Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, 1996, pp. 1530-1535A9626952, AIAA Paper 96-1484

Dynamic analysis of cable-driven flexible multibody systems and itsexperimental modification

Jin MitsugiNTT Corp., Wireless Systems Lab., Yokosuka, Japan

Yumi SenbokuyaNTT Corp., Wireless Systems Lab., Yokosuka, Japan

IN:AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and MaterialsConference and Exhibit, 37th, Salt Lake City, UT, Apr. 15-17, 1996, Technical Papers. Pt. 3(A96-26801 06-39), Reston, VA, American Institute of Aeronautics and Astronautics, 1996,

p. 1530-1535

We present a method to handle a cable element in a flexible multibody dynamics simulation. The formulation for thecable element successfully separates the material and geometric stiffness matrices, irrespective of the choice ofrotational parameter. The proposed formulation has been implemented in the previously developed flexible multibodydynamics computer program. An experiment on a simple deployable truss structure whose deployment is controlled bya cable reveals that the proposed method can yield results that are accurate enough for industrial applications. (Author)

Page 1

96-1484

A96-26952 AIAA-96-1484-CPDYNAMIC ANALYSIS OF CABLE-DRIVEN FLEXIBLE MULTIBODY

SYSTEMS AND ITS EXPERIMENTAL VERIFICATIONJin Mitsugi* and Yumi Senbokuyat

NTT Wireless Systems Laboratory1-2356 Take Yokosuka, 238-03 JAPAN

ABSTRACTThis paper presents a method to handle a cable element in the flexible multibody dynamics simula-tion. The formulation for the cable element successfully separates the material and geometric stiffnessmatrices irrespective to the choice of rotational parameter. The proposed formulation has been imple-mented in the previously developed flexible multibody dynamics computer program. An experimenton a simple deployable truss structure whose deployment is controlled by a cable reveals that theproposed method can yield results that are accurate enough for industrial applications.

1 Introduction

In a deployable space structure that has more than one de-ployment mechanism, the synchronization of deploymentmay be required to reduce the number of actuators and,therefore, weight. The synchronization can be done byusing driving cables, which are defined in this paper ascables routed in a structure by means of pulleys. Driv-ing cables have been and are expected to be extensivelyused in deployable space structures such as solar panelsand deployable antennas. The authors have developed adeployable space antenna reflector for an advanced ultra-multi-beam communication satellite [1]. The antenna iscomposed of a deployable truss structure and a metallicmesh reflector surface. The deployment force is furnishedby springs installed in the truss structure, and the deploy-ment is controlled by driving cable. Figure 1 shows theconfigurations when it is stowed and deployed.

Since the deployment of the reflector plays a crucial rolein the success of a spacecraft mission, pre-flight verifica-tion of deployment is indispensable. A ground deploymentexperiment, however, may be difficult with large and flex-ible reflectors. For such reflectors, a numerical simulationis one way to estimate the deployment characteristics inthe space environment. In the simulation, the tension inthe driving cable and the strains in the reflector during thedeployment, as well as the deployment kinematics, have tobe computed for accurate and optimal design. This type ofanalysis, mechanical motion including elastic deformation,

"Senior Research Engineer, Satellite Communication Systems De-partment, member AIAA

'Engineer, Satellite Communication Systems DepartmentCopyright ©1996 by the American Institute of Aeronautics and

Astronautics, Inc. All right reserved -| 530

Stowed configuration

Deployed configuration

Figure 1: The antenna reflector

has been investigated in the field of flexible multibodydynamics. The first author previously presented a flexi-ble multibody dynamics formulation for deployable trussstructures using a special coordinate partitioning methodand a co-rotational finite element method [2]. The for-mulation is implemented in a computer program calledSPADE (Simple Partitioning Algorithm based Dynamicsof finite Element). It yields a minimum set of differentialequations rather than differential and algebraic equations,yet preserves the sparsity of the mass and tangent stiffnessmatrices.

