Power calculation from torque 1

[BruceDiesel]

Hi All,

I hope this is the best forum for this question - because it is not directly related to electrical motors, but I believe this would be the best place to find the answer.

The problem I have actually relates to the calculation of power based on varying torque and angular velocity (on a bicycle of all places).

Background:
A power meter on a bicycle measures a cyclists power output by measuring torque (sampling it at around 70Hz and averaging those 70 samples) then multiplies this average by angular velicity derived from the rpm to provide a power reading once per second.

It is common that a cyclist experiences difficulty in achieving the same power outputs when riding on an indoor trainer (which uses a flywheel and applies a resistive load to the rear wheel) as apposed to riding outdoors.

My hypothesis is that the torque pattern changes between these two environments (do to the removal of the mass damping effect that the riders body mass provides) and that using an arithmetic mean to calculate the average torque per rotation is in fact not correct, and that the torque for the rotation should be calculated using a root-mean-square.

It's been 20 years since my undergraduate days so I'm dusting out the cobwebs

 

[Skogsgurra]

Yes, you are right. It is a mistake to average torque and then multiply. Torque is an "instant" variable and can change a lot in a tenth of a second. Speed is more "average". So, if you multiply instant torque with actual speed and multiply with time between samples and then accumulate these "energy chunks", you get total energy.

If you want average power, just divide energy into time. And if you want instantaneous power, just use your torque times speed products.

No root mean square should be used.

Gunnar Englund

http://www.gke.org

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100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...

 

[BruceDiesel]

Okay, so I am actually wrong in saying that rms should be used to calculate average torque, then multiply Trms by speed to get average power.

I don't know currently if the power meter is averaging torque first then multiplying by speed, or whether it is multiplying each torque sample by speed to get instantaneous power, then averaging instantaneous power.

We are just observing that the cyclist is not able to achieve the same power for the same duration when riding indoors - and as far as I can see, the only physical difference between the two systems is a change in mass damping.

 

[Skogsgurra]

Yes, I think that RMS is wrong in this case. As you know, RMS is used to calculate power from an intensity unit (like voltage, current, etcetera) where power is equal to value squared times a constant. In your case, power is not equal to torque squared but to torque times speed (or rather radians/second). RMS, useful as it is in other applications, can not be used here.

Gunnar Englund

http://www.gke.org

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100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...

 

[cswilson]

Skogs is correct (as usual). Let's go back to basics to see why.

Mechanical power delivered at any instant is torque times angular velocity. Both are signed quantities, so the direction depends on whether they have the same sign or different signs. In your case, you are only worried about the case where both are "positive". Energy transfered over a time interval is the power integrated over that time. Average power delivered in that time interval is the total energy divided by the time. Fundamentally, that's it -- period.

Let's look at an electrical analogy to see how you can get confused. You have a "black box" electrical load with two leads. You can measure the voltage across the leads and the current into a lead. Electrical power delivered at any instant is (signed) voltage times (signed) current. Energy transferred is power integrated over time, and average power transferred is this energy divided by the time. This is true for any pattern of voltage and current - AC or DC - and any electrical load (but not all three are independent, of course). You don't even have to know what the load is.

Now let's look at the case where you've got a fixed (frequency and voltage) AC input and a simple resistive load. Instantaneous current is simply instantaneous voltage divided by the resistance. In this case, voltage and current change sign at the same time, so power transfer is in the same direction all the time. If you have voltage and current measurements (and can process it fast enough), you can calcuate instantaneous power, net energy, and average power as explained above.

However, for steady-state linear AC circuits, RMS calculations simplify things enormously. If

V = A sin(wt)

then

I = (A/R) sin(wt)

and

P = (A^2/R) sin^2(wt)

With the time-averaged value of sin^2 being 0.5,

Pavg = 0.5 (A^2/R)

If the AC quantities are described by their RMS values, you can simplify these calculations. The RMS value of a sinusoid is the peak divided by sqrt(2). So

Vrms = A/sqrt(2)

and

Irms = A/R/sqrt(2)

so

Vrms*Irms = A^2/R/2 = Pavg

Remember that this is just a convenience to shorten the calculations in these cases.

