In fine art, one of the main tasks is to convey movement. Movement visible to the eye is distinguished by the richness and variety of positions in space, directions, tilts and rotations of bodies or their parts in relation to each other (Fig. 1). Rest or balance is only a fixed moment of movement.

Fig 1. Examples of the movement of shapes in nature

Using visual means in one drawing it is impossible to convey any movement in space that takes place in a certain period of time from beginning to end; it is possible to convey only one moment from a whole series that makes up the movement. Therefore, it is necessary to find such a characteristic moment that would reveal this entire movement as fully as possible and would give an idea of ​​its beginning and end. Different genres of fine arts require the transfer of different aspects and types of movement.
In objects of architectural and construction practice, through proportions, the sequence of arrangement of volumes in vertical and horizontal directions, symmetry and asymmetry, color and texture, a certain rhythm of architectural forms, a sense of movement is conveyed (up, to the center, in depth, to the left, to the right), which has a greater value for creating an artistic image of a structure or ensemble. So, for example, the schematic drawing shows a fragment of a complex of structures with the main compositional direction of movement along the street, which is “disturbed” by the recess of the courtyard (court d'honneur) perpendicular to the street with a structure rising in the depths. A spectator on the street involuntarily turns his gaze to a new direction. inside the court d'honneur and upward, while experiencing a certain change of impressions (Fig. 2, a). The schematic drawing shows examples of interior space solutions. In Fig. 2,(5 the main compositional movement is directed along the space, to the center and upward.

Fig 2. Spatial direction of movement a - along the street, across and up: b - inside the building

The transfer of various types of movement in the visual arts requires high visual and general culture. The task of educational drawing is to give the basic simple concepts of movement and teach how to depict it.
For those beginning to study drawing on motionless or at rest bodies, it is important to determine the nature of the direction of the bodies and their parts relative to the ground, i.e. vertical and horizontal, as well as the direction of the parts in relation to each other. It should be noted that the concept of movement is also closely related to the concept of gravity: the weight and location of the center of gravity in relation to the support determine the stable or unstable state of an object.

Figure 3. Stable and unstable state of bodies depending on the center of gravity and support - amorphous, cube, cylinders, ball, camus and hemispheres

Schematic drawings (Fig. 3) illustrate the simplest types of movement that can be depicted: stable and unstable states, movement forward, backward, sideways, up, down, and various turns that occur during rotation.
The drawings of simple geometric bodies show examples of stable and unstable states depending on the position of the center of gravity in relation to the support. An amorphous body is at rest if the resultant force of gravity passes through the support. The cube is depicted in three positions. In the case of support on the entire face, the position is stable; in the case of support on an edge line or corner point, the position is unstable. In addition, stability depends on a number of additional factors: for example, of two vertically standing cylinders or cones with identical bases, the one whose height is smaller will be more stable. With the same height and base, a cone is more stable than a cylinder, etc. With a small support area, such as, for example, a ball lying on a plane, it is very easy to remove the body from a stable position; with a large support area this is more difficult to do.
If the body is in an unstable position, the feeling of instability will be stronger the further the resultant force of gravity passes from the support. The concept of stable and unstable position is associated with the concept of material work (Fig. 4).

Figure 4. Examples of structures whose stability is ensured by compression and tension of individual elements

The figures show various examples of the simplest structures in connection with the work of the material in compression and tension. In one case, stability is created by compressing structural elements (pillars and ceiling, arch and its prototype of two inclined beams). In other cases, a stable state is ensured by stretching the structural elements - cables (cable-stayed structures). In the body of a living person, the role of rigid structural elements is performed by bones, and the role of flexible elements is played by muscles. Muscle contraction changes the position of the bones in relation to each other. These internal movements, subject to the laws of statics and dynamics, determine the movement of individual parts and the entire human figure as a whole and determine changes in the visible muscle cover and bones. In complex structural bodies, where each element can change its position in relation to others, the general movement inevitably causes corresponding internal changes in each component part. When considering the human figure in various positions, this process becomes most clear (Fig. 5).