This paper presents a method to handle driving cablesin the flexible multibody dynamics formulation. Experi-mental verification of the cable element is also reported.This paper is organized as follows. Section 2 overviews theflexible multibody dynamics formulation used in SPADEfor the sake of the completeness of the paper. Section 3shows how the internal force and stiffness matrix of thecable element are derived. Section 4 presents the experi-mental verification of the cable element in the multibodydynamics simulation.

2 Flexible Multibody DynamicsFormulation

2.1 Overview

In the formulation, each body is modelled by an finite el-ement or a group of finite elements and the internal forceand stiffness matrix of the body are derived by meansof the co-rotational finite element method. This methodassumes large displacement and small strain kinematics.The effect of large displacement, particularly the effectof the finite rotation of the body, is represented by a co-rotation frame, which is determined by the displaced con-figuration. The internal, inertia! and external forces ofeach body (element) are assembled into the whole multi-body system's equilibrium equation by means of a spe-cial coordinate partitioning method referred to as DirectCoordinate Partitioning (DCP). The details of DCP arepresented in the next subsection. Since the equilibriumequations are nonlinear, the combination of an implicit in-tegration and Newton's method is employed to evolve timesteps. For this purpose, the stiffness matrices of elementare derived element by element and assembled into thesystem's stiffness matrix by means of DCP. The flowchartof the analysis is shown in Fig. 2.

Time integration loop

Newton' s method loop

Variable Update

Solve for Update

Nodal Degree ofFreedom Partitioning

Internal, Inertial andExternal ForceComputation

Stiffness MatrixComputation

Construction of GlobalEquilibrium Equation

Figure 2: Flowchart of flexible multibody dynamics anal-ysis

2.2 Direct Coordinate Partitioning(DCP)

DCP partitions the independent and dependent nodal de-gree of freedom (DOF) hinge by hinge. One node in ahinge is termed a parent node and the other nodes aretermed child nodes. Each node has a position in the iner-tial frame and is rotated with respect to the un-deformedstate. The independent DOF comprises the parent nodemotion and the relative motions of child nodes with re-spect to the parent node. It should be noted that theprincipal variables are the nodal translation and rotationaldisplacement of all nodes rather than the those of onlyparent nodes and relative motions. In the rest of this sub-section, DCP is explained using a cylindrical hinge as anexample.

Suppose we have two nodes (referred to as node P andC) residing on different elements connected by a cylin-drical hinge. A cylindrical hinge can be considered as ahinge that allows a relative sliding motion and a relativerotational motion about a revolute axis as shown in Fig.3.The position vectors of nodes P and C are denoted as Xp

and Xc, respectively. The initial rotational matrices withrespect to the inertial frame are denoted as Rp0 and Rco.The rotational matrices from the initial orientation are de-noted as Rp and Rc. The sliding axis is denoted as pp inthe node P coordinate and the revolute axis is denoted asRc in the node C coordinate. The relative displacementcan be written as a scalar multiple of the sliding axis di-

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3 Cable Element

Revolute axis

Slide axis

Figure 3: Cylindrical Hinge

rection such that

Xc — Xp = (1)

Diiferentiating Eq.l with respect to the virtual transla-tional displacements of node P and C (6Xp, 6XC) and therotational displacements (6dp, 60C), the following relationbetween virtual displacements is obtained:

6XC - 6XP = 56 (2)

where 60p denotes the virtual rotational degree of freedomfor node P and ~ is a skew symmetric matrix. Denotingthe relative rotation from node P to node C as Ra, therelations between rotational matrices are

(3)

Differentiating Eq.3, the relation between virtual rota-tional displacement can be derived as follows.

