EN3: Introduction to Engineering and Statics

 

 

                      

 

   Division of Engineering

    Brown University

 

 

12. Friction

 

Friction forces act wherever two solids touch.  It is a type of contact force – but rather more complicated than the contact forces we’ve dealt with so far.

 

It’s worth reviewing our earlier discussion of contact forces.  When we first introduced contact forces, we said that the nature of the forces acting at a contact depends on three things:

(1)   Whether the contact is lubricated, i.e. whether friction acts at the contact

(2)   Whether there is significant rolling resistance at the contact

(3)   Whether the contact is conformal, or nonconformal.

 

We have so far only discussed two types of contact (a) fully lubricated (frictionless) contacts; and (b) ideally rough (infinite friction) contacts.

 

Remember that for a frictionless contact, only one component of force acts on the two contacting solids, as shown in the picture on the left below.  In contrast, for an ideally rough (infinite friction) contact, three components of force are present as indicated on the figure on the right.

             

 

(a) Reaction forces at a frictionless contact           (b) Reaction forces at an ideally rough contact

 

All real surfaces lie somewhere between these two extremes.  The contacting surfaces will experience both a normal and tangential force.  The normal force must be repulsive, but can have an arbitrary magnitude.  The tangential forces can act in any direction, but their magnitude is limited.  If the tangential forces get too large, the two contacting surfaces will slip relative to each other. 

 

This is why it’s easy to walk up a dry, rough slope, but very difficult to walk up an icy slope.  The picture below helps understand how friction forces work. The picture shows the big MM walking up a slope with angle , and shows the forces acting on M and the slope.  We can relate the normal and tangential forces acting at the contact to Mickey’s weight and the angle  by doing a force balance

Omitting the tedious details, we find that

Note that a tangential force  must act at the contact.  If the tangential force gets too large, then Mickey will start to slip down the slope.

 

When we do engineering calculations involving friction forces, we always want to calculate the forces that will cause the two contacting surfaces to slip.  Sometimes (e.g. when we design moving machinery) we are trying to calculate the forces that are needed to overcome friction and keep the parts moving.   Sometimes (e.g. when we design self-locking joints) we need to check whether the contact can safely support tangential force without sliding.

 

 

12.1 Experimental measurement of friction forces

 

To do both these calculations, we need to know how to determine the critical tangential forces that cause contacting surfaces to slip.  The critical force must be determined experimentally.  Leonardo da Vinci was apparently the first person to do this – his experiments were repeated by Amontons and Coulomb about 100 years later.  We now refer to the formulas that predict friction forces as Coulomb’s law or Amonton’s law (you can choose which you prefer!).

 

 

 

 

The experiment is conceptually very simple – it’s illustrated below

 

 

We put two solids in contact, and push them together with a normal force N.  We then try to slide the two solids relative to each other by applying a tangential force T.  The forces could be measured by force transducers or spring scales.   A simple equilibrium calculation shows that, as long as the weight of the components can be neglected, the contacting surfaces must be subject to a normal force N and a tangential force T.

 

In an experiment, a normal force would first be applied to the contact, and then the tangential force would be increased until the two surfaces start to slip.  We could measure the critical tangential force as a function of N, the area of contact A, the materials and lubricants involved, the surface finish, and other variables such as temperature.

 

You can buy standard testing equipment for measuring friction forces – one configuration is virtually identical to the simple experiment described above – a picture (from http://www.plint-tribology.fsnet.co.uk/cat/at2/leaflet/te75r.htm ) is shown below.  This instrument is used to measure friction between polymeric surfaces.

 

 

 There are many other techniques for measuring friction. One common configuration is the `pin on disk’ machine. Two examples are shown below.  The picture on the left is from www.ist.fhg.de/leistung/gf4/ qualitaet/bildgro4.html , and shows details of the pin and disk.  The picture on the right, from www.ulg.ac.be/tribolog/ test.htm shows a pin on disk experiment inside an environmental chamber.  In this test, a pin is pressed with a controlled force onto the surface of a rotating disk.  The force required to hold the pin stationary is measured.

