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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 rigid body - 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:
  • focused- 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, by 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 is analyzed, a comparative analysis of various beam cross sections is carried out.

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

body name free, if its movements are not limited by anything. A body whose movement is limited by other bodies is called not free, and the bodies that limit the movement of this body, - connections.At the points of contact, interaction forces arise between the given body and the bonds. The forces with which the bonds act on a given body are called bond reactions.

The principle of release: any non-free body can be considered as free if the action of the bonds is replaced by their reactions applied to the given body. In statics, the reactions of the bonds can be completely determined using the conditions or equations of equilibrium of the body, which will be established later, but their directions in many cases can be determined from an examination of the properties of the bonds. As a simple example, in Fig. 1.14, but a body is represented, the point M of which is connected to the fixed point O with the help of a rod, the weight of which can be neglected; the ends of the rod have hinges allowing freedom of rotation. In this case, the rod OM serves as a link for the body; constraint on the freedom of movement of the point M is expressed in the fact that it is forced to be at a constant distance from the point O. The force of action on such a rod should be directed along the straight line OM, and according to axiom 4, the counteraction force of the rod (reaction) R should be directed along the same straight line . Thus, the direction of the reaction of the rod coincides with the direct OM (Fig. 1.14, b). Similarly, the reaction force of a flexible inextensible thread must be directed along the thread. On fig. 1.15 shows a body hanging on two threads and the reactions of the threads R 1 and R 2 . The forces acting on a non-free body are divided into two categories. One category is formed by forces that do not depend on the bonds, and the other is the reactions of the bonds. At the same time, the reactions of the bonds are passive in nature - they arise because the forces of the first category act on the body. The forces that do not depend on the bonds are called active, and the reactions of the bonds are called passive forces. On fig. 1.16, and at the top are two active forces F 1 and F 2 equal in absolute value, stretching the rod AB, below are the reactions R 1 and R 2 of the stretched rod. On fig. 1.16, b, the active forces F 1 and F 2 compressing the rod are shown at the top, the reactions R 1 and R 2 of the compressed rod are shown below.

We agree to consider the body free , if its movements are not limited by anything. A body whose movement is limited by other bodies is called not free , and the bodies that limit the movement of this body, connections . At the points of contact, interaction forces arise between the given body and the bonds. The forces with which the bonds act on a given body are called bond reactions . When listing all the forces acting on a given body, these contact forces (reactions of bonds) must also be taken into account.

In mechanics, they take the following position, sometimes called the principle of liberation: any non-free body can only be considered as free if the action of the bonds is replaced by their reactions applied to the given body.

In statics, the reactions of the bonds can be completely determined using the conditions or equations of equilibrium of the body, but their directions in many cases can be determined from an examination of the properties of the bonds. As a simple example, consider a body, a point M which is connected to a fixed point O using a rod, the weight of which can be neglected; the ends of the rod have hinges allowing freedom of rotation. In this case, a rod serves as a link for the body. OM. Restriction of freedom of movement of a point M is expressed in the fact that it is forced to be at a constant distance from the point O. But, as we saw above, the force acting on such a rod must be directed in a straight line OM. According to axiom 4, the reaction force of the rod (reaction) R should be in the same straight line. Thus, the direction of the reaction of the rod coincides with the straight line OM. (In the case of a curved weightless rod - along a straight line connecting the ends of the rod).

Similarly, the reaction force of a flexible inextensible thread must be directed along the thread. On fig. A body hanging on two threads and the reactions of the threads are shown. R1 and R2.

In the general case, the forces acting on a non-free body (or on a non-free material point) can be divided into two categories. One category is formed by forces that do not depend on the bonds, and the other category is formed by the reactions of the bonds. At the same time, the reactions of bonds, in essence, are passive in nature. They arise only insofar as certain forces of the first category act on the body. Therefore, forces that do not depend on constraints are called active forces (sometimes called given ), and the bond reactions passive forces.

On fig. 1.16 at the top shows two active forces equal in modulus F1 and F2, stretching the rod AB, reactions are shown below R1 and R2 stretched rod. On fig. showing active forces F1 and F2, compressing the rod, the reactions are shown below R1 and R2 compressed rod.

Let us consider some more typical types of bonds and indicate the possible directions of their reactions. Reaction modules are determined by active forces and cannot be found until the latter are specified in a certain way. In this case, we will use some simplified representations that schematize the actual properties of real connections.

1. If a rigid body rests on a perfectly smooth (without friction) surface, then the point of contact of the body with the surface can freely slide along the surface, but cannot move along the normal to the surface. The reaction of an ideally smooth surface is directed along the common normal to the contacting surfaces.

