Complex point movement. Addition of accelerations during translational portable motion Relative portable

Complex point movement

Basic Concepts

In many problems, the motion of a point must be considered relative to two (or more) reference systems moving relative to each other.

In the simplest case, the complex motion of a point consists of relative And portable movements. Let's define these movements.

Let's consider two reference systems moving relative to each other. One frame of reference O 1 x 1 y 1 z 1 will be taken as the main and stationary one. Second frame of reference Oxyz will move relative to the first one.

Motion of a point relative to a moving reference frame Oxyz called relative. The characteristics of this movement, such as trajectory, speed and acceleration, are called relative. They are designated by the index r.

Motion of a point relative to the main fixed frame of reference O 1 x 1 y 1 z 1 is called absolute (or complex). The trajectory, speed and acceleration of this movement are called absolute. They are designated without an index.

Portable The motion of a point is the movement that it makes together with a moving frame of reference, as a point rigidly attached to this system at the moment in time under consideration. Due to the relative motion, the moving point at different times coincides with different points of the body S, to which the moving reference frame is attached. Portable speed and portable acceleration is the speed and acceleration of that point of the body S from which at this moment the moving point coincides. Portable speed and acceleration are denoted by an index e.

If the trajectories of all points of the body S, attached to a moving frame of reference, are depicted in the figure, then we obtain a family of lines - a family of trajectories of the portable movement of point M. Due to the relative movement of point M at each moment of time, it is on one of the trajectories of the portable movement.

One and the same absolute motion, choosing different moving frames of reference, can be considered to consist of different portable and, accordingly, relative motions.

Speed addition

Let us determine the speed of the absolute movement of point M if the speeds of the absolute and portable movements of this point are known.

In a short period of time along the trajectory, point M will make a relative movement determined by the vector. The curve itself, moving together with the moving axes, will move to a new position in the same period of time. At the same time, the point of the curve with which point M coincided will perform a portable movement. As a result, the point will move.

Dividing both sides of the equality by and passing to the limit, we get

Addition of accelerations during translational motion.

Let us determine the acceleration of the absolute motion of a point in the special case of translational translational motion.

The theorem is true. If the moving frame of reference moves translationally relative to the fixed one, then all points of the body attached to this system have the same speeds and accelerations, equal to the speed and acceleration of the origin of the moving frame O. Therefore, for the speed and acceleration of the portable motion we have

Let's express the relative speed in Cartesian coordinates

Substituting the values of portable and relative speeds into the theorem on the addition of velocities, we obtain

A-priory

So far we have studied the motion of a point or body in relation to one given reference frame. However, in a number of cases, when solving problems of mechanics, it turns out to be advisable (and sometimes necessary) to consider the movement of a point (or body) simultaneously in relation to two reference systems, of which one is considered the main or conditionally stationary, and the other moves in a certain way in relation to the first. The movement performed by the point (or body) is called composite or complex. For example, a ball rolling along the deck of a moving steamship can be considered to be performing a complex motion relative to the shore, consisting of rolling relative to the deck (moving frame of reference), and moving together with the deck of the steamship in relation to the shore (fixed frame of reference). In this way, the complex motion of the ball is decomposed into two simpler and more easily studied ones.

Fig.48

Consider the point M, moving relative to the moving reference system Oxyz, which in turn somehow moves relative to another reference system, which we call the main or conditionally stationary (Fig. 48). Each of these reference systems is associated, of course, with a specific body, not shown in the drawing. Let us introduce the following definitions.

1. Movement made by a point M in relation to the moving reference system (to the axes Oxyz), called relative movement(such movement will be seen by an observer associated with these axes and moving with them). Trajectory AB described by a point in relative motion is called a relative trajectory. Point speed M in relation to the axes Oxyz is called relative velocity (denoted by ), and acceleration is called relative acceleration (denoted by ). From the definition it follows that when calculating and it is possible to move the axes Oxyz do not take into account (consider them as motionless).

