Fundamentals of technical mechanics lectures. Topics for self-study in theoretical mechanics with lighting examples

DEPARTMENT OF EDUCATION AND SCIENCE OF THE KOSTROMA REGION

Regional state budget professional educational institution

"Kostroma Energy College named after F.V. Chizhov"

METHODOLOGICAL DEVELOPMENT

For vocational teacher

Introductory lesson on the topic:

"BASIC CONCEPTS AND AXIOMS ​​OF STATICS"

discipline "Technical mechanics"

O.V. Guryev

Kostroma

Annotation.

Methodical development designed to carry out introductory lesson in the discipline "Technical Mechanics" on the topic "Basic concepts and axioms of statics" for all specialties. Classes are held at the beginning of the study of the discipline.

Lesson hypertext. Therefore, the objectives of the lesson include:

educational -

Educational -

Educational -

Approved by the Subject Cycle Commission

Teacher:

M.A. Zaitsev

Protocol No. 20

Reviewer

INTRODUCTION

Methodology for conducting a lesson on technical mechanics

Routing classes

Hypertext

CONCLUSION

BIBLIOGRAPHY

Introduction

"Technical mechanics" is an important subject of the cycle of mastering general technical disciplines, consisting of three sections:

theoretical mechanics

resistance of materials

machine parts.

The knowledge studied in technical mechanics is necessary for students, as it provides the acquisition of skills for setting and solving many engineering problems that will be encountered in their practical activities. For the successful assimilation of knowledge in this discipline, students need good preparation in physics and mathematics. At the same time, without knowledge of technical mechanics, students will not be able to master special disciplines.

The more complex the technique, the more difficult it is to fit it into the framework of the instructions, and the more often specialists will encounter non-standard situations. Therefore, students need to develop independent creative thinking, which is characterized by the fact that a person does not receive knowledge in ready-made and independently applies them to solving cognitive and practical problems.

Skills play an important role in this independent work. At the same time, it is important to teach students to determine the main thing, separating it from the secondary, to teach them to make generalizations, conclusions, and creatively apply the foundations of theory to solving practical problems. Independent work develops abilities, memory, attention, imagination, thinking.

In the teaching of the discipline, all the principles of education known in pedagogy are practically applicable: scientific, systematic and consistent, visibility, awareness of the assimilation of knowledge by students, accessibility of learning, the connection of learning with practice, along with an explanatory and illustrative methodology, which was, is and remains the main one in the lessons of technical mechanics. Engaged learning methods are applied: quiet and loud discussion, brainstorming, analysis case study, question answer.

The topic "Basic concepts and axioms of statics" is one of the most important in the course "Technical Mechanics". She has great importance in terms of course study. This topic is an introductory part of the discipline.

Students perform work with hypertext, in which it is necessary to put questions correctly. Learn to work in groups.

Work on the assigned tasks shows the activity and responsibility of students, the independence of solving problems that arise in the course of the task, gives the skills and abilities to solve these problems. The teacher, by asking problematic questions, makes students think practically. As a result of working with hypertext, students draw conclusions from the topic covered.

Methodology for conducting classes in technical mechanics

The construction of classes depends on what goals are considered the most important. One of the most important tasks educational institution- teach to learn. Passing on practical knowledge students need to be taught to learn on their own.

- to captivate with science;

- interest in the task;

- to instill skills in working with hypertext.

Exceptionally important are such goals as the formation of a worldview and the educational impact on students. Achieving these goals depends not only on the content, but also on the structure of the lesson. It is quite natural that in order to achieve these goals, the teacher must take into account the characteristics of the contingent of students and use all the advantages of a living word and direct communication with students. In order to capture the attention of students, to interest and captivate them with reasoning, to accustom them to independent thinking, when building classes, it is necessary to take into account four stages of the cognitive process, which include:

1. statement of the problem or task;

2. proof - discourse (discursive - rational, logical, conceptual);

3. analysis of the result;

4. retrospection - establishing links between newly obtained results and previously established conclusions.

When starting a presentation of a new problem or task, it is necessary Special attention devote to staging it. It is not enough to confine ourselves to the formulation of the problem. This is well confirmed by the following statement of Aristotle: knowledge begins with surprise. It is necessary to be able to draw attention to a new task from the very beginning, to surprise, and therefore, to interest the student. After that, you can move on to solving the problem. It is very important that the statement of the problem or task is well understood by the students. They should be perfectly clear about the need to study a new problem and the validity of its formulation. When posing a new problem, strictness of presentation is necessary. However, it should be borne in mind that many questions and methods of solving are not always clear to students and may seem formal, unless special explanations are given. Therefore, each teacher should present the material in such a way as to gradually lead students to the perception of all the subtleties of a strict formulation, to an understanding of those ideas that make it quite natural to choose a certain method for solving a formulated problem.

Routing

TOPIC "BASIC CONCEPTS AND AXIOMS ​​OF STATICS"

Lesson Objectives:

educational - Learn three sections of technical mechanics, their definitions, basic concepts and axioms of statics.

