**6. Numerical simulation**

target marker. This modification worked well to identify the target marker and its displace‐ ment. Fig. 8 is a result based on the revised algorithm. In this revised algorithm, the areas to be considered target makers are decided based on the brightness level of each pixel and also on the size of the areas considered to be target markers. When an assumed target marker is too small, then the threshold level of brightness is automatically adjusted to meet the possi‐

Finally, Fig. 9 shows the motion of a target marker as based on the revised algorithm. We

can see that the estimated motion of the solar array paddle has less dispersion.

Case-0 Case 1

ble size of the target markers.

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**Figure 8.** Target markers estimated by the revised algorithm

**Figure 9.** Displacement of GOSAT's solar array paddle when going into eclipse

In order to verify whether the observation results described above are correct, and to under‐ stand the features of thermal snap on the solar array paddle, we conducted numerical simu‐ lation of the solar array paddle. There are many studies analyzing the thermal snap (Thornton, 1996; Boley, 1972; Johnston, 1998; Lin, 2004; Xue2007), we develop the method using a thermal model and a structural model and revising these models by applying the observed data. This section presents the analytical system that we constructed, the result of a thermal-structural analysis and its problems. To solve the problems, we have developed a new model in considering the effects of hinge/latch mechanisms and friction. The result of using new model is also introduced.

## **6.1. Thermal snap analysis procedure**

This section describes the thermal snap analysis procedure. Fig. 12 shows a flowchart of thermal snap analysis. The analysis can be broken down into to three parts: construction of the structural model, calculation of temperature distribution, and thermal snap analysis for the penumbra.

In the first part, a structural model of the solar array paddle is constructed for thermal snap analysis. To verify the structural model, we conducted modal analysis to obtain the natural frequency and vibrational mode of the solar array paddle model. These results will be com‐ pared with an on-orbit preliminary experiment, and if necessary, we will then revise the model (see Section 6.2 for details). In the second part, a thermal model is developed for the solar array paddle. Thermal analysis for the whole orbit is then conducted to verify the ther‐ mal model. The thermal analysis results will be applied to GOSAT's trajectory information, and also compared with data obtained by GOSAT, in order to verify accuracy. After accura‐ cy is verified, thermal analysis will focus on GOSAT during its integration and testing. In order to improve the accuracy of thermal analysis, a profile of thermal input was deter‐ mined based on the brightness of the solar paddles (see Section 6.3 for details).

In the third part, the thermal snap analysis is conducted using the structural FEM model of GOSAT.

cell paddles, followed by the induced vibration thereof. An earlier paper in this series re‐

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ported the measurement results.

**Figure 12.** Flowchart of thermal snap analysis using observed data

sis during development.

In this experiment, we severely shook the solar paddle with the 20N thrusters, in order to observe its behavior. The FFT data obtained from this experiment on solar paddle vibration is available. Fig. 13 shows an example of the obtained FFT data. From these data, the natural frequencies of the actual solar array paddle were assumed to be 0.215 [Hz] (out-of-plane) and 0.459 [Hz] (in-plane). These natural frequencies are identical to those obtained by analy‐

Fig. 10 and Fig. 11 below show the structural model and the thermal model, respectively.

## **6.2. Improvement of the structural model**

In constructing the structural model of GOSAT's two solar cell paddles, we were not al‐ lowed to access the detailed satellite design data. Therefore, the structural model was based on partly assumed data. Moreover, it is difficult to estimate the production errors on GO‐ SAT. We thus compared the structural model design data and the observation data. The pre‐ liminary observation of solar cell paddle motion was made as GOSAT conducted orbitraising maneuvers using the 20 Newton Gas Jet thrusters. This maneuvering applied relatively large force to the satellite's main body, thereby causing large bending of the solar cell paddles, followed by the induced vibration thereof. An earlier paper in this series re‐ ported the measurement results.

