Will the 12V cooling fan be damaged if it falls below the rated voltage?

The commonly used cooling fan voltage is usually 12V, but like temperature controlled cooling fans, the voltage setting for the cooling fan is relatively low at low temperatures, resulting in slower fan speed and lower noise. In addition, some customers have designed three different levels of air flow in order to design three different voltages below 12V. Will the cooling fan be damaged if it is not operated within the rated voltage range for a long time?
Usually, for a 12V cooling fan, as long as the supplier can ensure normal operation at low voltage and smooth start-up, there is no problem with a voltage lower than 12V. If a high speed 12V cooling fan can operate at a voltage as low as 5V, or even 4.5V, the sound will be quieter at this time
However, a 12V cooling fan should not be used at a voltage greater than 10% of its rated voltage. For example, if working at 13.2V, it is likely to exceed the power and cause the cooling fan to burn out. Especially for some high air volume and high speed cooling fans, if it is possible to exceed the rated voltage, this requirement needs to be provided to the cooling fan supplier to increase the voltage range.