Fig 5. Examples of movement of the human eye, head, body

All four positions of the human figure shown in the figure are statically stable, however, the location of the center of gravity of the entire figure and its parts in relation to the support causes movements of the structural parts inside the figure that are characteristic for each case. Without understanding this, an image of the general movement of the human figure cannot be created. With simultaneous support on both legs, the resultant force from the center of gravity passes within the limits of the support of both legs, while all parts of the figure are located symmetrically relative to the midline. When supporting on one leg, the skew of the pelvis and the curvature of the spine allow the parts of the body to be positioned in such a way that the center of gravity is projected onto the area of ​​the footprint of the supporting leg. Double support - on the legs and the tree trunk - causes even more complex displacements within the human figure, associated with the location of the center of gravity, supports and the internal work of the muscles. Rice. 5 illustrates various examples of movement of the head changing its position in relation to the body - upright position, tilting forward, backward and turning. It also shows the different positions of the pupil of the eye when the direction of gaze changes. The examples given convince us that without a comprehensive understanding of movement, it is impossible to fully solve the problems of educational drawing, and even more so the complex creative problems of architectural and construction practice.

1. Mechanical movement - change in the position of the body or its individual parts in space over time.

The internal structure of moving bodies and their chemical composition do not affect mechanical movement. To describe the motion of real bodies depending on the conditions of the problem, they use various models: material point, absolutely rigid body, absolutely elastic body, absolutely inelastic body, etc.

A material point is a body whose dimensions and shape can be neglected in the conditions of this problem. In what follows, instead of the term “material point” we will use the term “point”. The same body can be reduced to a material point in one problem, and it is necessary to take into account its dimensions in the conditions of another problem. For example, the movement of an airplane flying over the Earth can be calculated by considering it as a material point. And when calculating the air flow around the wing of the same aircraft, it is necessary to take into account the shape and dimensions of the wing.

Any extended body can be considered as a system of material points.

An absolutely rigid body (a.r.t.) is a body whose deformation can be neglected under the conditions of a given problem. A.t.t. can be considered as a system of material points rigidly interconnected, because the distance between them does not change during any interactions.

Absolutely elasticbody - a body whose deformation obeys Hooke's law (see § 2.2.2.), and after the cessation of force action it completely restores its original size and shape.

An absolutely inelastic body is a body that, after the cessation of the force applied to it, does not recover, but completely retains its deformed state.

2. To determine the position of a body in space and time, it is necessary to introduce the concept reference systems. The choice of reference system is arbitrary.

A reference system is a body or a group of bodies that are considered to be conditionally motionless and equipped with a time-keeping device (clock, stopwatch, etc.), relative to which the movement of a given body is considered.

A stationary body (or group of bodies) is called reference body and for the convenience of describing the movement, it is associated with coordinate system(Cartesian, polar, cylindrical, etc.).

Let us choose the Cartesian rectangular XYZ system as the coordinate system (see details). The position of point C in space can be determined by coordinates x, y, z (Figure 1).

However, the position of the same point in space can be specified using a single vector quantity
r = r(x, y, z), called the radius vector of point C (Figure 1).

3. The line that a body describes during its movement is called a trajectory. Based on the type of movement trajectory, it can be divided into straight and curved. The trajectory depends on the choice of reference system. Thus, the trajectory of movement of the aircraft propeller points relative to the pilot is a circle, and relative to the Earth is a helical line. Another example: what is the trajectory of the tip of the turntable relative to the record? player body? pickup bodies? The answers are: spiral, circular arc, state of rest (the needle is motionless).

2.1.2. Kinematic equations of motion. Path length and displacement vector

1. When a body moves relative to the selected coordinate system, its position changes over time. The motion of a material point will be completely determined if continuous and single-valued functions of time t are given:

x = x(t), y = y(t), z = z(t).

These equations describe the change in the coordinates of a point over time and are called kinematic equations of motion.

2. A path is part of the trajectory traversed by a body over a certain period of time. The moment of time t 0 from which its counting begins is called the initial moment of time, usually t 0 =0 due to the arbitrary choice of the starting point of time.

The length of the path is the sum of the lengths of all sections of the trajectory. The path length cannot be a negative value; it is always positive. For example, a material point moved from trajectory point C first to point A, and then to point B (Figure 1). The length of its path is equal to the sum of the lengths of arc CA and arc AB.

