Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
Lecture 15.
Friction
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Matthew T. Mason
Mechanics of Manipulation
Spring 2012
Some practical
friction tricks.
Today’s outline
Lecture 15.
Friction
Motivating friction.
Motivating friction.
Coulomb’s Law.
Coulomb’s Law.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Friction angle, friction cone.
Moment labeling of friction cone.
Static equilibrium problems.
Some practical friction tricks.
Some practical
friction tricks.
How do you move things around?
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
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Kinematics, kinematic constraint.
Force.
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Force of constraint;
Gravity;
Friction;
Momentum.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
How do you know where things are?
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
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You put them somewhere, or
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you look at them. And then,
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they stay put.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Theorem (Liouville’s theorem)
In a Hamiltonian system (including an energy-conserving
passive mechanical system) the uncertainty, measured
as a probability distribution in phase space, remains
constant as the system evolves.
Energy loss is essential.
Friction and plastic impact are essential.
Some practical
friction tricks.
An alternative reality
Lecture 15.
Friction
Motivating friction.
What would it be like to live in a world without friction?
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Grabbing anything would be like grabbing a bar of
soap.
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No tables. Bowls? Tables with edges?
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Walking wouldn’t work. Drill footholds everywhere?
(You’re stranded in the middle of a frozen pond,
perfectly flat and frictionless. Q: How do you get to
shore? A: Throw one of your shoes to the opposite
shore.)
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What would feet and hands look like?
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With no gravity? Where do you look for lost items?
On the space shuttle, they find lost articles at the AC
intake grille.
Coulomb’s Law.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
Overview
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
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Friction is complex and difficult to model.
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We will focus on simple approximations: generally,
Coulomb’s law of sliding friction, with known uniform
coefficient of friction.
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Applied with common sense, this model is good
enough to explore problems, and perform useful
manipulation.
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But, know the limits.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
Lecture 15.
Friction
An experiment, in the style of Coulomb
Motivating friction.
Coulomb’s Law.
Clean surfaces, but not too
clean—dry, unlubricated.
Pull on string with force fa ,
ramping up from 0.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Friction force ff will balance fa , up
to a point.
Max ff when not moving: µs mg.
fa
ff
mg
Some practical
friction tricks.
ff
µ s mg
µ d mg
Max ff when moving: µd mg.
fa
From now on we will assume
µs = µd = µ.
Static equilibrium
problems.
Lecture 15.
Friction
Coulomb’s observations
Coulomb conducted hundreds of experiments, and over a
broad range of conditions he observed:
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Frictional force is approximately independent of
contact area.
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
Moment labeling of
friction cone.
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Frictional force is approximately independent of
velocity magnitude.
Static equilibrium
problems.
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Coefficient of friction depends on pairs of materials.
Some practical
friction tricks.
Materials
metal on metal
rubber on concrete
plastic wrap on lettuce
Leonardo’s number
µ
0.15–0.6
0.6–0.9? 1.0–2.0?
∞
0.25
(Do not believe these numbers!)
Apply Coulomb’s law with care
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
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It holds over a broad range, but not nearly
everywhere.
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It is approximate.
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Coefficients of friction tables are terrible.
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How can you use something so unreliable?
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But, how can you not use it?
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
Lecture 15.
Friction
Contact modes
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
We can write Coulomb’s law:
ẋ
ẍ
fn
<0
ft = µfn
left sliding
>0
ft = −µfn right sliding
x
= 0 < 0 ft = µfn
left sliding
= 0 > 0 ft = −µfn right sliding
= 0 = 0 |ft | ≤ |µfn | rest
and define “contact mode” to be the right column—“left
sliding”, etc.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
ft
Lecture 15.
Friction
Friction angle
Block at rest on plane with angle
α:
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
fn = mg cos α
Moment labeling of
friction cone.
ft = mg sin α
fn
Static equilibrium
problems.
At rest |ft | ≤ µfn . Maximum α:
ft
ft = µfn
Substituting,
mg sin α = µmg cos α
α = tan−1 µ
Sometimes called the friction
angle or the angle of repose.
mg
Some practical
friction tricks.
Lecture 15.
Friction
Friction cone
Motivating friction.
Definition
Coulomb’s Law.
Define the friction cone to be the set of
all wrenches satisfying Coulomb’s law
for an object at rest, i.e. satisfying
2 tané 1 µ
Friction angle,
friction cone.
fn
|ft | ≤ µ|fn |
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The friction cone is a polyhedral convex cone in
wrench space.
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We can restate Coulomb’s law using contact modes
Left sliding:
fn + ft ∈ right edge of friction cone
Right sliding: fn + ft ∈ left edge of friction cone
Rest:
fn + ft ∈ friction cone
Moment labeling of
friction cone.
Static equilibrium
problems.
ft
Some practical
friction tricks.
Moment labeling of friction cone
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
Friction cone is positive linear
span of left edge unit vector and
right edge unit vector.
Moment labeling
Static equilibrium problems.
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
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Given a mobile body, several frictional contacts, and
an applied force, is equilibrium possible?
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Approach: use moment labelling to identify all
feasible contact wrenches; test whether one of them
balances the applied force.
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Caveat A: Presence of balancing wrench in the cone
does not imply nature will select that wrench. See
the wedged plank example (static indeterminacy).
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Caveat B: Even if equilibrium is attained, stability
may not be. Stability requires more careful analysis,
depending on hypothesized disturbances.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
Lecture 15.
Friction
Pipe clamp design problem
Motivating friction.
Coulomb’s Law.
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Why does pipe clamp work?
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Let diameter be 2 cm.
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Let length be 2 cm.
Moment labeling of
friction cone.
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Assume µ of 0.25.
Static equilibrium
problems.
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Find min moment arm.
Some practical
friction tricks.
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Extend to woodpecker toy?
Friction angle,
friction cone.
fc1
f1
fa
f2
fc2
Block on table
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
Wedged plank and piranha
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
Triangle and three fingers.
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.
Friction tricks
Lecture 15.
Friction
Motivating friction.
Coulomb’s Law.
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Measuring the coefficient of friction.
Driving
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Coulomb’s law doesn’t work so well for rubber on
concrete;
Theory suggests you don’t spin your wheels. Almost
always spinning is bad.
But sometimes, in snow or mud, spinning is a good
idea.
You can push a car sideways, if the wheels are
spinning, turning sliding friction into viscous friction.
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Reducing imprecision due to friction: dithering.
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Reducing imprecision due to friction: impact.
In theory, there is no difference between theory and
practice.
In practice, there is.—Anonymous
Friction angle,
friction cone.
Moment labeling of
friction cone.
Static equilibrium
problems.
Some practical
friction tricks.