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

1. Read log-scale plot

2. Use Bode plot to determine magnitude and phase of output signal

3. Find closed-form steady-state outputs for linear systems with simple harmonic input

4. Use normalized Bode plot to design systems to operate away from resonance

5. Understand sensor dynamic range from their Bode plot

6. Design simple analog LP and HP filters

This section on harmonic input builds your frequency-response toolkit for analyzing how linear systems respond to sinusoidal excitation. You will learn how to read log-scale plots and use a Bode plot to determine the magnitude and phase of an output signal across frequency, then connect those plots to closed-form steady-state solutions for simple harmonic inputs. The goal is practical engineering judgment: you should be able to predict when a system will amplify motion near resonance, when it will attenuate inputs, and how phase lag changes what you measure. The lessons move from an analytic gain and phase example into the Bode diagram, then through multiple worked examples for finding steady-state output and interpreting second-order system response to harmonic input. You will also study transmissibility and vibration isolation using a relative-frequency view, which is the same framing used to keep machines, structures, and instruments operating away from resonant peaks. The section closes by tying frequency response to sensors and filters, including how a sensor Bode plot reveals dynamic range limitations and how to design simple analog low-pass and high-pass filters, plus higher-order filter behavior. Use the table of contents below to jump between examples, multi-degree-of-freedom extensions, and the lecture code.