Showing posts with label gyroscopic flight instruments. Show all posts
Showing posts with label gyroscopic flight instruments. Show all posts

Wednesday, September 10, 2008

Precession

From Wikipedia, the free encyclopedia

Precession of a gyroscope
Precession of a gyroscope

Precession refers to a change in the direction of the axis of a rotating object. In physics, there are two types of precession, torque-free and torque-induced, the latter being discussed here in more detail. In certain contexts, "precession" may refer to the precession that the Earth experiences, the effects of this type of precession on astronomical observation, or to the precession of orbital objects.

Torque-free

Torque-free precession occurs when the axis of rotation differs slightly from an axis about which the object can rotate stably: a maximum or minimum principal axis. Poinsot's construction is an elegant geometrical method for visualizing the torque-free motion of a rotating rigid body. For example, when a plate is thrown, the plate may have some rotation around an axis that is not its axis of symmetry. When an object is not perfectly solid, internal vortices will tend to damp torque-free precession.

Torque-induced

Torque-induced precession (gyroscopic precession) is the phenomenon in which the axis of a spinning object (e.g. a part of a gyroscope) "wobbles" when a torque is applied to it. The phenomenon is commonly seen in a spinning toy top, but all rotating objects can undergo precession. If the speed of the rotation and the magnitude of the torque are constant the axis will describe a cone, its movement at any instant being at right angles to the direction of the torque. In the case of a toy top, if the axis is not perfectly vertical the torque is applied by the force of gravity tending to tip it over.

The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.
The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.

The device depicted on the right here is gimbal mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis and the vertical pivot.

To distinguish between the two horizontal axes, rotation around the wheel hub will be called 'rolling', and rotation around the gimbal axis will be called 'pitching.' Rotation around the pivot axis is called 'spinning'.

First, imagine that the device is spinning around the pivot axis. Then some rotation around the wheelhub is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a torque around the gimbal axis.

In the picture, a section of the wheel has been named 'dm1'. When the rolling starts, section dm1 is at the perimeter of the spinning motion. Section dm1 has a lot of velocity and as it is forced closer to the center of rotation, it tends to move in the direction of the top-left arrow in the diagram (shown at 45o in the direction of rolling). Section dm2 of the wheel starts out at the center of rotation, and thus initially has zero velocity before the wheel is rolled. A force would be required to increase section dm2's velocity to the velocity at the perimeter of the pivot axis' plane. If that force is not provided then section dm2's inertia will make it move in the direction of the top-right arrow. Note that both arrows point in the same direction.

The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some rolling is added to rotation around a vertical axis.

It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous.

In the discussion above, the setup was kept unchanging by preventing rotation around the gimbal axis. In the case of a spinning top, when the spinning top is tilting, gravity exerts a torque. Instead of rolling over, the spinning top pitches. The pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity elicits gyroscopic precession rather than causing the spinning top to fall to its side.

Precession or gyroscopic considerations have an effect on bicycle performance at high speed. Precession is also the mechanism behind gyrocompasses.

Gyroscopic precession also plays a large role in the flight controls on helicopters. Since the driving force behind helicopters is the rotor disk (which rotates), gyroscopic precession comes into play. If the rotor disk is to be tilted forward (to gain forward velocity), its rotation requires that the downward net force on the blade be applied roughly 90 degrees (depending on blade configuration) before, or when the blade is to one side of the pilot and rotating forward.

To ensure the pilot's inputs are correct, the aircraft has corrective linkages which vary the blade pitch in advance of the blade's position relative to the swashplate. Although the swashplate moves in the intuitively correct direction, the blade pitch links are arranged to transmit the pitch in advance of the blade's position.

Physics

Classical (Newtonian)

The torque caused by the two opposing forces Fg and -Fg causes a change in the angular momentum L in the direction of that torque. This causes the top to precess.
The torque caused by the two opposing forces Fg and -Fg causes a change in the angular momentum L in the direction of that torque. This causes the top to precess.

Precession is the result of the angular velocity of rotation and the angular velocity produced by the torque. It is an angular velocity about a line which makes an angle with the permanent rotation axis, and this angle lies in a plane at right angles to the plane of the couple producing the torque. The permanent axis must turn towards this line, since the body cannot continue to rotate about any line which is not a principal axis of maximum moment of inertia; that is, the permanent axis turns in a direction at right angles to that in which the torque might be expected to turn it. If the rotating body is symmetrical and its motion unconstrained, and if the torque on the spin axis is at right angles to that axis, the axis of precession will be perpendicular to both the spin axis and torque axis.

