**Acknowledgements**

This study was completed with the financial support by the Vietnam National Foundation for Science and Technology Development (NAFOSTED).

## **Author details**

Nguyen Van Khang\* and Nguyen Phong Dien

\*Address all correspondence to: nvankhang@mail.hut.edu.vn

Department of Applied Mechanics, Hanoi University of Science and Technology, Vietnam

## **References**

**Figure 20.** Dynamic moment acting on the driving shaft of the mechanical adjustment unit

**5. Concluding remarks**

328 Advances in Vibration Engineering and Structural Dynamics

this conclusion is only true for linear systems.

varying coefficients.

To verify the calculating results using the numerical methods, the dynamic load moment of the mechanical adjustment unit was measured on the driving shaft (see also Figure 16). A typical record of the measured moment is plotted in Figure 20, together with the curves calculated from the dynamic model by using the WKB-method [18, 34], the kinesto-static calculation and the proposed numerical procedures based on Newmark method and Runge-Kutta method. Comparing the curves displayed in this figure, it can be observed that the calculating result using the numerical methods is more closely in agreement with the experi‐ mental result than the results obtained by the WKB-method and the kinesto-static calculation.

The calculation of dynamic stable conditions and periodic vibrations of elastic mecha‐ nisms and machines is an important problem in mechanical engineering. This chapter deals with the problem of dynamic modelling and parametric vibration of transmission mecha‐ nisms with elastic components governed by linearized differential equations having time-

Numerical procedures based on Runge-Kutta method and Newmark integration method are proposed and applied to find periodic solutions of linear differential equations with timeperiodic coefficients. The periodic solutions can be obtained by Newmark based procedure directly and more conveniently than the Runge-Kutta method. It is verified that the compu‐ tation time with the Newmark based procedure reduced by about 60%-65% compared to the procedure using the fourth-order Runge-Kutta method (see also Figures 7 and 15). Note that

The numerical methods and algorithms are demonstrated and tested by three dynamic models of elastic transmission mechanisms. In the last two examples, a good agreement is obtained between the model result and the experimental result. It is believed that the pro‐ posed approaches can be successfully applied to more complicated systems. In addition, the proposed numerical procedures can be used to estimate approximate initial values for the

shooting method to find the periodic solutions of nonlinear vibration equations.


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**Chapter 13**

**Vibration of Satellite Solar Array Paddle Caused by**

**Thermal Shock When a Satellite Goes Through the**

Remote sensing satellites take images of the earth's surface in observing various activities by humans or nature. In order to obtain precise and high-resolution images from a satellite in Low Earth Orbit (LEO) at an attitude of 500 to 900 kilometers, the satellite's attitude must be stable when the onboard camera sensors take images of surface activities on the earth. Should the attitude stability of the satellite be disturbed while such images are being taken,

The fact that images taken when a satellite goes into or out from an eclipse do not provide good accuracy—due to degraded altitude stability at such timings—has been known for years. Such phenomena has long been attributed to the deformation and vibration of the sat‐ ellite's solar array paddle that occurs when the satellite go into or out from an eclipse, along with the instantaneous change in solar energy received by the satellite. Several trials were conducted in the past to identify the phenomena, but all failed to achieve reasonable results.

The reason why past trials failed to observe the phenomena described above might be that the motion of the solar array paddle is too small or slow to be observed by such onboard sensors as an accelerometer. Therefore, JAXA decided to measure the phenomena by using an onboard camera that was originally mounted on the satellite to monitor solar array pad‐

> © 2012 Oda et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 Oda et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

Mitsushige Oda, Akihiko Honda, Satoshi Suzuki and

Additional information is available at the end of the chapter

**Eclipse**

Yusuke Hagiwara

**1. Introduction**

http://dx.doi.org/10.5772/52626

poor image quality would probably result.

**2. Measurement of solar array paddle motion**
