[r-t] period of swing
Stuart Flockton
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
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
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