Free Gyroscope: A free gyroscope consists of a mass in the form of a rotor or wheel which is suspended in such a way that it is free to spin about an axis passing through its centre of mass and perpendicular to the plane of the rotor. Thus, it is a mass in the shape of wheel or so, symmetrical about an axis & can freely rotate about that axis. If such rotor is so suspended that it is free to point its spin axis in any direction.
This is achieved as follows:
- Freedom to spin about its own axis.
- Freedom to tilt about a horizontal axis.
- Freedom to turn in azimuth about a vertical axis.
Application of force can cause linear motion. Momentum is the product of mass and the velocity of the object. Any object moving with mass possesses momentum. Moment or torque on the other hand can cause rotary motion. Mass equivalent in rotary motion in moment of inertia. Angular momentum is the rotational equivalent of linear momentum.
If one gets on a bicycle and tries to balance without the stand, he or she will surely fall. But upon pedaling, when the wheels pick up angular momentum, the inertia is going to resist change, balancing becomes easier. Nice example to understand inertia.
Concept of angular momentum
Note: The angular momentum of a particle spinning about an axis is the product of its linear momentum and the perpendicular distance of the particle from the axle. Thus, angular momentum = mv x r, where r = rotor radius. Also, since v = rΩ, Ω being the angular velocity, the angular momentum = mr2Ω.
Angular momentum is defined as the property of any rotating object given by moment of inertia times angular velocity. The SI unit of angular momentum is Kg.m2.s-1.
H = IΩ
H = angular momentum
Ω = angular velocity
I = moment of inertia of the cross section about the spin axis.
Properties of a free gyroscope
A free gyroscope has two inherent properties:
The first inherent property is gyroscope inertia. A free gyroscope, once set spinning will always remain pointed in the same direction i.e. to a fixed point in space i.e. the spin axis will point in a constant direction with respect to space regardless of how the gyro support system has moved. The spin axis can therefore by the virtue of pointing at a fixed star on the celestial sphere will thus, follow its apparent motion i.e. it will follow the movement of that star on the celestial sphere. It is this property which is known as gyroscopic inertia or rigidity in space.
The amount of inertia, a gyro possess is dependent on:
mass of the spinning wheel & the distribution of mass around spin axis; and angular velocity of the rotor (faster the spin greater the inertia).
The gyroscope inertia of a rotor may be expressed quantitatively by its angular momentum. H = IΩ.
It can now be stated that gyroscopic inertia depends upon the momentum of the spinning rotor. The momentum of such a rotor is proportional to MK2 & thus, depends upon three main factors:
- the total mass, M of the rotor (for all particles)
- the radius r summed as the constant K (for all the particles) where K is the radius of gyration
- the angular velocity .
Most of the mass of rotor is concentrated at its outer edge. Normally, the rotor gas a large radius and spins fast. To spin freely the its center of gravity is kept at the intersection of the three axes) and its mounting bearings are maintained friction-free. A precisely controlled servo system maintains steady gyroscopic inertia.
The second property of a free gyroscope is to precess. Due to precession, when a torque is applied to the axles of a free gyroscope, it precesses in a direction perpendicular to the plane of torque.
Understanding the precession: Precession causes axle of a free gyroscope to move when a torque is applied. Torque is a force or a couple of force applied at a distance from the reference axis.
Precession will always caused in a direction right angles to the plane of torque. If a torque is applied about the horizontal axis trying to move the spin axis up or down the gyro spin axis will precess sideways i.e. about the vertical axis. If a torque is applied about the vertical axis the gyro will precess up or down i.e. about a horizontal axis.
Rate of precession of the gyro axis is proportional to the applied torque. It is inversely proportional to the gyroscopic inertia of the rotor which is expressed by the angular momentum possessed by the rotor.
Direction of precession depends on
- Direction of applied torque.
- Direction of rotation of wheel (clock wise or anticlockwise).
In the diagram below, at end A a vertically downward force (// to plane of paper) is applied. To find the direction of precession paste a parallel arrow on the disc in radially outward direction. After a 90º turn this arrow will point in a direction that emerges out of plane of paper. This is also the direction in which end A will precess.
1. If direction of applied force is reversed at A the direction of precession will also be reversed.
2. If rotational direction of disc is reversed the direction of precession will be reversed.
3. By choosing the top heavy / bottom heavy arrangement or clockwise / anticlockwise rotation we can achieve the precession in desired direction.
4. End B will precess in a direction opposite as end A.
This property, causing an axle to follow a fixed star in space results in two sub properties viz.
- Drifting, causing a change of azimuth.
Rate of drifting per hour = 15.041o Cos altitude x Sin latitude
- Tilting, causing a change of altitude.
Rate of tilting per hour = 15.041o Sin azimuth x Cos latitude
Tilt and tilting: the spin axis of a free gyro is said to be tilting when it moves in a vertical plane either in an upward direction (when east of meridian) or in a downward direction (when west of meridian). The tilt is equivalent of altitude whereas rate of change of altitude refers to ‘tilting’. The movement is similar to a star changing its altitude. The tilting or rate of change of altitude we know is zero on meridian. Any star, weather in lower meridian area or upper meridian area, has upward tilting (Tg) while it is to the east of meridian and downward tilting while it is to the west of meridian. This rule will follow in all the drawings showing the trace of axle. Since, rate of tilting per hour = 15.041o Sin azimuth x Cos latitude, the rate of tilting in a given latitude increases with the azimuth and in a trace the Tg is maximum on the east or west extremity.
The angle the spin axis is elevated above or below the horizontal denotes the tilt. The maximum angle of tilt occurs when the spin axis is aligned in the meridian, just as the maximum altitude of a star occurs when it is on the meridian. Minimum altitude occurs at lower meridian passage. The equivalent depiction of axle on the tilt axis of a trace will be seen at the lowermost point.
Azimuth and Drifting: The spin axis is said to be drifting when it changes its azimuth, which means it turns about a vertical axis. Rate of drifting means the rate at which the spin axis is drifting at any given time. Since, rate of drifting per hour = 15.041o Cos altitude x Sin latitude, the drifting is maximum at poles. In an ellipse or a spiral trace the maximum tilt is of just a small value only, Cos altitude nears unity, which means that for a given latitude the rate of drifting should be shown constant throughout the trace. In the trace drift refers to the angular distance from the meridian to the east or west and is equivalent to the azimuth.
Direction of drifting: When referring to the direction of drift it is always with reference to the North end of the spin axis when the angles of tilt is either nil or very small. i.e. when the spin axis is horizontal or near horizontal. It can be seen in the observer’s RH diagram that in north latitude the north end of spin axis drifts to east. Also, in south latitude the north end drifts to west.
Transforming a free gyroscope into a marine gyro compass (North seeking).
Due to the earth rotation the spin axis of a free gyroscope will trace out a circular path as it remains pointed in a fixed direction in space. The requirement of a gyro compass is that the spin axis should point in a fixed direction on earth and the chosen direction is true north.
To convert a free gyroscope into a gyroscope the free gyroscope must be made.
- North seeking i.e. seek the meridian.
- North settling i.e. must point in or near the meridian.
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