In the third part, the thermal snap analysis is conducted using the structural FEM model of

In constructing the structural model of GOSAT's two solar cell paddles, we were not al‐ lowed to access the detailed satellite design data. Therefore, the structural model was based on partly assumed data. Moreover, it is difficult to estimate the production errors on GO‐ SAT. We thus compared the structural model design data and the observation data. The pre‐ liminary observation of solar cell paddle motion was made as GOSAT conducted orbitraising maneuvers using the 20 Newton Gas Jet thrusters. This maneuvering applied relatively large force to the satellite's main body, thereby causing large bending of the solar

Fig. 10 and Fig. 11 below show the structural model and the thermal model, respectively.

GOSAT.

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**Figure 10.** Structural model

**Figure 11.** Thermal model

**6.2. Improvement of the structural model**

**Figure 12.** Flowchart of thermal snap analysis using observed data

In this experiment, we severely shook the solar paddle with the 20N thrusters, in order to observe its behavior. The FFT data obtained from this experiment on solar paddle vibration is available. Fig. 13 shows an example of the obtained FFT data. From these data, the natural frequencies of the actual solar array paddle were assumed to be 0.215 [Hz] (out-of-plane) and 0.459 [Hz] (in-plane). These natural frequencies are identical to those obtained by analy‐ sis during development.

thermal input profile is needed, but estimating changes from trajectory information alone is difficult due to atmospheric effects. We thus estimated the value of solar light incident (i.e. dominant factor in thermal input) from the brightness value of images taken in observation. By using gray-scale processed images, we assume that if the average gray scale value of an image is max, then the value of solar light incident will also be max. If it is minimum, then the value will be minimum. Fig. 15 shows the estimated solar light incident and total ther‐ mal input profile. From the result, we can see that GOSAT's actual optical environment changes gently three times, as long as the value is forecast based on geometric calculation of

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**Figure 14.** Modal shapes of the structural model (Out of Plane and In-Plane Vibration)

By using the thermal input profile thus revealed, we conducted thermal analysis for the pe‐ numbra. Fig. 16 shows the simulation results of solar paddle temperature and the values of thermometers. The thermometers are attached at two points. However, no temperature sen‐ sor was attached on the front (solar cell; optical incidence) side of the solar paddle, but one was attached only on a backside plane (radiation plane; indicated as Bottom CFRP in Fig. 16). The tendency of simulated solar paddle temperature is similar, however, to the actual data. We therefore conclude that our thermal analysis can sufficiently simulate actual ther‐

the satellite's trajectory.

mal changes.

**Figure 13.** Example of FFT data on out-of-plane solar array paddle vibration as obtained from observation

Next, we conducted modal analysis of the structural model of the solar array paddle to cal‐ culate the natural frequency. In case of any differences between the analysis results and ob‐ servation results, we adjusted the structural data around the root of the solar cell paddles.

By iterating this process, the structural model was revised to generate data similar to that obtained from the solar array paddle in orbit.

Tab. 1 below lists shows the final modal analysis results of natural frequency; Fig. 14 shows the modal shapes of the structural model. From these data, the modal analysis results of the revised structural model match the observed data. Hence, we expect that the revised model could sufficiently simulate the actual solar array paddle.


**Table 1.** Modal analysis results

#### **6.3. Detailed thermal input profile from observed data**

During the time when passing the boundary between the sunlight area and umbra area, the thermal environment changes gradually. To conduct thermal snap simulation, a detailed thermal input profile is needed, but estimating changes from trajectory information alone is difficult due to atmospheric effects. We thus estimated the value of solar light incident (i.e. dominant factor in thermal input) from the brightness value of images taken in observation. By using gray-scale processed images, we assume that if the average gray scale value of an image is max, then the value of solar light incident will also be max. If it is minimum, then the value will be minimum. Fig. 15 shows the estimated solar light incident and total ther‐ mal input profile. From the result, we can see that GOSAT's actual optical environment changes gently three times, as long as the value is forecast based on geometric calculation of the satellite's trajectory.

**Figure 14.** Modal shapes of the structural model (Out of Plane and In-Plane Vibration)

**Figure 13.** Example of FFT data on out-of-plane solar array paddle vibration as obtained from observation

obtained from the solar array paddle in orbit.

