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Technical mechanics

Modern production, determined by high mechanization and automation, offers the use of a large variety of machines, mechanisms, instruments and other devices. The design, manufacture, operation of machines is impossible without knowledge in the field of mechanics.

Technical mechanics - a discipline that includes the main mechanical disciplines: theoretical mechanics, strength of materials, theory of machines and mechanisms, machine parts and design fundamentals.

Theoretical mechanics - a discipline that studies the general laws of mechanical motion and mechanical interaction of material bodies.

Theoretical mechanics belongs to the fundamental disciplines and forms the basis of many engineering disciplines.

Theoretical mechanics is based on laws called the laws of classical mechanics or Newton's laws. These laws are established by summarizing the results of a large number of observations and experiments. Their validity has been verified by centuries of practical human activity.

Statics - section of theoretical mechanics. in which forces are studied, methods for converting systems of forces into equivalent ones, and the conditions for the balance of forces applied to solids are established.

Material point - a physical body of a certain mass, the dimensions of which can be neglected when studying its motion.

System of material points or mechanical system - this is such a set of material points in which the position and movement of each point depend on the position and movement of other points of this system.

Solid is a system of material points.

Absolutely solid - a body in which the distances between two arbitrary points of it remain unchanged. Assuming the bodies are absolutely rigid, they do not take into account the deformations that occur in real bodies.

Force F- a quantity that is a measure of the mechanical interaction of bodies and determines the intensity and direction of this interaction.

The SI unit of force is the newton (1 N).

As for any vector, for a force, you can find the projections of the force on the coordinate axes.

Force types

internal forces call the forces of interaction between points (bodies) of a given system

Outside forces called the forces acting on the material points (bodies) of a given system from the side of material points (bodies) that do not belong to this system. External forces (load) are active forces and coupling reactions.

Loads divided into:

• voluminous- distributed over the volume of the body and applied to each of its particles (self-weight of the structure, magnetic attraction forces, inertia forces).
• superficial- applied to the surface areas and characterizing the direct contact interaction of the object with the surrounding bodies:
  • concentrated- loads acting on the site, the dimensions of which are small compared to the dimensions of the structural element itself (pressure of the wheel rim on the rail);
  • distributed- loads acting on the site, the dimensions of which are not small compared to the dimensions of the structural element itself (the tractor caterpillars press on the bridge beam); the intensity of the load distributed along the length of the element, q N/m.

Axioms of statics

Axioms reflect the properties of the forces acting on the body.

1.Axiom of inertia (Galilean law).
Under the action of mutually balanced forces, a material point (body) is at rest or moves uniformly and rectilinearly.

2.Axiom of balance of two forces.
Two forces applied to a rigid body will be balanced only if they are equal in absolute value and directed along one straight line in the opposite direction.

The second axiom is the equilibrium condition for a body under the action of two forces.

3.Axiom of adding and dropping balanced forces.
The action of this system of forces on an absolutely rigid body will not change if any balanced system of forces is added to or removed from it.
Consequence. Without changing the state of an absolutely rigid body, the force can be transferred along its line of action to any point, keeping its modulus and direction unchanged. That is, the force applied to an absolutely rigid body is a sliding vector.

4. Axiom of the parallelogram of forces.
The resultant of two forces that intersect at one point is applied at the point of their section and is determined by the diagonal of the parallelogram built on these forces as sides.

5. Axiom of action and reaction.
For every action there is an equal and opposite counteraction.

6. The axiom of the balance of forces applied to a deformable body during its solidification (the principle of solidification).
The balance of forces applied to a deformable body (changeable system) is preserved if the body is considered to be solidified (ideal, unchanged).

7. Axiom of liberation of the body from bonds.
Without changing the state of the body, any non-free body can be considered as free, if we discard the connections, and replace their action with reactions.

Connections and their reactions

free body called a body that can carry out arbitrary movements in space in any direction.

connections bodies that restrict the movement of a given body in space are called.

