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Chapter 8 : Dry Friction

8.1 Dry vs. Wet Friction
8.2 Law of Static Friction
8.3 Law of Kinematic Friction
8.4 General Procedure of Analysis
8.5 Two-point Friction
8.6 Friction and Tilting
8.7 Square-threaded Screw
8.8 Flat-Belt Drive

8.1 Dry vs. Wet Friction
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Friction occurs whenever two bodies are in conctact with each other and are either moving with respect to each other or are due to forces acting on the bodies liable to move. When studying friction one has to distinguish between at least two major phenomena.

Dry Friction occurs when the surfaces in contact with each other are free of any lubricants. Motions of the the two bodies in a direction parallel to the touching surfaces is prevented (or hampered) due to molecular adhesion and/or irregularities on the involved surfaces often too minute to discern with the nacked eye.

In this course we are dealing only with dry friction.

Wet friction occurs when the surfaces of two solid bodies are not directly in contact but separated by a thin film of lubricants. Again friction will try to hamper the motion but the underlying physics is related to fluid mechanics which is not part of this course.

Above distinction is that of two extreme case with mixed friction inbetween which often occurs during start-up of motion when the separating lubricant film is not fully developed.

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8.2 Law of Static Friction
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Figure 8.2a
Block resting on surface
In the figure 8.2a a block of weight W is shown resting on a surface and subject to a force P. Assume both, P and W are given. If the block is not moving then the surface underneath has to exert onto the block a force which exactly counteracts the forces P and W. It is best to represent this force by two forces : force N which is perpendicular and force F which is parallel to the surface the block rests upon.

Force N, often called the normal force (normal=perpendicular), is directed perpendicular to the direction of possible motion (of the block) and can be thought of as the resistance of the surface against penetration by the block. The resistance of course is spread in some fashion over the entire contact area and hence N is a replacement force for this resistance.

The force F is called the friction force. It is parallel to the direction of possible motion and acting onto the block in the direction opposing that of possible motion (the motion the block would undertake if friction were absent).

If the block is at rest the equilibrium equations
are satisfied.

In the example of Fig.8.2a that means :

N = W + P sin(α)
F = P cos(α)

irrespective of how the block manages to stay put (glue,friction,nails etc.).

If friction is holding the block in place we know from experience that if the horizontal component of force P is getting too strong the block will ultimately start to slide. The critical case at which sliding just is about to set in is referred to as the case of impending motion and the value of the friction force in that case is called the maximum friction force, Fm.

Many experiments have been conducted to find out as to what influences the value of Fm. Luckily, nature is kind to us in the case of dry friction. As it turns out Fm does only depend on the normal force N and the type of materials involved and the surface roughnesses. Furthermore, the relationship between Fm and N is linear for given surfaces :

(8.2a)       Fm = μs N

Equat. (8.2a) is referred to as the law of dry friction and μs as the coefficient of static friction. Its value is listed below for a variety of combinations of materials for the block and for the supporting surface.

Coefficients of Static Friction
steel and steel 0.8
steel and brass 0.35
steel and graphite 0.1
steel and teflon 0.04
aluminum and aluminum 0.33
wood and wood (dry) 0.2-0.6
wood and leather 0.3-0.4
leather and metal 0.4-0.6
stone and stone 0.4-0.7
concrete and rubber 0.6-0.9

Here is a fairly simple experiment to measure the coefficient of static friction.


Figure 8.2b : Angle of Friction
A block is placed on an inclined plate and the angle of inclination, δ, is slowly increased from zero until sliding set in. The angle at which sliding is impending is referred to as the angle of (static) friction.

The analysis of the Free Body Digram of Fig. 8.2b (sum of the force along the incline and sum of the force perpendicular to that) gives the relationship :

(8.2b)       μs = tan(δ)

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8.3 Law of Kinematic Friction
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Figure 8.3a
Block at constant velocity
Although this course is dedicated to statics for reasons of completeness let's consider the case of motion of the block at constant, low velocity. Again the equilibirum equations hold (at constant velocity the accelearation is zero, right ?) and the horizontal component of the pushing force P is counteracted by the friction force F.
Again, nature is kind to us and a linear relationship between the friction force F and the normal force N holds :

(8.3a)       Fkin = μk N

where μk is the coefficient of kinematic friction. Again the value of this coefficient depends on the involved materials and surface properties. At least for moderate velocities it is independent of velocity.

For all materials I know of we observe :

(8.3b)       μk < μs

which reflects our experience when pushing a piece of furniture from here to there. The hardest part is to get it moving. And the best strategy is to keep it moving until it arrives at its final resting place.

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8.4 General Procedure of Analysis
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8.5 Two-point Friction
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This and the next three sections are dealing with specific application of the law of dry friction.
In this section we have a device ( a horizontal lever in the figure) which experiences friction with another item ( the vertical rod in the figure ) at two distinct points, A and B.
Possible questions in such problems revolve around whether the device (the lever) will move under influence of external forces (the load P) or whether friction will hold it in place. Often one needs to investigate what are minimum or maximum values of design parameters of your device ( the height h or the distance L for example ) which either make motion possible or prevent it.

Figure 8.5a
2-point Friction


The crucial element in such examples is that when gliding sets in it takes place at both points of contact simultaneously. Hence, at all points of contact the law of dry friction, Equation (8.2a), holds in the case of impending motion.

As usual you will have to draw the relevant Free Body Diagram and establish the equilibrium equation and invoke the law of dry friction at each of the contact points.

