I’ll help you solve an Ordinary Differential Equation (ODE) using $Python$.
Let’s solve a spring-mass-damper system, which is a classic second-order ODE problem.
1 | import numpy as np |
Let me explain the problem and solution in detail:
The Problem: Spring-Mass-Damper System
- Mathematical form: $m(d²x/dt²) + c(dx/dt) + kx = 0$
- Where:
- $x$ is displacement
- $m$ is mass
- $k$ is spring constant
- $c$ is damping coefficient
Solution Method:
- Convert 2nd order ODE to system of two 1st order ODEs
- Use scipy’s
odeint
for numerical integration - System parameters:
- Mass = $1.0 kg$
- Spring constant = $10.0 N/m$
- Damping coefficient = $0.5 Ns/m$
Visualization (four plots):
Position vs Time:
- Shows oscillatory motion with damping
- Amplitude decreases over time due to damping
Velocity vs Time:
- Shows velocity oscillations
- Amplitude decreases faster than position
Phase Portrait:
- Shows system trajectory in position-velocity space
- Spiral pattern indicates damped oscillation
Energy vs Time:
- Shows potential and kinetic energy exchange
- Total energy decreases due to damping
Physical Interpretation:
- System starts at $x = 1m$ with zero velocity
- Oscillates back and forth while losing energy
- Motion gradually decreases due to damping
- System eventually comes to rest at equilibrium ($x = 0$)
Energy Analysis:
- Energy constantly transfers between potential and kinetic
- Total energy decreases exponentially due to damping
- Potential energy is maximum at maximum displacement
- Kinetic energy is maximum at zero displacement