Educational - improve students' independent work skills.

Educational - consolidation of group work skills, the ability to listen to the opinion of comrades, to discuss in a group.

Lesson type- explanation of new material

Technology- hypertext

Stages	Steps	Teacher activity	Student activities	Time
I Organizational	Theme, goal, work order	I formulate the topic, goal, work order in the lesson: “We work in the hypertext technology - I will pronounce the hypertext, then you will work with the text in groups, then we will check the level of assimilation of the material and summarize. At each stage, I will give instructions for work.	Listen, watch, write down the topic of the lesson in a notebook
II Learning new material	Pronunciation of hypertext	Each student has hypertext on their desks. I propose to follow me through the text, listen, look at the screen.	Looking at printouts of hypertext
		Speak hypertext while showing slides on screen	Listen, watch, read
III Consolidation of the studied	1 Drafting a text plan	Instruction 1. Divide into groups of 4-5 people. 2. Break the text into parts and title them, be ready to present your plan to the group (when the plan is ready, it is drawn up on whatman paper). 3. Organize a discussion of the plan. Compare the number of parts in the plan. If there is something different, we turn to the text and specify the number of parts in the plan. 4. We agree on the wording of the names of the parts, choose the best. 5. Summing up. We write down final version plan.	1. Divide into groups. 2. Head the text. 3. Discuss making a plan. 4. Clarify 5. Write down the final version of the plan
	2. Drawing up questions on the text	Instruction: 1. Each group to make 2 questions to the text. 2. Be prepared to ask group questions in sequence 3. If the group cannot answer the question, the questioner answers. 4. Organize a "Question Spinner". The procedure continues until repetitions begin.	Make questions, prepare answers Asking questions, answering
IV. Checking the assimilation of the material	control test	Instruction: 1. Perform the test individually. 2. In conclusion, check the test of your desk mate by comparing the correct answers with the slide on the screen. 3. Rating according to the specified criteria on the slide. 4. We hand over works to me	Perform the test Checking Appreciate
V. Summing up	1. Summing up the goal	I analyze this test in terms of the level of assimilation of the material
	2. Homework	Compile (or reproduce) a reference abstract on hypertext I would like to draw your attention to the fact that the task for a higher grade is located in the Moodle remote shell, in the "Technical Mechanics" section	Write down the task
	3. Lesson reflection	I propose to speak on the lesson, for help I show a slide with a list of prepared initial phrases	Choose phrases, speak out

1. Organizing time

1.1 Getting to know the group

1.2 Mark present students

1.3 Acquaintance with the requirements for students in the classroom.

3. Presentation of the material

4. Questions to consolidate the material

5. Homework

Hypertext

Mechanics, along with astronomy and mathematics, is one of the most ancient sciences. The term mechanics comes from Greek word"Mechane" - a trick, a machine.

In ancient times, Archimedes - the greatest mathematician and mechanic ancient greece(287-212 BC). gives an exact solution to the problem of the lever and created the doctrine of the center of gravity. Archimedes combined ingenious theoretical discoveries with remarkable inventions. Some of them have not lost their significance in our time.

A major contribution to the development of mechanics was made by Russian scientists: P.L. Chebeshev (1821-1894) - laid the foundation for the world-famous Russian school of the theory of mechanisms and machines. S.A. Chaplygin (1869-1942). developed a number of issues of aerodynamics that are of great importance for the modern speed of aviation.

Technical mechanics is a complex discipline that sets out the main provisions on the interaction of solids, the strength of materials and methods for calculating the structural elements of machines and mechanisms for external interactions. Technical mechanics is divided into three large sections: theoretical mechanics, strength of materials, machine parts. One of the sections of theoretical mechanics is divided into three subsections: statics, kinematics, dynamics.

Today we will begin the study of technical mechanics with a subsection of statics - this is a section of theoretical mechanics in which the conditions for the equilibrium of an absolutely rigid body under the action of forces applied to them are studied. The main concepts of statics are: Material point

a body whose dimensions can be neglected under the conditions of the tasks set. Absolutely rigid body - a conditionally accepted body that does not deform under the action of external forces. IN theoretical mechanics absolutely rigid bodies are studied. Strength- a measure of the mechanical interaction of bodies. The action of a force is characterized by three factors: the point of application, the numerical value (modulus), and the direction (force - vector). Outside forces- forces acting on the body from other bodies. internal forces- forces of interaction between particles of the given body. Active forces- forces that cause the body to move. Reactive forces- forces that prevent the movement of the body. Equivalent Forces- forces and systems of forces that produce the same effect on the body. Equivalent forces, systems of forces- one force equivalent to the considered system of forces. The forces of this system are called constituents this resultant. Balancing force- a force equal in magnitude to the resultant force and directed along the line of its action in the opposite direction. Force system - set of forces acting on a body. Systems of forces are flat, spatial; converging, parallel, arbitrary. Equilibrium- such a state when the body is at rest (V = 0) or moves uniformly (V = const) and rectilinearly, i.e. by inertia. Addition of forces- determination of the resultant according to the given component forces. Decomposition of forces - replacement of force by its components.

