Theoretical mechanics- This is a branch of mechanics, which sets out the basic laws of mechanical motion and mechanical interaction of material bodies.
Theoretical mechanics is a science in which the movements of bodies over time (mechanical movements) are studied. It serves as the basis for other sections of mechanics (the theory of elasticity, resistance of materials, the theory of plasticity, the 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 such 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 theoretical mechanics, which deals with the problem of equilibrium of rigid 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.
- loose body is a body, on the movement of which no restrictions are imposed.
- Non-free (bound) body is a body whose movement is restricted.
- Connections- these are bodies that prevent the movement of the object under consideration (a body or a system of bodies).
- Communication reaction is a force that characterizes the action of a bond on a rigid body. If we consider the force with which a rigid body acts on a bond as an action, then the reaction of the bond is a counteraction. 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 set of interconnected bodies or material points.
- Solid can be considered as a mechanical system, the positions and distance between the points of which do not change.
- Force is a vector quantity characterizing the mechanical action of one material body on another.
Force as a vector is characterized by the point of application, the direction of action and the absolute value. The unit of measure for the modulus of force is Newton. - line of force is the straight line along which the force vector is directed.
- Concentrated Power is the force applied at one point.
- Distributed forces (distributed load)- these are forces acting on all points of the volume, surface or length of the body.
The distributed load is given 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 considered mechanical system.
- inner strength is the force acting on a material point of a mechanical system from another material point belonging to the system under consideration.
- Force system is the totality of forces acting on a mechanical system.
- Flat system of forces 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.
- Converging force system 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 systems of forces- these are systems of forces, the replacement of which one for another does not change the mechanical state of the body.
Accepted designation: . - Equilibrium A state in which a body remains stationary or moves uniformly in a straight line under the action of forces.
- Balanced system of forces- this is a system of forces that, being applied to a free solid body, does not change its mechanical state (does not unbalance it).
. - 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 value that characterizes the rotational ability of the force.
- Power couple is a system of two parallel equal in absolute value oppositely directed forces.
Accepted designation: .
Under the action of a couple of forces, the body will perform a rotational motion. - 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 on a Plane is a vector on a plane enclosed between the 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 a motion by inertia. Under the state of equilibrium of a material point and solid body understand not only the state of rest, but also the movement by inertia. For a rigid body, there are different kinds motion by inertia, for example, the 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 balanced.
In general, forces are said to be balanced if the rigid body to which these forces are applied is at rest. - Law 3. Without violating the state (the word "state" here means the state of motion or rest) of a rigid body, one can add and discard balancing forces.
Consequence. Without disturbing the state of a rigid 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 another without disturbing the state of the rigid body. - Law 4. The resultant of two forces applied at one point is applied at the same point, is equal in absolute value to the diagonal of the parallelogram built on these forces, and is directed along this
diagonals.
The modulus 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 one straight line.
It should be borne in mind that action- force applied to the body B, and opposition- force applied to the body BUT, are not balanced, since they are attached to different bodies. - Law 6 (the law of hardening). 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 rigid body, are necessary but insufficient for the corresponding non-rigid body. - Law 7 (the law of release from bonds). A non-free solid body can be considered as free if it is mentally freed from bonds, replacing the action of bonds with the corresponding reactions of bonds.
- Connections and their reactions
- Smooth surface restricts movement along the normal to the support surface. The reaction is directed perpendicular to the surface.
- Articulated movable support limits the movement of the body along the normal to the reference plane. The reaction is directed along the 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 termination counteracts any movement and rotation in the plane. Its action can be replaced by a force presented in the form of two components and a pair of forces with a moment.
Kinematics
Kinematics- a section of theoretical mechanics, which considers 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
- The law of motion of a point (body) is the dependence of the position of a point (body) in space on time.
- Point trajectory is the locus of the positions of a point in space during its movement.
- Point (body) speed- this is a characteristic of the change in time of the position of a point (body) in space.