2. Movement performed by a moving frame of reference Oxyz(and all points of space invariably associated with it) in relation to the fixed system, is for the point M portable movement.

The speed is invariably associated with the moving axes Oxyz points m, with which the moving point coincides at a given moment in time M, is called the transfer speed of the point M at this moment (denoted by ), and the acceleration of this point m- portable acceleration of a point M(denoted by ). Thus,

If you imagine that relative motion points occur on the surface (or inside) of a solid body, to which the movable axes are rigidly connected Oxyz, then the portable speed (or acceleration) of the point M at a given moment of time there will be the speed (or acceleration) of that point m of the body with which the point coincides at this moment M.

3. The movement made by a point in relation to a fixed frame of reference is called absolute or complex. Trajectory CD of this movement is called the absolute trajectory, the speed is called absolute speed (denoted by ) and acceleration is called absolute acceleration (denoted by ).

In the above example, the motion of the ball relative to the deck of the steamship will be relative, and the speed will be the relative speed of the ball; the movement of the steamer in relation to the shore will be a portable motion for the ball, and the speed of that point on the deck that the ball touches at a given moment in time will be its portable speed at that moment; finally, the motion of the ball relative to the shore will be its absolute motion, and the speed will be the absolute speed of the ball.

When studying the complex movement of a point, it is useful to apply the “Stopping Rule”. In order for a stationary observer to see the relative movement of a point, the portable movement must be stopped.

Then only relative motion will occur. Relative motion will become absolute. And vice versa, if you stop the relative movement, the portable one will become absolute and a stationary observer will see only this portable movement.

In the latter case, when determining the portable movement of a point, one very important circumstance is revealed. The portable movement of a point depends on the moment at which the relative movement is stopped, on where the point is on the medium at that moment. Since, generally speaking, all points of the medium move differently. Therefore, it is more logical to determine portable movement of a point as the absolute movement of that point in the environment with which the moving point currently coincides.

22.Theorem of addition of speeds.

Let some point M makes a movement relative to the reference system Oxyz, which itself moves in an arbitrary manner with respect to the stationary frame of reference , (Fig. 49).

Of course, the absolute movement of a point M determined by the equations

Relative motion - in moving axes by equations

Rice. 10.3.

There cannot be any equations that determine the portable motion of a point. Since, by definition, the portable movement of a point M– this is the movement relative to the fixed axes of that point of the system with which the point coincides at the moment. But all points of the moving system move differently.

The position of the moving reference frame can also be determined by specifying the position of the point ABOUT radius vector drawn from the origin of the fixed reference system, and the direction of the unit vectors of the moving axes Ox, Oy, Oz.

Fig.49

Arbitrary portable motion of a moving frame of reference is composed of translational motion with the speed of a point ABOUT and movements around the instantaneous axis of rotation OR passing through the point ABOUT, with instantaneous angular velocity. Due to the portable motion of the moving reference frame, the radius vector and the directions of the unit vectors change. If the vectors are given as a function of time, then the portable motion of the moving reference frame is completely defined.

Point position M with respect to the moving reference frame can be determined by the radius vector

where are the coordinates x, y, z points M change over time due to the movement of a point M relative to the moving frame of reference. If the radius vector is specified as a function of time, then the relative motion of the point M, i.e. the movement of this point relative to the moving frame of reference is given.

The position of point M relative to a fixed reference system can be determined by the radius vector. From Fig. 49 it is clear that

If the relative coordinates x,y,z points M and the vectors are defined as a function of time, then the composite motion of the point, consisting of relative and translational motions M, i.e. the movement of this point in relation to a fixed frame of reference must also be considered given.

Speed of compound point movement M, or the absolute speed of this point, is obviously equal to the derivative of the radius vector of the point M by time t

Therefore, differentiating equality (1) with respect to time t, we get

Let us divide the terms on the right side of this equality into two groups according to the following criterion. The first group includes those terms that contain derivatives only of relative coordinates x,y,z, and to the second - those terms that contain derivatives of vectors, i.e. from quantities that change only due to the portable movement of the moving reference frame

Each of the groups of terms, denoted by and , represents, at least in dimension, a certain speed. Let us find out the physical meaning of the velocities and .

