Vibration Displacement to Velocity Calculator

This calculator converts vibration displacement (amplitude) into vibration velocity (and vice versa) for a given frequency. You enter the displacement in microns (µm), peak-to-peak or zero-to-peak, and the frequency in Hz, and it returns the equivalent velocity in mm/s RMS, along with the peak velocity. It is based on the standard sinusoidal relationship v = 2πf·d, assuming a single-frequency, sinusoidal vibration.

Formula

The relationship between displacement and velocity for sinusoidal motion is:

v_peak = 2π × f × d

Where:

• v_peak = peak velocity (mm/s)
• f = frequency (Hz, cycles per second)
• d = displacement zero-to-peak (mm, not µm)

To convert to RMS (root mean square) velocity, which is the most common measurement in vibration analysis, divide the peak value by √2:

v_rms = v_peak / √2

If you have peak-to-peak displacement (often written as µm p-p), divide it by 2 to get zero-to-peak before using the formula.

Worked Example

Using the default values from the calculator: displacement = 25 µm zero-to-peak, frequency = 50 Hz.

1. Convert displacement to mm: 25 µm = 0.025 mm
2. Calculate peak velocity: v_peak = 2π × 50 × 0.025 = 7.85 mm/s
3. Convert to RMS: v_rms = 7.85 / √2 = 5.55 mm/s

The calculator shows 5.55 mm/s RMS and 7.85 mm/s peak, matching the steps.

Why Convert Displacement to Velocity?

Vibration measurements are taken in different units depending on the industry and the machine. Displacement (µm) is often used for low-frequency vibrations, such as shaft movement in rotating machinery. Velocity (mm/s) is preferred for mid-range frequencies because it correlates well with machine condition and is the basis for ISO 10816 severity charts. Acceleration (m/s²) is used for high-frequency vibrations like gear mesh or bearing defects.

Converting displacement to velocity lets you compare readings taken with different instruments or standards. For example, a vibration reading of 25 µm at 50 Hz might be acceptable on one machine, but the equivalent 5.55 mm/s might be in a different severity zone on the ISO chart. The conversion is essential when you have a displacement probe (like an eddy current probe) but need to check against a velocity limit.

How to Use the Calculator

Enter the displacement and frequency, then read the velocity. The calculator assumes a sinusoidal (single-frequency) vibration, which is typical for steady-state rotating machinery. If you have a peak-to-peak displacement value (common from a vibration meter), divide it by 2 before entering it, or use the formula with the peak-to-peak value and then divide the result by 2 for RMS.

For example, if a vibration meter shows 50 µm peak-to-peak at 1200 RPM (which is 20 Hz), enter 25 µm (zero-to-peak) and 20 Hz. The calculator gives 2.22 mm/s RMS. If you enter 50 µm directly, you would get 4.44 mm/s RMS, which is double the correct value.

Common Mistakes and Misconceptions

• Forgetting to convert µm to mm: The formula needs consistent units. If you use µm directly, the velocity comes out in µm/s, not mm/s. Always divide by 1000.
• Peak vs. RMS confusion: Vibration standards often specify RMS velocity, but some instruments display peak or peak-to-peak. Always check which one you have.
• Using the wrong frequency: The frequency must be the dominant vibration frequency, not the rotational speed if there are harmonics. For a machine running at 3000 RPM, the fundamental frequency is 50 Hz, but a gear mesh frequency might be higher.
• Assuming sinusoidal motion: Real vibrations often contain multiple frequencies. This conversion is only valid for a single frequency; for broadband vibration, you need a spectrum analyzer.

Limits of the Formula

The formula v = 2πf·d is exact for pure sinusoidal motion. In practice, vibration signals are rarely pure, but for a dominant frequency component, the conversion is accurate enough. The formula fails when:

• The vibration is not sinusoidal (e.g., impact or random vibration).
• There are multiple significant frequencies – you would need to convert each component separately.
• The displacement is measured at a different point than the velocity (e.g., displacement on a shaft, velocity on a bearing housing).

Also, the conversion does not account for phase or direction; it only relates magnitudes. For condition monitoring, always use the same measurement parameter and units consistently.

Frequently Asked Questions

How do I convert mm/s to µm?
Rearrange the formula: d = v_peak / (2πf). If you have RMS velocity, multiply by √2 first to get peak, then divide by 2πf. For example, 5.55 mm/s RMS at 50 Hz gives d = (5.55 × √2) / (2π × 50) = 0.025 mm = 25 µm zero-to-peak.

What is the difference between peak and RMS velocity?
Peak velocity is the maximum instantaneous velocity during a cycle. RMS velocity is the root mean square value, which is about 0.707 times the peak for a sine wave. RMS is used in most standards because it represents the energy content better.

Why do vibration meters show displacement, velocity, and acceleration?
Different parameters are sensitive to different frequency ranges. Displacement is best for low frequencies (below 100 Hz), velocity for mid-range (10 Hz to 1 kHz), and acceleration for high frequencies (above 1 kHz). Using the right parameter for the frequency of interest gives a clearer picture of the vibration.

Can I use this calculator for non-sinusoidal vibration?
No. The formula assumes a single sine wave. For complex vibrations, you need to analyze the frequency spectrum and convert each component separately, or use a vibration meter that measures velocity directly.

What units should I use for displacement?
The calculator uses microns (µm) for input, which is common in vibration analysis. If your measurement is in mils (1 mil = 25.4 µm), convert to µm first by multiplying by 25.4. The output is in mm/s, which is the standard for velocity.

How does frequency affect the conversion?
Velocity is directly proportional to frequency for a given displacement. Doubling the frequency doubles the velocity. This is why high-frequency vibration with small displacement can still produce significant velocity.

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in d | Displacement (zero-to-peak) | µm | 25 | 0
in f | Frequency | Hz | 50 | 0
out vel_rms | Velocity (RMS) | mm/s | (2 * pi * f * d / 1000) / sqrt(2) | 2
out vel_peak | Velocity (peak) | mm/s | 2 * pi * f * d / 1000 | 2