- Point (body) acceleration- this is a characteristic of the change in time of the speed of a point (body).
- Determination of the kinematic characteristics of a point
- Point trajectory
In the vector reference system, the trajectory is described by the expression: .
In the coordinate reference system, the trajectory is determined according to the law of point motion and is described by the expressions z = f(x,y) in space, or y = f(x)- in the plane.
In a natural reference system, the trajectory is predetermined. - 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 the time interval is called the average value of the speed in this time interval: .
Taking the time interval as an infinitesimal value, the value of the speed at a given moment of time (the instantaneous value of the speed) is obtained:
.
The average velocity vector is directed along the vector in the direction of the point movement, the instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point movement.
Conclusion: the speed of a point is a vector quantity equal to the derivative of the law of motion with respect to time.
Derivative property: the time derivative of any value determines the rate of change of this value. - Determining the speed of a point in a coordinate reference system
Rate of change of point coordinates:
.
The module of the full speed 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 steering 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 a natural reference system is defined as a derivative of the law of motion of a point: .
According to the previous conclusions, the velocity vector is directed tangentially to the trajectory in the direction of the point 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) task of movement and determination of the kinematic characteristics of the body as a whole;
2) determination of the kinematic characteristics of the points of the body. - Translational motion of a rigid body
Translational motion is a motion in which a straight line drawn through two points of the body remains parallel to its original position.
Theorem: in translational motion, all points of the body move along the same trajectories and at each moment of time have the same speed and acceleration in absolute value and direction.
Conclusion: the translational motion of a rigid body is determined by the motion of any of its points, and therefore, the task and study of its motion is reduced to the kinematics of a point. - Rotational motion of a rigid body around a fixed axis
The 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 an angle is radians. (A radian is the central angle of a circle whose arc length is equal to the radius, the full angle of the circle contains 2π radian.)
The law of rotational motion of a body around a fixed axis.
The angular velocity and angular acceleration of the body will be determined by the differentiation method:
— angular velocity, rad/s;
— angular acceleration, rad/s².
If we cut the body by a plane perpendicular to the axis, choose a point on the axis of rotation With and an arbitrary point M, then the point M will describe around the point With radius circle R. During dt there is an elementary rotation through the angle , while the point M will move along the trajectory for a distance
.
Linear speed module:
.
point acceleration M with a known trajectory is determined by its components:
,
where
.
As a result, we get formulas
tangential acceleration:
;
normal acceleration:
.
Dynamics
Dynamics- This is a branch of theoretical mechanics, which studies the mechanical movements of material bodies, 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 is a body with a mass, the dimensions of which are neglected in solving this problem.
- Center of mass of a mechanical system — 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 is the 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 about the axis is a quantitative measure of inertia during rotational motion.
The moment of inertia of a material point about the axis is equal to the product of the mass of the point and the square of the distance of the point from the axis:
.
The moment of inertia of the system (body) about the axis is equal to the arithmetic sum of the moments of inertia of all points:
- The force of inertia of a material point is a vector quantity equal in absolute value to the product of the mass of a point and the module of acceleration and directed opposite to the acceleration vector:
- Force of inertia of a material body is a vector quantity equal in absolute value to the product of the body mass and the module 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. - Elemental Force Impulse is a vector quantity equal to the product of the force vector by an infinitesimal time interval dt:
.
The total impulse of force for Δt is equal to the integral of elementary impulses:
. - Elementary work of force is a scalar dA, equal to the scalar
Kinematics
Kinematics of a material point
Determination of the speed and acceleration of a point according to the given equations of its motion
Given: Equations of motion of a point: x = 12 sin(πt/6), cm; y= 6 cos 2 (πt/6), cm.
Set the type of its trajectory and for the moment of time t = 1 s find the position of a point on the trajectory, its velocity, full, tangential and normal accelerations, as well as the radius of curvature of the trajectory.
