Damped Harmonic Oscillator | Deriving the Equation of Motion

👉 Full course on Udemy: Fundamentals of Dynamics & Oscillations https://www.udemy.com/course/simxacad...

📋 What you will learn in the course:

Model linear mechanical rigid-body 1-DOF systems using the impulse and angular-momentum theorem
Derive equations of motion systematically from physical principles
Analyze undamped and damped oscillations and interpret their physical meaning
Determine eigenvalues and derive homogeneous solutions of linear differential equations
Solve harmonic and periodic excitations in the frequency domain using Laplace transforms and transfer functions
Analyze arbitrary excitations using impulse response functions and the convolution integral
Develop confidence in interpreting vibration behavior of mechanical systems
Apply dynamics concepts reliably instead of relying on memorized formulas or procedural guessing

00:00 Intro
00:44 Free-Body-Diagram
03:24 Force Equilibrium
04:50 Natural Angular Frequency

Intro

Most errors in vibration analysis are not caused by solving the differential equation, but by setting it up incorrectly. This lecture shows how to derive the equation of motion of a damped single-degree-of-freedom (SDOF) oscillator in a clean, systematic, and physically consistent way.

About this lecture

In this video, the equation of motion of a linear mass–spring–damper system is derived step by step using a free-body diagram and Newton’s law. Special attention is given to force directions, sign conventions, and the origin of the eigenvalue. Full theoretical background, extended derivations, and additional system classes are covered in detail in the complete Udemy course.

What you will learn in this video

How to draw a correct free-body diagram for a vibrating mechanical system
How to set up the equation of motion of a damped SDOF oscillator
How to handle spring, damper, gravity, and external harmonic forces consistently
How the undamped natural frequency Ω₀ = √(k/m) emerges from the model
How to avoid typical sign and modeling errors in vibration analysis
How to interpret the physical meaning of each term in the differential equation

Who this video is for

Mechanical engineering students struggling with vibration modeling
Engineers who want a rigorous but practical refresher in dynamics
Learners preparing for exams in technical dynamics or mechanical vibrations
Anyone who wants to understand equations of motion instead of memorizing formulas

Why SimX Academy?

SimX Academy empowers engineers to Build Engineering Confidence by combining rigorous theory with practical exercises. Our courses bridge the gap between abstract mathematics and real-world applications, enabling learners to become autonomous problem-solvers rather than mere software operators.

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