Modelling and Simulation of the Damped Harmonic Oscillator: Analytical and Numerical Study
Rs 25000.00Rs 24500.00
Overview
This project studies the damped harmonic oscillator, a system that oscillates under the combined effect of a restoring force and a damping force. The governing second-order differential equation is derived and solved for three regimes: underdamped, critically damped, and overdamped.
Both analytical solutions and numerical simulations (Runge–Kutta 4, RK4) are implemented. A worked example demonstrates step-by-step calculations of displacement, velocity, and energy. Graphs of displacement and energy illustrate the effect of damping on oscillations and energy loss.
Suitable For
BS Physics – oscillations, damping, and kinematics
BS Mathematics / Applied Mathematics – solving differential equations
BS Computer Science / Software Engineering – Python-based simulations and visualization
Technologies Used
Language: Python
Libraries: NumPy, Matplotlib, SciPy
Techniques: Analytical ODE solving, RK4 method, parameter analysis
Visualization: Displacement vs time, energy decay plots
Features
Derivation of the damped oscillator equation
Analytical solutions for three damping regimes
Numerical example with displacement, velocity, and energy
RK4 simulation for oscillator motion
Comparative plots across damping cases
Documented Python scripts and LaTeX report
Deliverables
Python source code with explanations
Example plots and simulations
CSV of computed values
LaTeX report template with derivations and figures
Optional presentation slides
Benefits for Students
Practice in solving and simulating ODEs
Integration of mathematics and computation
Skills in Python, visualization, and reporting
Deeper understanding of oscillations and damping
Ready-to-present academic project
Highlights of Novelty
Side-by-side study of underdamped, critical, and overdamped motion
Combination of analytical and numerical approaches
Visualization of displacement and energy decay
Step-by-step worked example for clarity