Z. angew. Math. Phys. 50 (1999) 999–10050044-2275/99/060999-7 $ 1.50+0.20/0c© 1999 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Stability of nonlinear vibrations of a deploying flexiblebeam from a spacecraft

Li Junfeng and Wang Zhaolin

Abstract. In this paper, stability of a complex spacecraft is studied by the use of two newmodels. The first, is a rigid-flexible model, which consists of a rigid body and a deploying flexiblebeam; while the second, includes a liquid filled rigid body with a deploying flexible beam. Inparticular, the deploying beam in both models is assumed to have a finite deflection profile.It has been proved that nonlinear transverse vibrations of a beam are stable when undergoinguniform extension or retrieval.

Mathematics Subject Classification (1991). 70J25.

Keywords. Spacecraft, deploying flexible beam, rigid-flexible coupled system, partial stability.

1. Introduction

Recently, there has been a great interest in the stability of vibrations in flexibleappendages (beam, plates) deploying out of spacecraft [1-4]. Under the assumptionof infinitesimal deflection of flexible appendages, two spacecraft models are oftenused. One assumes a fixed rigid body with a deploying flexible beam [1]. Thiscorresponds with the case of the presence of orbital and attitude control. Anothermodel consists of a free rotating rigid body with a deploying flexible beam [2],which represents the case without attitude control. However, the appendages of themodern large scale spacecraft have more flexibility. In such case, the assumption ofinfinitesimal deformation is no longer true. Moreover, it is well known that a realspacecraft is often filled with liquid fuel for orbital and attitude control, precludingthe use of the rigid body models. In this paper, we consider the appendage as afinite deflection beam. Consequently, two new models, namely, fixed center modeland generalized model, are proposed for stability study of nonlinear transversevibrations of the deploying beam.

1000 Li Junfeng and Wang Zhaolin ZAMP

Figure 1. Geometry of the system

2. Fixed centre model

We begin with a simple planar model for a spacecraft. Consider a rigid bodywith a finite deflection beam (as depicted in Fig.1). The centre, C, of the rigidbody is fixed, and around which the body may rotate freely as the beam deploys.Let CX0Y0 denote the fixed inertial coordinate system, and OXY be the bodyaxes fixed to the rigid body at O, which are used to describe the deploymentand vibrations of the beam. The orientation of axis OX relative to axis OX0 isrepresented by α(t) . The natural state of the beam is assumed to coincide withOX , whilst the length of the beam be L(t) at any instant. If Q is any point ofthe beam having coordinates (x, 0) and (X(x, t), Y (x, t)) (in undeformed state anddeformed state, respectively), then, the tangent to the centroid of the beam at thepoint Q will form an angle of θ(x, t) with the OX axis. Finally, assuming that

vectors→i and

→j denote the unit vectors of axes OX and OY , respectively, the

position vector of the point Q can thus be written as

→r = (R +X(x, t))

→i + Y (x, t)

→j (1)

whereR = |OC| . The vector of absolute velocity is therefore given by

·→r = (

·X − Y ·α)

→i + (

·Y + (R +X)

·α)→j (2)

and the vector of absolute acceleration is

··→r = (

··X − 2

·Y·α− Y ··α− (R+X)

·α2)→i + (

··Y + 2

·X·α+ (R+X)

··α− Y ·α2)

→j (3)

Vol. 50 (1999) Nonlinear vibrations of a deploying flexible beam 1001

Hence the angular moment of the beam with respect to the point Cis

→G = ρA

∫ L

0

→r ×

·→r dx

where ρ is the density of the beam, and A is the area of cross section. If the beamhas a constant cross-sectional area with moment of inertia I, then the equationsof the motion can be obtained as follows [5]

∂X

∂x= cos θ (4)

∂Y

∂x= sin θ (5)

∂M

∂x= P sin θ − S cos θ (6)

EI∂θ

∂x= M (7)

∂P

∂x= ρA(

··X − 2

·Y·α− Y ··α− (R+X)

·α2) (8)

∂S

∂x= ρA(

··Y + 2

·X·α+ (R+X)

··α− Y ·α2) (9)

