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Program 1

2nd Order ODE (RK2 Method)

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

Output 1

Program 2

2nd Order ODE (RK2 Method)

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

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()

Output 2

Ordinary Differential Equation-2

2nd Order ODE (RK2 Method)

Damped Harmonic Oscillator ODE Solution

2nd Order ODE (RK2 Method)