IN THE NAME OF ALLAH MOST GRACIOUS MOST MERCIFUL.

 AZHAR IQBAL 2008-EP-21  IMRAN ZAHID 2008-EP-51  MUHAMMAD ASHRAF 2008-EP-30  FAISAL ALI YOUNAS 2008-EP-04

PRESENTER: AZHAR IQBAL Reg # 2008-EP-21

A Gyroscope is a device for measuring or maintaining orientation, based on the principles of conservation of angular momentum.

A Gyroscope is essentially 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.

Precessional Angular Motion

After a short interval of time δt, let the disc be spinning about the new axis of spin ‘OX’with an angular velocity (ω + δω). Using the right hand screw rule, initial angular velocity of the disc (ω) is represented by vector ox; and the final angular velocity of the disc (ω + δω) is represented by vector ox ′.

The vector xx ′ represents the change of angular velocity in time δt i.e. the angular acceleration of the disc. This may be resolved into two components, one parallel to ox and the other perpendicular to ox. Component of angular acceleration in the direction of ox;

Since δθ is very small, therefore substituting cos δθ = 1, we have; In the limit, when δt →0, Component of angular acceleration in the direction perpendicular to ox,

Since δθ in very small, therefore substituting sin δθ = δθ, we have; In the limit when δt → 0, ∴ Total angular acceleration of the disc = vector xx′ = vector sum of α t and α c

Presenter: IMRAN ZAHID Reg#:2008-EP-51

A device consisting of a spinning mass, typically a disk or wheel, mounted on a base so that its axis can turn freely in one or more directions and thereby maintain its orientation regardless of any movement of the base. The turning moment which opposes any change of the inclination of the axis of rotation of a gyroscope.

Consider a disc spinning with an angular velocity ω rad/s about the axis of spin OX, in anticlockwise direction when seen from the front; ∴ Change in angular momentum; and rate of change ofangular momentum; In the limit when δt → 0,

 Thin Rod mL ² / 12  Thin Ring or Hoop mr ²  Solid Disc or Cylinder mr ² / 2  Sphere 2 mr ²/5

Effect of the Gyroscopic Couple on an Aeroplane The top and front view of an aeroplane. Let engine or propeller rotates in the clockwise direction when seen from the rear or tail end and the aeroplane takes a turn to the left. Let ω = Angular velocity of the engine in rad/s, m = Mass of the engine and the propeller in kg, k = Its radius of gyration in metres, I = Mass moment of inertia of the engine and the propeller in kg-m ² = m.k ², v = Linear velocity of the aeroplane in m/s, R = Radius of curvature in metres, and ω P = Angular velocity of precession =v/R rad/s ∴ Gyroscopic couple acting on the aeroplane, C = I.ω.ω P

NUMERICAL PROBLEM An aeroplane runs at 600 km / h. The rotor of the engine weighs 4000 N with radius of gyration of 1 metre. The speed of rotor is 3000 r.p.m. in anticlockwise direction when seen from rear side of the aeroplane. If the plane takes a loop upwards in a curve of 100 metres radius, find : 1. gyroscopic couple developed; 2. effect of reaction gyroscopic couple developed on the body of aeroplane.

PRESENTER: MUHAMMAD ASHRAF Reg #: 2008-EP-30

GYROSCOPE IN NAVAL SHIP  The fore end of the ship is called bow.  The rear end is known as stern or aft.  The left hand is called port.  The right hand sides of the ship, star-board. Effect of Gyroscope in  Steering  Rolling  Pitching

Effect of Gyroscopic Couple on a Naval Ship during Steering Steering is the turningof a complete ship in a curve towards left or right,while it movesforward. Consider the ship taking a left turn, and rotor rotates in the clockwise direction.

When the rotor of the ship rotates in the clockwise direction when viewed from the stern, it will have its angular Momentum vector in the direction ox as shown in Fi g. (a) When the ship steers to the right under Similar conditions as discussed above, the effect of the reactive gyroscopic couple, as shown in Fi g. (b), will be to raise the stern and lower the bow..

Effect of Gyroscopic Couple on a Naval Ship during Rolling ` The effect of gyroscopic couple to occur, the axis of precession should always be perpendicular to the axis of spin. If, however, the axis of precession becomes parallel to the axis of spin, there will be no effect of the gyroscopic couple acting on the body of the ship In case of rolling of a ship, the axis of precession (i.e. longitudinal axis) is always parallel to the axis of spin for all positions. Hence, there is no effect of the gyroscopic couple acting on the body of a ship.

PRESENTER: FAISAL ALI YOUNAS Reg # 2008-EP-04

GYROSCPIC COUPLE ON NAVAL SHIP DURING PITCHING  PITCHING is the movement of a ship up and down in a vertical plane about transverse axis.  The transverse axis is the axis of precession.  The pitching of the ship is assumed to take place with the Simple Harmonic Motion.

Gyroscopic Couple

 Effect of gyroscopic couple Is always given on specific Specific position of the axis Of spin.  The pitching of a ship act Horizontally on bearings and Perpendicular to the motion of The ship.

Angular Acceleration The angular acceleration of a ship during pitching is given by the double derivative of the angular displacement as follows: The angular acceleration is maximum,if sin ω 1 t= 1 So, maximum angular acceleration during pitching is;

Gyroscopic couple will be:  Effective when axis of precession will be perpendicular to the axis of spin as in the case of pitching.  Ineffective when axis of precession will be parallel to the axis of spin of ship rolling. GYROSCPIC COUPLE ON NAVAL SHIP DURING ROLLING

NUMERICAL PROBLEM Each paddle wheel of a steamer have a mass of 1600kg and a radius of gyration of 1.2 m. The streamer turns to port in a circle of 160m radius at 24 km/h,the speed of paddles being 90 r.p.m. Find the magnitude of the gyroscopic couple acting on streamer.

An aeroplane makes a complete half circle of 50 m radius,towards left, when flying at 200 km/h. The rotary engine and the propeller of the plane has a mass of 400 kg and the radius of gyration of 0.3 m.The engine rotates at 2400 r.p.m clockwise when vied from the rear.Find the gyroscopic couple on the aircraft and state it’s effect on it.