Basic axioms of statics. 1. axiom. Under the action of a balanced system of forces, the body is at rest or moves uniformly and in a straight line. 2. axiom. The principle of attachment and rejection of a system of forces equivalent to zero. The action of this system of forces on the body will not change if balanced forces are applied to or removed from the body. 3 axiom. The principle of equality of action and reaction. In the interaction of bodies, to every action there corresponds an equal and oppositely directed reaction. 4 axiom. Theorem about three balanced forces. If three non-parallel forces lying in the same plane are balanced, then they must intersect at one point.

Relationships and their reactions: Bodies whose movement is not limited in space are called free. Bodies whose movement is limited in space are called non free. Bodies that prevent the movement of non-free bodies are called bonds. The forces with which the body acts on the bond are called active. They cause the body to move and are denoted F, G. The forces with which the bond acts on the body are called reactions of bonds or simply reactions and are denoted R. To determine the reactions of the bond, the principle of release from bonds is used or section method. The principle of release from bonds lies in the fact that the body is mentally freed from bonds, the actions of bonds are replaced by reactions. Section method (ROZU method) lies in the fact that the body mentally is cut in pieces, one piece discarded, the action of the discarded part is replaced forces, for the determination of which are drawn up equations balance.

Main types of connections smooth plane- the reaction is directed perpendicular to the reference plane. Smooth surface- the reaction is directed perpendicular to the tangent drawn to the surface of the bodies. Angle support the reaction is directed perpendicular to the plane of the body or perpendicular to the tangent drawn to the surface of the body. Flexible connection- in the form of a rope, a cable, a chain. The reaction is directed by communication. Cylindrical joint- this is the connection of two or more parts using an axis, a finger. The reaction is directed perpendicular to the axis of the hinge. Rigid rod with hinged ends reactions are directed along the rods: the reaction of a stretched rod - from the node, compressed - to the node. When solving problems analytically, it can be difficult to determine the direction of rod reactions. In these cases, the rods are considered stretched and the reactions are directed away from the nodes. If, when solving problems, the reactions turned out to be negative, then in reality they are directed in the opposite direction and compression takes place. Reactions are directed along the rods: the reaction of a stretched rod - from the node, compressed - to the node. Articulated non-movable support- prevents vertical and horizontal movement of the end of the beam, but does not prevent its free rotation. Gives 2 reactions: vertical and horizontal force. Articulated support prevents only vertical movement of the end of the beam, but not horizontal, nor rotation. Such a support under any load gives one reaction. Rigid termination prevents vertical and horizontal movement of the end of the beam, as well as its rotation. Gives 3 reactions: vertical, horizontal forces and couple of forces.

Conclusion.

Methodology is a form of communication between a teacher and an audience of students. Each teacher is constantly looking for and testing new ways of revealing the topic, arousing such interest in it, which contributes to the development and deepening of students' interest. The proposed form of the lesson allows you to increase cognitive activity, as students independently receive information throughout the lesson and consolidate it in the process of solving problems. This makes them active in the classroom.

"Quiet" and "loud" discussion when working in micro groups gives positive results when assessing students' knowledge. Elements of "brainstorming" activate the work of students in the classroom. The joint solution of the problem allows less prepared students to understand the material being studied with the help of more “strong” comrades. What they could not understand from the words of the teacher can be explained to them again by more prepared students.

Some problematic questions asked by the teacher bring learning in the classroom closer to practical situations. This allows you to develop the logical, engineering thinking of students.

Evaluation of the work of each student in the lesson also stimulates his activity.

All of the above suggests that this form of lesson allows students to gain deep and solid knowledge on the topic under study, to actively participate in the search for solutions to problems.

LIST OF RECOMMENDED LITERATURE

Arkusha A.I. Technical mechanics. Theoretical mechanics and resistance of rials.-M high school. 2009.

Arkusha A.I. Guide to solving problems in technical mechanics. Proc. for secondary prof. textbook institutions, - 4th ed. correct - M Higher. school ,2009

Belyavsky SM. Guidelines for solving problems in the strength of materials M. Vyssh. school, 2011.

Guryeva O.V. Collection of multivariate tasks in technical mechanics..

Guryeva O.V. Toolkit. To help students of technical mechanics 2012

Kuklin N.G., Kuklina G.S. Machine parts. M. Engineering, 2011

Movnin M.S., et al. Fundamentals of engineering mechanics. L. Engineering, 2009

Erdedi A.A., Erdedi N.A. Theoretical mechanics. Material resistance M Higher. school Academy 2008.

Erdedi A A, Erdedi NA Machine parts - M, Higher. school Academy, 2011

Topic No. 1. STATICS OF A SOLID BODY

Basic concepts and axioms of statics

Static subject.static called a branch of mechanics in which the laws of the addition of forces and the conditions for the equilibrium of material bodies under the influence of forces are studied.

