Schuler's period

The Schuler period is a geodetic constant that is of great importance when applying gyroscopic theory to navigation problems . The period is equal to the period of oscillation of a hypothetical pendulum with a pendulum length (6371 km) corresponding to the earth's radius R _E in a homogeneous gravitational field with the gravitational acceleration prevailing on the earth's surface (g = 9.81 m / s ² ). The investigations go back to Maximilian Schuler .

${\displaystyle T=2\pi {\sqrt {\frac {R_{E}}{g}}}=84,4}$ Minutes (Schuler period)

Remarks

Conditions in the hypothetical school pendulum

• Such a pendulum (see picture) is hypothetical in every respect. Its greatness cannot be realized, nor can it be guided through the earth to its center. Furthermore, there is a radial field around the center of the earth and not a homogeneous field (on the earth's surface the radial field can be viewed approximately as homogeneous).

• This hypothetical pendulum should not be confused with the Schuler pendulum , which was developed by M. Schuler for high-precision pendulum clocks.

Explanation

A fundamental problem with the navigation of a vehicle on or above the surface of the earth is the determination of the vertical direction (direction to the center of the earth) or the plane perpendicular to it ( artificial horizon ). For a stationary vehicle or a vehicle moving at a constant speed, this can be done in a simple manner with a spherical pendulum (which can be deflected in all directions) of any length. It always points to the center of the (rotating) earth (see picture).

However, if the vehicle is subject to acceleration (change in speed), the pendulum is deflected and a magnetic declination occurs. E.g. When the vehicle starts up, the pendulum body will remain behind the suspension point of the pendulum, which is moving with the vehicle, against the direction of travel, and when a constant speed is reached (decaying) it will swing back and forth.

In his publication from 1923, Max Schuler investigated which condition a pendulum must meet so that it shows the perpendicular direction even when its suspension point accelerates. The Schuler period resulted for the period of oscillation. Since a pendulum with this period cannot be realized, systems are required instead with which such an extremely long period of oscillation (and thus insensitivity to acceleration) can be achieved. These are gyroscopic devices that move with the vehicles to be navigated on or near the surface of the earth in the (approximately homogeneous) gravitational field that prevails there. The Schuler period is therefore of far-reaching importance, especially for the theory of the (vibratory) gyroscopic devices used for navigation purposes.

Derivation of the condition according to Schuler

Vehicle resting on the surface of the earth

Movement of a vehicle on the earth's surface

Acceleration of a vehicle on the surface of the earth

A spherical pendulum in a vehicle resting on the earth's surface or moving at a constant speed always points to the center of the earth. When the vehicle is at rest, the pendulum rotates in a certain time by an angle which corresponds to the rotation angle of the earth in the same time. The earth and pendulum rotate at the same angular velocity , with the pendulum resting relative to the vehicle (i.e. not swinging, but always pointing to the center of the earth). The reference system to which both the rotation of the earth and the rotation of the pendulum are related (i.e. the angle information) does not rotate with the earth. After one revolution of the earth, the pendulum has turned once while it remains completely at rest relative to the vehicle.

A vehicle can of course move in any direction. At constant speed, the (constant) angular speeds of the earth and vehicle add vectorially , with the pendulum again resting relative to the vehicle.

The rotation of the earth need not be taken into account when examining acceleration effects. When the vehicle accelerates (change in angular velocity ω _F ), the suspension point of the pendulum is accelerated in the same way and the pendulum is given an angular velocity ω _P in the direction of acceleration, which inevitably moves on a circular path around its suspension point. The pendulum rests in front of the acceleration and points, as mentioned above, to the center of the earth. The vehicle and thus the suspension point move on a circular path around the center of the earth (in any direction), so that the following applies to the acceleration b _F (see figure):

${\displaystyle b_{\text{F}}=R_{\text{E}}{\frac {\mathrm {d} \omega _{\text{F}}}{\mathrm {d} t}}}$

The torque acts on the mass m of the (mathematical) pendulum

${\displaystyle M=b_{\text{F}}ma}$,

where ɑ is the pendulum length. This torque causes a change in the angular velocity of the pendulum ( moment of inertia  :) : ${\displaystyle \theta =ma^{2}}$