Under these circumstances the angular velocity of precession is given by:

\boldsymbol\omega_p = \frac{Q}{I_s\boldsymbol\omega_s}

In which Is is the moment of inertia, \boldsymbol\omega_s is the angular velocity of spin about the spin axis, and Q is the torque. Using \boldsymbol\omega = \frac{2\pi}{T}, we find that the period of precession is given by:

T_p = \frac{4\pi^2I_s}{QT_s}

In which Is is the moment of inertia, Ts is the period of spin about the spin axis, and Q is the torque. In general the problem is more complicated than this, however.

An informal explanation of Precession: In a classic beginning physics demonstration, the instructor stands on a swiveling platform and holds a spinning bicycle wheel at arm's length. The wheel is vertical and the instructor is standing still. The instructor then tilts the wheel toward horizontal. This causes the instructor to start spinning slowly on the platform. Bringing the wheel back to vertical and tilting it the other way makes the instructor spin the other way. Why?

Imagine the wheel as a collection of small particles. Particles want to move in a straight line. In order for them to move in a circle there must be a force accelerating the particles toward the center of the circle (acceleration is a change in speed or direction or both — in this case just direction). This force is ultimately provided by bonds between the atoms in the wheel and spokes.

What happens when the instructor turns the spinning wheel from vertical to horizontal? Consider a particle somewhere on the wheel. If the wheel weren't being tilted, it would be accelerated around the circle as always. But since the wheel is tilting, it now has to follow a new path. A change in path is an acceleration, which in turn requires force (from the instructor's hands, transmitted through the spokes to the rim). Now consider the particle opposite the first particle on the wheel. It also has to change path, but in the opposite direction. Since the forces on opposite sides are in opposite directions, the result is torque. Each pair of opposite particles on the wheel contributes to the torque that causes the instructor to turn on the platform.

Tilting the wheel the other direction produces torque in the opposite direction, slowing the instructor's spin and eventually reversing it.

Relativistic

The special and general theories of relativity give three types of corrections to the Newtonian precession, of a gyroscope near a large mass such as the earth, described above. They are:

  • Thomas precession a special relativistic correction accounting for the observer being in a rotating non-inertial frame.
  • de Sitter precession a general relativistic correction accounting for the schwarzschild metric of curved space near a large non-rotating mass.
  • Lense-Thirring precession a general relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.

Of the Earth's axis

Main article: Precession (astronomy)
Precessional movement.
Precessional movement.

The Earth goes through one complete precession cycle in a period of approximately 25,800 years, during which the positions of stars as measured in the equatorial coordinate system will slowly change; the change is actually due to the change of the coordinates. Over this cycle the Earth's north axial pole moves from where it is now, within 1° of Polaris, in a circle around the ecliptic pole, with an angular radius of about 23.5 degrees (or approximately 23 degrees 27 arcminutes [1]). The shift is 1 degree in 180 years, where the angle is taken from the observer, not from the center of the circle.

Discovery of the precession of the equinoxes is generally attributed to the ancient Greek astronomer Hipparchus (ca. 150 B.C.), though the difference between the sidereal and tropical years was known to Aristarchus of Samos much earlier (ca. 280 B.C.). It was later explained by Newtonian physics. The Earth has a nonspherical shape, being oblate spheroid, bulging outward at the equator. The gravitational tidal forces of the Moon and Sun apply torque as they attempt to pull the equatorial bulge into the plane of the ecliptic. The portion of the precession due to the combined action of the Sun and the Moon is called lunisolar precession.

Revolution of a planet in its orbit around the Sun is also a form of rotary motion. (In this case, the combined system of Earth and Sun is rotating.) So the axis of a planet's orbital plane will also precess over time.

The major axis of each planet's elliptical orbit also precesses within its orbital plane, partly in response to perturbations in the form of the changing gravitational forces exerted by other planets. This is called perihelion precession or apsidal precession (see apsis). Discrepancies between the observed perihelion precession rate of the planet Mercury and that predicted by classical mechanics were prominent among the forms of experimental evidence leading to the acceptance of Einstein's Theory of Relativity, which predicted the anomalies accurately.[2][3]

These periodic changes of Earth's orbital parameters, combined with the precession of the equinoxes and of the inclination of the Earth's axis on its orbit, are an important part of the astronomical theory of ice ages. For precession of the lunar orbit see lunar precession.

A phenomenon analogous to apsidal precession is nodal precession (see orbital node), which affects the orientation of the orbital plane.

