Methods of giving lectures on technical mechanics. Introductory lesson on technical mechanics “Basic concepts and axioms of statics
Introduction
Theoretical mechanics is one of the most important fundamental general scientific disciplines. It plays a significant role in the training of engineers of any specialization. General engineering disciplines are based on the results of theoretical mechanics: strength of materials, machine parts, theory of mechanisms and machines, and others.
The main task of theoretical mechanics is the study of the movement of material bodies under the influence of forces. An important particular task is the study of the equilibrium of bodies under the influence of forces.
Course of Lectures. Theoretical mechanics
The structure of theoretical mechanics. Basics of statics
Equilibrium conditions for an arbitrary system of forces.
Equilibrium equations for a rigid body.
Flat system of forces.
Special cases of rigid body equilibrium.
Balance problem for a beam.
Determination of internal forces in rod structures.
Fundamentals of point kinematics.
Natural coordinates.
Euler's formula.
Distribution of accelerations of points of a rigid body.
Translational and rotational movements.
Plane-parallel motion.
Complex point movement.
Basics of point dynamics.
Differential equations of motion of a point.
Particular types of force fields.
Fundamentals of the dynamics of a system of points.
General theorems on the dynamics of a system of points.
Dynamics of rotational motion of the body.
Dobronravov V.V., Nikitin N.N. Course of theoretical mechanics. M., graduate School, 1983.
Butenin N.V., Lunts Ya.L., Merkin D.R. Course of theoretical mechanics, parts 1 and 2. M., Higher School, 1971.
Petkevich V.V. Theoretical mechanics. M., Nauka, 1981.
Collection of tasks for coursework in theoretical mechanics. Ed. A.A. Yablonsky. M., Higher School, 1985.
Lecture 1. The structure of theoretical mechanics. Basics of statics
In theoretical mechanics, the movement of bodies relative to other bodies, which are physical reference systems, is studied.
Mechanics allows not only to describe, but also to predict the movement of bodies, establishing causal relationships in a certain, very wide range of phenomena.
Basic abstract models of real bodies:
material point – has mass, but no size;
absolutely rigid body – a volume of finite dimensions, completely filled with a substance, and the distances between any two points of the medium filling the volume do not change during movement;
continuous deformable medium – fills a finite volume or unlimited space; the distances between points in such a medium can vary.
Of these, systems:
System of free material points;
Connected systems;
An absolutely solid body with a cavity filled with liquid, etc.
"Degenerate" models:
Infinitely thin rods;
Infinitely thin plates;
Weightless rods and threads connecting material points, etc.
From experience: mechanical phenomena occur differently in different places physical reference system. This property is the heterogeneity of space, determined by the physical reference system. Here, heterogeneity is understood as the dependence of the nature of the occurrence of a phenomenon on the place in which we observe this phenomenon.
Another property is anisotropy (non-isotropy), the movement of a body relative to a physical reference system can be different depending on the direction. Examples: river flow along the meridian (from north to south - Volga); projectile flight, Foucault pendulum.
The properties of the reference system (inhomogeneity and anisotropy) make it difficult to observe the movement of a body.
Practically free from this - geocentric system: the center of the system is in the center of the Earth and the system does not rotate relative to the “fixed” stars). The geocentric system is convenient for calculating movements on Earth.
For celestial mechanics(for solar system bodies): heliocentric frame of reference, which moves with the center of mass solar system and does not rotate relative to the “fixed” stars. For this system not yet discovered heterogeneity and anisotropy of space
in relation to mechanical phenomena.
So, the abstract is introduced inertial frame of reference for which space is homogeneous and isotropic in relation to mechanical phenomena.
Inertial reference frame- one whose own motion cannot be detected by any mechanical experiment. Thought experiment: “a point alone in the whole world” (isolated) is either at rest or moving in a straight line and uniformly.
All reference systems moving relative to the original one rectilinearly and uniformly will be inertial. This allows the introduction of a unified Cartesian coordinate system. Such a space is called Euclidean.
Conventional agreement - take the right coordinate system (Fig. 1).
IN 
time– in classical (non-relativistic) mechanics absolutely, the same for all reference systems, that is, the initial moment is arbitrary. In contrast to relativistic mechanics, where the principle of relativity is applied.
The state of motion of the system at time t is determined by the coordinates and velocities of the points at this moment.
Real bodies interact and forces arise that change the state of motion of the system. This is the essence of theoretical mechanics.
How is theoretical mechanics studied?
The doctrine of the equilibrium of a set of bodies of a certain frame of reference - section statics.
Chapter kinematics: part of mechanics in which dependencies between quantities characterizing the state of motion of systems are studied, but the reasons causing a change in the state of motion are not considered.
After this, we will consider the influence of forces [MAIN PART].
Chapter dynamics: part of mechanics that deals with the influence of forces on the state of motion of systems of material objects.
Principles for constructing the main course – dynamics:
1) based on a system of axioms (based on experience, observations);
Constantly - ruthless control of practice. Sign of exact science – presence of internal logic (without it - a set of unrelated recipes)!
Static is called that part of mechanics where the conditions are studied that the forces acting on a system of material points must satisfy in order for the system to be in equilibrium, and the conditions for the equivalence of systems of forces.
Equilibrium problems in elementary statics will be considered using exclusively geometric methods based on the properties of vectors. This approach is used in geometric statics(in contrast to analytical statics, which is not considered here).
