The Motion of a Rigid Body with an Attached

Spring-Mass-Damper∗

Anne E. Chinnery† and Christopher D. Hall‡

Air Force Institute of Technology

Wright-Patterson Air Force Base, Ohio 45433

Journal of Guidance, Control, and Dynamics, Vol. 18, No. 6, 1995, pp. 1404-1409

Abstract

The stability of motion of a torque-free rigid body with a precession damper is in-vestigated. The equations of motion are presented and non-dimensionalized, and thewell-known conditions for asymptotic stability of the major axis spin are stated. At-tention is given to the cases where these conditions are not met. Results of numericalintegration are used to identify three distinct types of motion, including apparentlimit cycle-like solutions. Bifurcation diagrams are used to argue that the apparentlimit cycles are actually slowly approaching an equilibrium. The bifurcation diagramsalso identify regions where the jump phenomenon may occur. The Liapunov-Schmidtreduction technique is used to obtain a simple analytical relationship between thesystem parameters which determines whether the jump phenomenon is possible.

∗Presented as AIAA Paper 94-3715 at the AIAA/AAS Astrodynamics Conference, Scottsdale,AZ, Aug. 1–3, 1994. This paper is declared a work of the U. S. Government and is not subject tocopyright protection in the United States.

†Formerly Graduate Student, Astronautical Engineering. Presently Captain, Foreign TechnologyAnalyst, National Air Intelligence Center, Wright-Patterson AFB, Ohio 45433.

‡Assistant Professor of Aerospace and Systems Engineering. Senior Member AIAA, AAS.

1

Introduction

Spin stabilization of satellites depends on the effective use of energy dissipationto damp out coning motions caused by perturbing torques. The major axis rule1 isone well-known design criterion for spin-stabilized satellites with unspecified energydissipation, but this rule must be modified for specific damping mechanisms. Forexample, the major axis rule is not sufficient for stability of a satellite with a precessiondamper of the type studied here. The purpose of this paper is to present some newresults on the stability of motion of such systems.

This problem has been studied by numerous authors. Cloutier2 studied the re-lated problem of nutation dampers on both spin and dual-spin stabilized spacecraft,and identified the possibility of coexisting stable equilibria. Schneider and Likins3

prescribed the damper motion and used an approximate analysis to compare the ef-fectiveness of precession and nutation dampers. Spin and dual-spin spacecraft werestudied by Sarychev and Sazonov,4 and they obtained criteria for stability of thesteady spin, as well as for optimal damping of small cone angles. Cochran andThompson5 compared nutation and precession dampers and obtained an approximateanalytical formula for energy dissipation rates. Month and Rand6 used perturbationmethods and computer algebra to determine stability conditions of a rigid body withan attached spring-mass oscillator with prescribed motion. Levi7 identified multipleequilibria and non-steady solutions for a system with the damper on a principal axisand the undeformed mass center fixed. Kane and Levinson8 introduced a new dissi-pation mechanism for axisymmetric bodies, and gave a lengthy list of references toprevious work in this area. Chinnery9 used a Liapunov function analysis to obtainthe same stability conditions as Sarychev and Sazonov.4 Hughes1 used this problemas a textbook example with a detailed linearized stability analysis.

In this paper we study the dynamics of a rigid body with a precession damper,using simulation, numerical continuation, and analytical bifurcation theory. We be-gin by stating the equations of motion as developed by Hughes,1 which we non-dimensionalize.9 We then state the stability conditions1,4, 9 in terms of dimensionlessparameters. We are primarily interested in what happens when these conditions arenot met. Numerical simulation results are used to identify three distinct types ofmotion, including a persistent oscillation which appears to be a limit cycle. It isthis apparent limit cycle that motivates our bifurcation analysis. Continuation meth-ods10,11 are used to develop bifurcation diagrams which indicate that the “limit cycle”is actually slowly moving towards an equilibrium. There are three different types ofbifurcation diagrams, all of which exhibit multiple coexisting stable solutions, andtwo of which permit the jump phenomenon. Separating the three types of bifurcation

2

diagrams are two critical bifurcations: one a transcritical and one a degenerate pitch-fork. The Liapunov-Schmidt method12,13 is used to obtain an analytical relationshipbetween the system parameters defining the degenerate pitchfork.