A free body is a body whose movement in space is limited by other bodies (connections).

Coupling reaction (support) is the force with which the bond acts on a given body.

The reaction of the bond is always directed opposite to the direction in which the bond counteracts the possible movement of the body.

Active (given) force , is a force that characterizes the action of other bodies on a given one, and causes or can cause a change in its kinematic state.

Reactive force - a force that characterizes the action of bonds on a given body.

According to the axiom about the release of the body from bonds, any non-free body can be considered as free, freeing it from bonds and replacing their action with reactions. This is the principle of liberation from ties.

Converging force system

Converging force system is a system of forces whose lines of action intersect at one point.

A system of converging forces equivalent to one force - resultant , which is equal to the vector sum of forces and applied at the point of section of the lines of their action.

Methods for determining the resultant system of converging forces.

1. The method of parallelograms of forces - Based on the axiom of the parallelogram of forces, every two forces of a given system, sequentially, are reduced to one force - the resultant.
2. Construction of a vector force polygon - Sequentially, parallel transfer of each force vector to the end point of the previous vector, a polygon is formed, the sides of which are the vectors of the forces of the system, and the closing side is the vector of the resultant system of converging forces.

Conditions for the equilibrium of a system of converging forces.

1. The geometric condition for the equilibrium of a converging system of forces: for the equilibrium of a system of converging forces, it is necessary and sufficient that the vector force polygon built on these forces be closed.
2. Analytical conditions for the equilibrium of a system of converging forces: for the equilibrium of a system of converging forces, it is necessary and sufficient that the algebraic sums of the projections of all forces onto the coordinate axes equal zero.

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An example of the calculation of a spur gear
An example of the calculation of a spur gear. The choice of material, the calculation of allowable stresses, the calculation of contact and bending strength were carried out.

An example of solving the problem of beam bending
In the example, diagrams of transverse forces and bending moments are plotted, a dangerous section is found, and an I-beam is selected. In the problem, the construction of diagrams using differential dependencies was analyzed, comparative analysis different cross sections of the beam.

An example of solving the problem of shaft torsion
The task is to test the strength of a steel shaft for a given diameter, material and allowable stresses. During the solution, diagrams of torques, shear stresses and twist angles are built. Self weight of the shaft is not taken into account

An example of solving the problem of tension-compression of a rod
The task is to test the strength of a steel rod at given allowable stresses. During the solution, plots of longitudinal forces, normal stresses and displacements are built. Self weight of the bar is not taken into account

Application of the kinetic energy conservation theorem
An example of solving the problem of applying the theorem on the conservation of kinetic energy of a mechanical system

1. Smooth plane (surface) or support. A smooth surface is a surface on which the friction of a given body can be neglected in the first approximation. Such a surface does not allow the body to move only in the direction of the common perpendicular (normal) to the surfaces of the bodies in contact at the point of their contact (Fig. 7, A). Therefore, the reaction N smooth surface or support is directed along the common normal to the surfaces of the bodies in contact at the point of their contact and is applied at this point. When one of the contacting surfaces is a point (Fig. 7, b), then the reaction is directed along the normal to the other surface.

If the surfaces are not smooth, one more force must be added - friction force, which is directed perpendicular to the normal reaction in the direction opposite to the possible sliding of the body.

Rice. 7

2. Thread. The connection, made in the form of a flexible inextensible thread (Fig. 8), does not give the body M move away from the suspension point of the thread in the direction AM. Therefore, the reaction T stretched thread is directed along the thread from the body to the point of its suspension.

Rice. 8

3. Cylindrical joint (bearing). If two bodies are connected by a bolt passing through holes in these bodies, then such a connection is called a hinge or simply a hinge; the centerline of the bolt is called the hinge axis. Body AB, hinged to a support D(fig.9, A), can be rotated arbitrarily around the hinge axis (in the plane of the drawing); while the end A body cannot move in any direction perpendicular to the hinge axis. Therefore, the reaction R cylindrical hinge can have any direction in the plane perpendicular to the axis of the hinge, i.e. in plane A hu. For strength R in this case, neither its modulus is known in advance R, nor direction (angle ).