The positions of various material bodies will be related to the coordinate system, which we will take as stationary.
Ideal models of material bodies:
1) material point – a geometric point with mass.
2) an absolutely rigid body is a collection of material points, the distances between which cannot be changed by any actions.
By forces we will call objective causes that are the result of the interaction of material objects, capable of causing the movement of bodies from a state of rest or changing the existing movement of the latter.
Since force is determined by the movement it causes, it also has a relative nature, depending on the choice of reference system.
The question of the nature of forces is considered in physics.
A system of material points is in equilibrium if, being at rest, it does not receive any movement from the forces acting on it.
From everyday experience: forces have a vector nature, that is, magnitude, direction, line of action, point of application. The condition for equilibrium of forces acting on a rigid body is reduced to the properties of vector systems.
Summarizing the experience of studying the physical laws of nature, Galileo and Newton formulated the basic laws of mechanics, which can be considered as axioms of mechanics, since they have are based on experimental facts.
Axiom 1. The action of several forces on a point of a rigid body is equivalent to the action of one resultant force constructed according to the rule of vector addition (Fig. 2).
Consequence. The forces applied to a point on a rigid body add up according to the parallelogram rule.
Axiom 2. Two forces applied to a rigid body mutually balanced if and only if they are equal in size, directed in opposite directions and lie on the same straight line.
Axiom 3. The action of a system of forces on a rigid body will not change if add to this system or discard from it two forces of equal magnitude, directed in opposite directions and lying on the same straight line.
Consequence. The force acting on a point of a rigid body can be transferred along the line of action of the force without changing the equilibrium (that is, the force is a sliding vector, Fig. 3)
1) Active - create or are capable of creating the movement of a rigid body. For example, weight force.
2) Passive - do not create movement, but limit the movement of a solid body, preventing movement. For example, the tension force of an inextensible thread (Fig. 4).
Axiom 4. The action of one body on a second is equal and opposite to the action of this second body on the first ( action equals reaction).
We will call the geometric conditions limiting the movement of points connections.
Terms of communication: for example,
- rod of indirect length l.
- flexible non-stretchable thread of length l.
Forces caused by connections and preventing movement are called forces of reactions.
Axiom 5. The connections imposed on a system of material points can be replaced by reaction forces, the action of which is equivalent to the action of the connections.
When passive forces cannot balance the action of active forces, movement begins.
Two particular problems of statics
1. System of converging forces acting on a rigid body
A system of converging forces This is called a system of forces whose lines of action intersect at one point, which can always be taken as the origin of coordinates (Fig. 5).
Projections of the resultant:
;
;
.
If , then the force causes the motion of the rigid body.
Equilibrium condition for a converging system of forces:
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2. Balance of three forces
If three forces act on a rigid body, and the lines of action of the two forces intersect at some point A, equilibrium is possible if and only if the line of action of the third force also passes through point A, and the force itself is equal in magnitude and opposite in direction to the sum 
(Fig. 6).
Examples:
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Moment of force about point O let's define it as a vector, in size equal to twice the area of a triangle, the base of which is the force vector with the vertex at a given point O; direction– orthogonal to the plane of the triangle in question in the direction from which the rotation produced by the force around point O is visible counterclockwise. is the moment of the sliding vector and is free vector(Fig.9).
So:
or
,
Where 
;;
.
Where F is the force modulus, h is the shoulder (the distance from the point to the direction of the force).
Moment of force about the axis is the algebraic value of the projection onto this axis of the vector of the moment of force relative to an arbitrary point O taken on the axis (Fig. 10).
This is a scalar independent of the choice of point. Indeed, let us expand :|| and in the plane.
About moments: let O 1 be the point of intersection with the plane. Then:
a) from - moment => projection = 0.
b) from - moment along => is a projection.
So, moment about an axis is the moment of the force component in a plane perpendicular to the axis relative to the point of intersection of the plane and the axis.
Varignon's theorem for a system of converging forces:
Moment of resultant force for a system of converging forces relative to an arbitrary point A is equal to the sum of the moments of all component forces relative to the same point A (Fig. 11).
Proof in the theory of convergent vectors.
Explanation: addition of forces according to the parallelogram rule => the resulting force gives a total moment.
Security questions:
1. Name the main models of real bodies in theoretical mechanics.
2. Formulate the axioms of statics.
3. What is called the moment of force about a point?
Lecture 2. Equilibrium conditions for an arbitrary system of forces
From the basic axioms of statics, elementary operations on forces follow:
1) force can be transferred along the line of action;
2) forces whose lines of action intersect can be added according to the parallelogram rule (according to the rule of vector addition);
3) to the system of forces acting on a rigid body, you can always add two forces, equal in magnitude, lying on the same straight line and directed in opposite directions.
Elementary operations do not change the mechanical state of the system.
Let's call two systems of forces equivalent, if one from the other can be obtained using elementary operations (as in the theory of sliding vectors).
A system of two parallel forces, equal in magnitude and directed in opposite directions, is called a couple of forces(Fig. 12).
Moment of a couple of forces- a vector equal in size to the area of the parallelogram built on the vectors of the pair, and directed orthogonally to the plane of the pair in the direction from where the rotation imparted by the vectors of the pair is seen to occur counterclockwise.
, that is, the moment of force relative to point B.
