National Technical University of Ukraine
“Kyiv Polytechnic Institute”
The abstract on a theme
Gyroscope. Gyroscopic devices
The student of group VL-62 DASS
Bilych Ivan Oleksandrovich
Kyiv 2008 y.
Table of contents
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. The device is a spinning wheel or disk whose axleis free to take any orientation. This orientation changes much less in response to a given external torque than it would without the large angular momentum associated with the gyroscope's high rate of spin. Since external torque is minimized by mounting the device in gimbals, its orientation remains nearly fixed, regardless of any motion of the platform on which it is mounted.
Within mechanical systems or devices, a conventional gyroscope is a mechanism comprising a rotor journaled to spin about one axis, the journals of the rotor being mounted in an inner gimbal or ring, the inner gimbal being journaled for oscillation in an outer gimbal which in turn is journaled for oscillation relative to a support. The outer gimbal or ring is mounted so as to pivot about an axis in its own plane determined by the support. The outer gimbal possesses one degree of rotational freedom and its axis possesses none. The inner gimbal is mounted in the outer gimbal so as to pivot about an axis in its own plane, which axis is always perpendicular to the pivotal axis of the outer gimbal.
The axle of the spinning wheel defines the spin axis. The inner gimbal possesses two degrees of rotational freedom and its axis possesses one. The rotor is journaled to spin about an axis which is always perpendicular to the axis of the inner gimbal. So, the rotor possesses three degrees of rotational freedom and its axis possesses two. The wheel responds to a force applied about the input axis by a reaction force about the output axis.
The behaviour of a gyroscope can be most easily appreciated by consideration of the front wheel of a bicycle. If the wheel is leaned away from the vertical so that the top of the wheel moves to the left, the forward rim of the wheel also turns to the left. In other words, rotation on one axis of the turning wheel produces rotation of the third axis.
A gyroscope flywheel will roll or resist about the output axis depending upon whether the output gimbals are of a free- or fixed- configuration. Examples of some free-output-gimbal devices would be the attitude reference gyroscopes used to sense or measure the pitch, roll and yaw attitude angles in a spacecraft or aircraft.
The center of gravity of the rotor can be in a fixed position. The rotor simultaneously spins about one axis and is capable of oscillating about the two other axes, and thus, except for its inherent resistance due to rotor spin, it is free to turn in any direction about the fixed point. Some gyroscopes have mechanical equivalents substituted for one or more of the elements, e.g., the spinning rotor may be suspended in a fluid, instead of being pivotally mounted in gimbals. A control moment gyroscope (CMG) is an example of a fixed-output-gimbal device that is used on spacecraft to hold or maintain a desired attitude angle or pointing direction using the gyroscopic resistance force.
In some special cases, the outer gimbal (or its equivalent) may be omitted so that the rotor has only two degrees of freedom. In other cases, the center of gravity of the rotor may be offset from the axis of oscillation, and thus the center of gravity of the rotor and the center of suspension of the rotor may not coincide.
The earliest known gyroscope was made by Johann Bohnenberger in 1817, although he called it simply the 'Machine'. The French mathematician Pierre-Simon Laplace, working at the École Polytechnique in Paris, recommended the machine for use as a teaching aid, and thus it came to the attention of Léon Foucault. In 1852, Foucault used it in an experiment involving the rotation of the Earth. It was Foucault who gave the device its modern name, in an experiment to see (Greek skopeein, to see) the Earth's rotation (Greek gyros, circle or rotation), although the experiment was unsuccessful due to friction, which effectively limited each trial to 8 to 10 minutes, too short a time to observe significant movement.
In the 1860s, electric motors made the concept feasible, leading to the first prototype gyrocompasses; the first functional marine gyrocompass was developed between 1905 and 1908 by German inventor Hermann Anschütz-Kaempfe. The American Elmer Sperry followed with his own design in 1910, and other nations soon realized the military importance of the invention— in an age in which naval might was the most significant measure of military power— and created their own gyroscope industries. The Sperry Gyroscope Companyquickly expanded to provide aircraft and naval stabilizers as well, and other gyroscope developers followed suit.
In 1917, the Chandler Company of Indianapolis, Indiana created the "Chandler gyroscope", a toy gyroscope with a pull string and pedestal. It has been in continuous production ever since and is considered a classic American toy.
MEMS gyroscopes take the idea of the Foucault pendulum and use a vibrating element, known as a MEMS (Micro Electro-Mechanical System). The MEMS based gyro was initially made practical and produceable by Systron Donner Inertial (SDI). Today, SDI is a large manufacturer of MEMS gyroscopes.
In the first several decades of the 20th century, other inventors attempted (unsuccessfully) to use gyroscopes as the basis for early black box navigational systems by creating a stable platform from which accurate acceleration measurements could be performed (in order to bypass the need for star sightings to calculate position). Similar principles were later employed in the development of inertial guidance systems for ballistic missiles.
