Roational Inertia Of Bearing
2021年6月26日Register here: http://gg.gg/v5an7
*How To Determine Rotational Inertia
*Rotational Inertia Of Bearing Friction
The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. Bearings are a major component in any rotating system. With continually increasing speeds, bearing failure modes take new unconventional forms that often are not understood. Such measurements are.An Experiment to Measure Rotational Inertia
Paul J. Dolan, Jr.* and David Sturm, 5709 Bennett Hall, Dept. of Physics and Astronomy, University of Maine, Orono, ME 04469-5709, liquidhelium@hotmail.com.
Abstract
We describe an experiment that is performed in the first semester of the introductory course at the University of Maine in which students calculate, and then experimentally investigate, the rotational inertia of an irregular (and changeable) object. The experiment utilizes a Force Table as a base and the Pasco Smart Pulley© photogate as a sensing device.
Introduction
Rotational Inertia is one of the more interesting, and sometimes more difficult, topics that introductory Physics students encounter. Experiments in which students can quantitatively measure I, and also carefully measure how the value of I changes as the positions of the masses change, can greatly aid their understanding of the topic, and their appreciation for the fact that ‘physics works’. The experiment that is done here allows students to both calculate the rotational inertia of a somewhat irregular object, as the positions of the masses are changed, and to verify their calculations experimentally. Careful students can obtain a discrepancy of 5% or less in their final result.
The apparatus was first proposed by Professors Charles Smith and Gerald Harmon several years ago, prior to the introduction of computer-assisted data collection, and has undergone several iterations to reach its current form. The apparatus is shown in figure 1. It consists of a bearing inside a spool; the shaft around which the spool rotates fits into the center hole of the standard Force Table. So long as the friction in the bearing is reasonably constant, its actual value does not matter, as this will be directly measured in the experiment. An aluminum bar, with pairs of holes, spaced at 2 cm intervals out from the center, is attached to the top of the spool; the spacing of the holes has been chosen so that the actual rotational inertia of the bar is 95 % of the value of a solid bar of the same size and masses. Pairs of masses (about 400 g) fit into these holes. Moving the positions of these masses allows one to change I, without changing the total mass. Torque may be applied via string is wrapped around the spool (radius 3 cm), which is attached to a standard 50 g weight hangar. The string passes over the force table pulley; a small riser block is used to insure that the force is applied horizontally and tangent to the spool. Each setup has been made to be identical, so that a direct comparison of all students’ data is possible.
How To Determine Rotational Inertia
Theory
First, students are asked to calculate the total rotational inertia of the system: spool plus bar plus masses, for five possible positions of the masses (R). Critical dimensions are measured with a vernier caliper with a dial gauge, to an accuracy of .001 cm.
The rotational inertia of the bar is given by:
*IB = (½) mo (l2 + w2)
where mo is the mass of the bar, l its length, and w its width. The rotational inertia of the cylindrical masses is given by
*IC = 2(½)[ Mp^2 + 2MR^2]
where M is the mass of one cylinder, and ? is its diameter. The second term is the contribution from the parallel axis theorem, and is the one portion of I that will be varied experimentally; in fact, this comprises the major contribution to the total rotational inertia. The rotational inertia of the spool is not simple to calculate, by the students at this level, as the spool is non-uniform. This has been measured to have a value of
*IS = 2500 g-cm2The students sum these three contributions to the total rotational inertia, and plot I vs. R2.
Experiment
The angular velocity of the bar (w ) is measured with the photogate for a Pasco Smart Pulley, and is plotted as a function of time. The quantity in which we are interested is the angular acceleration, i.e., the slope of w vs. t. One might be concerned that friction could play a major role in the experiment. However, this contribution is explicitly measured. The students first measure w vs. t for a ‘freely spinning’ apparatus. As observed over several rotations, the apparatus will slow down, and thus they can find the angular acceleration due to friction (af) (which is, of course, negative). Having done this for each position of the cylindrical masses, the students proceed to apply a mass to the string (a 50 g weight hangar is sufficient), and to find the angular acceleration (a ) of the apparatus ‘under load’.