${\displaystyle ma^{2}{\frac {\mathrm {d} \omega _{\text{P}}}{\mathrm {d} t}}=b_{\text{F}}ma}$

so

${\displaystyle ma^{2}{\frac {\mathrm {d} \omega _{\text{P}}}{\mathrm {d} t}}=maR_{\text{E}}{\frac {\mathrm {d} \omega _{\text{F}}}{\mathrm {d} t}}}$

If the pendulum is to point to the center of the earth unchanged during acceleration, the changes in the (time-dependent) angles of the pendulum and the vehicle caused by the acceleration must be the same. This is the case when both changes in angular velocity are identical. In

${\displaystyle {\frac {\mathrm {d} \omega _{\text{P}}}{\mathrm {d} t}}={\frac {R_{\text{E}}}{a}}{\frac {\mathrm {d} \omega _{\text{F}}}{\mathrm {d} t}}}$

must therefore

${\displaystyle {\frac {R_{\text{E}}}{a}}=1}$

be valid. This means that the pendulum length must be equal to the radius of the earth:

${\displaystyle R_{\text{E}}=a}$

The period of oscillation of a pendulum with length L is given by ( g = acceleration due to gravity of 9.81 m / s²)

${\displaystyle T=2\pi {\sqrt {\frac {L}{g}}}=2\pi {\sqrt {\frac {R_{\text{E}}}{g}}}=84,4}$  min

This is the Schuler period.

Applications

Solder pendulum

A hypothetical cannonball orbiting the earth in the smallest possible orbit (D) would have the orbit duration of the Schuler period and the first cosmic speed .

Schuler was concerned with the optimization of gyroscopes used for navigation . He assumed that tuning such devices with a duration of 84 minutes would lead to a minimization of acceleration errors. A general proof he did not (and is not) succeeded, but he was able to apply to the Lotkreisel (centrifugal pendulum) and Gyro (north-seeking compass) in his aforementioned writing (eg. T. again) prove. The company Anschütz, for which Schuler worked, had already submitted a patent specification years earlier. The publication of 1923 was particularly important for the further development of the gyro compass.

The practical implementation of the Schuler coordination is linked to technical problems (extremely high speed required). At the time of its publication, Schuler had achieved a rotation time of T = 30 minutes.

The Schuler criterion forms the basis for the optimization (coordination) of the various devices used in gyro navigation (e.g. inertial platform, artificial horizon, etc.).

Gyro

A gyroscope consists of a rotationally symmetrical flywheel (driven by an electric motor) rotating at high speed with a vertical axis. The top is attached to a suspension point like a pendulum. The gravity acting on a top that is not in a vertical axis exerts a torque and tries to pull the top axis into the vertical. The gyroscope reacts to this, however, with what is known as precession , ie it gives way to the side and its axis moves at a constant angular velocity on a conical surface .

The duration of a revolution (period of oscillation) is given by (gyro theory):

${\displaystyle T=2\pi {\frac {J}{mga}}}$,

where J is the angular momentum .

Schuler has proven that with a rotation time of 84 minutes ( Schuler vote ) the plumb bob is largely insensitive to the effects of acceleration and maintains its vertical position, provided the axis is in a vertical position before the acceleration occurs (no precession). Before a vehicle starts moving, the vertical position of the roundabout must be ensured.

Gyrocompass

The gyro compass (horizontal gyro axis) can also oscillate, but in contrast to the pendulum around a vertical axis, and can be optimized with the Schuler adjustment (oscillation duration 84 minutes). In practice, the system is constructed so that the natural frequency coincides with the Schuler period.

Explanation:

The gyro compass (single gyro compass) is looking north. When the vehicle (ship) is at rest, the horizontal axis of the gyro adjusts itself automatically in the north-south direction (i.e. in the meridian direction). It takes several hours. The setting does not take place through a smooth transition, but is characterized by a decaying oscillation process (multiple oscillations of the gyro axis around the north-south direction). The vehicle can only start moving after the vibration has subsided, if the function of the compass is to be guaranteed.

