Precession of a gyroscope under the influence of external forces. Elementary theory

In order to keep the position of the axis of rotation of a solid body unchanged over time, bearings are used in which it is held. However, there are axes of rotation of bodies that do not change their orientation in space without the action of external forces on it. These axes are called free axles(or axes of free rotation). It can be proven that in any body there are three mutually perpendicular axes passing through the center of mass of the body, which can serve as free axes (they are called main axes of inertia body). For example, the main axes of inertia of a homogeneous rectangular parallelepiped pass through the centers of opposite faces (Fig. 30). For a homogeneous cylinder, one of the main axes of inertia is its geometric axis, and the remaining axes can be any two mutually perpendicular axes drawn through the center of mass in a plane perpendicular to the geometric axis of the cylinder. The main axes of inertia of the ball

are any three mutually perpendicular axes passing through the center of mass.

For the stability of rotation, it is of great importance which of the free axes serves as the axis of rotation.

It can be shown that rotation around the main axes with the largest and smallest moments of inertia turns out to be stable, and rotation around the axis with the average moment is unstable. So, if you throw a body in the shape of a parallelepiped, bringing it into rotation at the same time, then, as it falls, it will steadily rotate around the axes 1 And 2 (Fig. 30).

If, for example, a stick is suspended by one end of the thread, and the other end, attached to the spindle of a centrifugal machine, is brought into rapid rotation, then the stick will rotate in a horizontal plane about a vertical axis perpendicular to the axis of the stick and passing through its middle (Fig. 31) . This is the free axis of rotation (the moment of inertia at this position of the stick is maximum). If now the stick rotating around the free axis is freed from external connections (carefully remove the upper end of the thread from the spindle hook), then the position of the rotation axis in space is maintained for some time. The property of free axes to maintain their position in space is widely used in technology. The most interesting in this regard gyroscopes- massive homogeneous bodies rotating at high angular velocity around their axis of symmetry, which is a free axis.

Let's consider one of the types of gyroscopes - a gimbal-mounted gyroscope (Fig. 32). A disk-shaped body - a gyroscope - is fixed on an axis AA, which can rotate around a horizontal axis perpendicular to it BB, which, in turn, can rotate around a vertical axis D.D. All three axes intersect at one point C, which is the center of mass of the gyroscope and remains motionless, and the axis of the gyroscope can take any direction in space. We neglect the friction forces in the bearings of all three axes and the moment of impulse of the rings.

Since the friction in the bearings is low, while the gyroscope is motionless, its axis can be given any direction. If you start to quickly rotate the gyroscope (for example, using a rope wound around the axis) and turn its stand, then the gyroscope axis maintains its position in space unchanged. This can be explained using the basic law of rotational motion dynamics. For a free rotating gyroscope, the force of gravity cannot change the orientation of its axis of rotation, since this force is applied to the center of mass (the center of rotation C coincides with the center of mass), and the moment of gravity relative to the fixed center of mass is zero. We also neglect the moment of friction forces. Therefore, if the moment of external forces relative to its fixed center of mass is zero, then, as follows from equation (19.3), L =

Const, i.e. the angular momentum of the gyroscope retains its magnitude and direction in space. Therefore, together With it retains its position in space and the axis of the gyroscope.

In order for the gyroscope axis to change its direction in space, it is necessary, according to (19.3), for the moment of external forces to differ from zero. If the moment of external forces applied to a rotating gyroscope relative to its center of mass is different from zero, then a phenomenon called gyroscopic effect. It consists in the fact that under the influence of a pair of forces F, applied to the axis of a rotating gyroscope, the axis of the gyroscope (Fig. 33) rotates around the straight line O 3 O 3, and not around the straight line ABOUT 2 ABOUT 2 , how natural it would seem at first glance (O 1 O 1 And ABOUT 2 ABOUT 2 lie in the plane of the drawing, and O 3 O 3 and the forces F perpendicular to it).

The gyroscopic effect is explained as follows. Moment M pairs of forces F directed along a straight line ABOUT 2 ABOUT 2 . During the time dt the moment of impulse L the gyroscope will receive an increment d L = M dt (direction d L coincides with the direction M) and will become equal L"=L+d L. Vector direction L" coincides with the new direction of the axis of rotation of the gyroscope. Thus, the axis of rotation of the gyroscope will rotate around the straight line O 3 O 3. If the time of action of the force is short, then, although the moment of force M and large, change in angular momentum d L The gyroscope will also be quite small. Therefore, short-term action of forces practically does not lead to a change in the orientation of the gyroscope rotation axis in space. To change it, force must be applied over a long period of time.

