If a hub motor (or any motor for that matter) is rated at some wattage, how do we know how far we can push it and not let it overheat?
There are several limitations:
1 - Motor integrity. Spin it too fast, it will tear itself apart.
2 - Motor heat buildup. Get it too hot, the insulation in the wires will melt (probably first), continue adding heat and eventually the copper itself will melt and short. Continue adding heat somehow, eventually the iron will melt (almost impossible). Also, more heat will mean weaker epoxies and material strengths.
3 - Motor insulation arcing. This would be from overvoltage.
#3 is almost negligible. Most enameled wire insulation is good for hundreds of volts before it will electrically arc through this to short cut to the next wire or the frame. So, overvoltage is probably never an issue unless you're doing 10x or something crazy.
#1 is dependent on voltage. More voltage means higher speeds means more structural forces. As a bonus, more speed means the spinning part of the motor brushes past air faster, helping keep it cool. As a negative, that means more "windage" losses, (air resistance) but these are miniscule. On a given motor, you can probably get away with 3-5x voltage (and thereby RPM) without the motor pulling itself apart via centrifugal forces. But it will be delicate. And stress the bearings more. And if you got a lemon that was poorly built but still plenty strong enough in normal range, it may fail. At some point, your losses increase faster than your RPM increase (efficiency per watt drops). This may be as much as 20% at 3-5x.
#2 is really the only thing worth talking about and dominates the discussion, the other two I only mentioned for thoroughness. Heat buildup comes from how much current (amps) are flowing in the wires. Note that this is cumulative heat buildup. Just about any given motor could be run at 10x wattage (call it, 3x voltage, 3x amperage), for a minute or two. You'd put 10 minutes of heat into it in 1 or 2 minutes. After that, you best let it rest for 5-8 minutes (not just normal temp, like, shut it off, drag racing style). Heat is almost the only thing that kills anything electrical, and you can always make the "continuous vs. surge" tradeoff. The motor doesn't care. Think of it like a chunk of metal under a torch.
Almost everything is overengineered, to be *sure* that it could be run at the given current for the given duty cycle (for most of us, 100% duty cycle, continuous). For example, a bridge rated for 200 tons of traffic doesn't fall apart if you put 200 tons of cars on it and then drive an extra vehicle on. (Bridges are often rated at 1/10th their actual calculated breaking point). There is no such thing as a motor than can provide X amps. Only, a motor that someone has decided is okay for X amps in Y situation. It's a somewhat made up limitation. A blender motor is "rated" at 10 amps often, because in expected residential blender use, (few seconds of vroom vroom, 5 minutes to drink the margarita), that's fine. The same motor when used as a continuous fan motor on the other hand, might be rated for 1 amp. "Is it a 10 amp or a 1amp motor then?" is what you're thinking, and the answer is both, it's a made up limitation based on expected use. A fan motor is generally 10x as big as a blender motor for the same ratings.
Helping us is that we're putting these on moving vehicles, in airflow. Generally the more we want out of them, the faster we're going, and the more airflow/cooling we're giving them. This lets us cheat a bit. Drive the same motor at the same load but sitting on rollers in a room, the motor may melt. Note that this matters for hills. Driving a 500W motor at 1000W on the flats to get extra speed yields bonus cooling. Driving a 500W motor to 1000W on an uphill yields same or slower speeds and thus less bonus cooling. What will not kill a motor at high speed will kill a motor at low speed, with power draw equal. Not a problem when obeying rated current, maybe a problem when ignoring ratings.
Here's how to think about heat limitations:
Moving motor parts have X amount of metal mass (copper and iron) which have to stay under Y temperature. That's about it. For max temperature, there's 3 things to consider. One is max temp before insulation melts. That's fairly easy and you can look it up in a book. Two is max temp before the insulation starts weakening and accelerating its way towards failure, perhaps after dozens/hundreds of heating/cooling cycles. Three is max temp where the insulation is fine indefinitely, not really undergoing thermal stresses. If you mistake one of the former for the latter, you may find your motor fails quickly, or, the worst "It was working fine, I took it on a dozen rides and then all of a sudden it failed! What the hell?" That's that #2 in play.
Still... go weigh, lookup, or guess at the mass of metal in the motor. Then look up the efficiency of the motor or test it (if it's 80% efficient, that means 20% of your output is heat energy added to the motor). 1000watts at 80% efficiency is 200 watts of energy pouring into heat. Then look up, or guess, a max temp that won't quickly lead to failure. Then figure from a cold room temp start, how much energy (presume ZERO cooling for now) you have to dump into the motor to heat up that much metal that many degrees. Basic multiplication. Power (watts) is energy per second so you pick any amount of seconds you want to dump the energy you want and that'll tell you how hard you can drive the motor to get there. That'll tell you your drag racing potential. (Motors may take 30 minutes to cool back down to air temp). One thing to note, the tiny little compact RC motors can't be short-term abused as much as bigger motors because there is no way getting around the fact that by being small they are low mass and respond rapidly to heat buildup. There's simple less metal to heat up. A cup of water in a pot on the stove boils faster than a gallon in a pot.
Then, remove the assumption of zero cooling fudge those numbers based on air cooling and try to guess at what point the temperature will stabalize at for a given load/airflow. If the difference indicates it's slightly increasing, it may take 20 minutes to stop appreciably increasing in heat when being driven at that load. One thing that helps, the higher the temp difference between the motor and ambient, the more effective air cooling is because they will rush even faster to match temperatures. The same amount of cooler air in the same time period will heat up hotter and take away more heat from the motor. So, in this respect, the penalty for overloading your motor actually decreases the more you abuse it. There's a bit of a free lunch.
Calculating cooling power requires somewhat complicated math to calculate. Math I barely know or don't really know. Fluid dynamics and rates of exchange and airspeeds and surface areas and all that and in the end even an engineer would only have a ballpark guess. You can easily calculate drag racing energy/heat. You can not easily calculate temperature stabilizing levels. So, don't bother trying unless, well, if you knew how you wouldn't be asking the question you'd be explaining the answer.
So long answer summarized... how do we know? We don't know. It just seems reasonable that they could be overdriven, we tried it, and it worked. The more you want to abuse it, the more of a gamble you're taking. Just like overloading a bridge. The only guarantee is that it will work when used at or below continuous rating.
Note that all of this heat buildup stuff applies to amps only. If you want to jack up the voltage, you'll increase amps too, (and anything you do will lead to increased amps), but not as much as getting the same power out by overloading the motor directly. With voltage you only have to be worried about insulation and structural integrity.
Extra amps, you can probably cheat by 25-50% most of the time just to account for conservative ratings. Extra voltage, much as you want (3-5x, not that you'd ever give it that much), just observe how much extra current flows when you do it.
Bonus complication: the inside of the motor, where not exposed to air, will heat up much hotter than the surface windings and laminations. You will melt the internals while the outside may still feel cool enough to touch. The math and modelling for this is delightfully messy and mostly guesswork.
Easiest way of saving your motor... throw a temperature-controlled breaker onto it (several inside every microwave) so that the motor simply refuses to allow power to flow if it gets too hot. You'll figure out experimentally how much you can punish and abuse it on average.
Complicated way of saying "it depends". And that was without actually explaining any of the math.
Make sense, sort of?