2.1.3. Kinematic characteristics. Speed

1. To characterize the speed of movement of bodies in physics, the concept is introduced speed. Speed ​​is a vector, which means it is characterized by magnitude, direction, and point of application.

Let's consider movement along the X axis. The position of the point will be determined by the change in the X coordinate over time.

If during the time ∆ the point has moved to ∆r, then the value is the average speed of movement:
.

The average speed of a moving body is a vector equal to the ratio of the displacement vector to the amount of time during which this displacement occurred.

The average velocity module is a physical quantity numerically equal to the change in path per unit time.

2. To determine the speed at a given time, instantaneous speed, you need to consider the time interval ∆ t→0, then

Using the concept of derivative, we can write for speed

The speed of a body at a given time is called instantaneous speed ( or simply speed).

Vector V instantaneous speed is directed tangentially to the trajectory in the direction of body motion.

2.1.4. Kinematic characteristics. Acceleration

1. The rate of change of the velocity vector is characterized by a quantity called acceleration. Acceleration can occur both due to a change in the magnitude of the speed and due to a change in the direction of the speed.

Let the speed of the body at time t be equal to v 1 , and after a period of time ∆ t at time t + ∆ t is equal v 2 , velocity vector increment per ∆ t equals ∆ v.

Average acceleration bodies in the time interval from t to t + ∆ t is called a vector a Wed, equal to the ratio of the velocity vector increment ∆ v to a period of time ∆ t:

Average acceleration is a physical quantity numerically equal to the change in speed per unit time.

2. To determine the acceleration at a given time, i.e. instantaneous acceleration, we need to consider a small time interval ∆ t→0. Then instantaneous acceleration vector equal to the limit of the average acceleration vector as the time interval tends ∆ t to zero:

Using the concept of derivative, we can give the following definition for acceleration:
Acceleration(or instantaneous acceleration) of a body is called a vector quantity A, equal to the first time derivative of the body speedvor the second time derivative of the path.

3. When a point rotates around a circle, its speed can change in magnitude and direction (Figure 2)

In Figure 2, in position 1, the speed of the point v 1, in position 2 point speed v 2 . Speed ​​module v 2 more speed module v 1 , ∆v- velocity change vector ∆v = v 2 -v 1

The rotating point has tangential acceleration, equal to a τ =dv/dt, it changes the speed in magnitude and is directed tangentially to the trajectory; And normal acceleration, equal to a n = v 2 /R, it changes the direction of speed and is directed along the radius of the circle (R) (see Figure 3)

The total acceleration vector is equal to, i.e. it can be represented as the sum of tangential vectors aτ and normal a n accelerations. The total acceleration module is equal to:

2.1.5. Translational and rotational motion of an absolutely rigid body

1. So far we have been talking about the nature of the movement, the trajectory, the kinematic characteristics, but the moving body itself has not been considered. Example. The car is moving. He is a complex body. The movements of its body and wheels are different. If the body is complex, then the question arises: to the movement of which parts of the body do the concepts of path, speed, acceleration, introduced earlier, apply?

Before answering this question, it is necessary to identify the forms of mechanical movement. No matter how complex the movement of the body is, it can be reduced to two main ones: translational movement and rotation around fixed axis. Oscillatory motion will be considered separately. In the car example, the body of the car moves forward. The car itself is a body that can be considered using the absolutely rigid body model (a.r.t.). For brevity, we will call an absolutely rigid body simply a rigid body.

Translational motion of a rigid body is a motion in which any straight line drawn between its two points remains parallel to itself during motion.

Translational motion may not be linear motion.

Examples. 1) In the Ferris Wheel attraction, the cabins - cradles in which people sit, move progressively. 2) If a glass of water is moved along the trajectory shown in Figure 5 so that the surface of the water and the guide of the glass make a right angle, then the movement of the glass is not rectilinear, but translational. The straight line AB remains parallel to itself as the glass moves.

