Theoretical mechanics lectures 2nd year. Course of lectures technical mechanics. e) Moment of force about the axis

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

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

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

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

Rigid body statics

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

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

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

Kinematics

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

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

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

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

Dynamics

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

Basic concepts of dynamics
• Inertia- this is the property of material bodies to maintain a state of rest or uniform rectilinear motion until external forces change this state.
• Weight is a quantitative measure of the inertia of a body. The unit of mass is kilogram (kg).
• Material point- this is a body with mass, the dimensions of which are neglected when solving this problem.
• Center of mass of a mechanical system- a geometric point whose coordinates are determined by the formulas:
  
  Where m k , x k , y k , z k— mass and coordinates k-that point of the mechanical system, m— mass of the system.
  In a uniform field of gravity, the position of the center of mass coincides with the position of the center of gravity.
• Moment of inertia of a material body relative to an axis is a quantitative measure of inertia during rotational motion.
  The moment of inertia of a material point relative to the axis is equal to the product of the mass of the point by the square of the distance of the point from the axis:
  .
  The moment of inertia of the system (body) relative to the axis is equal to the arithmetic sum of the moments of inertia of all points:
  
• Inertia force of a material point is a vector quantity equal in modulus to the product of the mass of a point and the acceleration modulus and directed opposite to the acceleration vector:
• The force of inertia of a material body is a vector quantity equal in modulus to the product of the body mass and the modulus of acceleration of the center of mass of the body and directed opposite to the acceleration vector of the center of mass: ,
  where is the acceleration of the center of mass of the body.
• Elementary impulse of force is a vector quantity equal to the product of the force vector and an infinitesimal period of time dt:
  .
  The total force impulse for Δt is equal to the integral of the elementary impulses:
  .
• Elementary work of force is a scalar quantity dA, equal to the scalar proi

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• Statics
  • Basic concepts of statics
  • Types of forces
  • Axioms of statics
  • Connections and their reactions
  • System of converging forces
    • Methods for determining the resultant system of converging forces
    • Equilibrium conditions for a system of converging forces
  • Moment of force about the center as a vector
    • Algebraic value of moment of force
    • Properties of the moment of force relative to the center (point)
  • Force couple theory
    • Addition of two parallel forces directed in the same direction
    • Addition of two parallel forces directed in different directions
    • Force pairs
    • Couple force theorems
    • Equilibrium conditions for a system of force pairs
  • Lever arm
  • Arbitrary flat system of forces
    • Cases of reducing a plane system of forces to a simpler form
    • Analytical equilibrium conditions
  • Center of parallel forces. Center of gravity
    • Center of Parallel Forces
    • Center of gravity of a rigid body and its coordinates
    • Center of gravity of volume, plane and line
    • Methods for determining the position of the center of gravity
• Basics of strength racesets
  • Objectives and methods of strength of materials
  • Load classification
  • Classification of structural elements
  • Rod deformation
  • Basic hypotheses and principles
  • Internal forces. Section method
  • Voltages
  • Tension and compression
  • Mechanical characteristics of the material
  • Allowable stresses
  • Hardness of materials
  • Diagrams of longitudinal forces and stresses
  • Shift
  • Geometric characteristics of sections
  • Torsion
  • Bend
    • Differential dependencies during bending
    • Flexural strength
    • Normal voltages. Strength calculation
    • Shear stress during bending
    • Flexural rigidity
  • Elements of the general theory of stress state
  • Strength theories
  • Bending with torsion
• Kinematics
  • Kinematics of a point
    • Trajectory of a point's movement
    • Methods for specifying point movement
    • Point speed
    • Point acceleration
  • Rigid body kinematics
    • Translational motion of a rigid body
    • Rotational motion of a rigid body
    • Kinematics of gear mechanisms
    • Plane-parallel motion of a rigid body
  • Complex point movement
• Dynamics
  • Basic laws of dynamics
  • Dynamics of a point
    • Differential equations of a free material point
    • Two point dynamics problems
  • Rigid body dynamics
    • Classification of forces acting on a mechanical system
    • Differential equations of motion of a mechanical system
  • General theorems of dynamics
    • Theorem on the motion of the center of mass of a mechanical system
    • Momentum change theorem
    • Theorem on the change in angular momentum
    • Theorem on the change of kinetic energy
• Forces acting in machines
  • Forces in the engagement of a spur gear
  • Friction in mechanisms and machines
    • Sliding friction
    • Rolling friction
  • Efficiency
• Machine parts
  • Mechanical gears
    • Types of mechanical gears
    • Basic and derived parameters of mechanical gears
    • Gears
    • Gears with flexible links
  • Shafts
    • Purpose and classification
    • Design calculation
    • Check calculation of shafts
  • Bearings
    • Plain bearings
    • Rolling bearings
  • Connecting machine parts
    • Types of detachable and permanent connections
    • Keyed connections
• Standardization of norms, interchangeability
  • Tolerances and landings
  • Unified system of admissions and landings (USDP)
  • Deviation of shape and location