Curt Wilson
Delta Tau Data Systems

 

[LionelHutz]

I would believe that the torque and speed are both varying in a non-linear fashion over time so you can't sample for a period of time then use the average of one value times the other value to get power.

I'd think you need to calculate power on a sample by sample basis and then average out the samples. Maybe some type of moving window to calculate the power for, say, the last minute or 5 minutes. You don't sum up these values over time or use a delta time multipier - the rpm already includes the time.

RMS is a statistical meausre of the magnitude of a varying quantity or to put it simply it just calculates the area under the curve.

 

[itsmoked]

A frictional riding machine is very unlike a bicycle ride. The human pumping pedals on a bicycle has to do it thru a the cranks. As the cranks go around linear motion is converted to circular motion. This is NOT a smooth process as the leg geometry and pedal geometry are cyclically changing constantly. At certain points the cyclist has horrible leverage and at other points great leverage. For example when the pedals are up/down the rider has no leverage at all.

As one rides a bike their entire moving mass 150lbs (more unfortunately in my case) is there to provide an inertial average(storage) of the full pedal rotations. Even though the rider is not accelerating, or even able to accelerate twice per rotation, there is still the smooth pedal motion and the sensory feel of moving smoothly.

Now on an exercise machine you end up with a load that is totally non cyclic. It is there always! No variation. And there is much less inertia available. This means those two zero points in a pedal stroke result in rapid variations in rotary motion/feel. That's why the "machines" feel so hard to pedal.

Can your measurements be missing the fact that there are actually "zero points"?

Keith Cress
Flamin Systems, Inc.-

http://www.flaminsystems.com

 

[cswilson]

Lionel, you say, "RMS is a statistical meausre of the magnitude of a varying quantity or to put it simply it just calculates the area under the curve."

This is NOT correct! RMS is the (square) Root of the Mean of the Square of the signal in question. It is not the "area under the curve".

The mathematical expression for the voltage signal out of a North American residential socket is:

V(t) = 170 * sin(2*pi*60*t)

The average voltage (area under the curve divided by time)
is zero.

The RMS voltage is 170/sqrt(2) = 120Vrms

The RMS value is useful for things like power calculations. If you put a 144-ohm load across this, you can calculate the average power as:

Pavg = Vrms^2 / R = 14,400 / 144 = 100W

Using RMS values makes power calculations for these (zero-centered) AC signals into linear loads as easy as for DC signals. That is, 120Vdc through a 144-ohm load also would have 100W of power transfer.

Bruce, I don't think your method of sampling torque 70 times per second and averaging, then multiplying by a single speed measurement for that second (which is probably some kind of effective average for the second anyway) will cause significant errors, because these quantities will not change that quickly.

As a side note, I've never felt that stationary cycles capture the "feel" of a real bike's loading on my legs, and that I'm not as "efficient" on a stationary cycle.

Curt Wilson
Delta Tau Data Systems

 

[BruceDiesel]

The torque curve is quite close to sinusoidal, with the dead spots happening at 12 o'clock and 6 o'clock - naturally where the rider has the least leverage.

At this stage, the common theory to explain this phenomenon is rider motivation is not as good riding indoors as outdoors. However, our experiences show a fairly constant discrepency - even when this is done by professional cyclists who are focused athletes. The discrepencies can be as much as 10%.

Googling around for the term "RMS Torque" I see that it is used when rating motors that have varying torque cycles as the equivalent constant torque that causes the motor to heat up to the same temperature.

The only physical difference that I can see between the two environments is the difference in inertial load. We are actually using the same bicycle, with the same power meter, indoors and outdoors. The only impact I can see that a difference in inertial load can have, is on the shape of the torque wave during the pedalling rotation, hence my theory that we are actually measuring power incorrectly.

