GYROSCOPE(from the Greek gyreuo - I spin, I rotate and skopeo - I look, I observe) - a rapidly rotating symmetrical solid body, the axis of rotation (axis of symmetry) to-rogo can change its direction in space. Rotating celestial bodies, artillery shells, rotors of turbines installed on ships, aircraft propellers, and so on have properties of gravity. G.'s technique - basic. element of various gyroscopes. devices or instruments widely used for automatic control the movement of aircraft, ships, torpedoes, missiles, and in a number of other gyroscopic systems. stabilization, for navigation purposes (indicators of the course, turn, horizon, cardinal points, etc.), for measuring angular or incoming. speeds of moving objects (eg, rockets) and in many others. other cases (eg, during the passage of adit shafts, the construction of subways, while drilling wells).

So that the G.'s axis can freely rotate in space, G. is usually fixed in the so-called rings. gimbals (Fig. 1), in Krom axis vnutr. and ext. rings and G.'s axis intersect at one point, called. suspension center. Mounted in such a suspension, a gage has three degrees of freedom and can make any turn about the center of the suspension. If the center of gravity of G. coincides with the center of suspension, G. is called. balanced, or astatic. The study of the laws of motion of gravity is a task of the dynamics of a rigid body.

Rice. 1. Classic gimbals, a- outer ring b- inner ring in- rotor.

Rice. 2. Precession of the gyroscope. Angular velocity precession is directed so that the eigenvector angular momentum H tends to coincide with the moment vector M pair acting on the gyroscope.

Basic properties of a gyroscope. If a pair of forces ( P-F) with moment ( h- the shoulder of the force) (Fig. 2), then (against expectations) the G. will begin to additionally rotate not around the axis X, perpendicular to the plane of the pair, and around the axis at, lying in this plane and perpendicular to the proper. body z axis. This will complement. movement called precession. G.'s precession will occur in relation to inertial reference frame(to the axes directed to the fixed stars) with an angular velocity

Fig 13. Direction gyroscope.

A number of instruments also make use of the property of gas to precess evenly under the action of constantly applied forces. So, if by means of supplement. load to cause G.'s precession with an angular velocity numerically equal and opposite to the vertical component of the angular velocity of the Earth's rotation (where U- the angular velocity of the Earth, - the latitude of the place), then the axis of such a G. with varying degrees of accuracy will maintain an unchanged direction relative to the cardinal points. During several hours, until an error of 1-2 ° accumulates, such a G., called gyroazimuth, or G. direction (Fig. 13), can replace the compass (for example, on airplanes, in particular in polar aviation, where the readings of the magnetic compass unreliable). Similar to G., but with a significantly larger shift of the center of gravity from the axis of precession, it is possible to determine the flow. the speed of an object moving in the direction of an axis bb 1 , with any acceleration (Fig. 14). If we abstract from the influence of gravity, then we can assume that the moment of the transfer force of inertia acts on the G. Q, where t- mass G., l- shoulder. Then, according to formula (1), the gyroscope will precess around the axis bb 1 with angular velocity . After integrating the last equality, we obtain , where - beg. object speed. T. o., it turns out to be possible to determine the speed of an object v at any moment in time along the angle , on which the G. will turn around the axis by this moment bb one . To do this, the device must be equipped with a revolution counter and a device that subtracts from the total angle of rotation the angle by which the turbine will turn due to the action of the moment of gravity on it. Such a device (an integrator of longitudinal apparent accelerations) determines the vertical velocities. rocket takeoff; in this case, the rocket must be stabilized so that it does not have rotation around its axis of symmetry.

Rice. 14. Gyroscopic meter of rocket ascent rate. - lifting acceleration; g- acceleration free fall; P- the force of gravity, Q - inertia force, - own kinetic moment.