If the axis of the gyroscope is fixed by bearings, then due to the gyroscopic effect, so-called gyroscopic forces, acting on the supports in which the gyroscope axis rotates. Their action must be taken into account when designing devices containing rapidly rotating massive components. Gyroscopic forces make sense only in a rotating frame of reference and are a special case of the Coriolis inertial force (see §27).

Gyroscopes are used in various gyroscopic navigation devices (gyrocompass, gyrohorizon, etc.). Another important application of gyroscopes is maintaining a given direction of movement of vehicles, for example, a ship (autopilot) and an airplane (autopilot), etc. For any deviation from the course due to some influence (wave, gust of wind, etc.), the position of the axis The gyroscope in space is preserved. Consequently, the axis of the gyroscope, together with the gimbal frames, rotates relative to the moving device. Rotating the gimbal frames with the help of certain devices turns on the control rudders, which return the movement to a given course.

The gyroscope was first used by the French physicist J. Foucault (1819-1868) to prove the rotation of the Earth.

Experience shows that the precessional motion of a gyroscope under the influence of external forces is generally more complex than that described above within the framework of elementary theory. If you give the gyroscope a push that changes the angle (see Fig. 4.6), then the precession will no longer 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 vector of the total angular momentum L, instantaneous angular velocity of rotation and axis of symmetry of the gyroscope.

The exact theory of the gyroscope is beyond the scope of the general physics course. From the relation it follows that the end of the vector L moving towards 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 energy

(4.14)

Where - kinetic energy gyroscope Expressing in terms of Euler angles and their derivatives, we can, using Euler's equations, describe the movement of a body analytically.

The result of such a description is as follows: the angular momentum vector L describes a cone of precession motionless in space, and at the same time the axis of symmetry of the gyroscope moves around the vector L along the surface of the nutation cone. The apex of the nutation cone, like the apex of the precession cone, is located at the gyroscope attachment point, and the axis of the nutation cone coincides in direction with L and moves with him. The angular velocity of nutations is determined by the expression

(4.15)

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

Thus, the gyroscope axis is involved in two movements: nutational and precessional. The trajectories of the absolute movement of the top of the gyroscope are intricate lines, examples of which are presented 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, in addition, he was given some push forward, and in the case of Fig. 4.7v - push back along the precession. Curves in Fig. 4.7 are quite similar to cycloids described by a point on the rim of a wheel rolling along a plane without slipping or with slipping in one direction or another. And only by imparting to the gyroscope an initial push of a very specific magnitude and direction can it be achieved that the gyroscope axis will precess without nutations. The faster the gyroscope rotates, the greater the angular velocity of the nutations and the smaller their amplitude. With very fast rotation, nutations become almost invisible to the eye.

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

Answers to these questions can only be obtained within the framework of the exact theory of gyroscopes. In fact, the gyroscope actually begins to fall, and precessional motion appears as a consequence of the law of conservation of angular momentum. In fact, the downward deviation of the gyroscope axis leads to a decrease in the projection of the angular momentum in the vertical direction. This decrease must be compensated by the angular momentum associated with the precessional movement of the gyroscope axis. From an energy point of view, the kinetic energy of precession appears due to changes 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, this happens), then soon after the “launch” 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 gyro axis must lower slightly to be able to precess around the vertical axis.

Rice. 4.8.

Gyroscopic forces.

Let's turn to a simple experiment: take in our hands the shaft AB with the wheel C mounted on it (Fig. 4.9). As long as the wheel is not untwisted, it is not difficult to rotate the shaft in space in an arbitrary manner. But if the wheel is spinning, 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 takes significant physical effort to hold the shaft with the rotating wheel in a horizontal plane.

Let's spin the gyroscope around it around its axis of symmetry to a large angular velocity (angular momentum L) and begin to rotate the frame with the gyroscope mounted in it around the vertical axis OO" with a certain angular velocity as shown in Fig. 4.10. Angular momentum L, will receive an increment that must be provided by the moment of force M, applied to the axis of the gyroscope. Moment M, in turn, is created by a pair of forces that arise during forced rotation of the gyroscope axis and act 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 trying to turn the axis of a rotating wheel (Fig. 4.9).

where is the angular velocity of forced rotation (sometimes called forced precession). On the axle side, the opposite moment acts on the bearings

(4.)

Thus, the shaft of the gyroscope shown in Fig. 4.10, will be pressed upward 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 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.

