Mechanical connections with friction

Too much often regarded as a disturbing element for calculations, one realizes very quickly that the Frottement is quite simply essential: if the Screw fastening remain tight, the nail in place, the scales upright and the car S on the road, it is thanks to friction. It is also on this phenomenon which rests the operation of the Frein S and Embrayage S.

To solve a Static problem of , its consideration systematic is not obligatory. There exist mathematical model simple, and however precis which describe this phenomenon. The laws of Coulomb belong to these models. The Tribologie is the science of the contact which proposes other more pointed models to use according to the requirements of the study.

Perfect contact and contact with friction.

In a perfect contact, the transmissible mechanical action, by obstacle, between 2 solids cannot be, in any point that normal with the contact (perpendicular to the common tangent plan of the contact). So that this mechanical action can take another direction, it required there of a tangential component which will not be transmitted by obstacle but by friction or adherence.

The resulting force transmitted in this contact projects according to a normal, said component pressing effort , and a tangential component known as of force of friction .

Note:: in much of problems of mechanics with friction, the “decomposition” of the actions of contact is an often useless and likely operation, in certain cases, to complicate the continuation of the operations terribly.

Adherence and slip.

If one considers two bodies in specific contact with friction, there are 2 situations to observe according to whether the slip between the two bodies is proven or not.

If the bodies do not slip , the transmissible line of action can deviate from the normal of contact until a fixed limit (in red on the figure). The field thus delimited takes the form of a cone says “cone of static friction”. The half point angle is called angle of adherence. The study of the case at the border of the cone is called strict balance .

If relative speed between the bodies becomes nonnull , then the line of action takes a fixed slope (purple). One defines in the same way the “cone of sliding friction”.

In general, the cone of slip is included inside the cone of adherence. Each one made the experiment push a cupboard and realize that it is less hard to maintain its slip, once it took off. It is also this difference which explains the chattering of the chairs that one trails, the whistle of the brakes, or the generation of the sound of the Violon S and other instruments with bow.

For certain material couples, the adhesion coefficient is on the contrary lower than the coefficient of friction, this one increasing more or less notably with the speed of slip. This behavior is required for the design of many industrial mechanisms, to see on this subject the wikilivre of tribology and in particular the chapters devoted to the modeling of the actions of contact, to the clutches, the brakes, etc

Laws of Coulomb.

In the two preceding situations, the direction of the transmissible action is known:

either because the slip is proven
or because balance is proven. The static study then allows the preliminary determination of all the mechanical actions.

We can then write relation between the pressing effort (normal component NR) and the effort of friction (tangential component T): it must check the laws of Coulomb.

Case of the proven slip:

T = NR tg φ = NR F where F = tg φ is called coefficient of friction of slip.
the direction of T is such as it is opposed to the slip.

Case of proven balance:

T ≤ NR tg φ' = NR F' where F' = tg φ' is called coefficient of friction of adherence.
if the inequality is not checked, it is that the assumption of balance must be called in question (slip).
the direction of T is normally induced by the study. In the case of strict balance, it is arbitrarily selected to be opposed to the probable slip of the two parts.

Case study.

By considering a paving stone posed on a plane surface. If the ground is well on level then the balance of the paving stone is possible and the forces weight

\ vec {P}

and reaction of the ground

\ vec {R}

are cancelled (see static).

To move this paving stone, one exerts a side force $\ vec {F}$ . As long as the paving stone remains in place, its balance is always true; what imposes the relation of balance between the three forces (Basic principle of statics). It is noticed whereas the action of the ground is not normal any more with the contact. There is thus necessarily friction.

The problem remains whole then: Can one know the intensity of the force allowing to make move the paving stone, when the only data is the weight of the paving stone?

It will be also noticed that the point of application of $\ vec {F}$ modifies the position of the reaction on the ground. The more $\ vec {F}$ is applied high, the more the support on the ground is done forwards. In the case of a very high object, a cupboard for example, it may be that the reaction of the ground is carried outside (what is not possible, since that would imply local actions of opposite sign). Balance not being more possible, then there is swing.

Values of the adhesion coefficients and slip.

The value of these coefficients is not modélisable and can be established only by measurement. One will find many values directly applicable in the wikilivre of tribology.

According to a first simplistic but often satisfactory model, these values depend only on the material couple put in contact; the form and dimensions of surfaces do not intervene. The laws of Coulomb remain applicable as long as the possible speed of slip and the surface pressure remain moderate. Roughness or the surface pressure is influential for extreme values and without any measurement with the current problems. The complexity of the model must be adapted to the need. Tribology gives the whole of the factors to consider if a too simple model were not appropriate.

The difference between the two coefficients of friction being weak, the two cases are often confused. Here some numerical values for the adhesion coefficients (slip):

Steel/steel: 0,2 (0,15)
Steel/ice: 0,02 (0,02)
Steel/bronzes: 0,1 (0,05) with lubrication.
Steel/brake linings: 0,4 (0,25) (maximum pressure 20 MPa, T°<200°C)
Tire/road dries: 1 (0,5)
Tire/wet road: 0,7 (0,35)

Friction in the articulations

case of the pivot: to come

Solution to problem with friction.

In a problem of statics, the consideration of friction should not be taken into account if the phenomenon is regarded as negligible. With a coefficient lower than 0,1 the lines of action are deviated little (compared to the perfect model), of this fact the end result is not very disturbed.

However friction must be considered when one wishes to determine his influence on the studied problem, where when the perfect model does not bring a possible solution (paved which one pushes but who however remains motionless). In this case 2 types of study are to be considered:

1 - Assumption of connection with static friction : the study is undertaken initially to determine the whole of the forces. The assumption of connection with friction grants additional freedoms to the directions of the right-hand sides of action. Then in the second time, one checks that the connections with friction are in a situation compatible with the laws of Coulomb (right of action inside the cones of adherence). If it is not the case, balance is not possible. It is then necessary to re-examine the geometry of the problem.

2 - Assumption of strict balance : When the data are insufficient, it is interesting to pose the assumption of slip (contradictory with the concept of balance). One thus arbitrarily fixes the direction of certain mechanical actions. The result corresponds then to the limiting situation beyond which balance is not possible any more. This assumption is also natural when it is a question of calculating the performances of a or brake Embrayage, in this case accelerations are often supposed to be null what eliminates the inertial considerations.

the layout of the cone of friction is inevitable in both cases; However the characteristic given is the coefficient of friction (tangent of the angle). That is possible without calculation by using the following method (which is based on the relations in the right-angled triangle: tan = with dimensions opposite/with dimensions adjacent):

to go up from 1 unit on the normal of contact,
to turn of the value of the coefficient,
to trace the edge of the cone.

This method is more precise than that which consists in calculating the angle then to defer it, because the usual values of coefficient are close and the bevel protactors often limited in precision to the half degree.

See too

the wikilivre of tribology and more especially the chapter devoted to the modeling of the actions of contact.

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