Equations of Motion

The model we study is shown in Fig. 1, consisting of a rigid body B, and a massparticle P , which is constrained to move along a line n fixed in B. For a precessiondamper, n is parallel to e2, which is the nominal spin axis for the spacecraft. Thereference axes ei are system principal axes when P is in its rest position (ξ

∗ = 0).

The dimensional equations of motion as derived by Hughes1 are given below. Thesystem linear and angular momentum are denoted by p∗ and h∗, respectively. (Thesuperscript ∗ is used to denote dimensional quantities.) The linear momentum of theparticle in the n direction is p∗n, and the relative position and velocity of the particlein the n direction are ξ∗ and ζ∗. The angular velocity of the body frame is ω∗, andthe velocity of the point O is v∗

o. The position vector from O to P is r∗p = b∗ + ξ∗n,where b∗ = b∗e1. The mass of the particle is m

∗

p, and the total mass is m∗. The first

and second moments of inertia are c∗ = m∗

pξ∗n and

J∗ =

I∗1 +m∗

pξ∗2 −m∗

pb∗ξ∗ 0

−m∗

pb∗ξ∗ I∗2 0

0 0 I∗3 +m∗

pξ∗2

(1)

The spring has stiffness k∗, and the dashpot damper has damping coefficient c∗. Interms of these variables and parameters, the equations describing torque- and force-free motion are:

p∗ = m∗v∗

o − c∗×ω∗ +m∗

pζ∗n (2)

h∗ = c∗×v∗

o + J∗ω∗ +m∗

pζ∗b∗×n (3)

p∗n = m∗

p

(

nTv∗

o − nTb∗×ω∗ + ζ∗)

(4)

p∗ = −ω∗×p∗ (5)

h∗ = −ω∗×h∗ − v∗×

o p∗ (6)

p∗n = m∗

pω∗Tn×

(

v∗

o − r∗×p ω∗)

− c∗ζ∗ − k∗ξ∗ (7)

ξ∗ = ζ∗ (8)

The superscript × denotes the skew-symmetric matrix form of a vector.1

To non-dimensionalize the equations, we first note that since there are no externaltorques, the angular momentum vector has constant magnitude; i.e., ‖h∗‖ = h∗. We

3

now define dimensionless variables and parameters as follows:

p∗ = h∗m∗b∗

I∗2

p v∗

o =h∗b∗

I∗2

vo t∗ =I∗2

h∗t

h∗ = h∗h ω∗ = h∗

I∗2

ω b∗ = b∗b

p∗n =h∗m∗b∗

I∗2

pn ζ∗ = h∗b∗

I∗2

z J∗ = I∗2J

ξ∗ = b∗x

(9)

and introduce four dimensionless parameters:

ε = m∗

p/m∗ (10)

b = m∗b∗2/I∗2 (11)

c = c∗I∗2/(m∗h∗) (12)

k = k∗I∗22 /(m∗h∗2) (13)

The parameter ε represents the mass fraction of the damper relative to the total

system mass; we also use ε′4= 1− ε. The parameters c and k represent the damping

coefficient and spring constant, respectively, and b characterizes the distance of theprecession damper from the origin. b is closely related to the system radius of gyrationabout the e2 axis, k

∗

g , since

I∗2 = m∗k∗2g (14)

The value b = 1 puts the precession damper at the radius of gyration. The dimen-sionless form of the inertia matrix is

J =

I1 + εbx2 −εbx 0−εbx 1 00 0 I3 + εbx2

(15)

The non-dimensional equations are:

p = vo −1

bc×ω+ εzn (16)

h = c×vo + Jω+ εbzb×n (17)

pn = ε(

nTvo − nTb×ω+ z)

(18)

p = −ω×p (19)

h = −ω×h− bv×

o p (20)

pn = εωTn×(

vo − r×p ω)

− cz − kx (21)

x = z (22)

Since there are no external forces linear momentum is conserved, and we assumewithout loss of generality that p = 0. We then eliminate vo and ω from Eqs. (20–22),

4

obtaining a fifth order system:

h = h×K−1L (23)

pn = −εLTK−1n×[

(ε′xn+ b)×

K−1L+ εxn]

−cx− kx (24)

x =pn/ε+ nTb×K−1h

ε′ + εbnTb×K−1b×n(25)

where

L = h− εbxb×n (26)

K−1 =

1/D1 εbx/D1 0εbx/D1 D2/D1 00 0 1/D3

(27)

D1 = I1 + εb (ε′ − εb)x2 (28)

D2 = I1 + εε′bx2 (29)

D3 = I3 + εε′bx2 (30)

These equations are used in the numerical studies reported in this paper.

