Mechanical movement of a body (point) is the change in its position in space relative to other bodies over time.

Types of movements:

A) Uniform rectilinear motion of a material point: Initial conditions

. Initial conditions

G) Harmonic oscillatory motion. An important case of mechanical motion is oscillations, in which the parameters of a point’s movement (coordinates, speed, acceleration) are repeated at certain intervals.

ABOUT scriptures of the movement . There are various ways to describe the movement of bodies. With the coordinate method specifying the position of a body in a Cartesian coordinate system, the movement of a material point is determined by three functions expressing the dependence of coordinates on time:

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

This dependence of coordinates on time is called the law of motion (or equation of motion).

With the vector method the position of a point in space is determined at any time by the radius vector r= r(t) , drawn from the origin to a point.

There is another way to determine the position of a material point in space for a given trajectory of its movement: using a curvilinear coordinate l(t) .

All three methods of describing the motion of a material point are equivalent; the choice of any of them is determined by considerations of the simplicity of the resulting equations of motion and the clarity of the description.

Under reference system understand a reference body, which is conventionally considered motionless, a coordinate system associated with the reference body, and a clock, also associated with the reference body. In kinematics, the reference system is selected in accordance with the specific conditions of the problem of describing the motion of a body.

2. Trajectory of movement. Distance traveled. Kinematic law of motion.

The line along which a certain point of the body moves is called trajectorymovement this point.

The length of the trajectory section traversed by a point during its movement is called the path traveled .

The change in radius vector over time is called kinematic law :
In this case, the coordinates of the points will be coordinates in time: x= x(t), y= y(t) Andz= z(t).

In curvilinear motion, the path is greater than the displacement modulus, since the length of the arc is always greater than the length of the chord contracting it

The vector drawn from the initial position of the moving point to its position at a given time (increment of the radius vector of the point over the considered period of time) is called moving. The resulting displacement is equal to the vector sum of successive displacements.

During rectilinear movement, the displacement vector coincides with the corresponding section of the trajectory, and the displacement module is equal to the distance traveled.

3. Speed. Average speed. Velocity projections.

Speed - speed of change of coordinates. When a body (material point) moves, we are interested not only in its position in the chosen reference system, but also in the law of motion, i.e., the dependence of the radius vector on time. Let the moment in time corresponds to the radius vector a moving point, and a close moment in time - radius vector . Then in a short period of time
the point will make a small displacement equal to

To characterize the movement of a body, the concept is introduced average speed his movements:
This quantity is a vector quantity, coinciding in direction with the vector
. With unlimited reduction Δt the average speed tends to a limiting value called instantaneous speed :

Velocity projections.

A) Uniform linear motion of a material point:
Initial conditions

B) Uniformly accelerated linear motion of a material point:
. Initial conditions

B) Movement of a body along a circular arc with a constant absolute speed:

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 linearly depends 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.

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.

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 movement methods 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 the idle 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 р 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.