Types of communication technical mechanics. Non-free systems

Bodies in nature are free and unfree. Bodies whose freedom of movement is not limited by anything are called free. Bodies that limit the freedom of movement of other bodies are called connections in relation to them.

One of the main provisions of mechanics is the principle of liberation from bonds, according to which a non-free body can be considered as free if the bonds acting on it are discarded and replaced by forces - reactions of the bonds.

It is very important to correctly place the bond reactions, otherwise the written equations will be incorrect. Below are examples of replacing bonds with their reactions. Figures 1.1–1.8 show examples of replacing forces located in a plane with reactions.

a – body of weight G on a smooth surface;
b – the action of the surface is replaced by a reaction – force R;
c – at point A there is a connection “reference point” or edge;
d – reactions are directed perpendicularly
supported or supported planes

Figure 1.1

The reaction of a smooth surface is always directed normal to this surface (Figure 1.1). The reaction of a “weightless” cable (thread, chain, rod) is always directed along the cable (thread, chain, rod) (Figure 1.2).

Figure 1.6

Figure 1.7a shows a bi-slip seal. In a plane, this support allows translational movement of the rod both horizontally and vertically, but prevents rotation (in the plane). The reaction of such a support will be the moment M C(Figure 1.7, b).

Figure 1.7

The console (blind or rigid embedding) does not allow any movement of the part. The reaction of such a support is a force unknown in magnitude and direction R A with angle α (or X A And Y A) and moment M A(Figure 1.8).

Figure 1.8

Figures 1.9 – 1.15 show examples of replacing forces located in space with their reactions.

The hinged-fixed support, or spherical hinge (Figure 1.9, a), is replaced by a system of forces (Figure 1.9, b) X A, Y A And Z A, i.e. a force unknown in magnitude and direction.

Any free body in space has six degrees of freedom: it can move along three axes and rotate about these axes. Bodies are rarely in a free state; in most cases, their movement is limited by connections. Constraints are restrictions that exclude the possibility of a body moving in a certain direction. If active forces act on a fixed body, then reactive forces or reactions arise in the connections, complementing the system of active forces to an equilibrium one. The combination of active and reactive balanced forces determines the stressed state of the body and its deformation.

Bond reactions are found using equilibrium equations. In this case, the decision is carried out according to the following plan:

identify external active forces applied to a selected body or group of bodies;
the selected object (body) is freed from the bonds and reaction forces of the bonds are applied instead;
Having chosen the coordinate axes, they compose equilibrium equations and, having solved them, find the reaction forces of the bonds.

For a spatial system of forces, six equilibrium equations (13.7) can be compiled. Using these equations, six unknown reactions are determined.

Problems that can be solved only using static equilibrium equations are called statically definable. If a larger number of connections are imposed on the selected object, then the task becomes statically indeterminate and to solve it, in addition to the equilibrium equations, it is necessary to use additional equations compiled on the basis of deformation analysis. In general, securing or connecting two parts can eliminate from one to six degrees of freedom, i.e. impose from one to six connections. In accordance with this, from one to six reactions can occur in consolidation. The amount of reactive forces and their direction depend on the nature of the connections.

Here are the most common types of fastening and connecting parts.

1. Connections that exclude the possibility of movement in only one direction. In such compounds, only one reaction of a certain direction occurs. Connections of this type include:
- a) connection by touching two bodies at a point or along a line. When touched, a reaction occurs directed along the general normal to the touching surfaces (Fig. 13.5). Such a connection is called articulated-movable;

Rice. 13.5.

b) the connection made by a cable, thread, chain gives a reaction directed along a flexible connection, and such a connection can only work in tension (see Fig. 13.5, b);
c) a connection in the form of a rigid straight rod with hinged ends also gives a reaction directed along the axis of the rod (see Fig. 13.5, c) at but can work in both tension and compression.

Rice. 13.6.

In Fig. 13.5, G a body is shown with three constraints imposed on it; each connection excludes the possibility of movement in one direction and gives one reaction, the direction of which is known.

2. A fastening or connection that excludes movement in two directions and, accordingly, gives two reactions, is called a hinged-fixed support or a cylindrical hinge (Fig. 13.6).
3. A connection that excludes movement in three directions and gives three reactions is called a spatial or ball joint (Fig. 13.7).
4. Fastening that excludes all six degrees of freedom is called rigid fastening or embedding. Six reactive force factors can arise in the embedding - three reactive forces and three reactive moments (Fig. 13.8). When forces located in one plane act on a body with a rigid embedding, two reactive forces and one reactive moment arise in the embedding.