4. Ball joint and thrust bearing. This type of connection fixes some point of the body so that it cannot make any movements in space. An example of such connections is the ball foot, with which the camera is attached to the tripod (Fig. 9, b) and a bearing with an emphasis (thrust bearing) (Fig. 9, V). Reaction R ball joint or thrust bearing can have any direction in space. For her, neither the reaction modulus is known in advance R, nor the angles it forms with the axes x, y, z.

Rice. 9

5. Rod. Let a rod be a link in some construction AB, fixed at the ends with hinges (Fig. 10). Let us assume that the weight of the rod in comparison with the load it perceives can be neglected. Then only two forces applied in the hinges will act on the rod A And IN. But if the rod AB is in equilibrium, then, according to axiom 1, applied at the points A And IN forces must be directed along one straight line, i.e. along the axis of the rod. Consequently, a rod loaded at the ends, whose weight can be neglected in comparison with these loads, works only in tension or compression. If such a rod is a bond, then the reaction of the rod will be directed along the axis of the rod.

Fig.10

6. Movable hinged support (Fig. 11, support A) prevents the body from moving only in the direction perpendicular to the sliding plane of the support. The reaction of such a support is directed along the normal to the surface on which the rollers of the movable support rest.

7. Fixed articulated support (Fig. 11, support IN). The reaction of such a support passes through the hinge axis and can have any direction in the plane of the drawing. When solving problems, we will represent the reaction by its components and along the directions of the coordinate axes. If we, having solved the problem, find and , then the reaction will be determined thereby; modulo

Fig.11

The fixing method shown in Fig. 11 is used in order to AB no additional stresses occurred when its length changed due to temperature changes or bending.

Note that if the support A beams (Fig. 11) also be made immovable, then the beam, under the action of any flat system forces will be statically indeterminate, because then the three equilibrium equations will include four unknown reactions , , , .

8. Fixed pinching support or rigid termination (Fig. 12). In this case, a system of distributed reaction forces acts on the embedded end of the beam from the side of the support planes. Assuming these forces are brought to the center A, we can replace them with one unknown force applied in this center, and a pair with a moment unknown in advance. The force can, in turn, be represented by its components and . Thus, to find the reaction of a fixed pinching support, it is necessary to determine three unknown quantities , and . If under such a beam somewhere at a point IN add another support, the beam will become statically indeterminate.

Fig.12

When determining the coupling reactions of other structures, it is necessary to establish whether it allows it to move along three mutually perpendicular axes and rotate around these axes. If it prevents any movement - show the corresponding force, if it prevents rotation - a pair with the corresponding moment.

Sometimes it is necessary to investigate the equilibrium of non-rigid bodies. In this case, we will use the assumption that if this non-rigid body is in equilibrium under the action of forces, then it can be considered as a rigid body, using all the rules and methods of statics.

Example 1 A weightless three-hinged arch is affected by a horizontal force (Fig. 13). Determine the line of action of the reaction (the reaction of the connection at the point A).

Solution: Consider the right side of the arch separately. At points IN And WITH we apply the reaction forces of bonds and . A body under the influence of two forces is in equilibrium. According to the axiom of the balance of two forces, the forces and are equal in magnitude and act along one straight line in opposite directions. Thus, we know the direction of the force (along the line Sun).

Rice. 13

Consider the left side of the arch separately. At points A And WITH we apply the reaction forces of bonds and . Force, action is equal to reaction. Three forces act on the body, the directions of the two forces ( and .) are known. According to the three forces theorem, the lines of action of all three forces intersect at one point. Therefore, the force is directed along the line AD. directed along the line AE.

Final part

Recall that in this lesson the basic concepts of statics are considered: a pair of forces, a moment of a pair of forces, bonds, reactions of bonds.