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Russian language

Calculation example of a spur gear
An example of calculating a spur gear. The choice of material, calculation of permissible stresses, calculation of contact and bending strength have been carried out.

An example of solving a beam bending problem
In the example, diagrams of transverse forces and bending moments were constructed, a dangerous section was found and an I-beam was selected. The problem analyzed the construction of diagrams using differential dependencies and carried out a comparative analysis of various cross sections of the beam.

An example of solving a shaft torsion problem
The task is to test the strength of a steel shaft at a given diameter, material and allowable stress. During the solution, diagrams of torques, shear stresses and twist angles are constructed. The shaft's own weight is not taken into account

An example of solving a problem of tension-compression of a rod
The task is to test the strength of a steel bar at specified permissible stresses. During the solution, diagrams of longitudinal forces, normal stresses and displacements are constructed. The rod's own weight is not taken into account

Application of the theorem on conservation of kinetic energy
An example of solving a problem using the theorem on the conservation of kinetic energy of a mechanical system

Determining the speed and acceleration of a point using given equations of motion
An example of solving a problem to determine the speed and acceleration of a point using given equations of motion

Determination of velocities and accelerations of points of a rigid body during plane-parallel motion
An example of solving a problem to determine the velocities and accelerations of points of a rigid body during plane-parallel motion

Determination of forces in the bars of a flat truss
An example of solving the problem of determining the forces in the rods of a flat truss using the Ritter method and the method of cutting nodes

state autonomous institution

Kaliningrad region

professional educational organization

College of Service and Tourism

A course of lectures with examples of practical tasks

"Fundamentals of Theoretical Mechanics"

by disciplineTechnical mechanics

for students3 course

specialties02/20/04 Fire safety

Kaliningrad

I APPROVED

Deputy Director for SD GAU KO POO KSTN.N. Myasnikova

APPROVED

Methodological Council of GAU KO POO KST

REVIEWED

At the PCC meeting

Editorial team:

Kolganova A.A., methodologist

Falaleeva A.B., teacher of Russian language and literature

Tsvetaeva L.V.., Chairman of the PCCgeneral mathematics and natural sciences

Compiled by:

Nezvanova I.V. teacher GAU KO POO KST

Content

1. Theoretical information

1. Theoretical information

1. Examples of solving practical problems

Dynamics: basic concepts and axioms

1. Theoretical information

1. Examples of solving practical problems

Bibliography

Statics: basic concepts and axioms.

1. Theoretical information

Statics – a section of theoretical mechanics that examines the properties of forces applied to points of a rigid body and the conditions for their equilibrium. Main goals:

1. Transformation of force systems into equivalent force systems.

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

Material point called the simplest model of a material body

any shape, the dimensions of which are small enough and which can be taken as a geometric point having a certain mass. A mechanical system is any collection of material points. An absolutely rigid body is a mechanical system whose distances between its points do not change during any interactions.

Force is a measure of the mechanical interaction of material bodies with each other. Force is a vector quantity, as it is determined by three elements:

numerical value;

direction;

point of application (A).

The unit of force is Newton(N).

Figure 1.1

A system of forces is a set of forces acting on a body.

A balanced (equal to zero) system of forces is a system that, when applied to a body, does not change its state.

A system of forces acting on a body can be replaced by one resultant, acting in the same way as a system of forces.

Axioms of statics.

Axiom 1: If a balanced system of forces is applied to a body, then it moves uniformly and rectilinearly or is at rest (law of inertia).

