Lectures on technical mechanics 2 course. Topics for self-study in theoretical mechanics with lighting examples
The manual contains the basic concepts and terms of one of the main disciplines of the subject block "Technical Mechanics". This discipline includes such sections as "Theoretical Mechanics", "Strength of Materials", "Theory of Mechanisms and Machines".
The manual is intended to assist students in self-study of the course "Technical Mechanics".
Theoretical Mechanics 4
I. Statics 4
1. Basic concepts and axioms of statics 4
2. System of converging forces 6
3. Flat system of arbitrarily distributed forces 9
4. The concept of a farm. Truss calculation 11
5. Spatial system of forces 11
II. Kinematics of point and solid body 13
1. Basic concepts of kinematics 13
2. Translational and rotational motion of a rigid body 15
3. Plane-parallel motion of a rigid body 16
III. Dynamics of point 21
1. Basic concepts and definitions. Laws of Dynamics 21
2. General theorems of point dynamics 21
Strength of materials22
1. Basic concepts 22
2. External and internal forces. Section method 22
3. The concept of stress 24
4. Tension and compression of a straight beam 25
5. Shift and Collapse 27
6. Torsion 28
7. Cross bend 29
8. Longitudinal bend. The essence of the phenomenon of longitudinal bending. Euler formula. Critical stress 32
Theory of mechanisms and machines 34
1. Structural analysis of mechanisms 34
2. Classification of flat mechanisms 36
3. Kinematic study of flat mechanisms 37
4. Cam mechanisms 38
5. Gear mechanisms 40
6. Dynamics of mechanisms and machines 43
Bibliography45
THEORETICAL MECHANICS
I. Statics
1. Basic concepts and axioms of statics
The science of the general laws of motion and equilibrium of material bodies and of the interactions between bodies arising from this is called theoretical mechanics.
static called the branch of mechanics, which sets out the general doctrine of forces and studies the conditions for the equilibrium of material bodies under the influence of forces.
Absolutely solid body such a body is called, the distance between any two points of which always remains constant.
The quantity, which is a quantitative measure of the mechanical interaction of material bodies, is called force.
Scalars are those that are fully characterized by their numerical value.
Vector quantities - these are those that, in addition to a numerical value, are also characterized by a direction in space.
Force is a vector quantity(Fig. 1).
Strength is characterized by:
- direction;
– numerical value or module;
- point of application.
Straight DE along which the force is directed is called line of force.
The totality of forces acting on a rigid body is called system of forces.
A body that is not bonded to other bodies, which this provision can report any movement in space, called free.
If one system of forces acting on a free rigid body can be replaced by another system without changing the state of rest or motion in which the body is located, then such two systems of forces are called equivalent.
The system of forces under which a free rigid body can be at rest is called balanced or equivalent to zero.
The resultant - it is a force that alone replaces the action of a given system of forces on a rigid body.
A force equal to the resultant in absolute value, directly opposite to it in direction and acting along the same straight line, is called balancing force.
External called the forces acting on the particles of a given body from other material bodies.
internal called the forces with which the particles of a given body act on each other.
A force applied to a body at any one point is called 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 absolute value and directed along one straight line in opposite directions (Fig. 2).
Axiom 2. The action of one system of forces on an absolutely rigid body will not change if a balanced system of forces is added to or subtracted from it.
Consequence from the 1st and 2nd axioms. The action of a force on an absolutely rigid body will not change if the point of application of the force is moved along its line of action to any other point on the body.
Axiom 3 (axiom of the parallelogram of forces). Two forces applied to the body at one point have a resultant applied at the same point and depicted by the diagonal of a parallelogram built on these forces as on the sides (Fig. 3).
R = F 1 + F 2
Vector R, equal to the diagonal of the parallelogram built on the vectors F 1 and F 2 is called geometric sum of vectors.
Axiom 4. With every action of one material body on another, there is a reaction of the same magnitude, but opposite in direction.
Axiom 5(hardening principle). The balance of a changeable (deformable) body under the action of a given system of forces will not be disturbed if the body is considered to be solidified (absolutely rigid).
A body that is not fastened to other bodies and can perform any movement in space from a given position is called free.
A body whose movement in space is prevented by some other bodies fastened or in contact with it is called not free.