Rice. 13.7.

Rice. 13.8.

When making calculations, supports are schematized and conditionally divided into three main groups:

articulated and movable(Fig. 13.9, A), perceiving only one linear reaction /?;
articulated-fixed(Fig. 13.9, b), perceiving two linear reactions R And N.
pinching, or sealing(Fig. 13.9, V), perceiving linear reactions R And N and moment M.

Rice. 13.9.

When real bodies come into contact and during their relative motion, friction forces arise at the places of their contact, which can be considered as a special type of reactive forces. The friction force is located in the plane of contact of the bodies; when moving, it is directed in the direction opposite to the relative speed of the body.

Example. Shaft 1 with gear 2 attached to it is mounted in two bearings A And IN. A belt drive pulley 3 is mounted on the free end of the shaft (Fig. 13.10). The geometric dimensions are known. A, s, transmitting torque M, pulley diameter D, all parameters of the bevel gear, as well as the ratio of belt tension forces F a JF al= 2. It is necessary to determine the reaction of the supports and the tension force of the belt.

Rice. 13.10.

We carry out the solution in three steps.

1. We identify the active forces acting in the system. A spatially located force acts on a bevel gear, the components of which along the coordinate axes are designated accordingly F v F r And F a . Component F ( , called circumferential force, is determined by a given torque based on the equation of moments about the axis z

Radial component Fr and axial component F a determined by circumferential force F ( based on the specified geometry of the bevel gear.

2. We free the shaft (equilibrium object) from the connections and instead apply reaction forces X l U l, X c, Y B Z B .

Bearings A And IN should be considered as hinged supports, since they always have gaps. In support A two reactions occur X l And U l, since this support prohibits the movement of the shaft only in transverse directions. Three reactions occur in the right support X in, U in And Z B , since it limits the movement of the shaft also in the axial direction. Active and reactive forces together form a spatial system of balanced forces.

3. Select a coordinate system: axes X And at placed in a plane perpendicular to the axis of the shaft, and the axis z we direct along the axis of the shaft. We create six equilibrium equations using (13.7) and (13.8).

Using a given condition F al = 2F ii2 and solving the equilibrium equations, we find the forces F aV F a2 and support reactions

In the process of studying statics, which is one of the constituent branches of mechanics, the main role is given to axioms and basic concepts. There are only five basic axioms. Some of them are known from school physics lessons, since they are Newton's laws.

Definition of mechanics

To begin with, it is necessary to mention that statics is a subsection of mechanics. The latter should be described in more detail, since it is directly related to statics. At the same time, mechanics is a more general term that combines dynamics, kinematics and statics. All these subjects were studied in the school physics course and are known to everyone. Even the axioms included in the study of statics are based on those known from school years. However, there were three of them, while the basic axioms of statics were five. Most of them concern the rules for maintaining balance and rectilinear uniform movement of a certain body or material point.

Mechanics is the science of the simplest method of motion of matter - mechanical. The simplest movements are considered to be actions that can be reduced to moving a physical object in space and time from one position to another.

What does mechanics study?

In theoretical mechanics, the general laws of motion are studied without taking into account the individual properties of the body, except for the properties of extension and gravity (from this follow the properties of matter particles to attract each other or have a certain weight).

Basic definitions include mechanical force. This term refers to movement that is mechanically transmitted from one body to another during interaction. Based on numerous observations, it was determined that force is considered to be characterized by the direction and point of application.

According to the method of construction, theoretical mechanics is similar to geometry: it is also based on definitions, axioms and theorems. However, the connection does not end with simple definitions. Most of the drawings related to mechanics in general and statics in particular contain geometric rules and laws.

Theoretical mechanics includes three subsections: statics, kinematics and dynamics. The first studies methods for transforming forces applied to an object and an absolutely rigid body, as well as the conditions for the emergence of equilibrium. Kinematics considers simple mechanical motion that does not take into account the acting forces. In dynamics, the movements of a point, a system, or a rigid body are studied, taking into account the acting forces.

Axioms of statics

To begin with, we should consider the basic concepts, axioms of statics, types of connections and their reactions. Statics is a state of equilibrium with forces that are applied to an absolutely rigid body. Its tasks include two main points: 1 - the basic concepts and axioms of statics include the replacement of an additional system of forces that were applied to the body by another system equivalent to it. 2 - derivation of general rules under which a body, under the influence of applied forces, remains in a state at rest or in the process of uniform translational rectilinear motion.

