the math of hill climbing

[mschwett]

there was an interesting discussion here https://forums.electricbikereview.com/threads/another-new-tq-motor-hpr40.57935/page-13 about climbing hills on lightweight hub drive and mid-drive motors, especially the new TQ HPR40 which looks very, very promising, and @Yako is currently riding.

i haven't been riding much lately, but thought i'd revisit the question on my way home from work. i have a couple bikes - a very lightweight road bike (±14lb s-works aethos), a lightweight road e-bike (±24lb scott addict rc eride) and a 500w front-hub commuter. both the road bikes have 4iiii dual sided power meters on them, and i've ridden maybe 15k miles on them over the years, with basically every ride in strava with power, heart rate, cadence, speed, etc.

i live at the top of this hill, and have accurate-enough map and contour data. the average grade of the fat yellow segment (no stoplights) is 10.57 percent, and the distance is (obviously) also precisely known.

so, yesterday i made sure both the mahle smartBike app and my regular cycling app (which records the rider power at the cranks, heart rate, cadence, etc) were running and rode home up the hill slowly. i have health issues which require keeping my heart rate quite low, so i didn't push it, just an easy commute but under controlled conditions.

average speed was 5.99 mph. average rider power was 147 watts, and average power output from the motor was 145 watts. you can see in the following chart a couple things : the small hub drive of the x20 is unable to produce full power at very low speeds, with the maximum power limited to around 125 watts at 5mph. by 10mph, the power is very closely approaching the maximum 200 mechanical/output watts. the output at lower speeds comes very close to the theoretical maximum of the x20, which can be shortened to w = 28.8 x mph based on the relationship between torque, speed, and power. 5mph should yield as much as 144w, but 125 average is close enough. the spiky green rider power line (sampled every second) stays fairly close to the average regardless of grade. the slightly less spiky red motor power line (sampled twice every second) drops when speed drops and increases when speed increases. the little blips in speed are the flat spots at intersections. i'll try this again soon with a longer climb without intersections that i have similarly accurate data for, but i don't expect the result to be meaningfully different.

but, do we know if these power meter and mahle app values are correct!?!? let's plug the grade, total weight (i had on a backpack with a laptop, clothes, etc, so a little heavier than i'd normally be riding), estimates for drag and friction into the great cycling calculator that @Jeremy McCreary pointed me to. in this case we'll take the drivetrain loss (estimated) for the rider power out, since we shouldn't apply that to the hub motor which doesn't go through the chain and we'll reduce the rider power as input, for a total of 284w. guess what? near perfect match. i'm always a skeptic of things like this but when the main factor is gravity and weight and power, it's really super predictable. it all falls apart when you start going really fast, since aerodynamic drag is a much less forgiving mistress.

any meaningful conclusions from this? at slow speeds, you can very easily predict how fast you can go with a given combination of rider power and motor power if you know grade and weight. the science is really simple. if you know how fast you're going, you can very easily figure power, probably up to 12 or maybe 15 mph. if you know RIDER power and speed, you can easily figure out how much your motor is really putting out.... which brings me to the big question here, and one that is still a question for me. how efficienct is a small hub motor like this on steep, slow climbs? we've seen figures published in the 70-80% range. i'm not ready to make a claim here because the ride is too short to really trust the battery voltage and percentage indicators, and the battery was pretty low. i'm going to repeat this on some longer climbs at different charge levels and see if the data is meaningful. i'll say for now that based on this one, efficiency is lower than it should be.

 

[Chargeride]

With a hub you need enough power to help you keep it in its most efficient rpm, you start producing massive amounts of heat below this, the controller mosfets are turning power into heat, the phase wires are warming up and become more resistant to electrical flow , the stator coils are heating up, the magnets start to drop off in effect, the hub is getting less volts, you start to tire the rpm drops even less.
Im starting to sweat, the I feel sick, people are pointing and laughing.

This is my average ride to the shop.

 

[mschwett]

With a hub you need enough power to help you keep it in its most efficient rpm, you start producing massive amounts of heat below this, the controller mosfets are turning power into heat, the phase wires are warming up and become more resistant to electrical flow , the stator coils are heating up, the magnets start to drop off in effect, the hub is getting less volts, you start to tire the rpm drops even less.
Im starting to sweat, the I feel sick, people are pointing and laughing.

This is my average ride to the shop.

right, the question is are these small hub motors geared internally in a way to be efficient at, say, 6mph - which is around 75 rpm here. i assume it’s internally at least 5:1 on a motor this small so the range we’re asking it to be efficient in is roughly 350 to 1,150 RPM.

 

[Jeremy McCreary]

With a hub you need enough power to help you keep it in its most efficient rpm, you start producing massive amounts of heat below this, the controller mosfets are turning power into heat, the phase wires are warming up and become more resistant to electrical flow , the stator coils are heating up, the magnets start to drop off in effect, the hub is getting less volts, you start to tire the rpm drops even less.
Im starting to sweat, the I feel sick, people are pointing and laughing.

This is my average ride to the shop.

