A gyrocompass is a type of non-magnetic compass which is based on a fast-spinning disc and
rotation of the Earth to automatically find geographical direction. Although one important
component of a gyrocompass is a gyroscope, these are not the same devices; a gyrocompass
is built to use the effect of gyroscopic precession, which is a distinctive aspect of the general
gyroscopic effect. Gyrocompasses are widely used for navigation on ships, because they
have two significant advantages over magnetic compasses:
they find true north as determined by Earth's rotation, which is different from, and navigationally
more useful than, magnetic north, and they are unaffected by ferromagnetic materials,
such as ship's steel hull, which change the magnetic field
Operation A gyroscope, not to be confused with gyrocompass,
is a spinning wheel mounted on gimbal so that the wheel's axis is free to orient itself
in any way. When it is spun up to speed with its axis pointing in some direction, due to
the law of conservation of angular momentum, such a wheel will normally maintain its original
orientation to a fixed point in outer space. Since our planet rotates, it appears to a
stationary observer on Earth that a gyroscope's axis is completing a full rotation once every
24 hours. Such a rotating gyroscope is used for navigation in some cases, for example
on aircraft, where it is known as heading indicator, but cannot ordinarily be used for
long-term marine navigation. The crucial additional ingredient needed to turn such gyroscope into
a gyrocompass, so it would automatically position to true north, is some mechanism that results
in an application of 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 along a line of longitude. 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 axis orientation 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, but otherwise allow it to rotate freely within the horizontal 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 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 gyrocompass's north-seeking function depends on the rotation around the axis of
the Earth that causes torque-induced gyroscopic precession, it will not orient itself correctly
to true north if it is moved very fast in an east to west direction, thus negating the
Earth's rotation. However, aircraft commonly use heading indicators or directional gyros,
which are not gyrocompasses and do not position themselves to north via precession, but are
periodically aligned manually to true north. Mathematical model of a gyrocompass
We will consider here a gyrocompass, as a gyroscope which is free to rotate about one
of its symmetry axis, and the whole rotating gyroscope is also free to rotate on the horizontal
plane, about the local vertical, the zenith. Therefore there are two independent local
rotations. In addition to these rotations we will also consider the rotation of the
Earth about its North-South axis, and we will model the planet as a perfect sphere. We will
neglect friction and the rotation of the Earth about the Sun.
In this case a non-rotating observer located at the center of the Earth can be approximated
as being an inertial frame. We can set cartesian coordinates for such an observer, and the
barycenter of the gyroscope will be located at a distance from the center of the Earth.
First time-dependent rotation Let us consider another observer located at
the center of the Earth but rotating about the NS-axis by , then we set coordinates attached
to the observer as
so that the unit versor is mapped to the point . For the 2-O the Earth is not moving so as
the barycenter of the gyroscope. The rotation of 2-O, according to 1-O, is performed with
angular velocity . We will suppose that the axis denotes points with zero longitude.
Second and third fixed rotations We will now rotate about the axis, so that
the -axis will have the longitude of the barycenter. In this case we have
With the next rotation we will bring the axis along the local zenith of the barycenter.
This can be achieved by the following orthogonal matrix
so that the versor is mapped to the point .
Constant translation We now choose another coordinate basis whose
origin is located at the barycenter of the gyroscope. This can be performed by the following
translation along the zenith axis
so that the origin of the new system, is located at the point , and is the radius of the Earth.
Now the -axis points towards the south direction. Fourth time-dependent rotation
Now we rotate about the zenith -axis so that the new coordinate system is attached to the
structure of the gyroscope, so that for an observer at rest in this coordinate system,
the gyrocompass is only rotating about its own axis of symmetry. In this case we find
The axis of symmetry of the gyrocompass is now along the -axis.
Last time-dependent rotation The last rotation is a rotation on the axis
of symmetry of the gyroscope as in
Dynamics of the system Since the gyroscope is not moving the height
of its barycenter, its gravitational potential energy is constant. Therefore its Lagrangian
corresponds to its kinetic energy only. We have
where the tensor of inertia is given by
and
Therefore we find
The Lagrangian can be rewritten as
where
is the part of the Lagrangian responsible for the dynamics of the system. Then, since
, we find
Since the angular momentum of the gyrocompass is given by , we see that the constant , is
the component of the angular momentum about the axis of symmetry. Furthermore, we find
the equation of motion for the variable as
or
Particular case: the poles At the poles we find , and the equations of
motion become
This simple solution implies that the gyroscope is uniformly rotating with constant angular
velocity in both the vertical and symmetrical axis.