A pair of forces is completely characterized by its moment.
A pair of forces can be transferred by elementary operations to any plane parallel to the plane of the pair; change the magnitude of the forces of the pair in inverse proportion to the shoulders of the pair.
Pairs of forces can be added, and the moments of pairs of forces are added according to the rule of addition of (free) vectors.
Bringing a system of forces acting on a rigid body to an arbitrary point (center of reduction)- means replacing the current system with a simpler one: a system of three forces, one of which passes through a predetermined point, and the other two represent a pair.
It can be proven using elementary operations (Fig. 13).
A system of converging forces and a system of pairs of forces.
- resultant force.
Resulting pair.
That's what needed to be shown.
Two systems of forces will equivalent if and only if both systems are reduced to one resultant force and one resultant pair, that is, when the conditions are met:
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General case of equilibrium of a system of forces acting on a rigid body
Let us reduce the system of forces to (Fig. 14):
Resultant force through the origin;
The resulting pair, moreover, through point O.
That is, they led to and - two forces, one of which passes through a given point O.
Equilibrium, if the two on the same straight line are equal and opposite in direction (axiom 2).
Then it passes through point O, that is.
So, general conditions for the equilibrium of a solid body:
These conditions are valid for an arbitrary point in space.
Security questions:
1. List the elementary operations on forces.
2. What systems of forces are called equivalent?
3. Write the general conditions for the equilibrium of a rigid body.
Lecture 3. Equilibrium equations for a rigid body
Let O be the origin of coordinates; – resultant force; – moment of the resultant pair. Let point O1 be the new center of reduction (Fig. 15).
New power system:
When the reduction point changes, => only changes (in one direction with one sign, in the other direction with another). That is, the point: 
the lines match
Analytically: 
(colinearity of vectors)
; coordinates of point O1.
This is the equation of a straight line, for all points of which the direction of the resulting vector coincides with the direction of the moment of the resulting pair - the straight line is called dynamo.
If the dynamism => on the axis, then the system is equivalent to one resultant force, which is called resultant force of the system. At the same time, always, that is.
Four cases of bringing forces:
1.) ;- dynamism.
2.) ;- resultant.
3.) ;- pair.
4.) ;- balance.
Two vector equilibrium equations: the main vector and the main moment are equal to zero,.
Or six scalar equations in projections onto Cartesian coordinate axes:
Here: 
The complexity of the type of equations depends on the choice of the reduction point => the skill of the calculator.
Finding the equilibrium conditions for a system of solid bodies in interaction<=>the problem of the equilibrium of each body separately, and the body is acted upon by external forces and internal forces (the interaction of bodies at points of contact with equal and oppositely directed forces - axiom IV, Fig. 17).
Let us choose for all bodies of the system one adduction center. Then for each body with the equilibrium condition number:
,
, (= 1, 2, …, k)
where , is the resulting force and moment of the resulting pair of all forces, except internal reactions.
The resulting force and moment of the resulting pair of forces of internal reactions.
Formally summing by and taking into account the IV axiom
we get necessary conditions for the equilibrium of a solid body:
,
Example.
Equilibrium: = ?
Security questions:
1. Name all cases of bringing a system of forces to one point.
2. What is dynamism?
3. Formulate the necessary conditions for equilibrium of a system of solid bodies.
Lecture 4. Flat force system
A special case of the general delivery of the problem.
Let all the acting forces lie in the same plane - for example, a sheet. Let us choose point O as the reduction center - in the same plane. We obtain the resulting force and the resulting steam in the same plane, that is (Fig. 19)
Comment.
The system can be reduced to one resultant force.
Equilibrium conditions:
or scalar:
Very common in applications such as strength of materials.
Example.
With the friction of the ball on the board and on the plane. Equilibrium condition: = ?
The problem of the equilibrium of a non-free rigid body.
A rigid body whose movement is constrained by bonds is called unfree. For example, other bodies, hinged fastenings.
When determining equilibrium conditions: a non-free body can be considered as free, replacing bonds with unknown reaction forces.
Example.
Security questions:
1. What is called a plane system of forces?
2. Write the equilibrium conditions for a plane system of forces.
3. Which solid body is called non-free?
Lecture 5. Special cases of rigid body equilibrium
Theorem. Three forces balance a rigid body only if they all lie in the same plane.
Proof.
Let us choose a point on the line of action of the third force as the reduction point. Then (Fig. 22)
That is, the planes S1 and S2 coincide, and for any point on the force axis, etc. (Simpler: in the plane 
only there for balancing).
DEPARTMENT OF EDUCATION AND SCIENCE OF THE KOstroma REGION
Regional state budgetary professional educational institution
“Kostroma Energy College named after F.V. Chizhov"
METHODOLOGICAL DEVELOPMENT
For teachers of secondary vocational education
Introductory lesson on the topic:
“BASIC CONCEPTS AND AXIOMS OF STATICS”
discipline "Technical mechanics"
O.V. Guryev
Kostroma
Annotation.
Methodological development is intended for conducting an 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 studying the discipline.
Hypertext lesson. Therefore, the objectives of the lesson include:
Educational -
Developmental -
Educational -
Approved by the subject cycle commission
Teacher:
M.A. Zaitseva
Protocol No. dated 20
Reviewer
INTRODUCTION
Methodology for conducting a lesson on technical mechanics
Technological map classes
Hypertext
CONCLUSION
REFERENCES
Introduction
“Technical mechanics” is an important subject in 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. To successfully master 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 technology, the more difficult it is to fit it into the framework of 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 a ready-made form, but independently applies it to solving cognitive and practical problems.