What happens when you hit a hill?
;^}

 

[Chargeride]

There is quite a steep ramp up the pavement

 

[Jeremy McCreary]

there was an interesting discussion here https://forums.electricbikereview.com/threads/another-new-tq-motor-hpr40.57935/page-13 about climbing hills on lightweight hub drive and mid-drive motors, especially the new TQ HPR40 which looks very, very promising, and @Yako is currently riding.

i haven't been riding much lately, but thought i'd revisit the question on my way home from work. i have a couple bikes - a very lightweight road bike (±14lb s-works aethos), a lightweight road e-bike (±24lb scott addict rc eride) and a 500w front-hub commuter. both the road bikes have 4iiii dual sided power meters on them, and i've ridden maybe 15k miles on them over the years, with basically every ride in strava with power, heart rate, cadence, speed, etc.

i live at the top of this hill, and have accurate-enough map and contour data. the average grade of the fat yellow segment (no stoplights) is 10.57 percent, and the distance is (obviously) also precisely known.

View attachment 198044

so, yesterday i made sure both the mahle smartBike app and my regular cycling app (which records the rider power at the cranks, heart rate, cadence, etc) were running and rode home up the hill slowly. i have health issues which require keeping my heart rate quite low, so i didn't push it, just an easy commute but under controlled conditions.

average speed was 5.99 mph. average rider power was 147 watts, and average power output from the motor was 145 watts. you can see in the following chart a couple things : the small hub drive of the x20 is unable to produce full power at very low speeds, with the maximum power limited to around 125 watts at 5mph. by 10mph, the power is very closely approaching the maximum 200 mechanical/output watts. the output at lower speeds comes very close to the theoretical maximum of the x20, which can be shortened to w = 28.8 x mph based on the relationship between torque, speed, and power. 5mph should yield as much as 144w, but 125 average is close enough. the spiky green rider power line (sampled every second) stays fairly close to the average regardless of grade. the slightly less spiky red motor power line (sampled twice every second) drops when speed drops and increases when speed increases. the little blips in speed are the flat spots at intersections. i'll try this again soon with a longer climb without intersections that i have similarly accurate data for, but i don't expect the result to be meaningfully different.

View attachment 198046

but, do we know if these power meter and mahle app values are correct!?!? let's plug the grade, total weight (i had on a backpack with a laptop, clothes, etc, so a little heavier than i'd normally be riding), estimates for drag and friction into the great cycling calculator that @Jeremy McCreary pointed me to. in this case we'll take the drivetrain loss (estimated) for the rider power out, since we shouldn't apply that to the hub motor which doesn't go through the chain and we'll reduce the rider power as input, for a total of 284w. guess what? near perfect match. i'm always a skeptic of things like this but when the main factor is gravity and weight and power, it's really super predictable. it all falls apart when you start going really fast, since aerodynamic drag is a much less forgiving mistress.

View attachment 198045

any meaningful conclusions from this? at slow speeds, you can very easily predict how fast you can go with a given combination of rider power and motor power if you know grade and weight. the science is really simple. if you know how fast you're going, you can very easily figure power, probably up to 12 or maybe 15 mph. if you know RIDER power and speed, you can easily figure out how much your motor is really putting out.... which brings me to the big question here, and one that is still a question for me. how efficienct is a small hub motor like this on steep, slow climbs? we've seen figures published in the 70-80% range. i'm not ready to make a claim here because the ride is too short to really trust the battery voltage and percentage indicators, and the battery was pretty low. i'm going to repeat this on some longer climbs at different charge levels and see if the data is meaningful. i'll say for now that based on this one, efficiency is lower than it should be.

Strong work here! Glad to see that the Gribble calculator checks out against some credible real-world data. My own spreadsheet model of these things checks out with Gribble but generally not with the more often cited bikecalculator.com.

You're right: This stuff isn't rocket science. The well-established formulas are just algebra and readily available from Gribble's site or from Wilson & Schmidt's Bicycling Science among many other sources.

Put in realistic values for gross mass (rider+bike+cargo), Crr (coefficient of rolling resistance), CdA (drag area), air density, drivetrain efficiency, and gradient, and you should get a realistic estimate of steady-state speed from a known total mechanical power input or vice versa. Getting realistic input parameter values is the trick, but @mschwett seems to have that covered for the Scott road ebike at hand.

And as you noted, the calculations are greatly simplified at climbing speeds low enough that air resistance can be safely ignored. That could be below 10 mph on an upright commuter on hybrid tires in street clothes. But when it's a legit approximation, you have an opportunity to estimate Crr or in this case, electromechanical motor efficiency Em.

Once you know Em as a function of hub-drive wheel speed or mid-drive cadence, you can take the math all the way back to the battery on the motor side.

Here we're trying to use credible empirical ride data to get at low-speed Em in the hub- and mid-drive motors used in lightweight road ebikes. @mschwett is definitely the man for the job, and I can't wait to see what he finds.

 

[mschwett]

Strong work here! Glad to see that the Gribble calculator checks out against some credible real-world data. My own spreadsheet model of these things checks out with Gribble but generally not with the more often cited bikecalculator.com.

...

Here we're trying to use credible empirical ride data to get at low-speed Em in the hub- and mid-drive motors used in lightweight road ebikes. @mschwett is definitely the man for the job, and I can't wait to see what he finds.

i was surprised to see (after you pointed both of them out to me) that bikecalculator seemed to diverge from gribble, but the real issue is just not knowing all the values they use for the drop downs. some of the CdA values are higher than i would have guessed but not unreasonable. my own experience shows that i'm even less aerodynamic than i thought, lol, so this explains why the values diverged initially.

i wonder how reliable the battery state of charge info in these bikes is. i'm sure it's all voltage based, which probably means it's really unreliable at the extremes, right?