In reality, there are no absolutely smooth surfaces. All surfaces of bodies are rough to one degree or another. Therefore, the reaction force of a rough surface when the body is in equilibrium depends on the active forces not only in numerical value, but also in direction.

Let us decompose the reaction force of a rough surface into components: one of which we will direct along the common normal to the contact surface, and the other we will direct in the tangent plane to these surfaces.

Friction force sliding (or simply friction force) is the component of the bond reaction force that lies in the tangent plane to the surfaces of the contacting bodies.

By force of normal reaction bond is the component of the bond reaction force, which is directed along the common normal to the surfaces of the contacting bodies.

The nature of the friction force is very complex and We do not touch upon it. In theoretical mechanics, it is assumed that there is no lubricant between the surfaces of contacting bodies.

Dry friction called friction when there is no lubricant between the surfaces of contacting bodies.

We will consider two cases: friction when a body is at rest or in equilibrium, and sliding friction when one body moves along the surface of another with a certain relative speed.

At rest, the friction force depends only on the active forces. With the chosen direction of the tangent at the point of contact of the surfaces of the bodies, the friction force is calculated by the formula:

Similarly, with the chosen direction of the normal, the normal reaction is expressed in terms of the given forces:

When one body moves on the surface of another, the friction force is a constant value.

Engineering calculations are usually based on a number of experimentally established patterns that reflect the main features of the dry friction phenomenon with sufficient accuracy for practice. These laws are called the laws of sliding friction or Coulomb's laws.

Coulomb's Laws

1. The sliding friction force is located in the common tangent plane of the contacting surfaces of the bodies and is directed in the direction opposite to the direction of possible sliding of the body under the action of active forces. The friction force depends on the active forces, and its module is between zero and the maximum value, which is achieved at the moment the body leaves the equilibrium position, that is:

Called ultimate friction force .

2. The maximum sliding friction force, all other things being equal, does not depend on the area of contact of the rubbing surfaces. From this law it follows that in order to move, for example, a brick, it is necessary to apply the same force, regardless of which face it is placed on the surface, wide or narrow.

3. The limiting sliding friction force is proportional to the normal reaction (normal pressure), that is

where the dimensionless coefficient is called the sliding friction coefficient; it is independent of the normal reaction.

4. The coefficient of sliding friction depends on the material and physical state of the rubbing surfaces, that is, on the size and nature of roughness, humidity, temperature and other conditions. The friction coefficient is determined experimentally.

It is believed that the friction coefficient does not depend on the speed of movement.

Friction angle. Equilibrium conditions.

Many problems involve balancing a body on a rough surface, i.e. in the presence of friction, it is convenient to solve geometrically. To do this, we introduce the concept of angle and cone of friction.

The reaction of a real (rough) connection consists of two components: the normal reaction and the friction force perpendicular to it. Consequently, the bond reaction deviates from the normal to the surface by a certain angle. When the friction force changes from zero to maximum, the reaction force changes from zero to , and its angle with the normal increases from zero to a certain limiting value j.

Friction angle is the largest angle between the maximum reaction force of a rough bond and the normal reaction.

The angle of friction depends on the coefficient of friction.

Friction cone called a cone described by the maximum reaction force of a rough bond around the direction of the normal reaction.

Example.

If a force P is applied to a body lying on a rough surface, forming an angle with the normal, then the body will move only when the shear force  is greater than the limiting friction force  (if we neglect the weight of the body, then but the inequality

Executed only when , i.e. at ,

Consequently, no force forming an angle with the normal that is smaller than the friction angle  can move the body along a given surface.

For the equilibrium of a solid body on a rough surface, it is necessary and sufficient that the line of action of the resultant active forces acting on the solid body pass inside the friction cone or along its generatrix through its apex.

A body cannot be thrown out of balance by any modulus active force if its line of action passes inside the friction cone.

Example.

Let's consider a body that has a vertical plane of symmetry. The cross-section of the body of this plane has the shape of a rectangle. Body width is 2a.

