gyroscope called a massive axisymmetric body (symmetrical top), rapidly rotating around the axis of symmetry, and the axis of rotation can change position in space. The axis of symmetry is called the axis of the gyroscope figure.

Video 7.6. What is a gyroscope?

Rice. 7.17. Movement of the gyroscope system

The axis of symmetry is one of the main axes of the gyroscope. Therefore, its angular momentum coincides in direction with the axis of rotation.

In order to change the position in space, the position of the axis of the gyroscope figure, it is necessary to act on it by the moment of external forces.

Video 7.7. Gyroscopic forces: a large gyroscope breaks the rope

At the same time, a phenomenon called gyroscopic: under the action of forces that, it would seem, should have caused axis 1 to rotate around axis 2 (Fig. 7.19), the axis of the figure rotates around axis 3.

Rice. 7.19. Movement of the axis of the gyroscope figure under the action of the moment of external forces

Video 7.8. Gyroscope with overloads: direction and speed of precession, nutation

Gyroscopic phenomena manifest themselves wherever there are rapidly rotating bodies, the axis of which can rotate in space.

Rice. 7.20. The response of the gyroscope to external influences

Strange at first glance, the behavior of the gyroscope, fig. 7.19 and 7.20 is fully explained by the equation of dynamics rotary motion solid body

Video 7.9. "Loving" gyroscope: the axis of the gyroscope runs along the guide without leaving it

Video 7.10. The action of the moment of friction force: "Columbus" egg

If the gyroscope is brought into rapid rotation, it will have a significant moment of momentum. If an external force acts on the gyroscope for time , then the increment of the angular momentum will be

If the force acts for a short time, then

In other words, with short impacts (shocks), the momentum of the gyroscope practically does not change. Related to this is the remarkable stability of the gyroscope with respect to external influences, which is used in various devices, such as gyrocompasses, gyro-stabilized platforms, etc.

Video 7.11. Gyrocompass model, gyro stabilization

Video 7.12. Big gyrocompass

7.21. Orbital station gyro stabilizer

In gyroscopes used in aviation and astronautics, a cardan suspension is used, which allows you to maintain the direction of the axis of rotation of the gyroscope, regardless of the orientation of the suspension itself:

Video 7.13. Gyroscopes in the circus: riding on one wheel on a wire

Additional Information

http://www.plib.ru/library/book/14978.html Sivukhin D.V. General course Physics, Volume 1, Mechanics Ed. Science 1979 - pp. 245–249 (§ 47): Euler's kinematic theorem on rotations of a rigid body about a fixed point.

Consider the motion of a gyroscope with a fixed point of support, as shown in Fig. 7.22.

The movement of a gyroscope under the action of an external force is called forced precession.

Rice. 7.22. Forced gyroscope precession: 1 - general form; 2 - top view

Let's apply at a point BUT force . If the gyroscope does not rotate, then, naturally, the right flywheel will go down, and the left one will go up. Another situation will be if the gyroscope is first brought into rapid rotation. In this case, under the action of a force, the axis of the gyroscope will rotate with an angular velocity around the vertical axis. That is, the axis of the gyroscope acquires speed in the direction perpendicular to the direction of the acting force.

Thus, the precession of a gyroscope is a movement under the action of external forces, occurring in such a way that the axis of the figure describes a conical surface.

Rice. 7.23. To the derivation of the gyroscope precession formula.

The explanation for this phenomenon is as follows. Moment of force about a point 0 will

The increment of the angular momentum of the gyroscope over time is equal to

This increment perpendicular angular momentum and, therefore, changes its direction, but not its magnitude.

The angular momentum vector behaves like a velocity vector when a particle moves in a circle. In the latter case, the velocity increments are perpendicular to the particle velocity and equal in absolute value to

In the case of a gyroscope, the elementary increment of angular momentum

and equal modulo

In time, the angular momentum vector will rotate by an angle

The angular velocity of rotation of the plane passing through the axis of the cone described by the axis of the figure, and the axis of the figure, is called angular velocity of precession gyroscope.

The oscillations of the axis of the gyroscope figure arising under certain conditions in a plane passing through the axis of the cone indicated above and the axis of the figure itself are called nutations. Nutations can be caused, for example, by a short push of the axis of the gyroscope figure up or down (see Fig. 7.24):

Rice. 7.24. Gyro nutation

The angular velocity of precession in the case under consideration is equal to

We note an important property of the gyroscope - its inertia, which means that after the termination of the external force, the rotation of the axis of the figure stops.

Additional Information

http://www.plib.ru/library/book/14978.html Sivukhin D.V. General course of physics, volume 1, Mechanics Ed. Science 1979 - pp. 288–293 (§ 52): outlined the foundations of an exact theory of the gyroscope.

http://femto.com.ua/articles/part_1/0796.html - physical encyclopedia. A variety of mechanical gyroscopes are described, which are used for navigation - gyrocompasses.

http://femto.com.ua/articles/part_1/1901.html - physical encyclopedia. A laser gyroscope for the purposes of space navigation is described.

The effect of gyroscopic forces in technology is illustrated by the following figures.

Rice. 7.25. Gyroscopic forces acting on the aircraft during rotation of the propeller

Rice. 7.26. Flip top under the action of gyroscopic forces

Rice. 7.27. How to put an egg "on the butt"

Additional Information

http://kvant.mirror1.mccme.ru/1971/10/mehanika_vrashchayushchegosya.htm - Kvant magazine - top mechanics (S. Krivoshlykov).

http://www.pereplet.ru/nauka/Soros/pdf/9809_096.pdf - Soros Educational Journal, 1998, No. 9 - the article discusses the problems of the dynamics of rotating bodies (Celtic stones) in contact with a solid surface (A .P. Markeev).

http://ilib.mirror1.mccme.ru/djvu/bib-kvant/kvant_35.djvu - Mikhailov A.A. Earth and its rotation, Biblioteka Kvant, issue 35 pp. 50–56 - the planet Earth is a big top, its axis precesses in space.

Appendix

About the principle of the wheel

Since we have talked a lot in this chapter about the rotation of bodies, let us dwell on the greatest and important discovery humanity - the invention of the wheel. Everyone knows that dragging a load is much more difficult than transporting it on wheels. The question is why? The wheel plays a huge role in modern technology, is rightfully considered one of the most ingenious inventions of mankind.

Moving cargo with a roller. The prototype of the wheel was a roller placed under the load. Its first uses are lost in the mists of time. Before dealing with the wheel, we will understand the principle of the roller. To do this, consider an example.

Example. Load of mass M placed on a cylindrical roller with mass and radius , which can move on a flat horizontal deck. A horizontal force is applied to the load (Fig. 7.28). Find the acceleration of the load and the roller. Ignore the rolling friction force. Assume that the system moves without slippage.

Rice. 7.28. Moving cargo with a roller

We denote the friction force between the roller and the load and - between the roller and the deck. For the positive direction we take the direction of the external force . Then the positive values ​​and correspond to the directions of the friction forces shown in Figs. 7.28.

Thus, forces and act on the load, and forces and act on the roller. Denote a cargo acceleration and a 1- Roller acceleration. In addition, the roller rotates clockwise with angular acceleration .