Objects in such systems are usually called a material point - a body, the dimensions of which can be omitted under the given conditions. A set of points or bodies interconnected in some way is called a system. The forces of mutual influence between these bodies are called internal, and the forces influencing this system are called external.

The resultant force in a certain system is a force equivalent to the reduced system of forces. Those included in this system are called component forces. The balancing force is equal in magnitude to the resultant force, but is directed in the opposite direction.

In statics, when deciding on a change in the system of forces affecting a solid body, or on the balance of forces, the geometric properties of force vectors are used. From this the definition of geometric statics becomes clear. Analytical statics, based on the principle of permissible displacements, will be described in dynamics.

Basic concepts and axioms of statics

The conditions for a body to be in equilibrium are derived from several basic laws that are used without additional evidence, but have confirmation in the form of experiments, called axioms of statics.

Axiom I is called Newton's first law (axiom of inertia). Each body remains in a state of rest or uniform linear motion until external forces act on this body, removing it from this state. This ability of the body is called inertia. This is one of the basic properties of matter.
Axiom II - Newton's third law (axiom of interaction). When one body acts on another with a certain force, then the second body, together with the first, will act on it with a certain force, which is equal in magnitude and opposite in direction.
Axiom III is the condition for the equilibrium of two forces. To obtain equilibrium of a free body that is under the influence of two forces, it is enough that these forces are identical in magnitude and opposite in direction. This is also related to the next point and is included in the basic concepts and axioms of statics, the equilibrium of a system of converging forces.
Axiom IV. Equilibrium will not be disturbed if a balanced system of forces is applied or removed to a solid body.
Axiom V is the axiom of the parallelogram of forces. The resultant of two intersecting forces is applied at the point of their intersection and is represented by the diagonal of a parallelogram constructed on these forces.

Connections and their reactions

In theoretical mechanics, a material point, system and solid body can be given two definitions: free and non-free. The differences between these words are that if pre-specified restrictions are not imposed on the movement of a point, body or system, then these objects will be, by definition, free. In the opposite situation, objects are usually called non-free.

Physical circumstances leading to restriction of the freedom of these material objects are called connections. In statics there may be the simplest connections performed by various rigid or flexible bodies. The force of a connection on a point, system or body is called the reaction of the connection.

Types of connections and their reactions

In ordinary life, connection can be represented by threads, laces, chains or ropes. In mechanics, this definition is taken to be weightless, flexible and inextensible bonds. Reactions can accordingly be directed along a thread or rope. In this case, connections take place, the lines of action of which cannot be determined immediately. As an example of the basic concepts and axioms of statics, we can cite a fixed cylindrical hinge.

It consists of a stationary cylindrical bolt, onto which is mounted a sleeve with a cylindrical hole, the diameter of which does not exceed the size of the bolt. When fastening the body to the bushing, the first can only rotate along the hinge axis. In an ideal hinge (provided that the friction between the surface of the bushing and the bolt is neglected), a barrier appears to the displacement of the bushing in a direction perpendicular to the surface of the bolt and bushing. In this regard, the reaction in an ideal hinge is directed along the normal - the radius of the bolt. Under the influence of acting forces, the bushing is capable of pressing against the bolt at an arbitrary point. In this regard, the direction of reaction at a fixed cylindrical hinge cannot be determined in advance. From this reaction, only its location in a plane perpendicular to the hinge axis can be known.

When solving problems, the reaction of the hinge will be determined analytically by decomposing the vector. The basic concepts and axioms of statics include this method. The reaction projection values are calculated from the equilibrium equations. The same is done in other situations, including the impossibility of determining the direction of the bond reaction.

System of converging forces

Basic definitions include a system of forces that converge. The so-called system of converging forces will be called a system in which the lines of action intersect at a single point. This system leads to a resultant or is in a state of equilibrium. This system is also taken into account in the previously mentioned axioms, since it is associated with maintaining the balance of the body, which is stated in several positions at once. The latter indicate both the reasons necessary to create equilibrium, and factors that will not cause a change in this state. The resultant of a given system of converging forces is equal to the vector sum of the named forces.

Equilibrium of the system

In the basic concepts and axioms of statics, the system of converging forces is also included in the study. For the system to be in equilibrium, the mechanical condition is the zero value of the resultant force. Since the vector sum of forces is zero, the polygon is considered closed.

In analytical form, the condition for equilibrium of the system will be as follows: a spatial system of converging forces that is in equilibrium will have an algebraic sum of force projections on each of the coordinate axes equal to zero. Since in such an equilibrium situation the resultant will be zero, the projections on the coordinate axes will also be zero.