Stability Criteria

The desired motion corresponds to a steady spin about the e2 axis, h = (0, 1, 0), x =x = 0, which is easily shown to satisfy the equilibrium conditions h = 0, pn = x = 0.For a rigid body this is a marginally stable motion if e2 is the major or minor axis. Fora “quasi-rigid” body with very slow energy dissipation, the nominal spin is asymp-totically stable only if e2 is the major axis, and unstable otherwise. For the finiteenergy dissipation provided by the precession damper, the result is not as simple.Sarychev and Sazonov4 and Hughes1 used linearization and the Routh-Hurwitz cri-teria to obtain conditions for stability of the steady spin. Chinnery9 derived thesame conditions using a Liapunov function consisting of the rotational kinetic energyplus spring potential energy constrained by the constant magnitude of the angularmomentum vector. In terms of our dimensionless parameters, the stability criteriaare:

I2 > I3 (31)

I2 > I1 + ε2b/k (32)

5

Note that it is not sufficient that the body be oblate. Also, note that in our non-dimensionalization, we have I2 = 1, so the second condition may be written as

k > ε2b/(1− I1) (33)

As we subsequently demonstrate, this relationship between the parameters cor-responds to a pitchfork bifurcation. We assume throughout that the major axisconditions are met; i.e., I2 > I1, and I2 > I3.

Numerical Integration Results

As noted previously, we are particularly interested in the system response whenthe stability criteria are not satisfied. We would like to answer the question: Whatother limiting motions are possible as t→∞?

In this section, we present simulation results for stable and unstable cases, ob-tained by numerical integration of Eqs. (23–25). In Figs. 2–4 we plot the transverseangular momenta h3 vs. h1 for an asymmetric spacecraft with I1 = 0.6664, I3 = 0.5,and ε = 0.1. In all three plots, the initial conditions are h = (0.00999, 0.999, 0.00999),x = pn = 0. Note that the scales of the three plots are different.

The first case (Fig. 2) is the stable case; i.e., Eqs. (31–32) are satisfied. Asexpected, the small initial cone angle is diminished by the energy dissipation. Inthe second case (Fig. 3) the solution approaches a different stable equilibrium, withnon-zero limiting values of h1 and x. This motion corresponds to a steady spin abouta new major axis. In the third case (Fig. 4) the solution tends to a closed periodicorbit, which appears to be a limit cycle. This solution raises additional questions: Dolimit cycles exist? If so, what parameters and/or initial conditions give rise to limitcycles?

We remark that the unsteady motions found by Levi7 for a similar model areessentially limit cycles. Also dual-spin satellites exhibit limit cycle behavior for non-linear damping14 and nonlinear restoring force.15 Thus it would not be too surprisingto find limit cycles in the present problem. In the following section we use numericalbifurcation results to argue that the apparent limit cycle is actually a transient mo-tion, and eventually settles down to a steady spin about a major axis of a deformedsystem.

Numerical Bifurcation Results

6

To determine whether Fig. 4 is actually a limit cycle, a bifurcation analysis isperformed.11 In this section we obtain bifurcation diagrams numerically, using theAUTO10 continuation program. In addition to the pitchfork bifurcation described byEq. (33), we find branches of limit points, a transcritical bifurcation, and a degeneratepitchfork corresponding to a transition from subcritical to supercritical behavior. Inthe following section we obtain an analytical expression for this degeneracy.

The bifurcation analysis is based on setting Eqs. (23–25) equal to zero and findingsolutions to the resulting algebraic equations. It is straightforward to show that thereare two isolated branches of equilibria satisfying h1 = h2 = 0, h3 = ±1, and that theseare unstable if Eq. (31) is satisfied. We do not consider these branches further, andwe use the fact that all other equilibrium solutions satisfy h3 = pn = 0 to reduce theequilibrium equations to the following three equations for h1, h2, and x:

0 = εb[(

h21 − h2

2

)

x+ ε′h1h2x2]

+ (I1 − 1)h1h2 (34)

0 = ε [(ε′ − εb)h1x− I1h2] (h1 + εbh2x)− kxD21 (35)

0 = 1− h21 − h2

2 (36)

where D1 is defined at Eq. (28). Note that these equations are independent of I3.