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 (axes of possible rotation) of the gyroscope rotor are provided by two gimbal frames. If such a device is not affected by external disturbances, then the axis of the rotor’s own rotation maintains a constant direction in space. If a moment of 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 quickly rotating gyroscope, mounted on highly advanced bearings with insignificant friction, the moment of external forces is practically absent, so that the gyroscope for a long time retains its orientation in space almost unchanged. Therefore, it can indicate the angle of rotation of the base on which it is attached. This is how the French physicist J. Foucault (1819-1868) first clearly demonstrated the rotation of the Earth. If the rotation of the gyroscope axis is limited by a spring, then if it is installed appropriately, 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 compression or tension force of the spring is proportional to the angular velocity of the aircraft. This is the principle of operation of an aircraft turn indicator and many other gyroscopic devices. Because there is very little friction in the bearings, it doesn't take much energy to keep the gyroscope rotor spinning. To set it into rotation and to maintain rotation, a low-power electric motor or a jet of compressed air is usually sufficient.
Application. The gyroscope is most often used as a sensitive element of indicating gyroscopic devices and as a rotation angle or angular velocity sensor for automatic control devices. In some cases, for example in gyrostabilizers, gyroscopes are used as torque or energy generators.
see also FLYWHEEL. The main areas of application of gyroscopes are shipping, aviation and astronautics (see INERTIAL NAVIGATION). Almost every long-distance sea vessel is equipped with a gyrocompass for manual or automatic control of the vessel, some are equipped with gyrostabilizers. In naval artillery fire control systems there are many additional gyroscopes that provide a stable reference frame or measure angular velocities. Without gyroscopes, automatic control of torpedoes is impossible. Airplanes and helicopters are equipped with gyroscopic devices that provide reliable information for stabilization and navigation systems. Such instruments include an attitude indicator, a gyrovertical, and a gyroscopic roll and turn indicator. Gyroscopes can be either indicating devices or autopilot sensors. Many aircraft are equipped with gyro-stabilized magnetic compasses and other equipment - navigation sights, cameras with a gyroscope, gyro-sextants. In military aviation, gyroscopes are also used in aerial shooting and bombing sights. Gyroscopes for various purposes (navigation, power) are produced in different sizes depending on operating conditions and the required accuracy. In gyroscopic devices, the rotor diameter is 4-20 cm, with a smaller value for aerospace devices. The diameters of the rotors of ship gyrostabilizers are measured in meters.
BASIC CONCEPTS
The gyroscopic effect is created by the same centrifugal force that acts on a spinning top, 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 leverage of the centrifugal force. The influence of 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 relative to the axis of the rotor's own rotation. Kinetic moment is a vector quantity that has not only a numerical value, but also a direction. In Fig. 1 kinetic moment 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 torque are also characterized by vector quantities. The direction of the angular velocity vector of precession and the torque vector are related by the gimlet rule to the corresponding direction of rotation.
see also VECTOR.
GYROSCOPE WITH THREE DEGREES OF FREEDOM
In Fig. Figure 1 shows a simplified kinematic diagram of a gyroscope with three degrees of freedom (three axes of rotation), and the directions of rotation are shown on it by curved arrows. The kinetic moment 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 semi-axes fixed in the frame, which is connected to the base). According to Newton's laws, such a moment of force must 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 through, say, an electric motor), this requirement of Newton’s laws can only be fulfilled by rotating the axis of rotation (towards the vector of the external torque), leading to increasing the projection of the kinetic moment on this axis. This rotation is the precession discussed earlier. The precession rate increases with increasing external torque and decreases with increasing kinetic torque of the rotor.
Gyroscopic heading indicator. In Fig. Figure 2 shows an example of the use of a three-degree gyroscope in an aviation heading indicator (gyro-half-compass). The rotation of the rotor in ball bearings is created and maintained by a stream of compressed air directed at the grooved surface of the rim. The internal and external frames of the gimbal provide complete freedom of rotation of the axis of the rotor's own rotation. Using 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 device. The friction in the bearings is so insignificant that after this azimuth value is entered, the axis of rotation of the rotor maintains the specified position in space, and using the arrow attached to the base, the rotation of the aircraft can be controlled on the azimuth scale. Turn indications do not exhibit any deviations other than drift effects associated with imperfections in the mechanism, and do not require communication with external (eg, ground) navigation aids.

