Gdz on those mechanics. Basic laws and formulas for theoretical mechanics. Solution of examples. Integration of differential equations of motion of a material point under the influence of variable forces

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Kinematics

Material point kinematics

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 \u003d 12 sin (πt / 6), cm; y \u003d 6 cos 2 (πt / 6), cm.

Set the type of its trajectory and for the time moment t \u003d 1 sec find the position of a point on the trajectory, its speed, total, tangential and normal accelerations, as well as the radius of curvature of the trajectory.

Translational and rotational motion of a rigid body

Given:
t \u003d 2 s; r 1 \u003d 2 cm, R 1 \u003d 4 cm; r 2 \u003d 6 cm, R 2 \u003d 8 cm; r 3 \u003d 12 cm, R 3 \u003d 16 cm; s 5 \u003d t 3 - 6t (cm).

Determine at time t \u003d 2 the speeds of points A, C; angular acceleration of wheel 3; point B acceleration and staff 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 slide 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: the 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 link AB; positions of instant centers of speeds P 2 and P 3 of links 2 and 3 of the mechanism.

Determination of absolute speed and absolute point acceleration

The rectangular plate rotates around a fixed axis according to the law φ \u003d 6 t 2 - 3 t 3 ... The positive direction of the angle φ is shown in the figures with an arc arrow. Rotation axis OO 1 lies in the plane of the plate (the plate rotates in space).

Point M moves along the line BD on the plate. The law of its relative motion is given, i.e., the dependence s \u003d AM \u003d 40 (t - 2 t 3) - 40 (s - in centimeters, t - in seconds). Distance b \u003d 20 cm... In the figure, point M is shown in a position at which s \u003d 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 \u003d 1 s.

Dynamics

Integration of differential equations of motion of a material point under the influence of variable forces

A load D of mass m, having received the 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, a constant force T (its direction is shown in the figure) and the resistance force R of the medium act on the load (the modulus of this force R \u003d μV 2, the vector R is directed opposite to the speed V of the load).

The load, having finished its movement on section AB, at point B of the pipe, without changing the value of its velocity modulus, goes to section BC. In 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 movement on the BC section, i.e. x \u003d f (t), where x \u003d BD. Disregard the friction of the load on the pipe.

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The 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 the pulleys; the thread sections are parallel to the corresponding planes. The roller (solid uniform cylinder) rolls on the reference plane without sliding. The radii of the steps of the pulleys 4 and 5 are, respectively, R 4 \u003d 0.3 m, r 4 \u003d 0.1 m, R 5 \u003d 0.2 m, r 5 \u003d 0.1 m. The mass of each pulley is considered uniformly distributed along its outer rim ... The support planes of weights 1 and 2 are rough, the sliding friction coefficient for each load is f \u003d 0.1.

Under the action of the force F, the modulus of which changes according to the law F \u003d F (s), where s is the displacement of the point of its application, the system starts moving from a state of rest. When the system moves, resistance forces act on the 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 the pulley 4 at that moment in time when the displacement s of the point of application of the force F becomes equal to s 1 \u003d 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 the mechanical system, determine the linear acceleration a 1. Assume that the masses of blocks and rollers are distributed along the outer radius. Cables and belts are considered weightless and inextensible; there is no slippage. Neglect 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

The vertical shaft AK, rotating uniformly with an angular velocity ω \u003d 10 s -1, is fixed by a thrust bearing at point A and a cylindrical bearing at point D.

A weightless rod 1 with a length of l 1 \u003d 0.3 m is rigidly attached to the shaft, at the free end of which there is a load with a mass of m 1 \u003d 4 kg, and a homogeneous rod 2 with a length of l 2 \u003d 0.6 m and a mass of m 2 \u003d 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 \u003d BD \u003d DE \u003d EK \u003d b, where b \u003d 0.4 m. Take the load as a material point.

By neglecting the mass of the shaft, determine the reaction of the thrust bearing and bearing.

Theoretical mechanics - this is a section 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 movement of bodies over time (mechanical movements) is studied. It serves as the basis for other branches of mechanics (theory of elasticity, resistance 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 such an interaction, as a result of which mechanical movement changes or the relative position of body parts changes.