By equilibrium we will understand the state of rest of the body in relation to other material bodies. If the body, in relation to which the equilibrium is being studied, can be considered motionless, then the equilibrium is conditionally called absolute, and otherwise, relative. In statics, we will study only the so-called absolute equilibrium of bodies. In practice, in engineering calculations, equilibrium with respect to the Earth or to bodies rigidly connected to the Earth can be considered absolute. The validity of this statement will be substantiated in dynamics, where the concept of absolute equilibrium can be defined more strictly. The question of the relative equilibrium of bodies will also be considered there.

The equilibrium conditions of a body essentially depend on whether the body is solid, liquid, or gaseous. The equilibrium of liquid and gaseous bodies is studied in the courses of hydrostatics and aerostatics. In the general course of mechanics, usually only problems of the equilibrium of solids are considered.

All naturally occurring solids under the influence of external influences to some extent change their shape (deform). The values ​​of these deformations depend on the material of the bodies, their geometric shape and dimensions, and on the acting loads. To ensure the strength of various engineering structures and structures, the material and dimensions of their parts are selected so that the deformations under the acting loads are sufficiently small. As a result, when studying general conditions equilibrium, it is quite acceptable to neglect small deformations of the corresponding solid bodies and consider them as non-deformable or absolutely rigid.

Absolutely solid body such a body is called, the distance between any two points of which always remains constant.

In order for a rigid body to be in equilibrium (at rest) under the action of a certain system of forces, it is necessary that these forces satisfy certain equilibrium conditions this system of forces. Finding these conditions is one of the main tasks of statics. But in order to find the conditions for the equilibrium of various systems of forces, as well as to solve a number of other problems in mechanics, it turns out to be necessary to be able to add the forces acting on a rigid body, to replace the action of one system of forces with another system, and, in particular, to reduce this system of forces to the simplest form. Therefore, the following two main problems are considered in the statics of a rigid body:

1) addition of forces and reduction of systems of forces acting on a rigid body to the simplest form;

2) determination of the equilibrium conditions for systems of forces acting on a solid body.

Strength. The state of equilibrium or motion of a given body depends on the nature of its mechanical interactions with other bodies, i.e. from those pressures, attractions or repulsions that a given body experiences as a result of these interactions. A quantity that is a quantitative measure of the mechanical interactionaction of material bodies, is called in mechanics force.

The quantities considered in mechanics can be divided into scalar ones, i.e. those that are fully characterized by their numerical value, and vector ones, i.e. those that, in addition to the numerical value, are also characterized by the direction in space.

Force is a vector quantity. Its effect on the body is determined by: 1) numerical value or module strength, 2) towardsniem strength, 3) application point strength.

The direction and point of application of the force depend on the nature of the interaction of the bodies and their relative position. For example, the force of gravity acting on a body is directed vertically down. The pressure forces of two smooth balls pressed against each other are directed along the normal to the surfaces of the balls at the points of their contact and are applied at these points, etc.

Graphically, the force is represented by a directed segment (with an arrow). The length of this segment (AB in fig. 1) expresses the modulus of force on the selected scale, the direction of the segment corresponds to the direction of the force, its beginning (point BUT in fig. 1) usually coincides with the point of application of the force. Sometimes it is convenient to depict a force in such a way that the point of application is its end - the tip of the arrow (as in Fig. 4 in). Straight DE, along which the force is directed is called line of force. Force is represented by the letter F . The modulus of force is indicated by vertical lines "on the sides" of the vector. Force system is the totality of forces acting on an absolutely rigid body.

Basic definitions:

A body that is not bonded to other bodies, which this provision can report any movement in space, called free.

If a free rigid body under the action of a given system of forces can be at rest, then such a system of forces is called balanced.

If one system of forces acting on a free rigid body can be replaced by another system without changing the state of rest or motion in which the body is located, then such two systems of forces are called equivalent.

If this system force is equivalent to one force, then this force is called resultant this system of forces. In this way, resultant - is the power that alone can replacethe action of this system, forces on a rigid body.

A force equal to the resultant in absolute value, directly opposite to it in direction and acting along the same straight line, is called balancing by force.

The forces acting on a rigid body can be divided into external and internal. External called the forces acting on the particles of a given body from other material bodies. Internal called the forces with which the particles of a given body act on each other.

A force applied to a body at any one point is called concentrated. Forces acting on all points of a given volume or a given part of the surface of a body are called feuddivided.

The concept of a concentrated force is conditional, since in practice it is impossible to apply a force to a body at one point. The forces that we consider in mechanics as concentrated are essentially the resultant of certain systems of distributed forces.

In particular, the force of gravity, usually considered in mechanics, acting on a given rigid body, is the resultant of the forces of gravity of its particles. The line of action of this resultant passes through a point called the center of gravity of the body.

Axioms of statics. All theorems and equations of statics are derived from several initial positions, accepted without mathematical proof and called axioms or principles of statics. The axioms of statics are the result of generalizations of numerous experiments and observations on the balance and movement of bodies, repeatedly confirmed by practice. Some of these axioms are consequences of the basic laws of mechanics.

