Determination of the velocities of points of a plane figure. In the direction of speed

Another simple and visual method for determining the velocities of points of a flat figure (or a body in plane motion) is based on the concept of an instantaneous center of velocities.

The instantaneous velocity center (IVC) is the point of a flat figure whose velocity at a given moment in time is zero.

If a figure moves non-progressively, then such a point at each moment of time t exists and, moreover, is the only one. Let at a moment in time t points A And IN the planes of the figure have speeds and , non-parallel to each other (Fig. 2.21.). Then point R, lying at the intersection of perpendiculars Ahh to the vector and Bb to the vector , and will be the instantaneous velocity center, since .

Figure 2.21

In fact, if , then according to the velocity projection theorem, the vector must be both perpendicular and AR(since ), and VR(since) which is impossible. From the same theorem it is clear that no other point of the figure at this moment in time can have a speed equal to zero.

If now at the moment of time t take a point R behind the pole. Then the speed of the point A will

and so on for any point of the figure.

It also follows from this that and , then

= ,

(2.54)

those. What the velocities of the points of a flat figure are proportional to their distance from the instantaneous velocity center.

The results obtained lead to the following conclusions:

1. To determine the instantaneous center of velocities, you only need to know the directions of velocities, for example, And some two points A and B of a plane figure.

2. To determine the speed of any point of a flat figure, you need to know the magnitude and direction of the speed of any one point A of the figure and the direction of the speed of its other point B.

3. Angular velocity of a flat figure is equal at each moment of time to the ratio of the speed of any point of the figure to its distance from the instantaneous center of velocities P:

Let's consider some special cases of defining the MCS, which will help solve theoretical mechanics.

1. If plane-parallel motion is carried out by rolling without sliding of one cylindrical body along the surface of another stationary one, then the point R of a rolling body touching a stationary surface (Fig. 2.22), at a given moment of time, due to the absence of sliding, has a speed equal to zero (), and therefore is an instantaneous center of velocities.

Figure 2.22

2. If the speed of the points A And IN flat figures are parallel to each other, and the line AB is not perpendicular (Fig. 2.23, a), then the instantaneous center of velocities lies at infinity and the velocities of all points // . In this case, from the theorem on velocity projections it follows that, i.e. , in this case the figure has an instantaneous translational movement. which gives.

3.5.1. Pole method

Since the motion of a flat figure can be considered as a composite of translational, when all points of the figure move in the same way as the pole A with speed , and rotational motion around the pole, then the speed of any point IN we define the figures by the vector sum of velocities (Fig. 23).

, (65)

where is the speed of the pole of the point A;

Point speed IN when rotating a figure around the pole of a point A(if we consider it motionless) is numerically equal

IN perpendicular VA in the direction of rotation of the angular velocity (Fig. 23).

Numerical value of the point speed IN we determine by the cosine theorem

where is the angle between the vectors and , О .

Equality of projections is a consequence of the constant distance between points A And IN, belonging to a rigid body, therefore the equality will be valid for any motion of a rigid body.

3.5.2. Instantaneous velocity center (IVC) method

The instantaneous center of velocities is the point R a flat figure whose speed at a given time is zero. The velocities of all other points of a flat figure at a given time are determined as if the movement of the figure were rotational relative to the point R(Fig. 25).

Fig.25.

According to the pole method, point speed IN will be equal

. (69)

Since the pole speed (MPS) of the point R is equal to zero (), then

The velocity vector is directed from the point IN perpendicular VR in the direction of rotation of the angular velocity w.

A similar equality can be presented for all points of a plane figure, thus, the velocities of points of a plane figure are proportional to their distances to the MCS.

To determine the position (MCS) of a flat figure, you need to know the direction of the lines along which the velocity vectors of the points act A And IN( And ). The MCS for a given figure will be located at the intersection point of the perpendiculars restored to these lines.

To find the speed of a point IN, according to Fig. 25, you need to know the speed of the point A. Then the angular velocity of the figure at a given time will be

Where AR– point distance A to the point R, is determined according to the initial data.

Angular velocity under the influence of velocity relative to the pole of a point R directed clockwise.

Point speed IN at a given moment will be

Point velocity vector IN() is directed perpendicular to the line RV in the direction of rotation of the angular velocity w (Fig. 25).

3.5.2.1. The concept of centroids

The trajectory that the MCS describes together with a moving figure is called a moving centroid (for example, when a wheel moves along a surface without sliding (Table 2), the moving centroid is the outer circumference of the wheel).

Geometric locus of the MCS, point positions R on a stationary plane is called a stationary centroid (when a wheel moves on a surface without sliding (see Table 2), the stationary centroid is the stationary surface on which the wheel rolls).

3.5.2.2. Special cases of MCS

Table 2.

