gyroscope

a rapidly spinning solid body whose axis of rotation can change its orientation in space. The gyroscope has a number of interesting properties, which are observed in rotating celestial bodies, artillery shells, toy tops, turbine rotors on ships, and other devices. Gyroscopic properties form the basis of various devices and instruments that are widely applied in modern technology for the automatic guidance of airplanes, ships, rockets, torpedoes, and other objects; for the determination of the horizon or the geographic meridian; for the measurement of translational or angular velocities of moving objects (for example, rockets); and for many other purposes.

The properties of a gyroscope are manifested when two conditions are fulfilled: (1) the axis of rotation of the gyroscope must be capable of changing its orientation in space, and (2) the angular velocity of the gyroscope’s rotation about its axis must be very large compared to the angular velocity of the axis itself when it changes its orientation.

A toy top that is rotating rapidly about its axis OA (Figure 1) is the simplest gyroscope; the axis OA may change its direction in space, since its end A is not fixed. In the gyroscopes used in engineering, free rotation of the axis may be achieved by mounting it in the frames (rings) of the so-called gimbal suspension (see Figure 2), which permits the axis AB to assume any position in space. Such a gyroscope has three degrees of freedom: it may perform three independent rotations about its axes AB, DE, and GK, which intersect in the center of the suspension O, which remains immobile with respect to the base. If the center of gravity of the gyroscope coincides with the center O, then the gyroscope is called astatic (balanced), whereas in the contrary case, it is called heavy.

Figure 1. Toy top: OA is its axis, and P is the force of gravity

The first property of a balanced gyroscope with three degrees of freedom is that its axis persistently tends to preserve its initial orientation in space. If this axis has been initially aimed at some star, then after any random impacts or displacement of the instrument’s base, it will continue to point toward this star, while at the same time changing its orientation with respect to the earth’s axes. This property was utilized for the first time by the French scientist J. Foucault for the experimental proof of the earth’s rotation about its axis (1852). This is also the origin of the name “gyroscope” itself, which means “to observe rotation.”

The second property of the gyroscope is manifested when its axis (or frame) is subjected to a force or a pair of forces

Figure 2. Gyroscope in a gimbal suspension. The rotor C may, in addition to the rotation about its axis AB, also rotate together with frame 1 about the axis DE and, together with frame 2, about the axis GK; consequently, the rotor axis may assume any position in space. O is the center of suspension, which coincides with the center of gravity of the gyroscope.

tending to rotate the axis (that is, the forces generating a rotational moment with respect to the center of the suspension). Under the influence of the force P (Figure 3), the end A of the axis AB of the gyroscope will be tilted not in the direction of the action of the force, as would be the case with a rotor at rest, but at right angles to this force. Consequently, the gyroscope, as well as its frame, starts to rotate about the axis DE, but with a constant angular velocity rather than with acceleration. This rotation is called precession; the faster the rotation of the gyroscope itself about its axis AB, the slower the precession. If at some moment of time the action of the force ceases, then precession ceases simultaneously, and the axis AB comes to rest instantaneously—that is, the precessional movement of the gyroscope is noninertial.

Figure 3. Action of the force P on a gyroscope with a spinning rotor; the axis AB moves perpendicular to the direction of P.

The magnitude of the angular velocity of precession is determined from the equation

where M is the moment of the force P with respect to the center O, α = ‹ AOE, Ω is the angular velocity of the gyroscope’s own rotation about the axis AB, I is the moment of inertia of the gyroscope with respect to the same axis, and h = AO is the distance from the point of application of the force to the center of the gyroscope; the second equality is valid when the force P is parallel to the axis DE. Equation (1) indicates directly that the greater Ω, or more precisely, the greater the quantity H = IΩ, which is called the intrinsic moment of momentum of the gyroscope, the slower the precession. The direction of precession may be determined as indicated in Figure 4.