Everything that limits the movement of a given body in space is called communication.
The force with which this connection acts on the body, preventing one or another of its movements, is called bond reaction force or bond reaction.
Communication reaction directed in the direction opposite to that where the connection does not allow the body to move.
Axiom of connections. Any non-free body can be considered as free, if we discard the bonds and replace their action with the reactions of these bonds.
2. System of converging forces
converging are called forces whose lines of action intersect at one point (Fig. 4a).
The system of converging forces has resultant equal to geometric sum(main vector) of these forces and applied at the point of their intersection.
geometric sum, or main vector several forces is represented by the closing side of the force polygon constructed from these forces (Fig. 4b).
2.1. Projection of force on the axis and on the plane
The projection of the force on the axis is called a scalar quantity equal to the length of the segment, taken with the corresponding sign, enclosed between the projections of the beginning and end of the force. The projection has a plus sign if the movement from its beginning to the end occurs in the positive direction of the axis, and a minus sign if in the negative direction (Fig. 5).
Projection of Force on the Axis is equal to the product of the modulus of force and the cosine of the angle between the direction of the force and the positive direction of the axis:
F X = F cos.
The projection of force on a plane called the vector enclosed between the projections of the beginning and end of the force on this plane (Fig. 6).
F xy = F cos Q
F x = F xy cos= F cos Q cos
F y = F xy cos= F cos Q cos
Sum Vector Projection on any axis is equal to the algebraic sum of the projections of the terms of the vectors on the same axis (Fig. 7).
R = F 1 + F 2 + F 3 + F 4
R x = ∑F ix R y = ∑F iy
To balance the system of converging forces it is necessary and sufficient that the force polygon constructed from these forces be closed - this is the geometric condition of equilibrium.
Analytical equilibrium condition. For the equilibrium of the system of converging forces, it is necessary and sufficient that the sum of the projections of these forces on each of the two coordinate axes be equal to zero.
∑F ix = 0 ∑F iy = 0 R =
2.2. Three forces theorem
If a free rigid body is in equilibrium under the action of three non-parallel forces lying in the same plane, then the lines of action of these forces intersect at one point (Fig. 8).
2.3. Moment of force about the center (point)
Moment of force about the center is called a value equal to taken with the corresponding sign to the product of the modulus of force and the length h(Fig. 9).
M = ± F· h
Perpendicular h, lowered from the center ABOUT to the line of force F, is called shoulder of force F relative to the center ABOUT.
Moment has a plus sign, if the force tends to rotate the body around the center ABOUT counterclockwise, and minus sign- if clockwise.
Properties of the moment of force.
1. The moment of force will not change when the point of application of force is moved along its line of action.
2. The moment of force about the center is zero only when the force is zero or when the line of action of the force passes through the center (shoulder is zero).
Introduction
Theoretical mechanics is one of the most important fundamental general scientific disciplines. It plays an essential role in the training of engineers of all specialties. 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 motion of material bodies under the action of forces. An important particular problem is the study of the equilibrium of bodies under the action of forces.
Lecture course. Theoretical mechanics
The structure of theoretical mechanics. Fundamentals of statics
Conditions for the equilibrium of an arbitrary system of forces.
Rigid Body Equilibrium Equations.
Flat system of forces.
Particular cases of equilibrium of a rigid body.
The problem of equilibrium of a bar.
Determination of internal forces in bar structures.
Fundamentals of point kinematics.
natural coordinates.
Euler formula.
Distribution of accelerations of points of a rigid body.
Translational and rotational movements.
Plane-parallel movement.
Complicated point movement.
Fundamentals of point dynamics.
Differential equations of motion of a point.
Particular types of force fields.
Fundamentals of the dynamics of the system of points.
General theorems of the dynamics of a system of points.
Dynamics of rotational movement of the body.
Dobronravov V.V., Nikitin N.N. Course of theoretical mechanics. M., high 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 term papers in theoretical mechanics. Ed. A.A. Yablonsky. M., Higher School, 1985.
Lecture 1 The structure of theoretical mechanics. Fundamentals 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 dimensions;
absolutely rigid body - a volume of finite dimensions, completely filled with matter, 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 the points of such a medium can vary.
