Calculate GRMS from PSD

How to Calculate gRMS from PSD Correctly

Engineers working with random vibration frequently need to calculate gRMS from PSD. Whether you are preparing a qualification profile, verifying a shaker setup, comparing environmental conditions, or reviewing supplier test reports, this conversion is foundational. The acceleration power spectral density curve describes how vibration energy is distributed across frequency. The gRMS value summarizes the total energy into one overall level.

In practical terms, gRMS is a compact way to answer the question: “How severe is this random vibration environment overall?” PSD shows the shape; gRMS shows the integrated magnitude. You need both, because two PSD curves can produce similar gRMS values while having very different frequency content and therefore very different effects on a product.

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Formula and Units for PSD to gRMS

The standard relationship is straightforward:

gRMS = sqrt( integral from f_low to f_high of PSD(f) df )

When PSD is in g²/Hz and frequency is in Hz, integrating over frequency returns g². Taking the square root gives gRMS. This value can be converted to SI acceleration using 1 g = 9.80665 m/s².

Many vibration specifications are defined by breakpoints. For each segment between two points, the PSD may be approximated as linear in ordinary coordinates or as linear on a log-log plot. In random vibration work, log-log interpolation is often more representative because specifications are usually defined that way.

Integration Methods: Linear vs Log-Log

1) Linear trapezoidal integration

For each segment between frequencies f1 and f2 with PSD values S1 and S2, the area is:

Area = (S1 + S2) / 2 × (f2 - f1)

This is simple and often acceptable when frequency spacing is dense and the curve changes gradually.

2) Log-log power-law integration

For a segment modeled as S(f) = C f^n, where n = ln(S2/S1) / ln(f2/f1), the exact integrated area is:

Area = S1 f1 × [ ((f2/f1)^(n+1) - 1) / (n+1) ]

If n = -1, use:

Area = S1 f1 × ln(f2/f1)

Summing all segment areas gives total variance in g². The square root gives gRMS.

Worked Example: Calculate gRMS from a Breakpoint PSD Profile

Consider a profile with points at 20 Hz, 80 Hz, 350 Hz, and 2000 Hz. After integrating each segment and summing all contributions, suppose the total area is 8.10 g². The resulting overall level is:

gRMS = sqrt(8.10) = 2.846 gRMS

Equivalent SI acceleration is:

2.846 × 9.80665 = 27.90 m/s² RMS

This single number is convenient for quick comparisons, but always keep the original PSD shape available for engineering decisions. Resonance-sensitive components may respond very differently even at equal gRMS.

Common Mistakes When Converting PSD to gRMS

Using unsorted frequency points

Frequency must increase strictly from one breakpoint to the next. Duplicate or descending frequencies produce invalid segment widths and wrong integrated energy.

Mixing units

Do not combine g²/Hz data with SI inputs unless you convert consistently. A unit mismatch can introduce large errors.

Ignoring interpolation assumptions

The integration method changes the result. If your specification is defined on log-log axes, use a log-log model for each segment.

Rounding too early

Keep more precision through intermediate steps and round only the final outputs used for reporting.

Where This Calculation Is Used

Converting PSD to gRMS is part of day-to-day workflow in aerospace, defense, automotive, electronics, and industrial reliability. Typical uses include:

• Qualification and acceptance test profile setup
• Supplier test report verification
• Environmental severity comparison across programs
• Design margin studies and proto-to-production correlation
• Shaker controller cross-checks before expensive test time

During requirement reviews, teams often discuss both overall gRMS and spectral shape. Strong engineering practice keeps these linked rather than treating gRMS alone as the full requirement.

Frequently Asked Questions