The pendulum: liner or nonliner?

郭帅斐

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Summery:

As we all know, the single pendulum is a simplified liner model, which motion as a harmonic oscillator. But this model only suit for the conditions that the amplitude is very small. Actually, the real motion of pendulum is nonliner, where the acceleration is proportional to the $sin\theta$. We can find that the approximation is well suitable when the amplitude is small through this program. Also, we get the conclusion that the real pendulum has a longer period compared to the ideal pendulum.

The pseudocode:

I used the class function to form the structure of this progrem. In fact, this is a more clear and more handy way to write a founctional progrem. A new finding is that the 'plot' punction can be inserted into the class, as a result, we can show many object in one figure conveniently. As usual, the 'calculate' section is the most important. Here we use Euler-Cromer method as the correct method, the algorithm show as follows:

$$\omega_{i+1}=\omega_i-\frac{g}{l}\theta_i\Delta t$$
$$\theta_{i+1}=\theta_i+\omega_{i+1}\Delta t$$
To make comparation, I write the algorithm of Euler mathod as follows:
$$\omega_{i+1}=\omega_i-\frac{g}{l}\theta_i\Delta t$$
$$\theta_{i+1}=\theta_i+\omega_{i}\Delta t$$
I used both the methods to calculate the solution, and find the difference between the two mathods. The solution shows as the follow figure:

To keep the Energy conservation of the pendulum, we chose the Euler-Cromer mathod to do the following calculaton.

the result:

• Firstly, I make the $\theta_0$ as small as 0.2 rad, and gaind the figure as this:
  
  Actually, we can hardly find any difference between the two model. So the approximation is resonable when the amplitude is small.
• Secondly, I make the $\theta_0$ to be 0.5 rad, the difference can be easily noticed already.
  
• finally, I make the $\theta_0$ as large as 1 rad($57.3^\circ$), and find a more noticable distinction between the two model. The figure is as follows:
  
  We can find that the phase-difference is large enough to make the liner pendulum leave the nonliner pendulum a period behind every 5 period.
  To analyse the difference, a phrase diagram is drawn as follows:
  
  The phrase diagram of the liner pendulum is more circular than that of a nonliner pendulm.
  To analyse the perodic propertity of the nonliner pendulum, a Lissajous figure are ploted, we can easily count that the ratio of the liner and nonliner pendulum is about 4:5.
  

The conclusion:

From this simple progrem, we draw the conclusion as these: (1) In the perodic prossess, we'd better choose the Euler-Cromer mathod to keep the energy conserved. (2) The liner approximation of real pendulum in small amplitude is resonable. (3) The period of the real pendulum become longer compared with the liner pendulum. We can use the Lissajous to analyse the perodic propertity of the pendulum.