A feature of the translational motion of a rigid body is that all points of the body describe the same trajectory, passing over certain periods of time ∆ t are the same paths and have the same speeds at any given time. Therefore, the kinematic consideration of the translational motion of a rigid body is reduced to the study of the motion of any of its points. The translational motion of a body can be reduced to the motion of a material point. In dynamics, this point is usually taken to be body center of mass. Kinematic characteristics and kinematic equations introduced for a material point also describe the translational motion of a rigid body.

2. The movement of the car's wheels is different from the movement of the body. Points on the wheel located at different distances from its axis describe different trajectories, travel different paths and have different speeds. The further a point is from the wheel axis, the greater its speed, the greater the distance it travels in a certain period of time. The movement in which the wheels of a car participate is called rotational. It is clear that the model of a material point is not suitable for describing the rotation of a real body. But here, instead of a real body (for example, car wheels with deformable tires, etc.), a physical model is used - an absolutely rigid body.

Rotational motion of a rigid body is a movement when all points of the body describe circles, the centers of which lie on a straight line, called the axis of rotation and perpendicular to the planes in which the points of the body rotate(Figure 5).

Since for different points of a rotating body the trajectories, paths, and velocities are different, the question arises: is it possible to find physical quantities that would have the same values ​​for all points of the rotating body? Yes, it turns out there are such quantities, they are called corner.

A rigid body rotating around a fixed axis has one degree of freedom; its position in space is completely determined by the value of the rotation angle ∆φ from a certain initial position (Figure 5). All points of the rigid body will rotate during a period of time ∆ by an angle ∆φ.

For short periods of time, when the angles of rotation are small, they can be considered as vectors, although not quite ordinary ones. Vector of the elementary (infinitesimal) rotation angle ∆ φ directed along the axis of rotation along right gimlet rule, its module is equal to the rotation angle (Figure 5). The vector ∆φ is called angular movement.

Right gimlet rule is as follows:

If the handle of the right gimlet rotates together with the body (point), then the translational movement of the gimlet coincides with the direction ∆ φ .

Another wording of the rule: From the end of the vector ∆φ it is clear that the movement points (bodies) occurs counterclockwise.

The position of the body at any time t is determined kinematic equation rotational motion ∆φ = ∆φ(t).

3. Angular velocity is used to characterize the speed of rotation.

Average angular velocity is a physical quantity equal to the ratio of the angular movement to the period of time during which this movement occurred

The limit to which the average angular velocity tends at ∆→0 is called instantaneous angular velocity bodies at a given moment in time or simply angular rotation speed solid body (point).

Angular velocity is equal to the first derivative of angular displacement with respect to time. The direction of the instantaneous angular velocity is determined by the right gimlet rule and coincides with the direction ∆ φ (Figure 6). The kinematic equation of motion for angular velocity has the form ω = ω (t).

4. For characteristics rate of change angular velocity of the body during uneven rotation, a vector is introduced angular accelerationβ , equal to the first derivative of its angular velocity ω by time t.

The average angular acceleration is the magnitude of the ratio of the change in angular velocity ∆ω to a period of time∆t, during which this change occurred β av = ∆ ω /∆t

The angular acceleration vector is directed along the axis of rotation and coincides with the direction of angular velocity if the movement is accelerated, and is opposite to it if the rotation is slow (Figure 6).

5. During the rotational motion of a rigid body, all its points move so that the rotational characteristics (angular displacement, angular velocity, angular acceleration) are the same for them. And the linear characteristics of movement depend on the distance of the point to the axis of rotation.

The relationship between these quantities v, ω , r is given by the following relation:

v = [ω ∙r],

those. linear speed v any point C of a rigid body rotating around a fixed axis with angular velocity ω , is equal to the vector product ω to the radius vector r point C relative to an arbitrary point O on the axis of rotation.

A similar relationship exists between the linear and angular accelerations of a rotating point of a rigid body:

A= [β ∙r].

2.1.6. Relationship between kinematic characteristics for different types of movements

According to the dependence of speed and acceleration on time, all mechanical movements are divided into uniform, uniform(uniformly accelerated and equally decelerated) and uneven.

Let us consider the kinematic characteristics and kinematic equations introduced in the previous paragraphs for different types of movements.

1. Straight-line movement

Rectilinear uniform motion.

The direction of movement is set by the OX axis.