Moment of power

This definition means the vector product of the vector of the point of application of forces. The vector of the moment of force is directed perpendicular to the plane in which the force and the point lie, in the direction from which the rotation from the action of the force is seen occurring counterclockwise.

Couple of forces

This definition refers to a system consisting of a pair of parallel forces, equal in magnitude, directed in opposite directions and applied to a body.

The moment of a pair of forces can be considered positive if the forces of the pair are directed counterclockwise in a right-handed coordinate system, and negative if they are directed clockwise in a left-handed coordinate system. When transferring from the right coordinate system to the left, the orientation of the forces changes to the opposite. The minimum value of the distance among the lines of action of forces is called the shoulder. It follows from this that the moment of a pair of forces is a free vector, modulo equal to M = Fh and having a direction perpendicular to the plane of action, and from the top of this vector the forces were oriented positively.

Equilibrium in arbitrary systems of forces

The required equilibrium condition for an arbitrary spatial system of forces applied to a rigid body is considered to be the vanishing of the main vector and moment with respect to any point in space.

It follows from this that in order to achieve equilibrium of parallel forces located in one plane, it is required and sufficient that the resulting sum of the projections of forces onto a parallel axis and the algebraic sum of all components of the moments provided by the forces relative to a random point is equal to zero.

Center of gravity of the body

According to the law of universal gravitation, every particle located near the surface of the Earth is affected by attractive forces called gravity. With small body sizes, in all technical applications the gravity forces of individual particles of the body can be considered a system of almost parallel forces. If we consider all the gravitational forces of particles to be parallel, then their resultant will be numerically equal to the sum of the weights of all particles, i.e., the weight of the body.

Subject of kinematics

Kinematics is a section of theoretical mechanics that studies the mechanical motion of a point, a system of points and a rigid body, regardless of the forces influencing them. Newton, based on a materialist position, considered the objective nature of space and time. Newton used the definition of absolute space and time, but separated them from moving matter, so he can be called a metaphysician. Dialectical materialism considers space and time to be objective forms of existence of matter. Space and time cannot exist without matter. In theoretical mechanics it is said that the space that includes moving bodies is called three-dimensional Euclidean space.

Compared to theoretical mechanics, the theory of relativity is based on different ideas about space and time. This was helped by the emergence of a new geometry created by Lobachevsky. Unlike Newton, Lobachevsky did not separate space and time from vision, considering the latter to be a change in the position of some bodies relative to others. In his own work, he pointed out that in nature only movement is cognized by man, without which sensory representation becomes impossible. It follows from this that all other concepts, for example, geometric ones, are created artificially by the mind.

From this it is clear that space is considered as a manifestation of the connection between moving bodies. Almost a century before the emergence of the theory of relativity, Lobachevsky pointed out that Euclidean geometry relates to geometrically abstract systems, while in the physical world spatial relationships are determined by physical geometry, which differs from Euclidean geometry, in which the properties of time and space are combined with the properties of matter moving in space and time.

It does not hurt to note that advanced scientists from Russia in the field of mechanics consciously adhered to the correct materialist positions in the interpretation of all the main definitions of theoretical mechanics, in particular time and space. At the same time, the opinion about space and time in the theory of relativity is similar to the ideas about space and time of the supporters of Marxism, which were created before the appearance of works on the theory of relativity.

When working with theoretical mechanics when measuring space, the meter is taken as the main unit, and the second is taken as time. Time is the same in each reference system and is independent of the interleaving of these systems in relation to each other. Time is indicated by a symbol and is considered as a continuous variable value used as an argument. When measuring time, the definitions of a period of time, a moment in time, and initial time are used, which are included in the basic concepts and axioms of statics.

Technical mechanics

In practical application, the basic concepts and axioms of statics and technical mechanics are interconnected. In technical mechanics, both the mechanical process of motion itself and the possibility of using it for practical purposes are studied. For example, when creating technical and building structures and testing them for strength, which requires a brief knowledge of the basic concepts and axioms of statics. However, such a brief study is suitable only for amateurs. In specialized educational institutions, this topic is of considerable importance, for example, in the case of the system of forces, basic concepts and axioms of statics.

In technical mechanics, the above axioms are also used. To 1, the basic concepts and axioms of statics are related to this section. Despite the fact that the very first axiom explains the principle of maintaining equilibrium. In technical mechanics, an important role is played not only by the creation of devices, but also in the construction of which stability and strength are the main criteria. However, it will be impossible to create something like this without knowing the basic axioms.

