Lecture 15. Friction
Lecture 15.
Friction
Matthew T. Mason
Mechanics of Manipulation
Spring 2012
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Today’s outline
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
How do you move things around?
I Kinematics, kinematic constraint.
I Force.
I
I
I
Force of constraint;
Gravity;
Friction;
I Momentum.
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
How do you know where things are?
I You put them somewhere, or
I you look at them. And then,
I they stay put .
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.
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
An alternative reality
What would it be like to live in a world without friction?
I Grabbing anything would be like grabbing a bar of soap.
I No tables. Bowls? Tables with edges?
I 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.)
I What would feet and hands look like?
I With no gravity? Where do you look for lost items?
On the space shuttle, they find lost articles at the AC intake grille.
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Overview
I Friction is complex and difficult to model.
I We will focus on simple approximations: generally,
Coulomb’s law of sliding friction, with known uniform coefficient of friction.
I Applied with common sense, this model is good enough to explore problems, and perform useful manipulation.
I But, know the limits.
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
An experiment, in the style of Coulomb
Clean surfaces, but not too clean—dry, unlubricated.
Pull on string with force f a
, ramping up from 0.
Friction force f f to a point.
will balance f a
, up
Max f f when not moving: µ s mg .
Max f f when moving: µ d mg .
From now on we will assume
µ s
= µ d
= µ .
µ s mg
µ d mg f f f f m g f a
f a
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Coulomb’s observations
Coulomb conducted hundreds of experiments, and over a broad range of conditions he observed:
I Frictional force is approximately independent of contact area.
I Frictional force is approximately independent of velocity magnitude.
I Coefficient of friction depends on pairs of materials.
Materials µ metal on metal 0.15–0.6
rubber on concrete 0.6–0.9? 1.0–2.0?
plastic wrap on lettuce ∞
Leonardo’s number 0.25
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
(Do not believe these numbers!)
Apply Coulomb’s law with care
I It holds over a broad range , but not nearly everywhere.
I It is approximate.
I Coefficients of friction tables are terrible.
I How can you use something so unreliable?
I But, how can you not use it?
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Contact modes
We can write Coulomb’s law: f n
<
>
0
0 f t
= µ f n left sliding f t
= − µ f n right sliding
= 0 < 0 f t
= 0 > 0 f t
= µ f n
= − µ f n left sliding right sliding
= 0 = 0 | f t
| ≤ | µ f n
| rest x and define “contact mode” to be the right column—“left sliding”, etc.
f t
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Friction angle
Block at rest on plane with angle
α : f n
= mg cos α f t
= mg sin α
At rest | f t
| ≤ µ f n
. Maximum α : f t
= µ f n
Substituting, mg sin α = µ mg cos α
α = tan
− 1
µ
Sometimes called the friction angle or the angle of repose .
f n m g f t
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Friction cone
Definition
Define the friction cone to be the set of all wrenches satisfying Coulomb’s law for an object at rest, i.e. satisfying
| f t
| ≤ µ | f n
|
2 tan
é 1
µ f n
I The friction cone is a polyhedral convex cone in wrench space.
I We can restate Coulomb’s law using contact modes
Left sliding: f n
+ f t
∈ right edge of friction cone
Right sliding: f n
+ f t
∈ left edge of friction cone
Rest: f n
+ f t
∈ friction cone
f t
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Moment labeling of friction cone
Friction angle, friction cone.
Moment labeling of friction cone.
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.
I Given a mobile body, several frictional contacts, and an applied force, is equilibrium possible?
I Approach: use moment labelling to identify all feasible contact wrenches; test whether one of them balances the applied force.
I Caveat A: Presence of balancing wrench in the cone does not imply nature will select that wrench. See the wedged plank example (static indeterminacy).
I 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.
Some practical friction tricks.
Pipe clamp design problem
I Why does pipe clamp work?
I Let diameter be 2 cm.
I Let length be 2 cm.
I Assume µ of 0.25.
I Find min moment arm.
I Extend to woodpecker toy?
fc
1 f
1 f a f
2 fc
2
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Block on table
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Wedged plank and piranha
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Triangle and three fingers.
Friction angle, friction cone.
Moment labeling of friction cone.
Some practical friction tricks.
Friction tricks
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I
Measuring the coefficient of friction.
Driving
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I
I
I
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.
I Reducing imprecision due to friction: dithering.
I 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.
