Mechanical Vibration and Rotating Machinery Reliability Exercises

These exercises practise vibration and rotating-machinery calculations for order tracking, resonance screening, damping, unbalance, vibration isolation, gear mesh, sampling, fatigue damage, condition monitoring, and reliability review. The goal is not only to compute a frequency. The goal is to connect dynamic response to failure modes, maintenance decisions, operating limits, and validation evidence.

Assume linear vibration behaviour and simplified single-mode models unless an exercise states otherwise. Real machines should also check bearing stiffness, foundation flexibility, thermal growth, lubrication, misalignment, process forces, control interaction, sensor mounting, filtering, uncertainty, and duty-cycle exposure.

How to Use These Exercises

For each calculation, define:

1. the machine speed, load, and process state;
2. the forcing source: unbalance, gear mesh, blade passing, torque ripple, fluid pulsation, looseness, or control action;
3. the structural or rotor mode that may respond;
4. the consequence: fatigue, bearing damage, noise, trip, product quality, leakage, or unsafe operation;
5. the measurement plan needed to validate the conclusion.

The common mistake is treating vibration as a single number. A useful vibration review identifies frequency content, operating condition, sensor location, trend behaviour, and the engineering action linked to each alarm or limit.

Use the exercises as reliability gates: identify a forcing order, reject an operating speed near resonance, revise an avoidance band, validate an isolation design, challenge a spectrum setup, reduce vibration-fatigue exposure, or require maintenance action only when the measured feature, trend, and failure mode support the decision.

Exercise 1: Shaft Orders and Gear Mesh Frequency

A gearbox input shaft rotates at:

The pinion has:

Calculate the shaft 1x, 2x, and 3x frequencies and the gear mesh frequency.

Solution

Shaft rotational frequency:

Second and third orders:

Gear mesh frequency:

Engineering Comment

The order map tells the analyst where to look in a spectrum. A peak at 1x may suggest unbalance, while 2x may indicate misalignment or looseness. Gear mesh and sidebands require speed reference, tooth count, load state, and knowledge of which shaft carries the gear.

Exercise 2: Natural Frequency Inside an Operating Range

A machine frame can be approximated as a mass:

on equivalent stiffness:

The machine operates from 900 rpm to 1800 rpm. Estimate natural frequency and determine whether the 1x operating range crosses it.

Solution

Natural frequency:

Operating range:

The mode is inside the 1x operating range. The corresponding speed is:

Engineering Comment

This is a resonance warning, not a final dynamic model. The engineer should measure the actual mode shape and damping, then decide whether to add speed avoidance, increase stiffness, add damping, reduce forcing, change ramp rate, or modify the support.

Exercise 3: Damping Ratio from Logarithmic Decrement

A free-decay test gives two consecutive displacement peaks:

and:

Estimate logarithmic decrement and damping ratio.

Solution

Logarithmic decrement:

For an underdamped single-degree system:

The damping ratio is approximately:

Engineering Comment

The estimate assumes a clean single mode. Real machinery decay may include multiple modes, friction, looseness, fluid-film effects, sensor filtering, and nonlinear supports. Damping should be measured under conditions that resemble the operating machine.

Exercise 4: Unbalance Force at Speed

A rotor has residual unbalance equivalent to:

at eccentricity radius:

The rotor speed is:

Estimate rotating unbalance force.

Solution

Unbalance is:

Convert units:

Angular speed:

Unbalance force:

Engineering Comment

Unbalance force scales with speed squared. A rotor that is acceptable at low speed can become damaging at high speed. The final decision should consider bearing load, support stiffness, balance grade, operating speed range, startup ramp, and whether vibration is measured in displacement, velocity, or acceleration.

Exercise 5: Vibration Isolation Transmissibility

A machine runs at:

on isolators with natural frequency:

and damping ratio:

Estimate force transmissibility using:

where:

Solution

Frequency ratio:

Damping term:

Transmissibility:

The transmitted force is about:

of the excitation force in this simplified model.

Engineering Comment

Isolation can work well above resonance, but startup and shutdown still pass through the isolator natural frequency. The design must check transient motion, alignment, pipe strain, coupling tolerance, seismic or shock load, and whether added damping is beneficial or harmful at the operating frequency.