The “trivial” solution branch is h2 = ±1, h1 = x = 0, and is the nominal so-lution for the spinning body. The stability of this branch is governed by Eq. (33).We are interested in the bifurcation which occurs for this relationship between thesystem parameters. We use AUTO10 to trace branches of equilibria for constantvalues of k, while continuing in b. Figures 5–7 illustrate the three generic types ofbifurcation diagrams which exist for this problem. These plots are for ε = 0.1, andplot the equilibrium value of h1 vs. b for different values of k. Following conven-tion, solid curves represent branches of stable equilibria and dashed curves denoteunstable branches. Note that Eqs. (34–36) are invariant under the transformation(h1, h2, x) 7→ (h1,−h2,−x); thus we identify two branches of equilibria with eachbranch of h1 shown.

The three different types of bifurcation diagrams correspond to different rangesof k. For small k (Fig. 5), the pitchfork is subcritical ; i.e., it opens to the left, andthe bifurcating branches are unstable. There are also two isolated branches of stableequilibria, which can not be found by continuation in b from the trivial solution.These branches are found by continuation in k from the trivial solution with a fixedvalue of b. The interesting thing here is that for any parameter values there are stableequilibria in addition to the trivial solution.

For an intermediate range of k (Fig. 6), the isolated branches connect with thepitchfork branches in a limit point. There are also additional isolated branches withlimit points. These also can not be found by continuation in b from the trivial solution,

7

but rather are found by two-parameter continuation of the “connected” limit points.For this range of k, there is a range of b where the trivial solution is the only stableequilibrium, and two ranges of b where the trivial solution coexists with two otherstable branches of equilibria. There is a critical value of k corresponding to the pointwhere the connected limit points coalesce with the isolated limit points, giving riseto a transcritical bifurcation, which can be seen in Figs. 8–9.

For larger values of k (Fig. 7), the pitchfork is supercritical; i.e., it opens to theright, and the bifurcating branches are stable. For this range of k, the trivial solution,when stable, is the only stable equilibrium (except for very small b). There is a criticalvalue of k corresponding to the point where the subcritical pitchforks in Figs. 5 and 6transition to supercritical pitchforks as in Fig. 7.

Figures 8 and 9 are composite bifurcation diagrams for a range of k. Figure 8 plotsh1 vs. b, and Fig. 9 shows the value of x vs. b. The mass fraction is ε = 0.1 in bothfigures. Since this value of ε is very large for ordinary precession damper applications,we also show bifurcation diagrams using a more realistic value, ε = 0.00026, basedon the data in Ref.5. Bifurcation diagrams for this value are shown in Figs. 10–11.Note that these plots are qualitatively similar to those in Figs. 8–9, but have muchlarger values of b.

The reason for constructing these bifurcation diagrams is to try to determinewhether the “apparent” limit cycle in Fig. 4 is in fact a limit cycle. As Figs. 5 and 8show, for given small k, there are two branches of stable equilibria (actually four,since each h1 branch corresponds to two branches) with h1 ≈ 1. It is this equilibriumthat the limit cycle is tending to as t → ∞, as can be checked by careful long-time integration of the differential equations. The two interesting values of k arethose which correspond to the transcritical bifurcation and the degenerate pitchforkbifurcation. In the next section, we use Liapunov-Schmidt reduction to obtain theanalytical conditions for the occurrence of the degenerate pitchfork. There does notappear to be a straightforward way to obtain a similar result for the transcriticalbifurcation.

Analytical Bifurcation Results

In this section we give the results of applying the Liapunov-Schmidt reduction12,13

to Eqs. (34–36) to obtain the conditions for the bifurcation from subcritical to su-percritical pitchforks identified in the numerical studies of the previous section. Thedetails of the calculations are outlined in an appendix.