A gyroscope exhibits a number of behaviours including precession and nutation. Gyroscopes can be used to construct gyrocompasses which complement or replace magnetic compasses (in ships, aircraft and spacecraft, vehicles in general), to assist in stability (bicycle, Hubble Space Telescope, ships, vehicles in general) or be used as part of an inertial guidance system. Gyroscopic effects are used in toys like tops, yo-yos, and Powerballs. Many other rotating devices, such as flywheels, behave gyroscopically although the gyroscopic effect is not used.
The fundamental equation describing the behavior of the gyroscope is:
where the vectors and are, respectively, the torque on the gyroscope and its angular momentum, the scalar is its moment of inertia, the vector is its angular velocity, and the vector is its angular acceleration.
It follows from this that a torque applied perpendicular to the axis of rotation, and therefore perpendicular to , results in a rotation about an axis perpendicular to both and . This motion is called precession. The angular velocity of precession is given by the cross product:
Precession can be demonstrated by placing a spinning gyroscope with its axis horizontal and supported loosely (frictionless toward precession) at one end. Instead of falling, as might be expected, the gyroscope appears to defy gravity by remaining with its axis horizontal, when the other end of the axis is left unsupported and the free end of the axis slowly describes a circle in a horizontal plane, the resulting precession turning. This effect is explained by the above equations. The torque on the gyroscope is supplied by a couple of forces: gravity acting downwards on the device's centre of mass, and an equal force acting upwards to support one end of the device. The rotation resulting from this torque is not downwards, as might be intuitively expected, causing the device to fall, but perpendicular to both the gravitational torque (horizontal and perpendicular to the axis of rotation) and the axis of rotation (horizontal and outwards from the point of support), i.e. about a vertical axis, causing the device to rotate slowly about the supporting point.
As the second equation shows, under a constant torque, the gyroscope's speed of precession is inversely proportional to its angular momentum. This means that, for instance, if friction causes the gyroscope's spin to slow down, the rate of precession increases. This continues until the device is unable to rotate fast enough to support its own weight, when it stops precessing and falls off its support, mostly because friction against precession cause another precession that goes to cause the fall.
By convention, these three vectors, torque, spin, and precession, are all oriented with respect to each other according to the right-hand rule.
To easily ascertain the direction of gyro effect, simply remember that a rolling wheel tends, when entering a corner, to turn over to the inside.
There is a considerable quantity of various gyroscopic devices. All of them have found the application in various areas. We will consider the most known and widespread examples which are used in the aviation and space industry on example of Gyrocompass and Control moment gyroscope.
A gyrocompass is similar to a gyroscope. It is a compass that finds true north by using an (electrically powered) fast-spinning wheel and friction forces in order to exploit the rotation of the Earth. Gyrocompasses are widely used on ships. They have two main advantages over magnetic compasses:
- they find true north, i.e., the direction of Earth's rotational axis, as opposed to magnetic north,
- they are far less susceptible to external magnetic fields, e.g. those created by ferrous metal in a ship's hull.
A gyrocompass is essentially a gyroscope, a spinning wheel mounted on gimbals so that the wheel's axis is free to orient itself in any way. Suppose it is spin up with its axis pointing in some direction other than the celestial pole. Because of the law of conservation of angular momentum, such a wheel will maintain its original orientation. Since the Earth rotates, it appears to a stationary observer on Earth that a gyroscope's axis is rotating once every 24 hours. Such a rotating gyroscope cannot be used for navigation. The crucial additional ingredient needed for a gyrocompass is some mechanism that results in applied torque whenever the compass's axis is not pointing north.
One method uses friction to apply the needed torque: the gyroscope in a gyrocompass is not completely free to reorient itself; if for instance a device connected to the axis is immersed in a viscous fluid, then that fluid will resist reorientation of the axis. This friction force caused by the fluid results in a torque acting on the axis, causing the axis to turn in a direction orthogonal to the torque (that is, to precess) towards the north celestial pole (approximately toward the North Star). Once the axis points toward the celestial pole, it will appear to be stationary and won't experience any more frictional forces. This is because true north is the only direction for which the gyroscope can remain on the surface of the earth and not be required to change. This is considered to be a point of minimum potential energy.
Another, more practical, method is to use weights to force the axis of the compass to remain horizontal with respect to the Earth's surface, but otherwise allow it to rotate freely within that plane. In this case, gravity will apply a torque forcing the compass's axis toward true north. Because the weights will confine the compass's axis to be horizontal with respect to the Earth's surface, the axis can never align with the Earth's axis (except on the Equator) and must realign itself as the Earth rotates. But with respect to the Earth's surface, the compass will appear to be stationary and pointing along the Earth's surface toward the true North Pole.
Since the operation of a gyrocompass crucially depends on the rotation of the Earth, it won't function correctly if the vessel it is mounted on is moving fast in an east to west direction.
