The law of dynamics of rotational motion for a rigid body has the form:

A similar expression can be obtained if we consider the rotational motion of a mechanical system relative to a fixed axis. In this case - the total angular momentum of the system, - the total moment of external forces applied to the system.

If the total moment of all external forces acting on a physical object (system) is equal to zero, i.e. system is closed, then for a closed system .

Consequently: .

The last expression is law of conservation of angular momentum: the angular momentum of a closed system is conserved (does not change) over time.

This is a fundamental law of nature. It is associated with the symmetry property of space - its isotropy, i.e. with the invariance of physical laws with respect to the choice of the direction of the coordinate axes of the reference system (with respect to the rotation of a closed system in space by any angle).

In order to keep the position of the axis of rotation of a rigid body unchanged over time, bearings are used in which the axis is held. However, there are such 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 proved 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 bodies).

For example, the main axes of inertia of a homogeneous rectangular parallelepiped pass through the centers of opposite faces (Fig. 3.1).

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 of the body.

It can be shown that the rotation around the principal axes with the largest and smallest moments of inertia is stable, and the rotation around the axis with the average moment is unstable. So, if you throw up a body that has the shape of a parallelepiped, bringing it into rotation at the same time, then it, falling, will steadily rotate around axes 1 and 2 (Fig. 3.1).

The property of free axes to maintain their position in space is widely used in engineering. Most interesting in this respect gyroscopes- massive homogeneous bodies rotating at a high angular velocity around their axis of symmetry, which is a free axis.

In order for the axis of the gyroscope to change its direction in space, the moment of external forces must be different 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 the gyroscopic effect is observed. It consists in the fact that under the action of a pair of forces applied to the axis of a rotating gyroscope (Fig. 3.2), the axis deviates in a direction perpendicular to the direction of the forces. The gyroscopic effect is explained by the fact that the moment of forces is directed along the straight line O 2 O 2. During the time dt, the momentum of the gyroscope will receive an increment , co-directed with the momentum vector. The direction of the vector coincides with the new direction of the axis of rotation of the gyroscope. Thus, the axis of rotation of the gyroscope will turn around the straight line O 3 O 3 . The movement of the axis of the angular momentum of the gyroscope as a result of the action of external forces on it is called precession.

If the axis of the gyroscope is fixed with bearings, then due to the gyroscopic effect, gyroscopic forces arise that act on the supports. Gyroscopes are used in various gyroscopic navigational instruments (gyrocompass, gyrohorizon, etc.). Another important application of gyroscopes is maintaining a given orientation of an object in space (gyroscopic platforms).

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, but . 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 within 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 L B 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, like 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. 8.

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. 8, but 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. 8, b moreover, he was given a certain push forward, and in the case of fig. 8, 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 FROM (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. 10

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. 13

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. 15

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. 16

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, but, 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

The main errors of gyroscopes are own care, gimbal error, slope error And apparent withdrawal.

• Value own care determined by the friction and balancing of the moving parts of the gyroscope.
• gimbal error is the difference between the heading angle, measured in the horizontal plane, and the gyrocompass readings when the outer frame axis is tilted (roll or pitch) from the vertical position.
• Slope error appears during turns and occurs in connection with the operation of the correction device, which ensures that the position of the gyroscope rotor is perpendicular to the plane of the outer frame of the gyro unit. Unlike the gimbal error, the turn error continuously accumulates during the turn and does not disappear after it ends. To reduce it, turn off the horizontal correction of the gyroscope during turns.
• Apparent care is caused by the fact that a free three-degree gyroscope keeps the direction of its axis unchanged in space relative to fixed stars, but by no means relative to the Earth and its planes. The Earth itself moves in space, therefore, even an absolutely motionless gyroscope in space rotates relative to the Earth, creating a visible apparent movement of its axis. To understand this phenomenon, let us recall the Foucault pendulum. A swinging pendulum is a kind of gyroscope. Therefore, looking at it, we can observe (unless, of course, we are on the equator) the rotation of the Earth around its axis.

