Inquiry Experiment of The chaotic behavior of driven nonlinear pendulum
Jiang Yuxiang 040012006029 grade1 science of biology
Ocean University of china Qingdao
Abstract: inquire the effects of different quantities (driving frequency, driving amplitude,
damping amplitude and the initial conditions) have on the oscillator’ behavior. Have a better
understanding of the formation of chaos and the chaotic theory
Key words: chaos, driving frequency, driving amplitude, damping amplitude, initial conditions
EQUIPMENT NEEDED
INCLUDED:
1
Large Rod Stand
ME-8735
2
120 cm Long Steel Rod (2)
ME-8741
1
45 cm Long Steel Rod
ME-8736
2
Multi Clamps (2)
SE-9442
1
Chaos/Driven Harmonic Accessory
CI-6689A
1
Mechanical Oscillator/Driver
ME-8750
1
DC Power Supply
SE-9720
1
Rotary Motion Sensor
CI-6538
1
Photo gate Head
ME-9498A
NOT INCLUDED, BUT REQUIRED:
1
Science Workshop 500 Interface
1
Data Studio Software
Image of the device:
The chaotic behavior is nonlinear, so by usual methods it is difficult to analysis.
We use the PASCO Science Workshop to undertake this experiment . A real-time phase plot is made by
graphing the angular velocity versus the displacement angle of the oscillation.
The oscillator consists of an aluminum disk connected to two springs. A point mass on the edge of the
aluminum disk makes the oscillator nonlinear. This nonlinearity is required to cause chaotic motion.
Also, the disk is magnetically damped.
Purpose
Several quantities can be varied to cause regular motion to become chaotic. These variables are the
driving frequency（expressed by voltage）, driving amplitude(the length of the arm of force), damping
amplitude(expressed by the distance between the magnetic damp and the disk and the initial
conditions. Our purpose is to inquire the effects of these quantities have on the oscillator’ behavior.
Theory
Chaos
Usually the so-called“single pendulum” is in fact sublimated, ignoring
resistance ,small angle switch. And its movement is seasonal.
In fact, pendulum can demonstrate big angle switch, which is usually
resisted, named “chaotic driven nonlinear pendulum” The dynamic equation:
This equation is nonlinear, it is impossible to parse. Only can we
calculate with given parameter. Chaos’ movement is without regulation; it
is a much more complicated system and has demonstrated uncertainty in Newton
mechanics.
Procedure
1. build the device. Plug the Rotary Motion Sensor into Channels 1 and 2 on the Science Workshop500
interface. Plug the photo gate into Channel 3 on the interface. Open data studio.
2、Voltage
Voltage ranges from 2v, 3v to 4v. Gradually increase the driving frequency
by increasing the voltage on power supply. Give the pendulum time to respond
to the change in driving frequency. Increase the frequency until the motion
of the pendulum is slightly more complicated: (in this process, damping
amplitude is 0. Arm of force is set at 3.5cm. Course5, 6, 7 separately represent
2v, 3v, 4v. Start the oscillation, holding the point mass end at the top and
letting go when the driver arm is at its lowest point. In the experiments that
follow, if not mentioned, the quantities are same.
Angular Velocity (ω) vs. Angular Displacement (θ) ：
WE can see clearly from the picture below that how the motion of the pendulum become
chaotic.at the same Angular Displacement, as the voltage increased, so was the angular velocity.
Angular acceleration (a) vs. Angular Velocity (ω)
Angular acceleration (a) vs. Angular Displacement (θ) :
Angular acceleration (a) vs. time (t):
Angular Displacement (θ) vs. time：
To my surprise, when the voltage becomes big enough, the picture is repeated.that is to say ,the
moment go back to sinusoidal
4．Length of the arm of force
The length incourse1 is 5cm, in course2 is 3.5cm.
When the driving amplitude increased, the pendulum rolled faster ,what’s more ,the number
of apices also increased, that means it turned back more frequently.
Damping amplitude
1. Length of arm is 3.5cm. no power. Course 1, 2 correspond to the distance between the
magnetic damp and the disk is 0cm,1cm. holding the point mass end at the top and letting go when the
driver arm is at its lowest point.
2. Turn on the power, the voltage is 3v. the other conditions are the same with part1.
As the damping amplitude increased, the image becomes smaller, but the shape is similar.
Though both are nonlinear
The impact of original condition
in course 1,the point mass fall into the equilibrium position on the left side of the pendulum. Course
2, from the right side. And course3, from the highest point.
We found that the original condition’ little change can have big impact on the movement. That
agrees with the theory.
Conclusion
By drawing and analyzing pictures of the movement of pendulum affected by different quantities, I have
a better understanding of the formation of chaos and the chaotic theory. But restricted by knowledge, I
made no further study.
Referred materials:
Data studio starter manual
College physics (edition2), zhaojinfang, yanxiaohong, beijing Youdian University
ISBN 7-5635-0707-8