Axiom 1. If absolutely freea rigid body is acted upon by two forces, then the body cancan be in equilibrium if and onlywhen these forces are equal in absolute value (F 1 = F 2 ) and directedalong one straight line in opposite directions(Fig. 2).

Axiom 1 defines the simplest balanced system of forces, since experience shows that a free body, on which only one force acts, cannot be in equilibrium.

BUT
xioma 2. The action of a given system of forces on an absolutely rigid body will not change if a balanced system of forces is added to or subtracted from it.

This axiom states that two systems of forces that differ by a balanced system are equivalent to each other.

Consequence from the 1st and 2nd axioms. The point of application of a force acting on an absolutely rigid body can be transferred along its line of action to any other point of the body.

Indeed, let a force F applied at point A act on a rigid body (Fig. 3). Let's take an arbitrary point B on the line of action of this force and apply two balanced forces F1 and F2 to it, such that Fl \u003d F, F2 \u003d - F. This does not change the effect of the force F on the body. But the forces F and F2, according to axiom 1, also form a balanced system that can be discarded. As a result, only one force Fl equal to F, but applied at point B, will act on the body.

Thus, the vector representing the force F can be considered applied at any point on the line of action of the force (such a vector is called a sliding vector).

The result obtained is valid only for forces acting on an absolutely rigid body. In engineering calculations, this result can be used only when the external action of forces on a given structure is studied, i.e. when the general conditions for the equilibrium of the structure are determined.

H

For example, the rod AB shown in (Fig. 4a) will be in equilibrium if F1 = F2. When both forces are transferred to some point FROM rod (Fig. 4, b), or when the force F1 is transferred to point B, and the force F2 is transferred to point A (Fig. 4, c), the balance is not disturbed. However, the internal action of these forces in each of the cases considered will be different. In the first case, the rod is stretched under the action of applied forces, in the second case it is not stressed, and in the third case, the rod will be compressed.

BUT

xiom 3 (axiom of the parallelogram of forces). two forces,applied to the body at one point, have a resultant,represented by the diagonal of the parallelogram built on these forces. Vector TO, equal to the diagonal of a parallelogram built on vectors F 1 And F 2 (Fig. 5), is called the geometric sum of vectors F 1 And F 2 :

Therefore, axiom 3 can also be formulate as follows: resultant two forces applied to a body at one point is equal to the geomet ric (vector) sum of these forces and is applied in the same point.

Axiom 4. Two material bodies always act each otheron each other with forces equal in absolute value and directed alongone straight line in opposite directions(briefly: action equals reaction).

W

The law of the equality of action and reaction is one of the basic laws of mechanics. It follows that if the body BUT acts on the body IN with force F, then at the same time the body IN acts on the body BUT with force F = -F(Fig. 6). However, forces F And F" do not form a balanced system of forces, since they are applied to different bodies.

property of internal forces. According to axiom 4, any two particles of a solid body will act on each other with equal and oppositely directed forces. Since, when studying the general conditions of equilibrium, the body can be considered as absolutely rigid, then (according to axiom 1) all internal forces form a balanced system under this condition, which (according to axiom 2) can be discarded. Therefore, when studying the general conditions of equilibrium, it is necessary to take into account only the external forces acting on a given rigid body or a given structure.

Axiom 5 (hardening principle). If any changeremovable (deformable) body under the action of a given system of forcesis in equilibrium, then the equilibrium will remain even ifthe body will harden (become absolutely solid).

The assertion made in this axiom is obvious. For example, it is clear that the balance of a chain must not be disturbed if its links are welded together; the balance of a flexible thread will not be disturbed if it turns into a bent rigid rod, and so on. Since the same system of forces acts on a body at rest before and after solidification, axiom 5 can also be expressed in another form: at equilibrium, the forces acting on any variable (deforworldable) body, satisfy the same conditions as forabsolutely rigid bodies; however, for a mutable body, theseconditions, while necessary, may not be sufficient. For example, for the equilibrium of a flexible thread under the action of two forces applied to its ends, the same conditions are necessary as for a rigid rod (the forces must be equal in magnitude and directed along the thread in different directions). But these conditions will not be sufficient. To balance the thread, it is also required that the applied forces be tensile, i.e. directed as in Fig. 4a.

The solidification principle is widely used in engineering calculations. It allows, when compiling equilibrium conditions, to consider any variable body (belt, cable, chain, etc.) or any variable structure as absolutely rigid and apply the methods of rigid body statics to them. If the equations obtained in this way are not enough to solve the problem, then equations are additionally drawn up that take into account either the equilibrium conditions of individual parts of the structure, or their deformation.

Topic № 2. DYNAMICS OF THE POINT

The manual contains the basic concepts and terms of one of the main disciplines of the subject block "Technical Mechanics". This discipline includes such sections as "Theoretical Mechanics", "Strength of Materials", "Theory of Mechanisms and Machines".

The manual is intended to assist students in self-study of the course "Technical Mechanics".