A vertical force is applied to the body at point C, lying on the axis of symmetry, and at point A, lying at a distance h from the base, a horizontal force is applied. The reaction of the base plane (bond reaction) is reduced to the normal reaction and frictional force. The line of action of the force is unknown. Let us denote the distance from point C to the line of action of the force as x. (). Let's create three equilibrium equations:

According to Coulomb's law, i.e. . (1)

Since , then (2)

Let's analyze the results:

We will increase our strength.

1) If , then equilibrium will take place until the friction force reaches its limiting value, condition (1) will turn into equality. A further increase in force will cause the body to slide along the surface.

2) If , then equilibrium will take place until the friction force reaches the value , condition (2) will turn into equality. The value of x will be equal to h. A further increase in force will cause the body to tip around point B (there will be no sliding).

Rolling friction

Rolling friction is the resistance that occurs when one body rolls over the surface of another.

Consider a cylindrical roller of radius r on a horizontal plane. Reactions may occur under the roller and the plane at the point of their contact, preventing the roller from rolling along the plane through the action of active forces. Due to the deformation of surfaces, not only sliding, but also rolling.

The active forces acting on rollers in the form of wheels usually consist of gravity, a horizontal force applied to the center of the roller, and a couple of forces with a moment tending to roll the wheel. The wheel in this case is called follower-leader. If , a , then the wheel is called slave. If , a , then the wheel is called leading.

Let a body of weight P move under the action of force T along a rough surface. On the one hand, the surface does not allow the body to fall down under the influence of gravity P. On the other hand, the surface prevents the free movement of the body under the influence of force T. Thus, the friction force F also , like a normal reaction, is brought to life by the surface, i.e. the friction force is also a reaction. The normal reaction and the friction force add up to the total reaction R, which is deviated from the normal by an angle c. This angle is called the friction angle. Using Fig. It is easy to calculate what the tangent of the friction angle is equal to tgts=F/N=µN/N=µ, i.e. the tangent of the friction angle is numerically equal to the friction coefficient.

Now imagine that you rotate the total reaction around the surface normal. In this case, the force R describes a cone, which is called the cone of friction. It is interesting in that the area limited by the friction cone determines the equilibrium region for the body: if a force acts on the body inside the friction cone, it will not move the body, no matter how great it is; if a force acts on a body outside the friction cone, it moves the body, no matter how small it is (Fig. 19).

Rice. 19.

Let's see why this happens (Fig. 20).

Rice. 20.

If the force Q acts inside the friction cone, then the shear force Q 1 = Qsinb. Let's calculate the friction force:

F=µN=µQcosб=Qcosбtgts.

Safety factor F-Q 1 =Q(cosb tgts-sin b) = Qsin(ts-b)/costs. Thus, the safety margin is proportional to Q, since sin(c-b)/cosс is a constant value. The greater the force Q, the greater the holding force F-Q 1.

This is why you need to be able to build a friction cone.

Once a bridge collapsed in Munich, and the fault was not a hurricane wind, not a regiment of soldiers marching in step, but... a friction cone.

This bridge was secured at one end with a hinge, and at the other end it was placed on rollers (Fig. 21). The bridge is always secured in such a way that it does not become warped due to temperature fluctuations. The hinge was filled with paste, which protected it from corrosion. On a hot summer day, the paste melted and its viscosity became less. The nature of friction has changed - it has also decreased. The friction cone narrowed, and the pressure force on the support went beyond the cone.

Rice. 21.

The balance was broken and the bridge collapsed. Engineers often have to construct a cone of friction to determine whether a given structure will be in equilibrium or not. But engineers are not the only ones who deal with the friction cone. Each of us encounters this physical phenomenon every day.

To get to the exit on a crowded bus or trolleybus, you have to squirm like a snake. We do this unconsciously, without thinking that in this way we get out of the friction cones at the points of contact with other passengers.

Whether we are skating, going to work, or turning a page in a book, everywhere we encounter friction and, in particular, the cone of friction.