As with all good experiments, the students do each measurements several times, and use the average. Use of the Smart Pulley facilitates this.
Data & Analysis
When one constructs the free body diagrams for the forces and torques on the spool and on the weight hanger, one finds that the total moment of inertia is related to the mass of the hangar (m) as
(4) I = m(g - ra )r/(a - af)
where m is the applied ‘load’ (the weight hangar), and g is the acceleration due to gravity. We have found it convenient in this experiment to use CGS units.
Measurements of the physical parameters of the problem are given in Table I. Typical data for the average acceleration of the rotating object, both freely (af) and ‘under load’, are shown in Table II. Typical data, both theoretical and experimental, for the rotational inertia is also shown in Table II. One can see that there is remarkable close agreement between the values.
Discussion
Except for the smallest spacing of the masses, the discrepancy between the experimental and theoretical values is well under 5 %. There are two major contributions to the uncertainty in the experiment, beyond the usual measurement uncertainty, which is quite small. The first is the value of Is. This value is an average experimental value taken from all the apparati in use. As the moveable masses dominate I as R increases, one would expect any discrepancy in Is to be most noticeable at the lowest values of R. The other uncertainty enters in the measurement of af, which is assumed to be independent of w . In fact, this is not the case, and it is seen that af increases as w increases, which is to be expected. The variation in af between ‘slow’ and ‘fast’ rotation rates is also largest at the lowest value of R, so that once again this effect should cause a greater discrepancy at the lower R values. In contrast, the variation in the measured value of a ‘under load’ is very small, typically 0.01 rad/s2.
Conclusion
This experiment has proven to be very successful in allowing students to calculate and directly measure the rotational inertia of a (somewhat) irregular object, as the positions of the masses is varied. Students come away with a greater appreciation of this often difficult topic. The apparatus used is not especially difficult to construct, and has the added advantage that it makes good use of an underutilized piece of equipment that most physics departments own, the force table.
* Permanent address: Physics Dept., Northeastern Illinois University, 5500 N. St. Louis Ave., Chicago, IL 60625.Table I: Physical Parameters of the Rotational Apparatus
Bar: length(l) = 30.00 cm, width (w) = 2.551 cm, height = 0.965 cm, mass (mo) = 185.6 g
Ib = 13320 g-cm2
Spool: height = 5.30 cm, diameter = 2.999 cm, mass = 183.2 g
Is = 2500 g-cm2
Cylindrical masses: (average) mass (M) = 399.8 g, (average) radius (r ) = 2.5605 cm, height = 10.05 cm
‘Load’ (weight hangar), mass (m) = 50.3 g
Table II: Theoretical values of I, Experimental values of I and a .
Distance from center (R) (cm)(Average) Acceleration due to friction (af) (rad/s2)(Average) Acceleration ‘under load’ (a ) (rad/s2)Theoretical Rotational Inertia (g-cm2)Experimental Rotational Inertia (g-cm2)Percent Discrepancy3.0-0.19945.85825637239686.5 %5.0-0.14143.78338431372343.1 %7.0-0.09392.50057621565561.8 %9.0-0.061661.72683208822591.1 %11.0-0.045791.231115192115348-0.1 %
Let’s talk rotating weight. Rotating weight is a big issue for wheel builders. Why? We make choices that determine wheel weight, its total and location. Builders must understand this topic.
Rotating weight directly affects inertia, so the topic is really inertia. Inertia is the resistance of a mass to acceleration. Moment of inertia (MOI) characterizes this resistance and depends on rotating and non-rotating mass. Create slots in solidworks. Builders should be measuring wheel MOI.
I’ll show you how to measure moment of inertia (MOI), plus I’ll share a spreadsheet to shortcut the math supporting MOI measurement and its effect on riding. Plug in numbers and get usable wattage estimates.
