Solving harmonic oscillator problem in Runge-Kutta method | Python

Introduction

     Harmonic oscillators are prevalent in physics, and simulating their behavior is crucial for understanding dynamic systems. In this blog post, we'll explore a Python program that uses the Runge-Kutta method to simulate the motion of a harmonic oscillator.  

Code

    
     
        from pylab import *

        # Initial conditions
        kx1 = kx2 = kx3 = kx4 = 0
        kv1 = kv2 = kv3 = kv4 = 0
        xn = 1.0
        vn = 0.0
        h = 0.01
        t = 0.0
        time = []
        displacement = []
        w_square = 39.4384  # Adjust this value as needed
        
        # Runge-Kutta method for simulating harmonic oscillation
        while t <= 3:
            kx1 = vn
            kv1 = -1 * xn * w_square
            kx2 = vn + 0.5 * h * kv1
            kv2 = -1 * (xn + 0.5 * h * kx1) * w_square
            kx3 = vn + 0.5 * h * kv2
            kv3 = -1 * (xn + 0.5 * h * kv2) * w_square
            kx4 = vn + h * kv3
            kv4 = -1 * (xn + h * kx3) * w_square
        
            xn = xn + (h * (kx1 + 2 * kx2 + 2 * kx3 + kx4) / 6)
            vn = vn + (h * (kv1 + 2 * kv2 + 2 * kv3 + kv4) / 6)
        
            displacement.append(xn)
            time.append(t)
            t = t + h
        
        # Plotting the results
        title('Harmonic Oscillator Simulation')
        plot(time, displacement)
        xlabel('Time')
        ylabel('Displacement')
        grid(True)
        show()        

       
     

Output

Explanation

1. Constants and Initial Conditions:

    Set initial conditions, step size (h), time (t), and the square of the angular frequency (w_square).

2. Runge-Kutta Method:

    Implement the Runge-Kutta method to update the position (xn) and velocity (vn) of the harmonic     oscillator at each time step.

3. Plotting:

    Use pylab to visualize the simulation results by plotting the displacement over time.

Conclusion

     This Python program provides a straightforward example of taking user input, performing a basic arithmetic operation, and displaying the result. Understanding these fundamental concepts is essential for anyone starting their programming journey. Feel free to experiment with different input values and observe how the program accurately calculates the sum. Happy coding!