Solving Damped Driven Pendulum ODEs

In summary: I have gone through a lot of resources and I am now confused. I am hoping that this forum might clear any confusions I have on mind. Thank you so much!In summary, the equations presented may be related to the damped driven pendulum, but they are not standard differential equations. The forum may help you clarify your confusion.
  • #1
HansBu
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Homework Statement
Hello everyone! I am having some confusions with regards to my problem, whether or not the differential equations presented to me are inclined to damped driven pendulum. From the differential equations given, I can see that these equations are part of the convection equations made my Lorenz dubbed as Lorenz equations.
Relevant Equations
The relevant equations are presented in the attempt to the solution part.
Here are the nonlinear and coupling ordinary differential equations:
reference.png

I was given values of a, b, and c as well as some initial values for x, y, and z. If ever the equations above are related to the pendulum, I can think of a as the damping factor, b as the forcing amplitude, and c as the driving frequency. On the other hand, initial values of x and y represent initial angular displacement and initial angular velocity, respectively. Can you help me verify this thought of mine? I have gone through a lot of resources and I am now confused. I am hoping that this forum might clear any confusions I have on mind. Thank you so much! A sample output shown below might be a help.

NOTE: I was able to come up with the source code in order to print out the graph below. My only concern here is the concept behind it.

reference 2.png
 
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  • #2
Have you searched the forum with damped pendulum? Does anything help?
 
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  • #3
Frodo said:
Have you searched the forum with damped pendulum? Does anything help?
Yes sir, I did only to see the standard differential equation of a damped driven pendulum in normalized version. Now, I am really confused. The graph above, from the problem, says that those are chaotic solutions. I understand what the problem means, but chaotic solutions of what system?
 
  • #4
HansBu said:
Homework Statement:: Hello everyone! I am having some confusions with regards to my problem, whether or not the differential equations presented to me are inclined to damped driven pendulum. From the differential equations given, I can see that these equations are part of the convection equations made my Lorenz dubbed as Lorenz equations.
Relevant Equations:: The relevant equations are presented in the attempt to the solution part.

Can you help me verify this thought of mine?
To verify what? You have not formulated anything verifiable. If you for example present other system and make hypothesis on how its solutions approximate the Lorenz solutions then the matter of discussion will appear.
 
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  • #5
wrobel said:
To verify what? You have not formulated anything verifiable. If you for example present other system and make hypothesis on how its solutions approximate the Lorenz solutions then the matter of discussion will appear.
Hello, Wrobel! Can you enlighten me more on what you are trying to imply?
 

1. What is a damped driven pendulum?

A damped driven pendulum is a physical system that consists of a pendulum, a mass attached to a fixed point by a rod, that is subject to both a driving force and a damping force. The driving force is an external force that causes the pendulum to move, while the damping force is a resistance force that slows down the motion of the pendulum.

2. Why is it important to solve damped driven pendulum ODEs?

Solving damped driven pendulum ODEs allows us to understand and predict the behavior of this physical system. This is important in various fields such as physics, engineering, and robotics, where pendulums are commonly used in applications such as clocks, cranes, and stabilizers.

3. What is an ODE and how is it related to damped driven pendulums?

An ODE (ordinary differential equation) is a mathematical equation that describes the relationship between a function and its derivatives. Damped driven pendulum ODEs are equations that describe the motion of the pendulum based on its position, velocity, and acceleration, as well as the driving and damping forces acting on it.

4. How do you solve damped driven pendulum ODEs?

There are various methods for solving damped driven pendulum ODEs, including analytical methods such as the method of undetermined coefficients and numerical methods such as the Euler method and the Runge-Kutta method. The specific method used depends on the complexity of the equation and the desired level of accuracy.

5. What are some real-life applications of damped driven pendulums?

Damped driven pendulums have a wide range of applications, including in clocks, cranes, and stabilizers. They are also used in seismology to study the motion of earthquakes and in robotics to control the motion of robotic arms and legs. Additionally, damped driven pendulums are used in musical instruments such as pianos and guitars to produce sound.

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