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**Table 1.** Modal analysis results

could sufficiently simulate the actual solar array paddle.

**6.3. Detailed thermal input profile from observed data**

Next, we conducted modal analysis of the structural model of the solar array paddle to cal‐ culate the natural frequency. In case of any differences between the analysis results and ob‐ servation results, we adjusted the structural data around the root of the solar cell paddles. By iterating this process, the structural model was revised to generate data similar to that

Tab. 1 below lists shows the final modal analysis results of natural frequency; Fig. 14 shows the modal shapes of the structural model. From these data, the modal analysis results of the revised structural model match the observed data. Hence, we expect that the revised model

1 0.218 First order out-of-plane

2 0.454 First order in-plane

3 1.262 First order twist

During the time when passing the boundary between the sunlight area and umbra area, the thermal environment changes gradually. To conduct thermal snap simulation, a detailed

… … …

**Mode number Frequency [Hz] Modal shapes**

By using the thermal input profile thus revealed, we conducted thermal analysis for the pe‐ numbra. Fig. 16 shows the simulation results of solar paddle temperature and the values of thermometers. The thermometers are attached at two points. However, no temperature sen‐ sor was attached on the front (solar cell; optical incidence) side of the solar paddle, but one was attached only on a backside plane (radiation plane; indicated as Bottom CFRP in Fig. 16). The tendency of simulated solar paddle temperature is similar, however, to the actual data. We therefore conclude that our thermal analysis can sufficiently simulate actual ther‐ mal changes.

tion data shows that the solar array paddles will bend in quasi-static while the paddle is illuminated by the Sun. And from the deformation plot, it is assumed that the wires have a

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By using the models, we conducted thermal deformation analysis for the whole orbit—a task not possible through camera observation alone. Fig. 18 shows the results. It shows that

much larger deformation occurred in the penumbra than at any other time.

**Figure 18.** Solar array paddle deformation for two round orbit (The orbital period of GOSAT is 6000 [sec].)

large effect on a deformation.

**Figure 17.** Analysis results of deformation in penumbra

**Figure 15.** Estimated solar light incident and total thermal input profile from images

**Figure 16.** Estimated solar array paddle temperature and values of thermometers attached on a backside plane (Bot‐ tom CFRP)

#### **6.4. Results of thermal snap analysis**

Thermal snap analysis is conducted using the revised structural model and temperature dis‐ tribution. Fig. 17 shows the results of thermal snap analysis in the penumbra. The simula‐ tion data shows that the solar array paddles will bend in quasi-static while the paddle is illuminated by the Sun. And from the deformation plot, it is assumed that the wires have a large effect on a deformation.

By using the models, we conducted thermal deformation analysis for the whole orbit—a task not possible through camera observation alone. Fig. 18 shows the results. It shows that much larger deformation occurred in the penumbra than at any other time.

**Figure 17.** Analysis results of deformation in penumbra

**Figure 15.** Estimated solar light incident and total thermal input profile from images

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tom CFRP)

**6.4. Results of thermal snap analysis**

**Figure 16.** Estimated solar array paddle temperature and values of thermometers attached on a backside plane (Bot‐

Thermal snap analysis is conducted using the revised structural model and temperature dis‐ tribution. Fig. 17 shows the results of thermal snap analysis in the penumbra. The simula‐

**Figure 18.** Solar array paddle deformation for two round orbit (The orbital period of GOSAT is 6000 [sec].)

## **6.5. Detailed modeling for hinges**

As dynamic response offers the possibility of adding more awkward disturbances to atti‐ tude stability than quasi-static deformation, we decided to check the penumbra data in de‐ tail. The observed data obtained in the penumbra indicates that the solar array paddles have a much lower natural frequency. Fig. 19 shows the FFT data on out-of-plane deformation in the penumbra.

tain the open position of each panel. As each hinge will have some backlash and in order to pull each panel to the deployed position, wires are used to maintain the deployed position after each panel is deployed, and to connect the solar panels, resulting in only a small load

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**Figure 21.** Concept of a hinge modeled using a nonlinear torsional spring. In Fig. 21, (a) shows how to define the

Should a small load (e.g. thermal stress) be applied to the solar paddle, deformation will mainly occur in the hinges that have small gaps. Conversely, in case a large load (e.g. inertia

under microgravity conditions. Fig. 20 and 21 shows the concept.