Acceleration a = 0 (a n = 0, and τ = 0), speed v = const, path s = v∙t, coordinate x = x 0 v∙t, where x 0 is the initial coordinate of the body on the OX axis.

Path is always a positive quantity. The coordinate can be both positive and negative, therefore, in the equation that specifies the dependence of the coordinate on time, the value v∙t in the equation is preceded by a plus sign if the direction of the OX axis and the direction of velocity coincide, and a minus sign if they are in opposite directions.

Rectilinear uniform motion.

Acceleration a = a τ = const, a n = 0, speed ,

path , coordinate .

Before the value (at) in the kinematic equation for speed, the plus sign corresponds to uniformly accelerated motion, and the minus sign corresponds to uniformly slow motion. This remark is also true for the kinematic equation of the path; different signs in front of the quantities (at 2 /2) correspond to different types of uniform motion.

In the equation for the coordinate, the sign in front of (v 0 t) can be a plus if the directions of v 0 and the OX axis coincide, and a minus if they are directed in different directions.

Different signs in front of the quantities correspond to uniformly accelerated or uniformly decelerated movements.

Rectilinear uneven movement.

Acceleration a = a τ >≠ const, and n = 0,

speed , path .

2. Forward movement

To describe translational motion, you can use the laws given in §2.1.6. (clause 2) or §2.1.4. (point 3). The use of certain laws to describe translational motion depends on its trajectory. For a straight trajectory, the formulas from §2.1.6 are used. (point 2), for curvilinear - §2.1.4. (point 3).

3. Rotational movement

Note that the solution to all problems involving the rotational motion of a rigid body around a fixed axis is similar in form to problems involving the rectilinear motion of a point. It is enough to replace the linear quantities s, v x, a x with the corresponding angular quantities φ, ω, β, and we will obtain all the laws and relationships for a rotating body.

Uniform rotation around the circumference

(R is the radius of the circle) .

Acceleration: complete a = a n, normal ,

tangential and τ = 0, cornerβ = 0.

Speed: angular ω = const, linear v = ωR = const.

Angle of rotation∆φ = ∆φ 0 + ωt, ∆φ 0 - initial value of the angle. The rotation angle is a positive value (analogous to a path).

Rotation period is the time interval T during which a body, uniformly rotating with an angular velocity ω, makes one revolution around the axis of rotation. In this case, the body rotates through an angle of 2π.

Rotation frequency shows the number of revolutions made by a body per unit time during uniform rotation with angular velocity ω:

Uniform rotation around a circle

Acceleration: angularβ = const,

Graphical representation
uniform rectilinear motion

Speed ​​graph shows how the speed of a body changes over time. In rectilinear uniform motion, the speed does not change over time. Therefore, the graph of the speed of such movement is a straight line parallel to the abscissa axis (time axis). In Fig. Figure 6 shows graphs of the speed of two bodies. Graph 1 refers to the case when the body moves in the positive direction of the O x axis (the projection of the body's velocity is positive), graph 2 - to the case when the body moves against the positive direction of the O x axis (the projection of velocity is negative). Using the velocity graph, you can determine the distance traveled by the body (If the body does not change the direction of its movement, the length of the path is equal to the modulus of its displacement).

2.Graph of body coordinates versus time which is otherwise called traffic schedule

In Fig. graphs of the motion of two bodies are shown. The body whose graph is line 1 moves in the positive direction of the O x axis, and the body whose motion graph is line 2 moves in the opposite direction to the positive direction of the O x axis.

3.Path graph

The graph is a straight line. This line passes through the origin of coordinates (Fig.). The greater the speed of the body, the greater the angle of inclination of this straight line to the abscissa axis. In Fig. graphs 1 and 2 of the path of two bodies are shown. From this figure it is clear that during the same time t, body 1, which has a higher speed than body 2, travels a longer distance (s 1 > s 2).

Rectilinear uniformly accelerated motion is the simplest type of uneven motion, in which a body moves along a straight line, and its speed changes equally over any equal periods of time.

Uniformly accelerated motion is motion with constant acceleration.

