Precession of a gyroscope

Precession is a change in the orientation of the rotational axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle (nutation) is constant. In physics, there are two types of precession: torque-free and torque-induced.

In astronomy, "precession" refers to any of several slow changes in an astronomical body's rotational or orbital parameters, and especially to the Earth's precession of the equinoxes. See Precession (astronomy).

Torque-free[link]

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. This occurs because the angular momentum (L) is constant in absence of torques. Therefore, it will have to be constant in the external reference frame, but the moment of inertia tensor (I) is non-constant in this frame because of the lack of symmetry. Therefore, the spin angular velocity vector (Failed to parse (Missing texvc executable; please see math/README to configure.): \omega_s ) about the spin axis will have to evolve in time so that the matrix product Failed to parse (Missing texvc executable; please see math/README to configure.): L = I \omega_s

remains constant.

When an object is not perfectly solid, internal vortices will tend to damp torque-free precession, and the rotation axis will align itself with one of the inertia axes of the body.

The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\omega_p = (I_s/I_p) \omega_s (1-\cos(\alpha))

where Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\omega_p

is the precession rate, Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\omega_s
is the spin rate about the axis of symmetry, Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\alpha
is the angle between the axis of symmetry and the axis about which it precesses, Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol I_s
is the moment of inertia about the axis of symmetry, and Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol I_p
is moment of inertia about either of the other two perpendicular principal axes. They should be the same, due to the symmetry of the disk.[1]

For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol R

that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal moment of inertia tensor Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol I_0
and fixed external angular momentum Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol L

, the instantaneous angular velocity is Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\omega(\boldsymbol R) = \boldsymbol R \boldsymbol I_0^{-1} \boldsymbol R ^T \boldsymbol L . Precession occurs by repeatedly recalculating Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol \omega

and applying a small rotation vector Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol \omega dt
for the short time Failed to parse (Missing texvc executable; please see math/README to configure.): dt

, e.g. Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol R_{new} = \exp([\boldsymbol\omega(\boldsymbol R_{old})]_{\times} dt) \boldsymbol R_{old}

for the skew-symmetric matrix Failed to parse (Missing texvc executable; please see math/README to configure.): [\boldsymbol\omega]_{\times}

. The errors induced by finite time steps tend to increase the rotational kinetic energy, Failed to parse (Missing texvc executable; please see math/README to configure.): E(\boldsymbol R) = \boldsymbol \omega(\boldsymbol R) \cdot \boldsymbol L / 2

this unphysical tendency can be counter-acted by repeatedly applying a small rotation vector Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol v

perpendicular to both Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol \omega
and Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol L

, noting that Failed to parse (Missing texvc executable; please see math/README to configure.): E(\exp([\boldsymbol v]_{\times}) \boldsymbol R) \approx E(\boldsymbol R) + (\boldsymbol \omega(\boldsymbol R) \times \boldsymbol L) \cdot \boldsymbol v .

Another type of torque-free precession can occur when there are multiple reference frames at work. For example, the earth is subject to local torque induced precession due to the gravity of the sun and moon acting upon the earth’s axis, but at the same time the solar system is moving around the galactic center. As a consequence, an accurate measurement of the earth’s axial reorientation relative to objects outside the frame of the moving galaxy (such as distant quasars commonly used as precession measurement reference points) must account for a minor amount of non-local torque-free precession, due to the solar system’s motion.

Torque-induced[link]

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, which causes a distribution of force around the acted axis. 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 device depicted on the right 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 'spinning', and rotation around the gimbal axis will be called 'pitching.' Rotation around the vertical pivot axis is called 'rotation'.

First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (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. At the depicted moment in time, section dm₁ is at the perimeter of the rotating motion around the (vertical) pivot axis. Section dm₁, therefore, has a lot of angular rotating velocity with respect to the rotation around the pivot axis, and as dm₁ is forced closer to the pivot axis of the rotation (by the wheel spinning further), due to the Coriolis effect dm₁ tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis. Section dm₂ of the wheel starts out at the vertical pivot axis, and thus initially has zero angular rotating velocity with respect to the rotation around the pivot axis, before the wheel spins further. A force (again, a Coriolis force) would be required to increase section dm₂'s velocity up to the angular rotating velocity at the perimeter of the rotating motion around the pivot axis. 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 spinning 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 pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity - via the pitching motion - elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) 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 that 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.