Exercise 6: Speed Avoidance Band During Run-Up

A variable-speed machine ramps from:

to:

in:

A measured structural mode corresponds to:

The commissioning plan requires an avoidance band of \pm5\% around the corresponding 1x speed. Calculate the speed band and the time spent crossing it during a linear ramp.

Solution

Critical speed:

Avoidance band:

Ramp rate:

Time at lower band edge:

Time at upper band edge:

Crossing time:

Engineering Comment

Avoidance bands are useful only when the control system can enforce them. If the process must dwell in the band, the design needs a different response: higher damping, shifted stiffness, reduced excitation, modified operating range, or a validated limit based on measured amplitude and fatigue consequence.

Exercise 7: Gear Mesh and Sideband Frequencies

A motor shaft drives a 20-tooth pinion at:

The mating gear has 60 teeth. Calculate pinion speed frequency, gear mesh frequency, output gear speed, and the first pinion-speed sidebands around mesh frequency.

Solution

Pinion frequency:

Gear mesh frequency:

Gear ratio:

Output speed:

Output frequency:

First pinion-speed sidebands:

Engineering Comment

Gear spectra must be interpreted with tooth count, shaft speed, load, and sensor location. Sidebands can indicate modulation from eccentricity, tooth damage, torque variation, or shaft-speed-related effects, but diagnosis should be confirmed with trend data, inspection, and operating condition.

Exercise 8: Sampling Rate and Frequency Resolution

A vibration signal is sampled at:

for:

Find Nyquist frequency and FFT bin spacing. Determine whether the setup can observe a 600 Hz gear mesh and its third harmonic.

Solution

Nyquist frequency:

Frequency bin spacing:

The 600 Hz mesh frequency is below Nyquist. Its third harmonic is:

which is also below Nyquist.

Engineering Comment

The sampling setup can represent these frequencies, but that does not guarantee a valid diagnosis. The engineer should also check anti-alias filtering, sensor bandwidth, mounting stiffness, windowing, leakage, speed variation during the record, signal-to-noise ratio, and whether higher harmonics or bearing features exceed Nyquist.

Exercise 9: Fatigue Damage from Vibration Exposure

A shaft shoulder experiences vibration stress amplitude at one operating condition. The dominant vibration frequency is:

The machine operates at this condition for:

The S-N assessment estimates life at this stress amplitude as:

Estimate Miner damage for one shift.

Solution

Cycles accumulated in one shift:

Convert time to seconds:

Then:

Miner damage:

Engineering Comment

The damage fraction is substantial for one shift if the S-N estimate is credible. A small resonance can accumulate damaging cycles quickly. The decision should consider stress concentration, mean stress, corrosion, surface condition, duty variability, confidence in stress measurement, and whether speed avoidance or balancing can reduce amplitude.

Exercise 10: RPN Change After Condition Monitoring

A bearing failure mode is initially ranked with:

where S is severity, O is occurrence, and D is detection ranking. A vibration-monitoring and lubrication-control action reduces occurrence to 3 and detection ranking to 3. Calculate initial and revised RPN.

Solution

Risk priority number:

Initial value:

Revised value:

Reduction:

Engineering Comment

The RPN reduction is useful only if the monitoring action is real: sensor locations, alarm thresholds, response procedure, lubrication control, and inspection actions must be defined. A lower spreadsheet score does not reduce risk unless it changes detection, maintenance, or design behaviour in service.

Review Checklist

When reviewing a vibration or rotating-machinery calculation, ask:

• Are speed, order, forcing source, load state, ramp condition, and process condition explicit?
• Is the response tied to a structural mode, rotor mode, bearing support, foundation, gearbox, pump, fan, or driven load?
• Are damping, stiffness, mounting, alignment, runout, balance grade, lubrication, and thermal growth represented where they affect the result?
• Does the measurement setup have adequate sensor location, bandwidth, mounting stiffness, anti-alias filtering, sampling rate, record length, and speed reference?
• Are spectra interpreted with trend data, sidebands, operating condition, inspection evidence, and known machine geometry?
• Are vibration limits connected to fatigue, bearing life, leakage, product quality, trip risk, or unsafe operation?
• Does every alarm or RPN reduction define a response owner, inspection action, acceptance criterion, and closeout evidence?

Vibration evidence becomes valuable when it changes a design, operating, or maintenance decision. A spectrum without a linked failure mode and action plan is only a plot.