 

 

 

Another test configuration consists of two disks that are pressed into contact and then rotated with different speeds.  The friction force can be deduced by measuring the torque required to keep the disks moving.  An example (from http://www.ms.ornl.gov/htmlhome/mituc/te53.htm ) is shown below

 

If you want to see a real friction experiment visit Professor Tullis’ lab at Brown (you don’t actually have to go there in person; he has a web site with very detailed descriptions of his lab) – he measures friction between rocks, to develop earthquake prediction models. 

 

 

A friction experiment must answer two questions:

 

(i)                  What is the critical tangential force that will cause the surfaces to start to slide? The force required to initiate sliding is known as the static friction force.

(ii)                If the two surfaces do start to slip, what tangential force is required to keep them sliding?  The force required to maintain steady sliding is referred to as the kinetic friction force.

 

 

We might guess that the critical force required to cause sliding could depend on

(i)                  The area of contact between the two surfaces

(ii)                The magnitude of the normal force acting at the contact

(iii)               Surface roughness

(iv)              The nature of the crud on the two surfaces

(v)                What the surfaces are made from

 

We might also guess that once the surfaces start to slide, the tangential force needed to maintain sliding will depend on the sliding velocity, in addition to the variables listed.

 

In fact, experiments show that

 

(i) The critical force required to initiate sliding between surfaces is independent of the area of contact.  This is very weird.  In fact, when Coulomb first presented this conclusion to the Academy Francaise, he was thrown out of the room, because the academy thought that the strength of the contact should increase in proportion to the contact area.  We’ll discuss why it doesn’t below.

 

(ii) The critical force required to initiate sliding between two surfaces is proportional to the normal force.  If the normal force is zero, the contact can’t support any tangential force.  Doubling the normal force will double the critical tangential force that initiates slip.

 

(iii) Surface roughness has a very modest effect on friction.  Doubling the surface roughness might cause only a few percent change in friction force.

 

(iv) The crud on the two surfaces has a big effect on friction.  Even a little moisture on the surfaces can reduce friction by 20-30%.  If there’s a thin layer of grease on the surfaces it can cut friction by a factor of 10.  If the crud is removed,  friction forces can be huge, and the two surfaces can seize together completely. 

 

(v) Friction forces depend quite strongly on what the two surfaces are made from.  Some materials like to bond with each other (metals generally bond well to other metals, for example) and so have high friction forces.  Some materials (e.g. Teflon) don’t bond well to other materials.  In this case friction forces will be smaller.

 

(v) If the surfaces start to slide, the tangential force often (but not always) drops slightly.  Thus, kinetic friction forces are often a little lower than static friction forces.  Otherwise, kinetic friction forces behave just like static friction – they are independent of contact area, are proportional to the normal force, etc.

 

(vi) The kinetic friction force usually (but not always) decreases slightly as the sliding speed increases.  Increasing sliding speed by a factor of 10 might drop the friction force by a few percent.

 

Note that there are some exceptions to these rules.  For example, friction forces acting on the tip of an atomic force microscope probe will behave completely differently (but you’ll have to read the scientific literature to find out how and why!).  Also, rubbers don’t behave like most other materials.  Friction forces between rubber and other materials don’t obey all the rules listed above.

 

 

 

 

 

12.2 Definition of friction coefficient: the Coulomb/Amonton friction law

 

A simple mathematical formula known as the Coulomb/Amonton friction law is used to describe the experimental observations listed in the preceding section. 

 

Friction forces at 2D contacts

 

 

Friction forces at a 2D contact are described by the following laws:

 

(i) If the two contacting surfaces do not slide, then

(ii) The two surfaces will start to slip if

(iii) If the two surfaces are sliding, then

The sign in this formula must be selected so that  opposes the direction of slip. 

 

In all these formulas,  is called the `coefficient of friction’ for the two contacting materials.  For most engineering contacts, .  Actual values are listed below. 