Classical (Newtonian)[link]

The torque caused by the two opposing forces F_g and -F_g 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 that 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 that 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\omega_p = \frac{\ mgr}{I_s\boldsymbol\omega_s}

In which I_s is the moment of inertia, Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\omega_s

is the angular velocity of spin about the spin axis, and m*g and r are the force responsible for the torque and the perpendicular distance of the spin axis about the axis of precession.  The torque vector originates at the center of mass. Using Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\omega
= Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{2\pi}{T}

, we find that the period of precession is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): T_p = \frac{4\pi^2I_s}{\ mgrT_s}

In which I_s is the moment of inertia, T_s is the period of spin about the spin axis, and Failed to parse (Missing texvc executable; please see math/README to configure.): \boldsymbol\tau

is the torque. In general, the problem is more complicated than this, however.

Relativistic[link]

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's 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.

Astronomy[link]

In astronomy, precession refers to any of several gravity-induced, slow and continuous changes in an astronomical body's rotational axis or orbital path. Precession of the equinoxes, perihelion precession, and changes in the tilt of the Earth's axis to its orbit, and the eccentricity of its orbit over tens of thousands of years are all important parts of the astronomical theory of ice ages.

Axial precession (precession of the equinoxes)[link]

Precessional movement.

Precession of the equinox in relation to the distant stars

Axial precession is the movement of the rotational axis of an astronomical body, whereby the axis slowly traces out a cone. In the case of Earth, this type of precession is also known as the precession of the equinoxes, lunisolar precession, or precession of the equator. Earth goes through one such complete precessional cycle in a period of approximately 26,000 years or 1° every 72 years, during which the positions of stars will slowly change in both equatorial coordinates and ecliptic longitude. Over this cycle, 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.

Hipparchus is the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1º per century (which is not far from the actual value for antiquity, 1.38º).^[2] The precession of Earth's axis was later explained by Newtonian physics. Being an oblate spheroid, the Earth has a nonspherical shape, bulging outward at the equator. The gravitational tidal forces of the Moon and Sun apply torque to the equator, attempting to pull the equatorial bulge into the plane of the ecliptic, but instead causing it to precess.

Perihelion precession[link]

Planets revolving the Sun follow elliptical (oval) orbits that rotate gradually over time (apsidal precession). The eccentricity of this ellipse is exaggerated for visualization. Most orbits in the Solar System have a much smaller eccentricity, making them nearly circular. (click on image to see the animation)

The orbit of a planet around the Sun is not really an ellipse but a flower-petal shape because 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.

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 (in particular, his General Theory of Relativity), which accurately predicted the anomalies.^[3][4]

See also nodal precession. For precession of the lunar orbit see lunar precession.

See also[link]

Wikimedia Commons has media related to: Precession

• De Sitter precession
• Larmor precession
• Lense–Thirring precession
• Nutation
• Polar motion
• Precession (mechanical)
• Thomas precession
• Euler angles

References[link]

External links[link]

Wikibooks has a book on the topic of Rotational Motion

• Explanation and derivation of formula for precession of a top
• Precession and the Milankovich theory from educational web site From Stargazers to Starships

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (September 2009)

Graham Hancock
Born	(1950-08-02)August 2, 1950 (age 61) Edinburgh, Scotland, UK
Nationality	British
Citizenship	British
Alma mater	Durham University
Occupation	Author
Known for	Author, The Sign and the Seal, Fingerprints of the Gods, The Message of the Sphinx

Graham Hancock (born 2 August 1950 in Edinburgh, Scotland) is a British writer and journalist. Hancock specialises in unconventional theories ^[1] involving ancient civilizations, stone monuments or megaliths, altered states of consciousness, ancient myths and astronomical/astrological data from the past. One of the main themes running through many of his books is the possible global connection with a "mother culture" from which he believes all ancient historical civilizations sprang.^[2]

Biography[link]

Born in Edinburgh, Scotland, Hancock's formative years were spent in India, where his father worked as a surgeon. Having returned to the UK, he graduated from Durham University in 1973, receiving a First Class Honours degree in Sociology.

As a journalist, Hancock worked for many British papers, such as The Times, The Sunday Times, The Independent, and The Guardian. He was co-editor of New Internationalist magazine from 1976–1979 and East Africa correspondent of The Economist from 1981-1983.

