Definition of complex (composite) motion of a point. Determination of absolute, relative and portable motion, speed and acceleration. Proof of the theorem on the addition of velocities and the Coriolis theorem on the addition of accelerations. Coriolis (rotary) acceleration.

Content

Here we will show that in complex motion, absolute point speed equal to vector sum relative and portable speeds:
.
Absolute acceleration of a point equal to the vector sum of the relative, transport and Coriolis (rotary) accelerations:
,
where is the Coriolis acceleration.

An example of the application of the theory outlined below is given on the page “Complex point motion. An example of solving a problem.”

Complex (composite) motion of a point

There are often cases when a point makes a certain movement relative to some solid. And this body, in turn, moves relative to a fixed coordinate system. Moreover, the motion of a point relative to the body and the law of motion of the body relative to a fixed coordinate system are known or specified. It is required to find the kinematic quantities (velocity and acceleration) of a point relative to a fixed coordinate system.

This movement of a point is called complex or complex.

Complex or composite movement of a point is movement in a moving coordinate system. That is, the movement of a point is described in a coordinate system, which itself moves relative to a fixed coordinate system.

Further, for clarity of presentation, we will assume that the moving coordinate system is rigidly connected to some rigid body. We will consider the movement of a point relative to the body ( relative motion) and the movement of the body relative to a fixed coordinate system (transportable movement).

Relative motion of a point during complex motion is the motion of the point relative to the body (moving coordinate system), assuming that the body is at rest.

The portable movement of a point during complex movement is the movement of a point rigidly connected by a body, caused by the movement of the body.

The absolute movement of a point during complex movement is the movement of the point relative to a fixed coordinate system, caused by the movement of the body and the movement of the point relative to the body.

Difficult movement. Point M moves relative to a moving body.

Let Oxyz be a fixed coordinate system, O n x o y o z o be a moving coordinate system rigidly connected to the body. Let be unit vectors (orts) directed along the x o , y o , z o axes of the moving coordinate system. Then the radius vector of point M in a fixed system is determined by the formula:
(1) ,
where is the radius vector of the point O n - the origin of the moving coordinate system associated with the body.

Relative speed and acceleration

At relative motion the coordinates x o , y o , z o of the point relative to the body change. And vectors are constant, independent of time. Differentiating (1) in time, assuming constants, we obtain formulas for relative speed and acceleration:
(2) ;
(3) .

The relative speed of a point during complex movement is the speed of the point in a stationary position of the body (moving coordinate system), caused by the movement of the point relative to the body.

The relative acceleration of a point during complex movement is the acceleration of a point when the body is stationary, caused by the movement of the point relative to the body.

Transfer speed and acceleration

At portable movement the vectors that determine the position of the body change. The relative coordinates of the point x o , y o , z o are constant. Differentiating (1) in time, considering x o , y o , z o constant, we obtain formulas for portable speed and acceleration:
(4) ;
(5) .

The portable speed of a point during complex movement is the speed of a point rigidly connected to the body, caused by the movement of the body.

The portable acceleration of a point during complex movement is the acceleration of a point rigidly connected to the body, caused by the movement of the body.

The time derivatives of are the speed and acceleration of the origin of the moving coordinate system O n: ; .

Let's find formulas for the time derivatives of vectors. To do this, take two arbitrary points of a rigid body A and B. Their speeds are related by the relation:

(see page “Speed ​​and acceleration of points of a rigid body”). Consider a vector drawn from point A to point B. Then
.
We differentiate by time and apply the previous formula:
.
So, we have found a formula for the time derivative of a vector connecting two points of the body:
.
Since the vectors are rigidly connected to the body, their time derivatives are determined by this formula:
(6) , , .

Substitute in (4) :

.
So the expression (4) leads to a formula for the velocity of points of a rigid body.

Performing similar transformations on the formula (5) , we obtain a formula for the acceleration of points of a rigid body:
,
where is the angular acceleration of the body.

Absolute speed and acceleration

At absolute motion both the vectors that determine the position of the body and the relative coordinates of the point x o , y o , z o change.

The absolute speed of a point during complex motion is the speed of the point in a fixed coordinate system.

The absolute acceleration of a point during complex motion is the acceleration of the point in a fixed coordinate system.

