[r-t] period of swing

[Stuart Flockton]

On Tue, 14 Jun 2005, John David wrote:

> Is there a formula which will relate the period of swing of a bell to the 
> angle it is swinging through?
> 
> We were having adiscussion the other night as to how much one has to raise 
> the centre of gravity of a bell to increase the period to allow for a dodge 
> or change of direction.

The formula for the time to complete one cycle is:

tau = (2/pi)*K(sin^2(alpha/2))*tau_0

where: alpha is the maximum angle of swing;
       tau_0 is the period of the bell for small angles of swing;
       K() is the complete elliptic integral of the first kind (which can 
only be evaluated numerically).

Here is a table of values of the relative period as a function of angular
amplitude.

alpha     tau/tau_0
(degrees) 
    0    1.0000
   10    1.0019
   20    1.0077
   30    1.0174
   40    1.0313
   50    1.0498
   60    1.0732
   70    1.1021
   80    1.1375
   90    1.1803
  100    1.2322
  110    1.2953
  120    1.3729
  130    1.4698
  140    1.5944
  150    1.7622
  160    2.0075
  170    2.4394

Typical peal times correspond to the tenor striking at intervals
corresponding to a bit more than twice the small-angle time for half a
cycle, i.e. to values of alpha of around 165 degrees.

Dodging on 8 (ignoring the effect of the open lead) means varying the
time between successive blows from 9/8 of the average time to 7/8 of the
average (and vice versa), corresponding to angles of about 170 degrees and
155 degrees respectively.

This corresponds to a change in potential energy when the bell is
stationary at the top of the swing of 0.0785*mgl where m is the mass of
the bell, g is the acceleration due to gravity and l is the distance of
the centre of mass of the bell from the pivot axis.

Putting some numbers into this (weight half a ton, distance of centre of
mass from pivot equal to 0.5m) to get an idea of the scale of the force
required for a middling weight bell, gives a figure for the change of
energy when dodging of about 200 joules - i.e. corresponding roughly to a
force of about 20 kg exerted over a distance of 1 metre.

-- 
Best wishes,

Stuart Flockton