Theoretical Mechanics 4

I. Statics 4

1. Basic concepts and axioms of statics 4

2. System of converging forces 6

3. Flat system of arbitrarily distributed forces 9

4. The concept of a farm. Truss calculation 11

5. Spatial system of forces 11

II. Kinematics of point and rigid body 13

1. Basic concepts of kinematics 13

2. Translational and rotational motion of a rigid body 15

3. Plane-parallel motion of a rigid body 16

III. Dynamics of point 21

1. Basic concepts and definitions. Laws of Dynamics 21

2. General theorems of point dynamics 21

Strength of materials22

1. Basic concepts 22

2. External and internal forces. Section method 22

3. The concept of stress 24

4. Tension and compression of a straight beam 25

5. Shift and Collapse 27

6. Torsion 28

7. Cross bend 29

8. Longitudinal bend. The essence of the phenomenon of longitudinal bending. Euler formula. Critical stress 32

Theory of mechanisms and machines 34

1. Structural analysis of mechanisms 34

2. Classification of flat mechanisms 36

3. Kinematic study of flat mechanisms 37

4. Cam mechanisms 38

5. Gear mechanisms 40

6. Dynamics of mechanisms and machines 43

Bibliography45

THEORETICAL MECHANICS

I. Statics

1. Basic concepts and axioms of statics

The science of the general laws of motion and equilibrium of material bodies and of the interactions between bodies arising from this is called theoretical mechanics.

static called the branch of mechanics, which sets out the general doctrine of forces and studies the conditions for the equilibrium of material bodies under the action of forces.

Absolutely solid body such a body is called, the distance between any two points of which always remains constant.

The quantity, which is a quantitative measure of the mechanical interaction of material bodies, is called force.

Scalars are those that are fully characterized by their numerical value.

Vector quantities - these are those that, in addition to a numerical value, are also characterized by a direction in space.

Force is a vector quantity(Fig. 1).

Strength is characterized by:

- direction;

– numerical value or module;

- point of application.

Straight DE along which the force is directed is called line of force.

The totality of forces acting on a rigid body is called system of forces.

A body that is not fastened to other bodies, to which any movement in space can be communicated from a given position, is called free.

If one system of forces acting on a free rigid body can be replaced by another system without changing the state of rest or motion in which the body is located, then such two systems of forces are called equivalent.

The system of forces under which a free rigid body can be at rest is called balanced or equivalent to zero.

The resultant - it is a force that alone replaces the action of a given system of forces on a rigid body.

A force equal to the resultant in absolute value, directly opposite to it in direction and acting along the same straight line, is called balancing force.

External called the forces acting on the particles of a given body from other material bodies.

Internal called the forces with which the particles of a given body act on each other.

A force applied to a body at any one point is called focused.

Forces acting on all points of a given volume or a given part of the surface of a body are called distributed.

Axiom 1. If two forces act on a free absolutely rigid body, then the body can be in equilibrium if and only if these forces are equal in absolute value and directed along one straight line in opposite directions (Fig. 2).

Axiom 2. The action of one system of forces on an absolutely rigid body will not change if a balanced system of forces is added to or subtracted from it.

Consequence from the 1st and 2nd axioms. The action of a force on an absolutely rigid body will not change if the point of application of the force is moved along its line of action to any other point on the body.

Axiom 3 (axiom of the parallelogram of forces). Two forces applied to the body at one point have a resultant applied at the same point and depicted by the diagonal of a parallelogram built on these forces as on the sides (Fig. 3).

R = F 1 + F 2

Vector R, equal to the diagonal of the parallelogram built on the vectors F 1 and F 2 is called geometric sum of vectors.

Axiom 4. With every action of one material body on another, there is a reaction of the same magnitude, but opposite in direction.

Axiom 5(hardening principle). The balance of a variable (deformable) body under the action of a given system of forces will not be disturbed if the body is considered to be solidified (absolutely rigid).

A body that is not fastened to other bodies and can perform any movement in space from a given position is called free.

A body whose movement in space is prevented by some other bodies fastened or in contact with it is called not free.

Everything that limits the movement of a given body in space is called communication.

The force with which this connection acts on the body, preventing one or another of its movements, is called bond reaction force or bond reaction.

Communication reaction directed in the direction opposite to that where the connection does not allow the body to move.

Axiom of connections. Any non-free body can be considered as free, if we discard the bonds and replace their action with the reactions of these bonds.

2. System of converging forces

converging are called forces whose lines of action intersect at one point (Fig. 4a).

The system of converging forces has resultant equal to geometric sum(main vector) of these forces and applied at the point of their intersection.

geometric sum, or main vector several forces is represented by the closing side of the force polygon constructed from these forces (Fig. 4b).

2.1. Projection of force on the axis and on the plane

The projection of the force on the axis is called a scalar quantity equal to the length of the segment, taken with the corresponding sign, enclosed between the projections of the beginning and end of the force. The projection has a plus sign if the movement from its beginning to the end occurs in the positive direction of the axis, and a minus sign if in the negative direction (Fig. 5).