The equations of translational motion take the form:

The equation for the rotational motion of the roller is written as follows:

Let us now turn to the conditions for the absence of slippage. Due to the rotation of the roller, its lowest point has a linear acceleration and, in addition, participates in translational motion with acceleration. In the absence of slippage between the roller and the deck, the total acceleration at the bottom of the roller must be zero, so that

The upper point of the roller acquires due to rotation an oppositely directed linear acceleration and the same acceleration of translational motion. To avoid slippage between the roller and the load, the full acceleration of the top point must be equal to the acceleration of the load:

From the obtained equations for accelerations it follows that the acceleration of the roller is two times less than the acceleration of the load:

and correspondingly,

Everyone knows from firsthand experience that the roller does fall behind the load.

Substituting the ratios for accelerations into the equations of motion and solving them for the unknowns , , , we obtain the following expression for the acceleration of the load

Both forces of friction and turn out to be positive in this case, so that in Fig. 12 of their directions are chosen correctly:

As you can see, the radius of the roller does not play a special role: the ratio depends only on its shape. With a given mass and radius, the moment of inertia of the roller is maximum when the roller is a pipe: . In this case, there is no friction force between the roller and the deck ( = 0), and the equations for the acceleration of the load and the friction force between the load and the roller take the form:

With a decrease in the mass of the roller, the friction force decreases, the acceleration of the load increases - the load is easier to move.

In the case of a roller-cylinder (log) /2 and we find the friction forces

and acceleration of the load.

Comparing with the results for the roller-pipe, we see that the mass of the roller has effectively decreased, as it were: the acceleration of the load increases, all other things being equal.

The main result of the considered example: the acceleration is different from zero (that is, the load starts to move) for an arbitrarily small external force. When dragging a load along the flooring, at least force must be applied to move it.

The second conclusion: acceleration does not depend at all on the amount of friction between the parts of the given system. The coefficient of friction was not included in the solutions found, it will appear only in the absence of slippage, which boil down to the fact that the applied force should not be too large.

The result obtained, that the roller, as it were, completely “destroys” the friction force, is not surprising. Indeed, in the absence of relative movement of the contacting surfaces, the friction forces do no work. In fact, the roller “replaces” sliding friction with rolling friction, which we have neglected. AT real case the minimum force required to move the system is non-zero, although much less than when dragging a load along the deck. In modern technology, the principle of the roller is implemented in ball bearings.

Qualitative consideration of the operation of the wheel. Having dealt with the skating rink, let's move on to the wheel. The first wheel in the form of a wooden disk mounted on an axle appeared, apparently, in the 4th millennium BC. in civilizations ancient east. In the II millennium BC. the design of the wheel is being improved: spokes, a hub and a bent rim appear. The invention of the wheel gave a giant impetus to the development of crafts and transport. However, many do not understand the very principle of the wheel. In a number of textbooks and encyclopedias, one can find an incorrect statement that a wheel, like a skating rink, also gives a gain by replacing the sliding friction force with the rolling friction force. Sometimes one hears references to the use of lubrication or bearings, but this is not the case, since the wheel obviously appeared before they thought of lubrication (and, moreover, bearings).

The operation of the wheel is most easily understood in terms of energy considerations. Ancient wagons are simple: the body is attached to a wooden axle with a radius (the total mass of the body with the axle is M). Wheels with mass and radius are mounted on the axle R(Fig. 7.29).

Rice. 7.29. Moving the movement of the load with the help of the wheel

Suppose that such a wagon is being driven along a wooden deck (then we have the same coefficient of friction in all contiguous places). First, we will jam the wheels and, acting by force, we will drag the wagon a distance s. As the cart slides on the deck, the friction force reaches its maximum possible value.

The work done against this force is

(since usually the mass of the wheels is much less than the mass of the wagon<<M).

Let us now free the wheels and again drag the wagon the same distance. s. If the wheels do not slide on the deck, then the friction force does not do work at the bottom of the wheel. But sliding friction occurs between the axle and the wheel at the bottom of the axle with radius . There is also a force of normal pressure. It will differ somewhat from the previous one due to the weight of the wheels and other reasons, which we will discuss below, but with a small mass of wheels and a small coefficient of friction, it can be considered approximately equal to . Therefore, the same friction force acts between the axle and the wheel

We emphasize again: the wheel itself does not reduce the force of friction. But work A" against this force there will now be much less than in the case of dragging a wagon with jammed wheels. Indeed, when the wagon travels a distance S, its wheels make revolutions. This means that the surfaces rubbing against the axle of the wheel will move relative to each other by a smaller distance. Therefore, the work against the friction forces will also be a corresponding number of times less:

Thus, by putting the wheels on the axles, we reduce not the friction force, as in the case of the skating rink, but the path on which it acts. Let's say a wheel with a radius R= 0.5 m and axis radius = 2 cm reduces work by 96%. The remaining 4% is successfully handled by lubrication and bearings, which reduce the friction itself (lubrication, in addition, prevents wear on the running gear of the cart). Now it’s clear why old carriages and war chariots had such big wheels. Modern grocery carts in supermarkets can only roll thanks to bearings.

where r is the radius vector drawn from point O to point A, the location of the material point, p=m v is the momentum of the material point. Momentum vector modulus:

where a is the angle between the vectors r and p, l is the shoulder of the vector p with respect to the point O. The vector L according to the definition of the cross product is perpendicular to the plane in which the vectors lie r and p(or v), its direction coincides with the direction of the translational movement of the right screw when it rotates from r to p

Angular moment about the axis is called a scalar quantity equal to the projection onto this axis of the angular momentum vector, defined relative to an arbitrary point on this axis.

The moment of force M of a material point relative to the point O is called a vector quantity determined by the vector product of the radius vector r, drawn from the point O to the point of application of the force, by the force F: .

Fig.2.

Moment of force vector modulus:

where a is the angle between the vectors r and F, d \u003d r * sina - the shoulder of the force - the shortest distance between the line of action of the force and point O. Vector M(as well as L) - perpendicular to the plane in which the vectors lie r and F, its direction coincides with the direction of the translational motion of the right screw when it rotates from r to F the shortest distance as shown in the figure.

Moment of force about the axis called a scalar quantity equal to the projection onto this axis of the vector of the moment of force M defined with respect to an arbitrary point on this axis.

The basic law of the dynamics of rotational motion

To clarify the purpose of the above concepts, we consider a system of two material points (particles) and then generalize the result to a system of an arbitrary number of particles (ie, to a rigid body). Let on particles with masses m 1 , m 2 , whose momenta p1 and p2, external forces act F1 and F2. Particles also interact with each other by internal forces f 12 and f 21 .

Fig.3.

Let's write down Newton's second law for each of the particles, as well as the connection between internal forces arising from Newton's third law:

Multiply the vector equation (1) by r1, and equation (2) – on r2 and add the resulting expressions:

Let us transform the left parts of equation (4), taking into account that

.

The vectors and are parallel and their cross product is zero, so we can write

. (5)

The first two terms on the right in (4) are equal to zero, i.e.

insofar as f 21 = -f 12, and the vector r1-r2 directed along the same straight line as the vector f 12.

Taking into account (5) and (6) from (4) we obtain

or

where L=L 1 +L 2; M=M1+M2. Generalizing the result to a system of n particles, we can write L=L 1 +L 2 +…+L n = M=M 1 +M 2 +M n=

Equation (7) is a mathematical record of the basic law of the dynamics of rotational motion: the rate of change of the angular momentum of the system is equal to the sum of the moments of external forces acting on it. This law is valid for any fixed or moving at a constant speed point in an inertial frame of reference. From this follows the law conservation of angular momentum: if the moment of external forces M is equal to zero, then the angular momentum of the system is conserved (L= const).

