What time does the wings open and close? Or is it at random moments?

When the wind is below 20 miles an hour, the Burke Brise Soleil ("wings") opens when the Museum opens (10am Tuesday-Sunday), "flaps" (lowers and raises again) at noon, and closes when the Museum closes (5pm Tuesday, Wednesday, Friday-Sunday, 8pm Thursdays). They'll go up for special events at the Museum, such as MAM After Dark and weddings. When the wind is high, it automatically lowers or stays shut -- to protect the spines as well as to keep the glass in Windhover Hall safe. So if the weather's rough, it may seem a bit more random.

I think the wings open at 10 am and close at 5 pm