Ic··α+

dG

dt= Mc(t) (10)

where E is the modulus of elasticity of the beam, Icand Mc(t) denote the momentof inertia and controlling moment of the central rigid body, respectively, M(x, t),P (x, t) and S(x, t) represent bending moment, axial force and shear force of thebeam, respectively. The boundary conditions are

X(0, t) = L(t)− L(0), Y (0, t) = 0, θ(0, t) = 0,

M(L, t) = 0, P (L, t) = 0, S(L, t) = 0, P (0, t) = F (t) (11)

where F (t) is the driving force of the beam. When Mc(t) = 0, F (t) = 0, theequations of the motion have a particular solution. That is,

X(x, t) = x+ vt, Y (x, t) = 0, θ(x, t) = 0,·α = 0,

M(x, t) = 0, P (x, t) = 0, S(x, t) = 0 (12)

In this case, the beam undergoes uniform deployment or retrieval without deflec-tions when the rigid body does not rotate.

1002 Li Junfeng and Wang Zhaolin ZAMP

3. Stability analysis

For the sake of convenience in studying the stability of the above particular solu-tion, we take disturbed variations as follows

X(x, t) = x+ vt+ u(x, t), Y (x, t) = y(x, t), θ(x, t) = ϕ(x, t),·α = ω,

M(x, t) = m(x, t), P (x, t) = p(x, t), S(x, t) = s(x, t) (13)

Substituting them into Eqs. (4)-(10) , the equations of disturbed motion of thesystem can be obtained as well. Furthermore, the kinetic energy of the system is

T =12Ic·α2 +

12ρA

∫ L

0

·→r ·

·→r dx +

12

(m0 − ρAL)·X2 (14)

where m0 is the mass of the beam. Likewise, the potential energy is given by

V =12

∫ L

0EI(

∂θ

∂x)2dx (15)

Note that, when Mc(t) = 0, F (t) = 0, the energy integral can be expressed by

H = T + V = const (16)

With disturbed variations substituted in (16), we have

H =12Icω2 + ρA

∫ L

0[(v +

·u− yω)2 + ((R + x+ vt+ u)ω +

·y)2]dx

+∫ L

0EI(

∂ϕ

∂x)2dx+ (m0 − ρAL)(v +

·u)2

(17)

Obviously H > 0 as m0 > ρAL. In particular, the conditions of the total energyH becoming zero are

ω = 0,∂ϕ

∂x= 0,

·y + (R + x+ vt+ u)ω = 0 (18)

With reference to the boundary conditions in (11), we have ω = 0, ϕ = 0, y =·y =

0. Therefore, H is a positive-definite function about partial variations ω, ϕ, y,·y. Moreover, according to the partial stable theorem[6], the solution in (9) shouldbe partial stable about ω, ϕ, y,

·y. This implies that the angular velocity ω(t),

winding angel ϕ(t) and amplitude of vibration y(x, t) will not get larger when thebeam uniformly deploys, thereby, stability being proven.

Vol. 50 (1999) Nonlinear vibrations of a deploying flexible beam 1003

4. Generalised model

In this section, the spacecraft will be modeled as a central rigid body with a bankcontaining ideal liquid and a deploying finite deflection beam. Let C0 be the centerof the system, and the coordinate system C0X0Y0 is assumed to be inertial. Theposition vector of the bodys center of mass is denoted by

→r c, and the mass of the

rigid body by m0. Consequently,

→r c = −(m0 + ρAL)−1ρA

∫ L

0

→r dx.

Therefore the position, velocity and acceleration vectors of point are as follows

→rQ =

→r c +

→r ,

·→rQ =

·→r c +

·→r ,

··→rQ =

··→r c +

··→r ,

whilst the equations of motion and boundary conditions can be shown to be (4)-(11) except for (8)-(10), which are further expressed as follows [7]

∂P

∂x= ρA

··→rQ ·

→i (8’)

∂S

∂x= ρA

··→rQ ·

→j (9’)

I0··α+ If

·Ω +

d

dt(Gc +G) = Mc(t) (10’)

where I0 = Ic + I∗, If = I ′ − I∗, Ω is the uniform vortex of the ideal liquid in thebank, I∗ denotes the inertial tensor of the equivalent rigid body of liquid, I ′ andis the inertial tensor of the solidified liquid. As Mc(t) ≡ 0, a particular solutionshould be