 

Probably we need to explain statement (iii) in more detail.  Why is there a ?  Well, the picture shows the tangential force  acting to the right on body (1) and to the left on body (2).  If (1) is stationary and (2) moves to the right, then this is the correct direction for the force and we’d use .  On the other hand, if (1) were stationary and (2) moved to the left, then we’d use  to make sure that the tangential force acts so as to oppose sliding.

 

Friction forces at 3D contacts

 

3D contacts are the same, but more complicated.  The tangential force can have two components.  To describe this mathematically, we introduce a basis  with  in the plane of the contact, and  normal to the contact.  The tangential force  exerted by body (1) on body (2) is then expressed as components in this basis

 

(i) If the two contacting surfaces do not slide, then

(ii) The two surfaces will start to slip if

(iii) If the two surfaces are sliding, then

where  denotes the tangential force exerted by body (1) on body (2), and  is the relative velocity of body (1) with respect to body (2) at the point of contact.  The relative velocity can be computed from the velocities  and  of the two contacting solids, using the equation .

 

12.3 Experimental values for friction coefficient

 

The table below (taken from `Engineering Materials’ by Ashby and Jones, Pergammon, 1980) lists rough values for friction coefficients for various material pairs. 

 

 

These are rough guides only – friction coefficients for a given material can by highly variable.  For example, Lim and Ashby (Cambridge University Internal Report CUED/C-mat./TR.123 January 1986) have catalogued a large number of experimental measurements of friction coefficient for steel on steel, and present the data graphically as shown below.   You can see that friction coefficient for steel on steel varies anywhere between 0.0001 to 3. 

Friction coefficient can even vary significantly during a measurement.  For example, the picture below (from Lim and Ashby, Acta Met 37 3 (1989) p 767) shows the time variation of friction coefficient during a pin-on-disk experiment.

 

 

 

 

 

 

12.4 Static and kinetic friction

 

Many introductory statics textbooks define two different friction coefficients.  One value, known as the coefficient of static friction and denoted by , is used to model static friction in the equation giving the condition necessary to initiate slip at a contact

A second value, known as the coefficient of kinetic friction, and denoted by , is used in the equation for the force required to maintain steady sliding between two surfaces

 

I don’t like to do this (I’m such a rebel).  It is true that for some materials the static friction force can be a bit higher than the kinetic friction force, but this behavior is by no means universal, and in any case the difference between  and  is very small (of the order of 0.05).  We’ve already seen that  can vary far more than this for a given material pair, so it doesn’t make much sense to quibble about such a small difference.

 

The real reason to distinguish between static and kinetic friction coefficient is to provide a simple explanation for slip-stick oscillations between two contacting surfaces.  Slip-stick oscillations often occur when we try to do the simple friction experiment shown below.

 

 

 

If the end of the spring is moved steadily to the right, the block sticks for a while until the force in the spring gets large enough to overcome friction.  At this point, the block jumps to the right and then sticks again, instead of smoothly following the spring.   If  were constant, then this behavior would be impossible.  By using , we can explain it.  But if we’re not trying to model slip-stick oscillations, it’s much more sensible to work with just one value of .

 

In any case, there’s a much better way to model slip-stick oscillations, by making  depend on the velocity of sliding.   Most sophisticated models of slip-stick oscillations (e.g. models of earthquakes at faults) do this.

 

 

12.5 The microscopic origin of friction forces

 

Friction is weird.  In particular, we need to explain

(i)                  why friction forces are independent of the contact area

(ii)                why friction forces are proportional to the normal force.

 

Coulomb grappled with these problems and came up with an incorrect explanation.  A truly satisfactory explanation for these observations was only found 20 years or so ago.

 

To understand friction, we must take a close look at the nature of surfaces.  Coulomb/Amonton friction laws are due to two properties of surfaces:

(1)   All surfaces are rough;

(2)   All surfaces are covered with a thin film of oxide, an adsorbed layer of water, or an organic film. 

 

Surface roughness can be controlled to some extent – a cast surface is usually very rough; if the surface is machined the roughness is somewhat less; roughness can be reduced further by grinding, lapping or polishing the surfaces.  But you can’t get rid of it altogether.  Many surfaces can be thought of as having a fractal geometry. This means that the roughness is statistically self-similar with length scale – as you zoom in on the surface, it always looks the same.

