Harmonic Excitation and Resonance
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Harmonic Excitation and Resonance.
§3.1 Force excitation and dynamic amplification
**OpenSTEM Mechanical Engineering / Book 09 / Mechanical Vibrations** *Unit 3: Harmonic Excitation and Resonance | Unreviewed university-level educational manuscript.*
§3.2 Phase lag and complex compliance
**OpenSTEM Mechanical Engineering / Book 09 / Mechanical Vibrations** *Unit 3: Harmonic Excitation and Resonance | Unreviewed university-level educational manuscript.*
§3.3 Half-power bandwidth
**OpenSTEM Mechanical Engineering / Book 09 / Mechanical Vibrations** *Unit 3: Harmonic Excitation and Resonance | Unreviewed university-level educational manuscript.*
§3.4 Resonance crossing and run-up
**OpenSTEM Mechanical Engineering / Book 09 / Mechanical Vibrations** *Unit 3: Harmonic Excitation and Resonance | Unreviewed university-level educational manuscript.*
§3.5 Frequency-response testing
**OpenSTEM Mechanical Engineering / Book 09 / Mechanical Vibrations** *Unit 3: Harmonic Excitation and Resonance | Unreviewed university-level educational manuscript.*
Step-by-Step Mechanical Vibrations 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 **Force excitation and dynamic amplification** under steady operational loading.
Worked Engineering Derivation:
Return to the case: A 5 kg stage with k=5000 N/m, ζ=0.10 and F0=20 N is driven at r=1. The screening result is useful for sizing a measurement range, identifying a suspected mode or testing whether an operating frequency overlaps a predicted response band. It is not a certification result. State the inputs you can actually measure, the ones you have merely assumed, and the likely interval of uncertainty. If two configurations differ by less than measurement variation, choosing the numerically better result is unjustified. For a machine with significant temperature changes, test cold and hot states; for a rotating machine, compare the speed sweep in both directions; for a flexible frame, check whether the fixture and mounting method influence the identified mode. Introduce a second parameter set, changing one input by 10% while holding the others fixed. Predict the sign of the change before calculating. If the sign differs, identify whether the example crossed a resonance or a stability condition, used the wrong output definition, or violated the small-motion assumption. Record a baseline parameter table with source, unit, uncertainty and date. When the mathematical response is highly sensitive to a small change, refine the test plan before declaring that the first design works.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Phase lag and complex compliance** under steady operational loading.
Worked Engineering Derivation:
Return to the case: A shaker frequency sweep changes whether displacement follows or opposes the applied force. The screening result is useful for sizing a measurement range, identifying a suspected mode or testing whether an operating frequency overlaps a predicted response band. It is not a certification result. State the inputs you can actually measure, the ones you have merely assumed, and the likely interval of uncertainty. If two configurations differ by less than measurement variation, choosing the numerically better result is unjustified. For a machine with significant temperature changes, test cold and hot states; for a rotating machine, compare the speed sweep in both directions; for a flexible frame, check whether the fixture and mounting method influence the identified mode. Introduce a second parameter set, changing one input by 10% while holding the others fixed. Predict the sign of the change before calculating. If the sign differs, identify whether the example crossed a resonance or a stability condition, used the wrong output definition, or violated the small-motion assumption. Record a baseline parameter table with source, unit, uncertainty and date. When the mathematical response is highly sensitive to a small change, refine the test plan before declaring that the first design works.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Half-power bandwidth** under steady operational loading.
Worked Engineering Derivation:
Return to the case: A vibration spectrum shows an isolated resonance near 30 Hz with a 3 Hz half-power width. The screening result is useful for sizing a measurement range, identifying a suspected mode or testing whether an operating frequency overlaps a predicted response band. It is not a certification result. State the inputs you can actually measure, the ones you have merely assumed, and the likely interval of uncertainty. If two configurations differ by less than measurement variation, choosing the numerically better result is unjustified. For a machine with significant temperature changes, test cold and hot states; for a rotating machine, compare the speed sweep in both directions; for a flexible frame, check whether the fixture and mounting method influence the identified mode. Introduce a second parameter set, changing one input by 10% while holding the others fixed. Predict the sign of the change before calculating. If the sign differs, identify whether the example crossed a resonance or a stability condition, used the wrong output definition, or violated the small-motion assumption. Record a baseline parameter table with source, unit, uncertainty and date. When the mathematical response is highly sensitive to a small change, refine the test plan before declaring that the first design works.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Resonance crossing and run-up** under steady operational loading.
Worked Engineering Derivation:
Return to the case: A test rig operates above resonance but must accelerate through the critical speed at startup. The screening result is useful for sizing a measurement range, identifying a suspected mode or testing whether an operating frequency overlaps a predicted response band. It is not a certification result. State the inputs you can actually measure, the ones you have merely assumed, and the likely interval of uncertainty. If two configurations differ by less than measurement variation, choosing the numerically better result is unjustified. For a machine with significant temperature changes, test cold and hot states; for a rotating machine, compare the speed sweep in both directions; for a flexible frame, check whether the fixture and mounting method influence the identified mode. Introduce a second parameter set, changing one input by 10% while holding the others fixed. Predict the sign of the change before calculating. If the sign differs, identify whether the example crossed a resonance or a stability condition, used the wrong output definition, or violated the small-motion assumption. Record a baseline parameter table with source, unit, uncertainty and date. When the mathematical response is highly sensitive to a small change, refine the test plan before declaring that the first design works.
Problem Statement: An engineering practitioner analyzes a mechanical system governed by **Frequency-response testing** under steady operational loading.
Worked Engineering Derivation:
Return to the case: An impact hammer and accelerometer are used to identify a cabinet resonance. The screening result is useful for sizing a measurement range, identifying a suspected mode or testing whether an operating frequency overlaps a predicted response band. It is not a certification result. State the inputs you can actually measure, the ones you have merely assumed, and the likely interval of uncertainty. If two configurations differ by less than measurement variation, choosing the numerically better result is unjustified. For a machine with significant temperature changes, test cold and hot states; for a rotating machine, compare the speed sweep in both directions; for a flexible frame, check whether the fixture and mounting method influence the identified mode. Introduce a second parameter set, changing one input by 10% while holding the others fixed. Predict the sign of the change before calculating. If the sign differs, identify whether the example crossed a resonance or a stability condition, used the wrong output definition, or violated the small-motion assumption. Record a baseline parameter table with source, unit, uncertainty and date. When the mathematical response is highly sensitive to a small change, refine the test plan before declaring that the first design works.