Speed , as follows from equality (3), is calculated under the assumption that only the relative coordinates change x,y,z points M, but the vectors remain constant, i.e. moving reference frame Oxyz as if conventionally considered motionless. So speed is the relative speed of a point M.

Speed is calculated as if a point M did not move relative to the moving frame of reference, since the derivatives x,y,z are not included in equality (4). Therefore, the speed is the portable speed of the point M.

So, . (5)

This equality expresses the theorem for adding velocities in the case when the portable motion is arbitrary: the absolute speed of a point M equal to the geometric sum of the portable and relative velocities of this point.

Example 13. ring M moves along a rotating rod so that (cm) and (rad).

Fig.50

It was previously established that the trajectory of relative motion is a straight line coinciding with the rod, and this motion is determined by the equation. Trajectory of portable point movement M at a point in time t– circle of radius .

Therefore the relative speed is . And it is directed tangentially to the trajectory along the rod (Fig. 50). The transfer speed of the ring, as when rotating around an axis, is . The vector of this velocity is directed tangentially to the trajectory of the portable movement, perpendicular to the rod.

Absolute speed of the ring. Its size, because

23.Acceleration addition theorem. Coriolis acceleration.

Acceleration of compound motion of a point M, or the absolute acceleration of this point, is obviously equal to the derivative of the absolute speed of the point M by time t

Therefore, differentiating the equality with respect to time, we obtain

Let us divide the terms on the right side of this equality into three groups.

The first group includes terms containing only derivatives of relative coordinates x,y And z, but not containing derivatives of vectors:

The second group includes terms that contain only derivatives of vectors, but do not contain derivatives of relative coordinates x,y,z:

There remained one more group of terms that could not be classified as either the first or the second, since they contain derivatives of all variables x,y,z, . Let us denote this group of terms by:

Each of the selected groups represents, at least in dimension, some acceleration. Let's find out the physical meaning of all three accelerations: .

Acceleration, as can be seen from the equality, is calculated as if the relative coordinates x,y,z changed over time, but the vectors remained unchanged, i.e. moving reference frame Oxyz seemed to be at rest, but period M moved. Therefore, the acceleration is the relative acceleration of the point M. Since the acceleration (and speed) of relative motion is calculated under the assumption that the moving frame of reference is at rest, then to determine the relative acceleration (and speed) you can use all the rules set out earlier in the kinematics of a point.

Acceleration, as can be seen from the equality, is calculated under the assumption that the point itself M is at rest with respect to the moving frame of reference Oxyz(x=const, y=const, z=const) and moves along with this reference system in relation to the stationary reference system. Therefore, acceleration is the portable acceleration of the point M.

The third group of terms determines acceleration, which cannot be attributed to relative acceleration, since it contains in its expression derivatives not to portable acceleration, since it contains in its expression derivatives

Let us transform the right side of the equality, recalling that

Substituting these values of the derivatives into the equalities, we get

Here the vector is the relative speed of the point M, That's why

Acceleration is called Coriolis acceleration. Due to the fact that Coriolis acceleration appears in the case of rotation of a moving frame of reference, it is also called rotational acceleration.

From a physical point of view, the appearance of rotational acceleration of a point is explained by the mutual influence of portable and relative movements.

So, the Coriolis acceleration of a point is equal in magnitude and direction to twice the vector product angular velocity portable movement to the relative speed of the point.

An equality that can now be abbreviated as

presents the theorem for the addition of accelerations in the case when the translational movement is arbitrary: the absolute acceleration of a point is equal to the vector sum of the translational, relative and rotational accelerations. This theorem is often called the Coriolis theorem.