The gyrocompass was patented in 1885 by the Dutch Marinus Gerardus van den Bos; however, his device never worked properly. In 1889, Captain Arthur Krebs designed an electric pendular gyroscope for the experimental French submarine Gymnote. It allowed the Gymnote to force a naval blockade in 1890. In 1903, the German Hermann Anschütz-Kaempfe (Raytheon Anschütz GmbH) constructed a working gyrocompass and obtained a patent on the design. In 1908, Anschütz-Kaempfe and the American inventor Elmer Ambrose Sperry patented the gyrocompass in Germany and the US. When Sperry attempted to sell this device to the German navy in 1914, Anschütz-Kaempfe sued for patent infringement. Sperry argued that Anschütz-Kaempfe's patent was invalid because it did not significantly improve on the earlier van den Bos patent. Albert Einstein testified in the case, first agreeing with Sperry but then reversing himself and finding that Anschütz-Kaempfe's patent was valid and that Sperry had infringed by using a specific damping method. Anschütz-Kaempfe won the case in 1915.
A control moment gyroscope (CMG) is an attitude control device generally used in satellite attitude control systems. CMGs control the orientation of spacecraft. A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotor’s angular momentum. As the rotor tilts, the changing angular momentum causes a gyroscopic torque that rotates the spacecraft.
CMGs differ from reaction wheels. The latter applies torque simply by changing rotor spin speed, but the former tilts the rotor's spin axis without necessarily changing its spin speed. CMGs are also far more power efficient. For a few hundred watts and about 100 kg of mass, large CMGs have produced thousands of newton meters of torque. A reaction wheel of similar capability would require megawatts of power.
CMGs have been used for decades in large spacecraft, including the Skylab and Mir space stations and the International Space Station. The first generation of commercial imaging satellites (QuickBird and Ikonos) used reaction wheels for attitude control. Today, only Ball Aerospace's WorldView-1 satellite built for DigitalGlobe (which provides data to Google Earth) is flying GMG's for attitude control. Ball Aerospace's next in the series, WV-2 will also use CMG's thus enabling the satellite to collect nearly 1 million sq kilometers per day.
The most effective CMGs include only a single gimbal. When the gimbal of such a CMG rotates, the change in direction of the rotor's angular momentum represents a torque that reacts onto the body to which the CMG is mounted, e.g. a spacecraft. Except for effects due to the motion of the spacecraft, this torque is due to a constraint, so it does no mechanical work (i.e. requires no energy). Single-Gimbal CMGs exchange angular momentum in a way that requires very little power, with the result that they can apply very large torques for minimal electrical input.
Such a CMG includes two gimbals per rotor. As an actuator it is more versatile than a single-gimbal CMG because it is capable of pointing the rotor's momentum vector in any direction. However, the torque caused by one gimbal's motion often must be reacted by the other gimbal on its way to the spacecraft, requiring more power for a given torque than a single-gimbal CMG. If the goal is simply to store momentum in a mass-efficient way, as in the case of the International Space Station, dual-gimbal CMGs are a good design choice. Instead, if a spacecraft requires large output torque per available input power, single-gimbal CMGs are a better choice.
Most CMGs hold the rotor speed constant. Some academic research has focused on the possibility of spinning up and down the rotor as the CMG gimbals. These so-called variable-speed CMGs (VSCMGs) offer few practical advantages, mostly because the output torque from the rotor is likely orders of magnitude smaller than that caused by the gimbal motion. So, this effect adds nothing of practical value on the time scale of the motion typical of CMGs. However, thanks to the additional degree of freedom, the variable-speed CMGs can be used to avoid the geometric singularity, which is the most serious drawback of the conventional CMGs. The VSCMGs also can be used as the mechanical battery which stores excess of electric energy in a form of kinetic energy of the flywheels.
At least three CMGs are necessary for linear control of spacecraft attitude. However, no matter how many CMGs a spacecraft uses, gimbal motion can lead to relative orientations that produce no usable output torque along certain directions. These orientations are known as "singularities" and are related to the kinematics of robotic systems that encounter limits on the end-effector velocities due to certain joint alignments. Avoiding these singularities is naturally of great interest, and several techniques have been proposed. David Bailey and others have argued (in patents and in academic publications) that merely avoiding the "divide by zero" error that is associated with these singularities is sufficient. Two more recent patents summarize competing approaches.
The ISS employs a total of four CMGs as primary actuating devices during normal flight mode operation. The objective of the CMG flight control system is to hold the space station at a fixed attitude relative to the surface of the Earth. In addition, it seeks a Torque Equilibrium Attitude (TEA), in which the combined torque contribution of gravity gradient, atmospheric drag, solar pressure, and geomagnetic interactions are minimized. In the presence of these continual environmental disturbances CMGs absorb momentum in an attempt to maintain the space station at a desired attitude. The CMGs may eventually saturate (absorbing momentum to the point where they can absorb no more), resulting in loss of effectiveness of the CMG array for control. Some kind of momentum management scheme (MMS) is necessary to allow the CMGs to hold a desired attitude and at the same time prevent CMG saturation. Since the CMGs are momentum-exchange devices, external control torques must be used to desaturate the CMGs, that is, bring the momentum back to nominal value. Some methods for unloading CMG momentum include the use of magnetic torques, reaction thrusters, and gravity gradient torque. For the space station, the gravity gradient torque approach is preferred because it requires no consumables or external hardware and because the gravity-gradient torque on the ISS can be very high.
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