Gyroscopic Couple Calculation PDF

DEPARTMENT OF MECHANICAL ENGINEERING, RSET, KAKKANAD EXPT NO:5 CALCULATING THE GYROSCOPIC COUPLE AIM: Experimental justification of the equation T= I ω ωp, for calculating the gyroscopic couple by observation and measurement of results for independent variation in appli...

DEPARTMENT OF MECHANICAL ENGINEERING, RSET, KAKKANAD EXPT NO:5 CALCULATING THE GYROSCOPIC COUPLE AIM: Experimental justification of the equation T= I ω ωp, for calculating the gyroscopic couple by observation and measurement of results for independent variation in applied couple T and precession ωp. APPARATUS: Stop watch, Dead weights. THEORY: A gyroscope is a device for measuring or maintaining orientation, based on the principle of preserving angular momentum. Mechanical gyroscopes typically comprise a spinning wheel or disc in which the axle is free to assume any orientation. Although the orientation of the spin axis changes in response to an external torque, the amount of change and the direction of the change is less and in a different direction than it would be if the disk were not spinning. Applications of gyroscopes include inertial navigation systems where magnetic compasses would not work (as in the Hubble telescope) or would not be precise enough (as in intercontinental ballistic missiles), or for the stabilization of flying vehicles like radio- controlled helicopters or unmanned aerial vehicles. Due to their precision, gyroscopes are also used in gyrotheodolites to maintain direction in tunnel mining. Gyroscopes are installed in ships in order to minimize the rolling and pitching effects of waves. They are also used in aeroplanes, monorail cars, gyrocompasses etc. 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, as shown in figure a. Since the plane in which the disc is rotating is parallel to the plane YOZ, therefore it is called plane of spinning. The plane XOZ is a horizontal plane and the axis of spin rotates in a plane parallel to the horizontal plane about an axis OY. In other words, the axis of spin is said to be rotating or processing about an axis OY. In other words, the axis of spin is said to be MECHANICAL ENGINEERING LAB –ME431 Page 26 DEPARTMENT OF MECHANICAL ENGINEERING, RSET, KAKKANAD rotating or processing about an axis OY (which is perpendicular to both the axes OX and OZ) at an angular velocity ωp rad/s. This horizontal plane XOZ is called plane of precession and OY is the axis of precession. Let I = Mass moment of inertia of the disc about OX, and ω = Angular velocity of the disc. ∴ Angular momentum of the disc = I ω Since the angular momentum is a vector quantity, therefore it may be represented by the vector ox, as shown in Figure b. The axis of spin OX is also rotating anticlockwise when seen from the top about the axis OY. Let the axis OX is turned in the plane XOZ through a small angle δθ radians to the position OX ′ , in time δt seconds. Assuming the angular velocity ω to be constant, the angular momentum will now be represented by vector ox′ Change in angular momentum Rate of change of angular momentum = I ×ω × δθ/ dt Since the rate of change of angular momentum will result by the application of a couple to the disc, therefore the couple applied to the disc causing precession, MECHANICAL ENGINEERING LAB –ME431 Page 27 DEPARTMENT OF MECHANICAL ENGINEERING, RSET, KAKKANAD Where, ωp= Angular velocity of precession of the axis of spin or the speed of rotation of the axis of spin about the axis of precession OY. It may be noted that: 1. The couple I.ω.ωp, in the direction of the vector xx′ (representing the change in angular momentum) is the active gyroscopic couple, which has to be applied over the disc when the axis of spin is made to rotate with angular velocity ωp about the axis of precession. The vector xx′ lies in the plane XOZ or the horizontal plane. In case of a very small displacement δθ, the vector xx′ will be perpendicular to the vertical plane XOY. Therefore the couple causing this change in the angular momentum will lie in the plane XOY. The vector xx′, as shown in figure.b, represents an anticlockwise couple in the plane XOY. Therefore, the plane XOY is called the plane of active gyroscopic couple and the axis OZ perpendicular to the plane XOY, about which the couple acts, is called the axis of active gyroscopic couple. 2. When the axis of spin itself moves with angular velocity ωP, the disc is subjected to reactive couple whose magnitude is same (i.e. I. ω.ωP) but opposite in direction to that of active couple. This reactive couple to which the disc is subjected when the axis of spin rotates about the axis of precession is known as reactive gyroscopic couple. The axis of the reactive gyroscopic couple is represented by OZ′ in figure a. 3. The gyroscopic couple is usually applied through the bearings which support the shaft. The bearings will resist equal and opposite couple. SPECIFICATION: Radius of the disc, r = 13.5x10 -2 m Mass of disc, m = 5.2 kg Distance of weight from centre of rotation, L = 17 x 10 -2 m MECHANICAL ENGINEERING LAB –ME431 Page 28  DEPARTMENT OF MECHANICAL ENGINEERING, RSET, KAKKANAD PROCEDURE: 1. First switch on the motor connected to the disc on no load and bring it to a steady speed. 2. Bring the pointer of precision axis to zero degree. 3. Add a known weight gently to the loading arm. 4. Note the time taken for angular displacement of ‗θ‘ degrees. 5. Remove the weight and add a higher weight after setting back the pointer to zero degree. 6. Repeat the procedure No.4 and tabulate the readings for different weights. PRECAUTIONS: 1. Check all the fastenings to be tight before start. 2. Check balance of the rotor before start. 3. Lubricate the bearings periodically. 4. Keep the base over a levelled platform. 5. Do not increase rotor speed beyond 1500 rpm. OBSERVATION AND CALCULATION TABLE: Sl No. Load Speed Angle Time taken ω ωp Actual Theoretical (kg) turned in for angular torque Torque of disc (rad/sec) (rad/sec) degrees displacement Tact Ttheoretical (rpm) of ‗θ‘ degree (θ) (N-m) (N-m) (Sec) MECHANICAL ENGINEERING LAB –ME431 Page 29 DEPARTMENT OF MECHANICAL ENGINEERING, RSET, KAKKANAD SAMPLE CALCULATION FOR SET NO: 1) Actual torque Tact = W × L Where, W = Weight placed on the arm= mg Applied mass m = Kg W= = N L = distance of weight from centre of rotation = 17 x 10 -2 m Tact = = Nm 2) Theoretical torque Ttheoretical = I×ω×ωp Where, I = moment of inertia of the disc in Kg-m2 = mk2 r = radius of disc= m k = Radius of gyration= = m I= = kgm2 ω = angular velocity of spin in rad/sec = 2 π N/60 N = Speed of the disc = rpm ω= = rad/sec θ = angle turned= degrees t = time taken for θ 0 displacement= sec ωp= angular velocity of precision= (θ/t) x (π/180) = = rad/sec MECHANICAL ENGINEERING LAB –ME431 Page 30 DEPARTMENT OF MECHANICAL ENGINEERING, RSET, KAKKANAD T theoretical= = Nm RESULT: Sl W ω ωp Tact Ttheoretical No. (Kg) (rad/sec) (rad/sec) (Nm) (Nm) INFERENCE: MECHANICAL ENGINEERING LAB –ME431 Page 31