Translational and rotational motion of a rigid body
Given:
t = 2 s; r 1 = 2 cm, R 1 = 4 cm; r 2 = 6 cm, R 2 = 8 cm; r 3 \u003d 12 cm, R 3 \u003d 16 cm; s 5 \u003d t 3 - 6t (cm).
Determine at time t = 2 the velocities of points A, C; angular acceleration of wheel 3; point B acceleration and rack acceleration 4.
Kinematic analysis of a flat mechanism
Given:
R 1 , R 2 , L, AB, ω 1 .
Find: ω 2 .
The flat mechanism consists of rods 1, 2, 3, 4 and slider E. The rods are connected by means of cylindrical hinges. Point D is located in the middle of bar AB.
Given: ω 1 , ε 1 .
Find: speeds V A , V B , V D and V E ; angular velocities ω 2 , ω 3 and ω 4 ; acceleration a B ; angular acceleration ε AB of link AB; positions of instantaneous centers of speeds P 2 and P 3 of links 2 and 3 of the mechanism.
Determining the absolute speed and absolute acceleration of a point
A rectangular plate rotates around a fixed axis according to the law φ = 6 t 2 - 3 t 3. The positive direction of reading the angle φ is shown in the figures by an arc arrow. Rotation axis OO 1 lies in the plane of the plate (the plate rotates in space).
The point M moves along the straight line BD along the plate. The law of its relative motion is given, i.e., the dependence s = AM = 40(t - 2 t 3) - 40(s - in centimeters, t - in seconds). Distance b = 20 cm. In the figure, point M is shown in the position where s = AM > 0 (for s< 0 point M is on the other side of point A).
Find the absolute speed and absolute acceleration of point M at time t 1 = 1 s.
Dynamics
Integration of differential equations of motion of a material point under the action of variable forces
A load D of mass m, having received an initial velocity V 0 at point A, moves in a curved pipe ABC located in a vertical plane. On the section AB, the length of which is l, the load is affected by a constant force T (its direction is shown in the figure) and the force R of the resistance of the medium (the module of this force is R = μV 2, the vector R is directed opposite to the velocity V of the load).
The load, having completed its movement in section AB, at point B of the pipe, without changing the value of its velocity modulus, passes to section BC. On the section BC, a variable force F acts on the load, the projection F x of which on the x axis is given.
Considering the load as a material point, find the law of its motion on the section BC, i.e. x = f(t), where x = BD. Ignore the friction of the load on the pipe.
Theorem on the change in the kinetic energy of a mechanical system
The mechanical system consists of weights 1 and 2, a cylindrical roller 3, two-stage pulleys 4 and 5. The bodies of the system are connected by threads wound on pulleys; sections of threads are parallel to the corresponding planes. The roller (solid homogeneous cylinder) rolls along the reference plane without slipping. The radii of the steps of pulleys 4 and 5 are respectively R 4 = 0.3 m, r 4 = 0.1 m, R 5 = 0.2 m, r 5 = 0.1 m. The mass of each pulley is considered uniformly distributed along its outer rim . The supporting planes of weights 1 and 2 are rough, the coefficient of sliding friction for each weight is f = 0.1.
Under the action of force F, the modulus of which changes according to the law F = F(s), where s is the displacement of the point of its application, the system begins to move from a state of rest. When the system moves, resistance forces act on pulley 5, the moment of which relative to the axis of rotation is constant and equal to M 5 .
Determine the value of the angular velocity of pulley 4 at the moment when the displacement s of the point of application of force F becomes equal to s 1 = 1.2 m. 
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Application of the general equation of dynamics to the study of the motion of a mechanical system
For a mechanical system, determine the linear acceleration a 1 . Consider that for blocks and rollers the masses are distributed along the outer radius. Cables and belts are considered weightless and inextensible; there is no slippage. Ignore rolling and sliding friction. 
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Application of the d'Alembert principle to the determination of the reactions of the supports of a rotating body
A vertical shaft AK rotating uniformly with an angular velocity ω = 10 s -1 is fixed with a thrust bearing at point A and a cylindrical bearing at point D.