8

We denote the vector Eqs. (34–36) by

f(y, λ) = 0 (37)

with y = (h1, h2 − 1, x), and λ = b − β. The translations are introduced so thatf(0, 0) = 0 is a bifurcation point when k takes the bifurcation value k = ε2b/(1− I1),given by Eq. (33).

At ordinary points where f(y, λ) = 0, the implicit function theorem providesthe algorithm for tracing equilibrium branches y(λ) which form the main parts of thebifurcation diagrams of the previous section. Because (0, 0) is a bifurcation point, theJacobian of f at this point, denoted by df(0,0), is singular, and the implicit functiontheorem is not applicable. Bifurcation theory10–13 provides the necessary tools fortracing the bifurcating branches of equilibria. This theory is especially well developedfor the scalar case where

f(y, λ) = 0 −→ g(u, λ) = 0 (38)

with g a scalar function of a scalar variable u. The Liapunov-Schmidt techniqueprovides the algorithm for reducing f(y, λ) to the scalar function g(u, λ), where u isthe component of y in the direction of the null space of df(0,0).

Once the reduction is accomplished, well-known methods for the scalar problemmay be applied. For example, a normal form for the pitchfork bifurcation is

g(u, λ) = λu− u3 = 0 (39)

which is characterized by the value of g and its derivatives at (0, 0):

g = gu = guu = gλ = 0, guuu < 0, guλ > 0 (40)

Any function g(u, λ) which satisfies these conditions exhibits a supercritical pitchforkat (0, 0), as in Fig. 7. Alternatively, the function g(u, λ) = λu+u3 exhibits a subcriti-cal pitchfork at the origin, as in Figs. 5 and 6. It is easy to see that the only differencein the given conditions above is that guuu > 0 for the subcritical pitchfork. In thepresent problem, we are interested in determining conditions on the parameters forwhether the bifurcation is subcritical or supercritical. This amounts to finding whereguuu or guλ changes sign. Thus we do not require g(u, λ) explicitly; we only need tocheck the conditions on its derivatives, as given by Eq. (40).

We find the derivatives by applying Liapunov-Schmidt reduction, the details ofwhich are discussed in an appendix. We check that g = gu = guu = gλ = 0 are allsatisfied, and we obtain the following remarkably simple expressions for the higherderivatives:

guλ = ε2I21/(1− I1) (41)

guuu =12ε3b2I2

1 [ε′(1− I1) + εb(I1 − 2)]

(1− I1)3(42)

9

Since guλ > 0 is independent of b, the test for subcritical vs. supercritical depends onthe sign of guuu, which simplifies to

ε′(1− I1) + εb(I1 − 2)

< 0 ⇒ supercritical= 0 ⇒ degenerate> 0 ⇒ subcritical

(43)

Considering ε and I1 as given, the critical value of b where the transition occurs is

bcr = ε′(1− I1)/[ε(2− I1)] (44)

The critical value of the spring constant is then obtained using Eq. (33):

kcr = εε′/(2− I1) (45)

Thus, if k > kcr, then the bifurcation diagram is of the type shown in Fig. 7; otherwise,Fig. 5 or 6 is applicable. This leads to a design criterion for k and b: k > kcrinsures against the jump phenomenon, and b satisfying Eq. (33) insures a stabletrivial solution. The limit on b probably will not be restrictive in most designs, sinceb will necessarily be limited by placement inside the spacecraft. The complete pictureis given by Fig. 12, which shows the bifurcation values of b and k. This figure isqualitatively unchanged by varying ε and I1.

Conclusions

The motion of a torque-free rigid body with a precession damper has been charac-terized using a combination of numerical integration, numerical continuation, and an-alytical bifurcation techniques. Apparent limit cycles have been identified as transientmotions. A transcritical bifurcation, and subcritical and supercritical pitchforks havebeen located using bifurcation diagrams involving two parameters. The transitionfrom subcritical to supercritical pitchfork bifurcation has been identified analyticallyusing Liapunov-Schmidt reduction.