Projection of Force on the Axis is equal to the product of the modulus of force and the cosine of the angle between the direction of the force and the positive direction of the axis:

F X = F cos.

The projection of force on a plane called the vector enclosed between the projections of the beginning and end of the force on this plane (Fig. 6).

F xy = F cos Q

F x = F xy cos= F cos Q cos

F y = F xy cos= F cos Q cos

Sum Vector Projection on any axis is equal to the algebraic sum of the projections of the terms of the vectors on the same axis (Fig. 7).

R = F 1 + F 2 + F 3 + F 4

R x = ∑F ix R y = ∑F iy

To balance the system of converging forces it is necessary and sufficient that the force polygon constructed from these forces be closed - this is the geometric condition of equilibrium.

Analytical equilibrium condition. For the equilibrium of the system of converging forces, it is necessary and sufficient that the sum of the projections of these forces on each of the two coordinate axes be equal to zero.

∑F ix = 0 ∑F iy = 0 R =

2.2. Three forces theorem

If a free rigid body is in equilibrium under the action of three non-parallel forces lying in the same plane, then the lines of action of these forces intersect at one point (Fig. 8).

2.3. Moment of force about the center (point)

Moment of force about the center is called a value equal to taken with the corresponding sign to the product of the modulus of force and the length h(Fig. 9).

M = ± F· h

Perpendicular h, lowered from the center ABOUT to the line of force F, is called shoulder of force F relative to the center ABOUT.

Moment has a plus sign, if the force tends to rotate the body around the center ABOUT counterclockwise, and minus sign- if clockwise.

Properties of the moment of force.

1. The moment of force will not change when the point of application of force is moved along its line of action.

2. The moment of force about the center is zero only when the force is zero or when the line of action of the force passes through the center (shoulder is zero).

BRIEF COURSE OF LECTURES ON THE DISCIPLINE "FUNDAMENTALS OF TECHNICAL MECHANICS"

Section 1: Statics

Statics, axioms of statics. Bonds, reaction of bonds, types of bonds.

The fundamentals of theoretical mechanics consist of three sections: Statics, fundamentals of strength of materials, details of mechanisms and machines.

Mechanical movement is a change in the position of bodies or points in space over time.

The body is considered as a material point, i.e. geometric point and at this point the entire mass of the body is concentrated.

The system is a set of material points, the movement and position of which are interconnected.

Force is a vector quantity, and the effect of force on a body is determined by three factors: 1) Numerical value, 2) direction, 3) point of application.

[F] - Newton - [H], Kg / s = 9.81 N = 10 N, KN = 1000 N,

MN = 1000000 N, 1N = 0.1 Kg/s

Axioms of statics.

1Axiom– (Defines a balanced system of forces): the system of forces applied to material point, is balanced if, under its influence, the point is in a state of relative rest, or moves in a straight line and uniformly.

If a balanced system of forces acts on a body, then the body is either: in a state of relative rest, or moves uniformly and rectilinearly, or uniformly rotates around a fixed axis.

2 Axiom– (Sets the condition for the balance of two forces): two forces equal in absolute value or numerical value (F1=F2) applied to an absolutely rigid body and directed

in a straight line in opposite directions are mutually balanced.

A system of forces is a combination of several forces applied to a point or body.

The system of forces of the line of action, in which they are in different planes, is called spatial, if in the same plane, then flat. A system of forces with lines of action intersecting at one point is called convergent. If two systems of forces taken separately have the same effect on the body, then they are equivalent.

Consequence of 2 axioms.

Any force acting on a body can be transferred along the line of its action, to any point of the body without violating its mechanical state.

3Axiom: (Basis for force transformation): without disturbing the mechanical state is absolutely solid body a balanced system of forces can be applied to it or rejected from it.

Vectors that can be moved along their line of action are called moving vectors.

4 Axiom– (Defines the rules for adding two forces): the resultant of two forces applied to one point, applied at this point, is the diagonal of a parallelogram built on these forces.

- Resultant force =F1+F2 - According to the parallelogram rule

According to the triangle rule.

5 Axiom- (Establishes that in nature there cannot be a one-sided action of force) in the interaction of bodies, every action corresponds to an equal and oppositely directed counteraction.

Connections and their reactions.

Bodies in mechanics are: 1 free 2 non-free.

Free - when the body does not experience any obstacles to move in space in any direction.

Non-free - the body is connected with other bodies that restrict its movement.

Bodies that restrict the movement of a body are called bonds.

When a body interacts with bonds, forces arise, they act on the body from the side of the bond and are called bond reactions.

The reaction of the bond is always opposite to the direction in which the bond impedes the movement of the body.

Communication types.

1) Communication in the form of a smooth plane without friction.

2) Communication in the form of a contact of a cylindrical or spherical surface.

3) Communication in the form of a rough plane.

Rn is the force perpendicular to the plane. Rt is the friction force.

R is the bond reaction. R = Rn+Rt

4) Flexible connection: rope or cable.