Answer student questions.

Submit a self-study assignment.

v. Self-study task

1. Analyze the abstract material.

2. To study the questions: the main task of statics, analytical conditions for the equilibrium of an arbitrary system of forces.

VI. Literature

1. Butenin N.V., Lunts Ya.L., Merkin D.R. Theoretical course

mechanics in 2 volumes. - St. Petersburg: Lan, 2008, 736 p.

2. Yablonsky A.A., Nikiforova V.M. Course of theoretical mechanics. Part 1. Statics. Kinematics. M.: Higher. school, 2004

3. Tsyvilsky V.L. Theoretical mechanics. M.: Vyssh.shk., 2004. - 343 p.

Developed ____________________________________________________

(signature, position, surname, title)

"___" ______________2012

Note that all the provisions and the dependences obtained earlier are valid. for a free rigid body. However, in most engineering problems we encounter a non-free body, that is, a body whose movement in space is hindered by some other bodies.

Bodies that limit the movement of a given body in space are in relation to it connections. For example, for a book lying on a table, the link will be the plane of the table, preventing the book from moving vertically down; for the table, the link is the floor surface; for a door, the link is the hinges on which it is hung, etc.

The effect of the bond on a given body is expressed by some forces acting on the body from the side of the bond. These forces are called communication reactions.

The numerical values ​​of the bond reactions are, as a rule, unknown, and they are determined by the solution of the corresponding problem in mechanics. The direction of the reaction of the bonds is determined by the design features of the place of conjugation (contact) of the body under consideration and the body making the connection. The reaction of the connection is directed opposite to those displacements of the body under consideration, which this connection does not allow to be realized.

The main types of bonds and their reactions are presented in Table. 2.1. The left column shows the body under consideration and the bonds imposed on it, the right column shows the body under consideration, freed from the bonds, and the reactions of the bonds acting on it; active forces acting on the body are not shown.

Smooth surface(see Table 2.1). A smooth surface is a surface on which friction can be neglected. Reaction N applied at the point of contact of the body with the surface of the support, directed towards the body along the common normal to the surfaces of the contacting bodies. When one of the contacting surfaces is a point (as it will be, for example, at a support in the form of a ledge), then the reaction is directed along the normal to the other surface.

A thread. The term "thread" denotes cables, ropes, chains, which are considered flexible, inextensible and can only perceive tensile forces. The reaction of the thread is directed along it from the body ("inside" the thread).

A hinge is such a connection of bodies that allows them to rotate mutually. If the body under consideration is connected by a hinge to a fixed base, then this connection is called a fixed hinged support.

Cylindrical hinge (bearing) allows mutual rotation of bodies around its axis and sliding along it. By its design, a cylindrical hinge is a support of a cylindrical element of one body (in Fig. 2.2, A its section is shaded, the axis of the cylinder is perpendicular to the plane of the drawing) to the inner surface of the cylindrical hole of another body. The contact of these bodies occurs along some generatrix of a cylindrical surface, which in section (Fig.

2.2, b) perpendicular to the axis of the cylinder is projected to the "point of contact" TO. Communication reaction (in Fig. 2.2, A the left body is considered a connection for the right one) passes through the hinge axis and is located in a plane perpendicular to this axis. Since, depending on the acting forces, the “point of contact” of the cylindrical surfaces of the bodies will change, then for the reaction N in this case, neither its modulus (7V) nor its direction (angle cp) are known (Fig. 2.2, V). When solving problems, instead of two unknowns N, it is convenient to represent the reaction of a cylindrical hinge in the form of two components X A, Y A(Fig. 2.2, G).

Table 2.1. Bond reactions

	Bond reactions

The name of the connections and their designation on the diagrams	Bond reactions

Spherical (ball) joint allows the connected bodies to make spatial mutual rotations around their center. The reaction of such a hinge passes through its center and can have any direction in space. When solving problems, the reaction of a spherical hinge is represented as three components directed along the coordinate axes. The reactions of the thrust bearing (thrust bearing) are directed in a similar way.