Of these, systems:
System of free material points;
Systems with links;
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 proceed differently in different places physical reference system. This property is the inhomogeneity of space, determined by the physical reference system. Heterogeneity here 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 motion of a body relative to the physical reference system can be different depending on the direction. Examples: the course of the river along the meridian (from north to south - the Volga); projectile flight, Foucault pendulum.
The properties of the reference system (heterogeneity and anisotropy) make it difficult to observe the motion of a body.
Practically free from this geocentric system: the center of the system is at 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 the Earth.
For celestial mechanics(for solar system bodies): a heliocentric reference frame that moves with the center of mass solar system and does not rotate relative to "fixed" stars. For this system not found yet heterogeneity and anisotropy of space
in relation to the phenomena of mechanics.
So, we introduce an abstract inertial reference frame for which space is homogeneous and isotropic in relation to the phenomena of mechanics.
inertial frame of reference- one whose own movement cannot be detected by any mechanical experience. Thought experiment: "the point that is alone in the whole world" (isolated) is either at rest or moving in a straight line and uniformly.
All frames of reference moving relative to the original rectilinearly will be uniformly inertial. This allows you to introduce a single Cartesian coordinate system. Such a space is called Euclidean.
Conditional agreement - take the right coordinate system (Fig. 1).
IN 
time– in classical (non-relativistic) mechanics absolutely, which is 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 that 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 reference frame - section statics.
Chapter kinematics: a part of mechanics that studies the relationships between quantities that characterize the state of motion of systems, but does not consider the causes that cause a change in the state of motion.
After that, consider the influence of forces [MAIN PART].
Chapter dynamics: part of mechanics, which considers the influence of forces on the state of motion of systems of material objects.
Principles of building the main course - dynamics:
1) based on a system of axioms (based on experience, observations);
Constantly - ruthless control of practice. Sign of exact science - the presence of internal logic (without it - set of unrelated recipes)!
static that part of mechanics is called, where the conditions that must be satisfied by the forces acting on a system of material points are studied in order for the system to be in equilibrium, and the conditions for the equivalence of systems of forces.
Problems of equilibrium in elementary statics will be considered using exclusively geometric methods based on the properties of vectors. This approach is applied in geometric statics(as opposed to analytic statics, which is not considered here).
The positions of various material bodies will be referred to the coordinate system, which we will take as fixed.
Ideal models of material bodies:
1) material point - a geometric point with mass.
2) absolutely rigid body - a set of material points, the distances between which cannot be changed by any actions.
By the forces we will call objective reasons, which 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 the force is determined by the motion it causes, it also has a relative character, depending on the choice of the frame of reference.
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 are vector in nature, that is, magnitude, direction, line of action, point of application. The condition for the equilibrium of forces acting on a rigid body is reduced to the properties of systems of vectors.
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 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 addition of vectors (Fig. 2).
Consequence. The forces applied to a point of a rigid body are added according to the parallelogram rule.
Axiom 2. Two forces applied to a rigid body mutually balanced if and only if they are equal in magnitude, 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 drop 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 balance (that is, the force is a sliding vector, Fig. 3)
1) Active - create or are able to create the movement of a rigid body. For example, the force of weight.
2) Passive - not creating movement, but limiting the movement of a rigid body, preventing movement. For example, the tension force of an inextensible thread (Fig. 4).
Axiom 4. The action of one body on the second is equal and opposite to the action of this second body on the first ( action equals reaction).
The geometric conditions that restrict the movement of points will be called connections.
Communication conditions: for example,
- rod of indirect length l.
- flexible inextensible thread of length l.
Forces due to bonds and preventing movement are called reaction forces.
Axiom 5. The bonds imposed on the system of material points can be replaced by reaction forces, the action of which is equivalent to the action of the bonds.
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 such a system of forces is called, the lines of action of which intersect at one point, which can always be taken as the origin (Fig. 5).
Projections of the resultant:
;
;
.
If , then the force causes the motion of a rigid body.
Equilibrium condition for a convergent system of forces:
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||
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2. Balance of three forces
If three forces act on a rigid body, and the lines of action of 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 oppositely directed to the sum 
(Fig. 6).
