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

1. Define a transfer function as the Laplace-domain input-output relationship for an LTI system under zero initial conditions

2. Convert a linear differential equation into a transfer function by applying the Laplace transform and solving algebraically for Y(s)/U(s)

3. Derive a transfer function from a physical mechanical model by constructing a free-body diagram, writing the equation of motion, then transforming to s-domain

4. Identify poles and zeros from a transfer function and explain what they imply about stability and qualitative response behavior

5. Use transfer function to compute time responses for standard inputs (step and impulse) by solving in the s-domain and taking the inverse Laplace transform

This section focuses on transfer functions for mechanical systems and how they connect physical modeling to system response. You will start by building clear free body diagrams for linear and rotational elements, then use those diagrams to derive equations of motion with consistent sign conventions and input and output definitions. From the equations of motion, you will develop transfer functions that relate forces and displacements for spring mass damper systems, first for single degree of freedom models and then for multi degree of freedom systems. The emphasis is on a repeatable workflow that engineers actually use: define the input and output, write the governing differential equations, transform to the Laplace domain, solve for the transfer function, and check that the result makes physical sense.