# Acceleration time of unloaded motor

I want to use a motor to generate musical tones by spinning a disk. The disk will have optically printed sound waves on it.

I would like to know how fast the motor will be able to change speed from one note to another.

The motor will be driven by 3 phase variable frequency drive.

So for example how fast will it be able to double its rpm from 100 to 200 if I instantaneously double the drive frequency?

We can ignore the inertia of the disk for now because I don't know it yet. It would be good to just know a general way to solve this problem, then I calculate for a different inertia.

The motor I am thinking of using is 3 phase brushless slotless type

From the data sheet

Speed/torque gradient 5.21 rpm/mNm

Nominal torque (max. continuous torque) 94.6 mNm

Nominal voltage 24v

Nominal current (Max continuous current) 7.58A Stall torque 3220 mNm

Rotor inertia 33.3 gcm²

Torque constant 13.6 mNm/A

Speed constant 700 rpm/V

Mechanical time constant 1.82 ms

$$M=J\alpha$$
$$\alpha=\frac{M}{J} = \frac{0.0946}{3.36*10^{-6}}\approx 28000\;rad/s^2$$ $$dt=\frac{d\omega}{\alpha}= \frac{100*2\pi}{60*28000} \approx 0.4ms$$