If the vehicle is moving at a certain constant speed in a direction other than east-west (or vice versa), the compass has a driving error. It should be noted that speed is to be understood as the resulting speed resulting from the vectorial addition of the circumferential speed of the earth (on the latitude of the vehicle position) and the speed of the vehicle. The gyro axis no longer points exactly to the north, but is perpendicular to the resulting speed. When driving on latitude (i.e. across the meridian direction), no driving error occurs. The speed-dependent driving error can be calculated. It is recorded in tables and can therefore be corrected by the navigator (nowadays with appropriate computer programs).

If the vehicle is accelerated with a north or south component , a torque acts on the gyrocompass (which is suspended pendulum), to which the gyro reacts with a corresponding precession (i.e. a rotation around the vertical axis). If the gyroscope had not been tuned according to Schuler, it would slowly adjust itself to the new driving error corresponding to the changed speed after the acceleration was completed. The navigator could therefore not be sure when the reading can be trusted again. This serious disadvantage called the usability of gyro compasses into question before Schuler's publication. However, if the system is adjusted according to Schuler, the rotation of the gyro axis around the vertical caused by the precession corresponds exactly to the angle of the change in travel error. There is no vibration. The new course (with the driving error corresponding to the changed speed) can be read off immediately after the end of the acceleration phase.

Since the tuning condition depends on the latitude, the compass must be tuned to the respective operating area of ​​the vehicle. The compass can, for example, be constructed in such a way that this can be done manually by changing the (extremely short) pendulum length of the pendulum-mounted gyroscope and thus the Schuler period can be set.

The voting condition is:

${\displaystyle {\frac {J}{mga\cdot \omega _{E}\cdot \cos \phi }}={\frac {R_{E}}{g}}}$

${\displaystyle {\phi }}$ Latitude, a pendulum length to be varied

The calculations for the gyro compass can also be found in the Schuler publication mentioned above.

Subsequent research has shown that the Schuler criterion is not universally applicable to gyroscopes. Particularly in the case of rapidly changing accelerations (e.g. the roll and yaw movements of a ship, roll errors ), different adjustments are appropriate (see also) or dual gyro compasses are required. Nor can it be neglected that the gravitational field on the earth's surface is not, as assumed, homogeneous, but has deviations, albeit slight. This is particularly noticeable in satellite navigation.

Others

In addition to the features of gyroscopes, other phenomena are also subject to the 84-minute criterion. For example, the orbit time of a satellite on the lowest possible orbit is 84 minutes. For more examples see.

Individual evidence

1. ↑ Kurt Magnus: Kreisel: Theory and Applications . Springer publishing house.
2. ↑ Richard Grammel: The top, its theory and its applications: Second volume . Springer Verlag, 1950.
3. ↑ Max Schuler: Pendulum for timing purposes. (PDF) Retrieved on April 19, 2019 (click on "Load full document" after calling up). 
4. ↑ Maximilian Schuler: The disturbance of pendulum and gyroscopic apparatus by the acceleration of the vehicle. (PDF) In: Physikalische Zeitschrift 24, page 344 (1923). Retrieved September 22, 2018 . 
5. ↑ horizon horizon. (PDF) Retrieved on March 14, 2019 (click on "Load full document"!). 
6. ↑ Müller / Krauss: Manual for the navigation part B . Springer Verlag, Berlin 1983.
7. ↑ Terheyden, Zickwolff: Navigation Part B . Springer-Verlag, Berlin Heidelberg.
8. ↑ Kurt Magnus: The acceleration-insensitive coordination of navigation devices. (PDF) Retrieved July 4, 2020 . 
9. ↑ Kurt Magnus: The acceleration dependency of the vertical display of gravity pendulum and plumb top. (PDF) In: Engineer Archive. 1966, accessed July 4, 2020 . 
10. ↑ Multi- gyro compass. (PDF) Retrieved on March 13, 2019 (click on "Load full document"!). 
11. ↑ Kurt Magnus: Gyroscope as a versatile aid in aerospace. (PDF) In: Journal for Aviation Sciences and Space Research. July 1978, accessed July 4, 2020 .