Works[link]

Hancock sees himself as a journalist who asks questions based upon observation and as someone who provides a counterbalance to what he perceives as the "unquestioned" acceptance and support given to orthodox views by the education system, the media, and by society at large.^[3] His books include Lords of Poverty, The Sign and the Seal, Fingerprints of the Gods, Keeper of Genesis (released in the US as Message of the Sphinx), The Mars Mystery, Heaven's Mirror (with wife Santha Faiia), Underworld: The Mysterious Origins of Civilization, and Talisman: Sacred Cities, Secret Faith (with co-author Robert Bauval). In 1996 he appeared in The Mysterious Origins of Man.^[4] He also wrote and presented the documentaries Underworld: Flooded Kingdoms of the Ice Age (2002) and Quest for the Lost Civilisation (1998) ^[5] shown on Channel 4. In Hancocks's book Talisman: Sacred Cities, Secret Faith,^[6] co-authored with Robert Bauval, the two put forward "a version of the old Jewish-Masonic plot so beloved by ultra-right-wing conspiracy theorists."^[7] They suggest a connection between the pillars of Solomon's Temple and the Twin Towers, and between the Star of David and The Pentagon.^[8]

Hancock's most recent non-fiction book, Supernatural: Meetings With the Ancient Teachers of Mankind, was published in the UK in October 2005 and in the US in 2006. In it, Hancock examines paleolithic cave art in the light of David Lewis-Williams' neuropsychological model, exploring its relation to the development of the fully modern human mind.

His first novel, Entangled: The Eater of Souls, the first in a fantasy series, was published in the UK in April 2010 and in the US in October 2010. The novel makes use of Hancock's prior research interests and as he has noted, "What was there to lose, I asked myself, when my critics already described my factual books as fiction?"

Although his books have sold more than five million copies worldwide and have been translated into twenty-seven languages, his methods and conclusions have found little support among academics, his work being labelled "pseudoarchaeology".^[1][9][10]

Orion Correlation Theory[link]

Representation of the central tenet of the OCT - the outline of the Giza pyramids superimposed over a photograph of the stars in Orion's Belt. To achieve this concordance the pyramids have been rotated and scaled to suit. The validity of this match has been called into question by Hancock's critics, as noted in the text.

One of the many recurring themes in several of Hancock's works has been an exposition on the "Orion Correlation Theory" (or OCT),^[11][12] first put forward by Belgian writer Robert Bauval and then further expounded in collaborative works with Hancock, as well as in their separate publications.

[edit] BBC Horizon controversy

BBC 2's Horizon TV series broadcast a programme, Atlantis Reborn, on 4 November 1999 that challenged the ideas presented by Hancock. It detailed one of Hancock's claims that the arrangement of an ancient temple complex was designed to mirror astronomical features and attempted to demonstrate that the same thing could be done with perhaps equal justification using famous landmarks in New York. It also alleged that Hancock had selectively moved or ignored the locations of some of the temples to fit his own theories (see below).^[13]

Hancock claimed he was misrepresented by the programme, and he and Robert Bauval made complaints to the Broadcasting Standards Commission against the way Horizon had portrayed them and their work. Eight points were raised by Hancock, two by Bauval (one of which duplicated a complaint of Hancock's).^[14] This included the complaint that:

The programme attempted to create the impression that he [Hancock] was an intellectual fraudster who had put forward half baked theories and ideas in bad faith, and that he was incompetent to defend his own arguments.^[15]

The BSC dismissed all but one of the complaints. Overall, the BSC concluded that "the programme makers acted in good faith in their examination of the theories of Mr Hancock and Mr Bauval".^[16] The complaint which was upheld was that

The programme unfairly omitted one of their arguments in rebuttal of a speaker who criticised the theory of a significant correlation between the Giza pyramids and the belt stars of the constellation Orion (the "correlation theory")

which the Commission did find to be unfair. That speaker was the astronomer Edwin Krupp. Krupp accused Bauval of fudging the maps of Orion and the Pyramids by placing them upside down in order to make the theory work.^[17] In interviews filmed in the later and modified Horizon documentary Atlantis Reborn Again shown on 14 December 2000, Hancock and Bauval provided detailed rebuttals to Krupp and argued that the ancient Egyptians had made the Pyramids correlate in the most obvious and intuitive manner with the three stars of Orion's Belt, that Orion could only be seen at the highest point in the sky by the Egyptians looking in a southward direction and a work of symbolic and religious art. A counter argument was provided on Atlantis Reborn Again to Bauval and Hancock's responses to Krupp's statements.