Axiom 2: An absolutely rigid body is in equilibrium under the action of two forces if and only if these forces are equal in magnitude, act in one straight line and are directed in opposite directions. Figure 1.2

Axiom 3: The mechanical state of the body will not be disturbed if a balanced system of forces is added to or subtracted from the system of forces acting on it.

Axiom 4: The resultant of two forces applied to a body is equal to their geometric sum, that is, it is expressed in magnitude and direction by the diagonal of a parallelogram built on these forces as on the sides.

Figure 1.3.

Axiom 5: The forces with which two bodies act on each other are always equal in magnitude and directed along the same straight line in opposite directions.

Figure 1.4.

Types of connections and their reactions

Connections are any restrictions that prevent the movement of a body in space. A body, trying under the influence of applied forces to carry out a movement that is prevented by a constraint, will act on it with a certain force called force of pressure on the connection . According to the law of equality of action and reaction, the connection will act on the body with the same magnitude, but oppositely directed force.
The force with which this connection acts on the body, preventing certain movements, is called force of reaction (reaction) of connection .
One of the basic principles of mechanics is principle of emancipation : any unfree body can be considered as free if we discard connections and replace their action with reactions of connections.

The reaction of the connection is directed in the direction opposite to that in which the connection does not allow the body to move. The main types of bonds and their reactions are given in Table 1.1.

Table 1.1

Types of connections and their reactions

Name of connection

Symbol

1

Smooth surface (support) – a surface (support) on which the friction of a given body can be neglected.
When supported freely, the reactionis directed perpendicular to the tangent drawn through the pointA body contact1 with supporting surface2 .

2

Thread (flexible, inextensible). The connection, made in the form of an inextensible thread, does not allow the body to move away from the point of suspension. Therefore, the reaction of the thread is directed along the thread to the point of its suspension.

3

Weightless rod - a rod whose weight, compared to the perceived load, can be neglected.
The reaction of a weightless hingedly attached rectilinear rod is directed along the axis of the rod.

4

Movable hinge, articulated-movable support. The reaction is directed normal to the supporting surface.

7

Hard seal. There will be two components of the reaction in the plane of the rigid embedding, and the moment of a couple of forces, which prevents the beam from turning1 relative to the pointA .
Rigid embedding in space takes away all six degrees of freedom from body 1 - three movements along the coordinate axes and three rotations about these axes.
There will be three components to the spatial rigid seal, , and three moments of couples of forces.

System of converging forces

A system of converging forces is a system of forces whose lines of action intersect at one point. Two forces converging at one point, according to the third axiom of statics, can be replaced by one force -resultant .
Main vector of the force system – a value equal to the geometric sum of the forces of the system.

Resultant of a plane system of converging forces can be determinedgraphically And analytically.

Addition of a system of forces . The addition of a flat system of converging forces is carried out either by sequential addition of forces with the construction of an intermediate resultant (Fig. 1.5), or by constructing a force polygon (Fig. 1.6).

Figure 1.5Figure 1.6

Projection of force on the axis – an algebraic quantity equal to the product of the force modulus and the cosine of the angle between the force and the positive direction of the axis.
ProjectionFx(Fig. 1.7) forces on the axis Xpositive if angle α is acute, negative if angle α is obtuse. If strengthperpendicular to the axis, then its projection onto the axis is zero.

Figure 1.7

Projection of force onto a plane Ohoo– vector , enclosed between the projections of the beginning and end of the forceto this plane. Those. projection of force onto a plane is a vector quantity, characterized not only by its numerical value, but also by its direction in the planeOhoo (Fig. 1.8).

Figure 1.8

Then the projection module to the plane Ohoo will be equal to:

Fxy =F cosα,

where α is the angle between the direction of the force and its projection.
Analytical method of specifying forces . For the analytical method of specifying the forceit is necessary to select a coordinate axes systemOhhz, in relation to which the direction of the force in space will be determined.
Vector depicting strength, can be constructed if the modulus of this force and the angles α, β, γ that the force forms with the coordinate axes are known. DotA application of force is specified separately by its coordinatesX, at, z. You can set the force by its projectionsFx, Fy, Fzto the coordinate axes. The modulus of force in this case is determined by the formula:

and direction cosines:

, .