The reaction of a real rough bond will consist of two components: the normal reaction N and the force F perpendicular to it. Consequently, the total reaction R will be deviated from the normal to the surface by a certain angle. When measuring the friction force from zero to, the force R will change from N to, and its angle with the normal will increase from zero to a certain limiting value. The largest angle that the total reaction of a rough bond makes with the normal to the surface is called the friction angle. . Because , then from here we find the following relationship between the friction angle and the friction coefficient . At equilibrium, the total reaction R, depending on the shear forces, can take place anywhere within the friction angle. When equilibrium becomes limiting, the reaction will be deviated from the normal by an angle . If a force P is applied to a body lying on a rough surface, forming an angle with the normal, then the body will move only when the shear force there will be more . But inequality > , in which , is executed only when those. at . Consequently, no force forming an angle with the normal that is smaller than the friction angle can move the body along a given surface.

Rolling friction. Rolling friction coefficient. Moment of rolling friction forces. p. 102.

Rolling friction is the resistance that occurs when one body rolls over the surface of another. - moment of force. Bye , the roller is at rest; at rolling begins. The linear quantity k included in the formula is called the rolling friction coefficient. The k value is usually measured in centimeters. The value of the coefficient k depends on the material of the bodies and is determined experimentally. The ratio for most materials is significantly less than the static coefficient of friction. Rolling friction is the resistance that occurs when one body rolls over the surface of another. Let's imagine a wheel standing on a horizontal plane. Let P be the weight of the wheel and its line of action passes through the center O of the wheel. Let us apply a horizontal force T at this point. Under the action of a shear force T at the point of contact of the roller and the surface, a sliding friction force Ftr arises, preventing the roller from slipping. These door forces T and Ftr, equal in modulus, form a pair that tends to rotate the roller. Under the action of force P, deformation occurs at the point of contact, and the normal reaction N shifts towards the action of force T by a certain distance h. As a result, the forces Pi and N form another pair that interferes with the action of the pair (T, Ftr). The maximum value h = k, corresponding to the limiting equilibrium position, is called the rolling friction coefficient. In contrast to the dimensionless sliding friction coefficient f, the rolling friction coefficient k has the dimension of length. The value of T corresponding to the case of limit equilibrium is T=k/r. At T > Nk/r the roller will begin to roll. Note that rolling friction occurs only when elastic bodies roll. If the contacting bodies are absolutely solid, then there is no deformation and T = 0, that is, no force is required to roll an absolutely solid roller on an absolutely solid surface. Typically, the force T determined by the equation is significantly less than the maximum sliding friction force. Therefore, bodies overcome rolling friction much earlier than sliding begins. Due to their low resistance to movement, rolling bearings are widely used in technology. Sliding is possible at T > fN, and rolling begins at T > Nk / r. So Thus, if f > k / r, then sliding is not possible; if f = k / r, then both rolling and sliding occur simultaneously; iff< k / r.– качение невозможно.При решении задач действие трения качения учитывается моментом сил сопротивления качению Мс. Его величина, как и величина силы трения скольжения, изменяется от нуля до предельного значения: 0 ≤ M c≤ M пред, где M пред= Nk . Своегопредельного значения момент сил сопротивления качению достигает в состоянии движения, то есть при перекатывании колеса.

FRICTION ANGLE is the angle formed when the reaction forces of two bodies deviate from the common normal to their contact surface due to the presence of friction forces

(Bulgarian language; Български) - the point of their contact

(Bulgarian language; Български) - ъгъл in triene

(Czech language; Čeština) - úhel trení

(German; Deutsch) -Reibungswinkel

(Hungarian; Magyar) - súrlódási szög

(Mongolian) - Ureltiin Ontsog

(Polish language; Polska) - kąt tarda

(Romanian language; Român) - unghi de frecare

(Serbo-Croatian language; Srpski jezik; Hrvatski jezik) - ugao trenja

(Spanish; Español) - ángulo de rozamiento

(English language; English) - angle of friction

(French; Français) - angle de frottement

Construction dictionary.

