How low can the frequency dip , before appliance motors start to have problems ? At work , we set underfrequency relays for 59.3 hz but that is not a good indicator of what I'm asking . My question relates more to the use of portable generators and the stability of the frequency .
There is no simple answer to that.
If an appliance uses only resistive heating elements, such as a toaster or simple oven, the answer is never. Frequency is relatively unimportant to a purely resistive load. In fact it could even be 25Hz, or even DC. Ever notice that an incandescent light bulb does not say AC or DC?
If the appliance has an SMPS (Switch Mode Power Supply) to run electronics, most likely it was designed for use anywhere in the world, so 50Hz -10% would be typical as a lower threshold. Below that and the little transformer that is part of the SMPS would start to over heat.
Linear power supplies, found in many older TV sets and radios, would be frequency sensitive and likely over heat with low frequency. They will typically have a limit printed on the nameplate data.
If the appliance has a motor, then it will depend a lot on loading. Since mechanical HP (shaft kW) is a function of speed and torque, if your speed drops, so does the shaft HP. If the load is essentially shed with a drop in speed, i.e. a centrifugal pump on a washing machine, then probably the overall effectiveness of it's primary task may suffer, but no harm to the machine. But f the loading does not diminish with speed, such as the drum of a dryer, then the motor will more easily overload.