Rate of change of angular momentum Gyroscopic Couple Gyroscopic Couple effect in a ship Gyroscopic effect in a aero plane Angular momentum L Angular momentum L If a circular disc rotating about an axis passing through center of disc and perpendicular to circular plane ID: 1001310
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1. Gyroscope
2. OUTLINEConcept of angular momentum (L).Rate of change of angular momentum.Gyroscopic Couple.Gyroscopic Couple effect in a ship.Gyroscopic effect in a aero plane.
3. Angular momentum (L)Angular momentum (L): If a circular disc rotating about an axis passing through center of disc and perpendicular to circular plane.zyxABABFront viewSide viewAxis of rotation (Spin)ωωω is angular velocity of rotor. Sense of rotation is anticlockwise direction.Axis of rotation about z-axis.Angular velocity is vector quantity.Mass moment of inertia o f rotor (I):Mass moment of inertia about axis of rotationof rotor is I.For circular disc, Izz =2M R2M = Mass of rotor.R = Radius of rotor.If radius of gyration of rotor is k, then massMoment of inertia (Izz) = m k2Unit : Kg - m2Angular momentum about axis of rotation (L)L = Mass moment of inertia * Angular velocityL = I ω Direction of angular momentum is same as direction of angular velocity. Unit : kg m2 / sec
4. Angular Velocity representation+ zωRepresentation of angular velocity in vector form:Sense of rotation : Anticlockwise direction.Axis of rotation : + z – axisMagnitude of angular speed : ω (rad/s)ω- zRepresentation of angular velocity in vector form:Sense of rotation : Clockwise direction.Axis of rotation : - z – axisMagnitude of angular speed : ω (rad/s)+ zω- zωLL
5. Axis of precessionzyxABωω yωpωp = axis of precision.Axis of precession is rotation of rotor about y-axis.
6. Change of angular momentum (L)zω yωpTime t = t seczxz'dႴ = ωp = ωp dt Time t = t + dt secz‘ is axis of spin at time t = t + dtz is axis of spin at time t = t secIωIωkidႴ is small, = dႴ = 1
7. PROBLEM: Suppose a ship is moving in sea. At any instant of time it make a left turn with radius of curvature R.ωTOP VIEWRotor is rotating in anticlockwise direction.LROTORL = Angular momentum. = I ωI = Mass moment of inertia about axis of spin.t = 0 sCXYR
8. PROBLEM: Suppose a ship is moving in sea. At any instant of time it make a left turn with radius of curvature R.ωTOP VIEWAxis of SpinLfL = Angular momentum. = I ωI = Mass moment of inertia about axis of spin.time = dtCXYdႴLidႴ dt= ωpωp = angular velocity of precessionLiLfdႴ
9. PROBLEM: Suppose a ship is moving in sea. At any instant of time it make a left turn with radius of curvature R.TOP VIEWL = Angular momentum. = I ωI = Mass moment of inertia about axis of spin.CXYdႴdႴ dt= ωpωp = angular velocity of precessionLi1800 - dႴLfdႴVector for of ωp.ωp = ωp kLiInitial angular momentum at time t =0
10. XYLiLfdႴLi = Iω iXLf = Iω( cos(dႴ) i + Sin(dႴ)j )Lf - Li = Iω( cos(dႴ) i + Sin(dႴ)j ) – Iω iLf - Li = Iω( i + (dႴ)j ) – Iω I = I ω dႴ jLf - Li = I ω dႴdtdtTorque= ( I ω ωp ) jYXX = Spin axisYY = Active Gyroscopic couple
11. Numerical
12. NumericalThe turbine rotor of a ship has a mass of 2.2 tonnes and rotates at 180 rpm clockwise when viewed from the aft. Radius of gyration k = 320 mm, find (i) The ship turns at a radius of 250 m with a speed of 25 km/h. Find gyroscopic couple. Gyroscopic couple = I * ω * ωp . Mass moment of inertia (I) = m*k2 ( k = radius of gyration) I = 2200 kg * (0.32)2 m2 I = 225.28 kg – m2 Angular velocity of spin (ω ) = (2*3.14* N)/60 = (2*3.14* 180)/60 = 18.85 rad/sec. Angular velocity of precision (ωp) = velocity of ship (v) / radius (R) = (25*1000)/ (250* 3600) = 0.03 rad/sec. Gyroscopic couple = I * ω * ωp = 225.28 * 18.85 * 0.03 C = 127.4 N-m