Analytical method of adding forces : the projection of the sum vector onto some axis is equal to the algebraic sum of the projections of the summand vectors onto the same axis, i.e., if:

That , , .
Knowing Rx, Ry, Rz, we can define the module

and direction cosines:

, , .

Figure 1.9

For a system of converging forces to be in equilibrium, it is necessary and sufficient that the resultant of these forces be equal to zero.
1) Geometric equilibrium condition for a converging system of forces : for the equilibrium of a system of converging forces, it is necessary and sufficient that the force polygon constructed from these forces

was closed (end of the vector of the last term

force must coincide with the beginning of the vector of the first term of the force). Then the main vector of the force system will be equal to zero ()
2) Analytical equilibrium conditions . The module of the main vector of the force system is determined by the formula. =0. Because the , then the radical expression can be equal to zero only if each term simultaneously becomes zero, i.e.

Rx= 0, Ry= 0, R z = 0.

Consequently, for the equilibrium of a spatial system of converging forces, it is necessary and sufficient that the sums of the projections of these forces onto each of the three coordinates of the axes are equal to zero:

For the equilibrium of a plane system of converging forces, it is necessary and sufficient that the sums of the projections of the forces on each of the two coordinate axes be equal to zero:

The addition of two parallel forces directed in the same direction.

Figure 1.9

Two parallel forces directed in one direction are reduced to one resultant force, parallel to them and directed in the same direction. The magnitude of the resultant is equal to the sum of the magnitudes of these forces, and the point of its application C divides the distance between the lines of action of the forces internally into parts inversely proportional to the magnitudes of these forces, that is

B A C

R=F 1 +F 2

The addition of two parallel forces of unequal magnitude directed in opposite directions.

Two unequal antiparallel forces are reduced to one resultant force parallel to them and directed towards the larger force. The magnitude of the resultant is equal to the difference in the magnitudes of these forces, and the point of its application C, divides the distance between the lines of action of the forces externally into parts inversely proportional to the magnitudes of these forces, that is

A couple of forces and a moment of force about a point.

A moment of power relative to point O is called, taken with the appropriate sign, the product of the magnitude of the force and the distance h from point O to the line of action of the force . This product is taken with a plus sign if the strength tends to rotate the body counterclockwise, and with the sign -, if the force tends to rotate the body clockwise, that is . The length of the perpendicular h is calledshoulder of strength point O. The effect of force i.e. The angular acceleration of a body is greater, the greater the magnitude of the moment of force.

Figure 1.11

With a couple of forces is a system consisting of two parallel forces of equal magnitude directed in opposite directions. The distance h between the lines of action of forces is calledcouple's shoulder . The moment of a couple of forces m(F,F") is the product of the magnitude of one of the forces composing the pair and the shoulder of the pair, taken with the appropriate sign.

It is written like this: m(F, F")= ± F × h, where the product is taken with a plus sign if a pair of forces tends to rotate the body counterclockwise and with a minus sign if the pair of forces tends to rotate the body clockwise.

Theorem on the sum of moments of forces of a pair.

The sum of the moments of forces of a pair (F,F") relative to any point 0, taken in the plane of action of the pair, does not depend on the choice of this point and is equal to the moment of the pair.

Theorem on equivalent pairs. Consequences.

Theorem. Two pairs whose moments are equal to each other are equivalent, i.e. (F, F") ~ (P, P")

Corollary 1 . A pair of forces can be transferred to any place in the plane of its action, as well as rotated to any angle and change the arm and magnitude of the forces of the pair, while maintaining the moment of the pair.

Corollary 2. A pair of forces does not have a resultant and cannot be balanced by one force lying in the plane of the pair.

Figure 1.12

Addition and equilibrium condition for a system of pairs on a plane.

1. Theorem on the addition of pairs lying in the same plane. A system of pairs, arbitrarily located in the same plane, can be replaced by one pair, the moment of which is equal to the sum of the moments of these pairs.

2. Theorem on the equilibrium of a system of pairs on a plane.

In order for an absolutely rigid body to be at rest under the action of a system of pairs, arbitrarily located in one plane, it is necessary and sufficient that the sum of the moments of all pairs be equal to zero, that is

Center of gravity

Gravity – the resultant of the forces of attraction to the Earth distributed throughout the entire volume of the body.

