Rotating Mass...
Moderator: robbosmans
I can't beleive you guys are still going on about this. The easiest way to explain or see a MICRO acceleration is to watch olympic rowing. With each stroke of the rower(s) you can see small surges where the scull momentarilly gains on the one nearest to it. How does this apply to CYCLING you ask? Well a pedal also has a stroke. Who do you know that pedala a perfect contiuous circle? I thought not.
AEROLITUSdefender of the faith
And your point is? We keep going on about this, because the same error is repeated over and over again. When you "micro" accelerate you also "micro" decelerate. According to the laws of physics this means that "micro" accelerations of objects with greater inertia will be less and subsequently "micro" decelerations will be less as well. The resulting "micro" acceleration is no different for either scenario, unless you're too weak to accelerate a rim with 200 grams extra mass to begin with I guess.....
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 asphaltdude
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divve wrote:And your point is? We keep going on about this, because the same error is repeated over and over again. When you "micro" accelerate you also "micro" decelerate.
Ignorance is bliss I guess
Whow! That's a pretty damn nice garage door!
wally318 wrote:I can't beleive you guys are still going on about this. The easiest way to explain or see a MICRO acceleration is to watch olympic rowing. With each stroke of the rower(s) you can see small surges where the scull momentarilly gains on the one nearest to it. How does this apply to CYCLING you ask? Well a pedal also has a stroke. Who do you know that pedala a perfect contiuous circle? I thought not.
Spinning a crankset in a circle is definitely not the same as rowing a boat.
 TunedCannondaleR700
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 Location: California
the bottom line is when you have a heavier rim it will hold momemtum better, but the small benefit of that is greatly out weighed by the effort of constanant micro accelerations, and the the extra energy need just to push that extra weight at a constant speed . On a hill more weight will mean more work at the same speed. lighter wheel=faster
Cannondale is quite simply the best
 asphaltdude
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TunedCannondaleR700 wrote:the bottom line is when you have a heavier rim it will hold momemtum better, but the small benefit of that is greatly out weighed by the effort of constanant micro accelerations,
Whow! That's a pretty damn nice garage door!
The rowing image might work.
Imagine boat using only one rowing action.
A very light boat. You row and it slows very quickly.
A very heavy boat. You row and it maintains it's speed though that speed is slower than lighter boat.
While the light boat gets up to speed quicker it also loses more speed between strokes.
A heavier boat takes longer to get up to speed and maintains it's speed so requires less effort at race pace.
Imagine boat using only one rowing action.
A very light boat. You row and it slows very quickly.
A very heavy boat. You row and it maintains it's speed though that speed is slower than lighter boat.
While the light boat gets up to speed quicker it also loses more speed between strokes.
A heavier boat takes longer to get up to speed and maintains it's speed so requires less effort at race pace.
I can't believe that people keep arguing that rotating mass climbs slower than nonrotating mass under the same power. When you are working against gravity, mass is mass, it doesn't matter if it rotates or not. The idea that microaccelerations due to pedal force fluctations make a difference in the overall picture is a strawman. During pedal force fluctuations, accelerations are decelerations cancel out. All that really matters is average power output vs. gravity.
Since Ras11 complained that no math has been offered, I decided to set up a model to simulate the accelerations/decelerations due to pedal fluctuations. The equations and variable values were taken from the Analytic Cycling web page.
Pedaling force: The propulsion force (from pedaling) was modeled as a sinusoidal. Since it is assumed average power is constant, the nomimal drive force will vary inversely with velocity. So, the propulsion force is modeled as:
Fp = (P/V)(1+Sine(2RT))
Fp = Propulsion force (pedaling)
P = Average power
V = Velocity
R = Pedaling revolution rate
T = Time
(Note: The angle in the sine term is double the pedal revolution rate, since there are two power strokes per revolution)
The drag forces on the rider are aerodynamic drag, rolling resistance, and gravity. These three terms together are:
Fd = (1/2)CdRhoAV^2 + MgCrrCosine(S) + MgSin(S)
Fd = drag force
Cd = Coefficient of aerodynamic drag
Rho = Density of air
A = Frontal area
M = total mass of bike and rider
Crr= Coefficient of Rolling Resistance
g = Acceleration of gravity
S = Slope of road
The total force is thus:
F = Fp  Fd
From Newton's second law, the equation of motion is:
dV/dt = F/I
I = Inertia
Because there is both rotating and nonrotating mass, total mass and total inertial will not be the same. Because mass at the periphery of the wheel as twice the inertia as nonrotating weight, the total mass and inertia of a bike are:
M = Ms + Mr
I = Ms + 2Mr
Ms = Static mass
Mr = Rotating mass
The complete equation of motion is thus:
dV/dt = {(P/V)(1+sin(2RT))  [ (1/2)CdRhoAV^2 + (Ms+Mr)gCrrCosine(S) + (Ms+Mr)gSine(S) ] } / (Ms + 2Mr)
This equation is nonlinear, so I solved it numerically with a 4th order RungeKutta numerical differentiation.