In a number of modern structures use the so-called. float, or integrating, G. The rotor of such G. is placed in a casing - a float immersed in a liquid (Fig. 15). When the float rotates around its axis X moment will act on G. Mx viscous friction, proportional to the angular velocity of rotation . Thanks to this, it turns out that if G. will force him to report. rotation around an axis at, then the angular velocity of this rotation in accordance with equality (1) will be proportional to . As a result, the angle of rotation of the float around the axis X will, in turn, be proportional to the integral over time of (which is why the G. is called integrating). Additional electric and electromechanical. Devices make it possible either to measure the angular velocity of this G., or to make it an element of a stabilizing device. In the first case, special electromagnets create a moment about the axis X, directed against the rotation of the float; the magnitude of this moment is adjusted so that the float stops. Then a moment M1 how to replace the moment Mx forces of viscous friction and, therefore, according to f-le (1), the angular velocity will be proportional to the value M 1 determined by current strength flowing through the windings of the electromagnet. In the second case, when stabilizing, for example, around fixed axle at, the body of the integrating g. is placed on the platform, which can rotate around the axis at specialist. electric motor (Fig. 16). To explain the principle of stabilization, suppose that the base, on which the platform bearings are located, will itself rotate around the axis at to some corner. When the engine is off, the platform will turn in this case together with the base at the same angle, and the float will rotate around the axis X by an angle proportional to the angle . If now the engine will rotate the platform in the opposite direction until the float returns to its original position, then the platform will also return to its original position at the same time. You can continuously control the motor so that the angle of rotation of the float is reduced to zero, then the platform will be stabilized. The combination of two float floats in a common suspension with similarly controlled electric motors leads to the stabilization of a fixed direction, and three - to space. stabilization used, in particular, in inertial navigation schemes.

Rice. 15. Float integrating gyroscope: a- gyroscope rotor; b- float, in the body of which the bearing of the rotor axis is located; in- supporting fluid; G- frame; d- steel trunnions in stone supports; e- sensor of the angle of rotation of the float relative to the body; and- an electromagnetic device that applies a moment around the axis of the float.

Rice. 16. Stabilization around a fixed axis by means of a float gyroscope a- float gyroscope; b- amplifier, in- electric motor; G- platform, d- base.

Rice. 17. Power gyro frame: a- the actual frame; b- gyroscope; in- a partner; G- sensor of the angle of rotation of the gyroscope relative to the frame; d- sensor signal amplifier; e- stabilizing engine; and- torque sensor.

In the system of stabilization considered above, sensitivity plays a role. an element that detects deviations of an object from a given position, and the return to this position is performed by an electric motor that receives a corresponding signal. Similar gyroscopic systems. stabilization called. indicator (stabilizers of indirect action). Along with this, the so-called systems are used in technology. power gyroscope. stabilization (direct-acting stabilizers), in which the G. directly take on the efforts that interfere with the implementation of stabilization, and the engines play auxiliary. role, unloading partially or completely G. and thereby limiting the angles of their precession. Structurally, such systems are simpler than indicator systems. An example is a single-axis two-gyroscopic. frame (Fig. 17); rotors located in the frame rotate in different directions. Suppose that a force acts on the frame, tending to rotate it around the axis X and report the angular velocity . Then, according to the Zhukovsky rule, a pair will begin to act on casing 1, tending to align the rotor axis with the axis X. As a result, gravity will begin to precess around the axis y 2 with some angular velocity . casing 2 for the same reason will precess around the axis y 2 in opposite side. The angles of rotation of the casings will be the same, since the casings are connected by a gear clutch. Due to this precession on the casing bearings 1 a new pair will act, striving to align the rotor axis with the axis y one . The same pair will act on the casing bearings 2 . The moments of these pairs are directed oppositely (which follows from the Zhukovsky rule) and stabilize the frame, i.e. keep it from turning around the axis X. However, if the G.'s precessions are not limited, then, as can be seen from formula (3), when the casings are rotated around the axes y 1 , at 2 90° will stop stabilization. Therefore, on the axis of one of the casings there is a sensor that registers the angle of rotation of the casing relative to the frame and controls the stabilization engine. The torque arising from the engine is directed opposite to the moment tending to rotate the frame around the axis X; as a result, G.'s precession stops. The considered frame is stabilized with respect to rotation around the axis X. Rotate the frame around any axis perpendicular to X, you can freely, but the resulting gyroscopic. moment can cause mean. pressure on G. bearings and their casings. The combination of three such frames with mutually perpendicular axes leads to spaces. stabilization (eg, artificial satellite).

In power gyroscopic systems, in contrast to free G., due to the large moments of inertia of the stabilized masses, very noticeable oscillations arise. nutation movements. Specials must be accepted. measures to ensure that these oscillations are damped, otherwise self-oscillations occur in the system. In technology, other gyroscopes are also used. devices, the principles of operation of which are based on the properties of G.

Lit.: Bulgakov B. V., Applied Theory Gyroscopes, 3rd ed., M., 1976; Nikolay E. L., Gyroscope in gimbals, 2nd ed., M., 1964; Maleev P. I., New types of gyroscopes, L., 1971; Magnus K., Gyroscope. Theory and Application, trans. from German, M., 1974; Ishlinsky A. Yu, Orientation, gyroscopes and inertial navigation, M., 1976; him, Mechanics relative motion and forces of inertia, M., 1981; Klimov D. M., Kharlamov S. A., Dynamics of a gyroscope in a gimbal suspension, M., 1978; Zhuravlev V. F., Klimov D. M., Wave solid-state gyroscope, M., 1985; Novikov L. 3., Shatalov M. Yu., Mechanics of dynamically tuned gyroscopes, M., 1985.