See what "FRICTION ANGLE" is in other dictionaries:

friction angle- The angle formed when the reaction forces of two bodies deviate from the common normal to their contact surface due to the presence of friction forces [Terminological dictionary of construction in 12 languages (VNIIIS Gosstroy USSR)] Subjects of science technical other EN angle ...

friction angle

friction angle- angle of friction The largest possible angle formed by the reaction of two contacting bodies and the common normal to their surfaces at the point of contact. IFToMM code: 3.5.51 Section: DYNAMICS OF MECHANISMS... Theory of mechanisms and machines

static friction angle- The angle of deviation from the normal of the resultant force obtained by graphically summing up the static friction forces. Mechanical engineering topics in general... Technical Translator's Guide

friction angle at the soil-structure interface- d - [English-Russian dictionary for the design of building structures. MNTKS, Moscow, 2011] Topics building structures Synonyms d EN structure ground interface friction angle ... Technical Translator's Guide

reduced friction angle- The angle of friction when parts come into contact with inclined surfaces, equal to the arcsine of the reduced friction coefficient. Mechanical engineering topics in general... Technical Translator's Guide

limiting friction angle- trinties kampas statusas T sritis fizika atitikmenys: engl. angle of friction; angle of repose; limiting angle; limiting angle of friction vok. Gleitwinkel, m; Grenzwinkel, m; Grenzwinkel der Reibung, m; Reibungswinkel, m rus. limit angle, m; … Fizikos terminų žodynas

Angle of internal friction- parameter of the direct dependence of soil shear resistance on vertical pressure, defined as the angle of inclination of this straight line to the abscissa axis. Source: GOST 30416 96: Soils. Laboratory tests. General provisions original document... Dictionary-reference book of terms of normative and technical documentation

Angle of repose is the angle formed by the free surface of a loose rock mass or other bulk material with a horizontal plane. The term "angle of internal friction" may sometimes be used. Mate particles ... Wikipedia

Angle of repose Angle of repose is the angle formed by the free surface of a loose rock mass or other bulk material with a horizontal plane. The term "angle of internal friction" may sometimes be used. Particles... ...Wikipedia

Books

Improving methods for determining the strength properties of rocks and their deformability when using new technological processes in quarries and maintaining the stability of workings in projects, G. M. Eremin. The main provisions of existing methods for determining the strength properties of rocks are presented. Some disadvantages of these methods are pointed out and ways to eliminate them are given. It is shown that the main…

Theoretical mechanics (Statics)

Lecture7

Summary: Sliding friction. Coulomb's laws. Angle and cone of friction. Equilibrium conditions. Rolling friction.

FRICTION

Sliding friction

Experience shows that when one body tries to move along the surface of another in the plane of contact of the bodies, a resistance force arises to their relative sliding. This force is called the sliding friction force.

If a solid body is in equilibrium on an absolutely smooth surface of another body, then the bond reaction is directed normal to the surface.

In reality, there are no absolutely smooth surfaces. All surfaces of bodies are rough to one degree or another. Therefore the reaction force rough surface when the body is in equilibrium depends on active forces not only in numerical value, but also in direction.

Let's break down the reaction force rough surface into components: one of which we direct along the common normal to the contact surface, and the other let us direct it in a tangent plane to these surfaces.

Friction force sliding (or simply friction force) is the component of the bond reaction force that lies in the tangent plane to the surfaces of the contacting bodies.

By force of normal reaction bond is the component of the bond reaction force, which is directed along the common normal to the surfaces of the contacting bodies.

The nature of the friction force is very complex and We do not touch upon it. In theoretical mechanics, it is assumed that there is no lubricant between the surfaces of contacting bodies.

Dry friction called friction when there is no lubricant between the surfaces of contacting bodies.

We will consider two cases: friction when a body is at rest or in equilibrium and sliding friction when one body moves along the surface of another with a certain relative speed.

Similarly, with the chosen direction of the normal, the normal reaction is expressed in terms of the given forces:

When one body moves on the surface of another, the friction force is a constant value.

