For focal plane shutters (most common on DSLR), the shutter is two curtains. The topic for you to research is "focal plane shutter". One curtain over the sensor moves to open the exposure, and then the other one moves to close it again. The exposure time is the delay between the two curtains.
The idea is that both curtains move in the same direction, so that all areas of the sensor receive the same exposure. As opposed to in any simpler leaf shutter, one opening and then simply closing back again the other way, so that then one side of the sensor receives more light than the other side, which can be substantially more for a fast exposure time. For focal plane shutters, the sensor area is all equally exposed, but not at the same exact time (as this open slit moves down the frame).
And then the shutter speed timing does NOT depend on any mechanical movement, but only depends on the delay of the second curtain. This delay is timed by a crystal, like any digital clock.
For a slow shutter speed, one curtain opens, standing fully open for a bit, and some time later, the other curtain closes. The flash can only be fired when both curtains are fully open (called flash sync).
For a fast shutter, one curtain begins opening, and as it is still moving and still opening, the other curtain begins closing behind it. The actual exposure then is just the width of the narrow "slit" between the two curtains (a very fast shutter time in that slit). An fast flash cannot work into this narrow slit, it would Not be in sync to expose the full opening.
In your case, the speed of your curtain travel means that 1/180 second is the fastest possible shutter time that both curtains are fully open for flash. Any faster, and the second curtain has already began closing, so that then, only a narrow "slit" would be exposed at any one time by fast flash.
Incidentally, a speedlight at full power is slower than 1/1000 second. That time is conventionally expressed as a t.5 time, meaning measured as the time between half power points, but there is much more light in the other half. A t.1 measurement (when power is above 10%) is about three times longer. But flash duration is conventionally measured to t.5 half power points. It's an engineering thing, to measure vague things which are difficult to know when it decays to exactly zero. But it is not exactly what photographers would want to know.
But, speedlight means that the flash always fires at full power, and then the output is truncated or cut off abruptly to perform any lower level, like half power or 1/4 power. This means that duration of power levels less than full power are more accurate, more similar to t.1, but the one full power is t.5. This is why specs for speedlight half power and full power timings are about the same number, but with different meanings.
But regardless, for a focal plane shutter, both curtains have to be fully open when the any flash fires. Focal plane shutter are considered the best quality shutter, but have this one down side of flash sync.