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 implementing a set of educational visual aids on technical mechanics: 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 the teacher of didactic material in classes on technical mechanics - using an interactive whiteboard, multimedia projector and other computer demonstration systems. Unlike conventional electronic textbooks for self-study, these presentations on technical mechanics are designed specifically to display drawings, diagrams, tables at 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. Selective ordering of sections or individual sheets from the set is possible.
Printed posters and tables on technical mechanics
To decorate classrooms, we produce tablets on a rigid base and posters on technical mechanics of any size on paper or a polymer base with fastening elements and a 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. Equilibrium of a mechanical system
10. Axiom of action and reaction
11. Flat force system
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 of 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 drives
83. Belt drives
84. Detachable fixed connections
85. Link 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. Bevel 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
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.
The 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 in technical mechanics
Technological map of the lesson
Hypertext
CONCLUSION
BIBLIOGRAPHY
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 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.
In this case, independent work skills become of great importance. 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. Participatory teaching methods are used: silent and loud discussion, brainstorming, case study, question and answer.
The topic “Basic concepts and axioms of statics” is one of the most important in the “Technical Mechanics” course. It is of great importance from the point of view of studying the course. 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 completion of a task, and provides skills and abilities to solve these problems. The teacher, by asking problematic questions, 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 of an educational institution is to teach how 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, you need to pay special attention to its formulation. 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 a 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, every 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.
Routing
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 - consolidation of group work skills, the ability to listen to the opinions of comrades, 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 |
|||
| 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. We write down the final version of the 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 | Control test | 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 | I analyze this test according to the level of mastery of the material. | ||
| 2. Homework | Compose (or reproduce) a reference summary of hypertext Please note that the assignment for a higher grade is 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. Organizational moment
1.1 Meet the group
1.2 Mark students present
1.3 Familiarity 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 is the greatest mathematician and mechanic of 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 aerodynamics issues that are of great importance for modern aviation speed.
Technical mechanics is a complex discipline that sets out the basic principles about 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, theoretical mechanics, is divided into three subsections: statics, kinematics, dynamics.
Today we will begin the study of technical mechanics with the subsection of statics - this is a section of theoretical mechanics in which the conditions of equilibrium of an absolutely rigid body under the influence of forces applied to them are studied. 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. Force- 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 the same effect 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. Force system - 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 hinge- 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. Articulating 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 a 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 of 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. Toolkit. 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 Highest. school Academy 2008.
Erdedi A A, Erdedi NA Machine parts - M, Higher. school Academy, 2011
SHORT COURSE OF LECTURES ON THE DISCIPLINE "FUNDAMENTALS OF TECHNICAL MECHANICS"
Section 1: Statics
Statics, axioms of statics. Connections, reaction of connections, types of connections.
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 the entire mass of the body is concentrated at this point.
A system is a collection of material points whose movement and position 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, 1Н = 0.1 Kg/s
Axioms of statics.
1Axiom– (Defines a balanced system of forces): a system of forces applied to a material point is balanced if, under its influence, the point is in a state of relative rest, or moves rectilinearly 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 rotates uniformly around a fixed axis.
2 Axiom– (Sets the condition of equilibrium of two forces): two forces equal in magnitude or numerical value (F1=F2) applied to an absolutely rigid body and directed
along one straight line in opposite directions are mutually balanced.
A system of forces is a combination of several forces applied to a point or body.
A system of forces of lines of action in which they are in different planes is called spatial; if they are in the same plane, then they are 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.
Corollary to axiom 2.
Any force acting on a body can be transferred along the line of its action to any point of the body without disturbing its mechanical state.
3
Axiom: (Basis for transformation of forces): without disturbing the mechanical state of an absolutely rigid body, a balanced system of forces can be applied to it or rejected from it. 
Vectors that can be transferred along the line of their action are called sliding.
4 Axiom– (Defines the rules for the addition of 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– (It establishes that in nature there cannot be a unilateral action of force) when bodies interact, every action corresponds to an equal and oppositely directed reaction.
Connections and their reactions.
Bodies in mechanics are: 1 free 2 non-free.
Free - when the body does not experience any obstacles to moving in space in any direction.
Unfree - the body is connected to other bodies that limit its movement.
Bodies that limit the movement of a body are called connections.
When a body interacts with connections, forces arise; they act on the body from the side of the connection and are called connection reactions.
The reaction of the connection is always opposite to the direction in which the connection prevents the movement of the body.
Types of communication.
1) Connection in the form of a smooth plane without friction.