General remarks

The simplest forms of movement of solid bodies include translational and rotational movement of the body. In the kinematics of rigid bodies for different types of movements, the kinematic characteristics of the movement of its different points are taken into account. Rotational motion of a body around a fixed point is a motion in which a straight line passing through a pair of arbitrary points during the motion of the body remains at rest. This straight line is called the axis of rotation of the bodies.

The text above briefly summarized the basic concepts and axioms of statics. At the same time, there is a large amount of third-party information with which you can better understand statics. Do not forget the basic data; in most examples, the basic concepts and axioms of statics include an absolutely rigid body, since this is a kind of standard for an object that may not be achievable under normal conditions.

Then you should remember the axioms. For example, the basic concepts and axioms of statics, communications and their reactions are among them. Despite the fact that many axioms only explain the principle of maintaining balance or uniform motion, this does not negate their significance. Starting from the school course, these axioms and rules are studied, since they are Newton’s laws that are well known to everyone. The need to mention them is associated with the practical application of information from statics and mechanics in general. An example was technical mechanics, in which, in addition to creating mechanisms, it is necessary to understand the principle of constructing sustainable buildings. Thanks to such information, the correct construction of conventional structures is possible.

Basic concepts and axioms of statics

Statics is the study of forces and conditions of equilibrium of material bodies under the influence of forces.

Force– a measure of the mechanical interaction of bodies. The set of forces acting on an absolutely rigid body is called a system of forces.

Absolutely solid body- a set of points, the distances between the current positions of which do not change, no matter what influences the given body is subjected to.

Statically solved two tasks:

1. Addition of forces and reduction of systems of forces acting on the body to their simplest form;

2. Determination of equilibrium conditions for systems of forces acting on a body.

The two systems of forces are called equivalent, if they have the same mechanical effect on the body.

The system of forces is called balanced(equivalent to zero) if it does not change the mechanical state of the body (that is, the state of rest or motion by inertia).

Resultant a force is one force, if it exists, equivalent to some system of forces.

Forces whose lines of action intersect at one point are called convergent.

1. Axiom about the equilibrium of a system of two forces. Under the action of two forces applied to an absolutely rigid body, the body can be in equilibrium if and only if these forces are equal in magnitude and directed along the same straight line in opposite directions (Fig. 1.1).

Figure 1.1

2. Axiom about adding (discarding) a system of forces equivalent to zero. The action of this system of forces on an absolutely rigid body is not

will change if a balanced system of forces (i.e. equivalent to zero) is added or subtracted from it.

We have a system ; let's add 0

We get ( ; }.

Consequence: When a force is transferred along its line of action, the effect of this force on the body does not change. From this consequence it follows that the force applied to an absolutely rigid body is a sliding vector.

Let at the point A force is applied to a rigid body (Fig. 1.2). To this force on its line of action at the point IN in accordance with axiom II, we add a system of forces equivalent to zero, for which . Let's choose a force equal to the force.

Figure 1.2

The resulting system of three forces is equivalent, according to the axiom of adding an equilibrium system of forces, to a force, that is.

The system of forces, according to axiom 1, is equivalent to zero, and according to axiom 2 it can be discarded. The result is one force applied at a point IN, that is . Finally we get . Force applied at point A. It is equivalent in magnitude and direction to the force applied at the point IN, where is the point IN– any point on the line of action of the force. The theorem has been proven: the action of a force on a rigid body will not change from the transfer of force along the line of action. The force for a rigid body can be considered applied at any point in the line of action, that is, the force is a sliding vector. As a sliding vector, force is characterized by: numerical value (modulus); direction of force; the position of the line of action of the force on the body.

3.Axiom of parallelogram of forces. Two forces applied at one point of an absolutely rigid body have a resultant force applied at the same point and equal to the geometric (vector) sum of these forces (Fig. 1.3).

Figure 1.3

Consequence: Theorem about three non-parallel forces: If under the action of three forces a body is in equilibrium and the lines of action of the two forces intersect, then all the forces lie in the same plane and their lines of action intersect at one point.

Drawing. 1.4

Let us assume that the body is in equilibrium under the action of three forces, 3, applied at points A, B, C (Fig. 1.4). According to axiom 3, the resultant of the first two forces can be found according to the parallelogram rule, built on forces 1 and 2, transferred along the line of their action to the point O of the intersection of the latter, i.e. According to the first axiom of statics, for a body to be balanced, it is necessary and sufficient that force 3 balances the first two forces. This is only possible if the forces and 3 lie on the same straight line and have opposite directions. But then the lines of action of the forces , 3 will intersect at one point O. Any of the three given forces balances the other two. The derived condition for the equilibrium of three non-parallel forces is necessary, but not sufficient. If the lines of action of three forces intersect at one point, then it does not at all follow that these three forces represent a balanced system of forces.

