# Vibration characterization: how to convert acceleration noise density to displacement pk-pk value?

A spec characterizes the vibration stability by acceleration noise density

$$Na < \dfrac{1\text{ μg}}{\sqrt{Hz}}$$

for 1-100 Hz, where g=9.8 m/s2. But usually what I see is something like 3.5 nm pk-pk.

My question is, how to convert these two?

I tried this as below, but not sure if it's correct, any comment is appreciated.

Displacement noise density (is this correct?)

$$Nx = \frac{Na}{4\pi^2f^2}$$

Thus

\begin{align} x^2 &= \int (Nx)^2 df = \int \frac{(Na)^2}{16\pi^4f^4} df \\ &= \frac{(1\mu g)^2}{16\pi^4} \int_{1}^{100} \frac{1}{f^4} df \\ &\approx 2 \times 10^{-14} m^2 \end{align}

Conversion between $g/\sqrt{hz}$ and $m/\sqrt {hz}$ is possible. However, converting from that to m pk-pk is usually not possible. The reason is that spectral density is a unit used to quantify random vibration, and m pk-pk is a unit used to quantify sinusoidal or periodic vibration. Random and periodic vibration simply are not the same thing. For specific applications, there are ways to come up with equivalences between them. But in the most general sense it is apples and oranges. I have some tutorials on my webite that may help explain this concept in more detail http://mechanicalvibration.com/Power_spectral_density.html