Forces
Aerodynamic drag, mechanical friction, and inertial energy are the forces we face in cycling.
Aerodynamic drag is the most significant.
Mechanical friction was, historically, a bigger obstacle. Numerous inventions were needed to launch the cycling age.
MOI is less discussed, often passed over as insignificant. Yet all who pedal are aware of the effect of weight, especially wheel weight.
Wheel MOI has two parts:
1/ Rotational energy – required to make a wheel rotate. Rotating takes energy.
2/ Translational energy – needed to accelerate a mass. Mass takes energy to get moving.
Benefits of reduced MOI
1/ Sprinting
Lower rotating weight means quicker acceleration and sprinters value it most.Rotational Inertia Of Bearing Friction
2/ Climbing
Faster climbing is possible with lower weight. Gravity prioritizes weight so its feel is magnified.
Speed varies more on climbs and variations are a larger percentage. Momentum is smaller. A lapse in pedal force makes a larger speed change. Consequently, experts advise that maintaining tempo is an important advantage.
Climbers universally prefer lighter bikes and wheels.
3/ Pedal acceleration
Pedaling is pulse inputs. Micro analyzed, bicycle speed is a sine wave driven by a bumpy, two piston engine. Even smooth pedaling delivers more power on the down stroke. Resulting speed variations include many moments of acceleration. If wheels are lighter, these recoveries are less costly.
While lighter bikes and wheels increase the number and magnitude of speed variations, the competitor senses an advantage. If another climber increases tempo, it is urgent to match the pace, a pure and simple sprinting challenge. Frequent tempo changes in competition biases riders to lighter equipment.
When Campagnolo introduced their extra light Fluid Dynamic disk in 1987, they asserted a lighter wheel is faster because pedal strokes involve small accelerations affected by wheel weight. Download their discussion.
4/ Handling
Low MOI means less force to change angle or direction. Lighter bicycles feel more agile. For touring, fast handling can be distracting, but even sedate tours include moments of lively pedaling. In racing, low MOI is an asset.
Ultimately, so much is subjective. Some worship lightness, others ignore it. It’s often discounted because lightness is associated with weakness, unreliability, and higher maintenance. Yet no riders disagree, lighter weight is generally beneficial. While there are selective benefits to higher weight (stability, for one), they are usually outweighed by the sensation of speed, efficiency, and freedom.
An excellent, approachable discussion of wheel inertia is found at the French website Roues Artisanales. Check the inertia section of Adrien’s “Grand Test,” where he offers MOI for nearly 80 wheel sets.
Measuring MOI
Who measures MOI? For wheels, the relevant numbers include all components including tire. Wheel builders are often the only people in a position to measure and catalog MOI. Too few builders are checking MOI and using data to guide their choices and advise riders.
Consequently, some place too much emphasis on the advantages of low weight. Others ignore the issue in favor of aerodynamics or bearings. The full picture includes MOI.
Regardless of its relative importance, wheel engineers closely monitor MOI. As Lennard Zinn reported in 2006.
“Perhaps some of you remember when I did a test in VeloNews seven years ago (in the 6/28/99 issue) of wheel inertia by building a rotational pendulum in my garage. I was just at a Mavic tech seminar in Annecy, France, last week and saw a test machine set up virtually the same in the Mavic test lab.”
Thankfully, this is simple physics and anyone can build an accurate MOI machine. Here is a trifilar pendulum, the type of machine you should make and use.