Rigid body statics

Statics - This is a section of theoretical mechanics, which deals with the problems 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 solid (solid, 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 Is a body, the movement of which is not subject to any restrictions.
• Unfree (bound) body Is a body whose movement is restricted.
• Connections - these are bodies that prevent the movement of the object under consideration (body or system of bodies).
• Communication reaction Is a force that characterizes the effect 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 bond reaction is a reaction. In this case, the force - the action is applied to the bond, and the bond reaction is applied to the solid.
• 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 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 measure for the modulus of force is Newton.
• Force action line Is a straight line along which the force vector is directed.
• Concentrated power - force applied at one point.
• Distributed forces (distributed load) - these are the forces acting on all points of the volume, surface or length of the body.
  The distributed load is set by the force acting on a 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 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 system of forces Is a system of forces whose lines of action lie in one 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 force system Is a system of forces, the lines of action of which do not intersect at one point.
• Equivalent systems of forces - these are systems of forces, replacing them with one another does not change the mechanical state of the body.
  Accepted designation:.
• Equilibrium - this is a state in which the body under the action of forces remains stationary or moves uniformly in a straight line.
• Balanced system of forces Is a system of forces that, when applied to a free solid, does not change its mechanical state (does not unbalance).
  .
• Resultant force Is a force, the action of which on the body is equivalent to the action of the system of forces.
  .
• Moment of power Is a value that characterizes the rotational ability of a force.
• A couple of forces Is a system of two parallel, equal in magnitude, oppositely directed forces.
  Accepted designation:.
  Under the action of a pair of forces, the body will rotate.
• Axis force projection 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 line segment coincides with the positive direction of the axis.
• Force projection onto 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 evenly and rectilinearly.
  The uniform and rectilinear motion of a material point is motion by inertia. The state of equilibrium between 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 inertial motion, for example, uniform rotation of a rigid body around a fixed axis.
• Law 2. A solid body is in equilibrium under the action of two forces only if these forces are equal in magnitude and directed in opposite directions along the common line of action.
  These two forces are called balancing forces.
  In general, forces are called balancing if the rigid 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 drop counterbalancing forces.
  Consequence. Without violating the state of a rigid body, force can be transferred along its line of action to any point in the body.
  Two systems of forces are called equivalent if one of them can be replaced by another without violating the state of a rigid 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 the parallelogram built on these forces, and is directed along this
  diagonals.
  The modulus of the resultant is equal to:
  
• Law 5 (the 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 act - force applied to the body Band opposition - force applied to the body Aare not balanced, since they are attached 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, are necessary, but not sufficient for the corresponding non-solid.
• Law 7 (the law of release from ties). An unfree solid 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 constrains movement normal to the support surface. The reaction is directed perpendicular to the surface.
• Articulated movable support constrains 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 bar. The reaction will be directed along the line of the bar.
• Blind termination 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, which considers the general geometric properties of mechanical motion, as a process that occurs 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 geometric location 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 kinematic characteristics of a point
• Point trajectory
  In the vector frame of reference, the trajectory is described by the expression:.
  In the coordinate system, the trajectory is determined by the law of motion of a point and is described by the expressions z \u003d f (x, y) - in space, or y \u003d f (x) - in the plane.
  In the natural frame of reference, the trajectory is set 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 the time interval is called the average value of the speed in this time interval:.
  Taking the time interval as an infinitely small value, the speed value is obtained at a given time (instantaneous speed value): .
  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 quantity determines the rate of change of this quantity.
• Determining the speed of a point in a coordinate system
  Point coordinates change rates:
  .
  The modulus of the full speed of a point with a rectangular coordinate system will be:
  .
  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 frame of reference
  The speed of a point in the natural frame of reference 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 motion of the point and in the axes is determined by only one projection.

Rigid body kinematics
• In the kinematics of solids, two main tasks are solved:
  1) the task of movement and the determination of the kinematic characteristics of the body as a whole;
  2) determination of the kinematic characteristics of body points.
• The translational motion of a rigid body
  A translational motion is a motion in which a straight line drawn through two points of the body remains parallel to its original position.
  Theorem: during translational motion, all points of the body move along the same trajectories and at each moment of time have the same velocity and acceleration in magnitude and direction.
  Conclusion: the translational motion of a rigid body is determined by the motion of any of its points, in connection with which, the task and study of its motion is reduced to the kinematics of the point.
• Rotational movement of a rigid body around a fixed axis
  The rotational movement of a rigid body around a fixed axis is the movement 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 angle unit is radians. (Radian is the central angle of a circle whose arc length is equal to the radius, the total angle of the circle contains 2π radians.)
  The law of rotational motion of a body around a fixed axis.
  The angular velocity and angular acceleration of the body is determined by the differentiation method:
  - angular velocity, rad / s;
  - angular acceleration, rad / s².
  If you cut the body with a plane perpendicular to the axis, select the point on the axis of rotation FROM and an arbitrary point Mthen point M will describe around the point FROM circle radius R... During dt an elementary turn through an angle occurs, while 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 - this is a section of theoretical mechanics, which studies the mechanical movements of material bodies, depending on the reasons 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 body inertia. The unit of measure for mass is kilogram (kg).
• Material point Is a body with a mass, the dimensions of which are neglected when solving this problem.
• Center of gravity of the mechanical system - geometric point, the coordinates of which are determined by the formulas:
  