The accuracy of the coincidence of the center of gravity of the gyroscopic system with the point of suspension (balance), the magnitude of the friction force in the axes of the gimbal, weight, diameter and speed of rotation are the determining factors for the stability of the axis of the gyroscope. When external forces act on the gimbal system, the gyroscope axis moves in a plane perpendicular to the direction of the force. This movement of the gyroscope is called precession. Precession stops with the termination of the impact on the gyroscope. In artificial horizons, it is required to keep the gyroscope in a vertical position during evolutions and changes in the speed of the aircraft. To reduce the accumulated errors, it is necessary to correct the position of the gyroscope by mechanisms vertical correction. As a vertical sensor, pendulum correction systems are used, which ensure that the lower end of the gyroscope axis is directed towards the center of the Earth. Pendulum systems are subject to the influence of accelerations that occur during maneuvering. As an example, one can cite a phenomenon called "air horizon roll" (indication of non-zero pitch or roll values ​​in straight flight after the completion of the maneuver). Therefore, at the stages of maneuvering, the correction systems are turned off. The error of the gyroscope readings will depend on the correction speed, the speed of its own departure, and the parameters of the correction switch. On the first pneumatic artificial horizons, the correction was not turned off on the turn. Therefore, the correction rate was chosen to be very small so that the gyroscope drift would not be significant during the turn. Accordingly, the vertical recovery time increased. Later, the correction began to be turned off in a turn, and on some, even during accelerations (AGD -1). Currently, inertial gyro-verticals are used, in which accuracy is achieved by creating an artificial pendulum with a "length" equal to the radius of the Earth.

Compensated by apparent care the gyroscope is a pointer

Experience shows that the precessional motion of a gyroscope under the action of external forces is generally more complicated 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 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, like 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 giving the gyroscope an initial push of a quite definite 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.

In order to keep the position of the axis of rotation of a rigid body unchanged over time, bearings are used in which it is held. However, there are such 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 proved 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 bodies). 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 the rotation around the principal axes with the largest and smallest moments of inertia is stable, and the rotation around the axis with the average moment is unstable. So, if you toss a body that has the shape of a parallelepiped, bringing it into rotation at the same time, then it, falling, 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, fixed 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 a stick rotating around a free axis is released from external bonds (carefully remove the upper end of the thread from the spindle hook), then the position of the axis of rotation in space is preserved for some time. The property of free axes to maintain their position in space is widely used in engineering. Most interesting in this respect gyroscopes- massive homogeneous bodies rotating at a high angular velocity around their axis of symmetry, which is a free axis.

Consider one of the varieties of gyroscopes - a gyroscope on a gimbal suspension (Fig. 32). A disk-shaped body - a gyroscope - is fixed on an axis AA, which can rotate about a horizontal axis perpendicular to it VV, 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 stationary, 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 momentum of the rings.

Since the friction in the bearings is small, while the gyroscope is stationary, its axis can be given any direction. If you start to rotate the gyroscope quickly (for example, using a rope wound around the axis) and turn its stand, then the gyroscope axis retains its position in space unchanged. This can be explained using the basic law of rotational motion dynamics. For a free rotating gyroscope, 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 from it retains its position in space and the axis of the gyroscope.

In order for the axis of the gyroscope to change its direction in space, it is necessary, according to (19.3), that the moment of external forces be different 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 action of a pair of forces F applied to the axis of the 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 , as it would seem natural 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 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 . Over time dt angular momentum L gyroscope will be incremented d L = M dt (direction d L coincides with the direction M) and becomes equal to 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 small, then, although the moment of forces M and large, the change in angular momentum d L the gyroscope will also be quite 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.

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 axis of the gyroscope rotates. Their action must be taken into account when designing devices containing rapidly rotating massive components. Gyroscopic forces only make sense in a rotating frame of reference and are a special case of the Coriolis force of inertia (see §27).

Gyroscopes are used in various gyroscopic navigational instruments (gyrocompass, gyrohorizon, etc.). Another important application of gyroscopes is to maintain a given direction of movement of vehicles, for example, a vessel (autopilot) and an aircraft (autopilot), etc. With any deviation from the course due to some influences (waves, gusts of wind, etc.), the position of the axis gyroscope in space is preserved. Consequently, the axis of the gyroscope, together with the gimbal frames, rotates relative to the moving device. Turning 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.