From the formula it follows that the modulus of rotational acceleration will be

where is the angle between vector and vector . To determine the direction of rotational acceleration, you need to mentally transfer the vector to the point M and be guided by the rule of vector algebra. According to this rule, the vector must be directed perpendicular to the plane defined by the vectors and , and so that, looking from the end of the vector, the observer can see the shortest turn from to occurring counterclockwise (Fig. 30). at a given moment in time becomes zero.

In addition, the rotational acceleration of a point can obviously vanish if:

a) the vector of the relative velocity of the point is parallel to the vector of the angular velocity of the portable rotation, i.e. the relative movement of the point occurs in a direction parallel to the axis of portable rotation;

b) the point has no movement relative to the moving frame of reference or the relative speed of the point at a given time is zero ().

Example 14. Let the body spin around fixed axis z. A point moves across its surface M(Fig. 52). Of course, the speed of this point’s movement is the relative speed, and the speed of rotation of the body is the angular speed of the portable movement.

The Coriolis acceleration is directed perpendicular to these two vectors, according to the rule of direction of the vector of the cross product. So, as shown in Fig. 52.

Fig.52

It is not difficult to formulate a more convenient rule for determining the direction of the vector: you need to project the relative velocity vector onto a plane perpendicular to the axis of the portable rotation and then rotate this projection 90 degrees in the plane in the direction of the portable rotation. The final position of the vector projection will indicate the direction of the Coriolis acceleration. (This rule was proposed by N.E. Zhukovsky).

Example 15.(Let's go back to example 13). Let's find the absolute acceleration of the ring M

It moves relative to some reference system, and that, in turn, moves relative to another reference system. In this case, the question arises about the connection between the movements of the point in these two reference points.

Usually one of the reference points is chosen as the base one (“absolute”), the other is called “movable” and the following terms are introduced:

absolute motion- this is the movement of a point/body in the base SO.
relative motion- this is the movement of a point/body relative to a moving reference system.
portable movement- this is the movement of the second CO relative to the first.

The concepts of corresponding velocities and accelerations are also introduced. For example, portable speed is the speed of a point due to the movement of a moving reference frame relative to the absolute one. In other words, this is the speed of a point in a moving reference system that at a given moment of time coincides with a material point.

It turns out that when obtaining a connection between accelerations in different reference systems, it becomes necessary to introduce another acceleration due to the rotation of the moving reference system:

In further consideration, the base CO is assumed to be inertial, and no restrictions are imposed on the moving one.

Classical mechanics

Kinematics of complex point motion

Speed

The main tasks of the kinematics of complex motion are to establish dependencies between the kinematic characteristics of the absolute and relative movements of a point (or body) and the characteristics of the motion of a moving reference system, that is, portable motion. For a point, these dependencies are as follows: the absolute speed of the point is equal to the geometric sum of the relative and portable speeds, that is

Acceleration

The connection between accelerations can be found by differentiating the connection for speeds, not forgetting that the coordinate vectors of the moving coordinate system can also depend on time.

The absolute acceleration of a point is equal to the geometric sum of three accelerations - relative, portable and Coriolis, that is

Kinematics of complex body movement

For solid, when all composite (that is, relative and translational) motions are translational, the absolute motion is also translational with a speed equal to the geometric sum of the velocities of the constituent motions. If the component motions of a body are rotational about axes that intersect at one point (as, for example, in a gyroscope), then the resulting motion is also rotational about this point with an instantaneous angular velocity equal to the geometric sum of the angular velocities of the component motions. If the component movements of the body are both translational and rotational, then the resulting movement in the general case will be composed of a series of instantaneous screw movements.

You can calculate the relationship between the velocities of different points of a rigid body in different reference systems by combining the formula for adding velocities and Euler’s formula for relating the velocities of points of a rigid body. The connection between the accelerations is found by simply differentiating the resulting vector equality with respect to time.

Dynamics of complex point motion

When considering motion in a non-inertial reference frame, the first 2 Newton laws are violated. To ensure their formal implementation, additional, fictitious (not actually existing) inertial forces are usually introduced: centrifugal force and Coriolis force. Expressions for these forces are obtained from the connection between accelerations (previous section).

