1) Memo from the keen-grasp-of-the-obvious department:
I don't believe in miracles. The mass cannot miraculously
increase. This would violate the laws of physics. It would
invalidate the equations of motion ... so figuring out what
the subsequent motion would look like is quite impossible.
As others have pointed out repeatedly, if you allow some
mass to move across the boundary of the system, you need
to ask how it was done, and what else moves along with it:
how much energy, how much angular momentum, et cetera.
2) Even if you did manage to change the mass of a pendulum,
doing so is not analogous to changing the L in an LC
oscillator, despite the claims made in the message that
started this sub-thread.
The LC oscillator is analogous to a mass on a spring, in
ways that a pendulum is not.
Feynman said "The same equations have the same solutions."
However, the converse does not hold. Just because you
have the same solution (simple harmonic motion) does not
mean the equations of motion are the same.
3) As previously mentioned on multiple occasions, the
theory of what happens when you change the L or the C
in an LC circuit is exceedingly well understood. There
is a vast literature on _parametric amplifiers_.
You can modulate the parameters of a pendulum, too, if
you are willing to modulate the length rather than the
mass. You can demonstrate this in less time than it
takes to talk about it. I'm not talking about merely
"modeling" the system, but rather instantiating exactly
the system of interest, in every detail: a pendulum
with an adjustable length.
Also as previously described,
a) you can build a tabletop instance of this, or
b) you can go to the playground and instantiate it
by standing on a swingset and bending your knees at
selected times.
4) It is even possible to account for the damping in an
LRC oscillator -- a physically-correct quantitative
accounting -- which was first figured out relatively
recently. A relatively accessible treatment can be
found at http://ajp.aapt.org/resource/1/ajpias/v52/i12/p1099_s1