Appendix: Details of the Liapunov-Schmidt

Reduction

Here we outline the calculations used to obtain the formula for the critical value ofk, Eq. (45), following Ch. 1, § 3 of Golubitsky and Schaeffer.12 The extensive algebrawas performed using Mathematica.16

10

Several matrices and vectors are needed for the calculation. The Jacobian of f atthe bifurcation point is

df(0,0)4= D =

I1 − 1 0 −εb−εI1 0 ε2bI1/(I1 − 1)0 −2 0

(46)

The null space of D, N (D), is spanned by the vector

nD = (εb/(I1 − 1), 0, 1) (47)

The range of D, R(D), is spanned by the two vectors

r1 = (1, εI1/(1− I1), 0) (48)

r2 = (0, 0, 1) (49)

The projection onto R(D) is

E =1

D4

(I1 − 1)2 εI1(1− I1) 0

εI1(1− I1) ε2I21 0

0 0 D4

(50)

whereD4 = (1− I1)

2 + ε2I21 (51)

We require additional derivatives of f(y, λ), namely d2f , d3f , fλ, and dfλ, all of whichare evaluated at (0, 0). We do not reproduce all these results here, as they are easilycomputed. Note that d2f and d3f are third and fourth rank tensors, respectively.

We use two decompositions of IR3:

IR3 = N (D)⊕M (52)

IR3 = R(D)⊕N (53)

where M is spanned by

m1 = (0, 1, 0) (54)

m2 = (1, 0, εb/(1− I1)) (55)

and N is spanned by

nR = (εI1/(I1 − 1), 1, 0) (56)

Note that M and N are not unique, and that we have chosen their bases so that themi are orthogonal to N (D), and nR is orthogonal to R(D).

11

Although D is singular, it is a nonsingular transformation from M to R(D). Wedenote the inverse of this transformation by D−1, given by

D−1 =1

D5

(I1 − 1)3 −εI1(I1 − 1)

2 00 0 −D5/2

−εb(I1 − 1)2 ε2bI1(I1 − 1) 0

(57)

whereD5 =

[

(1− I1)2 + ε2b2

] [

(1− I1)2 + ε2I2

1

]

(58)

The necessary formulas for the higher derivatives of g are then

guλ = 〈nR, dfλnD − d2f(nD,D−1Efλ)〉 (59)

guuu = 〈nR, d3f(nD,nD,nD)

−3d2f(nD,D−1Ed2f(nD,nD)〉 (60)

These are the expressions used to compute the derivatives in Eqs. (41–42).

Acknowledgments

We are grateful to Captain Jeff Beck for many useful discussions on using AUTO.

References

[1]Hughes, P. C., Spacecraft Attitude Dynamics, Wiley, New York, 1986, Chs. 3, 4.

[2]Cloutier, G. J., “Nutation Damper Instability on Spin-Stabilized Spacecraft,”AIAA Journal, Vol. 7, No. 11, 1969, pp. 2110–2115.

[3]Schneider, C. C., Jr. and Likins, P. W., “Nutation Dampers vs. PrecessionDampers for Asymmetric Spinning Spacecraft,” Journal of Spacecraft and Rockets,Vol. 10, No. ?, 1973, pp. 218–222.

[4]Sarychev, V. A. and Sazonov, V. V., “Spin-Stabilized Satellites,” Journal of theAstronautical Sciences, Vol. 24, No. 4, 1976, pp. 291–310.

[5]Cochran, J. E., Jr. and Thompson, J. A., “Nutation Dampers vs. PrecessionDampers for Asymmetric Spacecraft,” Journal of Guidance and Control, Vol. 3,No. 1, 1980, pp. 22–28.

12

[6]Month, L. A. and Rand, R. H., “Stability of a Rigid Body with an OscillatingParticle: An Application of MACSYMA,” Journal of Applied Mechanics, Vol. 52,No. ?, 1985, pp. 686–692.

[7]Levi, M., “Morse Theory for a Model Space Structure,” Dynamics and Control ofMultibody Systems, Contemporary Mathematics, Vol. 97, American MathematicalSociety, Providence, Rhode Island, 1989, pp. 209–216.

[8]Kane, T. R. and Levinson, D. A., “A Passive Method for Eliminating Coning ofForce-Free, Axisymmetric Rigid Bodies,” Journal of the Astronautical Sciences,Vol. 40, No. 4, 1992, pp. 439–448.

[9]Chinnery, A. E., Numerical Analysis of a Rigid Body with an Attached Spring-Mass-Damper, MS Thesis, Graduate School of Engineering, Air Force Institute ofTechnology, Wright-Patterson AFB Ohio, December 1994.