Skills become of great importance 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 fundamentals of theory to solving practical problems. Independent work develops abilities, memory, attention, imagination, and thinking.
In teaching the discipline, all the principles of teaching known in pedagogy are practically applicable: scientific, systematic and consistent, visual, conscious assimilation of knowledge by students, accessibility of learning, connection of learning with practice, along with explanatory and illustrative methods, which were, are and remain the main ones in lessons on technical mechanics. Participative teaching methods are used: silent and loud discussion, brainstorming, analysis concrete example, question - answer.
The topic “Basic concepts and axioms of statics” is one of the most important in the “Technical Mechanics” course. She has great value from a course learning perspective. This topic is an introductory part of the discipline.
Students work with hypertext in which they need to pose questions correctly. Learn to work in groups.
Working on assigned tasks shows students’ activity and responsibility, independence in solving problems that arise during the task, and gives them the skills and abilities to solve these problems. Teacher asking problematic issues, forces students to think practically. As a result of working with hypertext, students draw conclusions about the topic covered.
Methodology for conducting classes in technical mechanics
The structure of classes depends on what goals are considered the most important. One of the most important tasks educational institution- teach to learn. By imparting practical knowledge to students, we need to teach them to learn independently.
− get interested in science;
− get interested in the task;
− to instill skills in working with hypertext.
Goals such as the formation of a worldview and educational influence on students are also extremely important. 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 needs to take into account the characteristics of the student population and use all the advantages of the living word and direct communication with students. In order to capture the attention of students, interest and captivate them with reasoning, and accustom them to independent thinking, when organizing classes, it is necessary to especially take into account the four stages of the cognitive process, which include:
1. statement of a problem or task;
2. evidence - discourse (discursive - rational, logical, conceptual);
3. analysis of the result obtained;
4. retrospection - establishing connections between newly obtained results and previously established conclusions.
When starting to present a new problem or task, it is necessary special attention devote it to staging. It is not enough to limit yourself to just the formulation of the problem. This is well confirmed by the following statement of Aristotle: knowledge begins with surprise. You must be able to attract attention to the new task from the very beginning, surprise, and therefore interest the student. After this, you can move on to solving the problem. It is very important that the statement of the problem or task is well understood by students. They should be completely clear about the need to study a new problem and the validity of its formulation. When posing a new problem, rigor of presentation is necessary. However, it must be taken into account that many questions and methods of solution are not always clear to students and may seem formal if special explanations are not given. Therefore, each teacher must 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 completely natural to choose a certain method for solving the formulated problem.
Technological map
TOPIC “BASIC CONCEPTS AND AXIOMS OF STATICS”
Lesson objectives:
Educational - Master the three sections of technical mechanics, their definitions, basic concepts and axioms of statics.
Developmental - improve students' independent work skills.
Educational - strengthening group work skills, the ability to listen to the opinions of comrades, and discuss in a group.
Lesson type- explanation of new material
Technology- hypertext
| Stages | Steps | Teacher's activities | Student activities | Time |
| I Organizational | Topic, purpose, order of work | I formulate the topic, goal, order of work in the lesson: “We are working in the hypertext technology - I will say hypertext, then you will work with the text in groups, then we will check the level of mastery of the material and summarize the results. 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 | Speak Hypertext | Every student has hypertext on their desks. I suggest following me through the text, listening, looking at the screen. | View hypertext printouts | |
| I speak hypertext while showing slides on the screen | Listen, watch, read |
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| III Consolidation of what has been learned | 1 Drawing up a text plan | Instructions 1. Divide into groups of 4-5 people. 2. Break the text into parts and title them, be prepared to present your plan to the group (When the plan is ready, it is drawn up on whatman paper). 3. I will organize a discussion of the plan. We compare the number of parts in the plan. If there are different things, we turn to the text and clarify the number of parts in the plan. 4. We agree on the wording of the names of the parts and choose the best. 5. I summarize. Recording final version plan. | 1. Divided into groups. 2. Title the text. 3. Discuss drawing up a plan. 4. Clarify 5. Write down the final version of the plan | |
| 2. Compiling questions based on the text | Instructions: 1. Each group should write 2 questions to the text. 2. Be prepared to ask questions group to group sequentially 3. If the group cannot answer the question, the one who asked answers. 4. I will organize a “Question Spindle”. The procedure continues until repetitions begin. | Make up questions and prepare answers Ask questions, answer | ||
| IV. Checking your understanding of the material | Instructions: 1. Perform the test individually. 2. Finally, check the test of your desk neighbor, checking the correct answers with the slide on the screen. 3. Give a rating based on the specified criteria on the slide. 4. We hand over the work to me | Perform the test Check Evaluate | ||
| V. Summing up | 1. Summing up the goal | Analyzing this test by level of material mastery | ||
| 2. Homework | Compose (or reproduce) a reference summary of hypertext Please note that the task is for more highest rating, located in the Moodle remote shell, in the “Technical Mechanics” section | Write down the task | ||
| 3. Lesson reflection | I invite you to speak on the lesson, for help I show a slide with a list of prepared starting phrases | Choose phrases and speak out |
1.1 Meet the group
1.2 Mark students present
1.3 Familiarization with the requirements for students in the classroom.
3. Presentation of the material
4. Questions to reinforce the material
5. Homework
Hypertext
Mechanics, along with astronomy and mathematics, is one of the most ancient sciences. The term mechanics comes from the Greek word “Mechane” - device, machine.
