Traction and Wheels

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Traction and Wheels
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Pneumatic Tire Basics

• Weve just discussed the pressure under a track
  as it affects soil strength. What about the
  pressure under a pneumatic tire?
• Whoa! This is going to be hard to describe with
  an equation .

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Pneumatic Tire Basics

• Note the Low pressure in the center of the
  contact patch
• Note higher pressures at the perimeter. Why?
• If we want to apply c p tan f to this
  contact patch were gonna be frustrated

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Pneumatic Tire Basics

• Normal and shear stresses in the soil at the
  interface of the tire and the soil
• Both stresses vary with angular location and
  between the centerline and the side of the tire
• Challenging to use in predicting traction

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Wheels and Traction Mechanics

• Lets look at the possible cases
• We can have a towed wheel (front or rear steering
  axle for example)
• Self propelled wheel (combine or Snoopy)
• Braked wheel
• Driving wheel (produces thrust in excess of self
  propelled need

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Towed Wheel In More Detail

• Towed wheel free body mechanics
• W Weight
• TF towing force
• r rolling radius
• Note G will run just behind the wheel center to
  cause it to rotate
• G can be divided into vertical (R) and TF
  (results from soil.)

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Self Propelled Wheel

• W Weight
• T is torque as we now must supply some to make it
  propel itself
• Soil resistance and reaction, G, acts ahead of
  axle line and offsets the torque
• No draft or force on the axle (H 0)

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Driven Wheel

• Now we produce draft, H
• H F (gross thrust) minus TF (the towing force
  for a towed wheel)
• R W
• Torque, T, is balanced by F-TF and R acting at
  their radii
• H is available to accelerate the vehicle or
  provide draft

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Wheel Slip

• Diagram relates wheel slip, Torque and Pull for
  conditions from braked to driven
• Note that we dont produce pull (H), even to
  self propel without some slip
• Slip for towed wheel is negative on soil and will
  be near zero on a hard surface

Pull Torque
Slip
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Traction from Pneumatic Tires

• You wish to be able to predict or estimate
  tractive performance of your vehicle or tractor
  under different conditions
• Hmmm.
• How would you go about this?
• Which parameters might affect tractive effort?
• Vehicle weight?
• Soil condition?
• Tire size? Inflation pressure?
• Amount of Slip?
• Oofda

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Traction Ratios

• Well find that dynamic Weight affects nearly
  everything and so we will express performance
  variables per unit of Weight
• We are interested in the Draft or horizontal pull
  that a wheel/tire can produce so our first Ratio
  is H / W, or net Draft divided by Weight
• This ratio is sometimes called the Pull to Weight
  ratio, and in our text is given the symbol µ. It
  is conceptually similar to a friction
  coefficient. Remember it.

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Traction Ratios

• A wheel/tire can produce an increasing thrust
  force up to the point the soil fails under the
  tire
• We call this force F, or gross thrust
• If we divide again by weight we get gross thrust
  per unit weight, F/ W which we will denote as µg
  and call the gross coefficient of traction
• Again, much like a coefficient of friction but a
  gross value, (which implies something to be
  subtracted) What to subtract?

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Traction Ratios

• Now we consider motion resistance of a wheel.
  How much is this? For a rigid bicycle wheel on
  concrete it is not much. But consider pulling a
  filled gravity wagon over soft soil. Now the
  Towed force, TF is not insignificant. This towed
  force is motion resistance. It comes from the
  soil, as well as flexing of the tires. We will
  use a weight ratio again
• Call it TF / W or ?, (rho)

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Now Combine Ratios

• F/ W TF/ W H / W or
• µg - ? µ , or µnet or
• The gross coefficient of traction minus the
  motion resistance ratio is equal to the net
  coefficient of traction

H
TF
F
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Tractive Efficiency

• We are interested in getting driveline power to
  drawbar power efficiently
• Axle power is rotory, drawbar power is linear

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Traction Test Plot
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Slip Happens Plot
(Slip)
(Pull / Weight Ratio)
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Modeling Tractive Performance

• Note that the previous graphs represented one
  tire size on one soil condition
• What happens if we change tire size?
• What happens if it rains?
• We need an approach to predicting performance
  that allows us to vary parameters that we can
  control, or might encounter

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Dimensional AnalysisYouve seen this before in
fluids

• Identify all of the variables that are involved
  in the parameter of interest
• Combine variables into dimensionless ratios
• Perhaps combine one or more of these into a
  larger dimensionless ratio of variables
• Relate the ratio that contains the parameter of
  interest to the others using test data
• Results in an empirical equation relating a ratio
  of interest to the other combined variables

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Our Traction Variables

• Tire diameter, call it d, (length)
• Tire section width, call it b, (length)
• Tire section height, call it h (length)
• Tire deflection (squish), call it d (length)
• Weight, or dynamic load, W (mass x L x t-2)
• Soil strength, measured as cone index (CI), has
  units like pressure m x L x t-2 L-2 m x t-2 x
  L-1

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Tire Nomenclature
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Dimensionless Variables for Traction

• Deflection ratio, d/h
• Width to Diameter ratio, b/d
• Wheel numeric, Cn , CI b d/W
• (F/W) or µg, gross coefficient of traction
• TF/W or ?, the motion resistance ratio
• Now well combine the first three into something
  called the mobility number, Bn

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Traction Prediction Equations

• Gross coef. of traction µg
• Motion resistance ratio, ?
• For radial tires these are modified to
• Note that µg is an exponential equation, not
  unlike the one we found for a track system

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Net Coefficient of Traction

• To calculate the draft we can extract from a
  given tire and soil we subtract the motion
  resistance ratio from the gross coefficient of
  traction
• The result is much like a coefficient of friction
  (force / unit of normal load) It is also called
  the Pull-to-Weight ratio

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Draft and Weight Transfer - Statics

• We would like to apply our tire/traction model to
  the paper design of our tractor
• However you may note that the model depends upon
  W, which is dynamic weight
• Draft changes dynamic weight
• So Draft changes dynamic weight, which changes
  mobility number, which changes traction, which
  changes draft, which changes. You get the
  picture

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Statics Forces on the Tractor

• mg the tractors static weight at c.g.
• Soil reaction forces. Hold up the tractor
• Gross tractive effort our wheel thrust
• Motion resistance discussed earlier
• Draft (defined as horizontal) Px
• Vertical component of draft Py
• Incline Yes, but well ignore for now
• Wind. Naaaaaaah

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Draft and Weight Transfer - Statics
Vertical Forces 1st
Now Horizontal Forces
mg
Now Draft
In Components
Ph
Pv
P
Fr
TFr
TFf
Rf
Rr
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Sum Forces

• X direction
• Y direction
• Now mark some dimensions and sum moments

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Draft and Weight Transfer - Statics
Vertical Forces 1st
Now Horizontal Forces
mg
Now Draft
In Components
Ph
Pv
P
Fr
TFr
TFf
X2
Rf
Rr
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Sum Forces

• X direction
• Y direction
• Sum moments about A
• Now sum about B
• Now combine and solve for Rr and Rf

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Meaning of it all

• What do these equations tell us?
• Look at the terms
• The weight subtracted from the front axle is
  added to the rear axle
• Called dynamic weight transfer

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Putting Pieces Together

• We have a model for traction and draft. It
  depends upon dynamic wheel loading
• We have a statics model of the vehicle reaction
  forces. It depends upon draft
• To solve a traction/draft/tire/soil situation for
  our vehicle will require an iterative solution
• Good to remember programming here.