Saab vibration

I have a 2004 Saab convertible with a strange problem. It sometimes starts and then runs with a rumbling vibration. It feels as if you are driving on a very low or flat tire, but the vibration occurs whether the car is moving or not. The vibration frequency increases with the speed of the engine. I thought at first that it was bearing in the alternator or water pump but the bearings are fine. Also it only happens about half the time. As soon as the car starts you can tell whether it is vibrating or not and will continue vibrating, if it is, until the next time it is started. The car runs fine when this happens, except that it idles at a slightly lower rpm and somewhat roughly. I’m thinking it is related to the vacuum system?

Based on the description it could be a very large number of things - i.e. the description leaves a lot out.

One probability is that the car is running poorly/idling roughly/misfiring. The first things to look at are the basics - spark plugs and wires, fuel filter, air filter. (That’s the kind of info people need). Then, if all of that is taken care of there are quite a few other things to look at such as idle air control valve, throttle body, mass airflow sensor, vacuum leaks etc. Is your check engine light on?

Then there are more structural things like engine and transmission mounts. This is actually a good possibility since it isn’t specifically associated with motion and goes w/ the engine speed. Of course some combination of this and the above is also a good possibility.

It could be a bad temp sensor or poor wire connection. The engine temp sensor drives the fuel mix until the o2 sensors heat up and come online. A bad temp would tell tell computer to mix the fuel wrong and lead to a misfire condition. Does this happen ever with a warm engine?

The check engine light has not come on, nor are there any codes associated with the problem. The car is not misfiring. The engine and transmission mounts are fine. The car has 72k miles and has had all regularly scheduled maintenance, most of it by the dealer. The dealer has not looked at this issue, however, but a mechanic I know and trust said he couldn’t figure it out despite experiencing the problem and using a stethoscope to try to fine the source.