mcwrote:
Am I right in thinking the RMS voltage should be proportional to the

frequency when this is done?

Yes, it should be, however the formula for this varies by the motor
itself and it's physical construction. What you need to do if you
intend to try this is to carefully watch current as you alter the
frequency, however there is only so far you can go before the
reactance of the stator core breaks down and the stator fails to
respond proportionally because magnetic saturation has a sort-of
flywheel effect....yes, electrons have mass too....

As mentioned before, only a switchless motor will respond to any
resonable degree because most induction motors are not meant to vary
their speeds. When you lower the frequency, you affect the magnetic
flux of each laminate in the stator and rotor cores, as well as
overall efficiency and it's power factor. Lower-frequency motors have
thicker plates because they can respond to the lower frequency and
saturate more slowly.

The best way to perform this is to find yourself a 400Hz induction
motor (see aircraft service) made to run at a comparable speed as
what you are replacing, and then work from there. Be sure to find the
highest service factor for this motor that you can find, 1.35 is an
ideal, with class H insulation (preferably class B rise with class
F-H insulation)...

While it may be entertaining to try varying the speed of an induction
motor, it is far from preferred practice. If you really want to vary
speed while taking advantage of the non-contact design of an
induction motor, go find a synchronous/stepper motor to run from, as
this will respond exactly as you would want it to up to a much wider
range than the standard induction motor, the more phases the better.
If you really wanna get trick, use a Hall-Effect motor/circuit,
alhough a bit excessive for a simple fan....

Still above and beyond is the use of a DC power motor. Brush
replacement is not nearly the factor that many think it is, as proper
maintenance/operation can provide over a decade of service. If brushes
are that much a biog deal, opt for a brushless DC motor...

If ya wanna try something fun, take an alternator and short it's
rotor's slip-rings and run it as a low-impedance 3-phase induction
motor after you remove all the DC components from the circuit. Over
80 amps is usually wound delta, so keep his in mind...

Whatever you do, best of luck to ya, but I still strongly recommend
the DC power motor, like the treadmill motor.