**Figure 20.** Concept of vibration affected by the hinge gap

stiffness of torsional springs; (b) shows the model for the solar paddle.

**Figure 19.** FFT analysis of out-of-plane deformation in the penumbra and same data obtained when maneuvering to raise orbital altitude using powerful 20N thrusters

Fig. 19 shows the data obtained when using the powerful 20N thrusters to raise GOSAT's orbital altitude, with vibration of 0.215 [Hz] being observed. When the satellite goes into eclipse, however, the data obtained did not reveal vibration of 0.215 [Hz], but instead showed a lower frequency of 0.094 [Hz]. Similar data was not reported during the satellite integration and qualification tests.

From these reasons, we assumed that the small load (e.g. solar pressure) acts on the solar panels, and that this load acts as rotation torque at each hinge. As all solar cell panels of GO‐ SAT are linked together to maintain the angle of deployment to the deployed position, the small motion generated at each hinge will also act on all the hinges.

Vibration will occur at each hinge that connects each solar cell panel. Hinges and wires are used to interconnect the solar cell panels, in order to deploy the solar cell panels and main‐ tain the open position of each panel. As each hinge will have some backlash and in order to pull each panel to the deployed position, wires are used to maintain the deployed position after each panel is deployed, and to connect the solar panels, resulting in only a small load under microgravity conditions. Fig. 20 and 21 shows the concept.

**6.5. Detailed modeling for hinges**

346 Advances in Vibration Engineering and Structural Dynamics

raise orbital altitude using powerful 20N thrusters

integration and qualification tests.

the penumbra.

As dynamic response offers the possibility of adding more awkward disturbances to atti‐ tude stability than quasi-static deformation, we decided to check the penumbra data in de‐ tail. The observed data obtained in the penumbra indicates that the solar array paddles have a much lower natural frequency. Fig. 19 shows the FFT data on out-of-plane deformation in

**Figure 19.** FFT analysis of out-of-plane deformation in the penumbra and same data obtained when maneuvering to

Fig. 19 shows the data obtained when using the powerful 20N thrusters to raise GOSAT's orbital altitude, with vibration of 0.215 [Hz] being observed. When the satellite goes into eclipse, however, the data obtained did not reveal vibration of 0.215 [Hz], but instead showed a lower frequency of 0.094 [Hz]. Similar data was not reported during the satellite

From these reasons, we assumed that the small load (e.g. solar pressure) acts on the solar panels, and that this load acts as rotation torque at each hinge. As all solar cell panels of GO‐ SAT are linked together to maintain the angle of deployment to the deployed position, the

Vibration will occur at each hinge that connects each solar cell panel. Hinges and wires are used to interconnect the solar cell panels, in order to deploy the solar cell panels and main‐

small motion generated at each hinge will also act on all the hinges.

**Figure 21.** Concept of a hinge modeled using a nonlinear torsional spring. In Fig. 21, (a) shows how to define the stiffness of torsional springs; (b) shows the model for the solar paddle.

Should a small load (e.g. thermal stress) be applied to the solar paddle, deformation will mainly occur in the hinges that have small gaps. Conversely, in case a large load (e.g. inertia force at maneuver) is applied, deformation of the solar paddle is typically larger than the gaps; therefore, such deformation is mainly caused by the elastic deformation of the panels.

To simulate the effect caused by hinges with a gap, we modeled the hinges with nonlinear torsional springs. In Fig. 21, (a) shows how we added nonlinear characteristics to the tor‐ sional springs. The length of *u* is used to define the gap size. And stiffness within the gap is defined by K1. When the whole dynamic response shown in Fig. 17 is assumed to have oc‐ curred within gap area, the gap width is easily defined. Then, the stiffness of the gap is de‐ fined as shown below.