I put together an example to try to confirm my hypothesis, using two extremes.

When riding on the road, with the mass damping effect of the rider, lets say the torque samples look like this

0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1

average torque for this wave is 2, rms torque is 2.34

Now, when riding on the indoor trainer lets say the samples look like:

0 0 0 4 4 4 4 0 0 4 4 4 4 0 0 0

average torque for this wave is 2, rms torque is 2.82

This is an extreme example to illustrate a point.

A 25% difference. Does this explain the difference that the athlete experiences between the two environments?

 

[Skogsgurra]

Hi Bruce,

Your goals and definitions need stiffening up a bit. Are you interested in calculating power or are you interested in studying the effect that a variable power has on the human body?

The former case is clear-cut and there are no doubts as to what methods to use. The latter case is something entirely different.

If losses or fatigue in the human body are proportional to developed power squared, then your example is probably valid. Using RMS will then be an acceptable and well established method. But, if losses follows some other function of develpoed power, then you have to find and use some other way of calculating "specific load" or whatever the name for that may be.



Gunnar Englund

http://www.gke.org

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100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...

 

[waross]

I think that you may be on the wrong path alltogether.
When a rider is on a bicycle his body tends to stay verticle and the bicycle oscilates from side to side. In some cases, when the rider is striving for maximum acceleration, the side to side movement of the bicycle is extreme.
The vector force of the arms on the handle bars will have some effect, but basically, when you put all your weight on one foot, If your body's center of gravity is not over your foot print, you will fall down. On a bicycle, if your center of gravity is not over the center of balance (the center line when going straight ahead) you and the bike will fall.
A bicycle rider has four ways to compensate for the transfer of pressure from one side to another;
1> By moving the bicycle from side to side. (Low mass involved)
2> By moving his body from side to side. (High mas involved)
3> By swerving slightly from side to side. (Almost no mass involved)
4> Varying the force on the handle bars from one side to the other.

On the machine, the rider may only compensate and get maximum pedal pressure by moving his body from side to side, and by varying pressure on the handle bars.
Much of the effort exerted on the handle bars may be to move the body from side to side.
When a rider is pumping hard, and swinging the bicycle from side to side, and swerving slightly, virtually all of the handle bar pressure is in line with the balance points and is making the maximum contribution to increased pedal pressure.
I humbly suggest that these effects may a significant part of the explanation as to why a cyclist is able to preform better on the road. His range of options to transfer energy efficiently to the pedals are limited on the machine.
Just the energy wasted moving the body from side to side on the machine may be a large part of what you are trying to resolve as a measuring error.
A question? Do you get a close correlation between the road bicycle and the machine at low levels of exertion when the rider is seated comfotably on the seat, and progresively greater discrepancies as the exertion level increases?
If so, I rest my case.
respectfully.
Ps; re; rms torque. For a constant speed, horsepower is related to torque, and rms horsepower is used to determine an electric motor's suitability to handle a cyclic load that at times may exceed the horsepower rating of the motor.

Edward H. Cowern P.E. of Baldor Motors has prepared an excellent series of papers on electric motors. He describes the RMS horsepower method of predicting motor heating under cyclic loads and overloads.
Available at

http://www.baldor.com/pdf/literature/PR2525.pdf

"COWERN PAPERS"

On the machine, the only

 

[BruceDiesel]

Hi waross

The discrepency is happening at what we call the Functional Threshold Power - which is the power that is sustainable for 60 minutes. Which is porbably about 20% - 30% of peak power - i.e. power produced in a sprint. At this power, the upper body is very still, and the arms are relaxed. In fact the same power can be achieved on a Time Trial bike - in which position the elbows are actually resting on the handlebars and there is very little if not zero side-to-side movement.

skogsgurra: I posted a lengthy discussion yesterday regarding the relevance of power etc - trying to answer your questions, but I see it is not on the forum? Not sure why, but I will post it again today!