The angular momentum of a perfectly rigid body about a fixed axis.

Consider the rotation of an absolutely rigid body around a fixed axis z. A solid body can be represented as a system of n material points (particles). During rotation, some considered point of the body (we denote it by the index i, and i=1…n) moves along a circle of constant radius R i with a linear speed v i around the z axis (Fig. 4). Her speed v i and momentum m i v i perpendicular to the radius R i. Therefore, the modulus of the angular momentum of a particle of a body relative to the point O, located on the axis of rotation:

where r i is the radius vector drawn from the point О to the particle.

Using the relationship between the linear and angular velocity v i =wR i , where R i is the distance of the particle from the axis of rotation, we obtain

The projection of this vector onto the rotation axis z, i.e. the angular momentum of a body particle relative to the z-axis will be equal to:

The angular momentum of a rigid body about the axis is the sum of the angular momentum of all parts of the body:

The value I z , equal to the sum of the products of the masses of the particles of the body and the squares of their distances to the z axis, is called the moment of inertia of the body about this axis:

From expression (8) it follows that the angular momentum of the body does not depend on the position of the point O on the axis of rotation, therefore, we speak of the angular momentum of the body relative to some axis of rotation, and not relative to the point

There is a similarity between the formulations of the basic law of rotational motion, the definitions of the moment of momentum, and the force with the formulations of Newton's second law and the definitions of momentum for translational motion.

Free axes and principal axes of inertia of the body

In order to maintain a fixed position in space of the axis of rotation of a rigid body, it is mechanically fixed, usually using bearings, i.e. influenced by external forces. However, there are such axes of rotation of bodies that do not change their orientation in space without the action of external forces on them. These axes are called free axes. It can be proved that any body has three mutually perpendicular axes passing through its center of mass, which are free. These axes are also called main axes of inertia of the body.

Gyroscopes

Currently, gyroscopes are called a very wide class of devices that use more than a hundred different phenomena and physical principles. In this laboratory work, a classical gyroscope is studied, in the future, just a gyroscope.

A gyroscope (or top) is a massive symmetrical body rotating at a high angular velocity around its axis of symmetry. We will call this axis the axis of the gyroscope. The axis of the gyroscope is one of the main axes of inertia (free axis). The angular momentum of the gyroscope in this case is directed along the axis and is equal to L=I w.

Consider a horizontally balanced gyroscope (whose center of gravity is above the fulcrum). Since the moment of gravity for it is equal to zero, then, according to the law of conservation of angular momentum L=I w= const, i.e. the direction of its axis of rotation does not change its position in space.

When trying to cause the axis of the gyroscope to rotate, a phenomenon is observed called gyroscopic effect. The essence of the effect: under the action of a force F applied to the axis of a rotating gyroscope, the axis of the gyroscope rotates in a plane perpendicular to this force. For example, under the action of a vertical force, the axis of the gyroscope rotates in the horizontal plane. At first glance, this seems counterintuitive.

The gyroscopic effect is explained as follows (Fig. 5). Moment M strength F directed perpendicular to its axis, because M=, r is the radius vector from the center of mass of the gyroscope to the point of force application.

Fig.5.

During the time dt, the angular momentum of the gyroscope L will receive an increment d L=M*dt (in accordance with the basic law of rotational motion), and directed in the same direction as M and become equal L+d L. Direction L+d L coincides with the new direction of the axis of rotation of the gyroscope. Thus, the axis of the gyroscope will rotate in a plane perpendicular to the force F at some angle dφ=|dL|/L=M*dt/L, with angular velocity

The angular velocity of rotation of the gyroscope axis W is called the angular velocity of precession, and such rotational motion of the gyroscope axis precession.

From (9) it follows

Vectors M, L, W mutually perpendicular, so we can write

M=.

This formula is obtained when the vectors M, L, W are mutually perpendicular, but it can be proved that it is valid in the general case.

Note that these arguments and the derivation of formulas are valid in the case when the angular velocity of the gyroscope is w>>W.

It follows from formula (9) that the precession velocity W is directly proportional to M and inversely proportional to the gyroscope angular momentum L. If the time of action of the force is short, the angular momentum L is large enough, then the precession velocity W will be small. Therefore, the short-term action of forces practically does not lead to a change in the orientation of the axis of rotation of the gyroscope in space. To change it, forces must be applied for a long time.

Practical application of gyroscopes

The properties of the gyroscope described above have found various practical applications. One of the first applications of the properties of gyroscopes was found in rifled weapons. After leaving the gun barrel, the air resistance force acts on the projectile, the moment of which can overturn the projectile and change its orientation relative to the trajectory in a random way, which negatively affects the flight range and accuracy of hitting the target. Screw rifling in the barrel of the gun imparts a rapid rotation around its axis to the emerging projectile. The projectile turns into a gyroscope and the external moment of the air resistance force causes only the precession of its axis around the direction of the tangent to the trajectory of the projectile. At the same time, a certain orientation of the projectile in space is preserved.

Another important application of gyroscopes is various gyroscopic instruments: gyrohorizon, gyrocompass, etc. Balanced gyroscopes are also used to maintain a given direction of aircraft movement (autopilot). To do this, the gyroscope is mounted on a cardan suspension, which reduces the effect of external moments of forces arising during aircraft maneuver. Due to this, the axis of the gyroscope maintains its direction in space, regardless of the movement of the aircraft. When the direction of aircraft movement deviates from the direction specified by the axis of the gyroscope, automatic commands appear that return to the specified direction.

The described behavior of the gyroscope is also the basis of the device called the gyroscopic compass (gyrocompass). This device is a gyroscope, the axis of which can freely rotate in a horizontal plane. If the axis of the gyroscope does not coincide with the direction of the meridian, then, due to the rotation of the Earth, a force arises that tends to rotate the axis in the direction perpendicular to the horizon. However, due to the gyroscopic effect, it rotates in a horizontal direction until the direction coincides with the meridian, pointing exactly north. A gyroscopic compass compares favorably with a compass with a magnetic needle in that its readings do not need to be corrected for the so-called magnetic declination (associated with the mismatch of the geographic and magnetic poles of the Earth), and it is also not necessary to take measures to compensate for the effects of magnetic interference from the body and equipment vessel.

Description of the experimental setup

The experimental setup (Fig. 6) consists of the following main units:

1. Gyro disk.

2. Lever with metric scale.

3. The load, by moving it along the lever 2, the value of the moment of force is set.

4. Disk with an angular scale for determining the angle of rotation of the gyroscope axis in the horizontal plane during precession.

5. Block of measurements and control.

1. Determine the modulus of the moment of gravity for several positions of the load z on the gyroscope lever:

,

where m is the mass of the load, z p is the coordinate of the load on the metric scale of the lever when the gyroscope is balanced.

2. For each position of the load, determine the time of rotation of the gyroscope axis Δ t to a given angle Δ φ and calculate the angular velocity of precession:

3. Calculate the value of the momentum of the gyroscope for each of the measurements:

4. Calculate the average value of the momentum of the gyroscope:

Where N is the number of measurements.