The acceleration of a body during its uniformly accelerated motion is a quantity equal to the ratio of the change in speed to the period of time during which this change occurred:

→ →
→ v – v 0
a = ---
t

You can calculate the acceleration of a body moving rectilinearly and uniformly accelerated using an equation that includes projections of the acceleration and velocity vectors:

v x – v 0x
a x = ---
t

SI unit of acceleration: 1 m/s 2 .

Speed ​​of rectilinear uniformly accelerated motion.

v x = v 0x + a x t

where v 0x is the projection of the initial velocity, a x is the projection of acceleration, t is time.

→
If at the initial moment the body was at rest, then v 0 = 0. For this case, the formula takes the following form:

Displacement during uniform linear motion S x =V 0 x t + a x t^2/2

Coordinate at RUPD x=x 0 + V 0 x t + a x t^2/2

Graphical representation
uniformly accelerated linear motion

Speed ​​graph

The speed graph is a straight line. If the body moves with a certain initial speed, this straight line intersects the ordinate axis at point v 0x. If the initial velocity of the body is zero, the velocity graph passes through the origin. The velocity graphs of rectilinear uniformly accelerated motion are shown in Fig. . In this figure, graphs 1 and 2 correspond to movement with a positive projection of acceleration on the O x axis (speed increases), and graph 3 corresponds to movement with a negative projection of acceleration (speed decreases). Graph 2 corresponds to movement without an initial speed, and graphs 1 and 3 to movement with an initial speed v ox. The angle of inclination a of the graph to the abscissa axis depends on the acceleration of the body. Using velocity graphs, you can determine the distance traveled by a body during a period of time t.

The path covered in rectilinear uniformly accelerated motion with an initial speed is numerically equal to the area of ​​the trapezoid limited by the velocity graph, the coordinate axes and the ordinate corresponding to the value of the body’s velocity at time t.

Graph of coordinates versus time (motion graph)

Let the body move uniformly accelerated in the positive direction O x of the chosen coordinate system. Then the equation of motion of the body has the form:

x=x 0 +v 0x t+a x t 2 /2. (1)

Expression (1) corresponds to the functional dependence y = ax 2 + bx + c (square trinomial), known from the mathematics course. In the case we are considering
a=|a x |/2, b=|v 0x |, c=|x 0 |.

Path graph

In uniformly accelerated rectilinear motion, the time dependence of the path is expressed by the formulas

s=v 0 t+at 2 /2, s= at 2 /2 (for v 0 =0).

As can be seen from these formulas, this dependence is quadratic. It also follows from both formulas that s = 0 at t = 0. Consequently, the graph of the path of rectilinear uniformly accelerated motion is a branch of a parabola. In Fig. shows the path graph for v 0 =0.

Acceleration graph

Acceleration graph – dependence of the projection of acceleration on time:

rectilinear uniform movement. Graphic performance uniform rectilinear movement. 4. Instantaneous speed. Addition...

Lesson Topic: "Material point. Reference system" Objectives: to give an idea of ​​kinematics

Lesson

Definition uniform straightforward movement. - What is called speed? uniform movement? - Name the unit of speed movement in... projection of the velocity vector versus time movement U (O. 2. Graphic performance movement. - At point C...

Methods of movement of units and their assessment

There are three main types of movement of units (in the direction of working strokes relative to the boundaries of the working area): driving (working strokes along one of the sides of the site), diagonal (at an angle, diagonally to the sides of the site, a diagonal-cross variety) and circular (working stroke along all sides of a plot or paddock, a distinction is made between circular movement towards the center or towards the periphery of the plot).

Circular modes of movement are presented in Figure 8.4. The circular movement is most often performed in a collapsing spiral, from the periphery to the center (Fig. 8.4a), in this case there is no need to mark the central part. The method (Fig. 8.4b) is distinguished by the presence of internal turning strips, which are either prepared in advance (mowed, removed) or sealed after processing the paddock or area. Method (Fig. 8.4c) - processing from the center, in this case you need to find the center and mark the location and length of the first pass.

Figure 8.4 – Varieties of circular motion methods:

a - with a coiled spiral without turning off the working parts and headlands; b - the same, but with internal turning lanes; c - in an unfolding spiral, envelope method

Figure 8.5 shows diagonal movement methods for working areas or pens with a shape close to a square. If the pen has the shape of an elongated rectangle, then it is divided into parts close to a square shape. If turning lanes are needed here, they are built along all sides of the site.

