Band brakes are fairly popular in Japan. Other articles about brakes may be reached through this table of contents.
A band brake is an inside-out drum brake. Instead of brake shoes that press outward against the inside of a drum, a band brake has a flexible band which wraps around the outside of the drum. The image below illustrates the basic concept. Pulling up on the lever tightens the band around the drum.
Image credit: Nordisk familjebok, public domain, from Wikimedia Commons
Band brakes have commonly been used as parking brakes in motor vehicles. Band brakes have also been used as clutches in automatic transmissions.
The rotation of a band brake's drum tends to pull the band around it. This results in a capstan effect, which multiplies the braking effect. You can experiment with the capstan effect yourself by wrapping a rope around a tree. Hold one end of the rope in each hand. Pull on one end hard and on the other lightly. If the rope is wrapped by 1/2 turn, you can get it to slip by releasing tension on the lightly-tensioned end. If it is wrapped by 2 1/2 turns or more, no matter how hard you pull on it, it will not slip.
A Wikipedia article gives a techincal explanation of the captstan effect, but here's a simpler explanation: where the lightly-tensioned end of the rope pulls away from the tree, it is in line with the side of the tree trunk, and it isn't pressing on the tree at all -- so the tree trunk isn't pulling on the rope there, at all. After a quarter turn, the tension on the rope is pressing it directly inward against the tree trunk, so friction allows the tree trunk to resist the tension on the rope. The friction, in turn, allows the rope to resist more tension after the next quarter turn, and so forth. So, starting at the lightly-tensioned end, the tension increases as the rope wraps farther and farther around the tree. If the rope is wrapped far enough, a light pull on one end of the rope, or even the weight of the rope itself, generates enough friction to resist a hard pull on the other end.
In the image above, pulling up on the lever b (at R) pulls on the band at at the lightly-tensioned end a2 and tightens the band against the drum. The other end of the band is attached at a1 on the other side of the lever's pivot but closer so the overall effect is to pull the band tighter. Its tightness increases as it passes counterclockwise around the drum. The attachment of the highly tensioned end at a1 makes the brake self-locking: if the brake is applied strongly enough, the band will hold the lever up and keep the drum from turning until the lever is actively pushed down. This is a differential band brake, similar to our example of wrapping the rope around the tree because both ends of the band are active.
A bicycle band brake is cable-operated. One end of the band is attached to a lever, and the other end to a fixed stop, so this is a simple band brake. The band is held inside a shell attached to the axle inside the left rear dropout. In the photo below, the drum is at the upper right and the shell with the band and lever assembly is at the lower left. We are looking at the side which faces the hub. The lever attaches at 3 o'clock in the photo, and the band passes countercolockwise around the drum to the stop at 4 o'clock. The two little adjuster bolts with locknuts at 12 o'clock and 8 o'clock hold the band close to the drum, avoiding wasted motion A clamp on the left chainstay prevents the shell of the brake from being carried around, same as with a coaster brake.
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A bicycle band brake is not intended to be self-locking, but it is self-actuating -- the length and coefficient of friction of the band are chosen to multiply the force from the cable pull by a calculated amount. The drum turns clockwise as seen in the photo, and so it also tries to pull the band clockwise. A light pull on the lever at 3 o'clock results in tightness of the band against the drum which increases clockwise all the way around the band from the stop to the lever.
To remove a band brake, remove the wheel from the bicycle, remove the locknut and the shell, and insert pins (metal rods) into a vise and into two holes opposite one another in the drum. Rotate the wheel backwards to unscrew the drum.
Wikipedia article about band brakes
Wikipedia article about bicycle band brakes
Mathematical analysis of a band brake
Video of mathematical analysis of a band brake