5. Calculate the moment of inertia of the gyroscope using the formula I = L/w (w is the angular velocity of the gyroscope, w = 2pn, n is the number of engine revolutions per unit time) and determine the absolute and relative errors in determining the moment of inertia of the gyroscope.

test questions

1. What is the angular momentum of a material point relative to a point?

2. The basic law of the dynamics of rotational motion.

3. What is the moment of force about a point?

4. Momentum of an absolutely rigid body.

5. Moment of inertia of a rigid body about a given axis.

6. Formulate the law of conservation of angular momentum.

7. What is a gyroscope?

8. What is the gyroscopic effect?

9. What is called gyroscope precession and under what conditions is it observed?

10. What is the angular velocity of precession?

Literature

1. Saveliev I.V. Course of general physics. Proc. allowance. In 3 volumes. T.1 Mechanics. Molecular physics. M.: Science. Chief editor phys.math. lit., 19873. -432 p.

2. Trofimova T.I. Physics course. Proc. allowance for universities. M.: Higher. Shk., 2003. -541 p.

The content of the article

GYROSCOPE, a navigation device, the main element of which is a rapidly rotating rotor, fixed so that its axis of rotation can be rotated. Three degrees of freedom (axis of possible rotation) of the gyroscope rotor are provided by two gimbal frames. If such a device is not affected by external perturbations, then the axis of proper rotation of the rotor retains a constant direction in space. If a moment of an external force acts on it, tending to rotate the axis of its own rotation, then it begins to rotate not around the direction of the moment, but around an axis perpendicular to it (precession).

In a well-balanced (astatic) and fairly fast rotating gyroscope, mounted on high-performance bearings with low friction, the moment of external forces is practically absent, so that the gyroscope retains its orientation in space for a long time almost unchanged. Therefore, it can indicate the angle of rotation of the base on which it is fixed. This is how the French physicist J. Foucault (1819-1868) first demonstrated the rotation of the Earth. If, however, the rotation of the axis of the gyroscope is limited by a spring, then when it is properly installed, say, on an aircraft performing a turn, the gyroscope will deform the spring until the moment of the external force is balanced. In this case, the force of compression or tension of the spring is proportional to the angular velocity of the aircraft. This is the principle of operation of the aviation direction indicator and many other gyroscopic instruments. Since there is very little friction in the bearings, it does not take much energy to keep the gyroscope rotor spinning. A low-power electric motor or a jet of compressed air is usually sufficient to bring it into rotation and to maintain rotation.

Application.

The gyroscope is most often used as a sensitive element of indicating gyroscopic instruments and as a sensor for the angle of rotation or angular velocity for automatic control devices. In some cases, for example, in gyrostabilizers, gyroscopes are used as generators of moment of force or energy. see also FLYWHEEL.

The main areas of application of gyroscopes are shipping, aviation and astronautics ( cm. INERTIAL NAVIGATION). Almost every seagoing vessel is equipped with a gyrocompass for manual or automatic control of the vessel, some are equipped with gyro stabilizers. The fire control systems of naval artillery have many additional gyroscopes that provide a stable frame of reference or measure angular velocities. Without gyroscopes, automatic control of torpedoes is impossible. Airplanes and helicopters are equipped with gyroscopic instruments that provide reliable information for stabilization and navigation systems. Such instruments include the artificial horizon, vertical gyro, gyroscopic roll and turn indicator. Gyroscopes can be either pointing devices or autopilot sensors. Many aircraft are provided with gyro-stabilized magnetic compasses and other equipment - navigation sights, cameras with a gyroscope, gyrosextants. In military aviation, gyroscopes are also used in aerial firing and bombing sights.

Gyroscopes for various purposes (navigation, power) are produced in different sizes depending on the operating conditions and the required accuracy. In gyroscopic instruments, the rotor diameter is 4–20 cm, with a smaller value for aerospace instruments. The diameters of the ship's gyro stabilizer rotors are measured in meters.

BASIC CONCEPTS

The gyroscopic effect is created by the same centrifugal force that acts on the top rotating, for example, on a table. At the point of support of the top on the table, a force and moment arise, under the influence of which the axis of rotation of the top deviates from the vertical, and the centrifugal force of the rotating mass, preventing a change in the orientation of the plane of rotation, forces the top to rotate around the vertical, thereby maintaining a given orientation in space.

With this rotation, called precession, the gyroscope rotor responds to the applied moment of force about an axis perpendicular to the axis of its own rotation. The contribution of the rotor masses to this effect is proportional to the square of the distance to the axis of rotation, since the larger the radius, the greater, firstly, the linear acceleration and, secondly, the shoulder of the centrifugal force. The influence of the mass and its distribution in the rotor is characterized by its "moment of inertia", i.e. the result of summing the products of all its constituent masses by the square of the distance to the axis of rotation. The full gyroscopic effect of a rotating rotor is determined by its "kinetic moment", i.e. the product of the angular velocity (in radians per second) and the moment of inertia about the axis of the rotor's own rotation.

Momentum is a vector quantity that has not only a numerical value, but also a direction. On fig. 1, the angular momentum is represented by an arrow (the length of which is proportional to the magnitude of the moment) directed along the axis of rotation in accordance with the “gimlet rule”: where the gimlet is fed if it is turned in the direction of rotation of the rotor.

Precession and moment of force are also characterized by vector quantities. The direction of the vector of the angular velocity of the precession and the vector of the moment of force is connected by the gimlet rule with the corresponding direction of rotation. see also VECTOR.

GYROSCOPE WITH THREE DEGREES OF FREEDOM

On fig. Figure 1 shows a simplified kinematic diagram of a gyroscope with three degrees of freedom (three axes of rotation), with the directions of rotation shown by curved arrows. The angular momentum is represented by a thick straight arrow directed along the axis of the rotor's own rotation. The moment of force is applied by pressing a finger so that it has a component perpendicular to the axis of the rotor's own rotation (the second force of the pair is created by vertical semiaxes fixed in a frame that is connected to the base). According to Newton's laws, such a moment of force should create a kinetic moment that coincides with it in direction and is proportional to its magnitude. Since the kinetic moment (associated with the rotor's own rotation) is fixed in magnitude (by setting a constant angular velocity by means of, say, an electric motor), this requirement of Newton's laws can only be met by rotating the axis of rotation (in the direction of the vector of the external moment of force), leading to an increase in the projection of the angular momentum on this axis. This turn is the precession discussed earlier. The precession speed increases with an increase in the external moment of force and decreases with an increase in the kinetic moment of the rotor.

Gyroscopic course indicator.

On fig. 2 shows an example of the use of a three-degree gyroscope in an aviation heading indicator (gyro-semi-compass). The rotation of the rotor in ball bearings is created and maintained by a jet of compressed air directed to the corrugated surface of the rim. The internal and external frames of the gimbals provide complete freedom of rotation of the rotor's own rotation axis. On the azimuth scale attached to the outer frame, you can enter any azimuth value by aligning the axis of the rotor's own rotation with the base of the instrument. Friction in the bearings is so insignificant that after this azimuth value is entered, the rotor rotation axis maintains a given position in space, and using the arrow attached to the base, the aircraft's turn can be controlled on the azimuth scale. Turn readings do not show any deviations, except for the effects of drift associated with imperfections in the mechanism, and do not require communication with external (for example, ground-based) navigation aids.