Wikimedia Commons has media related to:

See also

References

  1. ^ Cook, David R. (1999). "Tilt of Earth's Axis". Environmental Earth Science Archive, Ask A Scientist. United States Department of Energy. Retrieved on 2006-05-24.
  2. ^ Max Born (1924), Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)
  3. ^ An even larger value for a precession has been found, for a black hole in orbit around a much more massive black hole, amounting to 39 degrees each orbit.
  • "Moon and Spica", StarDate July 14, 2005, University of Texas McDonald Observatory, [1]

External links

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Gyroscope

From Wikipedia, the free encyclopedia

A gyroscope
A gyroscope
For other uses, see Gyroscope (disambiguation).

A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum[1][2]. The device is a spinning wheel or disk whose axle is 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.

Description and diagram

Diagram of a gyro wheel. Reaction arrows about the output axis (blue) correspond to forces applied about the input axis (green), and vice versa.
Diagram of a gyro wheel. Reaction arrows about the output axis (blue) correspond to forces applied about the input axis (green), and vice versa.

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.

Animation of a gyro wheel in action
Animation of a gyro wheel in action

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.

History

Gyroscope invented by Léon Foucault, and built by Dumoulin-Froment, 1852. National Conservatory of Arts and Crafts museum, Paris.
Gyroscope invented by Léon Foucault, and built by Dumoulin-Froment, 1852. National Conservatory of Arts and Crafts museum, Paris.

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. [3] 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 Company quickly expanded to provide aircraft and naval stabilizers as well, and other gyroscope developers followed suit.[4]

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.[5]

Properties

A gyroscope in operation with freedom in all three axes. The rotor will maintain its spin axis direction regardless of the orientation of the outer frame.
A gyroscope in operation with freedom in all three axes. The rotor will maintain its spin axis direction regardless of the orientation of the outer frame.

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 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:

\boldsymbol\tau={{d \mathbf{L}}\over {dt}}={{d(I\boldsymbol\omega)} \over {dt}}=I\boldsymbol\alpha

where the vectors \boldsymbol\tau and \mathbf{L} are, respectively, the torque on the gyroscope and its angular momentum, the scalar I\, is its moment of inertia, the vector \boldsymbol\omega is its angular velocity, and the vector \boldsymbol\alpha is its angular acceleration.

It follows from this that a torque \boldsymbol\tau applied perpendicular to the axis of rotation, and therefore perpendicular to \mathbf{L}, results in a rotation about an axis perpendicular to both \boldsymbol\tau and \mathbf{L}. This motion is called precession. The angular velocity of precession \boldsymbol\Omega_P is given by the cross product:

\boldsymbol\tau=\boldsymbol\Omega_P \times \mathbf{L}
Precession on a gyroscope
Precession on a gyroscope

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.

Gyrostat

A gyrostat is a variant of the gyroscope. The first gyrostat was designed by Lord Kelvin to illustrate the more complicated state of motion of a spinning body when free to wander about on a horizontal plane, like a top spun on the pavement, or a hoop or bicycle on the road. It consists of a massive flywheel concealed in a solid casing. Its behaviour on a table, or with various modes of suspension or support, serves to illustrate the curious reversal of the ordinary laws of static equilibrium due to the gyrostatic behaviour of the interior invisible flywheel when rotated rapidly.

U.S. Patents

In the USPTO classification scheme, the generic locus for gyroscope patents is Class 74, Machine element or mechanism, and Subclass 5R. Every rotating body has gyroscopic action, but such devices are not included unless at least one axis of oscillation is present. The combinations of gyroscopes with other devices are placed in subclass 5.22.

Numbers
Reissued

See also

References

  1. ^ "Gyroscope" by Sándor Kabai, The Wolfram Demonstrations Project.
  2. ^ "Total Angular Momentum", ScienceWorld
  3. ^ Wagner JF, "The Machine of Bohnenberger", The Institute of Navigation [1]
  4. ^ MacKenzie, Donald. Inventing Accuracy: A Historical Sociology of Nuclear Missile Guidance. Cambridge: MIT Press, 1990. pp 31-40. ISBN 0-262-13258-3
  5. ^ MacKenzie, pp 40-42.

External articles and further readings

Books
  • Felix Klein and Arnold Sommerfeld, "Über die Theorie des Kreisels" (Tr., About the theory of the gyroscope). Leipzig, Berlin, B.G. Teubner, 1898-1914. 4 v. illus. 25 cm.
  • Audin, M. Spinning Tops: A Course on Integrable Systems. New York: Cambridge University Press, 1996.
Websites
Papers
Lectures
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