Examples:
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||
Moment of force relative to point O define as a vector , in size equal to twice the area of a triangle, the base of which is a force vector with a vertex at a given point O; direction- orthogonal to the plane of the considered triangle in the direction from where the rotation produced by the force around the 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 modulus of force, h is the shoulder (distance from the point to the direction of the force).
Moment of force about the axis is called 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, we expand :|| and in the plane.
About moments: let О 1 be the point of intersection with the plane. Then:
a) from - moment => projection = 0.
b) from - moment along => is a projection.
So, the moment about the axis is the moment of the force component in the plane perpendicular to the axis about 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 components of 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 the total moment.
Test 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, one 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 name 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 pair 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 which the rotation reported by the vectors of the pair can be seen to occur counterclockwise.
, that is, the moment of force about point B.
A pair of forces is fully 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 inversely proportional to the shoulders of the pair.
Pairs of forces can be added, while the moments of pairs of forces are added according to the rule of addition of (free) vectors.
Bringing the system of forces acting on a rigid body to an arbitrary point (reduction center)- means replacing the current system with a simpler one: a system of three forces, one of which passes through in advance given point, and the other two represent a pair.
It is proved with the help of elementary operations (fig.13).
The system of converging forces and the system of pairs of forces.
- resulting force.
The resulting pair
Which is what needed to be shown.
Two systems of forces will are equivalent if and only if both systems are reduced to one resultant force and one resultant pair, that is, under the following conditions:
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General case of equilibrium of a system of forces acting on a rigid body
We bring the system of forces to (Fig. 14):
Resulting force through the origin;
The resulting pair, moreover, through the point O.
That is, they led to and - two forces, one of which passes through a given point O.
Equilibrium, if the other one straight line, are equal, directed oppositely (axiom 2).
Then passes through the point O, that is.
so, the general equilibrium conditions for a rigid body:
These conditions are valid for an arbitrary point in space.
Test questions:
1. List 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 Rigid Body Equilibrium Equations
Let O be the origin of coordinates; is the resulting force; is the moment of the resulting pair. Let the point O1 be a new reduction center (Fig. 15).
New force system:
When the cast point changes, => changes only (in one direction with one sign, in the other with another). That is the point: 
match the lines
Analytically: 
(colinearity of vectors)
; point O1 coordinates.
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 on the axis of the dynamas => , then the system is equivalent to one resultant force, which is called the resultant force of the system. In this case, always, that is.
Four cases of bringing forces:
1.) ;- dynamo.
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 art of the calculator.
Finding the equilibrium conditions for a system of rigid bodies in interaction<=>the problem of the balance of each body separately, and the body is affected by external forces and internal forces (the interaction of bodies at points of contact with equal and oppositely directed forces - axiom IV, Fig. 17).
We choose for all bodies of the system one referral center. Then for each body with the equilibrium condition number:
,
, (= 1, 2, …, k)
where , - the resulting force and the moment of the resulting pair of all forces, except for internal reactions.
The resulting force and moment of the resulting pair of forces of internal reactions.
Formally summing up and taking into account the IV axiom
we get necessary conditions for the equilibrium of a rigid body:
,
Example.
Equilibrium: = ?
Test questions:
1. Name all cases of bringing the system of forces to one point.
2. What is a dynamo?
3. Formulate the necessary conditions for the equilibrium of a system of rigid bodies.
Lecture 4 Flat system of forces
A special case of the general task delivery.
Let all the acting forces lie in the same plane - for example, a sheet. Let us choose the point O as the center of reduction - in the same plane. We get the resulting force and the resulting pair in the same plane, that is (Fig. 19)
Comment.
The system can be reduced to one resultant force.
Equilibrium conditions:
or scalars:
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 is called non-free, the movement of which is constrained by constraints. For example, other bodies, hinged fastenings.
When determining the conditions of equilibrium: a non-free body can be considered as free, replacing the bonds with unknown reaction forces.
Example.
Test questions:
1. What is called a flat system of forces?
2. Write the equilibrium conditions for a flat system of forces.
3. What kind of 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.
We choose a point on the line of action of the third force as the point of reduction. Then (fig.22)
That is, the planes S1 and S2 coincide, and for any point on the axis of force, etc. (Easier: in the plane 
just for balance).