5) Connection in the form of a rigid straight rod with hinged fastening of the ends.

6) The connection is carried out by an edge of a dihedral angle or a point support.

R1R2R3 - Perpendicular to the surface of the body.

Flat system of converging forces. Geometric definition resultant. The projection of the force on the axis. Projection of the vector sum onto the axis.

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

Flat system of forces - the lines of action of all these forces lie in the same plane.

The spatial system of converging forces - the lines of action of all these forces lie in different planes.

Converging forces can always be transferred to one point, i.e. at the point where they intersect along the line of action.

F123=F1+F2+F3=

The resultant is always directed from the beginning of the first term to the end of the last (the arrow is directed towards the bypass of the polyhedron).

If, when constructing a force polygon, the end of the last force coincides with the beginning of the first, then the resultant = 0, the system is in equilibrium.

not balanced

balanced.

The projection of the force on the axis.

An axis is a straight line to which a certain direction is assigned.

The vector projection is scalar value, it is determined by the segment of the axis cut off by perpendiculars to the axis from the beginning and end of the vector.

The projection of the vector is positive if it coincides with the direction of the axis, and negative if it is opposite to the direction of the axis.

Conclusion: The projection of the force on the coordinate axis = the product of the modulus of force and cos of the angle between the force vector and the positive direction of the axis.

positive projection.

Negative projection

Projection = o

Projection of the vector sum onto the axis.

Can be used to define a module and

the direction of the force, if its projections on

coordinate axes.

Output: The projection of the vector sum, or resultant, on each axis is equal to the algebraic sum of the projection of the terms of the vectors on the same axis.

Determine the modulus and direction of the force if its projections are known.

Answer: F=50H,

Fy-?F -?

Section 2. Strength of materials (Sopromat).

Basic concepts and hypotheses. Deformation. section method.

The strength of materials is the science of engineering methods for calculating the strength, rigidity and stability of structural elements. Strength - the properties of bodies not to collapse under the influence of external forces. Rigidity - the ability of bodies in the process of deformation to change dimensions within specified limits. Stability - the ability of bodies to maintain their original state of equilibrium after the application of a load. The purpose of science (Sopromat) is the creation of practically convenient methods for calculating the most common structural elements. Basic hypotheses and assumptions regarding the properties of materials, loads and nature of deformation.1) Hypothesis(Homogeneity and oversights). When the material completely fills the body, and the properties of the material do not depend on the size of the body. 2) Hypothesis(On the ideal elasticity of a material). The ability of the body to restore the pile to its original shape and dimensions after the elimination of the causes that caused the deformation. 3) Hypothesis(Assumption of a linear relationship between deformations and loads, Fulfillment of Hooke's law). Displacement as a result of deformation is directly proportional to the loads that caused them. 4) Hypothesis(Flat sections). The cross-sections are flat and normal to the beam axis before the load is applied to it and remain flat and normal to its axis after deformation. 5) Hypothesis(On the isotropy of the material). Mechanical properties material in any direction are the same. 6) Hypothesis(On the smallness of deformations). The deformations of the body are so small compared to the dimensions that they do not have a significant effect on mutual arrangement loads. 7) Hypothesis (Principle of independence of action of forces). 8) Hypothesis (Saint-Venant). The deformation of the body far from the place of application of statically equivalent loads is practically independent of the nature of their distribution. Under the influence of external forces, the distance between the molecules changes, internal forces arise inside the body, which counteract deformation and tend to return the particles to their previous state - elastic forces. Section method. The external forces applied to the cut off part of the body must be balanced with the internal forces arising in the section plane, they replace the action of the discarded part with the rest. Rod (beams) - Structural elements, the length of which significantly exceeds their transverse dimensions. Plates or shells - When the thickness is small compared to the other two dimensions. Massive bodies - all three sizes are approximately the same. Equilibrium condition.





NZ - Longitudinal internal force. QX and QY - Transverse internal force. MX and MY - Bending moments. MZ - Torque. When a planar system of forces acts on a rod, only three force factors can occur in its sections, these are: MX - Bending moment, QY - Transverse force, NZ - Longitudinal force. Equilibrium equation. The coordinate axes will always direct the Z-axis along the bar axis. The X and Y axes are along the main central axes of its cross sections. The origin of coordinates is the center of gravity of the section.

The sequence of actions to determine the internal forces.

1) Mentally draw a section at the point of interest to us design. 2) Discard one of the cut off parts, and consider the balance of the remaining part. 3) Compose an equilibrium equation and determine from them the values ​​and directions of internal force factors. Axial tension and compression - internal forces in cross section They can be closed by one force directed along the axis of the rod. Tension. Compression. Shear - occurs when, in the cross section of the rod, the internal forces are reduced to one, i.e. transverse force Q. Torsion - 1 force factor MZ occurs. MZ=MK Pure bend– A bending moment MX or MY occurs. To calculate structural elements for strength, rigidity, stability, first of all, it is necessary (using the section method) to determine the occurrence of internal force factors.