Ordinary Differential Equation-2 - 2nd Order ODE (RK2 Method)

Program 1

2nd Order ODE (RK2 Method)

Python 3.11 Ready

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Damped Harmonic Oscillator ODE Solution

This program uses numerical methods to solve the ordinary differential equation (ODE) for a damped harmonic oscillator. The ODE is given by:

$\frac{d^2x}{dt^2} + 2 \zeta \omega_n \frac{dx}{dt} + \omega_n^2 x = 0$

Initial Conditions: $x(t=0)=1 \quad \frac{dx}{dt}_{t=0}=0$

Program 2

2nd Order ODE (RK2 Method)

Python 3.11 Ready

import numpy as np
import matplotlib.pyplot as plt

# Parameters
m = 1.0    
b = 0.2    
k = 1.0    
F0 = 1.0   
omega = 1.0

def F(t):
    return F0 * np.sin(omega * t)

def rk2_step(x, v, t, dt):
    dxdt = v
    dvdt = (F(t) / m) - (b / m) * v - (k / m) * x
    
    k1_x = dxdt
    k1_v = dvdt
    
    x_mid = x + 0.5 * k1_x * dt
    v_mid = v + 0.5 * k1_v * dt
    
    dxdt_mid = v_mid
    dvdt_mid = (F(t + 0.5 * dt) / m) - (b / m) * v_mid - (k / m) * x_mid
    
    x_new = x + k1_x * dt
    v_new = v + dvdt_mid * dt
    
    return x_new, v_new

t_start = 0.0
t_end = 20.0
dt = 0.01
num_steps = int((t_end - t_start) / dt)

time = np.linspace(t_start, t_end, num_steps)
x = np.zeros(num_steps)
v = np.zeros(num_steps)

x[0] = 0.0  # Initial position
v[0] = 0.0  # Initial velocity

for i in range(1, num_steps):
    t = time[i - 1]
    x[i], v[i] = rk2_step(x[i - 1], v[i - 1], t, dt)

plt.plot(time, x, label='Position (x)')
plt.plot(time, v, label='Velocity (v)', color='orange')
plt.title('Forced Oscillation')
plt.xlabel('Time (s)')
plt.ylabel('Position (x) and Velocity (v)')
plt.legend()
plt.savefig('plot.png')
plt.show()

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Solve a forced oscillation differential equation using the RK2 (Runge-Kutta 2nd order) method, we consider a standard form of the equation:

$$ m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = F_0 \sin(\omega t) $$

Initial conditions:$$ x(0) = 0 \quad \frac{dx}{dt}\big|_{t=0} = 0 $$

Ch. 12.2 2nd Order ODE (RK2 Method)

2nd Order ODE (RK2 Method)

Mathematical Problem Formulation

Damped Harmonic Oscillator ODE Solution

2nd Order ODE (RK2 Method)

Mathematical Problem Formulation