According to Graham Hancock, Krupp's criticism was rejected by two prominent British astronomers, Dr. Percy Seymour of Plymouth University and Dr. Archie Roy, professor Emeritus at Glasgow University.^[18]

Cultural references[link]

In 2009, Roland Emmerich released his blockbuster disaster movie 2012 citing Fingerprints of the Gods in the credits as an inspiration for the film,^[19] stating: "I always wanted to do a biblical flood movie, but I never felt I had the hook. I first read about the Earth's Crust Displacement Theory in Graham Hancock's Fingerprints of the Gods." ^[20] Also in 2009, author Geoff Stray released his best-selling book "Beyond 2012: Catastrophe or Awakening?" in which he cites Graham Hancock as one of his inspirations and as a guidepost to point others to do their own investigation into the wonder of ancient societies.^[21]

Books[link]

• Hancock, Graham (1985). Ethiopia: The Challenge of Hunger. London: V. Gollancz. ISBN 0-575-03680-X. 
• Hancock, Graham; Enver Carim (1986). AIDS: The Deadly Epidemic. London: V. Gollancz. ISBN 0-575-03837-3. 
• Hancock, Graham (1989). Lords of Poverty: The Power, Prestige, and Corruption of the International Aid Business. Boston: Atlantic Monthly Press. ISBN 0-87113-253-2. 
• Hancock, Graham (1992). The Sign and the Seal: The Quest for the Lost Ark of the Covenant. New York: Crown. ISBN 0-517-57813-1. 
• Hancock, Graham (1995). Fingerprints of the Gods: The Evidence of Earth's Lost Civilization. New York: Crown Publishers. ISBN 0-517-59348-3. 
• Hancock, Graham; Robert Bauval (1996). The Message of the Sphinx: A Quest for the Hidden Legacy of Mankind. New York: Crown Publishers. ISBN 0-517-70503-6.  Published in the United Kingdom as Hancock, Graham; Robert Bauval (1996). Keeper of Genesis: A Quest for the Hidden Legacy of Mankind. London: Heinemann. ISBN 0-434-00302-6. 
• Hancock, Graham (1998). The Mars Mystery: A Tale of the End of Two Worlds. London: Michael Joseph. ISBN 0-7181-4314-0. 
• Hancock, Graham; Santha Faiia (1998). Heaven's Mirror: Quest for the Lost Civilization. New York: Crown Publishers. ISBN 0-517-70811-6. 
• Hancock, Graham; Santha Faiia (2001). Fingerprints of the Gods: The Quest Continues (New Updated Edition). New York: Crown Century. ISBN 0-7126-7906-5. 
• Hancock, Graham (2002). Underworld: The Mysterious Origins of Civilization. New York: Crown. ISBN 1-4000-4612-2. 
• Hancock, Graham; Robert Bauval (2004). Talisman: Sacred Cities, Secret Faith. Tisbury: Element Books. ISBN 0-00-719036-0. 
• Hancock, Graham (2005). Supernatural: Meeting with the Ancient Teachers of Mankind. London: Century. ISBN 1-84413-681-7. 
• Hancock, Graham (2010). Entangled: The Eater of Souls. New York: The Disinformation Company. ISBN 978-1-934708-56-9. 

See also[link]

• Archaeoastronomy
• Erich von Däniken
• Mythology
• Zahi Hawass

References[link]

External links[link]

• The Official Graham Hancock Website
• Broadcasting Standards Commission – Synopsis of adjudication. Horizon: Atlantis Reborn (November 4, 1999)
• Horizon: Atlantis Reborn and the Broadcasting Standards Commission The BBC's response to the Horizon programme debate.
• Brass, M., nd, An analysis of the quality of Graham Hancock's "science", The Antiquity of Man Web Site.
• Brass, M., 2002, Tracing Graham Hancock's Shifting Cataclysm, Skeptical Inquirer. vol. 26, no. 4: 45-49.
• Carroll, R. T., 2009, Atlantis, The Skeptic’s Dictionary website.
• Fagan, N., nd, An Answer to Graham Hancock and Analysis of Hancock's Position Statement on C-14 Dating, Papers, In the Hall of Ma’at website.
• Tonkin, S., nd, Some Astronomical Errors of Graham Hancock, The Astronomical Unit website.
• Flemming, N., nd, Review of Flooded Kingdoms of the Ice Age, Papers, In the Hall of Ma’at website.
• The Big Idea: Graham Hancock