4. Axiom about the equality of action and reaction forces. With any action of one body on another, there is a reaction that is the same in number, but opposite in direction (Newton’s III law). The forces of interaction between two bodies do not constitute a system of balanced forces, since they are applied to different bodies.

Figure 1.5

5. Axiom about connections. Material objects (bodies and points) that limit the freedom of movement of the rigid body in question are called constraints. The force with which a connection acts on a body, preventing its movement, is called the reaction of the connection. The coupling reaction is directed opposite to the possible movement of the body. The axiom of connections states that any connection can be discarded and replaced by a force or a system of forces (in the general case), that is, connection reactions.

6. Axiom of solidification. The equilibrium of a deformable body under the influence of a given system of forces will not be disturbed if the body is considered solidified (absolutely solid). If the deformable body was in equilibrium, then it will be in equilibrium even after it hardens.

Main types of bonds and their reactions

Let us give examples of connections for a plane system of forces and their replacement by forces of reaction reactions.

1. Smooth surface(Fig. 1.6, a). If a body rests on an ideally smooth surface, then the reaction of the surface is directed normal to the common tangent of the surfaces of the bodies at the point of contact.

2. Movable hinge support, movable hinge– support placed on rollers that do not interfere with the movement of the body parallel to the supporting plane. The reaction of the movable hinge is directed normal to the surface on which the hinge rollers rest (Fig. 1.6, b).

3. Fixed hinge support, fixed hinge- a combination of a stationary roller and a bushing mounted on it with a solid body rotating around an axis (bearing, hinge). The reaction of the fixed hinge passes through the axis of the roller, in an unknown direction, so its two components are determined, directed parallel to the coordinate axes perpendicular to the axis of the roller (Fig. 1.6, c).

4. Hard seal– rigidly fixed beam, rod. The tie prevents any movement of the end of the beam. To determine the reaction of a rigid embedding, it is necessary to determine the components of the main vector R A, directed parallel to the coordinate axes and the main moment M A of the embedding (Fig. 1.6, d).

5. Rod– a rigid, weightless rod, the ends of which are connected to other parts of the structure by hinges. The reaction is directed along a line drawn through the supporting hinges of the rod (Fig. 1.6, e).

6. Flexible connection- thread, chain, cable. The reaction is applied to the solid at the point of contact and is directed along the bond (Fig. 1.6, e).

Theoretical mechanics is a section of mechanics that sets out the basic laws of mechanical motion and mechanical interaction of material bodies.

Theoretical mechanics is a science that studies the movement of bodies over time (mechanical movements). It serves as the basis for other branches of mechanics (theory of elasticity, strength of materials, theory of plasticity, theory of mechanisms and machines, hydroaerodynamics) and many technical disciplines.

Mechanical movement- this is a change over time in the relative position in space of material bodies.

Mechanical interaction- this is an interaction as a result of which the mechanical movement changes or the relative position of body parts changes.

Rigid body statics

Statics is a section of theoretical mechanics that deals with problems of equilibrium of solid bodies and the transformation of one system of forces into another, equivalent to it.