Body center of gravity - this is a point invariably associated with this body through which the line of action of the force of gravity of a given body passes for any position of the body in space.

Methods for finding the center of gravity

1. Symmetry method:

1.1. If a homogeneous body has a plane of symmetry, then the center of gravity lies in this plane

1.2. If a homogeneous body has an axis of symmetry, then the center of gravity lies on this axis. The center of gravity of a homogeneous body of rotation lies on the axis of rotation.

1.3 If a homogeneous body has two axes of symmetry, then the center of gravity is at the point of their intersection.

2. Partitioning method: The body is divided into the smallest number of parts, the gravity forces and the position of the centers of gravity of which are known.

3. Negative mass method: When determining the center of gravity of a body that has free cavities, the partitioning method should be used, but the mass of the free cavities should be considered negative.

Coordinates of the center of gravity of a flat figure:

The positions of the centers of gravity of simple geometric figures can be calculated using known formulas. (Figure 1.13)

Note: The center of gravity of the symmetry of a figure is on the axis of symmetry.

The center of gravity of the rod is at the middle of the height.

1.2. Examples of solving practical problems

Example 1: The load is suspended on a rod and is in equilibrium. Determine the forces in the rod. (Figure 1.2.1)

Solution:

The forces generated in the fastening rods are equal in magnitude to the forces with which the rods support the load. (5th axiom)

We determine the possible directions of reactions of the “rigid rod” bonds.

The forces are directed along the rods.

Figure 1.2.1.

Let's free point A from connections, replacing the action of connections with their reactions. (Figure 1.2.2)

Let's start the construction with a known force, drawing a vectorFon some scale.

From the end of the vectorFdraw lines parallel to the reactionsR 1 AndR 2 .

Figure 1.2.2

When the lines intersect, they create a triangle. (Figure 1.2.3.). Knowing the scale of the constructions and measuring the length of the sides of the triangle, you can determine the magnitude of the reactions in the rods.

For more accurate calculations, you can use geometric relationships, in particular the sine theorem: the ratio of the side of a triangle to the sine of the opposite angle is a constant value

For this case:

Figure 1.2.3

Comment: If the direction of the vector (coupling reaction) in a given diagram and in the triangle of forces does not coincide, then the reaction in the diagram should be directed in the opposite direction.

Example 2: Determine the magnitude and direction of the resultant plane system of converging forces analytically.

Solution:

Figure 1.2.4

1. Determine the projections of all forces of the system onto Ox (Figure 1.2.4)

By adding the projections algebraically, we obtain the projection of the resultant onto the Ox axis.

The sign indicates that the resultant is directed to the left.

2. Determine the projections of all forces on the Oy axis:

By adding the projections algebraically, we obtain the projection of the resultant onto the Oy axis.

The sign indicates that the resultant is directed downward.

3. Determine the module of the resultant from the magnitudes of the projections:

4. Let us determine the value of the angle of the resultant with the Ox axis:

and the value of the angle with the Oy axis:

Example 3: Calculate the sum of moments of forces relative to point O (Figure 1.2.6).

OA= AB= IND=DE=CB=2m

Figure 1.2.6

Solution:

1. The moment of force relative to a point is numerically equal to the product of the module and the arm of the force.

2. The moment of force is zero if the line of action of the force passes through the point.

Example 4: Determine the position of the center of gravity of the figure presented in Figure 1.2.7

Solution:

We break the figure into three:

1-rectangle

A 1 =10*20=200cm 2

2-triangle

A 2 =1/2*10*15=75cm 2

3-circle

A 3 =3,14*3 2 =28.3cm 2

Figure 1 CG: x 1 =10cm, y 1 =5cm

Figure 2 CG: x 2 =20+1/3*15=25cm, y 2 =1/3*10=3.3cm

Figure 3 CG: x 3 =10cm, y 3 =5cm

Defined similarly With =4.5cm

Kinematics: basic concepts.

Basic kinematic parameters

Trajectory - a line that a material point outlines when moving in space. The trajectory can be straight or curved, flat or spatial.