Figure 4. Rule for determining the direction of precession: Viewing the rotor from the point of application of the force P, it is necessary to establish whether the rotor is turning clockwise or counterclockwise. The vector AP should then be mentally rotated 90° about the axis AB in the same direction (clockwise or counterclockwise, respectively); then it will indicate the direction of precession (in this case, AC).

In addition to precession, the axis of the gyroscope may also perform under the influence of a force called nutation, which consists of small but fast oscillations of the axis about its mean position (they are usually undetectable by the eye). The amplitudes of these oscillations are very small in the case of rapidly rotating gyroscopes, and because of the inevitable presence of resistances, these oscillations are subject to rapid decay. This makes it possible to neglect nutation in solving most engineering problems and to formulate the so-called elementary theory of gyroscopes taking into account only the precession, whose velocity is determined by equation (1). Precessional movement may be observed in the toy top (Figure 5,a), in which case the role of the center of suspension is played by the point of support O. If the axis of such a top is adjusted to the angle AOE to the vertical and released, then under the influence of the force of gravity P its axis will tilt not in the direction of the action of this force (that is, not downward) but at right angles, and the axis will begin to precess about the vertical. The precession of the top is also accompanied by visually undetectable nutational oscillations, which decay rapidly because of air resistance. Because of air friction, the proper rotation of the top gradually slows down, and the precession velocity ω correspondingly increases. When the angular velocity of the top becomes smaller than a certain minimum velocity, the top becomes unstable and falls. In the case of slowly rotating tops, the nutational oscillations may be quite noticeable and may, upon superimposition on the precession, substantially change the pattern of the top’s axial movement: the end A of the axis will describe a clearly visible wavy or looped curve, alternately departing from and approaching the vertical (Figure 5,b).

Figure 5. (a) Precession of a top under the influence of the force of gravity, (b) motion of the top’s axis during slow proper rotation.

Another example of precessional movement is the artillery shell or a bullet. In addition to the force of gravity, a moving shell is also affected by the forces of air resistance, whose resultant R is directed approximately opposite to the velocity of the center of gravity of the shell and is applied to a point above the center of gravity of the shell (Figure 6,a). A non-spinning shell will tumble under the influence of air resistance, and its flight will become disorderly (Figure 6,b). This will result in a significantly increased resistance to motion, the flight range will decrease, and the shell will not strike the target head on. A spinning shell, however, has all of the properties of a gyroscope, and the force of air resistance gives rise to its deviation in a perpendicular direction rather than in the direction of action of the force. As a result, the axis of the shell precesses slowly about the straight line of the direction of the velocity v_c—that is, about the tangent to the trajectory of the center of gravity of the shell (Figure 6,c)—which makes the flight regular and permits the descending shell to strike the target head on. The planet earth is also a gigantic gyroscope, which precesses.

If the axis AB of the gyroscope rotor is supported in one frame, which can rotate with respect to the base of the instrument about the axis DE (Figure 7), then the gyroscope will be able to rotate only about the two axes AB and DE, meaning that it will have only two degrees of freedom. Such a gyroscope does not have any of the properties of a gyroscope with three degrees of freedom, but it has another very interesting property: if the gyroscope base is subjected to induced rotation with angular velocity ω about the axis KL,

Figure 6. (a) Precession of an artillery shell, (b) diagram of the motion and trajectory of a nonspinning shell, (c) diagram of the motion and trajectory of a spinning shell.

which forms the angle a with the axis AB, then a pair of forces with the gyroscopic moment

(2) M_gyr = IΩsin α

will begin to act on the axis of the rotor from the direction of the bearings A and B. This pair of forces tends to establish the gyroscopic rotor in the shortest possible way in a position parallel to the axis KL in such a way that the rotor’s rotation, as well as the induced rotation, will occur in the same direction.