68C = RQT RCO pp8a (4)

Using Eqs.2 and 4, the virtual displacement of node C canbe written in terms of the virtual displacement of node Pand relative motion such that

= C,cp

5XP

69P

6a

60

(5)

where

IO

-~~-T/3Rp0Rppp O

PC, O

In this section a cable element is derived. The basic for-mulation for the element is identical to the cable elementformulation used for the cable network structure analy-sis [3]. In the formulation, only the elongation strain ofcable is considered. The geometry of cable element is ex-pressed by introducing node, joint and segment. A nodeis equivalent to the definition of node in the finite elementmethod. The analytical variables, translational displace-ment and the rotational matrix to denote the orientationof the node, are defined on each node. A joint is definedon node to analytically express the offset between nodecenter and the point to which a cable is attached. A ca-ble element that passes through more than two joints canbe handled by approximating the element by an articu-lated line. The straight portion of the line is defined asa segment and and articulated point as a joint. In theproceeding subsections, the elongation strain is expressedin terms of the nodal displacement and orientation. Thevirtual work of the cable internal force is then derived byusing the variation of strain. The incremental form ofthe internal force gives the stiffness matrices for Newton'smethod.

3.1 Strain derivation

Let us take a cable element that has n segments. Whenthe terminal joints of the k-th segment of the cable aredenoted as Xk and Xk+i, they can be derived from theposition of nodes and its orientation such that

= Xk + (7)

where Xk, Rko, Rk and pk denote the position of node towith the joint is defined, the rotational matrix describingthe initial orientation of the node, the rotational matrixdescribing the deformation of the node and the local coor-dinate of the joint, respectively. The k-th segment vectorSjt is derived as following.

£i _ -y -y t Q\

The elongation strain f is given as the difference betweenthe summation of segment lengths and cable length 10 nor-malized by the cable length such that

e = (9)

3.2 Internal force derivation

tQ\ The virtual work of the cable element SP can be written asthe product of the cross sectional area A, Young's modulus

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E, elongation strain f and the variation of the strain 6fintegrated over the element volume such that

= EALe6e. (10)

The variation of the strain 6f can be written according tothe variation of the segment vector such that

(ID

The variation of the segment vector can be decomposedinto the virtual displacement of node as following,

(12)

where matrix Ck is defined as

Ck = J (13)

and I3 is a three-dimensional identity matrix.

3.3 Stiffness matrix derivation

The incremental form of the internal force can be writtenas follows.

8"P = EA10 (SeSc + t^c) (14)

Equation 14 shows that the stiffness matrix related tothe strain (EAl0f62€) can be clearly separated from thestiffness matrix defined by the material and the geome-try (EAl0f>c6e) in this formulation. They are termed thegeometric and material stiffness matrices after Crivelli [4].

The core component of the geometric stiffness matrix isderived straight forwardly as follows:

1 [cT g T — s ST] SS +

where

(16)

4 Experimental VerificationThe validity of the cable element as well as the flexiblemultibody dynamics was examined by an experiment ona planar deployable truss structure. In subsection 4.1, theexperiment setup and the corresponding analytical modelare explained. The results of the experiment and the anal-ysis are compared and discussed in subsection 4.2.

4.1 Experimental Setup

The model truss structure is composed of a fixed axis, anupper and a lower beam, a sliding axis, a synchronizingbeam and a driving cable (Fig.4). The deploying force is

Synchronizing beam

Upper beamDC motor Drive cable

fSliding axis

627mmLower beam

Figure 4: Model deployable truss structure

provided by the translational spring installed in the slidingaxis. The speed of the deployment is controlled by reelingin and out the driving cable. The synchronizing beam is lo-cated parallel to the fixed axis so as to force the upper andlower beams to be parallel during deployment. This trussstructure is the basic mechanical element for the antennareflector shown in Fig.l. In the experiment, the tension ofthe driving cable and the elongation strain at the centerof the upper beam were measured with strain gauge in ac-cordance with the driving cable length. The length of thedriving cable was measured by a large displacement sensor.The experimental setup is show in Fig.5. The structuralproperties of the truss structure are summarized in Table1, where $ and t denotes diameter and thickness, respec-tively. All the beams are make of Aluminum. This modeltruss structure is analytically modeled as shown in Fig.6,where MWeld, Revolute and Cylindrical denote a hingethat connects multiple nodes rigidly, a revolute hinge, andthe cylindrical hinge explained in section 2.2, respectively.The upper beam is divided into four sections connectedby MWeld hinges in order to accurately recover the strainat the point where the strain gauge is attached. The de-ployment sequence is shown in Fig.7.