In ancient times, Archimedes was 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 brilliant theoretical discoveries with remarkable inventions. Some of them have not lost their significance in our time.
Russian scientists made a major contribution to the development of mechanics: P.L. Chebeshev (1821-1894) - laid the foundation for the world-famous Russian school of theory of mechanisms and machines. S.A. Chaplygin (1869-1942). developed a number of issues of aerodynamics that are of great importance for modern speed aviation.
Technical mechanics is a complex discipline that sets out the basic principles about the interaction of solids, the strength of materials and calculation methods structural elements machines and mechanisms on external interactions. Technical mechanics is divided into three large sections: theoretical mechanics, strength of materials, machine parts. One of the sections, theoretical mechanics, is divided into three subsections: statics, kinematics, dynamics.
Today we will begin the study of technical mechanics with the subsection statics - this is a section of theoretical mechanics in which the conditions of absolute equilibrium are studied solid under the influence of forces applied to them. The basic concepts of statics include: Material point
a body whose dimensions can be neglected under the conditions of the assigned tasks. Absolutely rigid body - a conventionally accepted body that does not deform under the influence 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). External forces - forces acting on a body from other bodies. Inner forces- interaction forces between particles of a given body. Active forces- forces causing body movement. Reactive forces- forces that prevent the movement of a body. Equivalent forces- forces and systems of forces that produce same action on the body. Equivalent forces, systems of forces- one force equivalent to the system of forces under consideration. The forces of this system are called components 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. System of forces - a set of forces acting on a body. Systems of forces are flat, spatial; convergent, parallel, arbitrary. Equilibrium- 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 of these component forces. Disintegration of forces - replacing force with its components.
Basic axioms of statics. 1. axiom. Under the influence of a balanced system of forces, the body is at rest or moves uniformly and in a straight line. 2. axiom. The principle of attaching and discarding a system of forces equivalent to zero. The action of a given system of forces on a body will not change if balanced forces are applied to or taken away from the body. 3rd axiom. The principle of equality of action and reaction. When bodies interact, every action corresponds to an equal and opposite reaction. 4th axiom. Theorem of three balanced forces. If three non-parallel forces lying in the same plane are balanced, then they must intersect at one point.
Connections and their reactions: Bodies whose movement is not limited in space are called free. Bodies whose movement is limited in space are called not free. Bodies that prevent the movement of non-free bodies are called connections. The forces with which the body acts on the connection are called active. They cause movement of the body and are designated F, G. The forces with which the connection acts on the body are called reactions of the connections or simply reactions and are designated R. To determine the reactions of the connection, the principle of release from bonds is used or the section method. The principle of liberation from ties lies in the fact that the body is mentally freed from connections, the actions of connections are replaced by reactions. Section method (ROZU method) is that the body is mentally is cut into parts, one part discarded, the action of the discarded part replaced forces, to determine 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. Corner 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 communication- in the form of a rope, cable, chain. The reaction is directed through communication. Cylindrical joint- this is the connection of two or more parts using an axis, a finger. The reaction is directed perpendicular to the hinge axis. Rigid rod with hinged ends reactions are directed along the rods: the reaction of a stretched rod is from a node, a compressed rod is to a node. When solving problems analytically, it can be difficult to determine the direction of reactions of the rods. In these cases, the rods are considered to be stretched and the reactions are directed away from the nodes. If, when solving problems, the reactions turn out to be negative, then in reality they are directed in the opposite direction and compression occurs. The reactions are directed along the rods: the reaction of a stretched rod is from a node, a compressed rod is to a 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 movable support prevents only vertical movement of the end of the beam, but not horizontal movement or rotation. Such a support gives one reaction under any load. Hard seal prevents vertical and horizontal movement of the beam end, as well as its rotation. Gives 3 reactions: vertical, horizontal forces and pair 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 student interest. The proposed form of conducting the lesson allows you to increase cognitive activity, since students independently receive information throughout the lesson and consolidate it in the process of solving problems. This forces them to actively work in class.
“Quiet” and “loud” discussion when working in micro groups gives positive results when assessing students’ knowledge. Elements of “brainstorming” activate students’ work in class. Solving a problem together allows less prepared students to understand the material being studied with the help of stronger friends. 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 students to develop logical and engineering thinking.
Assessing the work of each student in the lesson also stimulates his activity.
All of the above suggests that this form The lesson allows students to gain deep and lasting knowledge on the topic being studied and to actively participate in finding solutions to problems.
LIST OF RECOMMENDED LITERATURE
Arkusha A.I. Technical mechanics. Theoretical mechanics and resistance of rials.-M Higher School. 2009.
Arkusha A.I. Guide to solving problems in technical mechanics. Textbook for intermediate professionals textbook establishments, - 4th ed. corr. - M Higher school ,2009
Belyavsky SM. Guide to solving problems on the strength of materials M. Vyssh. school, 2011.
Guryeva O.V. Collection of multi-choice tasks on technical mechanics..