Velocity addition theorem

With compound motion, the absolute speed of a point is equal to the vector sum of the relative and translational speeds:
.

Proof

Let's differentiate (1) (2) And (4) .
(1) ;
(7)
.

Coriolis theorem on the addition of accelerations

In compound motion, the absolute acceleration of a point is equal to the vector sum of the relative, translational and Coriolis (rotary) accelerations:
,
Where
- Coriolis acceleration.

Proof

Let's differentiate (7) in time, applying the rules of differentiation of sum and product. Then we substitute (3) And (5) .
(7) .


.

In the last term we apply (6) And (2) .

.
Then
.

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 FR 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 geometric sum 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 a rigid body, 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 composite 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.

The general formulation of the relative motion problem is as follows: the motion of a point is determined by observers associated with two different coordinate systems (reference systems), and these systems move in a given way relative to each other. Each observer determines the kinematic elements of movement: trajectory, speed and acceleration in his own reference system. The task is posed: knowing the movement of one reference system in relation to another, find the connection between the kinematic elements of the movement of a point in relation to each system separately. Let us assume that the motion of the point M in space is considered in two coordinate systems moving relative to each other: Oxyz, And (Fig. 41). Depending on the content of the task facing us, one of these systems Oxyz Let us take it as the main one and call it an absolute system and all its kinematic elements absolute. Another system Let's call it relative and, accordingly, the movement in relation to this system, as well as its kinematic elements, relative. The terms “absolute” and “relative” have a conventional meaning here; when considering movements, it may be advisable to take first one or the other system as absolute. Elements of absolute motion will be denoted by the subscript " A ", and relative - with the index " r ».

Let us introduce the concept of portable motion, the elements of which will be denoted by the subscript “ e " We will call the portable movement of a point the movement (in relation to absolute system) that point of the relative system through which the moving point passes at the moment in time under consideration. The concept of portable movement needs clarification. It is necessary to clearly distinguish the point, the absolute and relative movement of which is being considered, from the point invariably associated with the relative system through which the moving point is currently passing. Usually both points are designated by the same letter M, since the drawing does not convey movement; they are actually two different points moving relative to each other.

Let us dwell on two illustrations of the concept of portable motion. If a person walks on a moving platform, then we can consider, firstly, the “absolute” movement of the person in relation to the ground, and secondly, his “relative” movement along the platform. In this case, the portable movement will be the movement in relation to the ground of the place of the platform along which a person is currently passing.

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 Oxy, 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's denote angular velocity rotation of the coordinate system Оξηζ through 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, in order to construct a vector of relative speed, you need to geometrically add the absolute speed with a vector equal in absolute value, but opposite in direction to the transfer speed.

So far we have studied the movement of a point or body in relation to one given system countdown. 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 compound 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. The ability to decompose, by introducing an additional (moving) reference system, the more complex motion of a point or body into simpler ones is widely used in kinematic calculations and determines the practical value of the theory of complex motion discussed in this and the next chapters. In addition, the results of this theory are used in dynamics to study the relative equilibrium and relative motion of bodies under the action of forces.

Let's consider a point M moving relative to a moving reference system, which in turn somehow moves relative to another reference system which we call the main or conventionally stationary one (Fig. 182). Each of these reference systems is associated, of course, with a certain body, not shown in the drawing. Let us introduce the following definitions.

1. The movement performed by the point M in relation to the moving reference system (to the axes) is called relative movement (such movement will be seen by an observer associated with these axes and moving with them).

The trajectory AB described by a point in relative motion is called a relative trajectory. The speed of point M in relation to the Oxyz axes is called relative velocity (denoted), and the acceleration is called relative acceleration (denoted). From the definition it follows that when calculating, the movement of the axes can be ignored (considered as stationary).

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

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

If we imagine that the relative motion of a point occurs on the surface (or inside) of a solid body, with which the moving axes Oxy are rigidly connected, then the portable speed (or acceleration) of point M at a given moment in time will be the speed (or acceleration) of that point of the body, with which point M coincides at this moment.

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

To solve the corresponding problems of kinematics, it is necessary to establish the relationships between relative, portable and absolute velocities and accelerations of a point, which we will move on to.