Weightless rod with hinges at the ends. If the rod is rectilinear, then its reaction is directed along the axis (if curvilinear, then along the straight line connecting the end hinges of the rod). Unlike a thread, a rod perceives both tensile and compressive forces.

Articulated support is a combination of a cylindrical hinge and a smooth surface along which the support can either slide or move on rollers. This circumstance determines the direction of the reaction R- to the body, perpendicular to the fixed reference plane.

Termination. Rigid embedding is such an introduction of a given body into another, in which there is no mutual movement (for example, a nail is driven into a wall, a balcony slab is embedded in a wall, a pole is dug into the ground, the connection of metal bodies by welding). The embedding reaction is a force distributed over the contact surface of the bodies. If a flat system of forces acts on the body under consideration (Fig. 2.3, A), then the reaction of rigid termination is represented in the form of two components X A, Y A and pairs of forces with a moment t A(Fig. 2.3, b) operating in the lightness of the load. The reaction of the sliding stop consists of the force Y A and pairs of forces with a moment t A.

If an arbitrary spatial system forces, then the reaction of rigid termination (force R and a couple of_ forces with a moment M) represent_in the form of three components of forces X, Y, Z and three constituent pairs M x, M y, M z(see problem 2.7 in § 2.3).

It should be noted that when solving problems, the directions of the reactions of the bonds (or their components) in the drawing should be depicted in accordance with the design of the connection (see Table 2.1), regardless of the directions and magnitudes of the acting active forces. The numerical values ​​of the reactions will be determined by the subsequent calculation; if the algebraic value of the reaction is obtained with a minus sign, then, therefore, the corresponding reaction has a direction opposite to that originally accepted.

When solving problems of the mechanics of non-free mechanical systems, the axiom of connections (the principle of liberability) is used, according to which any non-free mechanical system can be considered as free if it is mentally freed from bonds and reactions of bonds are applied to it.

The condition for the equivalence of these two systems is the fulfillment of equilibrium equations for a free mechanical system. Reactions of bonds will participate in the equilibrium equations along with other forces acting on the considered mechanical system. Thus, applying the axiom of connections, the problem of a non-free body is reduced to the problem of a free body.

Let us show how the axiom of constraints is realized when considering an equilibrium, for example, a beam LV(Fig. 2.4, A), anchored at a point A cylindrical hinge and supported at a point IN onto a smooth surface. An active force is applied to the beam F and a couple of forces with a moment M. In accordance with the axiom of connections (principle of

expectancy) mentally discard from the beam L V connection and we will consider it as a free body (Fig. 2.4, b) on which, in addition to a given load (force ^ Г and a pair of forces with a moment M) bond reactions work X A, Y A And N.

It should be borne in mind that when solving problems, the image of a body without connections (as done in Fig. 2.4, b) is not strictly required; sometimes the effect of bond reactions on a body is shown on the original design drawing, implying that this body is "free".

1. Smooth surface (flat) or support(Figure 1.10). A smooth plane is a plane or surface on which friction can be neglected. Such a bond does not prevent the body surface from sliding along it, but only prevents movement in the direction of the common normal to the body surface and to the bond, so the reaction force is directed along this normal. (Any two surfaces that are in contact at a point have a common normal and a common tangent plane through the point of contact). Such a reaction is called normal reaction force.

When one of the contacting surfaces is a point (Fig. 1.10, b), then the reaction is directed along the normal to the other surface.

Figure 1.10 - Reactions of smooth surfaces or supports

2. Thread. The connection, made in the form of a flexible inextensible thread (Figure 1.11), does not allow the body M to move away from the suspension point of the thread in the direction AM. Therefore, the reaction of a stretched thread is directed along the thread to the point of its suspension.