Trajectory equation for plane motion: y =f ( x)

Distance traveled. The path is measured along the trajectory in the direction of travel. Designation -S, units of measurement are meters.

Equation of motion of a point is an equation that determines the position of a moving point as a function of time.

Figure 2.1

The position of a point at each moment of time can be determined by the distance traveled along the trajectory from some fixed point, considered as the origin (Figure 2.1). This method of specifying movement is callednatural . Thus, the equation of motion can be represented as S = f (t).

Figure 2.2

The position of a point can also be determined if its coordinates are known depending on time (Figure 2.2). Then, in the case of motion on a plane, two equations must be given:

In the case of spatial motion, a third coordinate is addedz= f 3 ( t)

This method of specifying movement is calledcoordinate .

Travel speed is a vector quantity that characterizes the current speed and direction of movement along the trajectory.

Speed ​​is a vector, at any moment directed tangentially to the trajectory towards the direction of movement (Figure 2.3).

Figure 2.3

If a point travels equal distances in equal periods of time, then the motion is calleduniform .

Average speed along the way ΔSdefined:

WhereΔS- distance traveled in time Δt; Δ t- time interval.

If a point travels unequal paths in equal periods of time, then the motion is calleduneven . In this case, speed is a variable quantity and depends on timev= f( t)

The speed at the moment is determined as

Point acceleration - a vector quantity characterizing the rate of change in speed in magnitude and direction.

The speed of a point when moving from point M1 to point Mg changes in magnitude and direction. Average acceleration value for this period of time

Current acceleration:

Usually, for convenience, two mutually perpendicular components of acceleration are considered: normal and tangential (Figure 2.4)

Normal acceleration a n , characterizes the change in speed along

direction and is defined as

Normal acceleration is always directed perpendicular to the speed towards the center of the arc.

Figure 2.4

Tangential acceleration a t , characterizes the change in speed in magnitude and is always directed tangentially to the trajectory; when accelerating, its direction coincides with the direction of the velocity, and when decelerating, it is directed opposite to the direction of the velocity vector.

The total acceleration value is defined as:

Analysis of types and kinematic parameters of movements

Uniform movement - This is a movement at a constant speed:

For rectilinear uniform motion:

For curvilinear uniform motion:

Law of Uniform Motion :

Equally alternating motion – This is motion with constant tangential acceleration:

For rectilinear uniform motion

For curvilinear uniform motion:

Law of uniform motion:

Kinematic graphs

Kinematic graphs - These are graphs of changes in path, speed and acceleration depending on time.

Uniform movement (Figure 2.5)

Figure 2.5

Equally alternating motion (Figure 2.6)

Figure 2.6

The simplest motions of a rigid body

Forward movement call the movement of a rigid body in which any straight line on the body during movement remains parallel to its initial position (Figure 2.7)

Figure 2.7

During translational motion, all points of the body move equally: the speeds and accelerations are the same at each moment.

Atrotational movement all points of the body describe circles around a common fixed axis.

The fixed axis around which all points of the body rotate is calledaxis of rotation.

To describe the rotational motion of a body around a fixed axis, you can only useangular parameters. (Figure 2.8)

φ – body rotation angle;

ω – angular velocity, determines the change in the angle of rotation per unit time;

The change in angular velocity over time is determined by angular acceleration:

2.2. Examples of solving practical problems

Example 1: The equation of motion of a point is given. Determine the speed of the point at the end of the third second of movement and the average speed for the first three seconds.

Solution:

1. Speed ​​equation

2. Speed ​​at the end of the third second (t=3 c)

3. Average speed

Example 2: Based on the given law of motion, determine the type of motion, the initial speed and tangential acceleration of the point, and the time to stop.

Solution:

1. Type of movement: uniformly variable ()
2. When comparing the equations, it is obvious that

- the initial path traveled before the start of the countdown 10m;

- initial speed 20m/s

- constant tangential acceleration

- the acceleration is negative, therefore, the movement is slow, the acceleration is directed in the direction opposite to the speed of movement.

3. You can determine the time at which the speed of the point will be zero.

3.Dynamics: basic concepts and axioms

Dynamics – a section of theoretical mechanics in which a connection is established between the movement of bodies and the forces acting on them.