My platform is three, ½” x 3’ dowels, assembled in a triangle. Each corner is connected overhead with 30lb monofilament fishing line. The exact dimensions don’t matter but must be accurately recorded and entered into formulas used to determine MOI.These are the mechanics:
Inertia = T^2( (W/r)*a*b)/(4*π^2*h)*1000
Inertia = gm^2
T = Oscillation period
W = weight in g
r = gf/N
a = diameter of top anchor
b = diameter of bottom
h = height
To calculate work (accelerate from zero to 30kph) in joules of energy:
Total energy = Rotational energy + Translational energy
Total energy = 1/2(Iw^2) + 1/2(mv^2)
Total energy = 1/2(Tire Inertia + Wheel Inertia)w^2 + 1/2(wheel mass + tire mass)v^2
Wheel total energy = 40% + 60%
w^2 = rotational speed in rad/s (30km/h = 24.54rad/s)
v^2 = wheel velocity in m/s (30km/h = 8.33m/s)
m = weight in kg
I = inertia in kg/m^2
To spare you the math, here is an auto calculating spreadsheet. The yellow cells need your input. Enter dimensions of your own jig, the period of the pendulum motion, and weight and you get an accurate MOI for your wheel, rim, tire, or anything. Pre-existing numbers are from my machine and two recently tested wheels. The other cells calculate based on your entries. Download the sheet (excel) for long-term use.
Watch this movie to see how easy it is to count pendulum cycles. Here is my own unit, whose dimensions are entered in the embedded spreadsheet. Remember to replace those with your own, once you build it.
The results are indisputable. The costs and benefits of MOI can be predicted but real world variables are numerous. The lesson is to understand MOI and establish it for various wheels. Wheel design needs all useful input and, all else equal, lower MOI is almost always preferable.
We want to learn how MOI affects wheel feel and performance. With a dynamic structure in a complex context (rider, bike, terrain), data is our best tool. Besides, if two wheel designs have similar weight, strength, cost, and longevity but one has much better MOI, its design is better. This should be perceived, quantified, and recognized. As we aspire to higher performance standards, all measurable dynamics must be considered!
Have fun building your MOI machine, you won’t be disappointed. Do some tests, take notes, share with customers and others like me. The pace of fundamental learning may seem glacial at times but we can all do a part.
For further insight to the subject, download Jim Martin’s seminal 1998 paper on bicycle power. Thanks also to Adrien Gontier of Roues Artisanales and illustrious theoretician, Josh Deetz.
Register here: http://gg.gg/v5an7
https://diarynote.indered.space
*How To Determine Rotational Inertia
*Rotational Inertia Of Bearing Friction
The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. Bearings are a major component in any rotating system. With continually increasing speeds, bearing failure modes take new unconventional forms that often are not understood. Such measurements are.An Experiment to Measure Rotational Inertia
Paul J. Dolan, Jr.* and David Sturm, 5709 Bennett Hall, Dept. of Physics and Astronomy, University of Maine, Orono, ME 04469-5709, liquidhelium@hotmail.com.
Abstract
We describe an experiment that is performed in the first semester of the introductory course at the University of Maine in which students calculate, and then experimentally investigate, the rotational inertia of an irregular (and changeable) object. The experiment utilizes a Force Table as a base and the Pasco Smart Pulley© photogate as a sensing device.
Introduction
Rotational Inertia is one of the more interesting, and sometimes more difficult, topics that introductory Physics students encounter. Experiments in which students can quantitatively measure I, and also carefully measure how the value of I changes as the positions of the masses change, can greatly aid their understanding of the topic, and their appreciation for the fact that ‘physics works’. The experiment that is done here allows students to both calculate the rotational inertia of a somewhat irregular object, as the positions of the masses are changed, and to verify their calculations experimentally. Careful students can obtain a discrepancy of 5% or less in their final result.
The apparatus was first proposed by Professors Charles Smith and Gerald Harmon several years ago, prior to the introduction of computer-assisted data collection, and has undergone several iterations to reach its current form. The apparatus is shown in figure 1. It consists of a bearing inside a spool; the shaft around which the spool rotates fits into the center hole of the standard Force Table. So long as the friction in the bearing is reasonably constant, its actual value does not matter, as this will be directly measured in the experiment. An aluminum bar, with pairs of holes, spaced at 2 cm intervals out from the center, is attached to the top of the spool; the spacing of the holes has been chosen so that the actual rotational inertia of the bar is 95 % of the value of a solid bar of the same size and masses. Pairs of masses (about 400 g) fit into these holes. Moving the positions of these masses allows one to change I, without changing the total mass. Torque may be applied via string is wrapped around the spool (radius 3 cm), which is attached to a standard 50 g weight hangar. The string passes over the force table pulley; a small riser block is used to insure that the force is applied horizontally and tangent to the spool. Each setup has been made to be identical, so that a direct comparison of all students’ data is possible.