Figure 1.11 - Thread reaction

3. Cylindrical joint (bearing). One body can rotate relative to another around a common axis called hinge axis(for example, like two halves of scissors). If the body AB attached with such a hinge to a fixed support D(Fig. 1.12, A), then the point A the body cannot move in any direction perpendicular to the axis of the hinge. Therefore, the reaction of the cylindrical hinge can have any direction in the plane perpendicular to the axis of the hinge, i.e. in the Axy plane. For the force in this case, neither its modulus is known in advance R, nor direction (angle α). Often the reaction is replaced by its components along the x and y axes, as in Fig. 1.12, b, where is the hinge A fixed with two hinged rods.

A) b)

Figure 1.12 - Hinge reaction

4. Movable hinge. A rod fixed on a hinge can rotate around the hinge, and the attachment point can move along the guide. The reaction of the movable hinge is directed perpendicular to the supporting surface, since only movement across the supporting surface is not allowed.

5. Spherical joint and thrust bearing. Bodies connected by a spherical hinge can arbitrarily rotate one relative to the other around the center of the hinge. An example is attaching a camera to a tripod. Dot A bodies (Fig. 1.13 A), coinciding with the center of the hinge, cannot make any movement in space. Therefore, the reaction of the spherical hinge can have any direction in space. For her, neither her module R nor the angles with the axes are known in advance Axyz.

An arbitrary direction in space can also have the reaction of the thrust bearing (bearing with a stop), shown in Fig. 1.13 b.

Figure 1.13 – Spherical joint and thrust bearing

6. Weightless rod. A weightless rod is a rod whose weight, compared with the load it perceives, can be neglected. Let, for a body in equilibrium, such a rod attached at points A And B hinges, is a connection (examples in Fig. 1.14 A And b). Then only two forces will act on the rod, applied at the points A And B. At equilibrium, these forces must be directed along one straight line, i.e. along AB. But then, according to the law of action and reaction, the rod will act on the body with a force also directed along AB. Hence, reaction of a weightless pivotally attached rod is directed along the straight line connecting the hinges. In this case, the rod can be straight or curved, compressed or stretched.

Figure 1.14 - Connections in the form of weightless rods: direct stretched ( A) and curvilinear compressed ( b)

7. guide allows only rectilinear movement along its axis. Generates a reaction force in a direction perpendicular to the axis and a reaction moment (in a plane system of forces). (Fig. 1.15, A.) If one rail is mounted on another rail, then the body can move in the plane as desired, but cannot rotate. (Fig. 1.15, b.). In such a support there is a reaction in the form of a moment.

A) b)

Figure 1.15 - Reactions of the guides: A)- single and b)- double

8. Rigid termination(Figure 1.16, a, b). Even one rigid seal ensures the balance of the body under any kind of load. The embedding reaction force can have any direction, so the components are usually determined Rx,Ry,Rz. (in figure 1.16, A they are marked with and ) In addition, a rigid embedment prevents the body from turning, so the embedment moment acts on the body. It, as we will see below, can also be represented as the sum of the moments Mx, M y, Mz. In the case of the action of a flat system of forces, two components of the reaction arise (in Figure 1.16, b they are designated and ) and the moment of termination in the plane of action of forces (in Figure 1.16, b it is marked M h).

Figure 1.16 - Embedding reactions in the spatial (a) and flat (b) system of forces

9. Communication is carried out through a non-smooth fixed surface(Figure 1.17, a, b). So far, we have considered connections that are carried out by means of absolutely smooth surfaces. In reality, real surfaces are non-smooth (rough). A non-smooth surface not only prevents movement that breaks the connection, but also offers some resistance to movement along this surface. This resistance also represents a certain reaction directed along the tangent plane to the surface and is called sliding friction force. The force of sliding friction is directed in the direction opposite to that in which the body is moved or tends to be moved by the active forces applied to it. Like any reaction of the connection, the force of friction is determined by those active forces that act on the body in question*. Consequently, the reaction of a non-smooth immovable surface has two components: one is normal to the surface making a non-smooth bond, and the other is lying in a common tangent plane to the surface of the body and the surface making a non-smooth bond. The first component - the normal reaction force - in Figure 1.17 a, b denoted by and , and
the second component, the sliding friction force, is denoted by and in the same figures.

