Understanding Power Spectral Density
Definition: What is Power Spectral Density?
Power spectral density (PSD) is a representation of vibration energy distribution across frequency, expressed as energy per unit frequency bandwidth (units: (m/s²)²/Hz for acceleration, (mm/s)²/Hz for velocity). Unlike a standard amplitude spectrum which shows amplitude at each frequency, PSD shows how vibration power is distributed across frequency, with values normalized by the frequency resolution bandwidth. This normalization makes PSD independent of analysis bandwidth, allowing meaningful comparison between spectra measured with different resolution settings.
PSD is particularly important for analyzing random vibration (where energy is distributed continuously across frequency rather than concentrated at discrete peaks), for noise analysis, and for applications requiring bandwidth-independent spectral characterization such as vibration testing and environmental qualification.
PSD vs. Amplitude Spectrum
Amplitude Spectrum
- Shows vibration amplitude at each frequency
- Units: mm/s, m/s², mils, etc.
- Peak amplitudes at discrete frequencies (unbalance, bearing faults)
- Values depend on FFT resolution bandwidth
- Standard display for machinery diagnostics
Power Spectral Density
- Shows vibration power per Hz of bandwidth
- Units: (mm/s)²/Hz, (m/s²)²/Hz, etc.
- Energy distribution across frequency
- Independent of analysis bandwidth
- Standard for random vibration analysis
Relationship
- PSD = (Amplitude)² / Δf
- Where Δf = frequency resolution (bin width)
- Squaring emphasizes large amplitudes
- Normalization makes bandwidth-independent
Applications
1. Random Vibration Analysis
Primary PSD application:
- Random Processes: Turbulence, road vibration, seismic, acoustic
- Continuous Spectra: Energy distributed across frequency, not discrete peaks
- Statistical Description: PSD describes random process energy distribution
- Standard Format: Vibration testing specifications in PSD
2. Broadband Noise Characterization
- Cavitation noise in pumps
- Turbulent flow noise in fans
- Aerodynamic noise
- Bearing defect noise characterization
3. Bandwidth-Independent Comparison
- Compare spectra measured with different FFT settings
- Data from different instruments or resolutions
- Historical data with different analysis parameters
- PSD values directly comparable regardless of bandwidth
4. Environmental Testing
- Vibration test specifications given as PSD vs. frequency
- Shaker table control based on PSD
- Product qualification testing
- Shock and vibration standards
Calculating PSD
From FFT
- Compute FFT of vibration signal
- Square each amplitude value
- Divide by frequency resolution (Δf = Fmax / Number of lines)
- Result: PSD in (units)²/Hz
Units
- Acceleration PSD: (m/s²)²/Hz or g²/Hz
- Velocity PSD: (mm/s)²/Hz or (in/s)²/Hz
- Displacement PSD: (µm)²/Hz or (mils)²/Hz
- Often Plotted: Logarithmic scale (dB relative to reference)
Interpreting PSD Plots
Flat Spectrum (White Noise)
- Constant PSD across frequency
- Equal energy per Hz at all frequencies
- Characteristic of broadband random vibration
- Example: Ideal random vibration for testing
Sloped Spectrum (Colored Noise)
- PSD varies with frequency
- Rising slope: more energy at high frequencies
- Falling slope: more energy at low frequencies (common in machinery)
- Slope indicates frequency distribution of energy
Peaks in PSD
- Discrete frequency components appear as peaks above general level
- Resonances show as elevated PSD regions
- Can identify dominant frequencies contributing to energy
Relationship to RMS and Total Energy
Total Energy from PSD
- Integrate PSD across frequency range
- Result: Mean square value
- Square root gives RMS value
- RMS = √[∫ PSD(f) df]
Energy in Frequency Bands
- Integrate PSD over specific frequency range
- Gives energy in that band
- Useful for assessing contribution of different frequency ranges
Advantages of PSD
Resolution Independence
- PSD values comparable regardless of FFT resolution
- Enables comparison of historical data with different settings
- Standardizes analysis across different instruments
Energy Representation
- Directly represents vibration energy distribution
- Squared values emphasize dominant frequencies
- Natural for energy-based analysis
Statistical Framework
- PSD is foundation of random vibration theory
- Enables probabilistic analysis
- Supports fatigue life prediction from random loading
When to Use PSD
Use PSD When:
- Analyzing random vibration or noise
- Comparing data with different analysis bandwidths
- Following test specifications in PSD format
- Characterizing broadband processes
- Energy-based analysis required
Use Amplitude Spectrum When:
- Routine machinery diagnostics
- Identifying discrete fault frequencies
- Trending specific components
- Amplitude values directly meaningful
Power Spectral Density is a fundamental concept in random vibration analysis and provides bandwidth-independent spectral characterization. While less commonly used than amplitude spectra for routine machinery diagnostics, PSD is essential for random vibration applications, noise analysis, and any situation requiring comparison of spectra measured with different analysis parameters or from different instruments.
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