[10]Doedel, E., AUTO: Software for Continuation and Bifurcation Problems in Ordi-nary Differential Equations, California Institute of Technology, Pasadena, 1986.

[11]Seydel, R., From Equilbrium to Chaos: Practical Bifurcation and Stability Anal-ysis, Elsevier, New York, 1988, Chs. 1–5.

[12]Golubitsky, M. and Schaeffer, D. G., Singularities and Groups in BifurcationTheory, Vol. I, Springer-Verlag, New York, 1985, Chs. 1, 7.

[13]Rand, R. H. and Armbruster, D., Perturbation Methods, Bifucation Theory, andComputer Algebra, Springer-Verlag, New York, 1987, Ch. 7.

[14]Likins, P. W., Tseng, G.-T., and Mingori, D. L., “Stable Limit Cycles due toNonlinear Damping in Dual-Spin Spacecraft,” Journal of Spacecraft and Rockets,Vol. 8, No. 6, 1971, pp. 568–574.

[15]Mingori, D. L., Tseng, G.-T., and Likins, P. W., “Constant and Variable Ampli-tude Limit Cycles in Dual-Spin Spacecraft,” Journal of Spacecraft and Rockets,Vol. 9, No. 11, 1972, pp. 825–830.

[16]Wolfram, S., Mathematica: A System for Doing Mathematics by Computer, 2ndEdition, Addison-Wesley, Redwood City, California, 1991.

13

Figure Captions

Figure 1. System model.

Figure 2. Stable nominal motion: trajectory spirals into origin.

Figure 3. Unstable nominal motion: trajectory spirals to deformed equilibrium.

Figure 4. Unstable nominal motion: trajectory spirals out to an apparent limit cycle.

Figure 5. Bifurcation diagram: h1 vs. b for small k.

Figure 6. Bifurcation diagram: h1 vs. b for intermediate k.

Figure 7. Bifurcation diagram: h1 vs. b for large k.

Figure 8. Bifurcation diagram: h1 vs. b for a range of k.

Figure 9. Bifurcation diagram: x vs. b for a range of k.

Figure 10. Bifurcation diagram: h1 vs. b for a range of k. ε = 0.00026.

Figure 11. Bifurcation diagram: x vs. b for a range of k. ε = 0.00026.

Figure 12. Bifurcations in the bk plane. I1 = 0.6664, ε = 0.1.

14

Fi

e

e

2

3

B

rP

n^

ξO

e1

p

b^

ν

^

^

Chinnery& HallG4404Fig. 1

15

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

Angular Momentum, h(1)

Angu

lar M

omen

tum

, h(3

)

k = 0.07

b = 1.997c = 0.03

Chinnery& HallG4404Fig. 2

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Angular Momentum, h(1)

Angu

lar M

omen

tum

, h(3

)

k = 0.05

b = 2.0 c = 0.03

Chinnery& HallG4404Fig. 3

17

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Angular Momentum, h(1)

Angu

lar M

omen

tum

, h(3

)

k = 0.005

b = 0.5 c = 0.3

Chinnery& HallG4404Fig. 4

18

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Parameter b

Varia

ble

h(1)

k = 0.04

Chinnery& HallG4404Fig. 5

19

0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Parameter b

Varia

ble

h(1)

k = 0.05

Chinnery& HallG4404Fig. 6

20

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Parameter b

Varia

ble

h(1)

k = 0.08

Chinnery& HallG4404Fig. 7

21

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Parameter b

Varia

ble

h(1)

k=0.01 k=0.11

Chinnery& HallG4404Fig. 8

22

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

3

4

5

Parameter b

Varia

ble

x

Chinnery& HallG4404Fig. 9

23

0 200 400 600 800 1000 1200 1400 1600 1800 2000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Parameter b

Varia

ble

h(1)

k=0.00003 k=0.00027

Chinnery& HallG4404Fig. 10

24

0 200 400 600 800 1000 1200 1400 1600 1800 2000-5

-4

-3

-2

-1

0

1

2

3

4

5

Parameter b

Varia

ble

x

Chinnery& HallG4404Fig. 11

25

Chinnery& HallG4404Fig. 12

26