If a solid body has a smooth surface and rests on a point, then the reaction is directed along the normal to the surface of the body itself.

If a solid body rests with its tip against a corner, then the connection prevents the tip from moving both horizontally and vertically. Accordingly, the reaction R angle can be represented by two components - horizontal R x and vertical R, whose magnitudes and directions are ultimately determined by the given forces.

2. spherical joint called a device that makes a fixed point O of the considered body (the center of the hinge). If the spherical contact surface is ideally smooth, then the reaction of the spherical hinge has the direction of the normal to this surface. Therefore, the only thing known about the reaction is that it passes through the center of the hinge O. The direction of the reaction can be any and is determined in each specific case, depending on the given forces and the general scheme of fixing the body. Similarly, it is impossible to determine in advance reaction direction thrust bearing .

3. Cylindrical pivot bearing . The reaction of such a support passes through its axis, and the direction of the reaction of the support can be any (in the plane perpendicular to the axis of the support).

4. Cylindrical pivot bearing prevents the movement of the fixed point of the body along the perpendicular to the support plane. The reaction of such a support also has the direction of this perpendicular.

5. Thrust bearing. The thrust bearing is a connection of a cylindrical hinge with a reference plane. Such a connection allows the shaft to rotate around its axis and move along it, but only in one direction.

The thrust bearing reaction is the sum of the reaction of a cylindrical bearing lying in a plane perpendicular to its axis (in the general case, it can be decomposed into components R 1 and R 2), and the normal reaction of the reference plane R 3 .

Several bonds, possibly of different types, can be imposed on the same body at the same time. Three examples of this kind are shown in Fig. On fig. the corresponding systems of forces are shown. In accordance with the principle of liberability, bonds are discarded and replaced by reactions.

6. Rod reactions directed along the rods (upper diagram); it is assumed that the rods are weightless and are connected to the body and supports with the help of hinges.

Reactions of perfectly smooth bearing surfaces directed along the normal to these surfaces (two lower diagrams). In addition, the reaction of a cylindrical bearing at the point BUT(middle diagram) must, on the basis of the theorem on three non-parallel forces, pass through the point of intersection of the lines of action of the forces F and R2- point With.

7. Reaction R1 perfectly flexible, inextensible and weightless thread directed along the thread (lower diagram).

In mechanical systems formed by the articulation of several solid bodies, along with external connections (supports), there are internal communications . In these cases, one sometimes mentally dismembers the system and replaces the discarded not only external, but also internal connections with the corresponding reactions. An example of this kind in which two bodies are connected by a hinge With, shown in Fig. Note that the forces R2 and R3 equal to each other in absolute value, but oppositely directed (according to axiom 4).

Note that the forces of interaction between individual points of a given body are called internal , and the forces acting on a given body and caused by other bodies are called external . From this it follows that the reactions of bonds are external forces for a given body.

Let us agree to call a body free if its movements are not limited by anything. A body whose movements are limited by other bodies is called non-free, and the bodies that limit the movements of this body are called bonds. As already mentioned, at the points of contact, interaction forces arise between a given body and bonds. The forces with which the bonds act on a given body are called the reactions of the bonds.

The forces that do not depend on the bonds are called active forces (given), and the reactions of the bonds are called passive forces.

In mechanics, the following position is adopted, sometimes called the principle of liberation: any non-free body can be considered as free if the actions of the bonds are replaced by their reactions applied to the given body.

In statics, the reactions of the bonds can be completely determined using the conditions or equations of equilibrium of the body, which will be established later, but their directions in many cases can be determined from the consideration of the properties of the bonds:

The main types of connections:

1. If a rigid body rests on a perfectly smooth (without thorns) surface, then the point of contact of the body with the surface can slide freely along the surface, but cannot move in the direction along the normal to the surface. The reaction of an ideally smooth surface is directed along the common normal to the contacting surfaces.

If the body has a smooth surface and rests on a point, then the reaction is directed along the normal to the surface of the body itself.

2. Spherical joint.

3. Cylindrical hinge is called a fixed support. The reaction of such a support passes through its axis, and the direction of the reaction can be any (in a plane parallel to the axis of the support).

4. Cylindrical hinged - movable support.

MAIN TASKS OF STATICS.

1. The task of reducing the system of forces: how can this system be replaced by another, in particular the simplest one, equivalent to it?

2. The problem of equilibrium: what conditions must a system of forces applied to a given body satisfy in order for it to be a balanced system?