Of course no surface can be truly fractal: roughness can’t be smaller than the size of an atom and can’t be larger than the component; but most surfaces look fractal over quite a large range of lengths.  Various statistical measures are used to quantify surface roughness, but a discussion of these parameters is beyond the scope of this course.

 

Now, visualize what the contact between two rough surfaces looks like.  The surfaces will only touch at high spots (these are known in the trade as `asperities’) on the two surfaces.  Experiments suggest that there are huge numbers of these contacts (nobody has really been able to determine with certainty how many there actually are).  The asperity tips are squashed flat where they contact, so that there is a finite total area of contact between the two surfaces.  However, the true contact area (at asperity tips) is much smaller than the nominal contact area.

 

The true contact area can be estimated by measuring the surface roughness, and then calculating how the surfaces deform when brought into contact.  At present there is some uncertainty as to how this should be done – this is arguably the most important unsolved problem in the field.  The best estimates we have today all agree that:

 

The true area of contact between two rough surfaces is proportional to the normal force pressing them together.

 

At present, there is no way to measure or calculate the contact C accurately.

 

This is true for all materials (except for rubbers, which are so compliant that the true contact area is close to the nominal contact area), and is just a consequence of the statistical properties of surface roughness.  The reason that the true contact area increases in proportion to the load is that as the surfaces are pushed into contact, the number of asperity contacts increases, but the average size of the contacts remains the same, because of the fractal self-similarity of the two surfaces.

 

Finally, to understand the cause of the Coulomb/Amonton friction law, we need to visualize what happens when two rough surfaces slide against each other. 

 

 

Each asperity tip is covered with a thin layer of oxide, adsorbed water, or grease.  It’s possible to remove this film in a lab experiment – in which case friction behavior changes dramatically and no longer follows Coulomb/Amonton law – but for real engineering surfaces it’s always present.

 

The film usually has a low mechanical strength.  It will start to deform, and so allow the two asperities to slide past each other, when the tangential force per unit area acting on the film reaches the shear strength of the film .

 

The tangential friction force due to shearing the film on the surface of all the contacting asperities is therefore

 

Combining this with the earlier result for the true contact area gives

 

Thus, the friction force is proportional to the normal force.  This simple argument also explains why friction force is independent of contact area; why it is so sensitive to surface films, and why it can be influenced (albeit only slightly) by surface roughness. 

 

 

12.6 Example problems involving friction

 

Friction problems are solved exactly like any equilibrium problem involving constraint and reaction forces.  The friction laws merely provide a little additional information relating normal and tangential forces acting at contacts.

 

 

Example 1

 

Our first example is the simplest possible problem involving friction.  A block with weight W is placed on an inclined slope. Let  denote the coefficient of friction between the two surfaces. Calculate the normal and tangential components of force acting at the contact.  Find the angle of the slope beyond which the block cannot remain at rest on the surface.

 

We draw a free body diagram for the block, labeling reaction forces at the contact in the usual way.  When solving problems involving friction, it’s convenient (but not essential) to choose basis vectors normal and tangential to the contact. 

Since only three forces act on the block, we can calculate the reaction forces using only force equilibrium – moment balance is satisfied automatically (the forces must all act through the same point).  The resultant force acting on the block is

The equilibrium equations are

so that

 

Notice that we haven’t considered the friction law at all while calculating the reaction forces.  In particular, we did not put .  That’s because the correct friction law is .  You can only use  if you know the contact is slipping, and you are sure that you have the direction of T correct. Many high-school physics courses teach you to put   for all contacts, but this is not correct.

 

We do use the friction law to calculate the maximum angle for which the block will remain on the slope without slip.  The friction law says

 

The block is just on the point of slipping when .

 

This provides a quick way to estimate the friction coefficient between two surfaces – you can just put a sample of one material on top of a flat sheet of the other, and tilt the sheet until the sample starts to slide.  At this point we know that , so the friction coefficient can be calculated from the angle.

