Chaos in the Driven Nonlinear Pendulum

程熹 2013301020038 物基一班
Homework

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Abstract

This article investigate the linear,forced penndulum with friction and the nonlinear pendulum. We describe the the dependence of the resonant amplitude on the driving force angular frequence in numberical way with Euler-Cromer method.We try to find the existence of chaos in these relations.

Key Word

pendulum motion,driven force,chaos,euler-cromer method.

Introduction

A pendulum is a weight suspended from a pivot so that it can swing freely.When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.
From the first scientific investigations of the pendulum around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s. The pendulum clock invented by Christian Huygens in 1658 became the world's standard timekeeper, used in homes and offices for 270 years, and achieved accuracy of about one second per year before it was superseded as a time standard by quartz clocks in the 1930s. Pendulums are also used in scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in geophysical surveys, and even as a standard of length. The word "pendulum" is new Latin, from the Latin pendulus, meaning 'hanging'.

The simple gravity pendulum is an idealized mathematical model of a pendulum. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. When given an initial push, it will swing back and forth at a constant amplitude. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines.

Main Features

The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called the amplitude.It is independent of the mass of the bob. If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:

$${\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}\qquad \qquad \qquad }$$
where L is the length of the pendulum and g is the local acceleration of gravity.
In our discussion, we include the driving force F
We use the Euler-Cromer method to translate the equation of motion $$\frac{ d^{2 }\theta }{ dt^{2} } =- \frac{g}{l} \theta -q \frac{d \theta }{dt} + F_{D} sin( \Omega _{D} t)$$ into equation suitable to design a program.

Result

1.The driving force=0.5 and the initial angle of the pendulum is 0

2.The driving force=1.2 and the initial angle of the pendulum is 0

3.The driving force=0 and the initial angle of the pendulum is 0.2

4.The diagram of angle versus the angular acceleration in driving force=0.5 and the initial angle of the pendulum is 0

5.The diagram of angle versus the angular acceleration in driving force=1.2 and the initial angle of the pendulum is 0

6.The diagram plot only at time that are in phase with the driving force

Conclusion

1. When the driving force is well mathch with the origin system, which means it happen to overcome the drag force in the system, in this case, the pendulum can do the perfect Simple harmonic motion.
2. When the driving force can offset the drag force in the system, the motion of the pendulum will continue to desease until it is totally rest,like in the diagram 3.
3. When the driving force is even greater than the drag force in the system, there will be chaos in the system,it' hard to predict the future motion of the pendulum.
4. We can also see from the diagram 5 and 6 that the driving force greater that the drag force in the system will lead to a chaos result.

Acknowledgement

This program is finished by myself,but the introduction part I have use the introduction part of last report of my own.

Reference

Computational Physics(second edition) Nicholas J.Giordano and Hisao Nakanishi
Wikipedia Pendulum
https://en.wikipedia.org/wiki/Pendulum