Random Vibration Displacement Calculator

This calculator estimates the root-mean-square (RMS) displacement of a structure or component subjected to a random vibration environment, given the acceleration spectral density (ASD) profile and the system’s natural frequency. You enter the ASD level at the natural frequency, the frequency itself, and the damping ratio; you get back the RMS displacement, along with the RMS velocity and acceleration.

Formula

The Miles equation is the standard textbook formula for estimating the RMS response of a single-degree-of-freedom system to a flat random vibration spectrum. The RMS displacement is:

d_rms = sqrt( (ASD × g) / (2 × (2π × f_n)³ × ζ) )

Where:

• d_rms – root-mean-square displacement (inches, or meters if you use metric units consistently)
• ASD – acceleration spectral density at the natural frequency (g²/Hz)
• g – acceleration due to gravity (386.09 in/s² or 9.81 m/s²)
• f_n – natural frequency of the system (Hz)
• ζ – damping ratio (dimensionless, typically 0.01–0.10 for structures)

The same equation gives RMS velocity and acceleration by replacing the frequency exponent: velocity uses (2πf_n)¹, acceleration uses (2πf_n)³. The calculator above uses the Miles equation with the ASD level at the natural frequency, assuming a flat spectrum near that frequency.

Worked Example

Using the default values: ASD = 0.04 g²/Hz, f_n = 100 Hz, ζ = 0.05.

1. Compute the denominator for displacement: 2 × (2π × 100)³ × 0.05 = 2 × (628.32)³ × 0.05 = 2 × 248,050,000 × 0.05 = 24,805,000
2. Multiply ASD by g: 0.04 × 386.09 = 15.4436
3. Divide: 15.4436 / 24,805,000 = 6.226 × 10⁻⁷
4. Take the square root: √(6.226 × 10⁻⁷) = 0.000789 in (about 0.00079 in)

The calculator shows 0.000789 in, which rounds to 0.0008 in. That is the RMS displacement, meaning the actual displacement will be within ±1σ about 68% of the time, and within ±3σ (about 0.0024 in) about 99.7% of the time.

When to Use This Calculator

Random vibration analysis is common in aerospace, automotive, and electronics packaging. You use the Miles equation when you have a random vibration test specification (like MIL-STD-810 or IEC 60068) that gives an ASD profile, and you need a quick estimate of how much a component will move. This is useful for:

• Checking clearance between a PCB and an enclosure
• Estimating stress on solder joints or mounting brackets
• Preliminary design before a full finite element analysis
• Verifying that a test fixture will not resonate excessively

The calculator is only valid for a single-degree-of-freedom system, which is a reasonable approximation for many lightly damped structures near a single resonance.

How to Read the Results

The RMS displacement is the statistical average of the displacement over time. It is not the peak displacement. For a Gaussian random process, the peak displacement is typically 3 to 4 times the RMS value. If your design has a clearance of 0.001 inches and the RMS displacement is 0.0008 inches, the 3σ peak would be 0.0024 inches, which would likely cause contact. Always design for at least 3σ, and often 4σ or higher for critical components.

The RMS velocity and acceleration are also useful. RMS acceleration is often compared to the input ASD to verify the response is reasonable. If the RMS acceleration seems too high, the system may need damping or a change in natural frequency.

Common Mistakes and Misconceptions

• Using the wrong ASD value: The ASD varies with frequency. You must use the ASD level at the natural frequency, not the overall average or the maximum.
• Unit mix-ups: If you use metric units, the acceleration due to gravity is 9.81 m/s², and the displacement will be in meters. The calculator above uses inches for displacement and inches/second for velocity, which is common in US aerospace practice.
• Ignoring damping: The Miles equation is very sensitive to damping. A damping ratio of 0.02 vs 0.05 changes the displacement by a factor of 1.58. Use a realistic value for your structure; if unsure, use 0.03 for bolted structures and 0.01 for welded ones.
• Assuming flat spectrum: The formula assumes the ASD is flat near the natural frequency. If the spectrum has a peak or valley at f_n, the result is approximate.

Limits of the Miles Equation

The Miles equation is a simplification. It assumes a single-degree-of-freedom system, linear behavior, and a flat spectrum around the natural frequency. Real structures have multiple modes, and the response can be higher if modes are closely spaced. For complex systems, a finite element analysis or a multi-degree-of-freedom random response analysis is necessary. The equation also does not account for nonlinearity, such as gaps or friction, which can alter the response.

For most preliminary design purposes, the Miles equation is a fast and conservative estimate, but it is not a substitute for testing or detailed analysis.

Frequently Asked Questions

What is the difference between RMS displacement and peak displacement?

RMS displacement is the root-mean-square of the displacement time history. For a Gaussian random process, the instantaneous displacement follows a normal distribution with zero mean and standard deviation equal to the RMS. The peak displacement is the maximum value over a given time period, typically 3 to 4 times the RMS for a long duration. Designers often use a 3σ or 4σ peak for clearance checks.

Can I use this for a random vibration test specification?

Yes, if you have an ASD profile from a test standard, you can use the ASD value at the natural frequency of your component. However, the test profile is usually defined over a range of frequencies, and the natural frequency may shift due to temperature or mounting. Use the worst-case ASD level in the frequency range of interest.

What if my system has multiple natural frequencies?

The Miles equation applies to each mode independently, but the total response is the square root of the sum of the squares of the responses of each mode. For a quick estimate, calculate the RMS displacement for each significant mode and combine them using the SRSS (square root of the sum of the squares) method. This is an approximation because modes are not perfectly independent, but it is often adequate.

How do I convert the displacement to metric units?

If you want the displacement in millimeters, multiply the inches result by 25.4. For a fully metric calculation, use g = 9.81 m/s², frequency in Hz, and ASD in (m/s²)²/Hz. The displacement will be in meters. Note that ASD in metric is often given in (m/s²)²/Hz, not g²/Hz; to convert g²/Hz to (m/s²)²/Hz, multiply by (9.81)² ≈ 96.24.

Why is damping ratio important?

Damping dissipates energy and reduces the response at resonance. In random vibration, the displacement is inversely proportional to the square root of the damping ratio. A structure with damping ratio 0.01 will move about 2.24 times more than one with 0.05. Accurate damping values are hard to predict; testing or prior experience is often needed.

Is the Miles equation conservative?

Generally, the Miles equation gives a conservative (overestimated) response for a single mode because it assumes all the energy at the natural frequency is concentrated in that mode. For multi-modal systems, the actual response can be higher if modes are close, but the equation is still a good starting point. Always verify with testing or more detailed analysis for critical designs.

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in asd | ASD at fn | g^2/Hz | 0.04 | 0
in fn | Natural frequency | Hz | 100 | 0
in damp | Damping ratio | - | 0.05 | 0
out disp | RMS displacement | in | sqrt((asd * 386.09) / (2 * (2 * pi * fn)^3 * damp)) | 3
out vel | RMS velocity | in/s | sqrt((asd * 386.09) / (2 * (2 * pi * fn) * damp)) | 3
out acc | RMS acceleration | g | sqrt(pi / 2 * asd * fn * damp) | 3