 

Example 2: 2D ladder

 

The warning labels on long ladders instruct you to use the ladders at 70 degree angles, as indicated below.  Calculate the minimum value of friction coefficient for the ladder to be safe.  Neglect friction at B.  Is the ladder more likely to slip when you stand near the top, or near the bottom?

 

We will first calculate reaction forces, then use the friction laws to answer the questions.  Reaction forces are calculated using the standard procedure – free body diagram; force and moment balance; solution of equilibrium equations.

 

Here’s the free body diagram for the ladder & moron together.

 

 

A force and moment balance table is shown below

 

 

 

 

 

The equilibrium equations are

 

 

The solution is

 

Again, note that we have not used the friction law yet.  We apply the friction law to answer the final part.

 

The ladder is safe as long as

 

The ladder is evidently most likely to slip when x=d, i.e. when the moron stands on the top.  For the ladder to be safe under worst-case conditions, the friction coefficient must satisfy

This is quite a high value – friction coefficient could easily drop below 0.36 if the surfaces are greasy or wet.  Beware!

 

Example 3

 

Repeat the preceding calculation, but this time assume that friction acts at both A and B (with identical friction coefficients  at both points). 

 

The new free body diagram is shown

 

 

 

 

The equilibrium equations become

 

 

Now, we are stuck.  These equations have no unique solution, because we have 4 unknown reaction components, and only three equations.  The reaction forces cannot be calculated without further information.

 

The information we need comes from the friction laws.  We suppose that the friction coefficient is just high enough to prevent the ladder from falling.  Under these conditions both contacts A and B are just on the point of sliding, and the tangential forces must be positive to oppose slip at the contacts.  Then, we can put

This give two more equations, and one additional unknown (the friction coefficient).  It’s straightforward to solve them

The critical friction coefficient varies with .  For  we find .  As , we find

.  The worst case again occurs when the moron stands at the top of the ladder, and we find that friction at B makes no difference to the critical friction coefficient necessary for safety.

 

Example 3.

 

Now, a 3D example.  Recall the emergency relief shelter design described in Sect 11.6 of the e-notes.  The design is shown below

The spreadsheet below shows the member forces for a particular choice of tent design.  Recall that the design relies on friction to prevent joints A, B, C, D from moving.  What value of friction coefficient is necessary to hold the joints in place?

 

 

 

We must find the reaction forces at all the joints in order to find the friction coefficient.  Because of symmetry, we only need to consider joints A and B.

 

The picture below shows a cut-out view of joint A, showing forces acting at the joint.

 

 

 

 

 

 

 

It’s helpful to find unit vectors parallel to each member at the joint.  Simple geometry shows that

We can calculate the reaction forces by summing forces acting at the joint. The resultant force acting at the joint is

The member force table in the spreadsheet shows that

Therefore

 

So, finally

 

 

The 3D friction law tells us that the joint will not slip as long as

Thus, for safety

This is a very high value of friction coefficient.  Of course, the members AB and AD would help prevent slip, but even so, a design modification is probably in order.

A similar calculation can be used to check joint B.  The picture below shows a cut-out view of the joint

 

 

 

Unit vectors parallel to the joints are

The resultant force acting on the joint is

The member force table gives

Thus

so that

The joint will not slip as long as

so that

This joint is even worse!

 

 

 

 

Example 4

 

Friction can be used to design self-locking joints.  A very simple example (often used to hold ropes on a sailboat) is shown below.   Our mission is to show that, as long as the distance d is sufficiently small, the strip (1) can never be pulled out of the self-locking clamp.

 

 

To make a conservative calculation, we will assume that the bearing at C, and the contact between (1) and (3) are frictionless.  Friction does act at the contact between (1) and (2); we will use  to denote the coefficient of friction.

 

The picture below shows forces acting on components (1) and (2) in the system.

 

 

A force balance for (1) shows immediately that

A force and moment balance table for the circular cam is shown below

 

The moment balance equation shows that

 

The strip can only be pulled out of the clamp if it can slip relative to the cam at C.  The contact cannot slip if

so the joint will lock if