Relativistic mechanics

Speed

At velocities close to the speed of light, the Galilean transformations are not exactly invariant and the classical formula for adding velocities ceases to hold. Instead, the Lorentz transformations are invariant, and the relationship between the velocities in two inertial reference frames is as follows:

under the assumption that the velocity is directed along the x-axis of the system S. It is easy to see that in the limit of non-relativistic velocities, the Lorentz transformations are reduced to the Galilean transformations.

Literature

N. G. Chetaev. " Theoretical mechanics" M.: Science. 1987. 368 p.

COMPLEX MOVEMENTS OF THE POINT

§ 1. Absolute, relative and portable motion of a point

In a number of cases, it is necessary to consider the movement of a point in relation to the coordinate system O 1 ξηζ, which, in turn, moves in relation to another coordinate system Oxz, conventionally accepted as stationary. In mechanics, each of these coordinate systems is associated with a certain body. For example, consider rolling without sliding of a car wheel on a rail. We will connect the fixed coordinate system Ax with the rail, and we will connect the moving system Oξη with the center of the wheel and assume that it moves translationally. The movement of a point on the rim of a wheel is compound or complex.

Let us introduce the following definitions:

1. The movement of a point relative to the coordinate system Oxyz (Fig. 53) is called absolute.

2. Movement of a point relative to a moving coordinate system O 1 ξηζ called inhabited.

3. The translational movement of a point is the movement of that point of a body associated with a moving coordinate system O 1 ξηζ, relative to a fixed coordinate system with which the moving point in question currently coincides.

Thus, portable motion is caused by the movement of a moving coordinate system in relation to a fixed one. In the given example with a wheel, the portable movement of a point on the wheel rim is due to the translational movement of the coordinate system O 1 ξηζ in relation to the fixed coordinate system Axy.

We obtain the equations of absolute motion of a point by expressing the coordinates of the point x, y, z as a function of time:

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

The equations of relative motion of a point have the form

ξ = ξ (t), η = η (t), ζ = ζ (t).

In parametric form, equations (11.76) express the equations of the absolute trajectory, and equations (11.77) - respectively, the equations of the relative trajectory.

There are also absolute, portable and relative speeds and, accordingly, absolute, portable and relative accelerations of a point. Absolute speed is denoted by υ a, relative - υ r, portable - υ e Accordingly, accelerations are denoted by: ω a, ω r And ω e.

The main task of the kinematics of complex motion of a point is to establish the relationship between the velocities and accelerations of a point in two coordinate systems: stationary and moving.

To prove theorems on the addition of velocities and accelerations in the complex motion of a point, we introduce the concept of local or relative derivative.

Velocity addition theorem

Theorem . With complex (composite) motion of a point, its absolute speed υ a equal to the vector sum of the relative υ r and portable υ e speeds

Let point M make simultaneous movements in relation to the fixed and moving coordinate systems (Fig. 56). Let us denote the angular velocity of rotation of the coordinate system Оξηζ by ω . The position of point M is determined by the radius vector r.

Let us establish the relationship between the velocities of point M in relation to two coordinate systems - stationary and moving. Based on the theorem proven in the previous paragraph

From the kinematics of a point it is known that the first derivative of the radius vector of a moving point with respect to time expresses the speed of this point. Therefore = r = υ a- absolute speed, = υ r- relative speed,

A ω x r = υ e- portable speed of point M. Therefore,

υ a= υ r+υ e

Formula (11.79) expresses the velocity parallelogram rule. We find the absolute velocity modulus using the cosine theorem:

In some kinematics problems, it is necessary to determine the relative speed υ r. From (11.79) it follows

υ r= υ a +(- υ e).

Thus, to construct a relative velocity vector, you need to geometrically add the absolute velocity with a vector equal to absolute value, but opposite to the direction of the transfer speed.

