hammer inertia

[Stephen Birkett]

> When the hammer swing about its pivot, is its pivot still in the flange? Does
> this mean that we're observing both the hammer moment of inertia and the
> friction at its center? When this lever applies its when on the next below,
> is there some way of calculating the distribution of that moment of inertia
> on either side of the second arm of that lever (here the knuckle, and with
> the repetition, the felted profile)? We all hunger for more than an educated
> guess at a lever train's inertia.
>

Here's a few details on how to measure the moment of inertia with
respect to the pivot for a hammer/shank assembly.

A solid object allowed to swing on a pivot is known as a compound
pendulum. To apply this to the hammer/shank remove it from the
hammer rail and hold it by the flange, letting it swing freely
vertically. The pivot is then where the shank is attached to the
flange. The various parameters are:

  I = moment of inertia with respect to shank pivot
  T = period i.e. time for one repetition or swing (sec)
  L = length from pivot to centre of mass (hammer/shank assembly)
  m = mass of hammer/shank assembly
  g = grav. accel. 9.8 m/s^2

Provided the oscillations are kept reasonably small (less than about
1/4 the length of the shank) the period of oscillation is

  T = 2 pi sqrt(I/mgL)

where sqrt is square root. Re-arrange this to give the mom. of
inertia:

  I = (m g L T^2) / (4 pi^2)

This includes the effect of the hammer shank swinging about the
pivot, as well as the hammerhead. I've never measured this but I'd
expect the bass hammers behave pretty much like a simple pendulum,
with all the mass concentrated at the hammer centre. For treble
hammers the shank may make a difference to I.

Friction will not affect the evaluation of I. Acceleration of the
hammer/shank assembly due to application of a force at the knuckle,
will depend on the position of the knuckle and friction. Moment of
inertia depends only on the manner in which the hammer/shank mass is
distributed with respect to the pivot point.

Stephen Birkett (Fortepianos)
Authentic Reproductions of 18th and 19th Century Pianos
Waterloo, Ontario, Canada
tel: 519-885-2228
fax: 519-763-4686