Basic concepts and laws of statics

Absolutely rigid body(solid body, body) is a material body, the distance between any points in which does not change.
Material point is a body whose dimensions, according to the conditions of the problem, can be neglected.
Free body- this is a body on the movement of which no restrictions are imposed.
Unfree (bound) body is a body whose movement is subject to restrictions.
Connections– these are bodies that prevent the movement of the object in question (a body or a system of bodies).
Communication reaction is a force that characterizes the action of a bond on a solid body. If we consider the force with which a solid body acts on a bond to be an action, then the reaction of the bond is a reaction. In this case, the force - action is applied to the connection, and the reaction of the connection is applied to the solid body.
Mechanical system is a collection of interconnected bodies or material points.
Solid can be considered as a mechanical system, the positions and distances between points of which do not change.
Force is a vector quantity that characterizes the mechanical action of one material body on another.
Force as a vector is characterized by the point of application, direction of action and absolute value. The unit of force modulus is Newton.
Line of action of force is a straight line along which the force vector is directed.
Focused Power– force applied at one point.
Distributed forces (distributed load)- these are forces acting on all points of the volume, surface or length of a body.
The distributed load is specified by the force acting per unit volume (surface, length).
The dimension of the distributed load is N/m 3 (N/m 2, N/m).
External force is a force acting from a body that does not belong to the mechanical system under consideration.
Inner strength is a force acting on a material point of a mechanical system from another material point belonging to the system under consideration.
Force system is a set of forces acting on a mechanical system.
Flat force system is a system of forces whose lines of action lie in the same plane.
Spatial system of forces is a system of forces whose lines of action do not lie in the same plane.
System of converging forces is a system of forces whose lines of action intersect at one point.
Arbitrary system of forces is a system of forces whose lines of action do not intersect at one point.
Equivalent force systems- these are systems of forces, the replacement of which one with another does not change the mechanical state of the body.
Accepted designation: .
Equilibrium- this is a state in which a body, under the action of forces, remains motionless or moves uniformly in a straight line.
Balanced system of forces- this is a system of forces that, when applied to a free solid body, does not change its mechanical state (does not throw it out of balance).
.
Resultant force is a force whose action on a body is equivalent to the action of a system of forces.
.
Moment of power is a quantity characterizing the rotating ability of a force.
Couple of forces is a system of two parallel forces of equal magnitude and oppositely directed.
Accepted designation: .
Under the influence of a pair of forces, the body will perform a rotational motion.
Projection of force on the axis- this is a segment enclosed between perpendiculars drawn from the beginning and end of the force vector to this axis.
The projection is positive if the direction of the segment coincides with the positive direction of the axis.
Projection of force onto a plane is a vector on a plane, enclosed between perpendiculars drawn from the beginning and end of the force vector to this plane.
Law 1 (law of inertia). An isolated material point is at rest or moves uniformly and rectilinearly.
The uniform and rectilinear motion of a material point is motion by inertia. The state of equilibrium of a material point and a rigid body is understood not only as a state of rest, but also as motion by inertia. For a rigid body, there are various types of motion by inertia, for example, uniform rotation of a rigid body around a fixed axis.
Law 2. A rigid body is in equilibrium under the action of two forces only if these forces are equal in magnitude and directed in opposite directions along a common line of action.
These two forces are called balancing.
In general, forces are called balanced if the solid body to which these forces are applied is at rest.
Law 3. Without disturbing the state (the word “state” here means the state of motion or rest) of a rigid body, one can add and reject balancing forces.
Consequence. Without disturbing the state of the solid body, the force can be transferred along its line of action to any point of the body.
Two systems of forces are called equivalent if one of them can be replaced by the other without disturbing the state of the solid body.
Law 4. The resultant of two forces applied at one point, applied at the same point, is equal in magnitude to the diagonal of a parallelogram constructed on these forces, and is directed along this
diagonals.
The absolute value of the resultant is:
Law 5 (law of equality of action and reaction). The forces with which two bodies act on each other are equal in magnitude and directed in opposite directions along the same straight line.
It should be kept in mind that action- force applied to the body B, And opposition- force applied to the body A, are not balanced, since they are applied to different bodies.
Law 6 (law of solidification). The equilibrium of a non-solid body is not disturbed when it solidifies.
It should not be forgotten that the equilibrium conditions, which are necessary and sufficient for a solid body, are necessary but insufficient for the corresponding non-solid body.
Law 7 (law of emancipation from ties). A non-free solid body can be considered as free if it is mentally freed from bonds, replacing the action of the bonds with the corresponding reactions of the bonds.

Connections and their reactions

Smooth surface limits movement normal to the support surface. The reaction is directed perpendicular to the surface.
Articulated movable support limits the movement of the body normal to the reference plane. The reaction is directed normal to the support surface.
Articulated fixed support counteracts any movement in a plane perpendicular to the axis of rotation.
Articulated weightless rod counteracts the movement of the body along the line of the rod. The reaction will be directed along the line of the rod.
Blind seal counteracts any movement and rotation in the plane. Its action can be replaced by a force represented in the form of two components and a pair of forces with a moment.

Kinematics

Kinematics- a section of theoretical mechanics that examines the general geometric properties of mechanical motion as a process occurring in space and time. Moving objects are considered as geometric points or geometric bodies.

Basic concepts of kinematics

Law of motion of a point (body)– this is the dependence of the position of a point (body) in space on time.
Point trajectory– this is the geometric location of a point in space during its movement.
Speed of a point (body)– this is a characteristic of the change in time of the position of a point (body) in space.
Acceleration of a point (body)– this is a characteristic of the change in time of the speed of a point (body).