Figure 7. Gyroscope with two degrees of freedom

Finally, let us consider a rotor whose axis AB is directly attached to the base D (Figure 8). If this base is fixed, then the axis cannot change its orientation in space, and consequently, the rotor does not have any gyroscopic properties. However, if the base is rotated about some axis KL with the angular velocity ω, then in accordance with the preceding rule, the axis AB will tend to become parallel to the axis KL. This motion will be resisted by the bearings supporting the axis. Consequently, the rotor will press against the bearings A and B with forces F_l and F₂, which are called the gyroscopic forces.

Figure 8. Action of gyroscopic forces on the bearings supporting the axis upon rotation of the base of the device about the axis KL

Seagoing ships and propeller aircraft contain many rotating parts: the engine shaft, the turbine or dynamo rotor, ship’s screws and airscrews, and so on. Turning motions of the ship or aircraft, as well as rolling, subject the bearings supporting these rotating parts to the gyroscopic forces mentioned previously, which must be taken into account in the corresponding engineering calculations. The magnitudes of these forces may reach several tons and, if the bearing supports are not suitably designed, damage may occur.

The theory of gyroscopes is a very important section of the dynamics of, solids with a fixed point. The gyroscopic properties are the consequences of the laws governing the motion of such bodies. The first property of a gyroscope with three degrees of freedom is a manifestation of the law of conservation of the angular momentum, whereas the second property is a manifestation of one of the theorems of dynamics, according to which changes in time of the angular momentum of the body are equal to the force moment acting on the body.

Gyroscopes in engineering. Gyroscopes that are used in engineering applications are usually designed as flywheels with a thick rim, with a weight of several grams-force to several dozen kilograms-force, supported by a gimbal suspension. To impart rapid rotation to the gyroscope, it is made the rotor of a high-speed AC or DC electric motor. In aeronautics, gyroscopes are used with rotors in the form of an air turbine that is driven by the airflow. Sometimes gyroscopes are designed in spherical form suspended on an air film formed by a supply of compressed air. A number of designs use the float gyroscope, whose rotor is enclosed in a sleeve floating in a fluid; this takes the load off the bearings and significantly decreases the frictional moment.

The design of gyroscopic devices is based on certain properties of gyroscopes with two or three degrees of freedom. The property of gyroscopes with three degrees of freedom to preserve invariably the orientation of their axis in space is utilized in the construction of instruments for the automatic control of the motion of airplanes (for example, autopilots), rockets, ships, and torpedoes. The gyroscope in these instruments plays the role of a sensing element that records the deviation of a moving object from a given course. At the same time, the instrument contains a servomechanism, which receives the signal concerning the deviation, amplifies it, and transmits it to the power device (motor), which returns the object to the given course, usually with the aid of rudders. The second property of gyroscopes with three degrees of freedom—precession under the influence of an applied force—forms the basis for the directional gyroscope (course indicator) and for important navigational instruments: the gyrocompass, which determines the geographic meridian, and the vertical gyroscope (azimuth gyroscope), which determines the true vertical direction (horizon).

In launching rockets, the rate of the rocket’s vertical ascent must be known with a high degree of accuracy. This apparently very difficult task is also readily solved by a precessing gyroscope.

Gyroscopes with two degrees of freedom are also frequently utilized in gyroscopic instruments. These instruments include the turn indicator in aviation, as well as some types of gyrostabilizers, particularly devices for stabilizing objects in space (for example, an earth satellite).

Modern technology places very high precision requirements on gyroscopic devices, which causes great technical difficulties in their production. For example, in some instruments with a rotor weight on the order of 1 kilogram-force, the required precision limits the displacement of the center of gravity from the center of suspension to fractions of a micron; otherwise the gravity force moment may lead to an undesirable precession (drift) of the axis of the gyroscope. In addition, the accuracy of readings of instruments with gyroscopes suspended by gimbals is affected by axle friction. All these considerations have led to the development of gyroscopes based on other physical principles rather than on purely mechanical principles.