Guryeva O.V. Methodical manual. To help students of technical mechanics 2012
Kuklin N.G., Kuklina G.S. Machine parts. M. Mechanical Engineering, 2011
Movnin M.S., et al. Fundamentals of mechanical mechanics. L. Mechanical Engineering, 2009
Erdedi A.A., Erdedi N.A. Theoretical mechanics. Material resistance M High. school Academy 2008.
Erdedi A A, Erdedi NA Machine parts - M, Higher. school Academy, 2011
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 methodological manual is intended to assist students in self-studying 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 randomly located forces 9
4. The concept of a farm. Truss calculation 11
5. Spatial system of forces 11
II. Kinematics of a point and a rigid body 13
1. Basic concepts of kinematics 13
2. Translational and rotational motions 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 for the dynamics of a point 21
Strength of materials22
1. Basic concepts 22
2. External and internal forces. Section method 22
3. The concept of voltage 24
4. Tension and compression of straight timber 25
5. Shear and crushing 27
6. Torsion 28
7. Transverse bend 29
8. Longitudinal bending. The essence of the phenomenon longitudinal bending. Euler's formula. Critical voltage 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
References45
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 the resulting interactions between bodies is called theoretical mechanics.
Static is a branch of mechanics that sets out the general doctrine of forces and studies the conditions of equilibrium of material bodies under the influence of forces.
Absolutely solid body A body is called the distance between any two points of which always remains constant.
A quantity that is a quantitative measure of the mechanical interaction of material bodies is called by force.
Scalar quantities- these are those that are completely characterized by their numerical value.
Vector quantities – These are those that, in addition to their numerical value, are also characterized by 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 action of force.
The set of forces acting on any solid body is called system of forces.
A body that is not attached to other bodies, to which any movement in space can be imparted 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 the influence of which a free rigid body can be at rest is called balanced or equivalent to zero.
Resultant – this is the force that alone replaces the action of a given system of forces on a solid body.
A force equal to the resultant in magnitude, directly opposite to it in direction and acting along the same straight line is called balancing force.
External are the forces acting on the particles of a given body from other material bodies.
Internal are 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 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 magnitude and directed along the same 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 it or subtracted from it.
Corollary of 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 of the body.
Axiom 3 (parallelogram of forces axiom). Two forces applied to a body at one point have a resultant applied at the same point and represented 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 a parallelogram built on vectors F 1 and F 2, called geometric sum of vectors.
Axiom 4. With any action of one material body on another, there is a reaction of the same magnitude, but opposite in direction.
Axiom 5(hardening principle). The equilibrium of a changing (deformable) body under the influence of a given system of forces will not be disturbed if the body is considered hardened (absolutely solid).
A body that is not attached to other bodies and can make any movement in space from a given position is called free.
A body whose movements in space are prevented by some other bodies fastened or in contact with it is called unfree.
Everything that limits the movement of a given body in space is called communication.
The force with which a given connection acts on a body, preventing one or another of its movements, is called bond reaction force or communication reaction.
The communication reaction is directed in the direction opposite to the one where the connection prevents the body from moving.
Axiom of connections. Any unfree body can be considered as free if we discard the connections and replace their action with the reactions of these connections.
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 the geometric sum (principal vector) of these forces and applied at the point of their intersection.
Geometric sum, or main vector several forces, is depicted by the closing side of a force polygon constructed from these forces (Fig. 4b).
2.1. Projection of force on the axis and on the plane
Projection of force onto the axis is a scalar quantity equal to the length of the segment taken with the appropriate 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 the force and the cosine of the angle between the direction of the force and the positive direction of the axis:
F X = F cos.
Projection of force onto a plane is called the vector enclosed between the projections of the beginning and end of the force onto 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
Projection of the sum vector on any axis is equal to the algebraic sum of the projections of the summands of the vectors onto the same axis (Fig. 7).
R = F 1 + F 2 + F 3 + F 4
R x = ∑F ix R y = ∑F iy
To balance a system of converging forces It is necessary and sufficient that the force polygon constructed from these forces be closed - this is a geometric equilibrium condition.
Analytical equilibrium condition. For the system of converging forces to be in equilibrium, 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 solid 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 relative to the center (point)
Moment of force relative to the center is called a quantity equal to taken with the corresponding sign, the product of the force modulus and the length h(Fig. 9).
M = ± F· h
Perpendicular h, lowered from the center ABOUT to the line of action of the force F, called force arm F relative to the center ABOUT.
The 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 moment of force.
1. The moment of force will not change when the point of application of the force is moved along its line of action.
2. The moment of a 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 (the arm is zero).
A set of educational and visual aids in technical mechanics includes materials for the entire course of this discipline (110 topics). Didactic materials contain drawings, diagrams, definitions and tables on technical mechanics and are intended for demonstration by the teacher during lectures.
There are several options for designing a set of educational visual aids on technical mechanics: a presentation on disk, films for an overhead projector and posters for decorating classrooms.
Disc with electronic posters on technical mechanics (presentations, electronic textbooks)
The disk is intended for demonstration by teachers of didactic material in classes on technical mechanics - using interactive whiteboard, multimedia projector and other computer demonstration complexes. Unlike conventional electronic textbooks for self-study, these presentations on technical mechanics are designed specifically for displaying drawings, diagrams, tables in lectures. The convenient software shell has a table of contents that allows you to view the required poster. The posters are protected from unauthorized copying. A printed manual is included to help the teacher prepare for classes.