BRIEF COURSE OF LECTURES ON THE DISCIPLINE "FUNDAMENTALS OF TECHNICAL MECHANICS"
Section 1: Statics
Statics, axioms of statics. Bonds, reaction of bonds, types of bonds.
The fundamentals of theoretical mechanics consist of three sections: Statics, fundamentals of strength of materials, details of mechanisms and machines.
Mechanical movement is a change in the position of bodies or points in space over time.
The body is considered as a material point, i.e. geometric point and at this point the entire mass of the body is concentrated.
The system is a set of material points, the movement and position of which are interconnected.
Force is a vector quantity, and the effect of force on a body is determined by three factors: 1) Numerical value, 2) direction, 3) point of application.
[F] - Newton - [H], Kg / s = 9.81 N = 10 N, KN = 1000 N,
MN = 1000000 N, 1N = 0.1 Kg/s
Axioms of statics.
1Axiom– (Defines a balanced system of forces): the system of forces applied to material point, is balanced if, under its influence, the point is in a state of relative rest, or moves in a straight line and uniformly.
If a balanced system of forces acts on a body, then the body is either: in a state of relative rest, or moves uniformly and rectilinearly, or uniformly rotates around a fixed axis.
2 Axiom– (Sets the condition for the balance of two forces): two forces equal in absolute value or numerical value (F1=F2) applied to an absolutely rigid body and directed
in a straight line in opposite directions are mutually balanced.
A system of forces is a combination of several forces applied to a point or body.
The system of forces of the line of action, in which they are in different planes, is called spatial, if in the same plane, then flat. A system of forces with lines of action intersecting at one point is called convergent. If two systems of forces taken separately have the same effect on the body, then they are equivalent.
Consequence of 2 axioms.
Any force acting on a body can be transferred along the line of its action, to any point of the body without violating its mechanical state.
3
Axiom: (The basis for the transformation of forces): without violating 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 moved along their line of action are called moving vectors.
4 Axiom– (Defines the rules for adding two forces): the resultant of two forces applied to one point, applied at this point, is the diagonal of a parallelogram built on these forces.
- Resultant force =F1+F2 - According to the parallelogram rule
According to the triangle rule.
5 Axiom- (Establishes that in nature there cannot be a one-sided action of force) in the interaction of bodies, every action corresponds to an equal and oppositely directed counteraction.
Connections and their reactions.
Bodies in mechanics are: 1 free 2 non-free.
Free - when the body does not experience any obstacles to move in space in any direction.
Non-free - the body is connected with other bodies that restrict its movement.
Bodies that restrict the movement of a body are called bonds.
When a body interacts with bonds, forces arise, they act on the body from the side of the bond and are called bond reactions.
The reaction of the bond is always opposite to the direction in which the bond impedes the movement of the body.
Communication types.
1) Communication in the form of a smooth plane without friction.
2) Communication in the form of a contact of a cylindrical or spherical surface.
3) Communication in the form of a rough plane.
Rn is the force perpendicular to the plane. Rt is the friction force.
R is the bond reaction. R = Rn+Rt
4) Flexible connection: rope or cable.
5) Connection in the form of a rigid straight rod with hinged fastening of the ends.
6) The connection is carried out by an edge of a dihedral angle or a point support.
R1R2R3 - Perpendicular to the surface of the body.
Flat system of converging forces. Geometric definition resultant. The projection of the force on the axis. Projection of the vector sum onto the axis.
Forces are called convergent if their lines of action intersect at one point.
Flat system of forces - the lines of action of all these forces lie in the same plane.
The spatial system of converging forces - the lines of action of all these forces lie in different planes.
Converging forces can always be transferred to one point, i.e. at the point where they intersect along the line of action.
F123=F1+F2+F3= 
The resultant is always directed from the beginning of the first term to the end of the last (the arrow is directed towards the bypass of the polyhedron).
If, when constructing a force polygon, the end of the last force coincides with the beginning of the first, then the resultant = 0, the system is in equilibrium.
not balanced
balanced.
The projection of the force on the axis.
An axis is a straight line to which a certain direction is assigned.
The vector projection is scalar value, it is determined by the segment of the axis cut off by perpendiculars to the axis from the beginning and end of the vector.