A. Yu. Ishlinsky.

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 the rotation of axis 1 around axis 2 (Fig. 7.19), a rotation of the axis of the figure around axis 3 is observed.

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. Gyro stabilizer orbital station

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 AND 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.

Application

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? wheel playing 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 .

Equations forward movement 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. In a real case, the minimum force required to move the system is different from 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 the civilizations of the 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 wagon 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 be somewhat different 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.

transcript

1 Lecture 14 Gyroscopes. Gyro precession. one

2 What is a gyroscope. A gyroscope is a massive axially symmetric solid body capable of rotating around the axis of symmetry with a high angular velocity. The axis of symmetry of a gyroscope is called its own axis of the gyroscope or simply the axis of the gyroscope. She can change her position in space. Examples of gyroscopes: a top, flywheels of gyroscopic compasses, turbine rotors for various purposes, etc. The movement of a gyroscope is necessarily the movement of a rigid body with one fixed point, which is called the fulcrum of the gyroscope. In the absence of a fixed point, a rapidly rotating axially symmetrical body is called a top. 2

4 Balanced gyroscope Balanced or unloaded is a gyroscope, the axis of rotation of which is vertical and the moment M of all external forces relative to the fixed point of the gyroscope is equal to zero: M=0 In this case, the gyroscope behavior coincides with free rotation around the axis of symmetry of the central main axis: L(t) L (0) The axis of the gyroscope keeps its direction all the time If the axis of the gyroscope is in a vertical position, then the gyroscope can rotate in this position for quite a long time. 4

5 Gimbal Gyro Gimbal Gimbal Gyro Gimbal Gyro is in a balanced or unloaded state if the fixing point is aligned with the center of inertia of the system. Freely moving suspension rings keep the axis of the unloaded gyroscope in the same direction 5

6 AA axis of the rotor BB DD axis of rotation of the inner ring connected to the axis of the rotor axis of rotation of the outer ring mounted on a fixed base 6

7 Precession of a loaded gyroscope make a rotational movement on the surface of the cone. The precession of a gyroscope can be represented as a superposition of rotations around two axes: a fast rotation around its own axis and a relatively slow rotation around the vertical. The intersection of these axes of rotation gives the fixed point of the gyroscope. The angular velocity ω of rotation around its own axis is called the intrinsic angular velocity of the gyroscope. Ω The angular velocity of rotation around the vertical axis is called the angular velocity of the gyroscope precession: The greater the natural rotation frequency, the lower the precession frequency 1/ 7

8 Approximate theory of the gyroscope dl dt In the approximate theory it is assumed that the momentum vector L of the gyroscope is always oriented along the axis of the gyroscope and is equal to the angular momentum of its own rotation: I M LL Iω. 0 - moment of inertia of the gyroscope relative to its axis: I I If the precession speed is much lower than the own rotation speed, the deviation of the vector L from the gyroscope axis is insignificant and can be neglected. eight

9 External forces acting on the gyroscope The axis of the gyroscope is deflected from the vertical by an angle The moment of external forces relative to a fixed point is created only by the gravity of the gyroscope, applied to its center of mass, located on the axis of the gyroscope and remote from its fixed anguish at a distance M r,mg r- radius vector drawn from the fixed point O to the center of mass of the gyroscope Total external force acting on the gyroscope: N mg N F mg F F tr tr ts. page r This force rotates the center of mass of the gyroscope. nine

10 Calculation of the angular frequency of the forced precession of the gyroscope M r, mg M r L const L r dl L Ω dl dt Ω, L M Ω, L r, mg mg, r Isin rmg sin ; rm Ω g I

11 Direction of rotation of the gyroscope axis during forced regular precession due to the force of gravity of the gyroscope Ωg

12 Nutation of the gyroscope The solution obtained is exact under a certain initial condition: at the initial moment the gyroscope axis is given an angular velocity equal to the forced precession velocity Ω. In other cases, the symmetry axis of the top oscillates relative to the direction of gravity. / L The figure shows the trace of the end of the axial frame L at various ratios between the rates of forced precession and the nutation period. e L 12

13 Geometric interpretation of the occurrence of nutations Precession of the axis of a free top around a fixed angular momentum vector Decomposition of the motion of the gyroscope axis into two rotations 13