In dynamics, two types of problems are solved:

determine motion parameters based on given forces;

determine the forces acting on the body according to the given kinematic parameters of movement.

Undermaterial point imply a certain body that has a certain mass (i.e., containing a certain amount of matter), but does not have linear dimensions (an infinitesimal volume of space).
Isolated is considered a material point that is not affected by other material points. In the real world, isolated material points, like isolated bodies, do not exist; this concept is conditional.

During translational motion, all points of the body move equally, so the body can be taken as a material point.

If the dimensions of the body are small compared to the trajectory, it can also be considered as a material point, and the point coincides with the center of gravity of the body.

During the rotational motion of a body, the points may not move in the same way; in this case, some provisions of the dynamics can be applied only to individual points, and the material object can be considered as a collection of material points.

Therefore, dynamics is divided into the dynamics of a point and the dynamics of a material system.

Axioms of dynamics

The first axiom ( principle of inertia): in Every isolated material point is in a state of rest or uniform and linear motion until applied forces bring it out of this state.

This state is called the stateinertia. Bring the point out of this state, i.e. An external force can impart some acceleration to it.

Every body (point) hasinertia. The measure of inertia is body mass.

Mass calledthe amount of substance in the volume of the body, in classical mechanics it is considered a constant value. The unit of mass is kilogram (kg).

Second axiom (Newton's second law is the fundamental law of dynamics)

F=ma

WhereT - point mass, kg;A - point acceleration, m/s 2 .

The acceleration imparted to a material point by a force is proportional to the magnitude of the force and coincides with the direction of the force.

The force of gravity acts on all bodies on Earth; it imparts to the body an acceleration of free fall directed towards the center of the Earth:

G = mg,

Whereg- 9.81 m/s², free fall acceleration.

Third axiom (Newton's third law): cThe forces of interaction between two bodies are equal in size and directed along the same straight line in different directions.

When interacting, the accelerations are inversely proportional to the masses.

Fourth axiom (law of independence of forces): toEach force in a system of forces acts as it would act alone.

The acceleration imparted to a point by a system of forces is equal to the geometric sum of the accelerations imparted to the point by each force separately (Figure 3.1):

Figure 3.1

The concept of friction. Types of friction.

Friction- resistance that occurs when one rough body moves over the surface of another. When bodies slide, sliding friction occurs, and when they roll, rocking friction occurs.

Sliding friction

Figure 3.2.

The reason is the mechanical engagement of the protrusions. The force of resistance to movement when sliding is called the sliding friction force (Figure 3.2)

Laws of sliding friction:

1. The sliding friction force is directly proportional to the normal pressure force:

WhereR- normal pressure force, directed perpendicular to the supporting surface;f- coefficient of sliding friction.

Figure 3.3.

In the case of body movement along an inclined plane (Figure 3.3)

Rolling friction

Rolling resistance is associated with mutual deformation of the soil and the wheel and is significantly less than sliding friction.

For uniform rolling of the wheel, it is necessary to apply forceF dv (Figure 3.4)

The condition for the wheel to roll is that the moving moment must be no less than the moment of resistance:

Figure 3.4.

Example 1: Example 2: To two material points of massm 1 =2kg andm 2 = 5 kg equal forces applied. Compare the acceleration values.

Solution:

According to the third axiom, acceleration dynamics are inversely proportional to masses:

Example 3: Determine the work done by gravity when moving a load from point A to point C along an inclined plane (Figure 3.7). Body gravity is 1500N. AB = 6 m, BC = 4 m. Example 3: Determine the work done by the cutting force in 3 minutes. The rotation speed of the workpiece is 120 rpm, the diameter of the workpiece is 40 mm, the cutting force is 1 kN. (Figure 3.8)

Solution:

1. Rotary work:

2. Angular speed 120 rpm

Figure 3.8.

3. The number of revolutions for a given time isz=120*3=360 rev.

Angle of rotation during this time φ=2πz=2*3.14*360=2261rad

4. Work in 3 turns:W=1*0.02*2261=45.2 kJ

Bibliography

Olofinskaya, V.P. "Technical Mechanics", Moscow "Forum" 2011.

Erdedi A.A. Erdedi N.A. Theoretical mechanics. Strength of materials.- R-n-D; Phoenix, 2010