Thomas A. Alspaugh
Motion of bell and rope

Bells are large and massive, even the smallest ones; the treble at my home tower is 608 pounds. There's only a limited amount by which a ringer can change what the bell was about to do on its own. Any change has to be effected by pulling the bell rope, since ropes don't push. The most effective way to control it, say the experienced ringers I know, is to spread out the pull over the entire stroke, and control how high the bell rises as a means of controlling how soon the bell will ring. Pulling harder at the beginning of a stroke will indeed make the bell swing down slightly faster and strike slightly sooner, but more significantly will send the bell up higher at the end of its swing and delay the next strike.

Figure 1. Motion varying by how high the bell rises

Figure 1 graphs (based on observation and reasoning, not a formula) the motion of two bells, one (solid blue line) ringing up nearly to the balance and the other (dotted red line) ringing up lower. Each bell strikes just before it reaches the top of its swing, when the ringer's hands are rising past his/her face. See this informative animation.

At any specific height, the higher-swinging bell is moving faster than the lower-swinging one, because on its way down the higher-swinging bell has more distance over which gravity can pull on it and speed it up. When it reaches its down or lowest position, it is moving faster than the other bell, and on its way back up it continues to outspeed the lower-swinging bell at every height. But the lower-swinging bell, despite swinging more slowly, still rings at a faster rate because it spends less time at the top of its swing. In Figure 1, the lower-swinging bell is ringing almost twice as often as the higher-swinging one, even though the higher-swinging one is moving faster (steeper slope to its position curve) at every height the lower-swinging bell reaches.

Figures 2-5 shows how a bell rings a full stroke, making almost a full circle in each direction. For simplicity, the figure shows the bell swinging up to the balance each time (steps 1 and 9) but the same relationships hold below the balance too.

1: The bell is up at hand. If the bell is below the balance, it will begin to fall on its own, otherwise the ringer begins the stroke by pulling (gently) on the rope.

2: This is the last point at which the ringer has full leverage on the bell wheel with the rope for a while. The handstroke pull acts from steps 1 to 2, over only about 1/8 of a circle, so the work it can do on the bell is quite limited. The ringer releases the sally at about this point and the bell is on its own.

3: The rope is perpendicular to the wheel. For an instant, the rope is neither rising nor falling, although the bell is in motion and accelerating downward.

4: The ringer is preparing to take tension in the rope once the bight of rope looping down to the floor has been taken up by the bell's motion. The bell is swinging fast, nearing its maximum speed.

5: The bell is down and is swinging at its maximum speed. The rope is rising at its maximum speed.

6: Sometime about this point the ringer can, if desired, start applying a pull against the rising rope in order to choke the bell or reduce its rise. The rope is still rising quickly but slowing.

7: The bell has slowed due to gravity and the ringer can slow it further if desired.

8: The ringer can continue to slow it further if desired.

9: The bell is at the balance at this point. In practice, the ringer usually will have pulled appropriately so that the bell is near the balance but not quite there. In either case, for an instant the rope is neither rising nor falling.

10: The bell has begun to come back down. The ringer keeps tension on the rope, at a minimum, so that the rope drops safely in a vertical line. The ringer may also pull harder in order to speed up the bell; this will make the bell rise higher at the next handstroke, and delay the second-next sounding.

11: The bell continues to accelerate downward. Soon the ringer's hands will have reached the lowest point of their motion, after which the bell is on its own.

12: The bell continues to accelerate downward.

13: The bell is down, and swinging at its greatest speed (as at step 5, but now in the opposite direction). The rope is (briefly) moving at its greatest speed too, but downward at this step.

14: The bell is swinging up and slowing as it rises. The ringer's hands have risen and are preparing to catch the sallie.

15: The bell continues to swing up and slow as it rises, but the rope is motionless for an instant. The ringer's hands have risen and are catching the sallie.

16: The bell is coming up to the balance and has slowed due to the action of gravity. For a brief time the ringer can slow it further if desired.

Figure 6. A full stroke