Visual aids on technical mechanics on films (slides, folios, code banners)
Code transparencies, slides, folios on technical mechanics are visual aids on transparent films intended for demonstration using an overhead projector (overhead projector). The folios included are placed in protective envelopes and collected in folders. A4 sheet format (210 x 297 mm). The set consists of 110 sheets, divided into sections. It is possible to selectively order sections or separate sheets from the kit.
Printed posters and tables on technical mechanics
To decorate classrooms, we produce tablets on a rigid basis and posters on technical mechanics of any size on paper or polymer based with fastening elements and round plastic profile along the top and bottom edges.
List of topics on technical mechanics
1. Statics
1. The concept of power
2. The concept of moment of force
3. The concept of a couple of forces
4. Calculation of the moment of force about the axis
5. Equilibrium equations
6. Axiom of liberation from connections
7. Axiom of liberation from connections (continued)
8. Axiom of solidification
9. Balance mechanical system
10. Axiom of action and reaction
11. Flat system of forces
12. Flat system of forces. External and internal forces. Example
13. Ritter method
14. Spatial system of forces. Example
15. Spatial system of forces. Continuation of the example
16. Converging system of forces
17. Distributed loads
18. Distributed loads. Example
19. Friction
20. Center of gravity
2. Kinematics
21. Frame of reference. Kinematics of a point
22. Point speed
23. Point acceleration
24. Translational motion of a rigid body
25. Rotational motion of a rigid body
26. Plane motion of a rigid body
27. Plane motion of a rigid body. Examples
28. Complex point movement
3. Dynamics
29. Dynamics of a point
30. D'Alembert's principle for a mechanical system
31. Inertia Forces of an Absolutely Rigid Body
32. D'Alembert's principle. Example 1
33. D'Alembert's principle. Example 2
34. D'Alembert's principle. Example 3
35. Theorems about kinetic energy. Power theorem
36. Theorems about kinetic energy. Theorem of works
37. Theorems about kinetic energy. Kinetic energy of a solid
38. Theorems about kinetic energy. Potential energy of a mechanical system in a gravity field
39. Momentum Theorem
4. Strength of materials
40. Models and methods
41. Stress and strain
42. Hooke's law. Poisson's ratio
43. Stress at a point
44. Maximum shear stress
45. Hypotheses (theories) of strength
46. Stretching and Compression
47. Tension - compression. Example
48. The concept of static indetermination
49. Tensile test
50. Strength under variable loads
51. Shift
52. Torsion
53. Torsion. Example
54. Geometric characteristics flat sections
55. Geometric characteristics of the simplest figures
56. Geometric characteristics of standard profiles
57. Bend
58. Bend. Example
59. Bend. Comments for example
60. Strength of materials. Bend. Determination of bending stresses
61. Strength of materials. Bend. Strength calculation
62. Zhuravsky formula
63. Oblique bend
64. Eccentric tension - compression
65. Eccentric stretching. Example
66. Stability of compressed rods
67. Calculation of normal stresses critical for stability
68. Stability of rods. Example
69. Calculation of twisted cylindrical springs
5. Machine parts
70. Rivet joints
71. Welded joints
72. Welded joints. Strength calculation
73. Carving
74. Types of threads and threaded connections
75. Force relationships in threads
76. Force relationships in fastening joints
77. Load in fastening threaded connections
78. Strength calculation of a fastening threaded connection
79. Calculation of a sealing threaded connection
80. Screw-nut transmission
81. Friction gears
82. Chain transmissions
83. Belt drives
84. Detachable fixed connections
85. Linking theorem
86. Gears
87. Involute gearing
88. Parameters of the original contour
89. Determination of the minimum number of teeth
90. Parameters of involute gearing
91. Design calculation of a closed gear drive
92. Basic Stamina Stats
93. Determination of gear parameters
94. Gear overlap ratios
95. Helical spur gear
96. Helical gearing. Geometry calculation
97. Helical gearing. Load calculation
98. Conical gear. Geometry
99. Bevel gear. Effort calculation
100. Worm gear. Geometry
101. Worm gear. Force Analysis
102. Planetary gears
103. Conditions for selecting planetary gear teeth
104. Willis method
105. Shafts and axles
106. Shafts. Stiffness calculation
107. Couplings. Clutch
108. Couplings. Overrunning clutch
109. Rolling bearings. Load Definition
110. Selection of rolling bearings
Topic No. 1. STATICS OF A SOLID BODY
Basic concepts and axioms of statics
Static subject.Static is called the branch of mechanics in which the laws of addition of forces and the conditions of 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 equilibrium is studied can be considered motionless, then the equilibrium is conventionally called absolute, and otherwise - relative. In statics we will study only the so-called absolute equilibrium of bodies. In practical engineering calculations, equilibrium can be considered absolute in relation to the Earth or to bodies rigidly connected to the Earth. 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 depend significantly on whether the body is solid, liquid or gaseous. The equilibrium of liquid and gaseous bodies is studied in hydrostatics and aerostatics courses. In a general mechanics course, only problems on the equilibrium of rigid bodies are usually considered.
