Let's stick strictly to PM DC motors as that is what you asked about. Don't apply this to any other type.
The rated voltage is the figure the manufacturer intends you to use to obtain the speed and power specified on the label. You can run at other voltages as long as you accept limitations. You may not be able to increase volts much because you may exceed a safe speed and you may cause serious sparking on the brushes. Most can tolerate 20 to 50% over voltage with an increase in power, but you must not exceed the current rating as that determines how hot it gets.
Within reason you can use as low a voltage as you like as long as you stay within the current rating, this implies that the power out falls with voltage. At lower volts the winding resistance has greater effect and the speed drop on load will be progressively higher as you reduce volts. At a certain low voltage ( very low), the internal volt drop will be so great that it will fail to turn.
When using it as a generator much the same applies except that the internal drop causes a falling in voltage with load when driven at constant speed. You are again more or less limited to the rated current so power out is small at low speed and voltage. You have to accept a fall in rated output and efficiency as you use it at lower volts.
On no load the speed is fairly closely related to speed as a motor. As a generator the open circuit voltage is directly proportional to speed. These relationships are significantly modified on load by the winding resistance.
Flux