Coulomb's Laws

The sliding friction force is located in the common tangent plane of the contacting surfaces of the bodies and is directed in the direction opposite to the direction of possible sliding of the body under the action of active forces. The friction force depends on the active forces, and its module is between zero and the maximum value, which is achieved at the moment the body leaves the equilibrium position, that is:

- called ultimate friction force .

The maximum sliding friction force, all other things being equal, does not depend on the area of contact of the rubbing surfaces. From this law it follows that in order to move, for example, a brick, it is necessary to apply the same force, regardless of which face it is placed on the surface, wide or narrow.

The limiting sliding friction force is proportional to the normal reaction (normal pressure), that is

where is the dimensionless coefficient called the sliding friction coefficient; it is independent of the normal reaction.

The sliding friction coefficient depends on the material and physical state of the rubbing surfaces, that is, on the size and nature of roughness, humidity, temperature and other conditions. The friction coefficient is determined experimentally.

It is believed that the friction coefficient does not depend on the speed of movement.

Friction angle. Equilibrium conditions.

Reaction of real (rough) connection consists of two components: normal reaction and the friction force perpendicular to it . Therefore, the coupling reaction deviates from the normal to the surface by a certain angle. When the friction force changes from zero to maximum, the reaction force changes from zero to
, and its angle with the normal increases from zero to a certain limiting value  .

U friction glom called the largest angle between the ultimate reaction force of a rough bond
and normal reaction .

; ;.

Friction angle depends on the friction coefficient.

Friction cone called a cone described by the ultimate reaction force of a rough bond
around the direction of the normal reaction.

Example.

E If a force P is applied to a body lying on a rough surface, forming an angle with the normal, then the body will move only when the shear force
 will be greater than the ultimate friction force
 (if we neglect body weight, then
but inequality

Executed only when
, i.e. at
,

Consequently, no force forming an angle with the normal , less than the friction angle  the body cannot be moved along this surface.

A body cannot be thrown out of balance by any modulus active force if its line of action passes inside the friction cone.

Example.

Let's consider a body that has a vertical plane of symmetry. The cross-section of the body of this plane has the shape of a rectangle. Body width is 2a.

A vertical force is applied to the body at point C, lying on the axis of symmetry and at point A, lying at a distance h from the base, the horizontal force . The base plane reaction (bond reaction) is reduced to the normal reaction and friction force . Line of action of force unknown. Distance from point C to the line of force let's denote x. (
). Let's create three equilibrium equations:

According to Coulomb's law
, i.e.
. (1)

Because
, That
(2)

Let's analyze the results:

We will increase our strength .

Rolling friction

Rolling friction is the resistance that occurs when one body rolls over the surface of another.

The active forces acting on rollers in the form of wheels usually consist of gravity , horizontal force , applied to the center of the skating rink, and a pair of forces with a moment trying to roll the wheel. The wheel in this case is called follower-leader. If
, A
, then the wheel is called slave. If
, A
, then the wheel is called leading.

The contact of the roller with a stationary plane due to the deformation of the roller and the plane occurs not at a point, but along a certain line BD. Along this line, distributed reaction forces act on the roller. If we bring the reaction forces to point A, then at this point we get the main vector these distributed forces with components (normal reaction) and (sliding friction force), as well as a couple of forces with moment
.

R
Let's look at the equality of the skating rink. The system of forces is flat. Let us write down the equilibrium equations for the system of forces.

(M A)

Moment
is called the rolling friction moment. Highest value M is achieved at the moment the roller begins to roll on the plane.

The following approximate laws have been established for the largest moment of a pair of forces that prevent rolling.

1. The largest moment of a pair of forces preventing rolling does not depend, within a fairly wide range, on the radius of the roller.

2. Limit torque value
proportional to normal reaction
.

Proportionality factor k called rolling friction coefficient at rest. Dimension k is the dimension of length.

3. Rolling friction coefficient k depends on the material of the skating rink, the plane and the physical condition of their surfaces. As a first approximation, the rolling friction coefficient during rolling can be considered independent of the angular velocity of the roller and its sliding speed along the plane.

For carriage wheel on rail
mm.