How To Determine Rotational Inertia
Theory
First, students are asked to calculate the total rotational inertia of the system: spool plus bar plus masses, for five possible positions of the masses (R). Critical dimensions are measured with a vernier caliper with a dial gauge, to an accuracy of .001 cm.
The rotational inertia of the bar is given by:
*IB = (½) mo (l2 + w2)
where mo is the mass of the bar, l its length, and w its width. The rotational inertia of the cylindrical masses is given by
*IC = 2(½)[ Mp^2 + 2MR^2]
where M is the mass of one cylinder, and ? is its diameter. The second term is the contribution from the parallel axis theorem, and is the one portion of I that will be varied experimentally; in fact, this comprises the major contribution to the total rotational inertia. The rotational inertia of the spool is not simple to calculate, by the students at this level, as the spool is non-uniform. This has been measured to have a value of
*IS = 2500 g-cm2The students sum these three contributions to the total rotational inertia, and plot I vs. R2.
Experiment
The angular velocity of the bar (w ) is measured with the photogate for a Pasco Smart Pulley, and is plotted as a function of time. The quantity in which we are interested is the angular acceleration, i.e., the slope of w vs. t. One might be concerned that friction could play a major role in the experiment. However, this contribution is explicitly measured. The students first measure w vs. t for a ‘freely spinning’ apparatus. As observed over several rotations, the apparatus will slow down, and thus they can find the angular acceleration due to friction (af) (which is, of course, negative). Having done this for each position of the cylindrical masses, the students proceed to apply a mass to the string (a 50 g weight hangar is sufficient), and to find the angular acceleration (a ) of the apparatus ‘under load’.
As with all good experiments, the students do each measurements several times, and use the average. Use of the Smart Pulley facilitates this.
Data & Analysis
When one constructs the free body diagrams for the forces and torques on the spool and on the weight hanger, one finds that the total moment of inertia is related to the mass of the hangar (m) as
(4) I = m(g - ra )r/(a - af)
where m is the applied ‘load’ (the weight hangar), and g is the acceleration due to gravity. We have found it convenient in this experiment to use CGS units.
Measurements of the physical parameters of the problem are given in Table I. Typical data for the average acceleration of the rotating object, both freely (af) and ‘under load’, are shown in Table II. Typical data, both theoretical and experimental, for the rotational inertia is also shown in Table II. One can see that there is remarkable close agreement between the values.
Discussion
Except for the smallest spacing of the masses, the discrepancy between the experimental and theoretical values is well under 5 %. There are two major contributions to the uncertainty in the experiment, beyond the usual measurement uncertainty, which is quite small. The first is the value of Is. This value is an average experimental value taken from all the apparati in use. As the moveable masses dominate I as R increases, one would expect any discrepancy in Is to be most noticeable at the lowest values of R. The other uncertainty enters in the measurement of af, which is assumed to be independent of w . In fact, this is not the case, and it is seen that af increases as w increases, which is to be expected. The variation in af between ‘slow’ and ‘fast’ rotation rates is also largest at the lowest value of R, so that once again this effect should cause a greater discrepancy at the lower R values. In contrast, the variation in the measured value of a ‘under load’ is very small, typically 0.01 rad/s2.
Conclusion
This experiment has proven to be very successful in allowing students to calculate and directly measure the rotational inertia of a (somewhat) irregular object, as the positions of the masses is varied. Students come away with a greater appreciation of this often difficult topic. The apparatus used is not especially difficult to construct, and has the added advantage that it makes good use of an underutilized piece of equipment that most physics departments own, the force table.