 

[waross]

Hello BruceDiesel;
Thank you for your reply.
Can you tell us a little about your instrumentation? How and where does your power meter measure torque? Brand and model?
Have you tried varying the mass of the flywheel?
respectfully

 

[BruceDiesel]

skogsgurra, to try to anser your question:

Essentially we are measuring the training effect that riding at a certain power level has on the athletes body (i.e. fitness).

The Funtional Threshold Power (FTP) - or power that is sustainable for 60 minutes is the baseline that is used to calculate all other power levels. Each level has specific physiological adaptions that are targeted in that level, so the athlete would target a certain level when the goal is to train that particular adaption. The levels have been defined as:
Level 1: Active Recovery <55% FTP
Level 2: Endurance 56% - 75% of FTP
Level 3: Tempo 76% - 90% of FTP
Level 4: Lactate Threshold 91% - 105% of FTP
Level 5: VO2Max 106% - 120% of FTP
Level 6: Anearobic Capacity > 121% of FTP
Level 7: Neuromuscular Power Maximal Effort

See

http://www.cyclingpeakssoftware.com/power411/levels.asp

Most cyclists wish to switch between indoor and outdoor environments, it is a lot easier to control external influences such as wind, traffic, gradient etc when training indoors, whereas it is more enjoyeable training outdoors. A loot of cyclists would use the indoor trainer to do interval training targeting the higher levels.

Now, if the load exerted on the athlete is different between indoors and outdoors, power based training is compromised and will be less effective.

 

[BruceDiesel]

waross:

There are five types of power meatures currently available:
1. Powertap - uses strain gauges and is effectively a hub fitted to the rear wheel of the bicycle. Torque is sampled at a rate of 70Hz, and velocity once per revolution.
2. SRM - also uses strain gauges, and replaces the front crank and attaches to the front sprockets. I don't know the sampling rate etc, but this device is considered to best one on the market.
3. Ergomo - uses two perforated discs on either side of the axle connecting the left and right crack. An LED and photosensor are used to create a square wave on each disc - as the axle twists, the phase difference between the two waves is calculated, from which the torque is derived. Note that this device can only measure the left leg and doubles it to calculate overall power.
4. Polar - measures the vibration and speed of the chain and calculates the torque being applied. This device is difficult to calibrate correctly and is not very popular.
5. iBike - uses a wind speed sensor, an accelerometer, a tilt sensor, rider aerodynamic coefficient etc to calculate the velocity vectors etc. Using the principle that the power output of the rider is equal and opposite to the forces of gravity and wind resistance, the power is calculated. This device obviously cannot be used indoors.

 

[DaveScott]

BruceDiesel, I think your hypothesis is correct (mass damping).

If you draw the torque curve, and the speed curve, can you agree that the speed curve lags by (less than*) 90 degrees compared to the torque curve? If I replace the system with an equivalent resistor and inductor - the resistor is the friction, and the inductor is the flywheel of the indoor equipment - the angle between torque and speed gets larger if the inductance is increased.
* When the torque reaches a peak and falls slightly, the bike is still accelerating. the bike will stop accelerating when the reducing torque matches the torque required to keep the bike going at that slightly higher speed.

In effect, I think, you should try to get the 'power factors' for outside and inside, the same. Which means you must match friction and inertia.

 

[BruceDiesel]

So in other words, we need to figure out a way to test the impulse-response of the system in both environments, and get them to match.

 

[BruceDiesel]

Thinking about this further, one of the ways we could adjust the system to suit the rider is to put weights on the bicycle's wheel when it is mounted on the trainer. Changing the flywheel would not be very practical.

 

[DaveScott]

//Changing the flywheel would not be very practical.// But you could change the gearing?
A test is a good idea. Try a coast down test.

 

[BruceDiesel]

But surely the gearing would not impact the damping effect - since it is not introducing any form of energy storage.