Citation Captain reports compass heading errors on both pilots' heading indicators during a very turbulent departure from MDW; resulting in a 40 degree heading error. Once clear of the Class B; an INS position update corrects the compass errors.

Narrative

We departed Runway 4L at MDW with winds blowing out of the north at 25 KTS gusting to 45 KTS. There were no annunciators or flags and ATC requested a left turn to 250 after departing. All directional gyros (DG) were aligned with the runway. The extreme turbulence; rapid frequency changes; and cleaning up the aircraft associated with normal takeoff focus were all in play at the time. About 3 to 4 minutes into the flight; ATC asked what heading we were assigned and we said; '250.' We were asked to turn to a heading of 210 which we did. ATC notified us that although we were on a heading of 210; we were tracking about 250. At that time we notice our 'CMPTR WARN' light was on. We don't know exactly when it came on. There are 7 parameters that are measured and can set it off; one of them being 'Heading.' We were given 'degree turns' such as turn left 20 degrees; etc. until we were clear of the Chicago Class B. We decided to work the problem when we got further out as it was then VFR just to the west of the MDW area over Aurora. Again; there were no flags of any kind showing throughout the flight. Once we were clear of the area; we then confirmed that our directional gyros were showing an incorrect alignment. Once we flew over a known waypoint; we 'repositioned' the inertial reference units and the DGs properly aligned themselves. We do not know what caused the abnormality and we are talking with Honeywell to explain how a DG can turn or not turn during flight. Although we were never in conflict with any traffic (we have TCAS II on board and no conflict was alerted) and no emergency maneuvering had to take place; we felt it necessary to file this report still in the event we are questioned about our flight path and the reason why it had occurred.