Rice. – Reactions taking into account the force of friction

Bonds and Reactions of Bonds

All laws and theorems of statics are valid for a free rigid body.

All bodies are divided into free and bound.

Free bodies are bodies whose movement is not limited.

Connected bodies are bodies whose movement is limited by other bodies.

Bodies that restrict movement other bodies are calledconnections.

Forces acting from bonds and preventing movement are calledbond reactions.

The reaction of communication is always directed from that side,where you can't move.

Any bound body can be represented as free if the bonds are replaced by their reactions (the principle of liberation from bonds).

All connections can be divided into several types.

Communication - smooth support (without friction)

Picture 1

The support reaction is applied at the support point and is always directed perpendicular to the support (Fig. 1).

Flexible connection (thread, rope, cable, chain) The load is suspended on two threads (Fig. 2).

Figure 2

Rigid rod

In the diagrams, the rods are depicted with a thick solid line (Fig. 3).

Figure 3

The rod can be compressed or stretched. The reaction of the rod is directed along the rod. The rod works in tension or compression. The exact direction of the reaction is determined by mentally removing the rod and considering possible movements of the body without this connection.

Possible displacement point is called such an infinitesimal mental displacement, which is allowed at a given moment by the bonds imposed on it.

We remove the rod 1, in this case the rod 2 falls down. Therefore, the force from rod 1 (reaction) is directed upwards. We remove rod 2. In this case, the pointA descends, moving away from the wall. Therefore, the reaction of rod 2 is directed towards the wall.

articulated support

The hinge allows rotation around the anchor point. There are two types of hinges.

Movable hinge

The rod, fixed on the hinge, can rotate around the hinge, and the attachment point can move along the guide (platform) (Fig. 4).

Figure 4

The reaction of the movable hinge is directed perpendicular to the supporting surface, since only movement across the supporting surface is not allowed.

fixed hinge

The attachment point cannot be moved. The rod can freely rotate around the hinge axis. The reaction of such a support passes through the hinge axis, but is unknown in direction. It is usually depicted in the form of two components: horizontal and vertical.( Rx ; R y) (Fig. 5).

Figure 5

Pinching or "pinching"

Any movement of the attachment point is not possible.

Under the action of external forces in the support, a reactive force and a reactive moment M R , which prevents rotation (Fig. 6).

Figure 6

It is customary to represent the reactive force in the form of two components along the coordinate axes

Examples of problem solving

Example 1 The load is suspended on rods and ropes and is in balance (Fig. 7). Depict the system of forces acting on the hingeA.

Figure 7

Solution

1. The reactions of the rods are directed along the rods, the reactions of the flexible bonds are directed along the threads in the direction of tension (Fig. 7a).

2. To determine the exact direction of the forces in the rods, we mentally remove the rods 1 and 2 in succession. We analyze the possible displacements of the pointA.

We do not consider a fixed block with forces acting on it.

3. Remove rod 1, pointA rises and moves away from the wall, therefore, the reaction of rod 1 is directed towards the wall.

4. Remove rod 2, pointA rises and approaches the wall, therefore, the reaction of rod 2 is directed downwards from the wall.

5. The rope pulls to the right.

6. We get rid of bonds (Fig. 7b).

Example 2 The ball is suspended on a thread and rests on a wall (Fig. 8a). Determine the reactions of a thread and a smooth support (wall).

Figure 8

Solution

1. Thread reaction - along the thread to the pointIN up (Fig. 8b).

2. The reaction of a smooth support (wall) - along the normal from the surface of the support.

Control questions and tasks

4. Indicate the possible direction of reactions insupports(Fig. 9).

Figure 9