Determination of kinematic characteristics of a point

Point trajectory
In a vector reference system, the trajectory is described by the expression: .
In the coordinate reference system, the trajectory is determined by the law of motion of the point and is described by the expressions z = f(x,y)- in space, or y = f(x)- in a plane.
In a natural reference system, the trajectory is specified in advance.
Determining the speed of a point in a vector coordinate system
When specifying the movement of a point in a vector coordinate system, the ratio of movement to a time interval is called the average value of speed over this time interval: .
Taking the time interval to be an infinitesimal value, we obtain the speed value at a given time (instantaneous speed value): .
The average velocity vector is directed along the vector in the direction of the point’s movement, the instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point’s movement.
Conclusion: the speed of a point is a vector quantity equal to the time derivative of the law of motion.
Derivative property: the derivative of any quantity with respect to time determines the rate of change of this quantity.
Determining the speed of a point in a coordinate reference system
Rate of change of point coordinates:
.
The modulus of the total velocity of a point with a rectangular coordinate system will be equal to:
.
The direction of the velocity vector is determined by the cosines of the direction angles:
,
where are the angles between the velocity vector and the coordinate axes.
Determining the speed of a point in a natural reference system
The speed of a point in the natural reference system is defined as the derivative of the law of motion of the point: .
According to previous conclusions, the velocity vector is directed tangentially to the trajectory in the direction of the point’s movement and in the axes is determined by only one projection.

Rigid body kinematics

In the kinematics of rigid bodies, two main problems are solved:
1) setting the movement and determining the kinematic characteristics of the body as a whole;
2) determination of kinematic characteristics of body points.
Translational motion of a rigid body
Translational motion is a motion in which a straight line drawn through two points of a body remains parallel to its original position.
Theorem: during translational motion, all points of the body move along identical trajectories and at each moment of time have the same magnitude and direction of speed and acceleration.
Conclusion: the translational motion of a rigid body is determined by the movement of any of its points, and therefore, the task and study of its motion is reduced to the kinematics of the point.
Rotational motion of a rigid body around a fixed axis
Rotational motion of a rigid body around a fixed axis is the motion of a rigid body in which two points belonging to the body remain motionless during the entire time of movement.
The position of the body is determined by the angle of rotation. The unit of measurement for angle is radian. (A radian is the central angle of a circle, the arc length of which is equal to the radius; the total angle of the circle contains 2π radian.)
The law of rotational motion of a body around a fixed axis.
We determine the angular velocity and angular acceleration of the body using the differentiation method:
— angular velocity, rad/s;
— angular acceleration, rad/s².
If you dissect the body with a plane perpendicular to the axis, select a point on the axis of rotation WITH and an arbitrary point M, then point M will describe around a point WITH circle radius R. During dt there is an elementary rotation through an angle , and the point M will move along the trajectory a distance .
Linear speed module:
.
Point acceleration M with a known trajectory, it is determined by its components:
,
Where .
As a result, we get the formulas
tangential acceleration: ;
normal acceleration: .

Dynamics

Dynamics is a section of theoretical mechanics in which the mechanical movements of material bodies are studied depending on the causes that cause them.

Basic concepts of dynamics

Inertia- this is the property of material bodies to maintain a state of rest or uniform rectilinear motion until external forces change this state.
Weight is a quantitative measure of the inertia of a body. The unit of mass is kilogram (kg).
Material point- this is a body with mass, the dimensions of which are neglected when solving this problem.
Center of mass of a mechanical system- a geometric point whose coordinates are determined by the formulas:

Where m k , x k , y k , z k— mass and coordinates k-that point of the mechanical system, m— mass of the system.
In a uniform field of gravity, the position of the center of mass coincides with the position of the center of gravity.
Moment of inertia of a material body relative to an axis is a quantitative measure of inertia during rotational motion.
The moment of inertia of a material point relative to the axis is equal to the product of the mass of the point by the square of the distance of the point from the axis:
.
The moment of inertia of the system (body) relative to the axis is equal to the arithmetic sum of the moments of inertia of all points:
Inertia force of a material point is a vector quantity equal in modulus to the product of the mass of a point and the acceleration modulus and directed opposite to the acceleration vector:
The force of inertia of a material body is a vector quantity equal in modulus to the product of the body mass and the modulus of acceleration of the center of mass of the body and directed opposite to the acceleration vector of the center of mass: ,
where is the acceleration of the center of mass of the body.
Elementary impulse of force is a vector quantity equal to the product of the force vector and an infinitesimal period of time dt:
.
The total force impulse for Δt is equal to the integral of the elementary impulses:
.
Elementary work of force is a scalar quantity dA, equal to the scalar proi