The projection of the vector is positive if it coincides with the direction of the axis, and negative if it is opposite to the direction of the axis.
Conclusion: The projection of the force on the coordinate axis = the product of the modulus of force and cos of the angle between the force vector and the positive direction of the axis.
positive projection.
Negative projection
Projection = o
Projection of the vector sum onto the axis.
Can be used to define a module and
the direction of the force, if its projections on
coordinate axes.
Output: The projection of the vector sum, or resultant, on each axis is equal to the algebraic sum of the projection of the terms of the vectors on the same axis.
Determine the modulus and direction of the force if its projections are known.
Answer: F=50H,
Fy-?F 
-?
Answer: 
Section 2. Strength of materials (Sopromat).
Basic concepts and hypotheses. Deformation. section method.
The strength of materials is the science of engineering methods for calculating the strength, rigidity and stability of structural elements. Strength - the properties of bodies not to collapse under the influence of external forces. Rigidity - the ability of bodies in the process of deformation to change dimensions within specified limits. Stability - the ability of bodies to maintain their original state of equilibrium after the application of a load. The purpose of science (Sopromat) is the creation of practically convenient methods for calculating the most common structural elements. Basic hypotheses and assumptions regarding the properties of materials, loads and nature of deformation.1) Hypothesis(Homogeneity and oversights). When the material completely fills the body, and the properties of the material do not depend on the size of the body. 2) Hypothesis(On the ideal elasticity of a material). The ability of the body to restore the pile to its original shape and dimensions after the elimination of the causes that caused the deformation. 3) Hypothesis(Assumption of a linear relationship between deformations and loads, Fulfillment of Hooke's law). Displacement as a result of deformation is directly proportional to the loads that caused them. 4) Hypothesis(Flat sections). The cross-sections are flat and normal to the beam axis before the load is applied to it and remain flat and normal to its axis after deformation. 5) Hypothesis(On the isotropy of the material). Mechanical properties material in any direction are the same. 6) Hypothesis(On the smallness of deformations). The deformations of the body are so small compared to the dimensions that they do not have a significant effect on mutual arrangement loads. 7) Hypothesis (Principle of independence of action of forces). 8) Hypothesis (Saint-Venant). The deformation of the body far from the place of application of statically equivalent loads is practically independent of the nature of their distribution. Under the influence of external forces, the distance between the molecules changes, internal forces arise inside the body, which counteract deformation and tend to return the particles to their previous state - elastic forces. 
Section method. The external forces applied to the cut off part of the body must be balanced with the internal forces arising in the section plane, they replace the action of the discarded part with the rest. Rod (beams) - Structural elements, the length of which significantly exceeds their transverse dimensions. Plates or shells - When the thickness is small compared to the other two dimensions. Massive bodies - all three sizes are approximately the same. 
Equilibrium condition.
NZ - Longitudinal internal force. QX and QY - Transverse internal force. MX and MY - Bending moments. MZ - Torque. When a planar system of forces acts on a rod, only three force factors can occur in its sections, these are: MX - Bending moment, QY - Transverse force, NZ - Longitudinal force. Equilibrium equation. The coordinate axes will always direct the Z-axis along the bar axis. The X and Y axes are along the main central axes of its cross sections. The origin of coordinates is the center of gravity of the section.
The sequence of actions to determine the internal forces.
1) Mentally draw a section at the point of interest to us design. 2) Discard one of the cut off parts, and consider the balance of the remaining part. 3) Compose an equilibrium equation and determine from them the values and directions of internal force factors. Axial tension and compression - internal forces in cross section They can be closed by one force directed along the axis of the rod.
Stretching. 
Compression. Shear - occurs when, in the cross section of the rod, the internal forces are reduced to one, i.e. transverse force Q. 
Torsion - 1 force factor MZ occurs. 
MZ=MK Pure bend– A bending moment MX or MY occurs. 
To calculate structural elements for strength, rigidity, stability, first of all, it is necessary (using the section method) to determine the occurrence of internal force factors. Topic No. 1. STATICS OF A SOLID BODY
Basic concepts and axioms of statics
Static subject.static called a branch of mechanics in which the laws of the addition of forces and the conditions for the equilibrium of material bodies under the influence of forces are studied.
