We describe an optical phenomenon of unmovable moiré patterns in sliding (moving) grids and gratings. The phenomenon was observed visually in the planar straight movement of the black-and-white gratings with a period of several mm. This is a velocity-independent effect confirmed analytically and in a computer simulation based on the spatial averaging. We found the static directions of the moiré patterns in the regular grids, but our technique can be also applied to other objects. The orientation and period of the static moiré patterns are not obvious, especially in the presence of the distance effect. The phenomenon can be practically used in security applications.