rudder, helm, wheel
柁 centers on a device that steers a vessel: a rudder, helm, or wheel used to control direction.
柁 combines 木 (wood) with 它, which likely contributes the sound. The character originally referred to a wooden rudder or helm.
The left side is wood 木, and the right side 它 looks like a steering mechanism. Picture a wooden rudder made from a tree, guiding a ship.
For ダ, imagine a wooden rudder that goes 'da' as it hits the water: da -> ダ, steering the boat.