Borrowing the default values in the Analytic Cycling web page for "Speed given Power" page, the values used are:
P = 250 Watts, Cd = 0.5, Rho = 1.226 Kg/m^3, A = 0.5 m^2, Crr = 0.004, g = 9.806 m/s^2, S = 3% (= 1.718 deg.)
(http://www.analyticcycling.com/ForcesSpeed_Page.html)
For this simulation, the pedal revolution rate was selected as 540 deg/sec. (90 rpm cadence)
To solve this equation, a 4th order RungeKutta numerical differentiation was set up using an Excel spread sheet. Step size was selected at 0.01 sec., and the initial Velocity was 1 m/sec. The solution was calculated for 3 cases of equal total mass, but different distributions of static and rotating mass, calculated over a 200 second period, by which time each case had reached steady state. As expected, the velocity oscillated with the pedal strokes. The average, maximum, and minimum velocities during the oscillilations during stead state were:
Case 1:
Ms = 75 kg, Mr = 0 kg (0% rotating mass)
Average Velocity: 7.457831059 m/s
Maximum Velocity: 7.481487113 m/s
Minimum Velocity: 7.434183890 m/s
Speed fluctuation: 0.047303224 m/s
Case 2:
Ms = 70 kg, Mr = 5 kg (5.33% rotating mass)
Average Velocity: 7.457834727 m/s
Maximum Velocity: 7.480016980 m/s
Minimum Velocity: 7.435662980 m/s
Speed fluctuation: 0.044354000 m/s
Case 3:
Ms = 65 kg, Mr = 10 kg (10.67% rotating mass)
Average Velocity: 7.457837584 m/s
Maximum Velocity: 7.478718985 m/s
Minimum Velocity: 7.436967847 m/s
Speed fluctuation: 0.041751139 m/s
These results agree very strongly with the solution on the Analytic Cycling web page, which predicted an average speed with constant power of 7.46 m/s (16.7 mph)
The results show that as expected, the smaller the percentage of rotating mass, the greater the magnitude of the velocity oscillations (which are quite small). But a more interesting result is in the average speed. As the amount of rotating mass decreased, the more the average velocity _decreased_, not increased (at steady stage). This result is actually not unexpected. The drag forces are not constant, but vary with velocity, especially aerodynamic drage (Because aerodynamic drag increases with the square of velocity, power losses are increase out of proportion with speeds  so, for example, aerodynamic losses at 20 mph are 4 times as much as they would be at 10 mph). Because speed fluctuates as the propulsion force oscillations, in the cases of the low rotating mass, the maximum peak speeds reached are higher than for the cases with the high rotating mass. This means that when a lower percentage of rotating mass there will be greater losses during the speed peaks. Because of the total drag losses will be greater over the long run, the greater momentary accelerations with lower rotating mass actually results in a lower average speed.
To see what happens at a steeper slope, which will have a lower speed (and presumably larger speed oscillattions), I ran the model again with a 10% (5.7 deg.) slope. Here are the results:
Case 1:
Ms = 75 kg, Mr = 0 kg (0% rotating mass)
Average Velocity: 3.217606390 m/s
Maximum Velocity: 3.272312291 m/s
Minimum Velocity: 3.162540662 m/s
Speed fluctuation: 0.109771630 m/s
Case 2:
Ms = 70 kg, Mr = 5 kg (5.33% rotating mass)
Average Velocity: 3.217613139 m/s
Maximum Velocity: 3.268918539 m/s
Minimum Velocity: 3.165997726 m/s
Speed fluctuation: 0.102920813 m/s
Case 3:
Ms = 65 kg, Mr = 10 kg (10.67% rotating mass)
Average Velocity: 3.217618914 m/s
Maximum Velocity: 3.265921742 m/s
Minimum Velocity: 3.169047012 m/s
Speed fluctuation: 0.096874730 m/s
This data follows the same pattern as above. The speed oscillations (microaccelerations) are greater with the lower rotating mass, but the average speed is also slightly lower with lower rotating mass. So next time you want to claim that lower rotating mass allows faster accelerations, remember too that the greater speed fluctuations (due to greater accelerations) will also results in greater energy losses due to drag forces.
