TAILIEUCHUNG - The Inverted Pendulum

If an oscillating force is applied to an inverted pendulum it is possible to keep it completely vertical,that is r/2 degrees from the surface the pendulum lies on, with some stability depending on the frequency it osculation paper describes how and why this stability occurs a particular inverted pendulum created by Sarashid | The Inverted Pendulum Yigal Weinstein yweinste@ Abstract If an oscillating force is applied to an inverted pendulum it is possible to keep it completely vertical that is tr 2 degrees from the surface the pendulum lies on with some stability depending on the frequency it oscillates at. This paper describes how and why this stability occurs for a particular inverted pendulum created by Mr. Jahan Sarashid. 1 Introduction This paper is completely theoerical except for a small calculation to test for approximate similarities with what has been observed. That is this paper does not try to summarize large data sets but tries to create an intuitively understandable model for the phenomenon at hand. Approaching this problem in a completely theoretical fashion is interesting and possibly rewarding in that the math used in explaining this phenomenon may show how this phenomenon relates to other systems that are completely different on the surface but are fundamentally similar. A picture of the inverted pendulum under investigation is found in Figure 1. For this paper the simplification is made that the pendulum has a mass m at the end of a massless stick of length I. This simplification is acceptable for a rough approximation but may need to be refined for future calculations. The rod is attached to a vertically oscillating table the position of which is given by A cos w i again a simplification is made that this source frequency ws is uniform. A mechanical description of the simplified pendulum is found in Figure 2 x. Thus the task at hand is to show that if w is large enough the pendulum will not fall over and to find what big enough means. This paper is divided into 3 sections. The first section is this introduction. The second deals with the math model and the third tries to sum up the findings of the previous section and considers where thought should be put next to understand the phenomenon. 1Tliis figure was inspired yy the first page of 1 Figure 1 picture of

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