By equilibrium we will understand the state of rest of the body in relation to other material bodies. If the body, in relation to which the equilibrium is being studied, can be considered motionless, then the equilibrium is conditionally called absolute, and otherwise, relative. In statics, we will study only the so-called absolute equilibrium of bodies. In practice, in engineering calculations, equilibrium with respect to the Earth or to bodies rigidly connected to the Earth can be considered absolute. The validity of this statement will be substantiated in dynamics, where the concept of absolute equilibrium can be defined more strictly. The question of the relative equilibrium of bodies will also be considered there.
The equilibrium conditions of a body essentially depend on whether the body is solid, liquid, or gaseous. The equilibrium of liquid and gaseous bodies is studied in the courses of hydrostatics and aerostatics. In the general course of mechanics, usually only problems of the equilibrium of solids are considered.
All naturally occurring solids under the influence of external influences to some extent change their shape (deform). The values of these deformations depend on the material of the bodies, their geometric shape and dimensions, and on the acting loads. To ensure the strength of various engineering structures and structures, the material and dimensions of their parts are selected so that the deformations under the acting loads are sufficiently small. As a result, when studying general conditions equilibrium, it is quite acceptable to neglect small deformations of the corresponding solid bodies and consider them as non-deformable or absolutely rigid.
Absolutely solid body such a body is called, the distance between any two points of which always remains constant.
In order for a rigid body to be in equilibrium (at rest) under the action of a certain system of forces, it is necessary that these forces satisfy certain equilibrium conditions this system of forces. Finding these conditions is one of the main tasks of statics. But in order to find the conditions for the equilibrium of various systems of forces, as well as to solve a number of other problems in mechanics, it turns out to be necessary to be able to add the forces acting on a rigid body, to replace the action of one system of forces with another system, and, in particular, to reduce this system of forces to the simplest form. Therefore, the following two main problems are considered in the statics of a rigid body:
1) addition of forces and reduction of systems of forces acting on a rigid body to the simplest form;
2) determination of the equilibrium conditions for systems of forces acting on a solid body.
Strength. The state of equilibrium or motion of a given body depends on the nature of its mechanical interactions with other bodies, i.e. from those pressures, attractions or repulsions that a given body experiences as a result of these interactions. A quantity that is a quantitative measure of the mechanical interactionaction of material bodies, is called in mechanics force.
The quantities considered in mechanics can be divided into scalar ones, i.e. those that are fully characterized by their numerical value, and vector ones, i.e. those that, in addition to the numerical value, are also characterized by the direction in space.
Force is a vector quantity. Its effect on the body is determined by: 1) numerical value or module strength, 2) towardsniem strength, 3) application point strength.
The direction and point of application of the force depend on the nature of the interaction of the bodies and their relative position. For example, the force of gravity acting on a body is directed vertically down. The pressure forces of two smooth balls pressed against each other are directed along the normal to the surfaces of the balls at the points of their contact and are applied at these points, etc.
Graphically, the force is represented by a directed segment (with an arrow). The length of this segment (AB in fig. 1) expresses the modulus of force on the selected scale, the direction of the segment corresponds to the direction of the force, its beginning (point BUT in fig. 1) usually coincides with the point of application of the force. Sometimes it is convenient to depict a force in such a way that the point of application is its end - the tip of the arrow (as in Fig. 4 in). Straight DE, along which the force is directed is called line of force. Force is represented by the letter F . The modulus of force is indicated by vertical lines "on the sides" of the vector. Force system is the totality of forces acting on an absolutely rigid body.
Basic definitions:
A body that is not fastened to other bodies, to which any movement in space can be communicated from a given position, is called free.
If a free rigid body under the action of a given system of forces can be at rest, then such a system of forces is called balanced.
If one system of forces acting on a free rigid body can be replaced by another system without changing the state of rest or motion in which the body is located, then such two systems of forces are called equivalent.
If this system force is equivalent to one force, then this force is called resultant this system of forces. In this way, resultant - is the power that alone can replacethe action of this system, forces on a rigid body.
A force equal to the resultant in absolute value, directly opposite to it in direction and acting along the same straight line, is called balancing by force.
The forces acting on a rigid body can be divided into external and internal. External called the forces acting on the particles of a given body from other material bodies. internal called the forces with which the particles of a given body act on each other.