To estimate stiffness, the modeling method for the model of spring-connected plates is used (Kojima et al., 2004). The rotational motion equation for the model of spring-connected plates as shown in Fig. 21 (b) is:

$$I\theta + K\theta = 0\tag{1}$$

where, *I* denotes the inertia moment of plates and *K* the stiffness of torsional springs. Here, given harmonic vibration in which frequency is , the stiffness of gap area *K* is expressed as:

$$K = I\alpha^2\tag{2}$$

**Figure 22.** Frequency response analysis results of a structural model with a hinge gap. The analysis of response at the inner side of the gap simulates the dynamic regime in the penumbra. The analysis of response at the outer side of the gap simulates the dynamic regime in the preliminary experiment using the 20N thrusters of GOSAT (see Section 6.2). From the results, the new model with a nonlinear hinge demonstrated that it could simulate the peak shift in response

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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

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

showed in the penumbra (Fig. 19).

1 is 0.094 [Hz]. Therefore, we assumed the stiffness of gap area *K* as about 1.21 [Nm/rad] at first, and adjust the value until the vibration properties in penumbra coincide with the value of the observation results.

By using the new model with a detailed hinge, we conducted frequency response analysis. The analysis targets were both a small load condition and a large load condition shown as Tab.2. Fig. 22 shows the results. From the results, we can see that the new model could simu‐ late the vibration occurring in the 20N maneuver test and in the penumbra at the same time.


**Table 2.** Details of frequency response analysis

force at maneuver) is applied, deformation of the solar paddle is typically larger than the gaps; therefore, such deformation is mainly caused by the elastic deformation of the panels.

To simulate the effect caused by hinges with a gap, we modeled the hinges with nonlinear torsional springs. In Fig. 21, (a) shows how we added nonlinear characteristics to the tor‐ sional springs. The length of *u* is used to define the gap size. And stiffness within the gap is defined by K1. When the whole dynamic response shown in Fig. 17 is assumed to have oc‐ curred within gap area, the gap width is easily defined. Then, the stiffness of the gap is de‐

To estimate stiffness, the modeling method for the model of spring-connected plates is used (Kojima et al., 2004). The rotational motion equation for the model of spring-connected

where, *I* denotes the inertia moment of plates and *K* the stiffness of torsional springs. Here, given harmonic vibration in which frequency is , the stiffness of gap area *K* is expressed as:

1 is 0.094 [Hz]. Therefore, we assumed the stiffness of gap area *K* as about 1.21 [Nm/rad] at first, and adjust the value until the vibration properties in penumbra coincide with the value

By using the new model with a detailed hinge, we conducted frequency response analysis. The analysis targets were both a small load condition and a large load condition shown as Tab.2. Fig. 22 shows the results. From the results, we can see that the new model could simu‐ late the vibration occurring in the 20N maneuver test and in the penumbra at the same time.

Condition 1 0.4 [m/s2] inertia force

Frequency interval 0.1[Hz]

Number of nodes 7903 Number of elements 8292

**Table 2.** Details of frequency response analysis

BC Tip of york : Fixed

Condition 2 7.2 [µPa] pressure on top side

Solver General-purpose non-liner analysis software "MSC. Marc 2011"

<sup>2</sup> *K I* = w

+ = 0 (1)

(2)

*I K* q q

fined as shown below.

plates as shown in Fig. 21 (b) is:

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of the observation results.

Excitation force

**Figure 22.** Frequency response analysis results of a structural model with a hinge gap. The analysis of response at the inner side of the gap simulates the dynamic regime in the penumbra. The analysis of response at the outer side of the gap simulates the dynamic regime in the preliminary experiment using the 20N thrusters of GOSAT (see Section 6.2). From the results, the new model with a nonlinear hinge demonstrated that it could simulate the peak shift in response showed in the penumbra (Fig. 19).