  Where m k, x k, y k, z k - mass and coordinates k-th point of the mechanical system, m Is the mass of the system.
  In a homogeneous gravity field, the position of the center of mass coincides with the position of the center of gravity.
• The 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 point's mass by the square of the point's distance 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 magnitude to the product of the point mass by the acceleration modulus and directed opposite to the acceleration vector:
• The force of inertia of a material body Is a vector quantity equal in magnitude to the product of body mass by the modulus of acceleration of the center of mass of the body and directed opposite to the vector of acceleration of the center of mass:,
  where is the acceleration of the center of mass of the body.
• Elementary Force Impulse Is a vector quantity equal to the product of the force vector by an infinitely small time interval dt:
  .
  The total impulse of force for Δt is equal to the integral of elementary impulses:
  .
• Elementary power work Is a scalar dAequal to scalar proi

Many university students face certain challenges when teaching basic technical disciplines such as strength of materials and theoretical mechanics in their course of study. This article will cover one such subject - the so-called technical mechanics.

Technical mechanics is the science that studies various mechanisms, their synthesis and analysis. In practice, this means a combination of three disciplines - resistance 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 all the variety of problems in theory, most often you can find problems from the section of kinematics and statics. These are tasks for the equilibrium of a flat frame, determination of the laws of motion of bodies and kinematic analysis of the lever mechanism.

To solve problems on the equilibrium of a flat frame, it is necessary to use the equilibrium equation of a plane system of forces:

The sum of the projections of all forces on the coordinate axes is zero and the sum of the moments of all forces relative to any point is zero. Solving these equations together, we determine the magnitude of the reactions of all supports of the flat frame.

In the tasks of determining the basic kinematic parameters of the motion 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 motion of a point are given by the trajectory equations:

The projections of the velocity of a point onto the coordinate axes are found by differentiating the corresponding equations:

Differentiating the velocity equations, we find the projection of the point acceleration. The tangent and normal accelerations, the radius of curvature of the trajectory are found graphically or analytically:

The kinematic analysis of the linkage is carried out according to the following scheme:

1. Splitting the mechanism into Assur groups
2. Building for each of the groups of plans of speeds and accelerations
3. Determination of the speeds and accelerations of all links and points of the mechanism.

Section two. Strength of materials

Resistance of materials is a rather complicated section for understanding, with many different tasks, most of which are solved according to their own method. In order to make it easier for students to solve them, most often in the course of applied mechanics they give elementary problems for 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 of 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, using the formulas for the equilibrium of a rigid body, we determine the values \u200b\u200bof internal forces at the boundaries of the sections, and, taking into account the cross-sectional area, internal stresses.

Based on the data obtained, we build graphs - diagrams, taking the axis of symmetry of the structure as the axis of the graph.

Torsion problems are similar to bending problems, except that torques are applied to the body instead of tensile forces. Taking this into account, it is necessary to repeat the stages of the calculation - dividing into sections, determining the twisting moments and twisting angles, and plotting diagrams.

In bending problems, it is necessary to calculate and determine the shear forces and bending moments for the 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 efforts.

After that, the bar is divided into sections, the boundaries of which will be the points of application of external forces. By considering the equilibrium of each section separately, shear forces and bending moments at the boundaries of the sections are determined. Based on the data obtained, diagrams are built.

Cross-sectional strength check is carried out as follows:

1. The location of the dangerous section is determined - the section where the greatest bending moments will act.
2. From the bending strength condition, the moment of resistance of the cross-section of the bar is determined.
3. The characteristic size of the section is determined - diameter, side length or profile number.

Section three. Machine parts

The section "Machine parts" combines all the tasks for calculating mechanisms operating in real conditions - it can be a conveyor drive or a gear transmission. The task is greatly facilitated by the fact that all formulas and calculation methods are given in reference books, and the student only needs to choose those of them that are suitable for a given mechanism.

Literature

1. Theoretical mechanics: Methodical instructions and test tasks for correspondence students of engineering, construction, transport, instrument-making specialties of higher educational institutions / Ed. prof. SM Targa, - M .: Higher school, 1989, fourth edition;
2. A. V. Darkov, G. S. Shpiro. "Strength of materials";
3. Chernavsky S.A. Course design of machine parts: Textbook. manual for students of engineering specialties of technical schools / S. A. Chernavsky, K. N. Bokov, I. M. Chernin and others - 2nd ed., revised. and add. - M. Mechanical Engineering, 1988 .-- 416 p .: ill.

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