14 Trajectories of movement of the gyroscope axis during forced precession 14

15 Gyroscopic forces and moments of forces. OO When the gyroscope axis rotates around the vertical axis, additional gyroscopic forces will act on the gyroscope axis, creating a torque M - “gyroscopic moment” - along the direction of rotation of the gyroscope axis: M dl. These forces, in accordance with Newton's third law, correspond to an oppositely directed pair of forces acting on axle holders - for example, bearings. The gyroscopic effect is the appearance of additional pressure in the bearings due to gyroscopic forces and associated gyroscopic moments. This phenomenon is widespread in technology. It is observed in turbine rotors on ships during turns and pitching, on helicopters when performing turns, etc. The gyroscopic effect has negative consequences, since it leads to additional wear of the bearings, and with sufficient force, it can lead to the destruction of the mechanism. fifteen

16 Zhukovsky's rule Shaft AB with wheel C mounted on it. While the wheel is not spun, you can easily turn the shaft in space in an arbitrary way. When you try to slightly rotate the shaft with a rapidly spun wheel in the horizontal plane, the shaft tends to break out of your hands and turn in the vertical plane. It is required to apply a tangible physical effort to keep the shaft with a rotating wheel in a horizontal plane. The action of the shaft on the hands ("axle holders") is the gyroscopic effect created by the gyroscopic forces. 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 (ie the axis AB of rotation of the rotor) with the direction of the angular velocity of the forced turn. 16

17 The action of gyroscopic forces when turning a bicycle When making a left turn (along the direction of the bicycle), the cyclist shifts the center of gravity of his body to the left, “filling up” the bicycle. The resulting forced rotation of the bicycle with angular velocity leads to the appearance of gyroscopic forces with a moment M g. On the rear wheel, this moment will be extinguished in bearings rigidly connected to the frame. The front wheel, which has freedom of rotation in the steering column in relation to the frame, under the influence of a gyroscopic moment, will begin to turn just in the direction that was necessary for the left turn of the bicycle: L Mgt 17 Experienced cyclists make such turns "without hands".

18 Biker steers motorcycle without steering assistance 18

19 Calculation of the magnitude of gyroscopic forces Ω, L M M 2 r, F

20 Application of gyroscopes Projectile flight along a parabolic trajectory in airless space Projectile tumbling in air 20

21 Effect of air flow on projectile 21

22 Due to the forced precession caused by the resistance forces of the air medium in which the bullet flies, the longitudinal axis of the bullet, as it were, follows the trajectory, describing a conical surface around it 22

23 THE MAGNUS EFFECT A rotating object creates a whirlwind in the environment around it. On one side of the object, the direction of the vortex coincides with the direction of the flow around and, accordingly, the velocity of the medium from this side increases. On the other side of the object, the direction of the vortex is opposite to the direction of the flow, and the speed of the medium decreases. Due to this speed difference, a pressure difference arises, which generates a transverse force from the side of the rotating body on which the direction of rotation and the direction of flow are opposite, to the side on which these directions coincide. 23

24 Bullet flight deviation due to the Magnus effect The direction of the deviation coincides with the direction of the rifling of the barrel. 24

25 Stabilization of projectile flight by means of its rotation In order for the precessional rotation to have a stable character, it is necessary that the self-momentum of the projectile exceed a certain critical value or 25

26 Maintaining the orientation of the axis of rotation of a free gyroscope is used to correct (adjust) the course of movement of various vehicles: ships, aircraft, torpedoes, etc. In gyrocompasses, the gyroscope is used as a device for indicating direction. In order for the axis of the gyroscope to turn in the required direction, the gyroscope must experience a certain effect, i.e. it cannot be completely free. 26

27 Autopilots Gimbal-balanced gyroscope: center of mass coincides with suspension point. The gyroscope is in a (almost) free state and retains momentum directed along its axis. If the inherent moment of the gyroscope is large (the angular velocity of its own rotation is sufficiently large), and the friction forces are sufficiently small, then the moments of friction forces created when the ship turns, little change the direction of the gyroscope axis in space. When the direction of movement of the apparatus deviates from the direction specified by the axis of the gyroscope, the frames (rings) of the gimbal, together with the axis of the gyroscope, change their position relative to the apparatus. The rotation of the gimbal frames with the help of various mechanisms turns into commands that cause the rudders to deflect, returning the crew to a given direction. When moving in a plane, one gyroscope is sufficient. When moving in three-dimensional space (on an airplane), two gyroscopes are needed to set the orientation in the horizontal and vertical planes. 27