A force applied to a body at any one point is called concentrated. Forces acting on all points of a given volume or a given part of the surface of a body are called feuddivided.
The concept of a concentrated force is conditional, since in practice it is impossible to apply a force to a body at one point. The forces that we consider in mechanics as concentrated are essentially the resultant of certain systems of distributed forces.
In particular, the force of gravity, usually considered in mechanics, acting on a given rigid body, is the resultant of the forces of gravity of its particles. The line of action of this resultant passes through a point called the center of gravity of the body.
Axiom 1. If absolutely freea rigid body is acted upon by two forces, then the body cancan be in equilibrium if and onlywhen these forces are equal in absolute value (F 1 = F 2 ) and directedalong one straight line in opposite directions(Fig. 2).
Axiom 1 defines the simplest balanced system of forces, since experience shows that a free body, on which only one force acts, cannot be in equilibrium.
BUT 
xioma 2. The action of a given system of forces on an absolutely rigid body will not change if a balanced system of forces is added to or subtracted from it.
This axiom states that two systems of forces that differ by a balanced system are equivalent to each other.
Consequence from the 1st and 2nd axioms. The point of application of a force acting on an absolutely rigid body can be transferred along its line of action to any other point of the body.
Indeed, let a force F applied at point A act on a rigid body (Fig. 3). Let's take an arbitrary point B on the line of action of this force and apply two balanced forces F1 and F2 to it, such that Fl \u003d F, F2 \u003d - F. This does not change the effect of the force F on the body. But the forces F and F2, according to axiom 1, also form a balanced system that can be discarded. As a result, only one force Fl equal to F, but applied at point B, will act on the body.
Thus, the vector representing the force F can be considered applied at any point on the line of action of the force (such a vector is called a sliding vector).
The result obtained is valid only for forces acting on an absolutely rigid body. In engineering calculations, this result can be used only when the external action of forces on a given structure is studied, i.e. when the general conditions for the equilibrium of the structure are determined.
H
BUT
Therefore, axiom 3 can also be formulate as follows: resultant two forces applied to a body at one point is equal to the geomet ric (vector) sum of these forces and is applied in the same point.
Axiom 4. Two material bodies always act each otheron each other with forces equal in absolute value and directed alongone straight line in opposite directions(briefly: action equals reaction).
W
property of internal forces. According to axiom 4, any two particles of a solid body will act on each other with equal and oppositely directed forces. Since, when studying the general conditions of equilibrium, the body can be considered as absolutely rigid, then (according to axiom 1) all internal forces form a balanced system under this condition, which (according to axiom 2) can be discarded. Therefore, when studying the general conditions of equilibrium, it is necessary to take into account only the external forces acting on a given rigid body or a given structure.
Axiom 5 (hardening principle). If any changeremovable (deformable) body under the action of a given system of forcesis in equilibrium, then the equilibrium will remain even ifthe body will harden (become absolutely solid).
The assertion made in this axiom is obvious. For example, it is clear that the balance of a chain must not be disturbed if its links are welded together; the balance of a flexible thread will not be disturbed if it turns into a bent rigid rod, and so on. Since the same system of forces acts on a body at rest before and after solidification, axiom 5 can also be expressed in another form: at equilibrium, the forces acting on any variable (deforworldable) body, satisfy the same conditions as forabsolutely rigid bodies; however, for a mutable body, theseconditions, while necessary, may not be sufficient. For example, for the equilibrium of a flexible thread under the action of two forces applied to its ends, the same conditions are necessary as for a rigid rod (the forces must be equal in magnitude and directed along the thread in different directions). But these conditions will not be sufficient. To balance the thread, it is also required that the applied forces be tensile, i.e. directed as in Fig. 4a.
The solidification principle is widely used in engineering calculations. It allows, when compiling equilibrium conditions, to consider any variable body (belt, cable, chain, etc.) or any variable structure as absolutely rigid and apply the methods of rigid body statics to them. If the equations obtained in this way are not enough to solve the problem, then equations are additionally drawn up that take into account either the equilibrium conditions of individual parts of the structure, or their deformation.
Topic № 2. DYNAMICS OF THE POINT
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