Solving the Pendulum Problem with RK4 Method

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In summary, the problem with the simulation is that the acceleration is greater than what nature would provide, and the larger the angle, the more inaccurate the model.
  • #1
MASH4077
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Pendulum Problem...

Hi all,

I've written a little simulator (in C++) that demonstrates a simple pendulum swinging from right to left and back again. To simulate the motion I'm using the simple second order differential equation:

accelaration = -(g/L) * theta

and numerically integrating this using the RK4 method. However I'm having a problem in that the simulation is escaping really quickly. I just want to ask under what conditions would this happen?. I think I'm starting off the simulation with reasonable values and they are outlined below:

angle = 22.5, start angular_vel = 0.0f, init accel = calculated using above formula.

Any advice on why the simulation could be breaking so quickly is much appreciated.

Thanks.
 
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  • #2


I am not saying that this is the problem with your program but the
acceleration cannot be g/L *theta. Maybe g*theta.
 
  • #3


I may not understand the simulation, but I do have something that may be of interest. The expression you have, "acclereration = -(g/L)*theta" is an approximation. The true differential equation should have a sin(theta) rather than theta. The approximation only holds for small angles. Could this be the source of your "escaping"?

It would seem to me, that you are providing an acceleration that is greater than what nature would provide, and the larger the angle, the more inaccurate the model. Also, the approximation that sin(theta) ~ theta is only accurate if theta is given in radians. You're model therefore assumes you are using radians. Try a reasonable number for angle like 0.3 and see what you get. Or, you could change to sin(theta) and keep your numbers.
 
  • #4


Yeah, Nasu is right too, but that L shouldn't really affect anything qualitatively.
 
  • #5


jdog said:
It would seem to me, that you are providing an acceleration that is greater than what nature would provide, and the larger the angle, the more inaccurate the model.
Yup. I think that that is the problem for the simulation escaping. I realized this when I actually took another look at the implementation. I wasn't calculating the acceleration of the pendulum based on the initial conditions provided to the simulation, but rather I was just "feeding in" a value. I've just re-written that particular part to ensure that given the initial conditions, the correct value is assigned to the acceleration variable. Everything is now behaving as it should do.

Many Thanks.

:)
 

1. What is the RK4 method and how does it work?

The RK4 method, also known as the fourth-order Runge-Kutta method, is a numerical algorithm used to solve ordinary differential equations. It works by taking small time steps and using a weighted average of four different estimates to calculate the next point in the solution.

2. How accurate is the RK4 method in solving the pendulum problem?

The RK4 method is considered to be a highly accurate numerical method for solving the pendulum problem. It is a fourth-order method, meaning that the error decreases quadratically with smaller time steps. However, the accuracy also depends on the step size and the specific parameters of the pendulum system being solved.

3. What are the advantages of using the RK4 method for solving the pendulum problem?

The RK4 method has several advantages for solving the pendulum problem. It is a stable method, meaning that it will not produce wildly inaccurate solutions. It also has a high order of convergence, meaning that it can produce accurate solutions with relatively few time steps. Additionally, the RK4 method is relatively simple to implement and can handle a wide range of differential equations.

4. Are there any limitations to using the RK4 method for solving the pendulum problem?

While the RK4 method is a powerful tool for solving the pendulum problem, it does have some limitations. One limitation is that it can become computationally expensive for systems with a large number of variables. Additionally, the accuracy of the method can be affected by the choice of step size and the specific parameters of the pendulum system being solved.

5. How does the RK4 method compare to other numerical methods for solving the pendulum problem?

The RK4 method is considered to be one of the most accurate and efficient numerical methods for solving the pendulum problem. Other commonly used methods include Euler's method and the Verlet method. While these methods may be simpler to implement, they are generally less accurate and can produce less stable solutions compared to the RK4 method.

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