DOUBLE STAGE GYRO

Many gyroscopic devices use a simplified, two-stage version of the gyroscope, in which the outer frame of the three-stage gyroscope is eliminated, and the semiaxes of the inner are fixed directly in the walls of the housing, rigidly connected to the moving object. If in such a device the only frame is not limited by anything, then the moment of external force about the axis associated with the body and perpendicular to the axis of the frame will cause the axis of the rotor's own rotation to continuously precess away from this original direction. The precession will continue until the axis of its own rotation is parallel to the direction of the moment of force, i.e. in a position where there is no gyroscopic effect. In practice, this possibility is excluded due to the fact that conditions are set under which the rotation of the frame relative to the body does not go beyond a small angle.

If the precession is limited only by the inertial reaction of the frame with the rotor, then the angle of rotation of the frame at any time is determined by the integrated accelerating moment. Since the moment of inertia of the frame is usually relatively small, it reacts too quickly to forced rotation. There are two ways to remedy this shortcoming.

Counter spring and viscous damper.

Angular speed sensor.

The precession of the rotor rotation axis in the direction of the force moment vector directed along the axis perpendicular to the frame axis can be limited by a spring and a damper acting on the frame axis. The kinematic diagram of a two-stage gyroscope with a counteracting spring is shown in fig. 3. The axis of the rotating rotor is fixed in the frame perpendicular to the axis of rotation of the latter relative to the housing. The input axis of the gyroscope is the direction associated with the base, perpendicular to the axis of the frame and the axis of proper rotation of the rotor with an undeformed spring.

The moment of external force about the reference axis of rotation of the rotor, applied to the base at the time when the base does not rotate in inertial space and, therefore, the axis of rotation of the rotor coincides with its reference direction, causes the axis of rotation of the rotor to precess towards the input axis, so that the angle frame deviation starts to increase. This is equivalent to applying a moment of force to a counteracting spring, which is the important function of the rotor, which, in response to the occurrence of an input moment of force, creates a moment of force about the output axis (Fig. 3). At a constant input angular velocity, the gyroscope's output moment of force continues to deform the spring until the moment of force generated by it, acting on the frame, causes the rotor's rotation axis to precess about the input axis. When the speed of such precession, caused by the moment created by the spring, becomes equal to the input angular velocity, equilibrium is reached and the angle of the frame stops changing. Thus, the deflection angle of the gyroscope frame (Fig. 3), indicated by an arrow on the scale, makes it possible to judge the direction and angular velocity of rotation of a moving object.

On fig. 4 shows the main elements of the angular velocity indicator (sensor), which has now become one of the most common aerospace instruments.

Viscous damping.

Viscous damping can be used to dampen the output moment of force relative to the axis of the two-degree gyro unit. The kinematic diagram of such a device is shown in fig. 5; it differs from the diagram in Fig. 4 by the fact that there is no counteracting spring here, and the viscous damper is increased. When such a device rotates at a constant angular velocity about the input axis, the gyro node's output moment causes the frame to precess about the output axis. Excluding the effects of the inertial reaction (mainly only some response delay is associated with the inertia of the frame), this moment is balanced by the moment of viscous resistance forces created by the damper. The moment of the damper is proportional to the angular velocity of rotation of the frame relative to the body, so the output torque of the gyro is also proportional to this angular velocity. Because this output torque is proportional to the input angular velocity (for small output frame angles), the output frame angle increases as the body rotates about the input axis. The arrow moving along the scale (Fig. 5) indicates the angle of rotation of the frame. The readings are proportional to the integral of the angular velocity of rotation about the input axis in inertial space, and therefore the device, the diagram of which is shown in fig. 5 is called an integrating two-power gyro sensor.

On fig. 6 shows an integrating gyro sensor, the rotor (gyro motor) of which is enclosed in a hermetically sealed glass, floating in a damping liquid. The signal of the angle of rotation of the floating frame relative to the housing is generated by an inductive angle sensor. The position of the float gyro unit in the housing sets the torque sensor in accordance with the electrical signals it receives. Integrating gyroscopes are usually installed on elements equipped with a servo drive and controlled by the output signals of the gyroscope. With this arrangement, the output signal of the torque sensor can be used as a command to rotate the object in inertial space. see also GYRO-COMPASS.

Consider the situation when a force is applied to the axis of the gyroscope, the line of action of which does not pass through the anchor point O.

Gyro precession- this type of movement, when, as a result of the constant action of the moment of an external force, the axis of a free gyroscope rotates around the direction of a given external force.

It is known that precession ensures the stability of movement. An example of precession is the movement of the axis of a children's toy - a top with a pointed

Rice. 6.5.

end (Fig. 6.5), i.e. gyroscope with one point of support. The spinning top, untwisted around its axis and placed on a horizontal plane slightly inclined, begins to precess around the vertical axis under the action of the moment of a pair of gravity forces and the normal reaction of the support: M= / x mg, where / = OS. The end of the gyroscope axis will move in the direction of the vector m, which lies in the horizontal plane and is directed perpendicular to the axis of the top.

The speed at which the axis of rotation of the gyroscope moves relative to the vertical axis is called precession angular velocity Q.

It can be proved that for a rotating top the angular velocity of precession does not depend on the top's inclination angle 0; it is inversely proportional to the angular momentum of the top:

The faster the top spins, the greater the angular momentum and the slower it precesses. Moreover, the instantaneous disappearance of the moment of force, for example, gravity, leads to the instantaneous disappearance of the precession, i.e. precessional movement is inertialess.

If we consider the rolling of an inclined disk, then the overturning moment of gravity and the reaction of the support will act on it. A light disk will fall much faster than a massive one, due to the small value of the angular momentum (the precession speed is greater).

Rice. 6.6.

Let us consider the case when the precession of a gyroscope, which moves under the action of gravity, is accompanied by nutations- oscillations of the axis of the gyroscope's own rotation around the vector of the total angular momentum. On fig. Figure 6.6 shows how, as a result of imposing nutations on the precessional motion, the gyroscope vertex describes a complex trajectory with a variable nutation angle 0. The axis of the nutation cone coincides in direction with the vector L and moves with it. The top of the nutation cone, like the top of the precession cone, is at a fixed point O- the point of attachment of the gyroscope.

The faster the gyroscope rotates, the greater the angular velocity of nutation and the smaller its amplitude and period. With very fast rotation, nutations become almost invisible to the eye. Note that, due to friction, the nutation oscillations quickly decay, and then the gyroscope performs only precessional motion when the angle 0 between the vectors L ii remains constant. If during the movement of the gyroscope there is no nutation and the values ​​of the angular velocity of precession Q and the angular velocity of rotation around its own axis w are constant, then such a movement is called regular precession (uniform).

For the first time, evidence of the rotation of the Earth around its axis from west to east was obtained by the French physicist J.-B.-L. Foucault using the Foucault pendulum (1851) and in experiments with a gimbaled gyroscope (1852). The first Foucault pendulum in Belarus was installed at the Belarusian State Pedagogical University. Maxim Tank (September 2004, Minsk).

The properties of gyroscopes are possessed by rotating celestial bodies, aircraft propellers, etc. The areas of practical application of gyroscopes are dynamically expanding. For example, gyroscopic devices and devices are used in medicine, in rocket and space technology, for navigation purposes (pointers to the countries of the world, the horizon, etc.), in topographic and geodetic work, and in the construction of subways.

Lecture 11. Gyroscopes.

This lecture covers the following questions:

1. Gyroscopes. Free gyroscope.

2. Precession of a gyroscope under the action of external forces. Angular speed of precession. Nutations.

3. Gyroscopic forces, their nature and manifestation.

4. Tops. Stability of rotation of a symmetrical top.

The study of these issues is necessary in the discipline "Machine parts".