Instantaneous translational movement of the link AB	Movement of the wheel on the surface (without sliding)	Moving block movement
Dot IN moves in a straight line x-x, therefore, the speed V B directed along the axis, draw perpendicular to the axis x-x. Since perpendicular lines do not intersect, then the link AB is in instantaneous translational motion, the velocities of all points of this link are equal, the MCS is at infinity, .	The MCS is located at the point of contact of the wheel with a stationary surface on which the wheel is rolling, point R. The angular speed of the wheel will be . Point speeds IN, WITH	MCS (point R) is located at the intersection point of the segment AB and a straight line passing through the ends of the vectors and . Determining the position of a point R. Block angular velocity

5) Forward movement. Examples.

Determination of the rotational motion of a body around a fixed axis.

Equation of rotational motion.

- a movement in which all its points move in planes perpendicular to some fixed line and describe circles with centers lying on this line, called the axis of rotation.

The motion is given by the law of change in the dihedral angle φ (angle of rotation), formed by the fixed plane P passing through the axis of rotation and the plane Q rigidly connected to the body:

Angular velocity is a quantity that characterizes the speed of change in the angle of rotation.

Angular acceleration is a quantity characterizing the rate of change in angular velocity.

Determining the speed of any point on a flat figure.

1 way to determine speeds is through vectors. The speed of any point on a flat figure is equal to the geometric sum of the speed of the pole and the rotational speed of this point around the pole. Thus, the speed of point B is equal to the geometric sum of the speed of pole A and the rotational speed of point B around the pole:

2nd way to determine velocities - through projections. (velocity projection theorem) The projections of the velocities of the points of a plane figure onto the axis passing through these points are equal.

3) Formulas for calculating the speed and acceleration of a point using the natural method of specifying its movement.

Velocity vector; - Projection of velocity onto a tangent;

Components of the acceleration vector; -acceleration projections on the t and n axes;

Thus, the total acceleration of a point is the vector sum of two accelerations:

tangent directed tangent to the trajectory in the direction of increasing arc coordinate, if (otherwise - in the opposite direction) and

normal acceleration directed along the normal to the tangent towards the center of curvature (concavity of the trajectory): Total acceleration module:

4) Formulas for calculating the speed and acceleration of a point using the coordinate method of specifying its movement in Cartesian coordinates.

Components of the velocity vector: -Projections of velocity on the coordinate axes:

- components of the acceleration vector; -projections of acceleration on the coordinate axis;

5) Forward movement. Examples.

(slider, pump piston, pair of wheels of a steam locomotive moving along a straight path, elevator cabin, compartment door, Ferris wheel cabin). - this is a movement in which any straight line rigidly connected to the body remains parallel to itself. Usually translational motion is identified with the rectilinear motion of its points, but this is not so. Points and the body itself (the center of mass of the body) can move along curved trajectories, see, for example, the movement of the Ferris wheel cabin. In other words, this is movement without turns.

PLANE MOTION OF A RIGID BODY

Study questions:

1.Equations of plane motion of a rigid body.

2. Speed ​​of points of a plane figure

3. Instantaneous velocity center

4. Acceleration of points of a flat figure

1.Equations of plane motion of a rigid body

Plane motion of a rigid bodythey call thismovement in which all cross-sectional points of a body move in their own plane.

Let the rigid body 1 makes a flat motion.

Secant plane in body 1 forms a section P that moves in the secant plane .

If parallel to the plane perform other sections of the body, for example through points
etc., lying on the same perpendicular to the sections, then all these points and all sections of the body will move equally.

Consequently, the movement of the body in this case is completely determined by the movement of one of its sections in any of the parallel planes, and the position of the section is determined by the position of two points of this section, for example A And IN.

Section position P in the plane Ohoo determined by the position of the segment AB, carried out in this section. Position of two points on a plane A(
) And IN(
) characterized by four parameters (coordinates), which are subject to one limitation - the connection equation in the form of the length of the segment AB:

Therefore, the position of section P in the plane can be specified three independent parameters - coordinates
pointsA and angle, which forms a segment AB with axle Oh. Full stop A, chosen to determine the position of section P is called POLE.

When a body section moves, its kinematic parameters are functions of time

The equations are kinematic equations of plane (plane-parallel) motion of a rigid body. Now we will show that, in accordance with the obtained equations, a body in plane motion undergoes translational and rotational motion. Let in Fig. section of a body specified by a segment
in the coordinate system Ooh, moved from the initial position 1 to final position 2.

We will show two ways of possible movement of a body from a position 1 to position 2.

First way. Let's take the point as a pole .Move the segment
parallel to itself, i.e. progressively, along a trajectory ,until the points are combined And . We get the position of the segment . at an angle and we obtain the final position of the flat figure, specified by the segment
.

Second way. Let's take the point as a pole . Moving the segment
parallel to itself, i.e. progressively along the trajectory
until the points are combined And .Get the position of the segment
. Next, we rotate this segment around the pole on corner and we obtain the final position of the flat figure, specified by the segment
.

Let us draw the following conclusions.