X(x, t) = x+ vt, Y (x, t) = 0, θ(x, t) = 0,·α = 0,

M(x, t) = 0, P (x, t) = 0, S(x, t) = 0, Ω = 0 (12’)

In this case, the beam undergoes uniform deployment or retrieval without deflec-tions when the rigid body with liquid does not rotate. For stability analysis, thefollowing disturbed variations are employed

X(x, t) = x+ vt+ u(x, t), Y (x, t) = y(x, t), θ(x, t) = ϕ(x, t),·α = ω,

M(x, t) = m(x, t), P (x, t) = p(x, t), S(x, t) = s(x, t), Ω = Ω′ (13’)

Similarly, the kinetic energy of the system is

T =12

(I0·α2 + IfΩ2 +m0

·rc

2 + ρA

∫ L

0

·rQ

2dx+ (m0 − ρAL)·X2) (14’)

1004 Li Junfeng and Wang Zhaolin ZAMP

and the potential energy of the system is expressed as in (15). Note that the totalenergy of the system equals zero if

ω = 0, Ω′ = 0,∂ϕ

∂x= 0,

·rc = 0,

·rQ = 0 (18)

This is equivalent to ω = 0, Ω′ = 0, ϕ = 0, y =·y = 0, subject to the conditions

stated in (12). Hence, H is again a positive-definite function about partial vari-ations ω, Ω,ϕ, y,

·y. Finally, according to the partial stable theorem, the angular

velocity ω(t), winding angel ϕ(t) and amplitude of vibration y(x, t) must be stable.

5. Conclusion

The stability of nonlinear vibrations of a beam deploying was investigated in thispaper. In particular, two new models of spacecraft were proposed. In the firstmodel, we considered a central rigid body deploying a flexible beam, where thecentre of the rigid body is fixed, and around which the body can rotate freely. Thesecond model takes filled liquid into account. Moreover, for both new models, theappendage is assumed as a finite deflection beam with linear constitutive equations.In conclusion, it has been shown that nonlinear transverse vibrations of the beamare stable with uniform extension or retrieval.

Acknowledgement

This work was supported by the National Natural Science Foundation of China(No. 19702011). The authors would like to thank anonymous reviewers, Mr.Shields and Dr. Lu Jianhua for their valuable comments which helped to improvethe presentation of the paper.

References

[1] S. Kalaycioglu, A. K. Misra, Approximate solutions for vibrations of deploying appendages.J. Guidance, Control and Dynamics 14 (1991), 287-293.

[2] A. E. Ibrahim, A. K. Misra, Attitude dynamics of a satellite during deployment of largeplate- type structures. J. Guidance, Control and Dynamics 5 (1982), 442-447.

[3] V. J. Modi, A. E. Ibrahim, A general formulation for librational dynamics of spacecraftwith deploying appendages. J. Guidance, Control and Dynamics 7 (1984), 563-569.

[4] Li Junfeng, Wang Zhaolin, Attitude Dynamics of a Spacecraft with Deploying FlexibleAppendages. J. Tsinghua University 36 (1996), 35-40. (Chinese)

[5] Xiao Shifu, Chen Bin. Study on Modeling and Stability of a Rigid-Flexible Coupled System.Acta Mechanica Sinica 29 (1997), 439-447. (Chinese)

[6] V. V. Rumjantsev, A. S. Oziraner, Partial stability and stabilization of motion. Moscow,Nauka 1987. (Russian)

Vol. 50 (1999) Nonlinear vibrations of a deploying flexible beam 1005

[7] N. N. Moiseev, V. V. Rumjantsev, Dynamic stability of bodies containing fluid . Moscow,Nauka 1965. (Russian)

Li Junfeng, Wang ZhaolinDept. of Engineering MechanicsTsinghua UniversityBeijing 100084China(e-mail: Lijunf@mail.tsinghua.edu.cn)

(Received: January 13, 1997; revised: September 8, 1998. resp. January 8, 1999)