**7. Modeling of stick-slip motion between wires and pulleys**

The conventional simulator also had a problem about amplitude of the dynamic responses shown as Fig.17. The slowness of the temperature change in the paddle might be the reason for that. As shown in Fig.16, the significant decreasing of temperature began 5-10 seconds af‐ ter going into penumbra. While discussing about the dynamic responses, we take notice of some particular measured data (showed in Fig.23). In the data, the solar array paddle shows the characteristic triangular wave before eclipse and the smaller dynamic responses than ordi‐ nary one. These triangular waves are found commonly in cantilever vibration affected "Stick-Slip phenomenon" (Maekawa, et.al., 2008). The Stick-Slip is a phenomenon that it arise the "stick" and "slip" behavior continuously to objects shared sliding surfaces. The phenomenon arise on the ground that the change of friction coefficient. And in slip phase, the stored strain energy is released at once. The Stick-Slip phenomenon occurred at space systems had prece‐ dents in Hubble Space Telescope (Thomton, 1993). Then, we assumed that the Stick-Slip phe‐ nomenon was occurred at deploy-speed-control wires and pulleys. That is because that the wires have small heat capacity and will be great affected by thermal environment changes. And the wires were empirically-deduced that govern the deformation of solar array paddle. The picture of them is shown in Fig. 24, and the position relation is indicated in Fig. 25. When simulating the thermal snap, sometime these specific dynamic conditions are needed to be considered. For example, when the analysis for Hubble Space Telescope was conducted, the effect from specific cross-section shape of boom was estimated. (Foster, 1995) In case of ADEOS, the tension control mechanism was evaluated in detail. (Taniwaki, 2007)

Satellite

deformation.

**Figure 25.** Position relation of GOSAT's Deploy-speed-control wires and pulleys.

change while the transition to the slip phase was estimated like below.

To verify the effect of Stick-Slip phenomenon, the phenomenon was introduced to the struc‐ tural model. In that time, the detailed information of wires and pulleys were not available. Therefore the value from measured data was used to demonstrate the effect of Stick-Slip phenomenon. Using the Bowden and Leben's equation (Bowden & Leben, 1939), the force

Vibration of Satellite Solar Array Paddle Caused by Thermal Shock When a Satellite Goes Through the Eclipse

Where the *x*r is a displacement defined like showed in Fig.23, and the *k* is a stiffness of the

**Figure 26.** Time transient response simulation with detailed hinge model and stick-slip motion modeling.

From the Fig. 23, the released force was estimated like *F*<sup>d</sup> ≒ 0.029[N]. Using this value, the new simulator was constructed, and verification analysis was conducted. In this analysis, the damping element of paddles was not concerned. The details of analysis are showed in

Deploy Control Wire

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351

*d r F xk* = (3)

**Figure 23.** Particular example that solar array paddle shows unexpected behavior.

**Figure 24.** Similar product of GOSAT's Deploy-speed-control wires and pulleys.

**Figure 25.** Position relation of GOSAT's Deploy-speed-control wires and pulleys.

simulating the thermal snap, sometime these specific dynamic conditions are needed to be considered. For example, when the analysis for Hubble Space Telescope was conducted, the effect from specific cross-section shape of boom was estimated. (Foster, 1995) In case of

ADEOS, the tension control mechanism was evaluated in detail. (Taniwaki, 2007)

350 Advances in Vibration Engineering and Structural Dynamics

**Figure 23.** Particular example that solar array paddle shows unexpected behavior.

**Figure 24.** Similar product of GOSAT's Deploy-speed-control wires and pulleys.

To verify the effect of Stick-Slip phenomenon, the phenomenon was introduced to the struc‐ tural model. In that time, the detailed information of wires and pulleys were not available. Therefore the value from measured data was used to demonstrate the effect of Stick-Slip phenomenon. Using the Bowden and Leben's equation (Bowden & Leben, 1939), the force change while the transition to the slip phase was estimated like below.

$$F\_d = \mathbf{x}\_r k \tag{3}$$

Where the *x*r is a displacement defined like showed in Fig.23, and the *k* is a stiffness of the deformation.

**Figure 26.** Time transient response simulation with detailed hinge model and stick-slip motion modeling.

From the Fig. 23, the released force was estimated like *F*<sup>d</sup> ≒ 0.029[N]. Using this value, the new simulator was constructed, and verification analysis was conducted. In this analysis, the damping element of paddles was not concerned. The details of analysis are showed in Tab.3. And the result of analysis is in Fig.26. As shown in Fig.26., the revised simulator could indicate roughly same value of amplitude of dynamic responses with consideration of Stick-Slip phenomenon

mal snap analysis. The result indicated the gap of hinges and Stick-Slip phenomenon mainly

Vibration of Satellite Solar Array Paddle Caused by Thermal Shock When a Satellite Goes Through the Eclipse

As ongoing work, we will model the Stick-Slip phenomenon more in detail and introduce

, Satoshi Suzuki3

[1] B., A. Boley(1972), Approximate Analyses of Thermal Induced Vibrations of Beam

[2] Chijie Lin, Ramesh B. Malla(2004), Coupled Thermo-Structural Analysis of an Earth Orbiting Flexible Structure, in 45th AIAA/ASME/ASCE/AHS/ASC Structures, Struc‐

[3] Earl, A. Thomton, Yool A. Kim(1993): Thermally Induced Bending Vibrations of a Flexible Rolled-Up Solar Array, Journal of Spacecraft and Rockets, Vol.30, No.4,

[4] E., A. Thornton (1996), Thermal Structures for Aerospace Application, AIAA, pp.

[5] F. P. Bowden & L. Leben (1939): The Nature of Sliding and the Analysis of Friction,

[6] Foster, C. L., Tinker, G. S., Nurre, W. & Till, W. A. (1995). NASA Technical Paper,

[8] Japan Society of Mechanical Engineering (February 2007). Mechanical Engineers' Handbook Applications 11: Space Equipment and Systems, Japan Society of Mechan‐

[7] Array-Induced Disturbance of the Hubble Space Telescope Pointing System, 3556

and Plates, *Journal of Applied Mechanics*, vol. 39, no. 1, pp212-216

tural Dynamics & Materials Conference, AIAA2004-1793

*Proceedings of Royal Society London,* vol.A169, pp371-391

ical Engineering, ISBN 978-4-88898-154-5, Japan

and Yusuke Hagiwara4

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effect on dynamical response of thermal snap on GOSAT.

the damping element using observed data.

, Akihiko Honda2

1 JAXA and Tokyo Institute of Technologies, Japan

3 Advanced Engineering Services Co.Lted, Japan

Mitsubishi Heavy Industries, Ltd, Japan

2 Tokyo Institute of technology, Japan

**Author details**

Mitsushige Oda1

**References**

pp438-448

343-354

The Solar

44


**Table 3.** Details of thermal snap analysis
