Understanding Integration in Vibration Analysis

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Definition: What is Integration?

Integration in vibration analysis is the mathematical process of converting vibration measurements from one parameter to another by performing integration in the time domain or dividing by frequency in the frequency domain. Most commonly, integration converts acceleration (measured by accelerometers) to velocity, or velocity to displacement. Since acceleration, velocity, and displacement are related through calculus (velocity = ∫acceleration dt; displacement = ∫velocity dt), integration allows expressing vibration in the most appropriate parameter for the application and frequency range.

Integration is essential because different vibration parameters are optimal for different purposes: acceleration for high-frequency analysis (bearing defects), velocity for general machinery condition (ISO standards), and displacement for low-speed equipment and clearance assessment.

Mathematical Relationships

Time Domain Integration

• Velocity from Acceleration: v(t) = ∫ a(t) dt
• Displacement from Velocity: d(t) = ∫ v(t) dt
• Displacement from Acceleration: d(t) = ∫∫ a(t) dt dt (double integration)

Frequency Domain Integration

Simpler in frequency domain:

• Velocity from Acceleration: V(f) = A(f) / (2πf)
• Displacement from Velocity: D(f) = V(f) / (2πf)
• Result: Dividing by frequency, so low frequencies amplified, high frequencies reduced

Why Integration is Needed

Sensor Limitations

• Accelerometers are most versatile and common sensors
• But acceleration not always best parameter for analysis
• Integration allows using accelerometer for all parameter types
• More economical than multiple sensor types

Parameter Selection by Frequency

• High Frequency (>1000 Hz): Acceleration best (bearing defects)
• Medium Frequency (10-1000 Hz): Velocity best (general machinery, ISO standards)
• Low Frequency (< 10 Hz): Displacement best (low-speed equipment, clearances)
• Integration: Enables using optimal parameter for each frequency range

Standard Requirements

• ISO 20816 specifies RMS velocity
• If measuring acceleration, must integrate to velocity
• Proximity probe measurements in displacement must convert for velocity comparison

Integration Challenges

Low-Frequency Drift

The primary integration problem:

• Any DC offset or very low-frequency component
• Integration amplifies low frequencies (dividing by small numbers)
• Creates huge low-frequency errors
• Signal “drifts” off scale
• Solution: High-pass filter before integration (typically 2-10 Hz cutoff)

Noise Amplification

• Integration is 1/f operation (amplifies low frequencies)
• Low-frequency noise amplified more than signal
• Can degrade signal-to-noise ratio
• Solution: Filter noise before integration

Double Integration Compounds Errors

• Acceleration to displacement requires double integration
• Errors multiply
• Very sensitive to DC offset and low-frequency noise
• Aggressive high-pass filtering essential (10-20 Hz typical)

Proper Integration Procedure

Single Integration (Acceleration to Velocity)

1. Acquire Signal: Collect acceleration data with adequate sample rate
2. DC Removal: Remove any DC offset
3. High-Pass Filter: Apply HPF at 2-10 Hz to remove drift
4. Integrate: Perform integration (divide by 2πf in frequency domain)
5. Verify: Check result for reasonable values and no drift

Double Integration (Acceleration to Displacement)

1. Aggressive HPF: 10-20 Hz cutoff (higher than single integration)
2. First Integration: Acceleration → velocity
3. Verify Intermediate: Check velocity result
4. Second Integration: Velocity → displacement
5. Final Verification: Confirm displacement reasonable

Frequency Domain vs. Time Domain

Frequency Domain Integration (Preferred)

• Method: FFT → divide by 2πf → inverse FFT
• Advantages: Straightforward, no cumulative errors, easy to apply filtering
• Implementation: Standard in modern analyzers
• Result: Clean, accurate integration

Time Domain Integration

• Method: Numerical integration (trapezoidal rule, Simpson’s rule)
• Challenges: Cumulative errors, drift, more complex filtering
• Use: When frequency-domain not practical

Practical Applications

Standards Compliance

• Convert accelerometer measurements to velocity for ISO 20816 comparison
• Convert proximity probe displacement to velocity
• Ensures consistent comparison across sensor types

Low-Speed Machinery

• At low speeds (< 500 RPM), acceleration and velocity become small
• Displacement more meaningful
• Integrate acceleration to displacement for analysis

Multi-Parameter Analysis

• View same vibration as acceleration, velocity, AND displacement
• Each parameter emphasizes different frequency ranges
• Comprehensive understanding of vibration characteristics

Common Mistakes

Integration Without Filtering

• Results in drift and errors
• Unusable displacement values
• Always high-pass filter before integrating

Wrong Cutoff Frequency

• Too low: drift problems
• Too high: valid low frequencies removed
• Must balance drift prevention vs. signal preservation

Comparing Mixed Parameters

• Don’t compare acceleration to velocity directly
• Convert to same parameter before comparison
• Frequency content affects which parameter shows higher values

Integration is a fundamental signal processing operation in vibration analysis that enables conversion between acceleration, velocity, and displacement measurements. Proper integration technique—including appropriate high-pass filtering to prevent drift and understanding of frequency-domain implementation—is essential for accurate vibration parameter conversion, standards compliance, and comprehensive multi-parameter analysis of machinery condition.

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