Vibration Displacement Calculator: Formula & Examples

This calculator works out the displacement (peak amplitude) of a vibrating object from its frequency and either its velocity or acceleration. You enter the frequency and one measured motion value, and it returns the displacement in the unit you choose, along with the corresponding velocity or acceleration.

Formula

For simple harmonic motion, the relationship between displacement, velocity, acceleration, and frequency is:

Displacement (peak) = Velocity (peak) / (2 × π × Frequency)

Where:

• Displacement is the peak amplitude of motion, in meters (m) or millimeters (mm).
• Velocity is the peak speed of the vibrating object, in meters per second (m/s) or millimeters per second (mm/s).
• Frequency is the number of cycles per second, in hertz (Hz).
• π ≈ 3.14159.

If you start from acceleration instead, use: Displacement = Acceleration / (2 × π × Frequency)² (with consistent units).

Worked Example

Using the default values from the calculator above:

1. Frequency = 50 Hz, Velocity = 10 mm/s.
2. Convert velocity to m/s: 10 mm/s = 0.01 m/s.
3. Angular frequency ω = 2 × π × 50 = 314.159 rad/s.
4. Displacement = 0.01 / 314.159 = 3.183 × 10⁻⁵ m = 0.0318 mm.
5. The calculator gives 0.032 mm (rounded to 3 decimals).

When to Use This Calculator

Vibration displacement is a key parameter in machinery condition monitoring, structural health monitoring, and product design. You would use this calculator when:

• You have a vibration velocity reading from a sensor or data sheet and need the physical movement amplitude.
• You are comparing vibration limits that are specified in displacement (e.g., microns) but your instrument measures velocity.
• You are designing a mounting or enclosure that must accommodate a certain vibration amplitude without impact.
• You are troubleshooting excessive vibration and need to convert between units to match a standard.

The calculator assumes pure sinusoidal (single-frequency) vibration. Real-world vibration often contains multiple frequencies, so the result is only valid for the frequency you enter.

How to Read the Result

The displacement value is the peak amplitude—the maximum distance from the neutral position in one direction. Some standards use peak-to-peak (twice the peak), or RMS (root mean square, which is peak divided by √2 for a sine wave). Always check which convention your measurement or standard uses.

For example, a peak displacement of 0.032 mm corresponds to a peak-to-peak displacement of 0.064 mm and an RMS displacement of 0.0226 mm. Mixing these up is a common source of error.

Common Mistakes and Unit Mix-Ups

• Using peak when you need RMS: If your velocity reading is RMS, multiply it by √2 to get peak before using the formula (for a sine wave).
• Forgetting to convert mm to m: The formula works in any consistent unit, but if you mix mm/s with Hz, the result will be in mm only if you keep the same length unit. The calculator above handles the conversion.
• Using frequency in rpm instead of Hz: Divide rpm by 60 to get Hz.
• Assuming the formula works for non-sinusoidal vibration: For complex waveforms, the relationship is not exact; you need a spectrum analyzer or time-waveform analysis.

Limitations of the Formula

The simple harmonic motion relationship is exact only for single-frequency sinusoidal motion. In practice, rotating machinery often has harmonics and noise, so the displacement calculated from a single velocity reading may not represent the true peak displacement of the complex waveform. For accurate peak displacement of complex vibration, you should use a vibration meter that measures displacement directly, or integrate the acceleration signal with proper filtering.

Also, the formula gives the displacement at the specified frequency. If you have a broadband vibration signal, you must know the frequency content to interpret the result correctly.

Alternatives and Related Measurements

Instead of calculating displacement from velocity, you can measure it directly with a proximity probe (eddy current sensor) or a laser vibrometer. These are often used in rotating machinery to measure shaft vibration relative to the bearing housing. Another alternative is to measure acceleration and integrate twice to get displacement, but this requires careful high-pass filtering to remove low-frequency drift.

For most field work, vibration velocity is the preferred parameter because it correlates well with machine condition across a wide frequency range. Displacement is more useful at low frequencies (below 10 Hz) where velocity and acceleration values become very small.

Frequently Asked Questions

What is the difference between displacement, velocity, and acceleration in vibration?

Displacement is how far the object moves from its rest position (in mm or mils). Velocity is how fast it moves (in mm/s or in/s). Acceleration is how quickly the velocity changes (in m/s² or g). For a sine wave, these are related by the frequency: velocity = displacement × 2πf, and acceleration = velocity × 2πf.

Can I use this calculator for any type of vibration?

It works best for steady, single-frequency vibration (like an unbalanced rotor). For random or impact vibration, the relationship still holds for each frequency component, but you need to know the frequency spectrum to get a meaningful total displacement.

What is the typical displacement for a machine vibration?

It varies widely. A small electric motor might vibrate at 0.01 mm peak, while a large slow-speed pump could have 0.5 mm peak. Standards like ISO 10816 give limits in velocity (mm/s RMS), not displacement, so you often need to convert.

How do I convert displacement from mm to mils?

1 mil = 0.001 inch = 0.0254 mm. So to convert mm to mils, divide by 0.0254. For example, 0.032 mm = 0.032 / 0.0254 = 1.26 mils.

Why does my vibration meter show velocity but the specification says displacement?

Many vibration meters measure velocity because it is a good overall indicator, but some standards (especially for low-frequency vibration) specify displacement. Use the formula above to convert, but make sure you know whether your meter reads peak, RMS, or average.

What is the difference between peak and peak-to-peak displacement?

Peak displacement is the maximum distance from the center (zero) position. Peak-to-peak is the total distance from the most negative to the most positive extreme. For a symmetric sine wave, peak-to-peak is twice the peak. Some displacement sensors (like proximity probes) naturally measure peak-to-peak.

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in f | Frequency | Hz | 50 | 0
in v | Velocity (peak) | mm/s | 10 | 0
out disp | Displacement (peak) | mm | v / (2 * pi * f) * 1000 | 3
out accel | Acceleration (peak) | m/s² | v * 2 * pi * f / 1000 | 3