1. Plane motion, in full accordance with the equations, is a combination of translational and rotational motions, and the model of plane motion of a body can be considered as the translational motion of all points of the body together with the pole and rotation of the body relative to the pole.

2. The trajectories of translational motion of a body depend on the choice of pole . In Fig. 13.3 in the case considered, we see that in the first method of motion, when a point was taken as a pole ,trajectory of translational movement significantly different from the trajectory
for the other pole IN.

3. The rotation of the body does not depend on the choice of pole. Corner rotation of the body remains constant in magnitude and direction of rotation . In both cases considered in Fig. 13.3, the rotation occurred counterclockwise.

The main characteristics of a body in plane motion are: the trajectory of the pole, the angle of rotation of the body around the pole, the speed and acceleration of the pole, the angular velocity and angular acceleration of the body. Additional axes
during translational motion they move along with the pole A parallel to the main axes Ohoo along the trajectory of the pole.

The speed of the pole of a plane figure can be determined using time derivatives from the equations:

The angular characteristics of the body are determined similarly: angular velocity
;

angular acceleration

.

In Fig. at the pole A projections of the velocity vector are shown on the axis Oh, oh. Body rotation angle , angular velocity and angular acceleration shown by arc arrows around a point A. Due to the independence of the rotational characteristics of motion from the choice of pole, the angular characteristics ,,can be shown at any point of a flat figure with arc arrows, for example at point B.

Speed ​​of an arbitrary point M we define the figure as the sum of the velocities that the point receives during translational motion together with the pole and rotational motion around the pole.

Let's imagine the position of the point M as (Fig. 1.6).

Differentiating this expression with respect to time we get:

, because

.

At the same time, the speed v MA. which point M obtained by rotating a figure around a pole A, will be determined from the expression

v MA=ω · M.A.,

Where ω - angular velocity of a flat figure.

Speed ​​of any point M flat figure is geometrically the sum of the speed of the point A, taken as the pole, and the speed, point M when a figure rotates around a pole. The magnitude and direction of the velocity of this velocity are found by constructing a parallelogram of velocities.

Problem 1

Determine the speed of a point A, if the speed of the center of the roller is 5 m/s, the angular speed of the roller . Roller radius r=0.2m, corner . The roller rolls without slipping.

Since the body performs plane-parallel motion, the speed of the point A will consist of the pole speed (point WITH) and the speed received by the point A when rotating around a pole WITH.

,

Answer:

Theorem on the projections of the velocities of two points of a body moving plane-parallel

Let's consider some two points A And IN flat figure. Taking a point A per pole (Fig. 1.7), we get

Hence, projecting both sides of the equality onto the axis directed along AB, and given that the vector is perpendicular AB, we find

v B· cosβ=vA· cosα+ v V A· cos90°.

because v V A· cos90°=0 we obtain: the projections of the velocities of two points of a rigid body onto the axis passing through these points are equal.

Problem 1

Kernel AB slides down a smooth wall and a smooth floor, point speed A V A =5m/s, angle between floor and rod AB equals 30 0 . Determine the speed of a point IN.

Determining the velocities of points on a plane figure using the instantaneous velocity center

When determining the speeds of points of a flat figure through the speed of the pole, the speed of the pole and the speed of rotational motion around the pole can be equal in magnitude and opposite in direction, and there is a point P whose speed at a given moment in time is zero , call it the instantaneous center of velocities.

Instantaneous velocity center is a point associated with a plane figure whose speed at a given moment in time is zero.

The velocities of the points of a flat figure are determined at a given moment in time as if the movement of the figure were instantly rotational around an axis passing through the instantaneous center of velocities (Fig. 1.8).

vA=ω · PA; ().

Because v B=ω · P.B.; (), That w=vB/P.B.=vA/PA

The velocities of the points of a flat figure are proportional to the shortest distances from these points to the instantaneous center of velocities.

The results obtained lead to the following conclusions:

1) to determine the position of the instantaneous velocity center, you need to know the magnitude and direction of the velocity and the direction of the velocity of any two points A And IN flat figure; instantaneous velocity center P is located at the point of intersection of perpendiculars constructed from points A And IN to the velocities of these points;

2) angular velocity ω flat figure at a given moment of time is equal to the ratio of the speed to the distance from it to the instantaneous center R speeds: ω =v A/PA;

3) The velocity of the point relative to the instantaneous velocity center P will indicate the direction of the angular velocity w.

4) The speed of a point is directly proportional to the shortest distance from the point IN to the instantaneous velocity center R v A = ω·BP

Problem 1

Crank OA length 0.2m rotates uniformly with angular velocity ω=8 rad/s. To the connecting rod AB at the point WITH connecting rod is hinged CD. For a given position of the mechanism, determine the speed of the point D slider if the angle is .

Point movement IN limited by horizontal guides, the slider can only make translational movement along the horizontal guides. Point speed IN directed in the same direction as . Since two points of the connecting rod have the same direction of velocities, the body performs instantaneous translational motion, and the velocities of all points of the connecting rod have the same direction and value.