Gyroscopes.Free gyroscope.

A gyroscope is a massive axially symmetrical body rotating at a high angular velocity around its axis of symmetry.

In this case, the moments of all external forces, including the force of gravity, relative to the center of mass of the gyroscope are equal to zero. This can be realized, for example, by placing the gyroscope in the gimbals shown in Fig.1.

Fig.1

Wherein

and the angular momentum is conserved:

L= const(2)

The gyroscope behaves in the same way as a freer body of revolution. Depending on the initial conditions, two options for the behavior of the gyroscope are possible:

1. If the gyroscope is spun around the axis of symmetry, then the directions of the angular momentum and angular velocity coincide:

, (3)

and the direction of the axis of symmetry of the gyroscope remains unchanged. This can be verified by turning the stand on which the gimbal is located - with arbitrary turns of the stand, the axis of the gyroscope retains the same direction in space. For the same reason, the top, "launched" on a sheet of cardboard and thrown up (Fig. 2), maintains the direction of its axis during the flight, and, falling with the tip onto the cardboard, continues to rotate steadily until its kinetic energy is used up.

Fig.2

A free gyroscope, spun around the axis of symmetry, has a very significant stability. It follows from the basic equation of moments that the change in angular momentum

If the time interval small, then small, that is, under short-term effects of even very large forces, the movement of the gyroscope changes insignificantly. The gyroscope, as it were, resists attempts to change its angular momentum and seems to be "hardened".

Let us take a cone-shaped gyroscope resting on a support rod at its center of mass O (Fig. 3). If the body of the gyroscope does not rotate, then it is in a state of indifferent equilibrium, and the slightest push moves it from its place. If this body is brought into rapid rotation around its axis, then even strong blows with a wooden hammer will not be able to significantly change the direction of the gyroscope axis in space. The stability of a free gyroscope is used in various technical devices, for example, in an autopilot.

Fig.3

2. If a free gyroscope is spun in such a way that the instantaneous angular velocity vector and the axis of symmetry of the gyroscope do not coincide (as a rule, this mismatch is insignificant during fast rotation), then a movement is observed, described as "free regular precession". In relation to the gyroscope, it is called nutation. In this case, the axis of symmetry of the gyroscope, the vectors L and lie in the same plane, which rotates around the direction L= constwith an angular velocity equal to where - the moment of inertia of the gyroscope about the main central axis, perpendicular to the axis of symmetry. This angular velocity (let's call it the nutation rate) during the rapid proper rotation of the gyroscope turns out to be quite large, and the nutation is perceived by the eye as a small jitter of the gyroscope's symmetry axis.

Nutational motion can be easily demonstrated using the gyroscope shown in Fig. 3 - it occurs when a hammer strikes the rod of a gyroscope rotating around its axis. At the same time, the more the gyroscope is spun, the greater its angular momentum L - the greater the nutation rate and the "smaller" the jitter of the axis of the figure. This experience demonstrates another characteristic feature of nutation - over time, it gradually decreases and disappears. This is a consequence of the inevitable friction in the gyroscope bearing.

Our Earth is a kind of gyroscope, and nutation movement is also characteristic of it. This is due to the fact that the Earth is somewhat flattened from the poles, due to which the moments of inertia about the axis of symmetryand about an axis lying in the equatorial planediffer. Wherein, a . In the reference frame associated with the Earth, the axis of rotation moves along the surface of the cone around the axis of symmetry of the Earth with an angular velocity w 0, that is, it makes one revolution in about 300 days. In fact, due to the non-absolute rigidity of the Earth, this time turns out to be longer - it is about 440 days. At the same time, the distance of the point on the earth's surface through which the axis of rotation passes, from the point through which the axis of symmetry passes (the North Pole), is only a few meters. The nutational motion of the Earth does not die out - apparently, it is supported by seasonal changes occurring on the surface

Precession of a gyroscope under the action of external forces. elemental theory.

Let us now consider the situation when a force is applied to the axis of the gyroscope, the line of action of which does not pass through the anchor point. Experiments show that in this case the gyroscope behaves in a very unusual way.

If a spring is attached to the axis of a gyroscope hinged at point O (Fig. 4) and pulled upwards with a force F , then the axis of the gyroscope will move not in the direction of the force, but perpendicular to it, sideways. This movement is called the precession of the gyroscope under the action of an external force.

Fig.4

Empirically, it can be established that the angular velocity of precession depends not only on the magnitude of the force F (Fig. 4), but also on which point of the gyroscope axis this force is applied: with an increase F and her shoulder lrelative to the fixing point O, the precession rate increases. It turns out that the more the gyroscope is rotated, the lower the angular velocity of precession for given F and l.

As a force F , causing precession, the force of gravity can act if the fixing point of the gyroscope does not coincide with the center of mass. So, if a rod with a rapidly rotating disk is suspended on a thread (Fig. 5), then it does not go down, as one might assume, but performs a precessional movement around the thread. Observation of the precession of a gyroscope under the action of gravity is in a sense even more convenient - the line of action of the force "automatically" shifts along with the axis of the gyroscope, while maintaining its orientation in space.

Fig.5

Other examples of precession can be given - for example, the movement of the axis of a well-known children's toy - a top with a pointed end (Fig. 6). The spinning top, untwisted around its axis and placed on a horizontal plane slightly inclined, begins to precess around the vertical axis under the action of gravity (Fig. 6).

Fig.6

An exact solution to the problem of the motion of a gyroscope in the field of external forces is quite an expression for the angular velocity of precession can be easily obtained in the framework of the so-called elementary theory of the gyroscope. This theory assumes that the instantaneous angular velocity of the gyroscope and its angular momentum are directed along the axis of symmetry of the gyroscope. In other words, it is assumed that the angular velocity of rotation of the gyroscope around its axis is much greater than the angular velocity of precession:

so the contribution to L , due to the precessional motion of the gyroscope, can be neglected. In this approximation, the angular momentum of the gyroscope is obviously equal to

where - the moment of inertia about the axis of symmetry.

So, consider a heavy symmetrical gyroscope, in which the fixed point S (support point on the stand) does not coincide with the center of mass O (Fig. 7).

Fig.7

Moment of gravity about point S

where θ - the angle between the vertical and the axis of symmetry of the gyroscope. The vector M is directed along the normal to the plane containing the axis of symmetry of the gyroscope and the vertical through the point S (Fig. 7). The reaction force of the support passes through S, and its moment about this point is zero.

Change in angular momentum L is defined by the expression

dL= mdt(8)

At the same time and L , and the axis of the top precess around the vertical direction with an angular velocity. We emphasize once again that the assumption is made that condition (5) is satisfied and that L is constantly directed along the axis of symmetry of the gyroscope. From Fig. 95 it follows that

In vector form

(10)

Comparing (8) and (10), we obtain the following relationship between the moment of force M , angular momentum L and the angular velocity of precession:

(11)

This relation makes it possible to determine the direction of precession for a given direction of rotation of the top around its axis.

Note that M determines the angular velocity of the precession, and not the angular acceleration, so the instantaneous "switching off" of M leads to the instantaneous disappearance of the precession, that is, the precessional motion is inertialess.

