Euler Method in MATLAB: Solving a Double Pendulum

In summary, the Euler Method in MATLAB can be used to solve a double pendulum system by breaking down the problem into smaller, simpler steps. This method involves approximating the solution at each step using a linear approximation and then updating the values for the next step. By repeating this process, an accurate solution can be obtained for the motion of the double pendulum. Additionally, the MATLAB code for implementing this method is relatively simple and can be easily modified for different initial conditions and parameters. Overall, the Euler Method is a useful tool for solving complex dynamic systems, such as the double pendulum, in a computational environment.
  • #1
blue-steel
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Hi all

Im working on a systems dynamics problem which involves modelling a double pendulum, a chaotic system

I have a non linear system of 4 1st order differential equations which I need to solve using numerical methods in MATLAB

The methods are ode45, ode23s and euler method

Ive been able to model using the first 2 but am having problems with the euler method as this involves the variable 'h' in the euler algorithm and creating a unique function (as opposed to built in ones in matlab) and I am not sure how to use loops and feval syntax and such

So for a system such as

x1' = f(x1,x2,x3,x4)
x2' = f(x1,x2,x3,x4)
x3' = f(x1,x2,x3,x4)
x4' = f(x1,x2,x3,x4)

How do you solve in MATLAB using the euler algorithm, y(i+1) = y(i) + hf(y,i)
my ultimate goal is to solve and graph points x1 and x3 with time (as these represent angles of the 2 pendulum branches)

Id appreciate any ideas on how to start with coding the euler algorithm in MATLAB and how apply user made functions, I don't know where to start...
 
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  • #2
This question is best answered by MATLAB:

https://www.mathworks.com/matlabcentral/answers/278300-matlab-code-help-on-euler-s-method
Example Euler Method implemented in MATLAB:
% Euler's Method

% Initial conditions and setup
h = (enter your step size here);  % step size
x = (enter the starting value of x here):h:(enter the ending value of x here);  % the range of x
y = zeros(size(x));  % allocate the result y
y(1) = (enter the starting value of y here);  % the initial y value
n = numel(y);  % the number of y values

% The loop to solve the DE
for i=1:n-1
    f = the expression for y' in your DE
    y(i+1) = y(i) + h * f;
end

plot(x,y);

First decide on the range of x values and then on how fine a step between them.

In this example, you could define x as being from 0:10 and a step size of 0.1:

x=[0:0.1:10]

this will give you a 100 points to compute. The finer step sizes will give you more points but taking correspondingly longer to compute and in this MATLAB example will take more memory for the x and y arrays.

Once run, you will see your results in the generated plot.

Euler is good for monotonically increasing or decreasing solutions. It doesn't work so well for periodic solutions where injected error makes the integration go astray. Basically global error is estimated as the square of the step size so here with h=0.1 then global erro at each iteration is 0.01 or 1%.

Here's the wikipedia article on it:

https://en.wikipedia.org/wiki/Euler_method
Here's some more resources to look at:

http://people.math.sfu.ca/~ralfw/math467w03/matlab/euler_matlab.pdf
http://mandal.faculty.ku.edu/math320/euler.html
and some MATLAB videos on the topic:

https://www.mathworks.com/videos/solving-odes-in-matlab-1-euler-ode1-117526.html
 
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Likes Wrichik Basu

FAQ: Euler Method in MATLAB: Solving a Double Pendulum

1. What is the Euler Method and how does it work?

The Euler Method is a numerical method used to approximate the solution of a differential equation. It works by breaking down the differential equation into small steps and using the derivative at each step to calculate the next point of the solution.

2. How do I use the Euler Method in MATLAB to solve a double pendulum?

To use the Euler Method in MATLAB to solve a double pendulum, you will need to define the differential equations that describe the motion of the pendulum and then use a loop to calculate the solution at each time step. You can also use MATLAB's built-in ODE solver, ode45, which uses the Euler Method internally.

3. What are the limitations of the Euler Method?

The Euler Method may not always provide accurate solutions, especially for complex or oscillatory systems. It also has a tendency to accumulate errors over time, leading to a less accurate solution. Therefore, it is often used as a first approximation and may need to be combined with other methods for better accuracy.

4. How can I improve the accuracy of the Euler Method?

One way to improve the accuracy of the Euler Method is to use a smaller time step, which will result in smaller errors. Another approach is to use higher-order methods, such as the Runge-Kutta methods, which use additional information to calculate the solution and have smaller errors than the Euler Method.

5. Can I use the Euler Method to solve other types of differential equations?

Yes, the Euler Method can be used to solve various types of differential equations, including first-order and higher-order equations. However, it may not always provide accurate solutions for complex or highly oscillatory systems, and other methods may need to be used in these cases.

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