* Permanent address: Physics Dept., Northeastern Illinois University, 5500 N. St. Louis Ave., Chicago, IL 60625.Table I: Physical Parameters of the Rotational Apparatus
Bar: length(l) = 30.00 cm, width (w) = 2.551 cm, height = 0.965 cm, mass (mo) = 185.6 g
Ib = 13320 g-cm2
Spool: height = 5.30 cm, diameter = 2.999 cm, mass = 183.2 g
Is = 2500 g-cm2
Cylindrical masses: (average) mass (M) = 399.8 g, (average) radius (r ) = 2.5605 cm, height = 10.05 cm
‘Load’ (weight hangar), mass (m) = 50.3 g
Table II: Theoretical values of I, Experimental values of I and a .
Distance from center (R) (cm)(Average) Acceleration due to friction (af) (rad/s2)(Average) Acceleration ‘under load’ (a ) (rad/s2)Theoretical Rotational Inertia (g-cm2)Experimental Rotational Inertia (g-cm2)Percent Discrepancy3.0-0.19945.85825637239686.5 %5.0-0.14143.78338431372343.1 %7.0-0.09392.50057621565561.8 %9.0-0.061661.72683208822591.1 %11.0-0.045791.231115192115348-0.1 %
Let’s talk rotating weight. Rotating weight is a big issue for wheel builders. Why? We make choices that determine wheel weight, its total and location. Builders must understand this topic.
Rotating weight directly affects inertia, so the topic is really inertia. Inertia is the resistance of a mass to acceleration. Moment of inertia (MOI) characterizes this resistance and depends on rotating and non-rotating mass. Create slots in solidworks. Builders should be measuring wheel MOI.
I’ll show you how to measure moment of inertia (MOI), plus I’ll share a spreadsheet to shortcut the math supporting MOI measurement and its effect on riding. Plug in numbers and get usable wattage estimates.
Forces
Aerodynamic drag, mechanical friction, and inertial energy are the forces we face in cycling.
Aerodynamic drag is the most significant.
Mechanical friction was, historically, a bigger obstacle. Numerous inventions were needed to launch the cycling age.
MOI is less discussed, often passed over as insignificant. Yet all who pedal are aware of the effect of weight, especially wheel weight.
Wheel MOI has two parts:
1/ Rotational energy – required to make a wheel rotate. Rotating takes energy.
2/ Translational energy – needed to accelerate a mass. Mass takes energy to get moving.
Benefits of reduced MOI
1/ Sprinting
Lower rotating weight means quicker acceleration and sprinters value it most.Rotational Inertia Of Bearing Friction
2/ Climbing
Faster climbing is possible with lower weight. Gravity prioritizes weight so its feel is magnified.
Speed varies more on climbs and variations are a larger percentage. Momentum is smaller. A lapse in pedal force makes a larger speed change. Consequently, experts advise that maintaining tempo is an important advantage.
Climbers universally prefer lighter bikes and wheels.
3/ Pedal acceleration
Pedaling is pulse inputs. Micro analyzed, bicycle speed is a sine wave driven by a bumpy, two piston engine. Even smooth pedaling delivers more power on the down stroke. Resulting speed variations include many moments of acceleration. If wheels are lighter, these recoveries are less costly.
While lighter bikes and wheels increase the number and magnitude of speed variations, the competitor senses an advantage. If another climber increases tempo, it is urgent to match the pace, a pure and simple sprinting challenge. Frequent tempo changes in competition biases riders to lighter equipment.
When Campagnolo introduced their extra light Fluid Dynamic disk in 1987, they asserted a lighter wheel is faster because pedal strokes involve small accelerations affected by wheel weight. Download their discussion.
4/ Handling
Low MOI means less force to change angle or direction. Lighter bicycles feel more agile. For touring, fast handling can be distracting, but even sedate tours include moments of lively pedaling. In racing, low MOI is an asset.
