Frequency-Domain Analysis
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Frequency-Domain Analysis.
§4.1 Sinusoidal steady-state and frequency response
**Mechanical Engineering · Control Systems Engineering · Unit 4: Frequency-Domain Analysis.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
§4.2 Bode plots, slopes and break frequencies
**Mechanical Engineering · Control Systems Engineering · Unit 4: Frequency-Domain Analysis.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
§4.3 Nyquist criterion and encirclements
**Mechanical Engineering · Control Systems Engineering · Unit 4: Frequency-Domain Analysis.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
§4.4 Gain margin, phase margin and bandwidth
**Mechanical Engineering · Control Systems Engineering · Unit 4: Frequency-Domain Analysis.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
§4.5 Sensitivity, disturbances and sensor noise
**Mechanical Engineering · Control Systems Engineering · Unit 4: Frequency-Domain Analysis.** This is an unreviewed instructional chapter; none of its calculations alone justifies commissioning machinery.
Step-by-Step Control Systems Engineering Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Sinusoidal steady-state and frequency response** under steady operational loading.
Worked Engineering Derivation:
For G=1/(1+0.5s) at ω=4 rad/s, magnitude=1/√5≈0.447 and phase=−arctan 2≈−63.4°. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Bode plots, slopes and break frequencies** under steady operational loading.
Worked Engineering Derivation:
For a pole at 10 rad/s, ω=10 rad/s gives amplitude 1/√2≈0.707 and phase −45° relative to low-frequency gain. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Nyquist criterion and encirclements** under steady operational loading.
Worked Engineering Derivation:
For L(jω)=K/(1+jωτ), positive K and τ produce phase between 0 and −90°, so the simple unity-feedback loop has no crossing of −180°. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Gain margin, phase margin and bandwidth** under steady operational loading.
Worked Engineering Derivation:
At crossover, a measured open-loop phase of −135° implies about 45° phase margin. If extra delay adds 0.15 rad at crossover, the margin drops by ≈8.6°. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Sensitivity, disturbances and sensor noise** under steady operational loading.
Worked Engineering Derivation:
With L(jω)=9 at a very low frequency, S≈0.1 and T≈0.9 for this real loop value. Large high-frequency loop gain instead passes measurement noise into the output. **Reproduction protocol.** Write the givens and units in a table, compute any derived time scales or gains, substitute into the displayed model and calculate the result with more precision than required for the final reported value. Compare against a limiting case: zero disturbance, zero controller gain, vanishing sample time, or a stationary equilibrium as appropriate. If a calculation predicts infinite control force, negative dissipation, or motion through a hard stop, that is an indication to revisit assumptions, not a valid operating instruction.