The portable movement of a point is its movement at the considered moment in time together with the moving coordinate system relative to a fixed coordinate system.

The portable speed and portable acceleration of a point are indicated by the index e: ,.

Portable speed (acceleration ) point M at a given time is called a vector equal to the speed
(acceleration
) that pointmmoving coordinate system with which the moving point M currently coincides(Fig. 8.1).

Let's draw the radius vector of the origin of coordinates (Fig. 8.1). From the figure it is clear that

To find the portable speed of a point at a given point in time, it is necessary to differentiate the radius vector provided that the coordinates of the point x, y, z do not change at a given time:

The transfer acceleration is correspondingly equal to

Thus, to determine the transfer speed and portable acceleration at a given moment in time it is necessary to mentally stop at this moment in time the relative movement of the point, determine the point m a body invariably associated with a moving coordinate system where the point is located at a stopped moment M, and calculate the speed and acceleration of the point m a body undergoing portable motion relative to a fixed coordinate system.

Setting tasks for complex point motion

1.Direct task:

Based on the given portable and relative movements of the point, find the kinematic characteristics of the absolute movement of the point.

2. Inverse problem:

To represent some given motion of a point in a complex way, decomposing it into relative and portable, and to determine the kinematic characteristics of these movements. To solve this problem unambiguously, additional conditions are required.

Velocity addition theorem

Absolute point speed is determined by the theorem on the addition of velocities, according to which the absolute speed of a point performing a complex movement is equal to the geometric sum of the portable and relative speeds:

Proof:

To determine the absolute speed of a point, we differentiate the expression on the right (8.4) with respect to time, using the properties of the derivative of a vector with respect to a scalar argument:

(8.8)

In the last expression on the left, the first four terms in formula (8.5) represent the transfer speed , the last three terms in formula (8.1) are the relative speed . The theorem has been proven.

Theorem for addition of accelerations in portable translational motion

The absolute acceleration of a point performing a complex movement during portable translational motion is equal to the geometric sum of the relative and portable acceleration:

. (8.9)

Proof:

Let's return to Fig. 8.1. With portable translational movement of the orta
do not change not only in size, but also in direction, i.e. these are constant vectors, and since derivatives of constant vectors, and since derivatives of constant vectors are equal to zero, then according to formula (8.6)

. (8.10)

To determine the absolute acceleration of a point, we differentiate the radius vector twice (8.4) in time, taking into account the constancy of the unit vectors
:

In the last expression, the first term in formula (8.10) represents the portable acceleration , and the last three according to formula (8.2) are the relative acceleration . The theorem has been proven.

Theorem for addition of accelerations during arbitrary translational motion (Coriolis theorem)

The absolute acceleration of a point is determined by Coriolis theorem, according to which the absolute acceleration of a point performing a complex movement is equal to the geometric sum of the portable, relative and Coriolis accelerations:

. (8.11)

Coriolis acceleration calculated by the formula:

, (8.12)

where is the vector of the angular velocity of the portable movement, is the vector of the relative speed of the point. The direction of the Coriolis acceleration vector is determined by the vector product rule: the Coriolis acceleration will be directed perpendicular to the plane in which the vectors lie (Fig. 8.2), in the direction from which the shortest turn from the vector to the vector is seen to occur counterclockwise.

The modulus of Coriolis acceleration is equal to .

Let us prove the validity of the theorem for portable rotational motion.

Let the moving coordinate system Oxyz rotates around an axis l with angular velocity
(Fig. 8.3). During the entire movement, the radius vectors of the point are still connected by the dependence

Since by definition
, let us differentiate expression (8.8) with respect to time, taking into account the properties of the derivative of a vector with respect to the scalar argument:

In the last expression, the first four terms represent the portable acceleration , the next three terms represent the relative speed . We denote the remaining terms (*). In expression (*), the derivative of each unit vector with respect to time represents the linear velocity of the point for which this unit unit is a radius vector. For example, for Orta (Fig. 8.3) speed
points A its end is equal