The first main task is important not only in statics, but also in dynamics. The second problem is often posed in those cases where equilibrium certainly takes place. In this case, the equilibrium conditions establish a relationship between all the forces applied to the body. In many cases, using these conditions, it is possible to determine the support reactions. Although this does not limit the scope of interests of solid body statics, it must be borne in mind that the determination of the reactions of bonds (external and internal) is necessary for the subsequent calculation of the strength of structures.

By force called a measure of the mechanical interaction of material bodies.

Force F- vector quantity and its action on the body is determined by:

• module or numerical value force (F);
• direction forces (orthom e);
• application point force (point A).

The line AB along which the force is directed is called the line of action of the force.

The force can be given:

• in a geometric way, that is, as a vector with a known modulus F and a known direction determined by the vector e ;
• in an analytical way, that is, its projections F x , F y , F z on the axis of the chosen coordinate system Oxyz .

The force application point A must be given by its x, y, z coordinates.

Force projections are related to its modulus and direction cosines(cosines of the angles , , , which are formed by the force with the coordinate axes Ox, Oy, Oz) by the following relations:

F=(F x 2 +F y 2 +F x 2) ; ex=cos=Fx/F; e y =cos =F y /F; e z =cos =F z /F;

Strength F, acting on an absolutely rigid body, can be considered applied to any point on the line of action of the force (such a vector is called sliding). If a force acts on a rigid deformable body, then its point of application cannot be transferred, since this transfer changes the internal forces in the body (such a vector is called attached).

The unit of force in the SI system of units is newton (N); a larger unit 1kN=1000N is also used.

Material bodies can act on each other by direct contact or at a distance. Depending on this, forces can be divided into two categories:

• superficial forces applied to the surface of the body (for example, pressure forces on the body from the environment);
• volumetric (mass) forces applied to a given part of the volume of the body (for example, gravitational forces).

Surface and body forces are called distributed forces. In some cases, forces can be considered distributed along a certain curve (for example, the weight forces of a thin rod). Distributed forces are characterized by their intensity (density), that is, the total amount of force per unit length, area or volume. The intensity can be constant ( evenly distributed force) or variable.

If we can neglect the small dimensions of the area of ​​action of distributed forces, then we consider concentrated a force applied to a body at one point (a conditional concept, since in practice it is impossible to apply a force to one point of the body).

The forces applied to the body under consideration can be divided into external and internal. External forces are called forces that act on this body from other bodies, and internal are the forces with which parts of this body interact with each other.

If the movement of a given body in space is limited by other bodies, then it is called not free. The bodies that restrict the movement of a given body are called connections.

Axiom of connections: connections can be mentally discarded and the body considered free if the action of the connections on the body is replaced by the corresponding forces, which are called bond reactions.

Reactions of bonds by their nature differ from all other forces applied to the body, which are not reactions, which are usually called active forces. This difference lies in the fact that the reaction of the bond is not completely determined by the bond itself. Its magnitude, and sometimes also its direction, depend on the active forces acting on the given body, which are usually known in advance and do not depend on other forces applied to the body. In addition, active forces, acting on a body at rest, can communicate to it this or that movement; reactions of bonds do not possess this property, as a result of which they are also called passive forces.

4. Method of Sections. Internal force factors.
To determine and then calculate the additional forces in any section of the beam, we use the method of sections. The essence of the method of sections is that the beam is mentally cut across into two parts and the balance of any of them is considered, which is under the action of all external and internal forces applied to this part. Being internal forces for the whole body, they play the role of external forces for the selected part.

Let the body be in equilibrium under the action of forces: (Figure 5.1, a). Let's cut it flat S and discard the right side (Figure 5.1, b). The law of distribution of internal forces over the cross section, in the general case, is unknown. To find it in each specific situation, it is necessary to know how the body under consideration is deformed under the influence of external forces.

Thus, the section method makes it possible to determine only the sum of internal forces. Based on the hypothesis of a continuous structure of the material, we can assume that the internal forces at all points of a particular section represent a distributed load.

We bring the system of internal forces in the center of gravity to the main vector and the main moment (Figure 5.1, c). Having designed and on the coordinate axes, we get a general picture of the stress-strain state of the considered section of the beam (Figure 5.1, d).

5. Axial tension - compression

Under stretching (compression) understand this type of loading, in which only longitudinal forces arise in the cross sections of the rod, and other force factors are equal to zero.

Longitudinal force- internal force equal to the sum of the projections of all external forces, taken from one side of the section, on the axis of the rod. Let's accept the following sign rule for longitudinal force : tensile longitudinal force is positive, compressive force is negative