A weightless rod 1 with a length of l 1 = 0.3 m is rigidly attached to the shaft, at the free end of which there is a load of mass m 1 = 4 kg, and a homogeneous rod 2 with a length of l 2 = 0.6 m, having a mass of m 2 = 8 kg. Both rods lie in the same vertical plane. The points of attachment of the rods to the shaft, as well as the angles α and β are indicated in the table. Dimensions AB=BD=DE=EK=b, where b = 0.4 m. Take the load as a material point.
Neglecting the mass of the shaft, determine the reactions of the thrust bearing and the bearing.
Tasks for settlement-analytical and settlement-graphic works on all sections of the course of technical mechanics are given. Each task includes a description of the solution of problems with brief guidelines, examples of solutions are given. The annexes contain the necessary reference material. For students of construction specialties of secondary professional educational institutions.
Determination of reactions of ideal bonds in an analytical way.
1. Indicate the point whose equilibrium is being considered. In tasks for independent work, such a point is the center of gravity of the body or the point of intersection of all rods and threads.
2. Apply active forces to the considered point. In tasks for independent work, the active forces are the own weight of the body or the weight of the load, which are directed downward (more correctly, towards the center of gravity of the earth). In the presence of a block, the weight of the load acts on the considered point along the thread. The direction of this force is determined from the drawing. Body weight is usually denoted by the letter G.
3. Mentally discard connections, replacing their action with reactions of connections. In the proposed problems, three types of bonds are used - an ideally smooth plane, ideally rigid straight rods and ideally flexible threads - hereinafter referred to as a plane, a rod and a thread, respectively.
TABLE OF CONTENTS
Foreword
Section I. Independent and control work
Chapter 1. Theoretical mechanics. Statics
1.1. Analytical Determination of Ideal Bond Reactions
1.2. Determination of the support reactions of a beam on two supports under the action of vertical loads
1.3. Determination of the position of the center of gravity of the section
Chapter 2. Strength of materials
2.1. Selection of sections of rods based on strength
2.2. Determination of the main central moments of inertia of the section
2.3. Plotting Shear Forces and Bending Moments for a Simple Beam
2.4. Determination of the allowable value of the central compressive force
Chapter 3
3.1. Construction of diagrams of internal forces for the simplest single-circuit frame
3.2. Graphical determination of forces in truss rods by constructing a Maxwell-Cremona diagram
3.3. Determination of linear movements in the simplest cantilever frames
3.4. Calculation of a statically indeterminate (continuous) beam according to the equation of three moments
Section II. Settlement and graphic works
Chapter 4. Theoretical mechanics. Statics
4.1. Determination of forces in the rods of the simplest cantilever truss
4.2. Determination of support reactions of a beam on two supports
4.3. Determination of the position of the center of gravity of the section
Chapter 5
5.1. Determination of forces in bars of a statically indeterminate system
5.2. Determination of the main moments of inertia of the section
5.3. Selection of the section of a beam from a rolled I-beam
5.4. Selection of the section of the centrally compressed composite rack
Chapter 6
6.1. Determination of forces in sections of a three-hinged arch
6.2. Graphical determination of forces in the bars of a flat truss by constructing a Maxwell diagram - Cremona
6.3. Calculation of a statically indeterminate frame
6.4. Calculation of a continuous beam according to the equation of three moments
Applications
Bibliography.
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Many university students face certain difficulties when they begin to teach basic technical disciplines, such as strength of materials and theoretical mechanics, in their course of study. This article will consider one of these subjects - the so-called technical mechanics.
Technical mechanics is a science that studies various mechanisms, their synthesis and analysis. In practice, this means a combination of three disciplines - strength of materials, theoretical mechanics and machine parts. It is convenient in that each educational institution chooses in what proportion to teach these courses.
Accordingly, in most control works, the tasks are divided into three blocks, which must be solved separately or together. Let's consider the most common tasks.