2) Communication in the form of contact of a cylindrical or spherical surface.
3) Connection in the form of a rough plane.
Rn – force perpendicular to the plane. Rt – friction force.
R – bond reaction. R = Rn+Rt
4) Flexible connection: rope or cable.
5) Connection in the form of a rigid straight rod with hinged ends.
6) The connection is carried out by the edge of a dihedral angle or a point support.
R1R2R3 – Perpendicular to the surface of the body.
Plane system of converging forces. Geometric definition of the resultant. Projection of force onto the axis. Projection of a vector sum onto an axis.
Forces are called convergent if their lines of action intersect at one point.
A plane system of forces - the lines of action of all these forces lie in the same plane.
A 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 of their intersection 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 round 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.
Unbalanced
balanced.
Projection of force onto the axis.
An axis is a straight line to which a certain direction is assigned.
The projection of a vector is a scalar quantity; it is determined by the axis segment 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: Projection of force onto the coordinate axis = the product of the magnitude of the force and the cos of the angle between the force vector and the positive direction of the axis.
Positive projection.
Negative projection
Projection = o
Projection of a vector sum onto an axis.
Can be used to define a module and
direction of force, if its projections on
coordinate axes.
Conclusion: The projection of the vector sum, or resultant, onto each axis is equal to the algebraic sum of the projection of the summands of the vectors onto the same axis.
Determine the magnitude and direction of the force if its projections are known.
Answer: F=50H,
Fy-?F 
-?
Answer: 
Section 2. Strength of materials (Sopromat).
Basic concepts and hypotheses. Deformation. Section method.
Strength of materials is the science of engineering methods of calculation for the strength, rigidity and stability of structural elements. Strength - the properties of bodies not to collapse under the influence of external forces. Rigidity is the ability of bodies to change dimensions within specified limits during deformation. Stability is the ability of bodies to maintain their original state of equilibrium after applying a load. The goal of science (Sopromat) is to create practically convenient methods for calculating the most common structural elements. Basic hypotheses and assumptions regarding the properties of materials, loads and the 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 the material). The ability of a body to restore a pile to its original shape and size after eliminating the causes that caused the deformation. 3) Hypothesis(Assumption of linear relationship between deformations and loads, Execution of Hooke's law). Displacement resulting from deformation is directly proportional to the loads that caused them. 4) Hypothesis(Plane sections). The cross sections are flat and normal to the axis of the beam before a load is applied to it, and remain flat and normal to its axis after deformation. 5) Hypothesis(On the isotropy of the material). The mechanical properties of the material are the same in any direction. 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 the relative position of the loads. 7) Hypothesis (Principle of independence of the action of forces). 8) Hypothesis (Saint-Venant). The deformation of a body far from the place of application of statically equivalent loads practically does not depend on 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. External forces applied to the cut-off part of the body must be balanced with internal forces arising in the section plane; they replace the action of the discarded part on the rest. Rod (beams) – Structural elements whose length 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 about the same. 
Equilibrium condition.
NZ – Longitudinal internal force. QX and QY – Transverse internal force. MX and MY – Bending moments. MZ – Torque. When a plane system of forces acts on a rod, only three force factors can arise in its sections, these are: MX - Bending moment, QY - Transverse force, NZ - Longitudinal force. Equilibrium equation. Coordinate axes will always direct the Z axis along the axis of the rod. 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.
Sequence of actions to determine internal forces.
1) Mentally draw a section at the point of the structure that interests us. 2) Discard one of the cut off parts and consider the equilibrium of the remaining part. 3) Draw up an equilibrium equation and determine from them the values and directions of internal force factors. Axial tension and compression are internal forces in the cross section. They can be closed by one force directed along the axis of the rod.
Stretching. 
Compression. Shear - occurs when in the cross section of the rod internal forces are reduced to one, i.e. shear force Q. 
Torsion – 1 force factor MZ occurs. 
MZ=MK Pure bending – Bending moment MX or MY occurs. 
To calculate structural elements for strength, rigidity, and stability, first of all, it is necessary (using the section method) to determine the occurrence of internal force factors. 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 equilibrium conditions, it is quite acceptable to neglect small deformations of the corresponding solid bodies 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.
Force. 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 a given system of forces 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, usually considered in mechanics, acting on a given solid body 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 freesolid body is acted upon by 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.
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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 along 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.
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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).
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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
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 of arbitrarily 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 of 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
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 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 forces are called 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 onto the axis and onto 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).