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8.6 Friction and Tilting
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    In the figure to the left we show a crate with a weight of 100 lbs resting on two legs A and B which are 1m apart. A force P is intended to push the crate sidewise but several questions arise. If P is applied very low ( very small h ) the crate actually might start to slide once P reaches a critical value. For larger values of h sliding though might never set in : rather an applied P will cause the crate to tilt over to the right.
Inbetween there is a value of h for which tilting and sliding set in simultaneously. What is this value and how does it depend on the geometric parameters of the crate, the coefficient of static friction and maybe the weight ?

As usual our analysis starts with the Free Body Diagram

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8.7 Square-threaded Screw
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Figure 8.7a
Top and Side View of screw through a plate
Forces of thread of plate onto thread of screw

Figure 8.7b
Screw moving in direction of load P



Figure 8.7c
Screw moving against load P

Figure 8.7a displays the top and side view of a screw going through a plate of which only a small section is shown. The screw is subject to an axial load P [N] and a torque M [N m] which is turning the screw. Friction of course tries to prevent the turning. Subject of this section is to determine the torque M in terms of the load P, the geometry of the screw ( mean radius r and lead L as well as the coefficent of dry friction, μs, between the threads of the screw and the plate material.

If in Figure 8.7a a moment M is acting in clockwise direction, the shown screw will move downward in the same direction load P is pointing, if the moment is acting counter-clockwise the screw will be moving upwards against the direction of the external load P. In either case, the desired relationship for the moment M needed to achieve this can be determined from equilibrium considerations, specifically from looking at the sum of the forces along the axis and the sum of the moments around the axis of the screw.
Although the action of the thread in the plate onto the thread of the screw is distributed more or less evenly across the entire contact area we can represent this by a single normal force , N, acting perpendicular and a single friction force, F, acting parallel to the inclined surface of the thread.

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8.8 Flat-Belt Drive
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Figure 8.8a
Typical Belt Drive
Figure 8.8a shows a typical belt-drive with both pulleys rotating in the clockwise direction. One of the pulleys is usually connected to some motor and is referred to as the driving pulley, the other pulley is then the driven pulley. In this section we will investigate only the flat belt drive. It is rarely used in todays world but is accessible to analysis we can perform at this stage. Here is what Wikipedia has to offer (some nice pictures) and history.

For the sake of discussion let's assume that pulley A is driving and rotating in the clockwise direction as shown. Hence the torque the motor is exerting onto this pulley is in the clockwise direction as well.
NOTE : Although my figures show the driving pulley A to be the larger of the two pulleys involved it might be the smaller one in real life. That has no affect on the equations I give/derive below.


As a result, the tension T2 in the right section of the belt is higher than T1 in the left section in order to rotate pulley B in clockwise direction against the resistance of the machinery hooked up to this pulley. In Fig. 8.8b the moment MB, representing the action of the driven machinery onto the axle of the pulley, is therefore acting in counter-clockwise direction.

Figure 8.8b
Equilibrium of upper pulley


Equilibrium on pulley B :

(8.8a)       (T2 - T1 ) rB = MB
(8.8b)       (T2 + T1 ) cos(γ) + Py = 0
(8.8c)       (T2 - T1 ) sin(γ) + Px = 0

With : γ = 90° - β/2     Px,Py = support force of bearings onto axle of pulley B.

A similar analysis can be done on pulley A. Combing the two results from pulley A and B and observing the fact that the velocity of the belt as it leaves pulley A is identical to the velocity of the belt arriving at pulley B leads to the relationships :

(8.8d)    MB / MA = rB / rA

ωB / ωA = rA / rB

ωA , ωB = angular velocities


As far as friction is concerned, the only case of interest is that of impending slippage between belt and pulley. In that case an additional relationship between the tensions T1 and T2 can be established.


Figure 8.8c
Equilibrium of belt
    This relationship is derived from studying the equilibrium of the belt, here by considering pulley B (pulley A can be looked at as well with identical results). On a global basis the piece of belt running over the pulley B is subject to the tensions T1 and T2 and the pulley pressing outward against the belt on every sqare inch where pulley and belt are in contact. Accompanying this normal pressure is the friction with which the belt "drags" along the pulley in clockwise direction. The friction force the belt experiences is than of course in counter-clockwise direction.


Figure 8.8d
Equilibrium of belt
    In Fig. 8.8d we look at a small segment of the belt extending from Θ to Θ+ΔΘ. The angle Θ would be zero if the segment would start at the most counter-clockwise point at which belt and pulley are in contact. If the belt segment is located at the most clockwise point of contact we would have :
Θ+ΔΘ = β = angle of contact

N = normal force from pulley onto belt segment
F = μs N = friction force onto belt segment
T = tension at left end of segment
T + ΔT = tension at right end of segment

Note that between either of the two tensions and the friction force F we have the angle ΔΘ/2

For the x-y coordinate system as indicated in Fig. 8.8d we can write down now the sum of the forces equal to zero in the x- and for the y-direction, respectively.

-T cos(ΔΘ/2) - F + (T+ΔT) cos(&DeltaΘ/2) = 0

-T
sin(ΔΘ/2) + N - (T+ΔT) sin(&DeltaΘ/2) = 0

Solving the above 2 equations for forces F and N and substituing into the law of friction and letting the angle ΔΘ become very small we obtain a differential equation. This can be solved to finally give us :

(8.8e)    T2/T1 = eμsβ    e=2.71828...

The contact angle β has to be entered radians.

Equations (8.8a) and (8.8e) are usually used to determine the tensile force in each belt section necessary to avoid slippage of the belt on the smaller of the two pulleys for given torques.
Equations (8.8d) relates the torques and angular velocities of the two pulleys to each other, while Equation (8.8b) is being used to determine the support forces needed to hold the axis of a pulley in place.

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Zig Herzog, hgn@psu.edu
Last revised: 11/07/11