Section one. Theoretical mechanics
Of the variety of problems in theoretical mechanics, one can most often meet problems from the section of kinematics and statics. These are tasks on the balance of a flat frame, the definition of the laws of motion of bodies and the kinematic analysis of the lever mechanism.
To solve problems for the equilibrium of a flat frame, it is necessary to use the equilibrium equation for a flat system of forces:
The sum of the projections of all forces on the coordinate axes is equal to zero and the sum of the moments of all forces about any point is equal to zero. By solving these equations together, we determine the magnitude of the reactions of all supports of the flat frame.
In tasks for determining the main kinematic parameters of the movement of bodies, it is necessary, based on a given trajectory or the law of motion of a material point, to determine its speed, acceleration (full, tangential and normal) and the radius of curvature of the trajectory. The laws of point motion are given by the trajectory equations:
The projections of the point velocity on the coordinate axes are found by differentiating the corresponding equations:
By differentiating the velocity equations, we find the projections of the acceleration of the point. The tangential and normal accelerations, the radius of curvature of the trajectory are found graphically or analytically:
The kinematic analysis of the lever mechanism is carried out according to the following scheme:
- Partitioning the mechanism into Assur groups
- Construction for each of the groups of plans for speeds and accelerations
- Determination of speeds and accelerations of all links and points of the mechanism.
Section two. Strength of materials
The strength of materials is a section that is quite difficult to understand, with many different tasks, most of which are solved according to its own methodology. In order to make it easier for students to solve their problems, most often in the course of applied mechanics they are given elementary problems for the simple resistance of structures - moreover, the type and material of the structure, as a rule, depends on the profile of the university.
The most common problems are tension-compression, bending and torsion.
In tension-compression problems, it is necessary to construct diagrams of longitudinal forces and normal stresses, and sometimes also displacements of structural sections.
To do this, it is necessary to divide the structure into sections, the boundaries of which will be the places where the load is applied or the cross-sectional area changes. Further, applying the formulas for the equilibrium of a solid body, we determine the values of internal forces at the boundaries of the sections, and, taking into account the cross-sectional area, internal stresses.
According to the data obtained, we build graphs - diagrams, taking the axis of symmetry of the structure as the graph axis.
Torsion problems are similar to bending problems, except that instead of tensile forces, torques are applied to the body. With this in mind, it is necessary to repeat the calculation steps - partitioning into sections, determining the twisting moments and angles of twisting and plotting.
In bending problems, it is necessary to calculate and determine the transverse forces and bending moments for a loaded beam.
First, the reactions of the supports in which the beam is fixed are determined. To do this, you need to write down the equilibrium equations of the structure, taking into account all the acting forces.
After that, the beam is divided into sections, the boundaries of which will be the points of application of external forces. By considering the balance of each section separately, the transverse forces and bending moments at the boundaries of the sections are determined. Based on the data obtained, plots are built.
The cross-sectional strength test is carried out as follows:
- The location of the dangerous section is determined - the section where the greatest bending moments will act.
- From the condition of strength in bending, the moment of resistance of the cross section of the beam is determined.
- The characteristic section size is determined - diameter, side length or profile number.
Section three. Machine parts
The "Machine Details" section combines all the tasks for calculating mechanisms that work in real conditions - this can be a conveyor drive or a gear train. It greatly facilitates the task that all formulas and calculation methods are given in reference books, and the student only needs to choose those that are suitable for a given mechanism.
Literature
- Theoretical mechanics: Guidelines and control tasks for part-time students of engineering, construction, transport, instrument-making specialties of higher educational institutions / Ed. prof. S.M.Targa, - M.: graduate School, 1989 Fourth edition;
- A. V. Darkov, G. S. Shpiro. "Strength of materials";
- Chernavsky S.A. Course design of machine parts: Proc. manual for students of mechanical engineering specialties of technical schools / S. A. Chernavsky, K. N. Bokov, I. M. Chernin, etc. - 2nd ed., revised. and additional - M. Mashinostroenie, 1988. - 416 p.: ill.
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