Viscous damping. To dampen the output moment of force relative to the axis of a two-degree gyro unit, viscous damping can be used. The kinematic diagram of such a device is shown in Fig. 5; it differs from the diagram in Fig. 4 in that there is no counter spring and the viscous damper is increased. When such a device is rotated at a constant angular velocity around the input axis, the output moment of the gyroscope causes the frame to precess around the output axis. Subtracting the effects of inertial reaction (the inertia of the frame is mainly associated with only a slight delay in the response), this moment is balanced by the moment of the viscous resistance forces created by the damper. The damper moment is proportional to the angular velocity of rotation of the frame relative to the body, so the output moment of the gyro unit is also proportional to this angular velocity. Since this output torque is proportional to the input angular velocity (at small output frame angles), the output frame angle increases as the body rotates about the input axis. An 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 relative to the input axis in inertial space, and therefore the device, the diagram of which is shown in Fig. 5 is called an integrating two-degree gyro sensor.

Collier's Encyclopedia. - Open Society. 2000 .

1. Free axes of rotation. Let us consider two cases of rotation of a solid rod about an axis passing through the center of mass.

If you untwist the rod relative to the axis O.O. and leave it to itself, that is, free the axis of rotation from the bearings, then in the case of Fig. 71-a, the orientation of the axis of free rotation relative to the rod will change, since the rod, under the influence of a pair of centrifugal forces of inertia, will unfold into a horizontal plane. In the case of Fig. 71-b, the moment of a pair of centrifugal forces is zero, so the untwisted rod will continue to rotate around the axis OO and after her release.

The axis of rotation, the position of which in space is maintained without the action of any external forces, is called the free axis of a rotating body. Consequently, the axis perpendicular to the rod and passing through its center of mass is the free axis of rotation of the rod.

Any rigid body has three mutually perpendicular free axes of rotation, intersecting at the center of mass. The position of the free axes for homogeneous bodies coincides with the position of their geometric axes of symmetry (Fig. 72).

Free axes of rotation are also called main axes of inertia. When bodies rotate freely around the main axes of inertia, only rotations around those axes that correspond to the maximum and minimum values ​​of the moment of inertia are stable. If external forces act on the body, then rotation is stable only around the main axis to which the maximum moment of inertia corresponds.

2. Gyroscope(from Greek gyreuo- I spin and skopeo– I see) is a homogeneous body of rotation rapidly rotating around an axis of symmetry, the axis of which can change its position in space.

When studying the movement of a gyroscope, we assume that:

A. The center of mass of the gyroscope coincides with its fixed point O. This gyroscope is called balanced.

b. Angular velocity w the rotation of the gyroscope around an axis is much greater than the angular velocity W of the movement of the axis in space, that is w >> W.

B. Gyroscope angular momentum vector L coincides with the angular velocity vector w , since the gyroscope rotates around the main axis of inertia.

Let a force act on the gyroscope axis F during time D t. According to the second law of dynamics for rotational motion, so the change in the angular momentum of the gyroscope during this time, (26.1)

Where r – radius vector drawn from a fixed point O to the point of action of the force (Fig. 73).

A change in the angular momentum of the gyroscope can be considered as a rotation of the gyroscope axis through an angle with angular velocity . (26.2)

Here is the component of the force acting on it normal to the gyroscope axis.

Under force F applied to the axis of the gyroscope, the axis rotates not in the direction of the force, but in the direction of the moment of force M relative to a fixed point O. At any moment of time, the speed of rotation of the gyroscope axis is proportional in magnitude to the moment of force, and with a constant arm of force, it is proportional to the force itself. Thus, movement of the gyroscope axis is inertia-free. This is the only case of inertia-free motion in mechanics.

The movement of the gyroscope axis under the influence of an external force is called forced precession gyroscope (from the Latin praecessio - movement ahead).

3. Shock action on the gyroscope axis. Let us determine the angular displacement of the gyroscope axis as a result of a short-term force on the axis, that is, an impact. Let within a short time dt on the gyroscope axis at a distance r from the center ABOUT force acts F . Under the influence of the impulse of this force F dt the axis rotates (Fig. 74) in the direction of the moment of force impulse it creates M dt at some angle

dq = W dt=(rF/Iw)dt. (26.3)

If the point of application of the force does not change, then r= const and upon integration we obtain. q = .(26.4)

The integral in each case depends on the type of function ( t). Under normal conditions, the angular velocity of rotation of the gyroscope is very high, so the numerator is most often much smaller than the denominator, and therefore the angle q– small value. A rapidly rotating gyroscope is resistant to impact - the greater the greater its angular momentum.

