[r-t] period of swing
uhap016 at rhul.ac.uk
Wed Jun 15 22:18:03 UTC 2005
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
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.
More information about the ringing-theory