The force causing the precessional motion can be of any nature. To maintain this motion, it is important that the torque vector M rotates along with the axis of the gyroscope. As already noted, in the case of gravity, this is achieved automatically. In this case, from (11) (see also Fig. 7) one can obtain:

(12)

If we take into account that relation (6) is valid in our approximation, then for the angular velocity of precession we obtain

It should be noted thatdoes not depend on the angletilt of the gyroscope axis and vice versa proportional w, which is in good agreement with experimental data.

Precession of a gyroscope under the action of external forces. Departure from elementary theory. Nutations.

Experience shows that the precessional motion of a gyroscope under the action of external forces is generally more complicated than that described above in the framework of elementary theory. If we give the gyroscope a push that changes the angle(see Fig. 7), then the precession will cease to be uniform (often said: regular), but will be accompanied by small rotations and tremors of the gyroscope top - nutations. To describe them, it is necessary to take into account the mismatch of the total angular momentum vector L, instantaneous angular velocity of rotation w and axes of symmetry of the gyroscope.

The exact theory of the gyroscope is beyond the scope of a general physics course. From the relationdL= mdtit follows that the end of the vector L moving in the direction M, that is, perpendicular to the vertical and to the axis of the gyroscope. This means that the projections of the vector L to the vertical LB and on the axis of the gyroscope L0 remain constant. Another constant is the energy

(14)

where T is the kinetic energy of the gyroscope. expressing L B , L 0 and T through the Euler angles and their derivatives, it is possible, with the help of the Euler equations, to describe the motion of the body analytically.

The result of such a description is as follows: the angular momentum vector L describes a precession cone that is immobile in space, and the axis of symmetry of the gyroscope moves around the vector L along the surface of the nutation cone. The top of the nutation cone, as well as the top of the precession cone, is located at the point where the gyroscope is fixed, and the axis of the nutation cone coincides in direction with L and moves with it. The angular velocity of nutations is determined by the expression

where and - moments of inertia of the body of the gyroscope about the axis of symmetry and about the axis passing through the fulcrum and perpendicular to the axis of symmetry,- angular velocity of rotation around the axis of symmetry.

Thus, the axis of the gyroscope is involved in two movements: nutation and precession. The trajectories of the absolute motion of the gyroscope top are intricate lines, examples of which are shown in Fig. eight.

Fig.8

The nature of the trajectory along which the top of the gyroscope moves depends on the initial conditions. In the case of Fig. eight, a the gyroscope was spun around the axis of symmetry, mounted on a stand at a certain angle to the vertical, and carefully released. In the case of Fig. eight, b moreover, he was given a certain push forward, and in the case of fig. eight, in- push back in the course of precession. The curves in fig. 8 are quite similar to the cycloids described by a point on the rim of a wheel rolling on a plane without slipping or with slipping in one direction or another. And only by informing the gyroscope of an initial push of a well-defined magnitude and direction, it is possible to achieve that the axis of the gyroscope will precess without nutations. The faster the gyroscope rotates, the greater the angular velocity of nutations and the smaller their amplitude. With very fast rotation, nutations become almost invisible to the eye.

It may seem strange: why the gyroscope, being spun, set at an angle to the vertical and released, does not fall under the action of gravity, but moves sideways? Where does the kinetic energy of precessional motion come from?

Answers to these questions can only be obtained within the framework of an exact theory of gyroscopes. In fact, the gyroscope really starts to fall, and the precessional motion appears as a consequence of the law of conservation of angular momentum. Indeed, the downward deviation of the gyroscope axis leads to a decrease in the projection of the angular momentum on the vertical direction. This decrease must be compensated by the angular momentum associated with the precessional motion of the gyroscope axis. From an energy point of view, the kinetic energy of precession appears due to a change in the potential energy of the gyroscopes.

If, due to friction in the support, the nutations are extinguished faster than the rotation of the gyroscope around the axis of symmetry (as a rule, this happens), then soon after the “start” of the gyroscope, the nutations disappear and pure precession remains (Fig. 9). In this case, the angle of inclination of the gyroscope axis to the verticalturns out to be more than it was at the beginning, that is, the potential energy of the gyroscope decreases. Thus, the axis of the gyroscope must be lowered slightly in order to be able to precess around the vertical axis.

Fig.9

Gyroscopic forces.

Let us turn to a simple experiment: take a shaft AB with a wheel on it With (Fig. 10). As long as the wheel is not spun, it is not difficult to turn the shaft in space in an arbitrary way. But if the wheel is untwisted, then attempts to turn the shaft, for example, in a horizontal plane with a small angular velocitylead to an interesting effect: the shaft tends to escape from the hands and turn in a vertical plane; it acts on the hands with certain forces R A and R B (Fig. 10). It is required to apply a tangible physical effort to keep the shaft with a rotating wheel in a horizontal plane.

Rice. ten

Let us consider the effects arising from the forced rotation of the gyroscope axis in more detail. Let the axis of the gyroscope be fixed in a U-shaped frame, which can rotate around the vertical axis OO "(Fig. 11). Such a gyroscope is usually called not free - its axis lies in a horizontal plane and cannot leave it.

Rice. eleven

We spin the gyroscope around it around its axis of symmetry to a high angular velocity (momentum L) and begin to rotate the frame with the gyroscope fixed in it around the vertical axis OO "with a certain angular velocityas shown in fig. 11. Angular moment L, will receive an incrementdL which must be provided by the moment of forces M applied to the axis of the gyroscope. Moment M , in turn, is created by a pair of forcesarising from the forced rotation of the gyroscope axis and acting on the axis from the side of the frame. According to Newton's third law, the axis acts on the frame with forces(Fig. 11). These forces are called gyroscopic; they create gyroscopic moment. The appearance of gyroscopic forces is called the gyroscopic effect. It is these gyroscopic forces that we feel when we try to turn the axis of a spinning wheel (Fig. 10).

The gyroscopic moment is easy to calculate. Let us suppose, according to the elementary theory, that

(16)

where J is the moment of inertia of the gyroscope about its axis of symmetry, andω - angular velocity of own rotation. Then the moment of external forces acting on the axis will be equal to

(17)

where ω - the angular velocity of the forced turn (sometimes they say: forced precession). From the side of the axle, the opposite moment acts on the bearings

(18)

Thus, the shaft of the gyroscope shown in Fig. 11 will press up in bearing B and exert pressure on the bottom of bearing A.

The direction of gyroscopic forces can be easily found using the rule formulated by N.E. Zhukovsky: gyroscopic forces tend to combine the angular momentum L of the gyroscope with the direction of the angular velocity of the forced turn. This rule can be clearly demonstrated using the device shown in Fig. 12.

Rice. 12

The axis of the gyroscope is fixed in a ring, which can freely rotate in the cage. We bring the clip into rotation around the vertical axis with an angular velocity(forced turn), and the ring with the gyroscope will turn in the holder until the directions L andwon't match. Such an effect underlies the well-known magnetomechanical phenomenon - the magnetization of an iron rod when it rotates around its own axis - while the electron spins align along the axis of the rod (Burnett's experiment).

Gyroscopic forces are experienced by the bearings of the axes of rapidly rotating parts of the machine when the machine itself is turned (turbines on a ship, propellers on an airplane, etc.). At significant values ​​of the angular velocity of forced precessionand own rotationas well as large flywheel dimensions, these forces can even destroy bearings. Let us consider some examples of the manifestation of gyroscopic forces.