28 Gyrocompasses Gyrocompasses use the properties of a not completely free gyroscope, i.e. a gyroscope whose axis can only move in one fixed plane. The axis N of the gyroscope can only move in a plane orthogonal to the fixed axis OO. Let the support on which such a gyroscope is installed rotate with a constant angular velocity ω OO, ω t n t n ω n OO The gyroscope is insensitive to the component, since it is a free movement in a plane orthogonal to the fixed axis OO . Under the action of this moment, the axis of the gyroscope rotates until its direction coincides with the direction of the velocity (Zhukovsky's rule) ω t 28

29 Behavior of a gyrocompass influenced by the rotation of the Earth. OO Fixed direction Coincides with the direction of the plumb line The axis of the gyroscope can move only in the horizontal plane Under the influence of the angular velocity of the daily rotation of the Earth, the axis of the gyroscope will be set in the direction in the t direction of the meridian to the North. The gyroscope behaves like a compass. Gyrocompasses have a number of advantages over magnetic ones, since their readings are not affected by magnetic storms and iron deposits, they are less sensitive to motion, etc. Therefore, gyrocompasses play an important role in navigation. At the moment, satellite navigation devices have become widespread, which to a certain extent narrowed the area of ​​applicability of gyroscopes as navigation devices (in particular, autopilots). However, the operation of satellite navigation systems is significantly hampered in conditions of heavy cloudiness. Therefore, the role of gyroscopes in navigation remains very significant.

30 Movement of tops in the absence of a fixed point. Chinese spinning top (Thomson spinning top). thirty

31 Turning over a fast spinning top

32 32

33 Movement of a top in the absence of friction force.

34 The action of the sliding friction force on the top F tr The sliding friction force acts in the direction of the precession of the fulcrum and tends to accelerate this precession. The moment M tr of the friction force raises the center of mass of the top. Additional pressure on the fulcrum leads to an increase in the force of its reaction: N P, N P 0

35 The force of sliding friction raises the center of mass of the Chinese spinning top

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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, however, 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 highly advanced 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 (18191868) first clearly 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 pointing 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 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 420 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 a 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, 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) 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 inner half-axes 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. five; 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.

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 you give the gyroscope a push that changes the angle (see Fig. 4.6), then the precession will cease to be uniform (often said: regular), but will be accompanied by small rotations and tremors of the top of the gyroscope - nutations. To describe them, it is necessary to take into account the mismatch of the total angular momentum vector L, the instantaneous angular velocity of rotation, and the axis of symmetry of the gyroscope.

The exact theory of the gyroscope is beyond the scope of a general physics course. It follows from the relation 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 on the vertical and on the axis of the gyroscope remain constant. Another constant is the energy

(4.14)

where - kinetic energy gyroscope. Expressing and in terms of the Euler angles and their derivatives, we can use Euler equations, 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

(4.15)

where and are the moments of inertia of the gyroscope body about the axis of symmetry and about the axis passing through the fulcrum and perpendicular to the axis of symmetry, is the angular velocity of rotation around the axis of symmetry (cf. (3.64)).

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. 4.7.

Rice. 4.7.

The nature of the trajectory along which the top of the gyroscope moves depends on the initial conditions. In the case of Fig. 4.7a, 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. 4.7b, it was also given some push forward, and in the case of fig. 4.7c - push back along the precession. The curves in fig. 4.7 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 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, it happens), then soon after the "start" of the gyroscope, the nutations disappear and pure precession remains (Fig. 4.8). In this case, the angle of inclination of the gyroscope axis to the vertical turns out to be greater 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.

Rice. 4.8.

Gyroscopic forces.

Let us turn to a simple experiment: let's take the shaft AB with the wheel C mounted on it (Fig. 4.9). 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 velocity, lead 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 and (Fig. 4.9). It is required to apply a tangible physical effort to keep the shaft with a rotating wheel in a horizontal plane.

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 velocity as shown in Fig. 4.10. Angular moment L, will receive an increment 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 forces arising 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. 4.10). These forces are called gyroscopic; they create gyroscopic moment The appearance of gyroscopic forces is called gyroscopic effect. It is these gyroscopic forces that we feel when we try to turn the axis of a spinning wheel (Fig. 4.9).

where is 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

(4.)

Thus, the shaft of the gyroscope shown in Fig. 4.10 will press up in bearing B and exert pressure on the bottom of bearing A.

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 gyroscope with the direction of the angular velocity of the forced turn. This rule can be clearly demonstrated using the device shown in Fig. 4.11.