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Dynamic Signals — Introduction and Learning Goals

Short summary Time-domain and dynamical-system behavior: step responses, first/second order systems, and vibration-based measurements.

Learning objectives - Interpret first- and second-order system responses and key parameters (time constant, damping, natural frequency). - Extract physical quantities (e.g., mass from vibrations) from measured signals. - Link time-domain responses to frequency content.

Key concepts (brief) - Step response, log-decrement, and damping ratio estimation. - Modal interpretation for simple systems and measurement-driven parameter estimation. - Practical considerations: sensor dynamics and filtering.

Prerequisites Ordinary differential equations basics and elementary signal processing.

Ordered reading (suggested)

Follow this sequence to understand how measurement systems actually respond to inputs. The order moves from basic first-order dynamics, through second-order systems and damping, to practical applications and frequency-domain connections.

  1. 1st order dynamic system — first-order systems: time constant and exponential response (e.g., RC circuits, temperature sensors with lag).
  2. Step responses of dynamical systems — interpreting step and impulse responses; the fundamental tool for characterizing system behavior.
  3. Log decrement method — damped oscillation, natural frequency, damping ratio, and log-decrement estimation from noisy data.
  4. Lab Worksheet: Measurement of Unknown Mass using Vibrating Beam — extract unknown mass from measured vibration frequency using the system model.
  5. Example of pre-measurement design — apply system dynamics to instrument design: trade-offs between sensitivity, bandwidth, and natural frequency.
  6. Fast Fourier Transform (FFT) of a sum of two sine signals — bridge from time domain to frequency domain with two sines.
  7. Symbolic evaluation of Fourier series coefficients — analytical Fourier background for dynamic signals.
  8. Plot power spectrum of experimental data — spectrum of real turbulent-jet data.
  9. Spectrum example — additional spectrum interpretation example.

Rationale: Students first understand how simple systems lag behind inputs (first-order), then recognize oscillatory behavior and damping (second-order). The final notebooks show why this matters for real sensors, and how time-domain behavior links to frequency content.

What comes next?

Time-domain analysis reveals when signals occur and how fast they change. But to understand what frequencies are present in your signal, you need frequency-domain analysis. The Signal Processing chapter teaches Fourier transforms and FFT — the frequency-domain complement to the time-domain tools you've just learned.