The general and physically relevant case Let us suppose, now that , and that , that
is the axis of the gyroscope is approximately along the North-South line, and let us find
the parameter space, for which the system admits stable small oscillations about this
same line. If this situation occurs, the gyroscope will be always approximately aligned along
the North-South line, giving direction. In this case we find
Let us consider the case that
and, further, we allow for fast gyro-rotations, that is
Therefore, for fast spinning rotations, implies . In this case, the equations of motion further
simplify to
Therefore we find small oscillations about the North-South line, as , where the angular
velocity of this harmonic motion of the axis of symmetry of the gyrocompass about the North-South
line is given by
which corresponds to a period for the oscillations given by
Therefore is proportional to the geometric mean of the Earth and spinning angular velocities.
In order to have small oscillations we have required , so that the North is located along
the right-hand-rule direction of the spinning axis, that is along the negative direction
of the -axis, the axis of symmetry. As a side result, on measuring , one can deduce the
local colatitude . History
The first, not yet practical, form of gyrocompass was patented in 1885 by Marinus Gerardus van
den Bos. Usable gyrocompass was invented in 1906 in Germany by Hermann Anschütz-Kaempfe,
and after successful tests in 1908 became widely used in German Imperial Navy.
The gyrocompass was an important invention for nautical navigation because it allowed
accurate determination of a vessel’s location at all times regardless of the vessel’s
motion, the weather and the amount of steel used in the construction of the ship. In the
United States, Elmer Ambrose Sperry produced a workable gyrocompass system, and founded
the Sperry Gyroscope Company. The unit was adopted by the U.S. Navy, and played a major
role in World War I. The Navy also began using Sperry's "Metal Mike": the first gyroscope-guided
autopilot steering system. In the following decades, these and other Sperry devices were
adopted by steamships such as the RMS Queen Mary, airplanes, and the warships of World
War II. After his death in 1930, the Navy named the USS Sperry after him.
Meanwhile, in 1913, C. Plath developed the first gyrocompass to be installed on a commercial
vessel. C. Plath sold many gyrocompasses to the Weems’ School for Navigation in Annapolis,
MD, and soon the founders of each organization formed an alliance and became Weems & Plath.
Before the success of gyrocompass, several attempts had been made in Europe to use gyroscope
instead. By 1880, William Thomson tried to propose a gyrostat to the British Navy. In
1889, Arthur Krebs adapted an electric motor to the Dumoulin-Froment marine gyroscope,
for the French Navy. Giving the Gymnote submarine the ability to keep a straight line under
water during several hours, it allowed her to force a naval block in 1890.
Errors A gyrocompass is subject to certain errors.
These include streaming error, where rapid changes in course, speed and latitude cause
deviation before the gyro can adjust itself. On most modern ships the GPS or other navigational
aids feed data to the gyrocompass allowing a small computer to apply a correction. Alternatively
a design based on an orthogonal triad of fibre optic gyroscope or ring laser gyroscopes will
eliminate these errors, as they depend upon no mechanical parts, instead using the principles
of optical path difference to determine rate of rotation.
Patents U.S. Patent 1,279,471 : "Gyroscopic compass"
by E. A. Sperry, filed June, 1911; issued September, 1918
See also Acronyms and abbreviations in avionics
Heading indicator, also known as direction indicator, a lightweight gyroscope used on
aircraft HRG gyrocompass
Fluxgate compass Fibre optic gyrocompass
Inertial navigation system, a more complex system that also incorporates accelerometers
Schuler tuning Notes
References
Bibliography Trainer, Matthew. "Albert Einstein's expert
opinions on the Sperry vs. Anschütz gyrocompass patent dispute". World Patent Information
30: 320. doi:10.1016/j.wpi.2008.05.003. External links
Elmer A. Sperry case file at the Franklin Institute contains records concerning his
1914 Franklin Award for the gyroscopic compass "A Job Thought Impossible", the story of Chrysler
Corporation's mass-production of previously hand-made gyrocompasses for World War II naval
requirements. Errors of the Gyrocompass
Using the Gyrocompass Ring Laser Gyrocompass manufactured by Sonardyne