4. It is interesting that the force under which the gyroscope axis precesses does not do any work. This occurs because the point on the gyroscope to which a force is applied is at any moment displaced in a direction perpendicular to the direction of the force. Therefore, the scalar product of a force and a small displacement vector is always zero.

Forces in this manifestation are called gyroscopic. Thus, the Lorentz force acting on an electrically charged particle from the side of the magnetic field in which it moves is always gyroscopic.

5. CT equilibrium condition. For the CT to be in equilibrium, it is necessary that the sum of external forces and the sum of the moments of external forces be equal to zero:

. (26.5)

There are 4 types of equilibrium: stable, unstable, saddle-shaped and indifferent.

A. The equilibrium position of the TP is stable if, with small deviations from equilibrium, forces begin to act on the body, tending to return it to the equilibrium position.

Figure 75 shows situations of stable equilibrium of bodies in a gravity field. Gravity forces are mass forces, therefore the resultant of the gravity forces acting on the point elements of the TT is applied to the center of mass. In such situations, the center of mass is called the center of gravity.

A stable equilibrium position corresponds to the minimum potential energy of the body.

b. If, with small deviations from the equilibrium position, forces in the direction away from equilibrium begin to act on the body, then the equilibrium position is unstable. An unstable equilibrium position corresponds to a relative maximum of the potential energy of the body (Fig. 76).

V. A saddle-shaped equilibrium is when, when moving along one degree of freedom, the body's equilibrium is stable, and when moving along another degree of freedom, it is unstable. In the situation shown in Figure 77, the position of the body relative to the coordinate x is stable, and with respect to the coordinate y– unstable.

G. If, when a body deviates from the equilibrium position, no forces arise that tend to displace the body in one direction or another, then the equilibrium position is called indifferent. For example, a ball in a gravity field on an equipotential surface, a rigid body suspended at the center of mass point (at the center of gravity point) (Fig. 78).

§ 89. Free gyroscope and its basic properties

A gyroscope is a body rapidly rotating around its axis of symmetry, and the axis around which the rotation occurs can change its position in space. The gyroscope is a massive disk, which in almost all modern navigation devices is driven electrically, being the rotor of an electric motor.

Rice. 120.

A gyroscope that can rotate around the three specified axes is called a gyroscope with three degrees of freedom. The point where these axes intersect is called the gyroscope's suspension point. A gyroscope with three degrees of freedom, in which the center of gravity of the entire system, consisting of a rotor and cardan rings, coincides with the suspension point, is called balanced, or ac static, gyroscope.

A balanced gyroscope to which no external torques are applied is called free gyroscope.

Thanks to its rapid rotation, a free gyroscope acquires properties that are widely used in all gyroscopic devices. The main properties of a free gyroscope are the properties of stability and precession.

The first is that the main axis of a free gyroscope tends to maintain the direction initially given to it relative to world space. The stability of the main axis is greater, the more accurately the center of gravity of the system coincides with the suspension point, the less friction force in the axes of the gimbal, and the greater the weight of the gyroscope, its diameter and rotation speed. The quantity that characterizes the gyroscope from this qualitative aspect is called the kinetic moment of the gyroscope and is determined by the product of the moment of inertia of the gyroscope and its angular velocity of rotation, i.e.

Q is the angular velocity of rotation.

When designing gyroscopic devices, they strive to achieve a significant value of the kinetic moment H by giving the gyroscope rotor a special profile, as well as by increasing the angular speed of its rotation. Thus, in modern gyrocompasses, gyromotor rotors have a rotation speed of 6,000 to 30,000 rpm.

Rice. 121.

This property of a gyroscope was first demonstrated by the famous French physicist Leon Foucault in 1852. He also came up with the idea of ​​​​using a gyroscope as a device for determining the direction of movement and for determining the latitude of a ship at sea.

The property of precession is that, under the action of a force applied to the cardan rings, the main axis of the gyroscope moves in a plane perpendicular to the direction of the force (Fig. 121).

This movement of the gyroscope is called precessional. The precessional movement will occur during the entire time of action of the external force and stops when its action ceases. The direction of precessional motion is determined using the rule of poles, which is formulated as follows: when a moment of external force is applied to the gyroscope, the gyroscope pole tends to the force pole in the shortest way. The pole of a gyroscope is that end of its main axis, from which the rotation of the gyroscope is observed to occur counterclockwise. The force pole is that end of the gyroscope axis, relative to which an applied external force tends to rotate the gyroscope counterclockwise.

In Fig. The 121 precessional motion of the gyroscope is indicated by the arrow.

The angular velocity of precession can be calculated using the formula