But, in reality, the differences in speed fluctions and average speeds are really very small between all these cases. For all practical purposes, when climbing, it is only total mass that matters, not how it is distributed.
I'd be happy to send the Excel spreadsheet to anyone that is interested.
Since Ras11 complained that no math has been offered, I decided to set up a model to simulate the accelerations/decelerations due to pedal fluctuations. The equations and variable values were taken from the Analytic Cycling web page.
Pedaling force: The propulsion force (from pedaling) was modeled as a sinusoidal. Since it is assumed average power is constant, the nomimal drive force will vary inversely with velocity. So, the propulsion force is modeled as:
Fp = (P/V)(1+Sine(2RT))
Fp = Propulsion force (pedaling)
P = Average power
V = Velocity
R = Pedaling revolution rate
T = Time
(Note: The angle in the sine term is double the pedal revolution rate, since there are two power strokes per revolution)
The drag forces on the rider are aerodynamic drag, rolling resistance, and gravity. These three terms together are:
Fd = (1/2)CdRhoAV^2 + MgCrrCosine(S) + MgSin(S)
Fd = drag force
Cd = Coefficient of aerodynamic drag
Rho = Density of air
A = Frontal area
M = total mass of bike and rider
Crr= Coefficient of Rolling Resistance
g = Acceleration of gravity
S = Slope of road
The total force is thus:
F = Fp  Fd
From Newton's second law, the equation of motion is:
dV/dt = F/I
I = Inertia
Because there is both rotating and nonrotating mass, total mass and total inertial will not be the same. Because mass at the periphery of the wheel as twice the inertia as nonrotating weight, the total mass and inertia of a bike are:
M = Ms + Mr
I = Ms + 2Mr
Ms = Static mass
Mr = Rotating mass
The complete equation of motion is thus:
dV/dt = {(P/V)(1+sin(2RT))  [ (1/2)CdRhoAV^2 + (Ms+Mr)gCrrCosine(S) + (Ms+Mr)gSine(S) ] } / (Ms + 2Mr)
This equation is nonlinear, so I solved it numerically with a 4th order RungeKutta numerical differentiation.
Borrowing the default values in the Analytic Cycling web page for "Speed given Power" page, the values used are:
P = 250 Watts, Cd = 0.5, Rho = 1.226 Kg/m^3, A = 0.5 m^2, Crr = 0.004, g = 9.806 m/s^2, S = 3% (= 1.718 deg.)
(http://www.analyticcycling.com/ForcesSpeed_Page.html)
For this simulation, the pedal revolution rate was selected as 540 deg/sec. (90 rpm cadence)
To solve this equation, a 4th order RungeKutta numerical differentiation was set up using an Excel spread sheet. Step size was selected at 0.01 sec., and the initial Velocity was 1 m/sec. The solution was calculated for 3 cases of equal total mass, but different distributions of static and rotating mass, calculated over a 200 second period, by which time each case had reached steady state. As expected, the velocity oscillated with the pedal strokes. The average, maximum, and minimum velocities during the oscillilations during stead state were:
Case 1:
Ms = 75 kg, Mr = 0 kg (0% rotating mass)
Average Velocity: 7.457831059 m/s
Maximum Velocity: 7.481487113 m/s
Minimum Velocity: 7.434183890 m/s
Speed fluctuation: 0.047303224 m/s
Case 2:
Ms = 70 kg, Mr = 5 kg (5.33% rotating mass)
Average Velocity: 7.457834727 m/s
Maximum Velocity: 7.480016980 m/s
Minimum Velocity: 7.435662980 m/s
Speed fluctuation: 0.044354000 m/s
Case 3:
Ms = 65 kg, Mr = 10 kg (10.67% rotating mass)
Average Velocity: 7.457837584 m/s
Maximum Velocity: 7.478718985 m/s
Minimum Velocity: 7.436967847 m/s
Speed fluctuation: 0.041751139 m/s
These results agree very strongly with the solution on the Analytic Cycling web page, which predicted an average speed with constant power of 7.46 m/s (16.7 mph)
The results show that as expected, the smaller the percentage of rotating mass, the greater the magnitude of the velocity oscillations (which are quite small). But a more interesting result is in the average speed. As the amount of rotating mass decreased, the more the average velocity _decreased_, not increased (at steady stage). This result is actually not unexpected. The drag forces are not constant, but vary with velocity, especially aerodynamic drage (Because aerodynamic drag increases with the square of velocity, power losses are increase out of proportion with speeds  so, for example, aerodynamic losses at 20 mph are 4 times as much as they would be at 10 mph). Because speed fluctuates as the propulsion force oscillations, in the cases of the low rotating mass, the maximum peak speeds reached are higher than for the cases with the high rotating mass. This means that when a lower percentage of rotating mass there will be greater losses during the speed peaks. Because of the total drag losses will be greater over the long run, the greater momentary accelerations with lower rotating mass actually results in a lower average speed.