Example 1A light single-engine aircraft with a right propeller makes a left turn (Fig. 13). The gyroscopic moment is transmitted through bearings A and B to the aircraft body and acts on it, trying to align the propeller's own rotation axis (vector) with the forced precession axis (vector). The plane begins to turn its nose up, and the pilot must "give the handle away from himself", that is, lower the elevator down. Thus, the moment of gyroscopic forces will be compensated by the moment of aerodynamic forces.

Rice. thirteen

Example 2When the ship pitches (from bow to stern and back), the rotor of a high-speed turbine participates in two movements: in rotation around its axis with an angular velocityand in rotation around a horizontal axis perpendicular to the turbine shaft, with an angular velocity(Fig. 14). In this case, the turbine shaft will press on the bearings with forceslying in the horizontal plane. When rolling, these forces, like the gyroscopic moment, periodically reverse their direction and can cause the ship to "yaw" if it is not too large (for example, a tug).

Rice. fourteen

Let us assume that the mass of the turbinem=3000 kg its radius of gyrationRin= 0.5 m, turbine speedn\u003d 3000 rpm, maximum angular velocity of the ship's hull during pitching=5 deg/s, distance between bearingsl=2 m. The maximum value of the gyroscopic force acting on each of the bearings is

After substituting numerical data, we getthat is about 1 ton.

Example 3Gyroscopic forces can cause the so-called "shimmy" oscillations of the car wheels (Fig. 15) [V.A. Pavlov, 1985]. A wheel revolving around axis AA" with an angular velocity w at the moment of collision with an obstacle, the additional speed of the forced turn around an axis perpendicular to the plane of the figure is reported. In this case, a moment of gyroscopic forces arises, and the wheel will begin to rotate around the axis BB. "Acquiring the angular velocity of rotation around the axis BB", the wheel will again begin to rotate around an axis perpendicular to the plane of the figure, deforming the elastic elements of the suspension and causing forces that tend to return the wheel to its previous vertical position. Then the situation repeats itself. If special measures are not taken in the design of the car, the resulting shimmy vibrations can lead to the tire breaking off the wheel rim and to breakage of its fastening parts.

Rice. fifteen

Example 4We also encounter the gyroscopic effect when riding a bicycle (Fig. 16). Making, for example, a turn to the right, the cyclist instinctively shifts the center of gravity of his body to the right, as if he were dumping the bike. The resulting forced rotation of the bicycle with angular velocityleads to the appearance of gyroscopic forces with a moment. On the rear wheel, this moment will be extinguished in bearings rigidly connected to the frame. The front wheel, which, in relation to the frame, has freedom of rotation in the steering column, under the influence of a gyroscopic moment, will begin to turn just in the direction that was necessary for the right turn of the bicycle. Experienced cyclists make such turns, as they say, "hands-free".

Rice. sixteen

The question of the origin of gyroscopic forces can also be considered from another point of view. We can assume that the gyroscope shown in Fig. 11, participates in two simultaneous movements: relative rotation around its own axis with an angular velocity w and portable, forced rotation around a vertical axis with an angular velocity. Thus, the elementary masses, into which the gyroscope disk can be divided (small circles in Fig. 17), must experience Coriolis accelerations

(20)

These accelerations will be maximum for the masses that are currently on the vertical diameter of the disk, and equal to zero for the masses that are on the horizontal diameter (Fig. 17).

Rice. 17

In a reference frame rotating with angular velocity(in this frame of reference the axis of the gyroscope is fixed), to the massesCoriolis forces of inertia will act

(21)

These forces create the momentwhich tends to rotate the axis of the gyroscope in such a way that the vector aligned with . Moment must be balanced by the reaction force momentacting on the axis of the gyroscope from the side of the bearings. According to Newton's third law, the axle will act on the bearings, and through them on the frame in which this axle is fixed, with gyroscopic forces. Therefore, they say that the gyroscopic forces are due to the Coriolis forces.

The occurrence of Coriolis forces can be easily demonstrated if instead of a hard disk (Fig. 17) we take a flexible rubber petal (Fig. 18). When the shaft with the untwisted petal is rotated around the vertical axis, the petal bends when passing through the vertical position, as shown in Fig. eighteen.

Rice. eighteen

Tops.

Spinning tops are fundamentally different from gyroscopes in that, in the general case, they do not have a single fixed point. The arbitrary movement of tops has a very complex character: being spun around the axis of symmetry and placed on a plane, they precess, "run" along the plane, writing out intricate figures, and sometimes even turn over from one end to the other. Without going into the details of such an unusual behavior of tops, we only note that an important role here is played by the friction force that occurs at the point of contact between the top and the plane.

Let us dwell briefly on the question of the stability of the rotation of a symmetrical top of arbitrary shape. Experience shows that if a symmetrical top is brought into rotation around the axis of symmetry and placed on a plane in a vertical position, then this rotation, depending on the shape of the top and the angular velocity of rotation, will be either stable or unstable.

Let there be a symmetrical top shown in Fig. 19. Let's introduce the following designations: O - the center of mass of the top,h- distance from the center of mass to the fulcrum; K - the center of curvature of the top at the fulcrum,r- radius of curvature;- moment of inertia about the axis of symmetry,- the moment of inertia about the main central axis, perpendicular to the axis of symmetry.

A Fig. 21

It should be noted that in the process of turning the top, the resulting angular momentum retains its original direction, that is, the vector L is always directed vertically upwards. This means that in the situation depicted in Fig. 21, b, when the axis of the top is horizontal, there is no rotation around the axis of symmetry of the top! Further, when tipping onto the leg, the rotation around the axis of symmetry will be opposite to the original one (if you look all the time from the side of the leg, Fig. 21, in).

In the case of an egg-shaped top, the surface of the body in the vicinity of the fulcrum is not a sphere, but there are two mutually perpendicular directions for which the radius of curvature at the fulcrum takes extreme (minimum and maximum) values. Experiments show that in the case shown in Fig. 21, a, the rotation will be unstable, and the top takes a vertical position, spinning around the axis of symmetry and continuing stable rotation at the sharper end. This rotation will continue until the friction forces are extinguished. sufficiently kinetic energy of the top, the angular velocity will decrease (become lessω 0 ) and the top will fall.

Rice. 22

Questions for self-examination

What solid body is called a gyroscope?

What is the angular momentum of a rapidly rotating gyroscope and how is it directed relative to its fixed point?

What are the physical properties of a rapidly rotating gyroscope with three degrees of freedom?

What effect does the action of the same force applied to the axis of a stationary and rapidly rotating gyroscope with three degrees of freedom produce?

Derive a formula for calculating the angular velocity of the gyroscope axis precession.

What is the difference in the properties of gyroscopes with two and three degrees of freedom?

What is the physical essence of the gyroscopic effect and under what conditions is it observed?

What formulas are used to determine the dynamic reactions of bearings in which the frame of a rotating gyroscope rotates with two degrees of freedom?

Literature

1. A.N. Matveev. Mechanics and the theory of relativity. Moscow: Higher school, 1986.

2. S.P. Strelkov. Mechanics. Moscow: Nauka, 1975.

3. S.E. Khaikin. Physical foundations of mechanics. Moscow: Nauka, 1971.

4. D.V. Sivukhin. General course of physics. T.1. Mechanics. Moscow: Nauka, 1989.

5. R.V. Paul. Mechanics, acoustics and the doctrine of heat. Moscow: Nauka, 1971.

6. R. Feynman et al. Feynman Lectures on Physics. M.: Mir, 1977. applied mechanics Machine parts Theory of machines and mechanisms