Ultimately, so much is subjective. Some worship lightness, others ignore it. It’s often discounted because lightness is associated with weakness, unreliability, and higher maintenance. Yet no riders disagree, lighter weight is generally beneficial. While there are selective benefits to higher weight (stability, for one), they are usually outweighed by the sensation of speed, efficiency, and freedom.
An excellent, approachable discussion of wheel inertia is found at the French website Roues Artisanales. Check the inertia section of Adrien’s “Grand Test,” where he offers MOI for nearly 80 wheel sets.
Measuring MOI
Who measures MOI? For wheels, the relevant numbers include all components including tire. Wheel builders are often the only people in a position to measure and catalog MOI. Too few builders are checking MOI and using data to guide their choices and advise riders.
Consequently, some place too much emphasis on the advantages of low weight. Others ignore the issue in favor of aerodynamics or bearings. The full picture includes MOI.
Regardless of its relative importance, wheel engineers closely monitor MOI. As Lennard Zinn reported in 2006.
“Perhaps some of you remember when I did a test in VeloNews seven years ago (in the 6/28/99 issue) of wheel inertia by building a rotational pendulum in my garage. I was just at a Mavic tech seminar in Annecy, France, last week and saw a test machine set up virtually the same in the Mavic test lab.”
Thankfully, this is simple physics and anyone can build an accurate MOI machine. Here is a trifilar pendulum, the type of machine you should make and use.
My platform is three, ½” x 3’ dowels, assembled in a triangle. Each corner is connected overhead with 30lb monofilament fishing line. The exact dimensions don’t matter but must be accurately recorded and entered into formulas used to determine MOI.These are the mechanics:
Inertia = T^2( (W/r)*a*b)/(4*π^2*h)*1000
Inertia = gm^2
T = Oscillation period
W = weight in g
r = gf/N
a = diameter of top anchor
b = diameter of bottom
h = height
To calculate work (accelerate from zero to 30kph) in joules of energy:
Total energy = Rotational energy + Translational energy
Total energy = 1/2(Iw^2) + 1/2(mv^2)
Total energy = 1/2(Tire Inertia + Wheel Inertia)w^2 + 1/2(wheel mass + tire mass)v^2
Wheel total energy = 40% + 60%
w^2 = rotational speed in rad/s (30km/h = 24.54rad/s)
v^2 = wheel velocity in m/s (30km/h = 8.33m/s)
m = weight in kg
I = inertia in kg/m^2
To spare you the math, here is an auto calculating spreadsheet. The yellow cells need your input. Enter dimensions of your own jig, the period of the pendulum motion, and weight and you get an accurate MOI for your wheel, rim, tire, or anything. Pre-existing numbers are from my machine and two recently tested wheels. The other cells calculate based on your entries. Download the sheet (excel) for long-term use.
Watch this movie to see how easy it is to count pendulum cycles. Here is my own unit, whose dimensions are entered in the embedded spreadsheet. Remember to replace those with your own, once you build it.
The results are indisputable. The costs and benefits of MOI can be predicted but real world variables are numerous. The lesson is to understand MOI and establish it for various wheels. Wheel design needs all useful input and, all else equal, lower MOI is almost always preferable.
We want to learn how MOI affects wheel feel and performance. With a dynamic structure in a complex context (rider, bike, terrain), data is our best tool. Besides, if two wheel designs have similar weight, strength, cost, and longevity but one has much better MOI, its design is better. This should be perceived, quantified, and recognized. As we aspire to higher performance standards, all measurable dynamics must be considered!
Have fun building your MOI machine, you won’t be disappointed. Do some tests, take notes, share with customers and others like me. The pace of fundamental learning may seem glacial at times but we can all do a part.
For further insight to the subject, download Jim Martin’s seminal 1998 paper on bicycle power. Thanks also to Adrien Gontier of Roues Artisanales and illustrious theoretician, Josh Deetz.
Register here: http://gg.gg/v5an7
https://diarynote.indered.space
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