To see what happens at a steeper slope, which will have a lower speed (and presumably larger speed oscillattions), I ran the model again with a 10% (5.7 deg.) slope. Here are the results:
Case 1:
Ms = 75 kg, Mr = 0 kg (0% rotating mass)
Average Velocity: 3.217606390 m/s
Maximum Velocity: 3.272312291 m/s
Minimum Velocity: 3.162540662 m/s
Speed fluctuation: 0.109771630 m/s
Case 2:
Ms = 70 kg, Mr = 5 kg (5.33% rotating mass)
Average Velocity: 3.217613139 m/s
Maximum Velocity: 3.268918539 m/s
Minimum Velocity: 3.165997726 m/s
Speed fluctuation: 0.102920813 m/s
Case 3:
Ms = 65 kg, Mr = 10 kg (10.67% rotating mass)
Average Velocity: 3.217618914 m/s
Maximum Velocity: 3.265921742 m/s
Minimum Velocity: 3.169047012 m/s
Speed fluctuation: 0.096874730 m/s
This data follows the same pattern as above. The speed oscillations (microaccelerations) are greater with the lower rotating mass, but the average speed is also slightly lower with lower rotating mass. So next time you want to claim that lower rotating mass allows faster accelerations, remember too that the greater speed fluctuations (due to greater accelerations) will also results in greater energy losses due to drag forces.
But, in reality, the differences in speed fluctions and average speeds are really very small between all these cases. For all practical purposes, when climbing, it is only total mass that matters, not how it is distributed.
I'd be happy to send the Excel spreadsheet to anyone that is interested.
My nomination for the best post ever to a bicycle related listserve.
For certain parts stiffer is more important than lighter.
While we are counting every watt of energy, an equally small wattage due to the loss caused by greater speed fluctuations are the losses caused by the fact that during the act of pushing the heavier rotating mass wheel you effectivly experience more resistance and therefore put a minute amount of extra energy in the form of heat into the pedals, crank, chainrings, chain, sprockets, hub, spokes, etc.
For certain parts stiffer is more important than lighter.
Let me quote just for the heck of it
ras11 wrote:Thank you for your reply Divve. I think the forum needs more pious and closeminded posts like that. Let me break this down for your highschoolphysicslimitedconservative thoughts on this subject. You can go kick your dog after you’re done reading this.

 Posts: 1288
 Joined: Thu Nov 04, 2004 4:05 pm
awesome macca
one minor point about the rowing analogy which nobody else seems to have picked is that water is about 8 times thicker than air  so therefore all these micro accelerations and decelerations that have been mentioned are therefore accentuated....
additionally, the "dead time" between each rowing stroke is enormous compared to a bicycle pedal revolution
one minor point about the rowing analogy which nobody else seems to have picked is that water is about 8 times thicker than air  so therefore all these micro accelerations and decelerations that have been mentioned are therefore accentuated....
additionally, the "dead time" between each rowing stroke is enormous compared to a bicycle pedal revolution
 asphaltdude
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 Joined: Tue Aug 12, 2003 8:39 pm
 Location: Holland
 Contact:
yourdaguy wrote:pushing the heavier rotating mass wheel you effectivly experience more resistance and therefore put a minute amount of extra energy in the form of heat into the pedals, crank, chainrings, chain, sprockets, hub, spokes, etc.
Why would you have to push harder when your wheels have more rotating mass???
You just exert the same force to the pedals as you would always do, only the acceleration of the bike during the pedal stroke is smaller
(and accordingly, the decelerations between the pedal strokes is also smaller)
Whow! That's a pretty damn nice garage door!
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