# Converting PSD against spatial frequency to RMS against temporal frequency

I am putting together some simple ride comfort models that use road surface PSDs from ISO 8606 (road surface profiles) as inputs to a quarter car suspension model to produce body acceleration spectra. These are then weighted according to ISO 2631 (human exposure to whole body vibration).

The road surface data is vertical displacement PSDs referencing spatial frequency (wave number; inverse of wavelength). The suspension model is simply a gain of body displacement per unit of ground displacement over a range of temporal frequencies. The input to the weighting process needs to be an RMS acceleration with respect to temporal frequency. So, the problem is to convert the road surface profile equation from a PSD against spatial frequency to an RMS against temporal frequency.

The following is the process I am using:

1. Convert road surface displacement PSD to acceleration PSD (given in ISO 8608)
2. Convert road surface acceleration PSD to acceleration RMS (still with respect to spatial frequency)
3. Convert road surface acceleration RMS to displacement RMS by dividing by square of spatial frequency
4. Convert road surface displacement RMS to acceleration RMS by multiplying by square of temporal frequency
5. Multiply by suspension gain to get sprung mass acceleration
6. Weight sprung mass acceleration using filter described in ISO 2631.

However, the sprung mass acceleration results (pre or post weighting) are rather high, so I just wanted to check whether the above process was sound. The units seem to check out so that gives me some confidence.

Can anyone confirm for me that steps 2-4 in the above process are correct?

Thanks.

• There's a lot of information here but it's really not clear to me what you're asking. Can you perhaps edit this to distill down your question, or provide a brief summary of what you need to know? Feb 28, 2022 at 16:14
• Hello @MichaelSeifert. Sorry if it's long winded, I did try to compress it as much as possible but have obviously failed to get my point across. The crux of the problem is that I want to convert a PSD of road displacement against spatial frequency to RMS road acceleration against temporal frequency. I am using the process described above but in doing so I'm not convinced the units are correct, thereby suggesting the process isn't correct in some way. Does that help? Happy to explain further if needed. Simon. Mar 1, 2022 at 7:55
• @MichaelSeifert. I've greatly simplified the question now. I hope it makes more sense. Mar 15, 2022 at 9:07
• Wouldn't you need to assume a given vehicle speed in order to establish a time-base to calculate the temporal frequency? The speed (time) that you traverse the roughness profile of the road (the spatial frequencies) determines how hard the pothole or bump feels. Drive slow it's not bad; drive fast and it knocks your teeth out. Jul 14, 2022 at 1:26
• Consider to spell out acronyms. Jul 14, 2022 at 1:38