By the end of this section, you should be able to

1. Create FBDs for simple spring-mass-damper (SMD) systems

2. Create equations of motion (EOM) for single-element mechanical system

3. Find closed-form solutions for the displacement of mass in SMD system

This section introduces mechanical modeling for dynamic systems, using the spring mass damper system as the core example you will return to throughout the Modeling unit. Mechanical modeling starts with a clear physical picture, then turns that picture into math that can predict motion over time. Here you will practice building free-body diagrams with consistent sign conventions, identifying spring forces and damping forces, and applying Newton’s second law to derive equations of motion for simple single-degree-of-freedom mechanical systems. Once you can write the governing differential equation, you will use analytic techniques to find closed-form solutions for displacement, which helps you connect parameters like mass, stiffness, and damping to behaviors such as oscillation, decay, and steady response. These skills are a foundation for everything that follows, including transfer functions, initial and final value checks, and state space models, because all of those tools depend on having the right equations first. Use the table of contents below to move between lessons, slides, and videos as you build a repeatable workflow for modeling mechanical systems from a diagram to an equation to a solution you can interpret and validate.