I assume you think a DC motor that has
- ideal non-saturating (=linear) magnetic path
- perfect commutation
- permanent magnets
- some resistance R
Let's rewrite your idealized equation without divisions which are not handy without math typesetting:
U = I * R + A * W
where U=the input voltage, I =the current drawn by the motor, R = total circuit resistance, A = the structure and material constant of the motor, W = the rotation speed as radians/second.
Note: A * W is the induced voltage.
Elementary electrodynamics give to us another equation between the torque T and the current:
T = A * I
This torque is needed for 2 purposes:
- to accelerate the rotation speed of the motor and possible external mass joined to the axis
- to win the frictional losses inside the motor and in the load
If there's no frictional losses, the rotation speed increases until the induced voltage is as high as the input voltage. The motor draws no current and the rotation speed is constant (= U/A). R means now nothing because I = 0.
If you have some friction, the behaviour of the system depends greatly on how the friction is related with the rotation speed W.
If the friction is a constant torque Tf that does not depend on the speed, the motor takes a current I = Tf/A. The achieved speed is less than U/A, it's (U - I * R)/A
In practice the friction torque Tf is at first highest in the startup, then drops when the rotation has started and finally increases when the rotation speed increases. This makes it hard to calculate exactly the achieved speed and the current. But it's surely possible if the relations are known.
Practical motors have some internal tricks that try to reduce the speed loss caused by the friction and resistance. One elementary trick is to add an extra magnetization coil and put the motor current go through it. This makes A non-constant but designing it carefully the drop of the rotation speed can be kept substantially smaller at moderate torques.
It's well possible that the speed loss caused by external 3 Nm torque is so small that you have not noticed it. This is possible even if there's no tricks applied to reduce the loss of the speed. Unfortunately I have no proper motor specs to check it.