Figure 8.6 shows the most common rutting methods of movement. The method of moving by overlapping is loopless, however, it requires frequent marking of the field; it is better to use it when processing an already marked field (in the form of rows of plants, when you just need to count the required number of rows). The shuttle method of movement is monotonous and easy to perform. The waddling and waddling methods of movement are most common (alternating across paddocks) in plowing. Their combined use on one paddock allows you to obtain a loop-free method of movement when plowing.

Various methods of moving units are compared in terms of the quality of the technological operation, ease of maintenance, operational safety, and the cost of preparing the work area. All indicators are closely related to the work performed, the size of the work area, the composition of the unit and its kinematic characteristics. It is more convenient to consider all this when studying the technology of performing individual agricultural work.

Figure 8.6 – Rutting modes of movement:

a - overlap; b - shuttle; c - dump; g - waddle

One of the main assessments of movement methods that affect the performance of units is the coefficient of working strokes or the degree of path utilization

, (8.6)

where ΣL р and ΣL x - the total length of working and idle strokes in the paddock; n p and n x - the number of working and idle passes in the paddock.

For all rut modes of movement, L р =L uch -2E, and n р =n x =С/Вρ. The length of idle passes must include not only the length of the path on turns, but also additional passes associated with sealing headlands, passes with an incomplete working width, drives and crossings on the work site.

With loopless racing modes of movement, the average length of the idle stroke L x.av =1.14ρ y +0.5С+2 e and hence the coefficient of working strokes

. (8.7)

For loop modes of movement (dumping, waddling) in areas up to 2ρ y wide, loop turns take place, their number n loops = 2ρ y / B ρ. The length of the loop idle strokes on the paddock would be ΣL x loops = (2ρ y / B ρ)(6ρ y + e). If these turns were made without loops (with a section width of 2ρ y), then their total length ΣL xbesp =(1.14ρ y +2 e+ρ y)2ρ y /B ρ . Then the difference in the no-load length will be ΔL x =3.86ρ y 2ρ y B ρ ≈ 8ρ y 2 /B ρ. Taking into account (8.6) and relating ΔL x to the number of passes n p =C/8ρ y, we obtain the coefficient of working strokes for loop (dump, waddle) modes of movement

For the shuttle mode of movement, all idle strokes are the same L x =6ρ y +2 e and stroke ratio

. (8.9)

The optimal (in terms of productivity) paddock width C opt is determined from the condition of the minimum total length of idle strokes or the maximum coefficient of working strokes on the site.

The total length of idling strokes in the section S h.uch =ΣL x (C uch /C), then for the loop mode of movement, taking into account (8.7)

Let's take the first derivative for S x uch along the width of the pen C and equate it to zero

,

The minimum (if feasible) paddock width (C min) is applicable only to non-loop methods (for example, the overlap method, the tumble-wadle combination). Loopless turning is possible only with X≥2ρ y; if the paddock contains three or four such minimum plots, then the minimum width of the paddock for the loopless method of movement will be equal to six or eight conditional turning radii of the unit.

For loopless methods of movement, as a rule, the calculated value of C opt is less than C min and, therefore, physically cannot be implemented. Therefore, for loopless methods, C opt is usually not calculated, but is taken equal to C min.

The coefficient of working strokes for loop motion methods (C=C opt) is determined by the formula

, (8.12)

and for loopless modes of movement (С=С min) is equal to

. (8.13)

When choosing one or another method of movement, one must proceed primarily from agrotechnical requirements - quality of work, ease of maintenance, the possibility of reducing auxiliary operations, etc. If these conditions allow the use of different methods of movement, the one that gives a higher value of φ should be chosen.

L p has the greatest influence on the value of the working stroke coefficient. The larger the turning radius ρ y, the smaller φ. The width of the pen C has almost no effect on φ with the shuttle method of movement. Deviation from C opt and C min in the direction of increase in order to ensure a whole number of passes of the unit on the paddock, convenience of dividing into paddocks, etc. does not provide a significant reduction in φ. In case of deviation from C opt in the direction of decreasing the width of the paddock, the value of φ decreases significantly.

