Ex 2.4 Double Pendulum

7/29/2019 Ex 2.4 Double Pendulum

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INTRODUCTORY COMPUTATIONAL PHYSICS

NAME: AROOJ MUKARRAM

EXERCISE 2.4:

SOLUTION:


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Theta1/ Theta2 = 0.0, 0.1, 0.2, . , 1.0 (Angle vs Time)


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Theta1/ Theta2 = 0.0, -0.1, -0.2, . , -1.0 (Angle vs Time)


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Theta1/ Theta2 = 0.0, 0.1, 0.2, . , 1.0 (Lissajous curves)


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Theta1/ Theta2 = 0.0, -0.1, -0.2, . , -1.0 (Lissajous curves)


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Theta1/ Theta2 = 0.0, 0.1, 0.2, . , 1.0 (Trace)


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Theta1/ Theta2 = -0.1, -0.2, . , -1.0 (Trace)


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OUTPUT ANALYSIS:

For positive values of 1 / 2 :

From angle vs time graphs: At 0.7 both masses move together in same direction and reach their

maximum and minimum values at same time. At smaller values of 0.0 and 0.1, the motions of masses


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are very different. But as this ratio is increased, masses start moving in sync and at 0.7 they move

together in same directions.

From Lissajous figures: At 0.7 it shows, the Lissajous figure shows a straight line (positive slope). At

other times, the figures cover some area (with particular boundary shapes) in the configuration space. If

simulation is run for a much longer time, the space inside these graphs will be completely filled.

From trace graphs: At 0.7 the trace is a nice parabola. For other ratios, the trace is comparatively

complex.

This implies that 1 / 2 = 0.7 gives a normal mode (symmetric).

For negative values of 1 / 2 :

From angle vs time graphs: At 0.7 both masses move in opposite direction so that when one mass

reaches its maximum, the other is at minimum. At other ratios the motions of the masses have no such

relationship.

From Lissajous figures: At 0.7 it shows, the Lissajous figure shows a straight line (negative slope). At

other times, the figures cover some area (with particular boundary shapes) in the configuration space. If

simulation is run for a much longer time, the space inside these graphs will be completely filled.

From trace graphs: At 0.7 the trace is a parabola, that fills up some area in the figure. For other ratios,

the trace is comparatively complex.

This implies that 1 / 2 =- 0.7 gives a normal mode (anti-symmetric).


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EXERCISE 2.4 Part b

The program is run with following input values (Exercise 2.4-b.py):

2 = 0.0

1 = [1.57,1.58, 2.8, 3.8, 4.7, 4.8]

Keeping 2 constant and changing the 1 values from /2 to 3/2 give the following output for

Angle vs time graphs

Lissajous figures

Trace graphs


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OUTPUT ANALYSIS:

From the above graphs, we see that motion is chaotic for 1 between /2 and 3/2. The angles versustime graphs in this region do not show regular repetitive behavior. Also the Lissajous figures and trace

do not have a pattern and are rather messy figures.

Step size effect on evolution of the system:

If the programs are run by decreasing the step size (for chaotic and non-chaotic regimes), the Lissajous

figures and traces graphs are no longer smooth curves .


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Ex 2.4 Double Pendulum