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Figure 5: Experimental setup

Table 1: Structural property of model truss structure

Element PropertyFixed axisUpper beamLower beamSlide beam (Outer tube)Slide beam (Inner tube)Synchronizing beam

$ 30mm, t 1.0mm Tube$ 9.5mm, t 1.0mm Tube$ 9.5mm, t 1.0mm Tube$ 32mm, t 1.0mm Tube$ 29mm, t 1.0mm Tube15 x 8 mm prismatic bar

MWeldMWeld

Revolute MWeld Cylindrical

Figure 6: Analytical model

Stowed Halfly deployed

Deployed

Figure 7: Deployment sequence of model truss structure

4.2 Results and Discussion

In this subsection, the analytical and experimental dataare compared to examine the validity of the flexible multi-body dynamics simulation. The raw experimental straindata are biased such that the strain of the experimentaland analytical data matches at the deployed configurationbecause the strain gauge could not be calibrated to abso-lute zero strain.

Figure 8 shows the driving cable tension transition withrespect to the driving cable length. Experimental andanalytical results agree well. The small vibration at thedeployed configuration (driving cable length is around 900mm long) in the analytical result is due to the initial valuesetting of the multi-step integration scheme, 2nd orderGear[5]. At the completion of contraction for the stow-ing (the driving cable length is around 100 mm long), thetension of the driving cable increases suddenly. This is be-cause the shortening of drive cable length does not inducemechanical motion but directly affects its tension.

The strain profiles in upper beam are shown in Fig.9.The experimental data agrees well to the analytical data.The maximum bending strain is observed when the drivecable length is about 500 mm long, at which the configu-ration is neither the stowed nor deployed configuration.

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5 Conclusion

2.0(Kgf)

1.5 -

1.0 '

.03 0.5 1aoc

I o.o -

-0.5

Computation

Experiment

0.0 250.0 500.0 750.0 1000.0(mm)

Driving cable length

Figure 8: Tension transition during deployment

4.010

-4

c'a

3.0 -

2.0 -

1.0 -

0.0

<> Experiment..........Computation

0 250 500 750

Driving cable length

1000(mm)

A cable element that can be used in the flexible multi-body dynamics has been presented. Since the mathemat-ical derivation of the cable element employs virtual rota-tion displacement to denote the rotational degree of free-dom, the formulation successfully separates the materialand geometric stiffness matrices. The resultant equilib-rium equation and the stiffness matrices are independentof the choice of the rotational parameters. The methodwas examined by an experiment on a planar truss struc-ture. Results showed that the proposed flexible multibodydynamics including the cable element can predict strainduring deployment accurately enough to be used for in-dustrial applications.

References

[1] Watanabe.M., Meguro,A., Mitsugi.J., Tsunoda,H.,"Module Composition and Deployment Method onDeployable Modular Mesh Antenna Structures", 46thInternational Astronautical Congress, IAF-95-I.6.05,(1995)

[2] Mitsugi.J.,"Direct Coordinate Partitioning for Multibody Dy-namics based on Finite Element Method", AIAA-95-1442-CP, AIAA/ASME/ASCE/AHS/ACS 36thStructures, Structural Dynamics and Materials Con-ference, New Orleans, LA, (1995), pp. 2481-2487.

[3] Mitsugi,J., "Static Analysis of Cable Networks andtheir Supporting Structures", Computers and Struc-tures, Vol.51, no.l, (1994), pp.47-56.

[4] Crivelli,L.,A., "A Total- Lagrangian Beam Elementfor Analysis of Nonlinear Space Structures", Ph.D.Thesis, University of Colorado, Aerospace Engineer-ing Science Department, (1991).

[5] Gear,C.,W., "Simultaneous Numerical Solution ofDifferential- Algebraic Equations", IEEE Transac-tions on Circuit Theory, vol.CT-18, no.l, (1971),pp. 89-95.

Figure 9: Bending strain profile

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Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.