Questions for self-control of knowledge

1. What is meant by unit kinematics?

2. List the kinematic characteristics of the MTA and describe them.

3. What types of MTA turns do you know?

4. Write down the formula to calculate the length of the piriform turn.

5. Write down a formula to calculate the minimum headland width for different types of turns.

6. What types of MTA traffic do you know?

7. Name the methods of movement of the MTA during the rutting type of movement.

8. Draw the methods of MTA movement “overlap”, “shuttle”, “dump” and “waddle”.

9. Write down the formula for calculating the MTA working stroke ratio.

10. Write down the formula for calculating the optimal width of the corral for the loopless method of MTA movement.

For greater clarity, movement can be described using graphs. The graph shows how one quantity changes when another quantity on which the first depends changes.

To construct a graph, both quantities on the selected scale are plotted along the coordinate axes. If the time elapsed from the beginning of time is plotted along the horizontal axis (abscissa axis), and the coordinate values ​​of the body are plotted along the vertical axis (ordinate axis), the resulting graph will express the dependence of the body coordinates on time (it is also called a motion graph).

Let us assume that the body moves uniformly along the X axis (Fig. 29). At moments of time, etc., the body is respectively in positions measured by coordinates (point A), .

This means that only its coordinate changes. In order to obtain a graph of the body’s motion, we will plot the values ​​along the vertical axis, and the time values ​​along the horizontal axis. The motion graph is a straight line shown in Figure 30. This means that the coordinate depends linearly from time.

The graph of the body’s coordinates versus time (Fig. 30) should not be confused with the trajectory of the body’s movement - a straight line, at all points of which the body visited during its movement (see Fig. 29).

Motion graphs provide a complete solution to the problem of mechanics in the case of rectilinear motion of a body, since they allow one to find the position of the body at any moment in time, including at moments in time preceding the initial moment (assuming that the body was moving before the start of time). Continuing the graph shown in Figure 29 in the direction opposite to the positive direction of the time axis, we, for example, find that the body 3 seconds before it ended up at point A was at the origin of the coordinate

By looking at the graphs of the dependence of coordinates on time, one can judge the speed of movement. It is clear that the steeper the graph, i.e., the greater the angle between it and the time axis, the greater the speed (the greater this angle, the greater the change in coordinates at the same time).

Figure 31 shows several motion graphs at different speeds. Graphs 1, 2 and 3 show that bodies move along the X axis in the positive direction. A body whose motion graph is line 4 moves in the direction opposite to the direction of the X axis. From the motion graphs, one can find the movements of a moving body over any period of time.

From Figure 31 it is clear, for example, that body 3, during the time between 1 and 5 seconds, made a movement in the positive direction, equal in absolute value to 2 m, and body 4 during the same time made a movement in the negative direction, equal to 4 m in absolute value.

Along with motion graphs, speed graphs are often used. They are obtained by plotting the velocity projection along the coordinate axis

bodies, and the x-axis is still time. Such graphs show how speed changes over time, that is, how speed depends on time. In the case of rectilinear uniform motion, this “dependence” is that the speed does not change over time. Therefore, the speed graph is a straight line parallel to the time axis (Fig. 32). The graph in this figure is for the case where the body is moving towards the positive direction of the X-axis. Graph II is for the case where the body is moving in the opposite direction (since the velocity projection is negative).

Using the velocity graph, you can also find out the absolute value of the movement of a body over a given period of time. It is numerically equal to the area of ​​the shaded rectangle (Fig. 33): the upper one if the body is moving in the positive direction, and the lower one in the opposite case. Indeed, the area of ​​a rectangle is equal to the product of its sides. But one of the sides is numerically equal to time and the other - to speed. And their product is exactly equal to the absolute value of the body’s displacement.

Exercise 6

1. What movement does the graph shown by the dotted line in Figure 31 correspond to?

2. Using graphs (see Fig. 31), find the distance between bodies 2 and 4 at time sec.

3. Using the graph shown in Figure 30, determine the magnitude and direction of the velocity.