Nope, IIUC, nothing changes except the increased mass. Look at the ODE.
I still think you are seriously overinterpreting the ODE.
There are lots of different ways of writing "the" ODE.
There exists a set of ODEs with the following contrasting
properties:
A) They are all equivalent when it comes to representing
the time-evolution of the oscillator. You are free to
chose any one of them. That's fine as far as it goes.
B) They are *not* equivalent when it comes to modeling what
happens when you take a stroll through parameter space,
e.g. by changing the mass.
Specifically:
-- You could assume constant absolute energy loss per cycle.
-- You could assume constant percentage energy loss per cycle.
... etc. etc. etc. etc.
These assumptions are harmless during task (A), but will lead
you astray during task (B).
This is what we call equation hunting. The student complains,
"This equation was correct for last week's homework, how dare
you tell me it's not correct for this week's homework?!?!!!?"
Seriously: The ODE is not the physics. It is at best an imperfect
model of the physics. Just because you used a certain equation
successfully every day for the last ten years doesn't mean it
will hold up if you ask it to do something new.
Saying "nothing else changes" is a non-starter. You cannot
change one thing without affecting lots of other things. The
number of plausible variables vastly exceeds the number of
degrees of freedom.
In particular: It's not very polite to ask people to "look at
the ODE" and reverse-engineer it to figure out what assumptions
you are making. It would be better to be upfront about which
assumptions you intend to make, and then demonstrate that this
particular way of modeling things captures those assumptions.
This process of clarifying which assumptions you intend to make
may even allow you to answer your question for yourself.