions is plotted inFig. 8.

5. Conclusions

The nonlinear motion of a dissipative symmetric gyroscope has been investigated qualitatively and numerically. It has been shown by computer simulation that, for this two-degree-of-freedom system, quasi-periodic and chaotic behav-ior exists for certain values of the amplitude and the frequency of the base excitation. Quasi-periodic routes to chaos in the system have also been observed. The qualitative behaviors of the system have been studied by the center manifold theorem and the normal form theorem in Section 3. The co-dimension one bifurcation analysis for the Hopf bifurcation has been carried out. The pitchfork, Hopf, and the saddle connection bifurcation for co-dimension two bifurcation have been found. The quasi-periodic and chaotic behaviors have been described in the phase plane. In addition to the Poin-care´ maps and the Liapunov exponents have been computed numerically and the related fractal dimensions have been used to show the presence of chaotic behavior in the system. Besides, the effect of the spin speed of the gyroscope on its dynamic behavior has also been considered. It has been found that the higher spin speed of the gyroscope can quench the chaotic motion in this study. It can be effectively used in controlling chaos. Our findings can be quite beneficial to further understanding and utilization of the complex gyroscope.

Acknowledgment

This research is supported by the National Science Council, Republic of China, under Grant Number NSC88-2212-E-009-001.

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