All solid bodies found in nature, under the influence of external influences, change their shape (deform) to one degree or another. The magnitude of these deformations depends on the material of the bodies, their geometric shape and size, 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 existing loads are sufficiently small. As a result, when studying general conditions equilibrium, it is quite acceptable to neglect small deformations of the corresponding solids and consider them as non-deformable or absolutely solid.
Absolutely solid body A body is called the distance between any two points of which always remains constant.
In order for a solid body to be in equilibrium (at rest) under the influence of a certain system of forces, it is necessary that these forces satisfy certain equilibrium conditions of this system of forces. Finding these conditions is one of the main problems of statics. But to find the equilibrium conditions for 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 up the forces acting on a solid body, replace the action of one system of forces with another system and, in particular, reduce a given system of forces to its simplest form. Therefore, in rigid body statics the following two main problems are considered:
1) addition of forces and reduction of systems of forces acting on a solid body to their simplest form;
2) determination of equilibrium conditions for systems of forces acting on a solid body.
Strength. The state of equilibrium or movement of a given body depends on the nature of its mechanical interactions with other bodies, i.e. from the pressures, attractions or repulsions that a given body experiences as a result of these interactions. A quantity that is a quantitative measure of mechanical interactionaction of material bodies is called force in mechanics.
The quantities considered in mechanics can be divided into scalar ones, i.e. those that are completely characterized by their numerical value, and vector ones, i.e. those that, in addition to their numerical value, are also characterized by direction in space.
Force is a vector quantity. Its effect on the body is determined by: 1) numerical value or module strength, 2) directionniya strength, 3) point of application 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 downward. The pressure forces of two smooth balls pressed against each other are directed normal to the surfaces of the balls at the points of their contact and are applied at these points, etc.
Graphically, force is represented by a directed segment (with an arrow). The length of this segment (AB in Fig. 1) expresses the force modulus on the selected scale, the direction of the segment corresponds to the direction of the force, its beginning (point A in Fig. 1) usually coincides with the point of application of 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 V). Straight DE, along which the force is directed is called line of action of the force. Strength is represented by the letter F . The force module is indicated by vertical bars “on the sides” of the vector. System of forces is called a set of forces acting on some absolutely rigid body.
Basic definitions:
A body that is not attached to other bodies, to which any movement in space can be imparted from a given position, is called free.
If a free rigid body under the influence 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 of this system of forces. Thus, resultant - this is the power that alone can replacethe action of a given system of forces on a rigid body.
A force equal to the resultant in magnitude, directly opposite to it in direction and acting along the same straight line is called balancing by force.
Forces acting on a solid body can be divided into external and internal. External are the forces acting on the particles of a given body from other material bodies. Internal are 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 infightingdivided.
The concept of concentrated force is conditional, since it is practically impossible to apply force to a body at one point. The forces that we consider in mechanics as concentrated are essentially the resultants of certain systems of distributed forces.
In particular, the force of gravity acting on a given solid body, usually considered in mechanics, is the resultant of the gravitational forces of its particles. The line of action of this resultant passes through a point called the center of gravity of the body.
Axiom 1. If absolutely freea rigid body is subject to two forces, then the body cancan be in equilibrium if and onlywhen these forces are equal in magnitude (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.
A 
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 it or subtracted from it.
This axiom states that two systems of forces that differ by a balanced system are equivalent to each other.
Corollary of 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.
In fact, 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 = F, F2 = - F. This will not change the action of force F on the body. But the forces F and F2, according to axiom 1, also form a balanced system that can be rejected. As a result, only one force Fl will act on the body, equal to F, but applied at point B.
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 sliding).
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 equilibrium conditions of the structure are determined.
N
A
Therefore, axiom 3 can also be formulate this way: resultant two forces applied to a body at one point is equal to geomet ric (vector) sum of these forces and applied in the same point.
Axiom 4. Two material bodies always act togetheron each other with forces equal in magnitude and directed alongone straight line in opposite directions(briefly: action equals reaction).
Z
Property of internal forces. According to axiom 4, any two particles of a solid body will act on each other with forces equal in magnitude and oppositely directed. Since, when studying the general conditions of equilibrium, the body can be considered as absolutely solid, then (according to axiom 1) all internal forces under this condition form a balanced system, which (according to axiom 2) can be discarded. Consequently, when studying the general conditions of equilibrium, it is necessary to take into account only the external forces acting on a given solid body or a given structure.
Axiom 5 (solidification principle). If any changea flexible (deformable) body under the influence of a given system of forcesis in equilibrium, then equilibrium will remain even whenthe body will harden (become absolutely solid).
The statement expressed in this axiom is obvious. For example, it is clear that the balance of a chain should not be disturbed if its links are welded together; the balance of a flexible thread will not be disturbed if it turns into a curved rigid rod, etc. 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: in equilibrium, the forces acting on any variable (deformationrealizable) body, satisfy the same conditions as forabsolutely solid body; however, for a changeable 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. For the thread to be balanced, it is also required that the applied forces be tensile, i.e. directed as in Fig. 4a.
The principle of solidification is widely used in engineering calculations. When drawing up equilibrium conditions, it allows us to consider any variable body (belt, cable, chain, etc.) or any variable structure as absolutely rigid and apply rigid body statics methods to them. If the equations obtained in this way are not enough to solve the problem, then additional equations are drawn up that take into account either the equilibrium conditions of individual parts of the structure or their deformation.
Topic No. 2. DYNAMICS OF A POINT
